DYNAMICAL BREAKING OF SUPERSYMMETRY

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On Non-Perturbative Results in Supersymmetric Gauge Theories - A Lecture

a rXiv:h ep-th/9611152v12N ov1996CERN-TH/96-268hep-th/9611152ON NON-PERTURBATIVE RESULTS IN SUPERSYMMETRIC GAUGE THEORIES –A LECTURE 1Amit Giveon 2Theory Division,CERN,CH-1211,Geneva 23,Switzerland ABSTRACT Some notions in non-perturbative dynamics of supersymmetric gauge theories are being reviewed.This is done by touring through a few examples.CERN-TH/96-268September 19961IntroductionIn this lecture,we present some notions in supersymmetric Yang-Mills(YM) theories.We do it by touring through a few examples where we face a variety of non-perturbative physics eﬀects–infra-red(IR)dynamics of gauge theories.We shall start with a general review;some of the points we consider follow the beautiful lecture notes in[1].Phases of Gauge TheoriesThere are three known phases of gauge theories:•Coulomb Phase:there are massless vector bosons(massless photonsγ;no conﬁnement of both electric and magnetic charges).The behavior of the potential V(R)between electric test charges,separated by a large distance R,is V(R)∼1/R;the electric charge at large distance behaves like a constant:e2(R)∼constant.The potential of magnetic test charges separated by a large distance behaves like V(R)∼1/R,and the magnetic charge behaves like m2(R)∼constant,e(R)m(R)∼1 (the Dirac condition).•Higgs Phase:there are massive vector bosons(W bosons and Z bosons), electric charges are condensed(screened)and magnetic charges are conﬁned(the Meissner eﬀect).The potential between magnetic test charges separated by a large distance is V(R)∼ρR(the magnetic ﬂux is conﬁned into a thin tube,leading to this linear potential witha string tensionρ).The potential between electric test charges isthe Yukawa potential;at large distances R it behaves like a constant: V(R)∼constant.•Conﬁning Phase:magnetic charges are condensed(screened)and elec-tric charges are conﬁned.The potential between electric test charges separated by a large distance is V(R)∼σR(the electricﬂux is conﬁned1into a thin tube,leading to the linear potential with a string tensionσ).The potential between magnetic test charges behaves like a constant at large distance R.Remarks1.In addition to the familiar Abelian Coulomb phase,there are theorieswhich have a non-Abelian Coulomb phase[2],namely,a theory with massless interacting quarks and gluons exhibiting the Coulomb poten-tial.This phase occurs when there is a non-trivial IRﬁxed point of the renormalization group.Such theories are part of other possible cases of non-trivial,interacting4d superconformalﬁeld theories(SCFTs)[3,4]. 2.When there are matterﬁelds in the fundamental representation of thegauge group,virtual pairs can be created from the vacuum and screen the sources.In this situation,there is no invariant distinction between the Higgs and the conﬁning phases[5].In particular,there is no phase with a potential behaving as V(R)∼R at large distance,because the ﬂux tube can break.For large VEVs of theﬁelds,a Higgs description is most natural,while for small VEVs it is more natural to interpret the theory as“conﬁning.”It is possible to smoothly interpolate from one interpretation to the other.3.Electric-Magnetic Duality:Maxwell theory is invariant underE→B,B→−E,(1.1) if we introduce magnetic charge m=2π/e and also interchangee→m,m→−e.(1.2) Similarly,Mandelstam and‘t Hooft suggested that under electric-magnetic duality the Higgs phase is interchanged with a conﬁning phase.Con-ﬁnement can then be understood as the dual Meissner eﬀect associated with a condensate of monopoles.2Dualizing a theory in the Coulomb phase,one remains in the same phase.For an Abelian Coulomb phase with massless photons,this electric-magnetic duality follows from a standard duality transforma-tion,and is extended to SL(2,Z)S-duality,acting on the complex gauge coupling byτ→aτ+b2π+i4πBy eﬀective,we mean in Wilson sense:[modes p>µ]e−S=e−S ef f(µ,light modes),(2.3) so,in principle,L eff depends on a scaleµ.But due to supersymmetry,the dependence on the scaleµdisappear(except for the gauge couplingτwhich has a logµdependence).When there are no interacting massless particles,the Wilsonian eﬀec-tive action=the1PI eﬀective action;this is often the case in the Higgs or conﬁning phases.2.1The Eﬀective SuperpotentialWe will focus on a particular contribution to L eff–the eﬀective superpo-tential term:L int∼ d2θW eff(X r,g I,Λ)+c.c,(2.4) where X r=light chiral superﬁelds,g I=various coupling constants,and Λ=dynamically generated scale(associated with the gauge dynamics): log(Λ/µ)∼−8π2/g2(µ).Integrating overθ,the superpotential gives a scalar potential and Yukawa-type interaction of scalars with fermions.The quantum,eﬀective superpotential W eff(X r,g I,Λ)is constrained by holomorphy,global symmetries and various limits[9,1]:1.Holomorphy:supersymmetry requires that W eff is holomorphic in thechiral superﬁelds X r(i.e.,independent of the X†r).Moreover,we will think of all the coupling constants g I in the tree-level superpotential W tree and the scaleΛas background chiral superﬁeld sources.This implies that W eff is holomorphic in g I,Λ(i.e.,independent of g∗I,Λ∗).2.Symmetries and Selection Rules:by assigning transformation laws bothto theﬁelds and to the coupling constants(which are regarded as back-ground chiral superﬁelds),the theory has a large global symmetry.This implies that W eff should be invariant under such global symmetries.43.Various Limits:W eff can be analyzed approximately at weak coupling,and some other limits(like large masses).Sometimes,holomorphy,symmetries and various limits are strong enough to determine W eff!The results can be highly non-trivial,revealing interesting non-perturbative dynamics.2.2The Gauge“Kinetic Term”in a Coulomb Phase When there is a Coulomb phase,there is a term in L eff of the formL gauge∼ d2θIm τeff(X r,g I,Λ)W2α ,(2.5) where Wα=gauge supermultiplet(supersymmetricﬁeld strength);schemat-ically,Wα∼λα+θβσµνβαFµν+....Integrating overθ,W2αgives the term F2+iF˜F and its supersymmetric extension.Therefore,τeff=θeffg2eff(2.6)is the eﬀective,complex gauge coupling.τeff(X r,g I,Λ)is also holomorphic in X r,g I,Λand,sometimes,it can be exactly determined by using holomorphy, symmetries and various limits.2.3The“Kinetic Term”The kinetic term is determined by the K¨a hler potential K:L kin∼ d2θd2¯θK(X r,X†r).(2.7) If there is an N=2supersymmetry,τeff and K are related;for an N=2 supersymmetric YM theory with a gauge group G and in a Coulomb phase, L eff is given in terms of a single holomorphic function F(A i):L eff∼Im d4θ∂F2 d2θ∂2FA manifestly gauge invariant N =2supersymmetric action which reduces to the above at low energies is[10]Imd 4θ∂F2 d 2θ∂2F3Some of these results also appear in the proceedings [13]of the 29th International Symposium on the Theory of Elementary Particles in Buckow,Germany,August 29-September 2,1995,and of the workshop on STU-Dualities and Non-Perturbative Phe-nomena in Superstrings and Supergravity ,CERN,Geneva,November 27-December 1,1995.6N A supermultiplets in the adjoint representation,Φab α,α=1,...,N A ,and N 3/2supermultiplets in the spin 3/2representation,Ψ.Here a,b are fundamental representation indices,and Φab =Φba (we present Ψin a schematic form as we shall not use it much).The numbers N f ,N A and N 3/2are limited by the condition:b 1=6−N f−2N A −5N 3/2≥0,(3.1)where −b 1is the one-loop coeﬃcient of the gauge coupling beta-function.The main result of this section is the following:the eﬀective superpoten-tial of an (asymptotically free or conformal)N =1supersymmetric SU (2)gauge theory,with 2N f doublets and N A triplets (and N 3/2quartets)isW N f ,N A (M,X,Z,N 3/2)=−δN 3/2,0(4−b 1) Λ−b 1Pf 2N f X det N A (Γαβ)2 1/(4−b 1)+Tr N A ˜mM +1√4Integrating in the “glueball”ﬁeld S =−W 2α,whose source is log Λb 1,gives the non-perturbative superpotential:W (S,M,X,Z )=S log Λb 1S 4−b 1Here,the a,b indices are raised and lowered with anǫab tensor.The gauge-invariant superﬁelds X ij may be considered as a mixture of SU(2)“mesons”and“baryons,”while the gauge-invariant superﬁelds Zαij may be considered as a mixture of SU(2)“meson-like”and“baryon-like”operators.Equation(3.2)is a universal representation of the superpotential for all infra-red non-trivial theories;all the physics we shall discuss(and beyond) is in(3.2).In particular,all the symmetries and quantum numbers of thevarious parameters are already embodied in W Nf,N A .The non-perturbativesuperpotential is derived in refs.[11,12]by an“integrating in”procedure, following refs.[14,15].The details can be found in ref.[12]and will not be presented here5.Instead,in the next sections,we list the main results concerning each of the theories,N f,N A,N3/2,case by case.Moreover,a few generalizations to other gauge groups will be discussed.4b1=6:N f=N A=N3/2=0This is a pure N=1supersymmetric SU(2)gauge theory.The non-perturbative eﬀective superpotential is6W0,0=±2Λ3.(4.1) The superpotential in eq.(4.1)is non-zero due to gaugino(gluino)conden-sation7.Let us consider gaugino condensation for general simple groups[1].Pure N=1Supersymmetric Yang-Mills TheoriesPure N=1supersymmetric gauge theories are theories with pure superglue with no matter.We consider a theory based on a simple group G.The theorycontains vector bosons Aµand gauginosλαin the adjoint representation of G.There is a classical U(1)R symmetry,gaugino number,which is broken tosubgroup by instantons,a discrete Z2C2(λλ)C2 =const.Λ3C2,(4.2) where C2=the Casimir in the adjoint representation normalized such that, for example,C2=N c for G=SU(N c).This theory conﬁnes,gets a mass gap,and there are C2vacua associ-symmetry to Z2by gaugino ated with the spontaneous breaking of the Z2C2condensation:λλ =const.e2πin/C2Λ3,n=1,...,C2.(4.3) Each of these C2vacua contributes(−)F=1and thus the Witten index is Tr(−)F=C2.This physics is encoded in the generalization of eq.(4.1)to any G,givingW eff=e2πin/C2C2Λ3,n=1,...,C2.(4.4) For G=SU(2)we have C2=2.Indeed,the“±”in(4.1),which comes from the square-root appearing on the braces in(3.2)when b1=6,corresponds, physically,to the two quantum vacua of a pure N=1supersymmetric SU(2) gauge theory.The superpotentials(4.1),(4.3)can be derived byﬁrst adding fundamen-tal matter to pure N=1supersymmetric YM theory(as we will do in the next section),and then integrating it out.5b1=5:N f=1,N A=N3/2=0There is one case with b1=5,namely,SU(2)with oneﬂavor.The superpo-tential isΛ5W1,0=vacuum degeneracy of the classical low-energy eﬀective theory is lifted quan-tum mechanically;from eq.(5.1)we see that,in the massless case,there is no vacuum at all.SU(N c)with N f<N cEquation(5.1)is a particular case of SU(N c)with N f<N c(N f quarks Q i and N f anti-quarks¯Q¯i,i,¯i=1,...,N f)[1].In these theories,by using holomorphy and global symmetries,U(1)Q×U(1)¯Q×U(1)RQ:100¯Q:010(5.2)Λ3N c−N f:N f N f2N c−2N fW:002oneﬁnds thatW eff=(N c−N f) Λ3N c−N f N c−N f,(5.3) whereX i¯i≡Q i¯Q¯i,i,¯i=1,...,N f.(5.4) Classically,SU(N c)with N f<N c is broken down to SU(N c−N f).The ef-fective superpotential in(5.3)is dynamically generated by gaugino condensa-tion in SU(N c−N f)(for N f≤N c−2)8,and by instantons(for N f=N c−1).The SU(2)with N f=1ExampleFor example,let us elaborate on the derivation and physics of eq.(5.1).An SU(2)eﬀective theory with two doublets Q a i has one light degree of freedom: four Q a i(i=1,2is aﬂavor index,a=1,2is a color index;2×2=4)threeout of which are eaten by SU(2),leaving4−3=1.This single light degree of freedom can be described by the gauge singletX=Q1Q2.(5.5) When X =0,SU(2)is completely broken and,classically,W eff,class=0 (when X =0there are extra masslessﬁelds due to an unbroken SU(2)). Therefore,the classical scalar potential is identically zero.However,the one-instanton action is expected to generate a non-perturbative superpotential.The symmetries of the theory(at the classical level and with their cor-responding charges)are:U(1)Q=number of Q1ﬁelds(quarks or squarks),1=number of Q2ﬁelds(quarks or squarks),U(1)R={number of U(1)Q2gluinos}−{number of squarks}.At the quantum level these symmetries are anomalous–∂µjµ∼F˜F–and by integrating both sides of this equation one gets a charge violation when there is an instanton background I.The instanton background behaves likeI∼e−8π2/g2(µ)= Λ9For SU(N c)with N fﬂavors,the instanton background I has2C2=2N c gluino zero-modesλand2N f squark zero-modes q and,therefore,its R-charge is R(I)=number(λ)−number(q)=2N c−2N f.Since I∼Λb1and b1=3N c−N f,we learn thatΛ3N c−N f has an R-charge=2N c−2N f,as it appears in eq.(5.2).11and,therefore,W eff has charges:U(1)Q(5.9)1×U(1)Q2×U(1)RW eff:002Finally,because W eff is holomorphic in X,Λ,and is invariant under symme-tries,we must haveΛ5W eff(X,Λ)=c10This is reﬂected in eq.(3.2)by the vanishing of the coeﬃcient(4−b1)in front of the braces,leading to W=0,and the singular power1/(4−b1)on the braces,when b1=4, which signals the existence of a constraint.12At the classical limit,Λ→0,the quantum constraint collapses into the clas-sical constraint,Pf X=0.SU(N c)with N f=N cEquations(6.1),(6.2)are a particular case of SU(N c)with N f=N c.[1]In these theories one obtains W eff=0,and the classical constraint det X−B¯B=0is modiﬁed quantum mechanically todet X−B¯B=Λ2N c,(6.3) whereX i¯i=Q i¯Q¯i(mesons),B=ǫi1...i N c Q i1···Q i N c(baryon),¯B=ǫ¯i1...¯i N c¯Q¯i1···¯Q¯iN c(anti−baryon).(6.4)6.2N f=0,N A=1,N3/2=0The massless N A=1case is a pure SU(2),N=2supersymmetric Yang-Mills theory.This model was considered in detail in ref.[17].The non-perturbative superpotential vanishesW non−per.0,1=0,(6.5) and by the integrating in procedure we also get the quantum constraint:M=±Λ2.(6.6) This result can be understood because the starting point of the integrating in procedure is a pure N=1supersymmetric Yang-Mills theory.Therefore,it leads us to the points at the verge of conﬁnement in the moduli space.These are the two singular points in the M moduli space of the theory;they are due to massless monopoles or dyons.Such excitations are not constructed out of the elementary degrees of freedom and,therefore,there is no trace for them in W.(This situation is diﬀerent if N f=0,N A=1;in this case,monopoles are diﬀerent manifestations of the elementary degrees of freedom.)137b1=3There are two cases with b1=3:either N f=3,or N A=N f=1.In both cases,for vanishing bare parameters in(3.2),the semi-classical limit,Λ→0, imposes the classical constraints,given by the equations of motion:∂W=0; however,quantum corrections remove the constraints.7.1N f=3,N A=N3/2=0The superpotential isW3,0=−Pf X2Tr mX.(7.1)In the massless case,the equations∂X W=0give the classical constraints; in particular,the superpotential is proportional to a classical constraint: Pf X=0.The negative power ofΛ,in eq.(7.1)with m=0,indicates that small values ofΛimply a semi-classical limit for which the classical constraints are imposed.SU(N c)with N f=N c+1Equation(7.1)is a particular case of SU(N c)with N f=N c+1[1].In these theories one obtainsW eff=−det X−X i¯iB i¯B¯iThis is consistent with the negative power ofΛin W eff which implies that in the semi-classical limit,Λ→0,the classical constraints are imposed. 7.2N f=1,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW1,1=−Pf X2Tr mX+12TrλZ.(7.5)Here m,X are antisymmetric2×2matrices,λ,Z are symmetric2×2 matrices andΓ=M+Tr(ZX−1)2.(7.6) This superpotential was foundﬁrst in ref.[18].Toﬁnd the quantum vacua, we solve the equations:∂M W=∂X W=∂Z W=0.Let us discuss some properties of this theory:•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve:y2=x3+ax2+bx+c(7.7)(and some other equations),where the coeﬃcients a,b,c are functions of only theﬁeld M,the scaleΛ,the bare quark masses,m,and Yukawa couplings,λ.Explicitly,a=−M,b=Λ316,(7.8)whereα=Λ62Γ.(7.10)15•W1,1has2+N f=3vacua,namely,the three singularities of the elliptic curve in(7.7),(7.8).These are the three solutions,M(x),of the equations:y2=∂y2/∂x=0;the solutions for X,Z are given by the other equations of motion.•The3quantum vacua are the vacua of the theory in the Higgs-conﬁnement phase.•Phase transition points to the Coulomb branch are at X=0⇔˜m= 0.Two of these singularities correspond to a massless monopole or dyon,and are the quantum splitting of the classically enhanced SU(2) point.A third singularity is due to a massless quark;it is a classical singularity:M∼m2/λ2for large m,and thus M→∞when m→∞, leaving the two quantum singularities of the N A=1,N f=0theory.•The elliptic curve deﬁnes the eﬀective Abelian coupling,τ(M,Λ,m,λ), in the Coulomb branch:Elliptic Curves and Eﬀective Abelian CouplingsA torus can be described by the one complex dimensional curve in C2 y2=x3+ax2+bx+c,where(x,y)∈C2and a,b,c are complex parameters.The modular parameter of the torus isτ(a,b,c)= βdx αdxIn this form,the modular parameterτis determined(modulo SL(2,Z)) by the ratio f3/g2through the relation4(24f)3j(τ)=8b1=2There are three cases with b1=2:N f=4,or N A=1,N f=2,or N A=2.In all three cases,for vanishing bare parameters in(3.2),there are extra massless degrees of freedom not included in the procedure;those are expected due toa non-Abelian conformal theory.8.1N f=4,N A=N3/2=0The superpotential isW4,0=−2(Pf X)1Λ+12N c<N f<3N c the theory is in an interacting,non-AbelianCoulomb phase(in the IR and for m=0).In this range of N f the theory is asymptotically ly,at short distance the coupling18constant g is small,and it becomes larger at larger distance.However, it is argued that for32N c<N f<3N c,the IR theory is a non-trivial4d SCFT.The elementary quarks and gluons are not conﬁned but appear as in-teracting massless particles.The potential between external massless electric sources behaves as V∼1/R,and thus one refers to this phase of the theory as the non-Abelian Coulomb phase.•The Seiberg Duality:it is claimed[2]that in the IR an SU(N c)theory with N fﬂavors is dual to SU(N f−N c)with N fﬂavors but,in addition to dual quarks,one should also include interacting,massless scalars. This is the origin to the branch cut in W eff at X =0,because W eff does not include these light modes which must appear at X =0. The quantum numbers of the quarks and anti-quarks of the SU(N c) theory with N fﬂavors(=theory A)are11A.SU(N c),N f:The Electric TheorySU(N f)L×SU(N f)R×U(1)B×U(1)RQ:N f111−N cN f(8.2)The quantum numbers of the dual quarks q i and anti-quarks¯q¯i of the SU(N f−N c)theory with N fﬂavors theory(=theory B)and its mass-less scalars X i¯iareB.SU(N f−N c),N f:The Magnetic TheorySU(N f)L×SU(N f)R×U(1)B×U(1)R q:¯N f1N cN f ¯q:1N f−N c N fX:N f¯N f02 1−N ctheory A and theory B have the same anomalies:U(1)3B:0U(1)B U(1)2R:0U(1)2B U(1)R:−2N2cSU(N f)3:N c d(3)(N f)SU(N f)2U(1)R:−N2cN2f(8.5)Here d(3)(N f)=Tr T3f of the global SU(N f)symmetries,where T fare generators in the fundamental representation,and d(2)(N f)=Tr T2f2.Deformations:theory A and theory B have the same quantummoduli space of deformations.Remarks•Electric-magnetic duality exchanges strong coupling with weak cou-pling(this can be read oﬀfrom the beta-functions),and it interchanges a theory in the Higgs phase with a theory in the conﬁning phase.•Strong-weak coupling duality also relates an SU(N c)theory with N f≥3N c to an SU(N f−N c)theory.SU(N c)with N f≥3N c is in a non-Abelian free electric phase:in this range the theory is not asymptoti-cally ly,because of screening,the coupling constant becomes smaller at large distance.Therefore,the spectrum of the theory at large distance can be read oﬀfrom the Lagrangian–it consists of the elemen-tary quarks and gluons.The long distance behavior of the potential between external electric test charges isV(R)∼1R,e(R→∞)→0.(8.6)For N f≥3N c,the theory is thus in a non-Abelian free electric phase; the massless electrically chargedﬁelds renormalize the charge to zero21at long distance as e −2(R )∼log(R Λ).The potential of magnetic test charges behave at large distance R asV (R )∼log(R Λ)R ,⇒e (R )m (R )∼1.(8.7)SU (N c )with N f ≥3N c is dual to SU (˜N c )with ˜N c +2≤N f ≤3R ∼e 2(R )2˜N c ,the massless magnetic monopoles renormal-ize the electric coupling constant to inﬁnity at large distance,with a conjectured behavior e 2(R )∼log(R Λ).The potential of magnetic test charges behaves at large distance R asV (R )∼1R ⇒e (R )m (R )∼1.(8.9)•The Seiberg duality can be generalized in many other cases,includ-ing a variety of matter supermultiplets (like superﬁelds in the adjoint representation [20])and other gauge groups [21].8.2N f =2,N A =1,N 3/2=0In this case,the superpotential in (3.2)readsW 2,1=−2(Pf X )1ΛΓ+˜mM +1√•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve(7.7)(and some other equations),where the coeﬃcients a,b,c are functions of only theﬁeld M,the scaleΛ,the bare quark masses,m,and Yukawa couplings,λ.Explicitly[11,12],a=−M,b=−α4Pf m,c=α16detλ,µ=λ−1m.(8.12)•As in section7.2,the parameter x,in the elliptic curve(7.7),is given in terms of the compositeﬁeld:x≡1•As in section7.2,the negative power ofΛ,in eq.(8.10)with˜m= m=λ=0,indicates that small values ofΛimply a semi-classical limit for which the classical constraints are imposed.Indeed,for vanishing bare parameters,the equations∂W=0are equivalent to the classical constraints,and their solutions span the Higgs moduli space[22].•For special values of the bare masses and Yukawa couplings,some of the 4vacua degenerate.In some cases,it may lead to points where mutually non-local degrees of freedom are massless,similar to the situation in pure N=2supersymmetric gauge theories,considered in[3].For example,when the masses and Yukawa couplings approach zero,all the 4singularities collapse to the origin.Such points might be interpreted as in a non-Abelian Coulomb phase[1]or new non-trivial,interacting, N=1SCFTs.•The singularity at X=0(inΓ)and the branch cut at Pf X=0 (due to the1/2power in eq.(8.10))signal the appearance of extra massless degrees of freedom at these points;those are expected similar to references[2,20].Therefore,we make use of the superpotential only in the presence of bare parameters,whichﬁx the vacua away from such points.8.3N f=0,N A=2,N3/2=0In this case,the superpotential in eq.(3.2)readsdet MW0,2=±212The fractional power1/(4−b1)on the braces in(3.2),for any theory with b1≤2, may indicate a similar phenomenon,namely,the existence of conﬁnement and oblique24theory has two quantum vacua;these become the phase transition points to the Coulomb branch when det˜m=0.The moduli space may also contain a non-Abelian Coulomb phase when the two singularities degenerate at the point M=0[18];this happens when˜m=0.At this point,the theory has extra massless degrees of freedom and,therefore,W0,2fails to describe the physics at˜m=0.Moreover,at˜m=0,the theory has other descriptions via an electric-magnetic triality[1].9b1=1There are four cases with b1=1:N f=5,or N A=1,N f=3,or N A=2, N f=1,or N3/2=1.9.1N f=5,N A=N3/2=0The superpotential isW5,0=−3(Pf X)1Λ12Tr mX.(9.1)This theory is a particular case of SU(N c)with N f>N c+1.The discussion in section8.1is relevant in this case too.9.2N f=3,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW3,1=−3(Pf X)1Λ13+˜mM+1√conﬁnement branches of the theory,corresponding to the4−b1phases due to the fractional power.It is plausible that,for SU(2),such branches are related by a discrete symmetry.25•The equations∂W=0can be re-organized into the singularity condi-tions of an elliptic curve(7.7)(and some other equations),where the coeﬃcients a,b,c are[11,12]a=−M−α,b=2αM+α4Pf m,c=α64detλ,µ=λ−1m.(9.4) In eq.(9.3)we have shifted the quantumﬁeld M toM→M−α/4.(9.5)•The parameter x,in the elliptic curve(7.7),is given in terms of thecompositeﬁeld:x≡12.(9.6)Therefore,as before,we have identiﬁed a physical meaning of the pa-rameter x.•W3,1has2+N f=5quantum vacua,corresponding to the5singularities of the elliptic curve(7.7),(9.3);these are the vacua of the theory in the Higgs-conﬁnement phase.•From the phase transition points to the Coulomb branch,we conclude that the elliptic curve deﬁnes the eﬀective Abelian coupling,τ(M,Λ,m,λ), for arbitrary bare masses and Yukawa couplings.As before,on the sub-space of bare parameters,where the theory has N=2supersymmetry, the result in eq.(9.3)coincides with the result in[7]for N f=3.•For special values of the bare masses and Yukawa couplings,some of the 5vacua degenerate.In some cases,it may lead to points where mutually non-local degrees of freedom are massless,and might be interpreted as in a non-Abelian Coulomb phase or another new superconformal theory in four dimensions(see the discussion in sections7.2and8.2).26•The singularity and branch cuts in W3,1signal the appearance of extra massless degrees of freedom at these points.•The discussion in the end of sections7.2and8.2is relevant here too.9.3N f=1,N A=2,N3/2=0In this case,the superpotential in(3.2)reads[12]W1,2=−3(Pf X)1/32Tr mX+12TrλαZα.(9.7)Here m and X are antisymmetric2×2matrices,λαand Zαare symmetric 2×2matrices,α=1,2,˜m,M are2×2symmetric matrices andΓαβis given in eq.(3.3).This theory has3quantum vacua in the Higgs-conﬁnement branch.At the phase transition points to the Coulomb branch,namely, when det˜m=0⇔det M=0,the equations of motion can be re-organized into the singularity conditions of an elliptic curve(7.7).Explicitly,when ˜m22=˜m12=0,the coeﬃcients a,b,c in(7.7)are[12]a=−M22,b=Λ˜m21132 2detλ2.(9.8)However,unlike the N A=1cases,the equations∂W=0cannot be re-organized into the singularity condition of an elliptic curve,in general.This result makes sense,physically,since an elliptic curve is expected to“show up”only at the phase transition points to the Coulomb branch.For special values of the bare parameters,there are points in the moduli space where (some of)the singularities degenerate;such points might be interpreted as in a non-Abelian Coulomb phase,or new superconformal theories.For more details,see ref.[12].9.4N f=N A=0,N3/2=1This chiral theory was shown to have W non−per.0,0(N3/2=1)=0;[24]perturb-ing it by a tree-level superpotential,W tree=gU,where U is given in(3.4), may lead to dynamical supersymmetry breaking[24].2710b1=0There areﬁve cases with b1=0:N f=6,or N A=1,N f=4,or N A= N f=2,or N A=3,or N3/2=N f=1.These theories have vanishing one-loop beta-functions in either conformal or infra-red free beta-functions and, therefore,will possess extra structure.10.1N f=6,N A=N3/2=0This theory is a particular case of SU(N c)with N f=3N c;the electric theory is free in the infra-red[1].1310.2N f=4,N A=1,N3/2=0In this case,the superpotential in(3.2)readsW4,1=−4(Pf X)1Λb12+˜mM+1√β2 2α+1β2α13A related fact is that(unlike the N A=1,N f=4case,considered next)in the(would be)superpotential,W6,0=−4Λ−b1/4(Pf X)1/4+1。

