Two-loop Light Quark Corrections To The Top Width

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The Subleading Isgur-Wise Form Factor $chi_3(vcdot v')$ to Order $alpha_s$ in QCD Sum Rules

a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we ﬁnd that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark eﬀective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coeﬃcients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suﬃces to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the eﬀective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures theﬁnite mass diﬀerence between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)deﬁned by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus eﬀectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the eﬀective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have reﬁned the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such eﬀects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only brieﬂy sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”oﬀ-shell momenta of the heavy quarks.Due to the phase redeﬁnition of the eﬀective heavy quark ﬁelds in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coeﬃcient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which deﬁne the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the eﬀective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coeﬃcient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one ﬁrst writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coeﬃcient function.possible subtraction terms.Because of theﬂavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The eﬀective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suﬃces to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonﬁeld,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usﬁrst present the result for the nonperturbative power corrections.WeﬁndK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its eﬀect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we ﬁnd that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our ﬁnal result,there is one more eﬀect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the eﬀective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the eﬀective theory[16,18].Oneﬁnds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,weﬁnd stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourﬁnal result forχ3(y)is not very diﬀerent from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisﬁgure one can also deduce how changes in the values of the vacuum condensates would aﬀect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Weﬁnd that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonﬁeld and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesﬁnancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientiﬁc,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisﬁgure and are both represented by the narrow dots.11。

圆二色光谱英文

圆二色光谱英文Circular Bicolor SpectrumIntroductionThe circular bicolor spectrum, also known as the color wheel, is a visual representation of the entire range ofcolors in a circular format. It is a fundamental tool for artists, designers, and anyone working with colors. The spectrum showcases the relationships between different colors, allowing for the creation of harmonious color schemes and the understanding of color theory.History of the Color WheelThe concept of the color wheel dates back to the 18th century when Sir Isaac Newton first discovered that whitelight could be divided into the various colors of the spectrum. His experiments with prisms led to the creation ofthe first color wheel, which consisted of 7 colors - red, orange, yellow, green, blue, indigo, and violet.In the 19th century, artists and theorists began to delve deeper into the study of color and its relationships. Swiss painter Johannes Itten and American painter Albert Henry Munsell developed their own versions of the color wheel, each adding their own insights into color theory and harmony.In the 20th century, the color wheel became a standard tool in art and design education, with its principles being applied across various fields such as fashion, interior design, and graphic design.The Circular Bicolor SpectrumThe color wheel is typically divided into two main categories - warm and cool colors. Warm colors, such as red, orange, and yellow, are associated with energy and warmth. Cool colors, such as blue, green, and violet, are calming andsoothing. By arranging the colors in a circular format, the relationships between them become more apparent.Primary Colors: At the center of the color wheel are the three primary colors - red, blue, and yellow. These colors are considered the building blocks of all other colors and cannot be created by mixing other colors.Secondary Colors: Located between the primary colors are the secondary colors - orange, green, and violet. These colors are created by mixing equal parts of two primary colors. For example, mixing red and yellow creates orange.Tertiary Colors: The spaces between the primary and secondary colors are filled with the tertiary colors, which are a combination of a primary color and a secondary color. Examples of tertiary colors include red-orange, yellow-green, and blue-violet.Color Relationships: The placement of colors on the color wheel dictates their relationships and the resulting colorschemes. Complementary colors are located opposite each other on the wheel, such as red and green. These colors create contrast and vibrancy when used together. Analogous colors are located next to each other on the wheel, such as red, orange, and yellow. These colors create a harmonious and cohesive color scheme.Application of the Color WheelThe color wheel is a valuable tool for artists and designers in various applications. It is used in creating pleasing color combinations for paintings, illustrations, graphic designs, and more. Understanding the relationships between colors allows for the creation of visually appealing compositions.In interior design, the color wheel is used to create color schemes for different rooms and spaces. By using complementary or analogous colors, designers can evoke specific moods and atmospheres within a space.In fashion, the color wheel is used to create cohesive and stylish color combinations for clothing and accessories. It also helps designers understand the impact of color on the human psyche and how it can influence emotions and perceptions.ConclusionThe circular bicolor spectrum, or color wheel, is a fundamental tool for understanding the relationships between colors. It provides a visual representation of the entire range of colors and their interactions, allowing for the creation of harmonious color schemes in various artistic and design fields. Its enduring importance and application make it an essential tool for anyone working with colors.。

Phase diagram of dense neutral three-flavor quark matter

ular and gapless color-ﬂavor-locked (gCFL) phases [15] are favored. At nonzero temperature, T = 0, some other phases were proposed as well [16]. In this paper, we study the phase diagram of dense neutral three-ﬂavor quark matter as a function of the strange quark mass at zero and ﬁnite temperature. In order to allow for the most general ground state, we employ a nine-parameter ansatz for the gap function. The eﬀects of the strange quark mass are incorporated in our model by a shift of the chemical potential of strange quarks, i 2 µi s → µs − ms /(2µ) where i = r, g, b is the color index, ms is the strange quark mass, and µ is the quark chemical potential. This shift reﬂects the reduction of the Fermi momenta of strange quarks due to their mass. Such an approach is certainly reliable at small values of the strange quark mass. We assume that it is also qualitatively correct at large values of the strange quark mass. We note that the analysis of this paper is restricted to locally neutral phases only. This automatically excludes, for example, mixed [18] and crystalline [19] phases. Also, in the mean ﬁeld approximation utilized here, we cannot get any phases with meson condensates [20]. This paper is organized as follows. In Sec. II we present the details of our approach which is based on the Cornwall-Jackiw-Tomboulis (CJT) formalism [22]. There, we also derive the gap equations and the neutrality conditions, and obtain an expression for the pressure. In the next two sections, the gap equations and the neutrality conditions are studied by using numerical methods. The results at zero temperature are presented and discussed in Sec. III. There, the appearance of the CFL and gCFL phases is established in the case of small and moderately large values of the strange quark mass, respectively. The results at nonzero temperature are discussed in Sec. IV. Also, the phase diagram of quark matter in the T –m2 s /µ plane, as well as in the T –µ plane is presented. Finally, Sec. V concludes this paper with a summary of the results. Our units are = c = kB = 1. The metric tensor is gµν = diag (1, −1, −1, −1). Four-vectors are denoted by capital Latin letters, e.g., K µ = (k 