Can the Supersymmetric $mu$ parameter be generated dynamically without a light Singlet

a r X i v :h e p -p h /9404257v 1 12 A p r 1994DESY 94-061ISSN 0418-9833April 1994hep-ph/9404257Can the Supersymmetric µparameter be generated dynamically without a light Singlet?Ralf Hempfling Deutsches Elektronen-Synchrotron,Notkestraße 85,D-22603Hamburg,Germany ABSTRACTIt is generally assumed that the dynamical generation of the Higgs mass param-eter of the superpotential,µ,implies the existence of a light singlet at or below the supersymmetry breaking scale,M SUSY .We present a counter-example in which the singlet ﬁeld can receive an arbitrarily heavy mass (e.g.,of the order of the Planck scale,M P ≈1019GeV).In this example,a non-zero value of µis generatedthrough soft supersymmetry breaking parameters and is thus naturally of the order of M SUSY .The cancellation of quadratic divergences in the unrenormalized Green func-tions is one of the main motivations of supersymmetry(SUSY).It stabilizes any mass scale under radiative corrections and thus allows the existence of diﬀerent mass scales such as the electroweak scale,given by the Z boson mass,m z,and the Planck scale,M P.The minimal supersymmetric standard model(MSSM)is the most popular model of this kind due to its minimal particle content[1].In this model,the SU(2)L⊗U(1)Y symmetry breaking is driven by soft SUSY break-ing parameters.Thus,the SUSY breaking scale,M SUSY,has to be at or slightly above m z.For this mechanism to work it is also necessary that the SUSY Higgs mass parameter,|µ|<∼M SUSY.This parameter also determines the chargino and neutralino mass spectrum.From here one can deduce a experimental lower bound from LEP experiments of|µ|>∼m z/4independent of tanβ[2].The fact that in the MSSM theµ-parameter,which is a priori arbitrary,has to lie within the narrow range1will be evaded[5].We will demonstrate in the following that it is also possible to make N1heavy [say m N1=O(M P)]while keeping N1 =O(M SUSY)withoutﬁne-tuning.In this limit we recover the predictive Higgs sector of the MSSM[6]with its well deﬁned upper limit of the lightest Higgs boson mass[eq.(2)].First we need to extend the symmetry group of our Lagrangian in order to forbid the explicit Higgs mass term of the superpotential,W H=µH8(ξ+2N∗1N1−2N∗2N2−N∗3N3)2.(4)Here the inclusion of a Fayet-Iliopoulos term[7],ξ,is the easiest way of breaking the U(1)Y′gauge symmetry but one can envisage other alternatives[8].The VEVs are denoted byn1= N1 =0,n2= N2 =1λ+ λ2+4ξ , n3= N3 = λ.(5)The CP-even and CP-odd components of the scalarﬁeld N1are mass-degenerate mass-eigenstates with m N1=(m2+λ2n23)1/2.The gauge boson,g′,acquires a massm g′=g′(n22+n23/4)1/2via the Higgs mechanism.The masses of the remaining CP-,m g′(m N1,0;the zero mass eigenvalue corresponds even(CP-odd)scalars are m N1to the Goldstone boson which is absorbed to give mass to the gauge boson).Theand±m g′as required if mass eigenvalues of the fermionic components are±m N1SUSY is unbroken.Note that in addition to the gauge and the SUSY transformations the La-grangian is invariant under the global U(1)R-symmetry[9]which does not com-mute with SUSY.This symmetry transformsΦ→exp(inΦα)Φ,where nΦ= 2,0,0,0for the bosons and n Φ=1,−1,−1,1for the fermions(Φ=N1,N2,N3,g′). We now break SUSY explicitly in the standard fashion by including soft SUSY breaking terms[10]V soft=BmN1N2−AλN1N23+h.c.,(6) where A,B=O(M SUSY)are the soft SUSY breaking parameters.With these terms the R-symmetry is broken down to a discrete Z2symmetry(α=±π).If we minimize the full potential,V=V SUSY+V soft,weﬁndN1 ≈(A−B)mn2ξ.However,the condition N1 =0is protected by R-symmetry to all orders in perturbation theory and is only broken by adding soft SUSY breaking terms[eq.(6)].We now include in our model the full particle content of the MSSM.The Z2symmetry is equivalent to the usual R-parity that prevents baryon and lepton number violating interactions. The full superpotential can then be written asW=W N+W H+W Y,(8) where W H=κN1Hof W in eq.(8)and by requiring the absence of anomalies.These constraints can be satisﬁed by introducing additional pairs of SUSY multiplets T∼(n c,n w,Y,Y′1) and T c∼(¯n c,n w,−Y,Y′2).These representations have been included in pairs such√that below the U(1)Y′breaking scale,Useful conversations with W.Buchm¨u ller are gratefully ac-knowledgedREFERENCES1.H.P.Nilles,Phys.Rep.110,1(1984);H.E.Haber and G.L.Kane,Phys.Rep.117,75(1985);R.Barbieri,Riv.Nuovo Cimento11,1(1988).2.J.-F.Grivaz,in Proceedings of the Workshop on e+e−Collisions at500GeV:The Physics Potential,Munich,Annecy,Hamburg,DESY report DESY92-123B(1992).3.G.F.Guidice and A.Masiero,Phys.Lett.B206,480(1988);J.E.Kimand H.P.Nilles,Phys.Lett.B263,79(1991);J.A.Casas and C.Mu˜n oz, Phys.Lett.B306,288(1993).4.E.Witten,Phys.Lett.B105,267(1981);L.Ib´a˜n ez and G.G.Ross,Phys.Lett.B110,215(1982);P.V.Nanopoulos and K.Tamvakis,Phys.Lett.B113,151(1982).5.see,e.g.,J.Ellis,J.F.Gunion,H.E.Haber,L.Roszkowski and F.Zwirner,Phys.Rev.D39,844(1989).6.see,e.g.,J.F.Gunion,H.E.Haber,G.L.Kane,and S.Dawson,The HiggsHunter’s Guide,(Addison-Wesley,Redwood City,CA,1990).7.P.Fayet and J.Iliopoulos,Phys.Lett.B51,461(1974).8.e.g.,L.O’Raifeartaigh,Nucl.Phys.B96,331(1975).9.P.Fayet,Nucl.Phys.B90,104(1975);A.Salam and J.Strathdee,Nucl.Phys.B87,85(1975).10.L.Girardello and M.T.Grisaru,Nucl.Phys.B194,65(1982).。

A Systematic Approach to Confinement in N=1 Supersymmetric Gauge Theories

a r X i v :h e p -t h /9610139v 1 17 O c t 1996hep-th/9610139MIT-CTP-2581BUHEP-96-41A Systematic Approach to Conﬁnement in N =1Supersymmetric Gauge TheoriesCsaba Cs´a ki a ,Martin Schmaltz b and Witold Skiba aaCenter for Theoretical Physics Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology Cambridge,MA 02139,USA csaki@,skiba@ b Department of Physics Boston University Boston,MA 02215,USA schmaltz@ Abstract We give necessary criteria for N =1supersymmetric theories to be in a smoothly conﬁning phase without chiral symmetry breaking and with a dynamically generated ing our general arguments we ﬁnd all such conﬁning SU and Sptheories with a single gauge group and no tree level superpotential.Following the initial breakthrough in the works of Seiberg on exact results in N=1supersymmetric QCD(SQCD)[1],much progress has been made in extending these results to other theories with diﬀerent gauge and matterﬁelds[2-11].We now have a whole zoo of examples of supersymmetric theories for which we know results about the vacuum structure and the infrared spectrum.A number of theories are known to have dual descriptions,others are known to conﬁne with or without chiral symmetry breaking,and some theories do not possess a stable ground state.Unfortunately,we are still lacking a systematic and general approach that allows one to determine the infrared properties of a given theory.The results in the literature have mostly been obtained by an ingenious guess of the infrared spectrum.This guess is then justiﬁed by performing a number of non-trivial consistency checks which include matching of the global anomalies,detailed study of the moduli space of vacua, and the behavior of the theory under perturbations.In this letter,we will depart from the customary trial and error procedure and give some general arguments which allow us to classify a subset of supersymmetric theories. To be speciﬁc,we intend to answer the general question of which supersymmetricﬁeld theories may be conﬁning without chiral symmetry breaking and with a conﬁning superpotential.We present a few simple arguments which allow us to rule out most theories as possible candidates for conﬁnement without chiral symmetry breaking.For the most part,these arguments already exist in the literature but our systematic way of putting them to use is new.As a demonstration of the power of our arguments we give a complete list of all SU(N)and Sp(N)gauge theories with no tree level superpotential which conﬁne without chiral symmetry breaking,and we determine the conﬁned degrees of freedom and the superpotential describing their interactions (“conﬁning superpotential”).To begin,let usﬁrst explain what we mean by“smooth conﬁnement without chiral symmetry breaking and with a non-vanishing conﬁning superpotential”,which,from now on,we will abbreviate by s-conﬁnement.We will call a theory conﬁning when its infrared physics can be described exactly in terms of gauge invariant composites and their interactions.This description has to be valid everywhere on the moduli space of vacua.Our deﬁnition of s-conﬁnement also requires that the theory dynamically generates a conﬁning superpotential,which excludes models of the type presented in Ref.[11].Furthermore,the phrase“without chiral symmetry breaking”implies that the origin of the classical moduli space is also a vacuum in the quantum theory.In this vacuum,all the global symmetries of the ultraviolet remain unbroken.Finally,the conﬁning superpotential is a holomorphic function of the conﬁned degrees of freedom and couplings,which describes all the interactions in the extreme infrared.Note that this deﬁnition excludes theories which are in a Coulomb phase on a submanifold of the moduli space[2],or theories which have distinct Higgs and conﬁning phases with associated phase boundaries on the moduli space.Our prototype example for an s-conﬁning theory is Seiberg’s SQCD[1]with the number ofﬂavors F chosen to equal N+1,where N is the number of colors,and a“ﬂavor”is a pair of matterﬁelds in the fundamental and antifundamental represen-tations of SU(N).Seiberg argued that the matterﬁelds Q and¯Q are conﬁned into “mesons”M=Q¯Q and“baryons”B=Q N,¯B=¯Q N.At the origin of moduli space all components of the mesons and baryons are massless and interact via the conﬁning superpotential1W=1We normalize the index of the fundamental representation to1.integers.Therefore2, jµj−µ(G)=1or2,and for SU and Sp theories anomaly cancellation further constrainsjµj−µ(G)=2.(3)This formula constitutes a necessary condition for s-conﬁnement,it enables us to rule out most theories immediately.For example,for SQCD weﬁnd that the only candidate theory is the theory with F=N+1.Unfortunately,Eq.3is not a suﬃcient condition.An example for a theory which satisﬁes Eq.3but does not s-conﬁne is SU(N)with an adjoint superﬁeld and oneﬂavor.This theory is easily seen to be in an Abelian Coulomb phase for generic VEVs of the adjoint scalars and vanishing VEVs for the fundamentals.We could now simply examine all theories that satisfy Eq.3byﬁnding all in-dependent gauge invariants and checking if this ansatz for the conﬁning spectrum matches the anomalies.Apart from being very cumbersome,this method is also not very useful to demonstrate that a given theory satisfying Eq.3is not s-conﬁning.A better strategy relies on our second observation.An s-conﬁning theory with a smooth description in terms of gauge invariants at the origin must also be s-conﬁning everywhere on its moduli space.This is because the conﬁning superpotential at the origin which is a simple polynomial in theﬁelds is analytical everywhere,and no additional massless states are present anywhere on the moduli space.Therefore,the theory restricted to a particularﬂat direction must have a smooth description as well. This observation has two very useful applications.First,if we have a theory that s-conﬁnes and we know its conﬁned spectrum and superpotential,we can easilyﬁnd new s-conﬁning theories by going to diﬀerent points on moduli space.In the ultraviolet description,the gauge group is broken to a sub-group of the original group,some matterﬁelds are eaten by the Higgs mechanism, and the remaining ones decompose under the unbroken subgroup.The corresponding conﬁned description is obtained by simplyﬁnding the corresponding point on the moduli space of the conﬁned theory.The global symmetries will be broken in the same way,and someﬁelds may be massive and can be integrated out.This newly found conﬁned theory is guaranteed to pass all the standard consistency checks be-cause they are a subset of the consistency checks for the original theory.For example, the anomalies of the new s-conﬁning theory are guaranteed to match:the unbroken global symmetries are a subgroup of original global symmetries,and the anomalies under the subgroup are left unchanged–both in the infrared and ultraviolet descrip-tions–because the fermions which obtain masses give cancelling contributions to the anomalies.Second,the above observation can be turned around to provide another necessary condition for s-conﬁnement.If anywhere on the moduli space of a given theory we ﬁnd a theory which is not s-conﬁning or completely higgsed,we know that the original theory cannot be s-conﬁning either.Let us study some examples.Suppose we knew that SU(N)with N+1ﬂavors for some large N is s-conﬁning,then we could immediately conclude that the theories with n<N also s-conﬁne.We simply need to give a VEV to some of the quark-antiquark pairs to break SU(N)to any SU(n)subgroup.The quarks with vevs are eaten,leaving n+1ﬂavors and some singlets.We remove these singlets by adding “mirror”superﬁelds with opposite global charges and giving them a mass.We now identify the corresponding point on the moduli space of the conﬁned SU(N)theory. Someﬁelds obtain masses from the superpotential of Eq.1when we expand around the new point in moduli space.After integrating the massiveﬁelds and removing the ﬁelds corresponding to the singlets in the ultraviolet theory via masses with mirror partners,we obtain the correct conﬁned description of SU(n).A non-trivial example of a theory which can be shown to not s-conﬁne is SU(4) with three antisymmetric tensors and twoﬂavors.This theory satisﬁes Eq.3and is therefore a candidate for s-conﬁnement.By giving a VEV to an antisymmetric tensor we canﬂow from this theory to Sp(4)with two antisymmetric tensors and four fundamentals.VEVs for the other antisymmetric tensors let usﬂow further to SU(2) with eight fundamentals which is known to be at an interactingﬁxed point in the infrared.We conclude that the SU(4)and Sp(4)theories and all theories thatﬂow to them cannot be s-conﬁning either.This allows us to rule out the following chain of theories,all of which are gauge anomaly free and satisfy Eq.3SU(7)→SU(6)→SU(5)→SU(4)→Sp(4)(4)432232Note that a VEV for one of the quarkﬂavors of the SU(4)theory lets usﬂow to an SU(3)theory with fourﬂavors which is s-conﬁning.We must therefore be careful, when weﬁnd aﬂow to an s-conﬁning theory,it does not follow that the original theory is s-conﬁning as well.Theﬂow is only a necessary condition.However,we suspect that a theory with a single gauge group and no tree-level superpotential is s-conﬁning if it is found toﬂow to s-conﬁning theories in all directions of its moduli space.We do not know of any counter examples.Armed with formula in Eq.3and our observation onﬂows of s-conﬁning theories, we were able toﬁnd all s-conﬁning SU and Sp gauge theories with a single gauge group and no tree-level superpotential for arbitrary tensor representations.To achieve this, weﬁrst found all possible matter contents satisfying Eq.3.We list all these theories in Table1.We then studied the possibleﬂows of these theories and discarded all those withﬂows to theories which do not s-conﬁne.This process eliminated all except about a dozen theories for which we then explicitly determined the independent gaugeinvariants and matched anomalies toﬁnd the conﬁning spectra.These results are summarized in Table1.Six of the ten theories which s-conﬁne are new3:SU(N)with++3, SU(7)with2+6,SU(6)with2++4+4,and SU(5)with3+3.For the theories which do not s-conﬁne we indicated the method by which we obtained this result:either by noting that the theory has a branch with only unbroken U(1)gauge groups,or else byﬂowing along aﬂat direction to a theory with smaller non-Abelian gauge group which does not s-conﬁne.Detailed results on the new theories including the conﬁning spectra,superpoten-tials,variousﬂows,and consistency checks will be reported elsewhere[14].Here,we just point out a few salient features.Most of the new s-conﬁning theories contain vector-like matter.Perturbing these theories by adding mass terms for some of the vector-like matter,we easily obtain exact results on the theories with the matter integrated out.Among the theories that weﬁnd in this way are new theories which conﬁne with chiral symmetry breaking, theories with runaway vacua,and theories which conﬁne without chiral symmetry breaking and vanishing superpotentials.Since many of the new theories presented here are chiral,they can be used toﬁnd models of dynamical supersymmetry breaking along the lines of Refs.[15].Examples for such supersymmetry breaking theories will also be included in the detailed paper[14].Our s-conﬁning theories might be used for building extensions of the standard model with composite quarks and leptons[16].Finally,we comment on possible exceptions and generalizations of our arguments.A possible exception to our condition in Eq.3arises,when allµi andµ(G)have a common divisor.Then the superpotential Eq.2can be holomorphic even when jµj−µ(G)= 2.However,whereas Eq.3is preserved under mostﬂows,the property that allµ’s have a common divisor is not.Therefore,such theoriesﬂow to theories which are not s-conﬁning,and by our second necessary condition the original theory is not s-conﬁning either.Another possibility is that the conﬁning superpotential vanishes,and the conﬁned degrees of freedom are free in the infrared.This can only happen if there are no clas-sical constraints among the basic gauge invariant operators which satisfy the’t Hooft anomaly matching conditions,otherwise the quantum solution would not have the correct classical limit.Examples of theories which are believed to conﬁne in this way can be found in the literature[7,11,14].Generalizations to SO(N)groups are not completely straightforward because in the case of SO(N)theories“exotic composites”containing the chiral superﬁeld Wαmight appear in the infrared spectrum and superpotential,thus modifying our argu-ment and result of Eq.3.SU(N)(N+1)(s-conﬁningSU(N)+N+4s-conﬁningSU(N)++)+SU(4)Adj++)SU(4)4+SU(2):+4Coulomb branchSU(5)3(+)+4)SU(5)2++) SU(6)2+5+s-conﬁningSU(6)2+SU(4):3+2(+4(s-conﬁningSU(6)+2+ ++Sp(6):SU(6)2+)SU(7)2(+3)+4+2SU(6):SU(7)+Sp(6):Sp(2N)(2N+4)s-conﬁningSp(2N)+6s-conﬁningSp(2N)+2Coulomb branchSp(4)3+2SU(2):+4SU(2):2Sp(6)2+2Sp(4):2+4+5Sp(4):2+4++SU(2):+4Sp(8)2Generalizations to theories with more than one gauge group or tree level super-potentials are more diﬃcult.The additional interactions break some of the global symmetries which are now not suﬃcient to completely determine the functional form of the conﬁning superpotential.Another complication is that in these theories theﬂat directions of the quantum theory are sometimes diﬃcult to identify.Since our second argument only applies toﬂows in directions which are on the quantum moduli space, incorrect conclusions would be obtained fromﬂows along classicalﬂat directions which are notﬂat in the quantum theory.In summary,we have discussed general criteria for s-conﬁnement and used them toﬁnd all s-conﬁning theories with SU(N)or Sp(2N)gauge groups.It is a pleasure to thank P.Cho,A.Cohen,N.Evans,L.Randall,and R.Sundrum for useful discussions.We also thank B.Dobrescu,A.Nelson,and J.Terning for comments on the manuscript. C.C.and W.S.are supported in part by the U.S. Department of Energy under cooperative agreement#DE-FC02-94ER40818.M.S.is supported by the U.S.Department of Energy under grant#DE-FG02-91ER40676. 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On the Interactions of Light Gravitinos

On the Interactions of Light GravitinosT.E.Clark1,Taekoon Lee2,S.T.Love3,Guo-Hong Wu4Department of PhysicsPurdue UniversityWest Lafayette,IN47907-1396AbstractIn models of spontaneously broken supersymmetry,certain light gravitino processes are governed by the coupling of its Goldstino components.The rules for constructing SUSY and gauge invariant actions involving the Gold-stino couplings to matter and gaugeﬁelds are presented.The explicit oper-ator construction is found to be at variance with some previously reported claims.A phenomenological consequence arising from light gravitino inter-actions in supernova is reexamined and scrutinized.1e-mail address:clark@2e-mail address:tlee@3e-mail address:love@4e-mail address:wu@1In the supergravity theories obtained from gauging a spontaneously bro-ken global N=1supersymmetry(SUSY),the Nambu-Goldstone fermion, the Goldstino[1,2],provides the helicity±1degrees of freedom needed to render the spin3gravitino massive through the super-Higgs mechanism.For a light gravitino,the high energy(well above the gravitino mass)interactions of these helicity±1modes with matter will be enhanced according to the su-persymmetric version of the equivalence theorem[3].The eﬀective action de-scribing such interactions can then be constructed using the properties of the Goldstinoﬁelds.Currently studied gauge mediated supersymmetry breaking models[4]provide a realization of this scenario as do certain no-scale super-gravity models[5].In the gauge mediated case,the SUSY is dynamically broken in a hidden sector of the theory by means of gauge interactions re-sulting in a hidden sector Goldstinoﬁeld.The spontaneous breaking is then mediated to the minimal supersymmetric standard model(MSSM)via radia-tive corrections in the standard model gauge interactions involving messenger ﬁelds which carry standard model vector representations.In such models,the supergravity contributions to the SUSY breaking mass splittings are small compared to these gauge mediated contributions.Being a gauge singlet,the gravitino mass arises only from the gravitational interaction and is thus farsmaller than the scale √,where F is the Goldstino decay constant.More-2over,since the gravitino is the lightest of all hidden and messenger sector degrees of freedom,the spontaneously broken SUSY can be accurately de-scribed via a non-linear realization.Such a non-linear realization of SUSY on the Goldstinoﬁelds was originally constructed by Volkov and Akulov[1].The leading term in a momentum expansion of the eﬀective action de-scribing the Goldstino self-dynamics at energy scales below √4πF is uniquelyﬁxed by the Volkov-Akulov eﬀective Lagrangian[1]which takes the formL AV=−F 22det A.(1)Here the Volkov-Akulov vierbein is deﬁned as Aµν=δνµ+iF2λ↔∂µσν¯λ,withλ(¯λ)the Goldstino Weyl spinorﬁeld.This eﬀective Lagrangian pro-vides a valid description of the Goldstino self interactions independent of the particular(non-perturbative)mechanism by which the SUSY is dynam-ically broken.The supersymmetry transformations are nonlinearly realized on the Goldstinoﬁelds asδQ(ξ,¯ξ)λα=Fξα+Λρ∂ρλα;δQ(ξ,¯ξ)¯λ˙α= F¯ξ˙α+Λρ∂ρ¯λ˙α,whereξα,¯ξ˙αare Weyl spinor SUSY transformation param-eters andΛρ≡−i Fλσρ¯ξ−ξσρ¯λis a Goldstinoﬁeld dependent translationvector.Since the Volkov-Akulov Lagrangian transforms as the total diver-genceδQ(ξ,¯ξ)L AV=∂ρ(ΛρL AV),the associated action I AV= d4x L AV is SUSY invariant.The supersymmetry algebra can also be nonlinearly realized on the matter3(non-Goldstino)ﬁelds,generically denoted byφi,where i can represent any Lorentz or internal symmetry labels,asδQ(ξ,¯ξ)φi=Λρ∂ρφi.(2) This is referred to as the standard realization[6]-[9].It can be used,along with space-time translations,to readily establish the SUSY algebra.Under the non-linear SUSY standard realization,the derivative of a matterﬁeld transforms asδQ(ξ,¯ξ)(∂νφi)=Λρ∂ρ(∂νφi)+(∂νΛρ)(∂ρφi).In order to elim-inate the second term on the right hand side and thus restore the standard SUSY realization,a SUSY covariant derivative is introduced and deﬁned so as to transform analogously toφi.To achieve this,we use the transformation property of the Volkov-Akulov vierbein and deﬁne the non-linearly realized SUSY covariant derivative[9]Dµφi=(A−1)µν∂νφi,(3) which varies according to the standard realization of SUSY:δQ(ξ,¯ξ)(Dµφi)=Λρ∂ρ(Dµφi).Any realization of the SUSY transformations can be converted to the standard realization.In particular,consider the gauge covariant derivative,(Dµφ)i≡∂µφi+T a ij A aµφj,(4)4with a=1,2,...,Dim G.We seek a SUSY and gauge covariant deriva-tive(Dµφ)i,which transforms as the SUSY standard ing the Volkov-Akulov vierbein,we deﬁne(Dµφ)i≡(A−1)µν(Dνφ)i,(5) which has the desired transformation property,δQ(ξ,¯ξ)(Dµφ)i=Λρ∂ρ(Dµφ)i, provided the vector potential has the SUSY transformationδQ(ξ,¯ξ)Aµ≡Λρ∂ρAµ+∂µΛρAρ.Alternatively,we can introduce a redeﬁned gaugeﬁeldV aµ≡(A−1)µνA aν,(6) which itself transforms as the standard realization,δQ(ξ,¯ξ)V aµ=Λρ∂ρV aµ, and in terms of which the standard realization SUSY and gauge covariant derivative then takes the form(Dµφ)i≡(A−1)µν∂νφi+T a ij V aµφj.(7) Under gauge transformations parameterized byωa,the original gaugeﬁeld varies asδG(ω)A aµ=(Dµω)a=∂µωa+gf abc A bµωc,while the redeﬁnedgaugeﬁeld V aµhas the Goldstino dependent transformation:δG(ω)V aµ= (A−1)µν(Dνω)a.For all realizations,the gauge transformation and SUSY transformation commutator yields a gauge variation with a SUSY trans-formed value of the gauge transformation parameter,δG(ω),δQ(ξ,¯ξ)=δG(Λρ∂ρω−δQ(ξ,¯ξ)ω).(8) 5If we further require the local gauge transformation parameter to also trans-form under the standard realization so thatδQ(ξ,¯ξ)ωa=Λρ∂ρωa,then the gauge and SUSY transformations commute.In order to construct an invariant kinetic energy term for the gaugeﬁelds, it is convenient for the gauge covariant anti-symmetric tensorﬁeld strength to also be brought into the standard realization.The usualﬁeld strengthF a αβ=∂αA aβ−∂βA aα+if abc A bαA cβvaries under SUSY transformations asδQ(ξ,¯ξ)F aµν=Λρ∂ρF aµν+∂µΛρF aρν+∂νΛρF aµρ.A standard realization of thegauge covariantﬁeld strength tensor,F aµν,can be then deﬁned asF aµν=(A−1)µα(A−1)νβF aαβ,(9) so thatδQ(ξ,¯ξ)F aµν=Λρ∂ρF aµν.These standard realization building blocks consisting of the gauge singlet Goldstino SUSY covariant derivatives,Dµλ,Dµ¯λ,the matterﬁelds,φi,their SUSY-gauge covariant derivatives,Dµφi,and theﬁeld strength tensor,F aµν, along with their higher covariant derivatives can be combined to make SUSY and gauge invariant actions.These invariant action terms then dictate the couplings of the Goldstino which,in general,carries the residual consequences of the spontaneously broken supersymmetry.A generic SUSY and gauge invariant action can be constructed[9]asI eﬀ=d4x detA L eﬀ(Dµλ,Dµ¯λ,φi,Dµφi,Fµν)(10)6where L eﬀis any gauge invariant function of the standard realization basic building ing the nonlinear SUSY transformationsδQ(ξ,¯ξ)detA=∂ρ(ΛρdetA)andδQ(ξ,¯ξ)L eﬀ=Λρ∂ρL eﬀ,it follows thatδQ(ξ,¯ξ)I eﬀ=0.It proves convenient to catalog the terms in the eﬀective Lagranian,L eﬀ, by an expansion in the number of Goldstinoﬁelds which appear when covari-ant derivatives are replaced by ordinary derivatives and the Volkov-Akulov vierbein appearing in the standard realizationﬁeld strengths are set to unity. So doing,we expandL eﬀ=L(0)+L(1)+L(2)+···,(11)where the subscript n on L(n)denotes that each independent SUSY invariant operator in that set begins with n Goldstinoﬁelds.L(0)consists of all gauge and SUSY invariant operators made only from light matterﬁelds and their SUSY covariant derivatives.Any Goldstinoﬁeld appearing in L(0)arises only from higher dimension terms in the matter covariant derivatives and/or theﬁeld strength tensor.Taking the light non-Goldstinoﬁelds to be those of the MSSM and retaining terms through mass dimension4,then L(0)is well approximated by the Lagrangian of the mini-mal supersymmetric standard model which includes the soft SUSY breaking terms,but in which all derivatives have been replaced by SUSY covariant ones and theﬁeld strength tensor replaced by the standard realizationﬁeld7strength:L(0)=L MSSM(φ,Dµφ,Fµν).(12) Note that the coeﬃcients of these terms areﬁxed by the normalization of the gauge and matterﬁelds,their masses and self-couplings;that is,the normalization of the Goldstino independent Lagrangian.The L(1)terms in the eﬀective Lagrangian begin with direct coupling of one Goldstino covariant derivative to the non-Goldstinoﬁelds.The general form of these terms,retaining operators through mass dimension6,is given byL(1)=1[DµλαQµMSSMα+¯QµMSSM˙αDµ¯λ˙α],(13)Fwhere QµMSSMαand¯QµMSSM˙αcontain the pure MSSMﬁeld contributions to the conserved gauge invariant supersymmetry currents with once again all ﬁeld derivatives being replaced by SUSY covariant derivatives and the vector ﬁeld strengths in the standard realization.That is,it is this term in the eﬀective Lagrangian which,using the Noether construction,produces the Goldstino independent piece of the conserved supersymmetry current.The Lagrangian L(1)describes processes involving the emission or absorption of a single helicity±1gravitino.Finally the remaining terms in the eﬀective Lagrangian all contain two or more Goldstinoﬁelds.In particular,L(2)begins with the coupling of two8Goldstinoﬁelds to matter or gaugeﬁelds.Retaining terms through mass dimension8and focusing only on theλ−¯λterms,we can writeL(2)=1F2DµλαDν¯λ˙αMµν1α˙α+1F2Dµλα↔DρDν¯λ˙αMµνρ2α˙α+1F2DρDµλαDν¯λ˙αMµνρ3α˙α,(14)where the standard realization composite operators that contain matter and gaugeﬁelds are denoted by the M i.They can be enumerated by their oper-ator dimension,Lorentz structure andﬁeld content.In the gauge mediated models,these terms are all generated by radiative corrections involving the standard model gauge coupling constants.Let us now focus on the pieces of L(2)which contribute to a local operator containing two gravitinoﬁelds and is bilinear in a Standard Model fermion (f,¯f).Those lowest dimension operators(which involve no derivatives on f or¯f)are all contained in the M1piece.After application of the Goldstino ﬁeld equation(neglecting the gravitino mass)and making prodigious use of Fierz rearrangement identities,this set reduces to just1independent on-shell interaction term.In addition to this operator,there is also an operator bilinear in f and¯f and containing2gravitinos which arises from the product of det A with L(0).Combining the two independent on-shell interaction terms involving2gravitinos and2fermions,results in the eﬀective actionIf¯f˜G˜G =d4x−12F2λ↔∂µσν¯λf↔∂νσµ¯f9+C ffF2(f∂µλ)¯f∂µ¯λ,(15)where C ff is a model dependent real coeﬃcient.Note that the coeﬃcient of theﬁrst operator isﬁxed by the normaliztion of the MSSM Lagrangian. This result is in accord with a recent analysis[10]where it was found that the fermion-Goldstino scattering amplitudes depend on only one parameter which corresponds to the coeﬃcient C ff in our notation.In a similar manner,the lowest mass dimension operator contributing to the eﬀective action describing the coupling of two on-shell gravitinos to a single photon arises from the M1and M3pieces of L(2)and has the formIγ˜G˜G =d4xCγF2∂µλσρ∂ν¯λ∂µFρν+h.c.,(16)with Cγa model dependent real coeﬃcient and Fµνis the electromagnetic ﬁeld strength.Note that the operator in the square bracket is odd under both parity(P)and charge conjugation(C).In fact any operator arising from a gauge and SUSY invariant structure which is bilinear in two on-shell gravitinos and contains only a single photon is necessarily odd in both P and C.Thus the generation of any such operator requires a violation of both P and ing the Goldstino equation of motion,the analogous term containing˜Fµνreduces to Eq.(16)with Cγ→−iCγ.Recently,there has appeared in the literature[11]the claim that there is a lower dimensional operator of the form˜M2F2∂νλσµ¯λFµνwhich contributes to the single photon-102gravitino interaction.Here˜M is a model dependent SUSY breaking massparameter which is roughly an order(s)of magnitude less than √.¿Fromour analysis,we do notﬁnd such a term to be part of a SUSY invariant action piece and thus it should not be included in the eﬀective action.Such a term is also absent if one employs the equivalent formalism of Wess and Samuel [6].We have also checked that such a term does not appear via radiative corrections by an explicit graphical calculation using the correct non-linearly realized SUSY invariant action.This is also contrary to the previous claim.There have been several recent attempts to extract a lower bound on the SUSY breaking scale using the supernova cooling rate[11,12,13].Unfortu-nately,some of these estimates[11,13]rely on the existence of the non-SUSY invariant dimension6operator referred to ing the correct low en-ergy eﬀective lagrangian of gravitino interactions,the leading term coupling 2gravitinos to a single photon contains an additional supression factor ofroughly Cγs˜M .Taking√s 0.1GeV for the processes of interest and using˜M∼100GeV,this introduces an additional supression of at least10−12in the rate and obviates the previous estimates of a bound on F.Assuming that the mass scales of gauginos and the superpartners of light fermions are above the core temperature of supernova,the gravitino cooling of supernova occurs mainly via gravitino pair production.It is interesting to11compare the gravitino pair production cross section to that of the neutrino pair production,which is the main supernova cooling channel.We have seen that for low energy gravitino interactions with matter,the amplitudes for gravitino pair production is proportional to1/F2.A simple dimensional analysis then suggests the ratio of the cross sections is:σχχσνν∼s2F4G2F(17)where GF is the Fermi coupling and√s is the typical energy scale of theparticles in a supernova.Even with the most optimistic values for F,thegravitino production is too small to be relevant.For example,taking √F=100GeV,√s=.1GeV,the ratio is of O(10−11).It seems,therefore,thatsuch an astrophysical bound on the SUSY breaking scale is untenable in mod-els where the gravitino is the only superparticle below the scale of supernova core temperature.We thank T.K.Kuo for useful conversations.This work was supported in part by the U.S.Department of Energy under grant DE-FG02-91ER40681 (Task B).12References[1]D.V.Volkov and V.P.Akulov,Pis’ma Zh.Eksp.Teor.Fiz.16(1972)621[JETP Lett.16(1972)438].[2]P.Fayet and J.Iliopoulos,Phys.Lett.B51(1974)461.[3]R.Casalbuoni,S.De Curtis,D.Dominici,F.Feruglio and R.Gatto,Phys.Lett.B215(1988)313.[4]M.Dine and A.E.Nelson,Phys.Rev.D48(1993)1277;M.Dine,A.E.Nelson and Y.Shirman,Phys.Rev.D51(1995)1362;M.Dine,A.E.Nelson,Y.Nir and Y.Shirman,Phys.Rev.D53,2658(1996).[5]J.Ellis,K.Enqvist and D.V.Nanopoulos,Phys.Lett.B147(1984)99.[6]S.Samuel and J.Wess,Nucl.Phys.B221(1983)153.[7]J.Wess and J.Bagger,Supersymmetry and Supergravity,second edition,(Princeton University Press,Princeton,1992).[8]T.E.Clark and S.T.Love,Phys.Rev.D39(1989)2391.[9]T.E.Clark and S.T.Love,Phys.Rev.D54(1996)5723.[10]A.Brignole,F.Feruglio and F.Zwirner,hep-th/9709111.[11]M.A.Luty and E.Ponton,hep-ph/9706268.13[12]J.A.Grifols,R.N.Mohapatra and A.Riotto,Phys.Lett.B400,124(1997);J.A.Grifols,R.N.Mohapatra and A.Riotto,Phys.Lett.B401, 283(1997).[13]J.A.Grifols,E.Masso and R.Toldra,hep-ph/970753.D.S.Dicus,R.N.Mohapatra and V.L.Teplitz,hep-ph/9708369.14。

马蝇效应英语作文

马蝇效应英语作文The Butterfly Effect: A Captivating Exploration of Chaos TheoryThe concept of the Butterfly Effect has captivated the minds of scientists philosophers and the general public alike for decades Its core premise is deceptively simple yet profoundly impactful a single seemingly insignificant event can have far reaching and unpredictable consequences This idea challenges our intuitive understanding of causality and determinism revealing the inherent complexity and unpredictability of the natural worldAt the heart of the Butterfly Effect lies the field of chaos theory which explores the behavior of highly sensitive dynamical systems Chaos theory emerged in the 1960s when the meteorologist Edward Lorenz discovered that tiny changes in the initial conditions of a weather simulation could lead to radically different outcomes Lorenz famously illustrated this by suggesting that the flapping of a butterfly s wings in Brazil could theoretically alter the course of a tornado in Texas This striking analogy gave rise to the term Butterfly Effect which has since become a ubiquitous part of our culturallexiconThe Butterfly Effect arises from the nonlinear nature of many real world systems When a system is nonlinear small perturbations can be amplified through feedback loops causing the system to evolve in unpredictable ways This sensitivity to initial conditions is a hallmark of chaotic systems which exhibit a high degree of complexity and unpredictability over time For example the weather is a classic example of a chaotic system where minuscule changes in atmospheric conditions can lead to dramatically different weather patterns emerging days or weeks laterThe implications of the Butterfly Effect are far reaching and challenge our conventional understanding of causality and predictability In the realm of weather forecasting the Butterfly Effect means that beyond a certain time horizon weather predictions become increasingly unreliable as tiny uncertainties in the initial conditions are magnified Likewise in fields such as biology ecology and economics the Butterfly Effect suggests that complex systems can behave in highly unpredictable ways making it difficult to anticipate and plan for future outcomesYet the Butterfly Effect is not solely a scientific concept it also has profound philosophical and existential implications For instance the idea that small actions can have outsized consequences calls intoquestion the notion of free will and human agency Do we truly have control over our lives and the world around us or are we at the mercy of chaotic forces beyond our comprehension The Butterfly Effect also raises questions about the limits of human knowledge and the hubris of our attempts to understand and predict the worldFurthermore the Butterfly Effect has important ethical implications It suggests that our individual actions however small can have far reaching and unintended impacts on the lives of others and the environment This realization can foster a greater sense of responsibility and humility as we navigate the complex web of cause and effect that underlies the human experience The Butterfly Effect reminds us that we are all interconnected that the flap of a wing in one corner of the world can create a storm in anotherAt the same time the Butterfly Effect also offers a sense of hope and possibility Just as small actions can lead to large negative consequences the reverse is also true positive changes however modest can ripple outward and transform the world in profound ways This notion has inspired social and political movements that seek to harness the power of individual agency to create systemic change from the ground upIn conclusion the Butterfly Effect is a captivating and multifaceted concept that continues to captivate and challenge us It reveals theinherent complexity and unpredictability of the natural world while also offering profound insights into the human condition the limits of our knowledge and the ethical responsibilities we bear as individuals living in an interconnected world As we grapple with the implications of the Butterfly Effect we are reminded of our own fragility and the awesome power of the natural world but also of our capacity to create positive change through our individual actions no matter how small。

Gluino production in some supersymmetric models at the LHC

a rXiv:085.294v2[he p-ph]9J ul28Gluino production in some supersymmetric models at the LHC C.Brenner Mariotto and M.C.Rodriguez Universidade Federal do Rio Grande -FURG Departamento de F´ısica Av.It´a lia,km 8,Campus Carreiros 96201-900,Rio Grande,RS Brazil Abstract In this article we review the mechanisms in several supersymmetric models for producing gluinos at the LHC and its potential for discov-ering them.We focus on the MSSM and its left-right extensions.We study in detail the strong sector of both models.Moreover,we obtain the total cross section and diﬀerential distributions.We also make an analysis of their uncertainties,such as the gluino and squark masses,which are related to the soft SUSY breaking parameters.PACS numbers:12.60.-i ;12.60.Jv;13.85.Lg;13.85.Qk;14.80.Ly.1Introduction Although the Standard Model (SM)[1],based on the gauge symmetry SU (3)c ⊗SU (2)L ⊗U (1)Y describes the observed properties of charged leptons and quarks it is not the ultimate theory.However,the necessity to go beyond it,from the experimental point of view,comes at the moment only from neutrino data.If neutrinos are massive then new physics beyond the SM isneeded.Supersymmetry (SUSY)or symmetry between bosons (particles with in-teger spin)and fermions (particles with half-integer spin)has been introduced in theoretical papers nearly 30years ago [2].Since that time there appeared thousands of papers.The reason for this remarkable activity is the unique mathematical nature of supersymmetric theories,possible solution of vari-ous problems of the SM within its supersymmetric extentions as well as the opening perspective of uniﬁcation of all interactions in the framework of a single theory [3,4,5].1However supersymmetry seemed,in the early days,clearly inappropriate for a description of our physical world1,for obvious and less obvious reasons, which often tend to be somewhat forgotten,now that we got so accustomed to deal with Supersymmetric extensions of the Standard Model.We recall the obstacles which seemed,long ago,to prevent supersymmetry from possibly being a fundamental symmetry of Nature2[6].We know that bosons and fermions should have equal masses in a su-persymmetric theory.However,it even seemed initially that supersymmetry could not be spontaneously broken at all which would imply that bosons and fermions be systematically degenerated in mass,unless of course supersymmetry-breaking terms are explicitly introduced“by hand”.As a result,this lead to the question:•Is spontaneous supersymmetry breaking possible at all?Until today,the spontaneous supersymmetry breaking remains,in gen-eral,rather diﬃcult to obtain.Of course just accepting the possibilityof explicit supersymmetry breaking without worrying too much aboutthe origin of supersymmetry breaking terms,as is frequently done now,makes things much easier–but also at the price of introducing a largenumber of arbitrary parameters,coeﬃcients of these supersymmetrybreaking terms.These terms essentially serve as a parametrizationof our ignorance about the true mechanism of supersymmetry break-ing chosen by Nature to make superpartners heavy.In any case suchterms must have their origin in a spontaneous supersymmetry breakingmechanism.However,much before getting to the Supersymmetric Standard Model,and irrespective of the question of supersymmetry breaking,the crucialquestion,if supersymmetry is to be relevant in particle physics,is:•Which bosons and fermions could be related?But there seems to be no answer since known bosons and fermions donot appear to have much in common–except,maybe,for the photonand the neutrino.May be supersymmetry could act at the level of composite objects,e.g.as relating baryons with mesons?Or should it act at a fundamental level,i.e.at the level of quarks and gluons?(But quarks are color triplets,and electrically charged,while gluons transform as an SU(3) color octet,and are electrically neutral!)In a more general way the number of(known)degrees of freedom is signiﬁcantly larger for the fermions(now90,for three families of quarks and leptons)than for the bosons(27for the gluons,the photon and the W±and Z gauge bosons,ignoring for the moment the spin-2 graviton,and the still-undiscovered Higgs boson).And these fermions and bosons have very diﬀerent gauge symmetry properties!This leads to the question:•How could one deﬁne(conserved)baryon and lepton numbers,in a supersymmetric theory?Of course nowadays we are so used to deal with spin-0squarks and sleptons,carrying baryon and lepton numbers almost by deﬁnition,that we can hardly imagine this could once have appeared as a problem.Supersymmetry today is the main candidate for a uniﬁed theory beyond the SM.Search for various manifestations of supersymmetry in Nature is one of the main tasks of numerous experiments at colliders.Unfortunately, the result is negative so far.There are no direct indications on existence of supersymmetry in particle physics,however there are a number of theoretical and phenomenological issues that the SM fails to address adequately[7]:•Uniﬁcation with gravity;The point is that SUSY algebra being a gen-eralization of Poincar´e algebra[5,8,9]{Qα,¯Q˙α}=2σmα,˙αP m.(1) Therefore,an anticommutator of two SUSY transformations is a lo-cal coordinate translation.And a theory which is invariant under the general coordinate transformation is General Relativity.Thus,making SUSY local,one obtains General Relativity,or a theory of gravity,or supergravity[10].•Uniﬁcation of Gauge Couplings;According to hypothesis of Grand Uni-ﬁcation Theory(GUT)all gauge couplings change with energy.All3known interactions are the branches of a single interaction associated with a simple gauge group which includes the group of the SM.To reach this goal one has to examine how the coupling change with en-ergy.Considerating the evolution of the inverse couplings,one can see that in the SM uniﬁcation of the gauge couplings is impossible.In the supersymmetric case the slopes of Renormalization Group Equation curves are changed and the results show that in supersymmetric model one can achieve perfect uniﬁcation[11].•Hierarchy problem;The supersymmetry automatically cancels all quadratic corrections in all orders of perturbation theory due to the contributionsof superpartners of the ordinary particles.The contributions of the bo-son loops are cancelled by those of fermions due to additional factor (−1)coming from Fermi statistic.This cancellation is true up to the SUSY breaking scale,M SUSY,sincebosons m2− fermions m2=M2SUSY,(2)which should not be very large(≤1TeV)to make theﬁne-tuning natural.Therefore,it provides a solution to the hierarchy problem by protecting the eletroweak scale from large radiative corrections[12]. However,the origin of the hierarchy is the other part of the problem. We show below how SUSY can explain this part as well.•Electroweak symmetry breaking(EWSB);The“running”of the Higgs masses leads to the phenomenon known as radiative electroweak sym-metry breaking.Indeed,the mass parameters from the Higgs potentialm21and m22(or one of them)decrease while running from the GUT scaleto the scale M Z may even change the sign.As a result for some valueof the momentum Q2the potential may acquire a nontrivial minimum. This triggers spontaneous breaking of SU(2)symmetry.The vacuum expectations of the Higgsﬁelds acquire nonzero values and provide masses to fermions and gauge bosons,and additional masses to their superpartners[13].Thus the breaking of the electroweak symmetry is not introduced by brute force as in the SM,but appears naturally from the radiative corrections.SUSY has also made several correct predictions[7]:4•Supersymmetry predicted in the early1980s that the top quark would be heavy[14],because this was a necessary condition for the validity of the electroweak symmetry breaking explanation.•Supersymmetric grand uniﬁed theories with a high fundamental scale accurately predicted the present experimental value of sin2θW before it was measured[15].•Supersymmetry requires a light Higgs boson to exist[16],consistent with current precision measurements,which suggest M h<200GeV[17].Together these successes provide powerful indirect evidence that low energy SUSY is indeed part of correct description of nature.Certainly the most popular extension of the SM is its supersymmetric counterpart called Minimal Supersymmetric Standard Model(MSSM)[3, 18,19].Theﬁrst attempt to construct a phenomenological model was done in [20],where the author tried to relate known particles together(in particu-lar,the photon with a“neutrino”,and the W±’s with charged“leptons”, also related with charged Higgs bosons H±),in a SU(2)⊗U(1)electroweak theory involving two doublet Higgs superﬁelds now known as H1and H23. The limitations of this approach quickly led to reinterpret the fermions of this model(which all have1unit of a conserved additive R quantum num-ber carried by the supersymmetry generator)as belonging to a new class of particles.The“neutrino”ought to be considered as a really new parti-cle,a“photonic neutrino”,a name transformed in1977into photino;the fermionic partners of the colored gluons(quite distinct from the quarks)then becoming the gluinos,and so on.More generally this led one to postulate the existence of new R-odd“superpartners”for all particles and consider them seriously,despite their rather non-conventional properties: e.g.new bosons carrying“fermion”number,now known as sleptons and squarks,or Majorana fermions transforming as an SU(3)color octet,which are pre-cisely the gluinos,etc..In addition the electroweak breaking must be in-duced by a pair of electroweak Higgs doublets,not just a single one as in the SM,which requires the existence of charged Higgs bosons,and of severalneutral ones[3,18].We also want to stress that on reference[3]were intro-duced squarks and gluinos(color octet of Majorana fermions,which couple to squark/quark pairs within what is now known as Supersymmetric Quantum Chromodynamics(sQCD)),that is the main subject of this article.The still-hypothetical superpartners may be distinguished by a new quan-tum number called R-parity,ﬁrst deﬁned in terms of the previous R quan-tum number as R p=(−1)R,i.e.+1for the ordinary particles and−1for their superpartners.It is associated with a Z2remnant of the previous R-symmetry acting continuously on gauge,lepton,quark and Higgs superﬁelds as in[3],which must be abandoned as a continuous symmetry to allow masses for the gravitino[18]and gluinos[21].The conservation(or non-conservation)of R-parity is therefore closely related with the conservation (or non-conservation)of baryon and lepton numbers,B and L,as illus-trated by the well-known formula reexpressing R-parity in terms of baryon and lepton numbers,as(−1)2S(−1)3B+L[22].This may also be written as(−1)2S(−1)3(B−L),showing that this discrete symmetry may still be conserved even if baryon and lepton numbers are separately violated,as long as their diﬀerence(B−L)remains conserved,at least modulo2.Theﬁnding of the basic building blocks of the Supersymmetric Standard Model,whether“minimal”or not,allowed for the experimental searches for “supersymmetric particles”,which started with theﬁrst searches for gluinos and photinos,selectrons and smuons,in the years1978-1980,and have been going on continuously since.These searches often use the“missing energy”signature corresponding to energy-momentum carried away by unobserved neutralinos[3,22,23].A conserved R-parity also ensures the stability of the“lightest supersymmetric particle”,a good candidate to constitute the non-baryonic Dark Matter that seems to be present in the Universe.Massive neutrinos can also be naturally accommodated in R-parity vio-lating supersymmetric theories,in which neutrinos can mix with neutralinos so that they acquire small masses[24,25].However,the phenomenological bounds on B and/or L violation[25,26]can be satisﬁed by imposing B as a symmetry and allowing the lepton number violating couplings to be large enough to generate Majorana neutrino masses.However,the minimalistic extension of the MSSM is to introduce a gauge singlet superﬁeldˆN,this model is called“Next Minimal Supersymmetric Standard Model”(NMSSM)[8,27].It is mainly motivated by its potential to eliminate theµproblem of the6MSSM[28],where the origin of the theµparameter in the superpotentialW MSSM=µH1H2(3) is not understood.For phenomenological reasons it has to be of the order of the electroweak scale,while the“natural”mass scale would be of the order of the GUT or Planck scale.This problem is evaded in the NMSSM where theµterm in the superpotential is dynamically generated through the superpotential1W NMSSM=λˆH1ˆH2ˆN−4For the see-saw mechanism theﬁrst paper is[29]71.It incorporates Left-Right(LR)symmetry[34](on top of the quark-lepton symmetry mentioned above),which leads naturally to the spon-taneous breaking of parity and charge conjugation[34,37].2.It incorporates a see-saw mechanism for small neutrino masses[30,31].3.It predicts the existence of magnetic monopoles[38].4.It leads to rare processes such as K L→µ¯e through the lepto-quarkgauge bosons(with however a negligible rate for M P S≥10GeV)[34].5.In the case of single-step breaking,predicts the scale of quark-lepton(and Left-Right)uniﬁcation[39].6.It allows naturally for∆B=2process of n−¯n oscillations(withhowever a negligible rate unless there are light diquarks in the T eV mass region)[40].st but not least,it allows for implementation of the leptogenesisscenario,as suggested by the see-saw mechanism[41].On the technical side,the left-right symmetric model has a problem sim-ilar to that in the SM:the masses of the fundamental Higgs scalars diverge quadratically.As in the SM,the Supersymmetric Left-Right model(SU-SYLR)can be used to stabilize the scalar masses and cure this hierarchy problem.On the literature there are two diﬀerent SUSYLR models.They diﬀer in their SU(2)R breakingﬁelds:one uses SU(2)R triplets[42](SUSYLRT)and the other SU(2)R doublets[43](SUSYLRD).Another,maybe more important raison d’etre for SUSYLR is the fact that they lead naturally to R-parity conservation[44].Namely,Left-Right models contain a B−L gauge symmetry,which allows for this possibility[45]. All that is needed is that one uses a version of the theory that incorporates a see-saw mechanism[30,31]at the renormalizable level.As we said before,the supersymmetric particles have not yet been de-tected in the present machines such as HERA and Tevatron.By another hand there are many interesting supersymmetric models in the literature as we shown above.Therefore,if SUSY is detected in the Large Hadron Col-lider(LHC),one of the next steps will be to discriminate among the diﬀerent8SM extensions,scenarios and also toﬁnd the mass spectrum of the diﬀerent particles(which can be obtained theoretically in diﬀerent scenarios).To discriminate among the several possibilities,it is important to make predictions for diﬀerent observables and confront these predictions with the forthcoming experimental data.One important process which could be mea-sured at the LHC is the gluino production.The R-symmetry transformations act chirally on gluinos,so that an unbroken R-invariance would require them to remain massless,even after a spontaneous breaking of the supersymmetry!In the early days it was very diﬃcult to obtain large masses for gluinos,since:i)no direct gluino mass term was present in the Lagrangian density;and ii)no such term may be generated spontaneously,at the tree approximation,since gluino couplings involve colored spin-0ﬁelds.On this case,gluino remain massless,and we would then expect the ex-istence of relatively light“R-hadrons”[22,23]made of quarks,antiquarks and gluinos,which have not been observed.We know today that gluinos, if they do exist,should be rather heavy,requiring a signiﬁcant breaking of the continuous R-invariance,in addition to the necessary breaking of the supersymmetry.A third reason for abandoning the continuous R-symmetry could now be the non-observation at LEP of a charged wino–also called chargino–lighter than the W±,that would exist in the case of a continuous U(1) R-invariance[3,20].The just-discoveredτ−particle could tentatively beconsidered,in1976,as a possible light wino/chargino candidate,before get-ting clearly identiﬁed as a sequential heavy lepton.The gluino masses result directly from supergravity5Remember that the see-saw mechanism[29]did not attract attention at the time, however the article[46]on gluino masses discusses a see-saw mechanism for gluinos.9therefore their production is only feasible at a very energetic machine such as the LHC.Being the fermion partners of the gluons,their role and interactions are directly related with the properties of the supersymmetric QCD(sQCD).The aim of this paper is twofold.Theﬁrst one is to study the strong sector of some supersymmetric models and to show explicit that the Feynmann rules for the gluino production are the same in all supersymmetric extensions of the Standard Model here considered.After that,as the second aim of this article, we show predictions for the gluino production on these models at the LHC, for various SPS benchmark points.The outline of the paper is the following. In sections2and3,we obtain the relevant Feynman rules of the strong sector from the MSSM and SUSYLR models,respectivelly.Moreover,we see that the Feynman rules are indeed the same in these models.In section 4we consider the diﬀerent scenarious for the relevant SUSY parameters, which lead to diﬀerent gluinos and squark masses.In section5we consider gluino production in the studied models and present the relevant expressions, which are used to obtain the numerical results in section6.Conclusions are summarized in section7.2Minimal Supersymmetric Standard Model (MSSM).In the MSSM[3,18,19],the gauge group is SU(3)C⊗SU(2)L⊗U(1)Y.The particle content of this model consists in associate to every known quark and lepton a new scalar superpartner to form a chiral supermultiplet.Similarly, we group a gauge fermion(gaugino)with each of the gauge bosons of the standard model to form a vector multiplet.In the scalar sector,we need to introduce two Higgs scalars and also their supersymmetric partners known as Higgsinos(Our notation6is given at[47]).We also need to impose a new global U(1)invariance usually called R-invariance,to get interactions that conserve both lepton and baryon number(invariance).On this section we will derive the Feynman rules of the strong sector of this model.2.1Interaction from L GaugeWe can rewrite L Gauge,see[8,9],in the following wayL Gauge=L cin+L gaugino+L gaugeD.(5) whereL cin=L SU(3)cin +L SU(2)cin+L U(1)cin,L gaugino=L SU(3)gaugino +L SU(2)gaugino+L U(1)gaugino,L gauge D =L SU(3)D+L SU(2)D+L U(1)D,(6)theﬁrst part is given by:L SU(3) cin =−1λa C¯σm D mλa C,(9)whereD mλa C=∂mλa C−g s f abcλa C g c m.(10) The last term is given byL SU(3) D =12.1.2Gluino–Gluino–Gluon InteractionThis interaction is got from Eqs.(9,10)combining both equations to obtainL gaugino =L cin +L ˜g ˜g g ,(12)whereL cin =ı(∂mλa C ¯σm λb C g c m ,(13)the ﬁrst term gives the cinetic term to gluino,while the last one provides the gluino-gluino-gluon interaction.Considerating the four-component Majorana spinor for the gluino,given by Ψ(˜g a )= −ıλa C (x )ı2g s f bac ¯Ψ(˜g a )γm Ψ(˜g b )g c m .(15)Owing to the Majorana nature of the gluino one must multiply by 2to obtain the Feynman rule (or add the graph with ˜g ↔¯˜g )!The equation above induce the following Feynman rule,for the vertice gluino-gluino-gluon,given at Fig.(1)and it is the same result as presented˜g a˜g b−g s f bac γm Figure 1:Feynman rule for the vertice Gluino-Gluino-Gluon at the MSSM.at [8,9,19,48].122.2Interaction from L QuarksThe interaction of the strong sector are obtained from the following La-grangiansL qqg=g s¯Q¯σm T a Qg am+g s d c¯σm¯T a d c g amL˜q˜q g=ıg s ¯˜QT a∂m˜Q−˜QT a∂m¯˜Q g am+ıg s ˜u c g am +ıg s ˜d c g am,L q˜q˜g=−ı√λa C−¯˜QT a Qλa C −ı√u c¯T a˜u c˜u c¯T a u cλa C −ı√d c¯T a˜d c˜d c¯T a d cλa C ,L˜q˜q gg=−g2s¯˜QT a T b˜Qg a m g bm−g2s˜d c¯T a¯T b˜d c g a m g bm.(16) Where T a rs are the color triplet generators,then one must use¯T a=−T∗a rs=−T a sr,(17)rsfor the color anti-triplet generators.2.2.1Quark–Quark–Gluon InteractionThis interaction comes from theﬁrst Lagrangian given at Eq.(16),and can be rewritten asL qqg=g s(¯u r¯σm T a rs u s+¯d r¯σm T a rs d s+d c r¯σm¯T a rs d c s)g a m,(18) u and d are color triplets while u c and d c are color anti-triplets.We must recall that r and s are color ing Eq.(17)we can rewrite Eq.(18) asL qqg=g s(¯u r¯σm T a rs u s+¯d r¯σm T a rs d s+u c rσm T a rs d c s)g a m.(19) Now,if we take into account the four-component Dirac spinor of the quarks (q=u,d),given byΨ(q)= q L(x)we obtain the Feynman ruleL qqg =−g sq =u,d ¯Ψ(q r )γm T a rs Ψ(q s )g a m ,(21)for the vertice qqg drawn at Fig.(2)this results again,as expected,agreeq rq s−ıg s T a rs γm Figure 2:Feynman rule for the vertice Quark-Quark-Gluon at the MSSM.with the results given at [8,9,19,48].The gluino-gluon interaction is like the quark-gluon interaction,compare Fig.(1)with Fig.(2)7and this fact gives the idea of “R -hadrons”presented at [22,23],mentioned in our introduction.2.2.2Squark–Squark–Gluon InteractionThis interaction is obtained from the second line given at Eq.(16).This term can be rewritten in the following wayL ˜q ˜q g =ıg s (¯˜ur T a rs (∂m ˜u s )−(∂m ˜d r )T a rs ˜d s +˜u c r )¯T a rs ˜u c s +˜d c r )¯T a rs ˜d c s)g am .(22)Using the following identityA ↔∂mB ≡A (∂m B )−(∂m A )B (23)we can rewrite our Lagrangian in the following simple wayL ˜q ˜q g =ıg sq =u,d (˜q ∗Lr T a rs ↔∂m ˜q Ls −˜q ∗Rr T a rs ↔∂m ˜q Rs )g am .(24)Note the relative minus sign between the terms with ˜q L and ˜q R :This is due to the facts that ˜q R are colour anti–triplets and the anti–colour generator given at Eq.(17).Here,we use a similar notation as given at [19],it means that ˜q ∗creates the scalar quark ˜q ,while ˜q destroys the scalar quark ˜q .Including the generalization to six ﬂavors,we can writeL ˜q ˜q g =ıg s q =u,d 6 p =1(˜q ∗Lpr T a rs ↔∂m ˜q Lps −˜q ∗Rpr T a rs ↔∂m ˜q Rps )g am (25)The corresponding Feynman rule we obtain from˜q ∗j ↔∂m ˜q i =ı(k i +k j )m (26)where k i and k j are the four–momenta of ˜q i and ˜q j in direction of the charge ﬂow.This relations give us the following Feynman rules given at Fig.(3)and we conclude that this results,as expected,agree with the references˜q r˜q s−ıg s T a rs (k i +k j )mFigure 3:Feynman rule for the vertice Squark-Squark-Gluon at the MSSM.[8,9,19,48].2.2.3Squark-Squark-Gluon-Gluon InteractionThis interaction comes from the last line given at Eq.(16),which can be written asL ˜q ˜q gg =−g 2s (¯˜u r T a rs T b st ˜u t +¯˜d r T a rs T b st ˜d t +˜d c r ¯T a rs ¯T b st ˜d c t )g a m gbm .15(27) By using the following formula valid for SU(3)generatorsT a rs T b st=12(d abc+ıf abc)T c rt,(28)it allows us to rewrite our Lagrangian in the following way8L˜q˜q gg=−g2s3+d abc T c rt g mnFigure4:Feynman rule to Squark-Squark-Gluon-Gluon vertice at MSSM.2.2.4Gluino-Quark-Squark InteractionThis interaction is described by the third line at Eq.(16),writing this term asL˜q qg=−√λa C−λa C−u c r¯T a rs˜u c s˜u c r¯T a rs u c sλa C+λa C−8Including the generalization to sixﬂavors,see Eq.(25).16Using the Eqs.(20,14)and the usual chiral projectorsL=12(1−γ5),(31)we can rewrite our Lagrangian in the following wayL˜q qg=−√2g s(LT a rs−RT a rs)Figure5:Feynman rule for the vertice Gluino-Quark-Squarks at the MSSM.it is in concordance with[8,9,19,48].Before considerating the left-right models,we want to say that the color sector of the interesting models NMSSM and MSSM3RHN are the same asin the MSSM model,therefore the results presented above are still hold on these models.3Supersymmetric Left-Right Model(SUSYLR) The supersymmetric extension of left-right models is based on the gauge group SU(3)C⊗SU(2)L⊗SU(2)R⊗U(1)B−L.On the literature,as we saidat introduction,there are two diﬀerent SUSYLR models.They diﬀer in theirSU(2)R breakingﬁelds:one uses SU(2)R triplets[42](SUSYLRT)and the other SU(2)R doublets[43](SUSYLRD).Some details of both models are17described at[47].SUSYLR models have the additional appealing character-istics of having automatic R-parity conservation.In this article,we are interested in studying only the strong sector.As this sector is the same in both models,SUSYLRT and SUSYLRD,the results we are presenting in this section hold in both models.3.1Interaction from L GaugeWe can rewrite L Gauge,as done in the MSSM case,in the following wayL Gauge=L cin+L gaugino+L gaugeD.(33) whereL cin=L SU(3)cin +L SU(2)Lcin+L SU(2)Rcin+L U(1)cin,L gaugino=L SU(3)gaugino +L SU(2)Lgaugino+L SU(2)Rgaugino+L U(1)gaugino,L gauge D =L SU(3)D+L SU(2)LD+L SU(2)RD+L U(1)D,(34)theﬁrst part is given by:L SU(3) cin =−1λa C¯σm D mλa C,(36)while D C nλa C is deﬁned at Eq.(10).The last term is given byL SU(3) D =13.2Interaction from L QuarksIn terms of the doublets the Lagrangian of the strong sector can be rewritten asL qqg=g s¯Q¯σm T a Qg am+g s¯Q c¯σm¯T a Q c g amL˜q˜q g=ıg s ¯˜QT a∂m˜Q−˜QT a∂m¯˜Q g am+ıg s ¯˜Q c¯T a∂m˜Q c−˜Q c¯T a∂m¯˜Q c g am, L q˜q˜g=−ı√2g s ¯Q c¯T a˜Q c¯˜g a−¯˜Q c¯T a Q c˜g a , L˜q˜q gg=g2s¯˜QT a T b˜Qg a m g bm−g2s¯˜Q c¯T a¯T b˜Q c g a m g bm.(38) 3.2.1Quark–Quark–Gluon InteractionThis interaction is given by theﬁrst line given at Eq.(38).Using the doublets we can write our Lagrangian in the following wayL qqg=g s(¯u r¯σm T a rs u s+¯d r¯σm T a rs d s+d c r¯σm¯T a rs d c s)g a m,(39) which is the same as of the MSSM,see Eq.(18),therefore the Feynman rule is again given by Eq.(21).3.2.2Squark–Squark–Gluon InteractionThis interaction comes from the second line at Eq.(38),and can be rewritten asL˜q˜q g=ıg s(¯˜u r T a rs(∂m˜u s)−(∂m˜d r)T a rs˜d s +˜u c r)¯T a rs˜u c s+˜d c r)¯T a rs˜d c s)g am,(40) which is the same of the MSSM,see Eq.(22),and the Feynman rule is given by Eqs.(25,26),as expected.3.2.3Squark-Squark-Gluon-Gluon InteractionThis interaction is given by the last line of Eq.(38),and it is given byL˜q˜q gg=−g2s(¯˜u r T a rs T b st˜u t+¯˜d r T a rs T b st˜d t+˜d c r¯T a rs¯T b st˜d c t)g am g b m,(41)19which is the same as Eq.(27),and the Feynman rule is given at Eq.(29). 3.2.4Gluino-Quark-Squark InteractionThis interaction is given by the third line of Eq.(38),and can be rewritten asL˜q qg=−√λa C−λa C−u c r¯T a rs˜u c s˜u c r¯T a rs u c sλa C+λa C−。

Large radii and string unification

a rXiv:h ep-th/96929v213N ov1996CPTH–S465.0996IEM-FT-143/96hep-th/9609209Large radii and string uniﬁcation ⋆I.Antoniadis a and M.Quir´o s b a Centre de Physique Th´e orique,Ecole Polytechnique,†F-91128Palaiseau,France b Instituto de Estructura de la Materia,CSIC,Serrano 123,28006Madrid,Spain.Abstract We study strong coupling eﬀects in four-dimensional heterotic string models where supersymmetry is spontaneously broken with large internal dimensions,consistently with perturbative uniﬁcation of gauge couplings.These eﬀects give rise to thresholds associated to the dual theories:type I superstring or M-theory.In the case of one large dimension,we ﬁnd that these thresholds appear close to the ﬁeld-theoretical uniﬁcation scale ∼1016GeV,oﬀering an appealing scenario for uniﬁcation of grav-itational and gauge interactions.We also identify the inverse size of the eleventhdimension of M-theory with the energy at which four-fermion eﬀective operators be-come important.Large internal dimensions in string theories have been studied in connection with per-turbative breaking of supersymmetry[1]–[7].Their inverse size is proportional to the scale of supersymmetry breaking which is expected to be of the order of the electroweak scale. The existence of such large dimensions is consistent with perturbative uniﬁcation in a class of four-dimensional models which include some orbifold compactiﬁcations of the heterotic superstring[3].Present experimental limits have been obtained from an analysis of eﬀective four-fermion operators,yielding R−1>∼200GeV or1TeV in the case of one or two large dimensions,respectively[5].The main experimental signature of these models is the direct production of Kaluza-Klein excitations for gauge bosons which can be detected at future colliders[6].The presence of large internal dimensions implies that the ten-dimensional heterotic string is strongly coupled[8].In spite of this,in the class of models mentioned above,the radiative corrections to the four-dimensional couplings remain small[3].This happens for instance when the corresponding Kaluza-Klein excitations are organized in multiplets of N=4supersymmetry(possibly spontaneously broken to N=2[9])leading to cancellations among particles of diﬀerent spins.However,the fact that the ten-dimensional coupling is strong raises the question of possible large corrections to other quantities of the four-dimensional eﬀectiveﬁeld theory,such as non-renormalizable operators[5,10].Although this problem is diﬃcult to handle using perturbative methods,it can be studied using recent results on string dualities.There is a growing evidence that the strongly coupled heterotic string in ten dimen-sions is equivalent to the weakly coupled type I superstring[11]or to the eleven-dimensional M-theory[12].The corresponding duality relations imply the existence of diﬀerent thresh-olds associated to these dual theories where the eﬀective theory changes regime.These thresholds may also appear as energy scales at which non-renormalizable operators be-come important[10].An analysis of type I superstring and M-theory thresholds in modelswith six large internal dimensions reveals that these thresholds appear much below the compactiﬁcation scale R−1,implying for the latter a lower bound∼4×107GeV[10].In this letter we study strong coupling eﬀects in the class of models of refs.[3]–[5] with anisotropic compactiﬁcation space and where supersymmetry breaking is induced by the large internal dimension(s).Weﬁnd that the threshold of dual theories appear now much above the compactiﬁcation scale.Moreover,in the case of one large dimension at the TeV range,these thresholds are close to the experimentally inferred uniﬁcation scale∼1016 GeV,while the inverse size of the eleventh dimension of M-theory is at an intermediate scale ∼1013GeV.This oﬀers an alternative economical scenario for uniﬁcation of gravitational and gauge interactions in the context of open strings or M-theory1.In fact both the uniﬁcation and supersymmetry breaking scales,along with the electroweak one,can in principle be determined by a single dynamical calculation in the low energy theory[4].For type I strings,we establish the precise relation between the open string scale and the uniﬁcation mass in the low energy theory.We also analyze the role that non-renormalizable operators play concerning the diﬀerent thresholds.Weﬁnd that,while the dimension eight operators F4µνlead to the threshold of type I superstrings,the dimension six four-fermion operators reproduce the threshold of the eleventh dimension of M-theory,providing additional evidence for heterotic–M-theory duality.Type I superstring thresholdIn the heterotic string,the ten-dimensional string couplingλH and the string scale M H≡α′−1/2H are expressed in terms of four-dimensional parameters as:λH=2(αG V)1/2M3H M H= αG1For a diﬀerent approach see ref.[13].where(2π)6V is the volume of the six-dimensional internal manifold,αG is the gauge cou-pling at the uniﬁcation scale and M P=G−1/2is the Planck ing the experimentalNvalues M P=1.2×1019GeV andαG∼1/25(assuming minimal supersymmetric uniﬁca-tion),oneﬁnds M H∼1018GeV whileλH grows to huge values as the internal volume gets large.In ten dimensions,the heterotic SO(32)and type I strings are related by duality as [11]:1λI=22where r is the size of the ﬁve “small”internal radii in units of M P .For R −1=1TeV and r =O (1),one obtains M I ∼7×1015GeV which is very close to the gauge coupling uniﬁcation scale.In this way,when going up in energies,the physical picture is the following.Between the TeV and the uniﬁcation scale the eﬀective theory can be studied using perturbation theory in the heterotic string.It behaves as ﬁve dimensional but with peculiarities related to the orbifold character of the compactiﬁcation and the mechanism of supersymmetry breaking [3]–[5].In particular,chiral states (quarks and leptons)do not have Kaluza-Klein excitations,while the couplings run with the energy as in four dimensions.At the uniﬁcation scale,the theory becomes a genuine type I string weakly coupled.In order to make precise the relation of the open string scale M I with the uniﬁcation mass,one has to take into account the string threshold corrections to gauge couplings which can be computed on the type I side for any particular model.A direct one loop computation in the string theory gives [16]:4παG + dtt B i ≡lim Λ→∞ 43 1/4Λ2B i .(6)The integral (5)still has a logarithmic infrared divergence,since as t →∞,B i goes to a constant b i .This is a physical divergence which reproduces the correct low-energyrunning of the gauge couplingsαi with beta function coeﬃcients b i.It can be regularized by introducing an infrared cutoﬀat t=1/α′Iµ2.To compare the string expression with the ﬁeld theoretical couplings in a particular renormalization scheme,we have to add in the r.h.s.of eq.(5)an appropriate constant term[17].In theµ2−b i1/α′Iµ20dtαi =4πµ2+∆I i,(8)where the type I string threshold corrections∆I i in thet B i(t)+b i lnε−b i ln(πeγ).(9)Notice that the threshold corrections in the heterotic string have a similar expression as an integral over the complex modular parameterτof the world-sheet torus.The main diﬀerence is that the ultraviolet divergence is now regularized by the restriction to the fundamental domainΓof the modular group.The limitε→0in eq.(9)can then be taken easily by subtracting and adding b i to the integrand B i:∆H i= Γd2τImτ+lnε −b i ln(πeγ)= Γd2τπ√(6).However,eq.(9)provides a well deﬁned way for computing threshold corrections in type I string models.For the class of models we consider in this work,threshold eﬀects depend mainly on the size of the“small”dimensions r and can provide model dependent corrections to the uniﬁcation scale M I.M-theory thresholdThe strong coupling limit of the heterotic E8×E′8superstring in ten dimensions is believed to be described by the eleven-dimensional M-theory compactiﬁed on the semi-circle S1/Z2of radiusρ[12].The relations between the eleven-and ten-dimensional parameters are:M11=M H √λH 1/3ρ−1=12λH M H,(11) where we have deﬁned the eleven-dimensional scale M11=2π(4πκ2)−1/9[10].When the ten-dimensional heterotic coupling is large(λH≫1),the radius of the semi-circle is large and M-theory is weakly coupled on the world-volume.Using eq.(1),one can express M11andρin terms of the four-dimensional parameters: M11=(2αG V)−1/6ρ−1= 2αG 3/2r−5/2R−1/2M1/2P.(13) For R−1=1TeV and r=O(1),one obtains M11∼3×1016GeV which essentially coincideswith the gauge coupling uniﬁcation scale2.Moreover,the eleventh dimension threshold is at the intermediate scaleρ−1∼4×1013GeV.Thus,in the region between the TeV and the intermediate scale,the eﬀectiveﬁve dimensional theory has a perturbative heterotic string description(as in the case of type I strings below M I).Above the intermediate scale,strong coupling eﬀects are relevant and the eleventh dimension of M-theory opens up.Finally,at the uniﬁcation scale,gravity becomes important through M-theory interactions.The situation is reversed if there are more than two large dimensions in the internal volume V.The threshold of the eleventh dimensionρ−1is now below the compactiﬁcation scale R−1[19].In this case,as going up in energies,the theory wouldﬁrst becomeﬁve-dimensional at the scaleρ−1,while the other large dimensions will open up at a higher scale R−1<M11.Moreover,in all these regions the theory does not have a perturbative string description.For the case of two large dimensions R−1∼10−2ρ−1,while for the case of six R−1∼M11.One may ask the question whether the scale of the eleventh dimensionρ−1can appear on the heterotic side as a threshold at which some non-renormalizable eﬀective operators become important,in a similar way as the open string threshold M I was determined from an analysis of the dimension eight operators F4µν.In the following we will argue that the dimension six four-fermion operators are the relevant ones.Let us consider indeed the eﬀective interaction of four chiral fermions corresponding to twisted states in orbifold models with2≤d≤6large internal dimensions of common size R.At energies below R−1the(tree-level)result can be obtained directly in the eﬀectiveﬁeld theory by summing over all Kaluza-Klein excitations exchanged between the two fermion lines[5].In the case where all fermions arise at the sameﬁxed point of the orbifold,the coupling of two twisted states with one excited(untwisted)mode,labeled by the d-dimensional vector n,is[20]:g n=gδ− n2α′H/2R2,(14) where g is the four-dimensional string coupling and the value of the constantδ≥1depends on the orbifold.The strengthξ2of the corresponding eﬀective operator can be written as:ξ2=αG R2 { n} =0δ− n2α′H/R2from E8×E′8arises at the twoﬁxed points of the semi-circle and has only gravitational interactions with Kaluza-Klein states associated to the eleventh dimension[10].Therefore, the above four-fermion operator could not be used to extract information on the scaleρ−1.Finally,it would be interesting to consider,in the context of M-theory,the possibility of breaking spontaneously N=1supersymmetry using the radius of the eleventh dimension by a mechanism analogous to the Scherk-Schwarz compactiﬁcation[22].Here,there are two possibilities:•In the case where M-theory is compactiﬁed on a seven-dimensional space which does not contain the semi-circle as a product factor,the scale of supersymmetry breaking in the observable sector(m susy)would be generically proportional toρ−1∼1TeV.From eq.(12),the d additional large dimensions would then open up at an intermediate scale varying between106and1013GeV corresponding to d=3and d=6,respectively.•In the case where the internal manifold of M-theory is M6×S1/Z2,supersymmetry is broken only in the gravitational sector(at the lowest order)and will be communicated to the observable world by gravitational interactions,yielding m susy∼ρ−2/M P.As a result,the threshold of the eleventh dimensionρ−1should be at an intermediate scale∼1012GeV.Interestingly enough,for d=6the inverse size of the six-dimensional internal manifold M6is now of the order of the gauge coupling uniﬁcation mass∼1016 GeV.It is suggestive that this situation could describe ordinary gaugino condensation in the dual strongly coupled heterotic string[23].References[1]I.Antoniadis,C.Bachas,D.Lewellen and T.Tomaras,Phys.Lett.B207(1988)441.[2]C.Kounnas and M.Porrati,Nucl.Phys.B310(1988)355;S.Ferrara,C.Kounnas,M.Porrati and F.Zwirner,Nucl.Phys.B318(1989)75;C.Kounnas and B.Rostand, Nucl.Phys.B341(1990)641.[3]I.Antoniadis,Phys.Lett.B246(1990)377;Proc.PASCOS-91Symposium,Boston1991(World Scientiﬁc,Singapore)p.718;K.Benakli,Phys.Lett.B386(1996)106.[4]I.Antoniadis,C.Mu˜n oz and M.Quir´o s,Nucl.Phys.B397(1993)515.[5]I.Antoniadis and K.Benakli,Phys.Lett.B326(1994)69.[6]I.Antoniadis,K.Benakli and M.Quir´o s,Phys.Lett.B331(1994)313.[7]C.Bachas,hep-th/9503030;Proc.Topics in the Theory of Fundamental Interactions,Maynooth,Ireland(1995)p.127.[8]V.S.Kaplunovsky,Phys.Rev.Lett.55(1985)1036;M.Dine and S.Seiberg,Phys.Lett.B162(1985)299.[9]E.Kiritsis,C.Kounnas,P.M.Petropoulos and J.Rizos,hep-th/9606087.[10]E.C´a ceres,V.S.Kaplunovsky and I.M.Mandelberg,hep-th/9606036.[11]J.Polchinski and E.Witten,Nucl.Phys.B460(1996)525.[12]P.Hoˇr ava and E.Witten,Nucl.Phys.B460(1996)506;Nucl.Phys.B475(1996)94.[13]E.Witten,Nucl.Phys.B471(1996)135;[14]C.Bachas,unpublished;J.D.Lykken,Phys.Rev.D54(1996)3693.[15]A.A.Tseytlin,Phys.Lett.B367(1996)84;Nucl.Phys.B467(1996)383.[16]C.Bachas and C.Fabre,Nucl.Phys.B476(1996)418;I.Antoniadis,C.Bachas,C.Fabre,H.Partouche and T.R.Taylor,hep-th/9608012.[17]V.S.Kaplunovsky,Nucl.Phys.B307(1988)145and erratum,ibid.B382(1992)436.[18]I.Antoniadis,J.Ellis,caze and D.V.Nanopoulos,Phys.Lett.B268(1991)188.[19]T.Banks and M.Dine,hep-th/9605136.[20]S.Hamidi and C.Vafa,Nucl.Phys.B279(1987)465;L.Dixon,D.Friedan,E.Martinec,S.Shenker,Nucl.Phys.B282(1987)13.[21]C.Vafa and E.Witten,hep-th/9507050.[22]J.Scherk and J.H.Schwarz,Phys.Lett.B82(1979)60;P.Fayet,Phys.Lett.B159(1985)121;Nucl.Phys.B263(1986)649.[23]For a diﬀerent approach,see:P.Horava,hep-th/9608019(to appear in Phys.Rev.D).。

Beyond The Standard Model

Beyondห้องสมุดไป่ตู้The Standard Model
J. W. Moﬀat
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada (February 1, 2008) An overview of uniﬁed theory models that extend the standard model is given. A scenario describing the physics beyond the standard model is developed based on a ﬁnite quantum ﬁeld theory (FQFT) and the group G=SO(3, 1) ⊗ SU (3) ⊗ SU (2) ⊗ U (1). The ﬁeld theory is Poincar´ e invariant, gauge invariant, ﬁnite and unitary √ to all orders of perturbation theory and has a fundamental scale which is chosen to be ΛF = 1/ GF ∼ 300 GeV, where GF is the Fermi coupling constant. The physical Higgs particle is protected from acquiring a large mass beyond ∼ 1 TeV, removing the gauge hierarchy problem associated with the scalar Higgs ﬁeld. This avoids the need for a composite Higgs ﬁeld or supersymmetry. The coupling constants and the fermion masses can be calculated from a set of low-energy relativistic eigenvalue equations based on truncated Green’s functions and the FQFT, reducing the number of free parameters in the model without a grand uniﬁcation scheme. The proton is predicted to be stable. Quantum gravity is perturbatively ﬁnite and unitary to all orders.

A natural solution to the μ-problem

problems result in a brief that the high-energy physics should be described by supergravity. The hierarchy of mass scales can be naturally explained if supersymmetry is exact at high energies but becomes spontaneously broken, above MW , by a non-perturbative mechanism. At low energies, this mechanism should decouple from the observable physics and supersymmetry would appear to be broken by explicit soft terms in the eﬀective low-energy Lagrangian. ¿From the low-energy point of view, these soft terms - which include the masses of the super-partners of all known particles - are simply independent input parameters, just like the gauge and the Yukawa couplings of the Standard Model, but from the high-energy point of view, they are calculable in terms of the supergravity couplings. Because of its non-renormalizability, supergravity itself has to be thought of as an eﬀective theory, valid below the Planck scale Mpl . Currently, the best candidate for a consistent theory governing the physics of energies near the Planck scale is the superstring theory. However, in general, we have no reliable stringy mechanisms that lead to non-perturbative spontaneous breaking of supersymmetry. Instead, one generally assumes that the dominant non-perturbative eﬀects emerges at energies well below Mpl . Gaugino condensation in an asymptotically free hidden sector of the eﬀective supergravity is a prime example of this type of mechanisms. These models, however, face another problem of naturalness which we call the µ-problem where µ is the coeﬃcient of the H1 H2 term in the low energy superpotential and H1 and H2 denote the usual Higgs SU(2) doublet chiral superﬁelds. In the minimal supersymmetric standard model(MSSM) the matter content consists of three generations of quarks and lepton superﬁelds plus two higgs doublets H1 and H2 of opposite hypercharge. The most general eﬀective observable superpotential has the form: W M SSM = W 0 + W µ−term W0 =

Flavoring the gravity dual of N=1 Yang-Mills with probes

Flavoring the gravity dual of N = 1 Yang-Mills with probes
´ Carlos N´ un ˜ ez † 1 , Angel Paredes ∗
† 2
and Alfonso V. Ramallo ∗
3
Center for Theoretical Physics, Massachusetts Institute of Technology Cambridge, MA 02139, USA
US-FT-3/03 MIT-CPT/3441 hep-th/0311201
November 2003
1 2
nunez@ angel@c.es 3 alfonso@c.es
1
Introduction
The gauge/string correspondence, an old proposal due to ’t Hooft [1], is now well understood in the context of maximally supersymmetric super Yang-Mills (SYM) theories. Indeed, the so-called AdS/CFT correspondence is a conjectured equivalence between type IIB string theory on AdS5 × S 5 and N = 4 SYM theory [2]. In the large ’t Hooft coupling limit, the N = 4 SYM theory is dual to the type IIB supergravity background corresponding to the near-horizon geometry of a stack of parallel D3-branes, whose metric is precisely that of the AdS5 × S 5 space. There are nowadays a lot of non-trivial tests of this duality (for a review see [3]). The extension of the gauge/string correspondence to theories with less supersymmetries is obviously of great interest. A possible way to obtain supergravity duals of SYM theories with reduced supersymmetry is to consider branes wrapping supersymmetric cycles of CalabiYau manifolds [4]. In order to preserve some supersymmetry the normal bundle of the cycle within the Calabi-Yau space has to be twisted [5]. Gauged supergravities in lower dimensions provide the most natural framework to implement this twisting. In these theories the gauge ﬁeld can be used to ﬁber the cycle in which the brane is wrapped in such a way that some supersymmetries are preserved. In this paper we will restrict ourselves to the case of the supergravity dual of N = 1 SYM. This background, which corresponds to a ﬁvebrane wrapping a two-cycle, was obtained in ref. [6] from the solution found in ref. [7] representing non-abelian magnetic monopoles in four dimensions. The geometry of this background is smooth and leads to conﬁnement and chiral symmetry breaking. Actually, if only the abelian part of the vector ﬁeld of seven dimensional gauged supergravity is excited, one obtains a geometry which is singular at the origin and coincides with the smooth one at large distances, i.e. in the UV. Therefore, the singularity at the origin is resolved by making the gauge ﬁeld non-abelian, in complete analogy to what happens with the resolution of the Dirac string by the ’t Hooft-Polyakov monopole. Moreover, as argued in ref. [8], the same mechanism that de-singularizes the supergravity solution also gives rise to gaugino condensation. Based on this observation, the NSVZ beta function can be reproduced at leading order [9, 10, 11]. Other aspects of this supergravity dual have been studied in ref. [12](for a review see [13]). Most of the analysis carried out with the background of [6] do not incorporate quarks in the fundamental representation which, in a string theory setup, correspond to open strings. In order to introduce an open string sector in a supergravity dual it is quite natural to add Dbrane probes and see whether one can extract some information about the quark dynamics. As usual, if the number of brane probes is much smaller than those of the background, one can assume that there is no backreaction of the probe in the bulk geometry. In this paper we follow this approach and we will probe with D5-branes the supergravity dual of N = 1 SYM. The main technique to determine the supersymmetric brane probe conﬁgurations is kappa symmetry [14], which tells us that, if ǫ is a Killing spinor of the background, only those embeddings for which a certain matrix Γκ satisﬁes Γκ ǫ = ǫ . (1.1)

An Extension for Direct Gauge Mediation of Metastable Supersymmetry Breaking

Abstract We study the direct mediation of metastable supersymmetry breaking by a Φ2 (Φ is the meson ﬁeld) deformation to the ISS model and extend it by splitting Φ into two parts and gauging the ﬂavor symmetry. We ﬁnd that with such an extension the enough longlived metastable vacua can be obtained and the proper gaugino masses can be generated. Further, this allows for constructing a kind of models which can avoid the Landau pole problem even if the supersymmetry breaking scale is as low as ∼ 107 GeV.
Dec 2007
1
Introduction
Dynamical supersymmetry breaking is a convincing scenario to solve the gauge hi-
erarchy problem, but it seemed an exceptional phenomenon because its realistic models in general have to statisfy many theoretical requirements. On the other hand, the phenomenological considerations are also complex when these dynamical SUSY breaking eﬀects are mediated to the visible sector. Recently, Intriligator, Seiberg and Shih (ISS) [1] discovered the meta-stable supersymmetry breaking in a surprising context of vector-like theory. Their model is just the N = 1

On the No-Go Theorem of Supersymmetry Breaking

a r X i v :h e p -t h /0308094v 1 14 A u g 2003On the No-Go Theorem of Supersymmetry BreakingWung-Hong Huang*Department of PhysicsNational Cheng Kung UniversityTainan,70101,Taiwan ABSTRACT It is proved that,even if the gauge symmetry has been broken spontaneously at tree level,supersymmetry would never break through any ﬁnite orders of perturbation if it is not broken classically.*E-mail:whhwung@.twPhysics Letters B179(1986)92Supersymmetry[1],which is the only graded Lie algebra of symmetries of the S-matrix that is consistent with the relativistic quantumﬁeld theory[2],could exhibit improved ultraviolet behavior[3,4]and provide us with a solution of the gauge hierarchy problem[5]. It has been shown that,in a broad class ofﬁeld theories[6],if supersymmetry is not broken at the classical level then it is not broken by radiative corrections.Those are the models with only chiral superﬁelds and models whose gauge symmetry and supersymmetry are unbroken classically.Recently Ovrut and Wess[7]have examined a class of supersymmetric theories whose internal symmetry is completely broken spontaneously.After extending the Rζgauge-ﬁxing [8]procedure to the supersymmetric theories they calculated the one-loop corrections to the auxiliaryﬁelds.It is found that the invariance of the superpotential under the complexiﬁca-tion of the internal symmetry group allows the vacuum expectation values of scalarﬁelds to be adjusted to make the corrected D terms vanish while keeping the F terms zero.Hence, the supersymmetry is not broken.In this paper,without explicitly calculating the quantum correction,we will prove that this another no-go theorem is always true in any supersym-metric gauge theory.Furthermore,the internal symmetry is allowed not to be spontaneously broken completely.To break the supersymmetry inﬁnite orders of perturbation,one must get an expectation value of the F or Dﬁing the supergraphic techniques[4]it can be easily seen that the Fﬁeld does not give an expectation value if supersymmetry is unbroken at the classical level.Also,there is no induced expectation value of the D term if both gauge symmetry and supersymmetry are unbroken classically[9].In other words,the Coleman-Weinberg mecha-nism[10]is not possible.This is a no-go theorem that has been well established.When the gauge symmetry is already broken spontaneously at the classical level,quantum corrections for the D terms(but not F terms)are induced.In the following sections we shall show that, for any induced Dﬁeld,through some suitable transformations of complexiﬁcation of the internal symmetry group one can alwaysﬁnd vacuum expectation values of the scalarﬁelds to make the D terms vanish.(Recall that the superpotential is invariant under the complex-iﬁed internal symmetry group so that the F terms remain zero.)Hence,supersymmetry isnot broken as claimed by Ovrut and Wess[7].Considerﬁrst the supersymmetric theory with abelian gauge symmetry group.Let Φ+i,Φ−j,andΦ0k be chiral superﬁelds with positive charge q+i,negative charge q−j and zero charge,respectively.In order to preserve the supersymmetry in the n-th order of per-turbation,there must exist the vacuum expectation values a(n)+i,a(n)−j and a(n)0k of scalarﬁelds in the chiral superﬁelds,Φ+i,Φ−j,andΦ0k,which satisfyi q+i|a(n)+i|2+ j q−j|a(n)−j|2=C(n)(a(n)+i,a(n)−j,a(n)0k),(1) where the functional form of C(n)depends on the order of perturbation.At tree level,the C(n)term does not show up.After the quantum correction,it is induced.Toﬁnd the solution of(1)we set,in the spirit of perturbation theory,the values of a(n)+i,a(n)−j,and a(n)0k in the argument of the function C(n)to the(n−1)th order only and the equation which we need to solve becomesi q+i|a(n)+i|2+ j q−j|a(n)−j|2=C(n)(a(n−1)+i,a(n−1)−j,a(n−1)0k),(2)The existence of a solution for the above equation can be proved by the following observa-tions.First,when the gauge group is already broken classically then there must exist at least one nonzero vacuum expectation value among,a(0)+i or a(0)−j which satisﬁesi q+i|a(0)+i|2+ j q−j|a(0)−j|2=0.(3) Furthermore,because q+i is positive while q−j is negative,there must exist at least one nonzero value a(0)+i accompanied with at least one nonzero value a(0)+i in order to satisfy the above equation.Next,consider the one-loop correction.We see that the C(1)term is only a function of tree-level vacuum expectation values,so it can be regarded as aﬁxed number when we want toﬁnd the one-loop vacuum expectation values a(1)+i,a(1)−j,a(1)0k which satisfy eq.(2).Therefore,regardless of the value of C(1),it is easily seen that we can alwaysﬁnd a real parameterθsuch that a(1)+i and a(1)−j,which we are searching now,are just the transformed values of a(0)+i and a(0)−j under the complexiﬁed internal symmetry group,i.e.a(1)+i=a(0)+i exp(q+iθ),a(1)−j=a(0)−j exp(q−jθ).(4) For example,if C(1)is positive(negative)thenθmust be positive(negative).The point is that there exists at least one nonzero value a(0)+i accompanied with at1east one nonzero value a(0)−j.For higher loop corrections,the above discussion can still be used.We next examine supersymmetric theories with non-abelian internal symmetry which is broken spontaneously.But the complete breaking is not necessary.Let Tαbe the generators of the gauge group and the chiral superﬁeldΦbe its representation.(Extension to many diﬀerent representations is straightforward.)The vacuum expectation value of the scalar ﬁeld inΦis denoted as a.To preserve the supersymmetry in the n th order of perturbation there must exist an a(n)to satisfya(n)†Tαa(n)=C(n)α(a(n)),(5) where the functional form of C(n)αdepends on the order of the perturbation.In the spirit of perturbation theory,as in the abelian case,we let a(n)in the argument of C(n)αbe a(n−l) and(5)becomesa(n)†Tαa(n)=C(n)α(a(n−1)),(6) To solve a(n)in the above equation we can regard a(n−l)asﬁxed ing this property,we prove that there always exists a solution a(n)which satisﬁes(6)regardless of the value of C(n)αwhich depends on the model and the order of perturbation.Hence, supersymmetry is not broken through quantum corrections.Consider a system whose gauge symmetry has been spontaneously broken at tree level; there exists at least a generator T r which satisﬁesT r a(0)=0,(7)where T r belongs to the Cartan subalgebra with diagonal elements only,i.e.=t r iδij.(8)T rijEq.(7)then tells us that there must exist at least one nonzero value t r i a(0)i.Also,eq.(6) becomesa(0)†T r a(0)= i t r i|a(0)i|2=0.(9)This shows that there must exist at least one positive and one negative value among t r i|a(0)i|2.If we rotate a(0)in Cartan subspace of the complexiﬁed internal symmetry group1a(0)→a(0)exp(,(11)2 r t r iθrthen it is obvious that the slopes of L(θr)are just a(0)′†T r a(0)′and can take on any value.It is this property that,no matter what the value of α|C(n)α|2,we can alwaysﬁnd a set of θr to satisfyα|a(0)′†T r a(0)′|2= α|C(n)α|2.12) Finally,with the fact that the a(0)′†T r a(0)′are transformed as vectors in the regular representation of the internal symmetry group,we can rotate them to satisfy(6).Therefore, the solution of(6)is found and supersymmetry is still preserved at the quantum level.REFERENCES1.J.Wess and B.Zumino,Nucl.Phys.B70(1974)39.2.R.Hagg,J.Lopuszanski and M.Sohnius,Nucl.Phys.B88(1975)257.3.J.Wess and B.Zumino,Phys.Lett.B49(1974)52;J.Iliopoulos and B.Zumino,Nucl.Phys.B79(1974)310.4.M.T.Grisaru,W.Siegel and M.Rocek,Nucl.Phys.B159(1979)429.5.E.Witten,Phys.Lett.B105(1981)267.6.B.Zumino,Nucl.Phys.B89(1975)535;ng,Nucl.Phys.B114(1976)123;S.Weinberg,Phys.Lett.B62(1976)111.7.B.A.Ovrut and J.Wess,Phys.Rev.D25(1982)409.8.K.Fujikawa,B.W.Lee and A.I.Sanda,Phys.Rev.D6(1972)2923.9.E.Witten,Nucl.Phys.B185(1981)513.10.S.Coleman and E.Weinberg,Phys.Rev.D7(1973)888.。

Dynamical SUSY Breaking in Meta-Stable Vacua

a r X i v :h e p -t h /0602239v 3 1 A p r 2006UCSD-PTH-06-03Dynamical SUSY Breaking in Meta-Stable VacuaKenneth Intriligator 1,2,Nathan Seiberg 2and David Shih 31Department of Physics,University of California,San Diego,La Jolla,CA 92093USA 2School of Natural Sciences,Institute for Advanced Study,Princeton,NJ 08540USA 3Department of Physics,Princeton University,Princeton,NJ 08544USA Dynamical supersymmetry breaking in a long-lived meta-stable vacuum is a phenomeno-logically viable possibility.This relatively unexplored avenue leads to many new models of dynamical supersymmetry breaking.Here,we present a surprisingly simple class of mod-els with meta-stable dynamical supersymmetry breaking:N =1supersymmetric QCD,with massive ﬂavors.Though these theories are strongly coupled,we deﬁnitively demon-strate the existence of meta-stable vacua by using the free-magnetic dual.Model buildingchallenges,such as large ﬂavor symmetries and the absence of an R-symmetry,are easily accommodated in these theories.Their simplicity also suggests that broken supersymmetry is generic in supersymmetric ﬁeld theory and in the landscape of string vacua.February 20061.Introduction1.1.General RemarksAtﬁrst glance,dynamical supersymmetry breaking appears to be a rather non-generic phenomenon in supersymmetric gauge theory.The non-zero Witten index of N=1Yang-Mills theory immediately implies that any N=1supersymmetric gauge theory with mas-sive,vector-like matter has supersymmetric vacua[1].So theories with no supersymmetric vacua must either be chiral,as in the original examples of[2,3],or if they are non-chiral, they must have massless matter,as in the examples of[4,5].The known theories that sat-isfy these requirements and dynamically break supersymmetry look rather complicated, and applications to realistic model building only compounds the complications.The result has been a literature of rather baroque models of dynamical supersymmetry breaking and mediation.For reviews and references,see e.g.[6].We point out that new model building avenues are opened up by abandoning the prej-udice that models of dynamical supersymmetry breaking must have no supersymmetric vacua.This prejudice is unnecessary,because it is a phenomenologically viable possibility that we happen to reside in a very long lived,false vacuum,and that there is a super-symmetric vacuum elsewhere inﬁeld space.Meta-stable supersymmetry breaking vacua have been encountered before in the literature of models of supersymmetry breaking and mediation;some examples are[7-9].Indeed,even if the supersymmetry breaking sector has no supersymmetric vacua,there is a danger that the mediation sector will introduce supersymmetric vacua elsewhere.Such encounters of meta-stable supersymmetry break-ing are generally accompanied with a(justiﬁed)apology for the aesthetic defect and,in favorable cases,it is shown that the lifetime can nevertheless be longer than the age of the Universe.The novelty here is that we accept meta-stable vacua from the outset,even in the supersymmetry breaking sector.This approach leads us immediately to many new and much simpler models of supersymmetry breaking.Classic constraints,needed for hav-ing no supersymmetric vacua,no longer constrain models of meta-stable supersymmetry breaking.For instance,theories with non-zero Witten index and/or with no conserved U(1)R symmetry[3,10]can nevertheless have meta-stable supersymmetry breaking vacua.A condition for supersymmetry breaking that does still apply in the meta-stable context is the need for a massless fermion to play the role of the Goldstino.But even this condition can be subtle:the massless fermion can be present in the low-energy macroscopic theory,1even if it is not obvious in the original,ultraviolet,microscopic theory.This happens in our examples.Phenomenologically,we would like the lifetime of our meta-stable state to be longer than the age of the Universe.Moreover,the notion of meta-stable states is meaningful only when they are parametrically long lived.It is therefore important for us to have a dimensionless parameter,ǫ,whose parametric smallness guarantees the longevity of the meta-stable state.In our examples,ǫis given by a ratio of a mass and a dynamical scale,µmǫ≡1.2.Our main exampleOur main example of meta-stable dynamical supersymmetry breaking in this paper is surprisingly simple:N=1supersymmetric SU(N c)QCD,with N f massive fundamental ﬂavors.In order to have control over the theory in the IR,we take N f in the free magnetic range[12-14],N c+1≤N f<3mN c,(1.3)arises as the infrared free,low-energy2eﬀective theory of the magnetic dual[13],with N=N f−N c andµ∼√potential for the pseudo-moduli can be computed using the one-loop correction to the vacuum energy,V(1) eff =1Λ2≡1Λ2−Tr m4F logm2F1The ultraviolet cutoﬀΛin(1.4)can be absorbed into the renormalization of the coupling constants appearing in the tree-level vacuum energy V0.In particular,STr M4is independent of the pseudo-moduli.4group,taking N f>3N(which becomes N f<32.The Macroscopic Model:Part IIn this section we discuss our macroscopic theory(1.3)without the gauge interactions. This is a Wess-Zumino model with global symmetry groupSU(N)×SU(N f)2×U(1)B×U(1)′×U(1)R(2.1) (later we will identify N=N f−N c),with N f>N and the following matter contentSU(N)SU(N f)SU(N f)U(1)B U(1)′U(1)RΦ11110˜ϕThe vacua of maximal unbroken global symmetry are(up to unbrokenﬂavor rotations)Φ0=0,ϕ0=˜ϕ0=µ1I N,(2.7) This preserves an unbroken SU(N)D×SU(N f−N)×U(1)B′×U(1)R,as well as a discrete charge conjugation symmetry that exchangesϕand˜ϕ.We now examine the one-loop eﬀective potential of the classical pseudo-ﬂat directions around the vacua(2.7).To simplify the presentation,we will expand around(2.7)and show that the classical pseudo-moduli there get positive mass-squared.To see what the lightﬁelds are,we expand around(2.7)using the parametrizationΦ= δYδZ T δ˜Zδ Φ ,ϕ= µ+12(δχ++δχ−)12(δρ++δρ−) ,˜ϕT=µ+12(δχ+−δχ−)12(δρ+−δρ−)(2.8)(HereδY andδχ±are N×N matrices,andδZ,δ˜Z,andδρ±are(N f−N)×N matrices.) The potential from(2.4)gives most of theﬁelds tree-level masses∼|hµ|.There are also massless scalars,some of which are Goldstone bosons of the broken global symmetries:µ∗|µ|δρ+,Im µ∗|µ|δχ−+h.c.(2.10) These pseudo-moduli acquire masses,starting at one-loop,from their couplings to the massiveﬁelds.The eﬀective theory for the pseudo-moduli has the formL eff=Tr∂(δ Φ)†∂(δ Φ)+1The one-loop eﬀective potential for the pseudo-moduli can be computed from the one-loop correction (1.4)tothe vacuumenergy,in thebackground where the pseudo-moduli have expectation values.Expanding to quadratic order around the vacua (2.7),the eﬀective potential for the pseudo-moduli must be of the formV (1)eff = h 4µ2 18π2(N f −N ),b =log 4−18π2 12Equivalently,it is easily veriﬁed that only planar diagrams contribute at one loop.83.The Macroscopic Model:Part II–Dynamical SUSY RestorationWe now gauge the SU(N)symmetry of the previous section.We are interested in the case N f>3N,where the SU(N)theory is IR free instead of asymptotically free.Thus the theory has a scaleΛm,above which it is strongly coupled.(The subscript m onΛm is for“macroscopic.”)The running of the holomorphic gauge coupling of SU(N)is given bye−8π2/g2(E)+iθ= Eg2 A(Trϕ†T Aϕ−Tr˜ϕT A˜ϕ†)2.(3.2)2The D-term potential(3.2)vanishes in the vacua(2.7),so(2.7)remains as a minimum of the tree-level potential.The SU(N)gauge symmetry is completely Higgsed in this vacuum.Through the super-Higgs mechanism,the SU(N)gaugeﬁelds acquire mass gµ, the erstwhile Goldstone bosons Im(µ∗δχ−/|µ|)′are eaten(the prime denotes the traceless part),and the erstwhile pseudo-moduliδ χ′=Re(µ∗δχ−/|µ|)′get a non-tachyonic,tree-level mass gµfrom(3.2).3Thus,theﬁeldsδ Φand Trδ χremain as classical pseudo-moduli.We should compute the leading quantum eﬀective potential for these pseudo-moduli, as in the previous section,to determine whether the vacua(2.7)are stabilized,or develop tachyonic directions.Actually,no new calculation is needed:the eﬀect of the added SU(N)gaugeﬁelds drops out in the leading order eﬀective potential for the pseudo-moduli. The reason is that the tree-level spectrum of the massive SU(N)vector supermultiplet is supersymmetric,so its additional contributions to the supertrace of(1.4)cancel.To see this,note that the SU(N)gaugeﬁelds do not directly couple to the supersymmetrybreaking:the D-terms(3.2)vanish on the pseudo-ﬂat space,and the non-zero expectation values ofϕand˜ϕ,which give the SU(N)gaugeﬁelds their masses,do not couple directly to any non-zero F terms.We conclude that the leading order eﬀective potential(2.14)for the pseudo-moduli is unaﬀected by the gauging of SU(N).The vacua are as in(2.7),with broken supersymmetry and no tachyonic directions.Though gauging the SU(N)does not much aﬀect the supersymmetry breaking vacua (2.7),it does have an important eﬀect elsewhere inﬁeld space:it leads to supersymmetric vacua.To see this,consider givingΦgeneral,non-zero expectation values.By the super-potential(2.4),this gives the SU(N)fundamentalﬂavors,ϕand˜ϕ,mass hΦ .Below the energy scale hΦ ,we can integrate out these massiveﬂavors.The low-energy theory is then SU(N)pure Yang-Mills,with holomorphic coupling given bye−8π2/g2(E)+iθ= ΛLΛN f−3N m E3N.(3.3) In the last equality,we matched the running coupling to that above the energy scale hΦ , as given in(3.1).The low-energy theory has superpotentialW low=N(h N fΛ−(N f−3N)m detΦ)1/N−hµ2TrΦ,(3.4) where theﬁrst term comes from SU(N)gaugino condensation,upon using(3.3)to relate ΛL toΛm.We stress that the appearance ofΛm in(3.4)does not signify that we are including any eﬀects coming from physics at or above the ultraviolet cutoﬀΛm.Rather, it appears because we have expressed the infrared free coupling g as in(3.1).Extremizing the superpotential(3.4),weﬁnd N f−N supersymmetric vacua at.(3.5) hΦ =Λmǫ2N/(N f−N)1I N f=µ1ΛmNote that,for|ǫ|≪1,|µ|≪| hΦ |≪|Λm|.(3.6) Because hΦ is well below the Landau pole atΛm,this analysis in the low-energy,macro-scopic theory is justiﬁed and reliable.As we will discuss in section7,|µ|≪| hΦ |also guarantees the longevity of the meta-stable,non-supersymmetric vacua(2.7).We see here an amusing phenomenon:dynamical supersymmetry restoration,in a theory that breaks supersymmetry at tree-level.ForΛm→∞withµﬁxed,the theorybreaks supersymmetry.ForΛm large butﬁnite(corresponding to small but nonzeroǫ),a supersymmetric vacuum comes in from inﬁnity.The relevant non-perturbative eﬀect arises in an IR free gauge theory,and it can be reliably computed.The existence of these supersymmetric vacua elsewhere inﬁeld space implies that the non-supersymmetric vacua of the previous section become only meta-stable upon gauging SU(N).The model with gauged SU(N)therefore exhibits meta-stable supersymmetry breaking.We shall realize it dynamically in section5.We note that our conclusions are in complete accord with the connection of[3,10]be-tween the existence of a U(1)R symmetry and broken supersymmetry.The theory of the previous section has a conserved U(1)R symmetry,and it has broken supersymmetry.In the theory of this section,there is no conserved U(1)R symmetry,because it is anomalous under the gauged SU(N);this breaking is explicit in(3.4).Correspondingly,there are supersymmetric vacua.For Φ near the origin,the SU(N)gauge theory is IR free,so the U(1)R symmetry returns as an accidental symmetry of the infrared theory.So super-symmetry breaking in our meta-stable vacuum near the origin is related to the accidental R-symmetry there.4.Eﬀects from the underlying microscopic theoryThe theory we discussed in the previous sections is IR free and therefore it cannot be a complete theory.It breaks down at the UV scale|Λm|where its gauge interactions becomelarge.(The coupling h in(2.4)also has a Landau pole;for simplicity we discuss only a single scale|Λm|.)In this section we will examine whether our results above depend on the physics at the scale|Λm|which we do not have under control.The only dimensionful parameter of the low energy theory isµand therefore,we will assume|ǫ|= µ|Λm|2Tr(Φ†Φ)2+...,(4.2)with c being a dimensionless number of order one.The standard decoupling argument is based on the fact that such high dimension operators are suppressed by inverse powers of |Λm|and therefore they do not aﬀect the dynamics of the low energy theory.Let us explore in more detail this fact and its relation to the one-loop computation of the eﬀective potential described in section2.There,we calculated the eﬀect of super-symmetry breaking mass terms on the low energy eﬀective potential of the pseudo-ﬂat directions.In that computation we focused on the lightﬁelds,whose mass is of orderµ(for simplicity,we set h=1),and we neglected the modes with mass of orderΛm.Can the eﬀect of these modes,whose masses are also split by supersymmetry breaking,change our conclusion about the eﬀective potential?Our one-loop eﬀective potential(2.14)is proportional to|µ2|,and is thus not real analytic in the parameterµ2appearing in the superpotential.This non-analyticity is because the modes that we integrated out become massless asµ→0,so their contribution to the eﬀective potential is singular there.On the other hand,corrections from heavier modes,whose masses are of orderΛm,are necessarily real analytic inµ2.In particular,the leading correction from the microscopic theory to the mass of the pseudo-modulus must have coeﬃcient|µ2|2/|Λm|2=|µ2ǫ2|≪|µ2|.Such corrections are much smaller than our result from the low-energy macroscopic theory.One way to see that is to integrate out the massive modes forµ=0and summarize the eﬀect in a correction to the K¨a hler potential as in(4.2).Then we can use this corrected K¨a hler potential with the tree level superpotential toﬁnd the eﬀect on the pseudo-ﬂat directions.These corrections are∼|µ2ǫ2|,and are negligible.This fact is signiﬁcant.Without knowing the details of the microscopic theory,we cannot determine these loop eﬀects involving modes with mass∼Λm.We cannot even determine the sign of the dimensionless coeﬃcients like c in(4.2),and therefore we cannot determine whether they bend the pseudo-ﬂat directions upward or downward.Fortunately, these eﬀects which we cannot compute are smaller than the one loop eﬀects in the low energy theory which we can compute.The latter have the eﬀect of stabilizing our vacuum.Of course,this discussion about the irrelevance of irrelevant operators which are sup-pressed by powers ofΛm is obvious and trivial.However,in equation(3.4)we took into account a nonperturbative eﬀect which leads to a superpotential which is suppressed by powers ofΛm.We are immediately led to ask two questions.First,how come this non-renormalizable interaction is reliably computed even though it depends onΛm?Second,given that we consider this interaction,why is it justiﬁed to neglect other terms as in (4.2)which are also suppressed bypowers of Λm ?Let us ﬁrst address the ﬁrst question.As in (3.1),Λm appears as a way to parameterize the infrared free gauge coupling g ,at energy scales below |Λm |.This is conceptually diﬀerent from the appearance of |Λm |in (4.2),which has to do with eﬀects from the microscopic theory,above the Landau pole scale.The superpotential (3.4)is generated by low energy eﬀects and therefore it is correctly computed in the low energy eﬀectivetheory.As a check,the resulting expectation value of Φ(3.5)is much smaller than Λm and therefore it is reliably calculated.Let us now turn to the second question,of how we can neglect higher order corrections to the K¨a hler potential while keeping the superpotential (3.4).The leading contribution of such terms comes from corrections in the K¨a hler potential (4.2)of the schematic form |Φ|4/|Λm |2.The leading eﬀect of such corrections in the scalar potential are,schematically,∆K V eff ∼ µ2ΦΛm ,and are clearly negligible for |Φ|≪|Λm |.The correction (4.3)should be comparedwith the correction to the tree level potential from the superpotential (3.4),which is of the form∆W V eff ∼ µ2ΦN f −N ΛN f −3N N f −3N|the correction due to the superpotential (4.4)is more importantthan the correction due to the K¨a hler potential (4.3).For smaller values of Φboth correc-tions are negligible.This answers our second question.We conclude that the corrections due to the high energy theory and other modes at the scale Λm do not invalidate our conclusions.Our perturbative computations in section 2and the nonperturbative computations in section 3are completely under control and lead to the dominant contributions to the low energy dynamics.5.Meta-stable Vacua in SUSY QCDIn the preceding sections,we have gradually assembled the tools necessary for ana-lyzing supersymmetry breaking in SQCD.Now let us put these tools to work.The model of interest is SU(N c)SQCD with scaleΛcoupled to N f quarks Q f,˜Q g,f,g=1,...,N f (for a review,see e.g.[14]).We take for the tree-level superpotentialW=Tr mM,where M fg=Q f·˜Q g,(5.1) and m is a non-degenerate N f×N f mass matrix.This theory has N c supersymmetric ground states withM = Λ3N c−N f det m 1m(5.2) All these supersymmetric ground states preserve baryon number and correspondingly the expectation values of all the baryonic operators vanish.The mass matrix m can be diagonalized by a bi-unitary transformation.Its diagonal elements can be set to real positive numbers m i.We will be interested in the case where the m i are small and of the same order of magnitude.More precisely,we explore the parameter rangem i≪|Λ|;m im j∼1the expectation values M in(5.2)approach the origin.The region around the origin can be studied in more detail using the duality of[13] between our electric SU(N c)SQCD and a magnetic SU(N f−N c)gauge theory with scale Λ,coupled to N2f singlets M fg and N f magnetic quarks q f and˜q f in the fundamental and anti-fundamental representation of SU(N f−N c).We will mostly limit ourselves to the free magnetic range N f<3βTr(q†q+˜q†˜q)+1they are not associated with the holomorphic information in the theory.Our quantitative answers will depend onαandβ,but our qualitative conclusions will not.The superpotential of the dual SU(N f−N c)theory is[13]W dual=1Λ2) and are negligible.Therefore,the theory based on the K¨a hler potential(5.4)and the superpotential(5.5)is the same as the model studied in section3,with the parameters andﬁelds related by the dictionaryϕ=q,˜ϕ=˜q,Φ=M αΛ,h=√Λ,µ2=−m0 Λ,Λm= Λ,N=N f−N c(5.7)Here we have chosenβ=1and expressed our answers as functions of Λand Λ.As a consistency check,notice that(5.2)becomes identical to the supersymmetric vacuum(3.5) discussed at the end of section3,after applying the dictionary(5.7)and the identity(5.6).An interesting special case is N f=N c+1,where the magnetic gauge group is trivial. Here it is not natural to setβ=1.Instead,we scale q and q such that they are the same as the baryons B=Q N c and B= Q N c of the electric theory.Then,we should replace the kinetic term for the magnetic quarks in(5.4)with1(˜B T MB−det M)+Tr mM(5.8)Λ2N c−1(Note the additional determinant term.)For N c>2the determinant interaction is neg-ligible near the origin and this theory is the same as the N=1version of the theory in section2.We can now essentially borrow all our results from sections2and3.We thus conclude that,for N f in the range N c+1≤N f<3h Tr ϕΦ˜ϕ−hN f i =1µ2i Φi i ,where µ2i =−m i Λ.We order the m i so that m 1≥m 2...≥m N f >0.The meta-stable vacuum is then given byΦ=0,ϕ= ϕT = ϕ00 ,ϕ0=diag(µ1,µ2,...,µN ).(5.9)In this vacuum,the non-vanishing F-terms are F Φi i for i =N +1,...N f ,and the vacuum energy is V 0= N f i =N +1|hµ2i |.For the vacuum (5.9)to be (meta)stable,it is crucial that the ϕ0expectation values in (5.9)are set by the N largest masses m i .Replacing one of the ϕ0entries µi ≤N in (5.9)with a µi>N does not yield a (meta)stable vacuum –the treelevel spectrum contains an unstable mode,sliding down to the vacuum (5.9).What happens for m i large compared with |Λ|?Clearly,our approximations can no longer be trusted.In particular,if all m i ≫|Λ|we have no reason to believe that such a meta-stable state exists.However,let us try to make one of the masses,m N f large whilekeeping the other masses small.For m N f ≫|Λ|we can integrate out the heavy quark and reduce the problem to that of smaller number of ﬂavors.As long as the number of lightﬂavors Nf satisﬁes N f ≥N c +1,our eﬀective Lagrangian argument shows that such a meta-stable vacuum exists.Let us try to go one step further and ﬂow down from N f =N c +1→N c .We start with N c +1light ﬂavors with m i =1...N c ≪m N c +1≪|Λ|and ﬁnd a meta-stable state which up to symmetry transformations has B i =˜B i =0,for all i =1...N c ,and B N c +1=˜B N c +1=0.If we can trust this approximation as m N c +1≫|Λ|,we ﬁnd the following picture for the N f =N c problem.For m =0the low energy theory is characterized by the modiﬁedmoduli space of vacua [12]det M −B ˜B =Λ2N c (5.10)and the K¨a hler potential on that space is smooth.Consider the theory at the vicinity of the points related toM =0,B =˜B =i ΛN c (5.11)by the action of the global baryon number symmetry.The K¨a hler potential around that point depends on the ﬁelds which are tangent to the constraint (5.10)K =1βb †b +...(5.12)where B =i ΛN c e b ,˜B=i ΛN c e −b ,and again αand βare dimensionless real and positive numbers which we cannot compute.Turning on the superpotential m 0Tr M leaves un-lifted,to leading order,the pseudo-ﬂat directions labelled by M and b .These pseudo-ﬂatdirections are lifted by the higher order terms in(5.12)which we cannot compute.(Note that unlike the case with moreﬂavors,where the loops of massive but lightﬁelds give the dominant correction to the pseudo-ﬂat directions,here there are no such lightﬁelds which can lead to a reliable conclusion.)Although we cannot prove it in this case,motivated by theﬂow from the problem with one moreﬂavor,we suggest that the states(5.11)might also be meta-stable.So far we have restricted attention to N f<3N c<N f<3N c the theoryﬂows to a nontrivial2ﬁxed point[13].We can again use the magnetic description whichﬂows to the sameﬁxed point.However,the analysis above in the magnetic theory should be modiﬁed in this case.The duality is still valid only belowΛ,but unlike the free magnetic case,here the magnetic theory is interacting in this range.A closely related fact is that,for nonzero M, the dynamically generated superpotential is[16,14,17]:W dyn=(N c−N f) det M N f−N c(5.13)(One can check that this is the same as(3.4)after using(5.7)and(5.6).)For M near theN forigin,this scales like MN c is more subtle because the magnetic theory is IR free only because2of its two loop beta function.Here the superpotential(5.13)scales like M3and again it cannot be neglected near the origin.It is interesting that in this case(5.13)is independent ofΛand in terms of the magnetic variables the superpotential(5.13)is independent of Λm.To summarize,we have demonstrated in this section that SU(N c)SQCD with N c+1≤N f<3If we take the masses m i to all be equal,there is a vector-like U(N f)∼=SU(N f)×U(1)B global symmetry.This symmetry is unbroken in the supersymmetric vacua(5.2),which is consistent with their mass gap.In the meta-stable,dynamical supersymmetry breaking vacua,the U(N f)global symmetry is spontaneously broken to S(U(N f−N c)×U(N c)) (plus there is an accidental U(1)R symmetry).The meta-stable dynamical supersymmetry breaking vacua is thus a compact moduli space of vacua,M c∼=U(N f)mΛ≪Λ;this includes the magnetic gaugeﬁelds and gauginos,which are Higgsed.The pseudo-moduli have masses which are smaller,suppressed by a loop factor of the IR free Yukawa coupling of the magnetic dual. There are massless scalars:the Goldstone bosons of the vacuum manifold(5.14).There are also massless fermions(including the Goldstino):the N2c fermionic partners of the pseudo-moduliΦ0,i.e.the fermionsψM in the null space of both q and q .We also note that the non-trivial topology of the vacuum manifold(5.14)means that there are topological solitons,whose lifetime is expected to be roughly the same as that of the meta-stable vacuum.In4d,there are p-brane topological solitons ifπ3−p(M c)is non-trivial.In particular,the vacuum manifold(5.14)leads to solitonic strings.6.SO(N)and Sp(N)GeneralizationsIn this section,we give the generalizations of our models to SO(N)and Sp(N)groups. The SO(N)theory(or more precisely,Spin(N),so we can introduce sources in the spinor representation)exhibits a new phenomenon:the meta-stable,non-supersymmetric vacua are in the conﬁning phase,whereas the supersymmetric vacua are in a diﬀerent phase,the oblique conﬁning phase.These diﬀerent phases occur in this case because the dynamical matter is in an unfaithful representation of the center of the gauge group,leaving Z2×Z2electric and magnetic order parameters which can not be screened.The order parameters determine whether Wilson and’t Hooft loops in the spinor representation of the SO(N) group have area or perimeter law.We will argue that,in the meta-stable vacua with broken supersymmetry,the’t Hooft loop with magnetic Z2charge has perimeter law,while that with oblique electric and magnetic Z2charges has area law.In the supersymmetric vacua the situation is reversed:the oblique charged loop has perimeter law,and the magnetic charged loop has area law.6.1.The SO(N)macroscopic theoryConsider a model with global symmetry and matter contentSO(N)SU(N f)U(1)′U(1)R(6.1)Φ1−22ϕ10The K¨a hler potential is taken to be canonical,K=Trϕ†ϕ+TrΦ†Φ(6.2) (BecauseΦis a symmetric matrix,the K¨a hler potential has an extra factor of2for the oﬀ-diagonal components ofΦ.This will be properly taken into account in the following analysis.)The superpotential is taken to beW=h TrϕTΦϕ−hµ2TrΦ.(6.3)Forµ=0,the SU(N f)×U(1)′global symmetry is broken to SO(N f).For N f>N andµ=0,supersymmetry is spontaneously broken as the rank condition again prevents FΦfrom all vanishing.Up to global symmetries,the potential is minimized byΦ= 000Φ0 ,ϕ= ϕ00 ,withϕT0ϕ0=µ21I N(6.4) whereΦ0is an arbitrary(N f−N)×(N f−N)symmetric matrix,andϕ0is an N×N matrix subject to the condition in(6.4).All vacua on this space of classical pseudo-ﬂat directions have degenerate vacuum energy densityV min=(N f−N)|h2µ4|.(6.5)20We can use the SU(N)result of section2to show that(6.5)is indeed the absolute minimum of the potential.The classical potential of this SO(N)theory satisﬁes V SO(N)≥|hϕϕT−hµ2|2≥(N f−N)|h2µ4|,where for theﬁrst inequality we simply setΦ=0and in the second we used the SU(N)result,restricted to the smaller space where˜ϕ=ϕT.We now show that perturbative quantum eﬀects lift the above classical vacuum degen-eracy,and that a local minimum of the one-loop eﬀective potential is(up to symmetries)Φ0=0,ϕ0=µ1I N.(6.6) Of the classical vacua(6.4),this has maximal unbroken global symmetry,with SO(N)×SO(N f)×U(1)R→SO(N)D×SO(N f−N)×U(1)R.We will focus on the leading perturbative corrections to the eﬀective potential,expanded around the vacuum(6.6).Expanding around(6.6),we write theﬁelds asΦ= δYδZ T δZδ Φ ,ϕ= µ+δχA+δχSδρ .(6.7)whereδχA andδχS denote the antisymmetric and symmetric part,respectively,of δχA+δχS.The Goldstone bosons of the broken global symmetry are Re µ∗|µ|δρ .The former are in the adjoint of SO(N)×SO(N)F/SO(N)D∼=SO(N) (with SO(N)F⊂SO(N f)),and hence they are antisymmetric;the latter are in SO(N f)/SO(N)F×SO(N f−N).There are also the classically massless pseudo-moduliﬁelds,δ Φandδ χ≡Im µ∗2a Trδ χTδ χ+b Trδ Φ†δ Φ +...(6.9) for some numerical coeﬃcients a and b.These coeﬃcients are computed in appendix B; the calculation is very similar to the SU(N)case.The result isV(1)eff=|h4µ2|(log4−1)。

高中英语作文《兄弟姐妹的争吵与和解》

Growing up with siblings can be a rollercoaster of emotions, filled with laughter, arguments, and moments of deep connection. The dynamics between siblings are complex and everchanging, often characterized by a mix of rivalry and camaraderie. This essay delves into the common phenomenon of sibling quarrels and the subsequent reconciliations, exploring the underlying emotions and the impact these interactions have on our lives.One of the most vivid memories from my childhood is the seemingly endless bickering with my younger brother. It was a typical afternoon the sun was setting, casting a warm glow through the living room window, and the aroma of moms cooking wafted through the air. Yet, amidst this idyllic scene, a storm was brewing. My brother and I were locked in a heated argument over a board game, our voices rising with each passing minute. The dispute was trivial, a matter of whose turn it was or who had cheated, but it felt like a battle of epic proportions at the time.Arguments between siblings often stem from a desire for fairness and recognition. As children, we are quick to perceive any perceived slight or injustice, and our reactions can be intense. The clash of wills and the struggle for attention can lead to moments of high tension. Yet, these confrontations are not just about the immediate issue at hand they are a reflection of the deeper need for validation and understanding.Despite the intensity of our quarrels, there was always a moment of reconciliation. It might have been a shared joke, a mutual realization of the absurdity of our argument, or simply the silent understanding that ourbond was stronger than our differences. These moments of peace were as sweet as the arguments were bitter, serving as a reminder of the unbreakable bond between siblings.One such moment stands out in my memory. After a particularly heated exchange, my brother and I found ourselves in the backyard, each nursing our bruised egos. The sky had turned a deep shade of indigo, and the first stars were beginning to twinkle in the evening sky. We sat on the old wooden swing, our feet dragging in the damp grass, neither of us speaking. The silence was heavy, but it was also a silence of understanding. Without words, we acknowledged the hurt we had caused each other and the love that remained despite our differences.Reconciliation between siblings is often a silent process, a mutual decision to let go of the anger and embrace the love that binds us. Its a recognition of the fact that our relationship is more important than the argument, and that our shared history and future together are worth preserving.The impact of these quarrels and reconciliations on our lives is profound. They teach us about conflict resolution, empathy, and the value of compromise. They also highlight the importance of communication and the power of forgiveness. Sibling relationships are a microcosm of the larger social dynamics we encounter in life, providing us with a safe space to learn and grow.In conclusion, the ebb and flow of sibling relationships, marked by arguments and reconciliations, are an integral part of our personaldevelopment. They shape our understanding of relationships, conflict, and resolution. While the path may be bumpy, it is these shared experiences that ultimately strengthen the bond between siblings, creating a foundation of love and understanding that lasts a lifetime.。

什么激励着奋斗的年轻人英语作文

什么激励着奋斗的年轻人英语作文全文共5篇示例，供读者参考篇1The Fires that Fuel the Driven YouthIn every generation, there are those who seem driven by an insatiable hunger - a burning desire to achieve, to overcome, to leave an indelible mark upon the world. While the ambitions may differ, the coals that stoke the flames of purpose in these impassioned young souls share common traits. What kindles such ferocity in the hearts and minds of the daring youth who refuse to accept the ordinary?For some, it is a longing for transcendence that propels them ever forward. A craving to etch their names amongst the immortals of history's halls, undaunted by the immensity of the task before them. They seek brilliance in their chosen field, be it the arts, sciences, philosophy or elsewhere - spurred on by visions of legacy that could echo through the ages. Whether to inspire generations anew with a masterwork or to reshape mankind's understanding through ground-breaking theories, these individuals are consumed by an ambition bordering oncelestial. Complacency is abhorred, replaced by an inextinguishable thirst for the rarified air of true greatness that so few have breathed.Yet not all are driven by a quest for immortality's embrace. Many of the ambitious arise each dawn renewed by that most basic of human conditions - an intense desire to prove their worth to the world. Wrought from the coals of rejection, marginalization or underestimation by those around them, an inferno ignites within. They are lifted by ambitions born of defiance - of shattering prejudices, beliefs and assumptions through sheer, unwavering force of will and accomplishment. For these fierce talents, each triumph is a repudiation of a life once forced upon them, a decisive step towards sculpting their own destiny through the strength of their convictions.Nor should the unifying power of a valiant cause be overlooked as a profound source of motivation. When the youths' passions become entwined with a virtuous mission - to abolish injustice, defy oppression or uplift the downtrodden - their dedication transcends the personal. They are consumed by a vision of a better reality, one that can only be achieved through the crucible of their toils and sacrifices. Though the path is fraught with hardship, the sense of higher purpose fortifies theirresolve. For they understand that complacency and inaction are unforgivable when one has been granted the power to inspire lasting change.Then there are those whose gaze is not transfixed upon some far-flung horizon, but upon their own cherished dreams laid before them. Poverty, discrimination, disability - the very challenges that crush so many naive ambitions instead fuel these unshakable individuals. Each harrowing obstacle is perceived as another step towards the summit of self-actualization, another reason to fight even harder to achieve their definition of a life well-lived. Armed with steadfast self-belief, they refuse to lend the hardships they've endured the power to dictate their destinies.Ultimately, within every driven young person burns the soul of a nonconformist. A scorching refusal to be constrained by the boundaries of tradition, circumstance and expectation. Theirs is a life attuned to the harmonies and clamors of passion, wherein the established paths of the ordinary are rejected in favor of forging new trailheads into the unknown. Whether in pursuit of greatness, salvation, truth or something more primal and visceral, their journeys are fueled by the resolute belief that they alone have the power to define their universe.For as long as the human spirit has endured, it has sparked these beacons of conviction - each one's story a unique and indelible testimony to the limitless potential that lies within us all. The paths, the triumphs, the hardships - all may differ. But their light, their fire, their unwillingness to let possibility remain untapped potential - that binds them all across the endless seas of time. They are the ones who stoke the flames of progress, who dare the rest of humanity to dream bigger, to expect more from tomorrow than what was considered possible today. They are the driven youth, and their eternal fires are what light our way through the darkness of stagnation.篇2What Drives the Ambition of Today's Youth?As an older millennial, I often find myself pondering the motivations and aspirations of the generation coming up behind me. Gen Z and their younger counterparts seem to approach life with a unique blend of idealism, pragmatism, and a relentless drive to achieve. But what exactly fuels this fire? What pushes ambitious young people today to pour themselves into their passions and pursuits with such fervor?To begin, I think we have to look at the world they were born into - a world of rapid technological advancement, global interconnectivity, and unprecedented access to information. This environment has bred a generation of hyper-aware individuals who can see both the vast potential and the monumental challenges that lie before them. They are the first truly global generation, exposed from a young age to diverse cultures, radical ideas, and a heightened consciousness of the world's injustices and inequalities.This awareness, coupled with the self-actualization values deeply ingrained in modern Western society, has created a potent mixture of existential angst and personal drive. Today's youth are robbed of the blind optimism of generations past, yet they retain an empowered belief in their ability to create positive change. Their ambition is thus rooted in an urgent desire to find purpose and to make their mark on the world before it's too late.Of course, ambition alone is not enough - it requires immense dedication, sacrifice, and daily hard work to translate vision into reality. Here, many of the consistent hallmarks of Gen Z come into play, including their proclivity toward hustle culture, their prioritization of passion over traditional securities, and their remarkable tech savviness. Armed with an entrepreneurialmindset and enabled by digital resources their predecessors could scarcely imagine, this generation possesses unique tools to relentlessly push forward on their desired paths.Take the influx of teenage entrepreneurs, content creators, and social activists as an example. Platforms like YouTube, Instagram, and TikTok have allowed young visionaries to bypass traditional gatekeepers, build a following, and even generate income around their creative pursuits from an early age. Stories of high schoolers running six-figure businesses or activists catalyzing global movements are no longer anomalies, but representative of the immense potential this generation wields with dedication and an internet connection.Alternatively, you have those whose ambition is channeled into more conventional streams of achievement like academics or athletics with hopes of attending elite universities and eventually assuming roles that allow them to enact change from within prominent organizations and institutions. Tremendous pressure exists, both internal and external, to secure the right bullet points and credentials to maximize future impact.It's also worth examining the role modern parenting plays. Whereas youth ambition was once commonly stoked by overbearing or living vicariously through their children, today'saspiring young people seem to be driven more by being loosed into the world with immense opportunity, but minimal handrails. Liberal philosophies around concepts like self-directed learning, passion-based uncovering, and rejecting rigid life roadmaps have become the norm in many circles. This hands-off approach breeds remarkable self-starters, but also a ambition bordering on desperation to rapidly define oneself and one's purpose.Ultimately, I believe ambition in today's youth boils down to an intoxicating blend of privilege and burden. They have been empowered with unprecedented tools, agency, and a vision of what's possible. But they also carry the immense psychological weight of generational responsibility - the prospect of quite literally having to save the world their poorly-prepared ancestors left them.While the sources of motivation are complex and varied, make no mistake: ambitious young people today are fully engaged in a relentless pursuit of impact. Disillusioned by the status quo and enabled to an unprecedented degree, their appetite for challenging outdated norms and creating a better future is utterly ravenous. How they'll reshape society remains to be seen, but if their ambition is matched by adaptability and wisdom, I'm cautiously optimistic about the world they'll build.篇3The Driving Force Behind AmbitionWhat is it that fuels the unrelenting ambition and drive of today's young go-getters? The fervent desire to leave an indelible mark on the world? The intrinsic hunger for success and achievement? The aspirations of fame and fortune? Or could it be something more profound - an innate need for purpose, meaning, and self-actualization?For many ambitious youths, the quest for achievement is an all-consuming fire that burns within. It is a potent cocktail of passion, dedication, and an unwavering belief in their ability to surmount any obstacle. This combustible combination propels them forward, undeterred by setbacks or skeptics who question their lofty dreams.At the core of this ambition often lies a deep-seated hunger to prove one's worth – to themselves and to the world. These driven individuals seek validation that they are capable of greatness, that their talents and efforts can translate into tangible success. Every achievement, no matter how small, serves as a potent affirmation, stoking the flames of ambition ever higher.For others, ambition is intrinsically linked to a relentless pursuit of knowledge and personal growth. They are insatiable learners, perpetually seeking to expand their horizons and push the boundaries of what is possible. Each accomplishment is not an endpoint, but a stepping stone towards grander goals and deeper self-discovery.The allure of wealth and material trappings cannot be discounted as a motivating factor for some. The promise of financial freedom, luxury, and status can be a powerful siren call, beckoning the ambitious to scale greater heights. However, for those truly driven by ambition, money is often a means to an end – a tool to fund grander visions and leave an enduring legacy.Yet, beneath the surface of material desires, there often lies a more profound yearning – the quest for purpose and meaning. Ambitious individuals crave to make a lasting impact, to contribute something of value to the world. They seek to transcend the mundane and forge a path that resonates with their deepest values and beliefs.This existential pursuit is perhaps the most potent catalyst for ambition. It is the desire to lead a life of significance, to create something that outlives them and resonates through generations. Whether through groundbreaking innovations,artistic masterpieces, or social advocacy, the ambitious seek to etch their names into the annals of history.Underpinning this drive is a profound sense of self-belief and an unwavering conviction in one's abilities. Ambitious individuals possess an unshakable faith that they have what it takes to defy the odds and conquer seemingly insurmountable challenges. This self-assurance is not born of arrogance, but rather a deep understanding of their strengths, weaknesses, and an indomitable will to succeed.Moreover, ambition is often fueled by a fierce competitive spirit – a desire to outshine peers, to be the best in one's chosen field. This competitive drive pushes individuals to constantly raise the bar, to strive for excellence, and to never settle for mediocrity.Yet, true ambition is not solely an individualistic pursuit. It is also driven by a desire to inspire and uplift others. Ambitious individuals seek to be role models, to blaze trails that others can follow, and to create opportunities for those who come after them.In the end, ambition is a multifaceted force that defies simple explanations. It is a complex tapestry woven from threads of passion, purpose, self-belief, and an insatiable hunger forgrowth and impact. And for the ambitious youth of today, this driving force is the engine that propels them towards greatness, fueling their relentless pursuit of dreams that may seem audacious to some, but utterly attainable to those who possess the unwavering fire of ambition.篇4What Drives the Ambition of Youth?Drive, tenacity, restlessness - the willingness to sacrifice the comforts of the present for greater future rewards. These are the hallmarks of ambition, that powerful motivating force which spurs countless young people to forgo an easy life in pursuit of loftier goals. But what is the wellspring of such ambition? The reasons are as diverse as the ambitious individuals themselves.For many, ambition arises from a desire to transcend one's humble beginnings. Coming from meager means or a family of limited opportunities can ignite an internal fire to rise above those circumstances. The visions of a better life, of greater respect and security, of providing more for oneself and loved ones - these become powerful catalysts driving one forward with relentless determination. The ambitious see challenges as opportunities to prove their mettle and defy expectations.Others are propelled by dreams writ large - visions of changing the world through pioneering innovations, brilliant artistic expression, or transformative philanthropic work. Youth is the time for boldness, when constraints feel illusory and the grandest ambitions seem achievable through sheer force of will and effort. An idealistic young mind can be inspired by the examples of visionaries and achievers who left indelible marks on society. If they could reshape the world, why not me? Such soaring dreams light the flame of ambition.Ego and status also factor for some. The ambitious seek the admiration and envy of their peers through conspicuous accomplishments. A prestigious career path, renown for professional excellence, accumulation of wealth as tangible validation of one's worth - such extrinsic motivations can propel extreme ambition and drive someone to immerse themselves in endless work and sacrifice. Ego ambition has fueled many remarkable achievements, even if less palatable driving forces lay beneath.For the lucky few, ambition stems from an inborn fire, an innate restlessness and dissatisfaction with the ordinary. These ambitious souls find idle complacency utterly unfulfilling; they must always have a goal ahead to strive for lest they feel adriftand unstimulated. Their ambition is an intractable part of their character rather than being drawn from external forces. This relentless quality enables them to persist in the face of setbacks that might thwart those with more transient motivations.Then there are those motivated by a sense of責任personal conviction. They seek to honor the sacrifices of those who came before, to make the most of opportunities their ancestors never had. Being the first in their family to attend university, to achieve professional credentials, or to accumulate generational wealth - these can bestow a profound duty to maximize one's talents through tireless effort. Such ambition is driven by a moral calling to justify the faith placed in one's potential.Of course, ambition is frequently stoked by a confluence of these factors. A university student juggling studies with a job to defray costs could be propelled by a combination of desire to rise above financial hardship, dreams of personal accomplishment, and the pressure to make ancestors' sacrifices worthwhile. Ambition's causes are rarely singular.The enticements of an unambitious life are manifold - the siren calls of leisure, nights untroubled by work's burdens, and a carefree disregard for the judgment of others. Rejecting these comforts in favor of ambition's endless striving requiresformidable drive and sacrifice. Those who can persevere deserve admiration, for their ambition enriches society through innovation, economic growth, and inspiration to others.The first flushes of youthful ambition may be idealistic, but ambition's ultimate crucible is sustaining that initial fervor through inevitable setbacks and wearying trials. It is one thing to dream of changing the world and another to endure the thankless grind of working toward that dream unabated for decades. The ambitious must develop resilience and grit or see their fire sputter out, resigning to join the ranks of the contented ordinary.So while we may never fully agree on the singular wellspring of ambition - whether it be circumstance, innate character, or personal conviction - we can celebrate its fascinating varieties and the essential role it plays in human achievement and progress. The boundless ambition of the young is one of our species' most renewable and vital resources. May it never be extinguished amidst the petty comfort of complacency and stagnation.篇5What Drives the Ambitious Youth?As the world rapidly evolves, a new generation of driven and ambitious youth is emerging, fueled by a complex blend of aspirations, dreams, and a burning desire to leave their mark on society. This cohort, brimming with energy and determination, defies conventional norms and challenges the status quo, propelling themselves forward with an unwavering sense of purpose. But what lies at the core of this relentless pursuit? What ignites the fire within these young souls, compelling them to push boundaries and redefine success?For many, the pursuit of personal growth andself-actualization serves as a powerful catalyst. The desire to unlock their full potential, to continually evolve and transcend perceived limitations, burns brightly within their hearts. They crave knowledge, seek out new experiences, and embrace challenges as opportunities for growth. Each obstacle surmounted, each skill acquired, becomes a testament to their resilience and fortitude, further fueling their ambition.Others find their motivation in the desire to create lasting impact and leave a positive footprint on the world around them. Driven by a sense of social responsibility, they channel their energy into addressing global issues, advocating for change, and seeking innovative solutions to longstanding problems. Whetherthrough entrepreneurial endeavors, artistic expression, or scientific discoveries, they aim to contribute something meaningful, something that will resonate long after they are gone.Yet, for some, the fire within is stoked by a profound desire for financial freedom and security. Witnessing the struggles of previous generations, they are determined to break free from the shackles of economic constraints, to forge their own paths, and to build a life of abundance and stability. This pursuit is not fueled by greed but by a deep-rooted belief in the power of financial independence to unlock limitless possibilities and the freedom to pursue their passions without compromise.Underlying these diverse motivations is a common thread: a profound belief in their own capabilities and an unwavering determination to defy the odds. These young, driven individuals possess an innate confidence, a self-assurance that propels them forward even in the face of adversity. They refuse to be defined by societal expectations or limitations imposed by others, instead forging their own unique paths, fueled by an unshakable faith in their ability to shape their destinies.Yet, the journey is not without its challenges. Setbacks, rejections, and failures are inevitable companions on the road tosuccess. But what sets these ambitious youth apart is their resilience, their ability to bounce back from disappointments, and their unwavering commitment to learning from their mistakes. Each stumbling block becomes a stepping stone, a catalyst for growth and self-reflection, fueling their determination to rise above and conquer the next hurdle.In a world that often celebrates instant gratification and fleeting success, the driven youth stand as a beacon of perseverance and long-term vision. They understand that true greatness is not achieved overnight but through a relentless pursuit of excellence, a steadfast commitment to their goals, and an unwavering dedication to continuous improvement.As we bear witness to the remarkable achievements of these ambitious individuals, we are reminded of the incredible potential that lies within the human spirit. Their stories inspire us, challenge us, and compel us to reflect on our own aspirations and motivations. Perhaps, in their tireless pursuit of greatness, we too can find the courage to embrace our dreams, to push beyond our self-imposed limitations, and to leave an indelible mark on the world around us.The driven youth of today are not merely the leaders of tomorrow; they are the catalysts of change, the architects of abrighter future. Their unwavering ambition, their relentless pursuit of excellence, and their commitment to making a lasting impact will undoubtedly shape the course of history, inspiring generations to come with their remarkable tales of determination and triumph.。

New Tools for Low Energy Dynamical Supersymmetry Breaking

truly weakly coupled, in the sense that perturbation theory should be good for the Kahler potential. It is hard to see how this can be consistent with the expected behavior of the dilaton potential. Alternatively, non-Abelian avor symmetries may play some role 8]. Other issues include a variety of cosmological problems, perhaps the most severe being the moduli problem 9]. One solution to the latter problem is that the dilaton and moduli are stabilized by nonperturbative physics at high energies and play no role in the breaking of supersymmetry 9]; another possible solution is weak scale in ation 10,11]. Alternatively, models are known in which supersymmetry is broken without at directions 1]. In such cases, one does not require the intervention of complicated stabilization mechanisms. As in the case of at directions described above, one can imagine breaking supersymmetry at a scale intermediate between MW and Mp. This idea, however, turns out to be fraught with di culties, particularly with obtaining appreciable gluino masses 1,9,12]. Alternatively, one can imagine breaking supersymmetry at comparatively low energies, of order 10's{1000's of TeV. In this case, gauge interactions can serve as the \messengers" of supersymmetry breaking. Apart from the fact that the physics of supersymmetry breaking is potentially accessible, such a scheme has an immediate bonus: there is automatically su cient squark and slepton degeneracy to understand the absence of avor changing neutral currents. Early e orts to build models along these lines su ered from a number of di culties. The most severe of these were that SU (3)C typically became strong a few decades above the scale of supersymmetry breaking, and that the known models all possessed (astrophysically) dangerous light axions from a spontaneously broken R symmetry. Nelson and Seiberg noted that dimension ve operators expected from Planck scale physics could explicitly break the R symmetry and give the axion a su ciently large mass so that it would not be produced in stars 13] without restoring supersymmetry. Bagger, Poppitz and Randall pointed out that when R symmetry and supersymmetry break at the same scale, cancellation of the cosmological constant within the framework of supergravity by adding a constant to the superpotential 14] necessarily requires such explicit R symmetry breaking but also does not restore supersymmetry. Solutions to the rst problem were provided in refs. 15,16]. Here it was suggested that a new gauge symmetry, referred to as 2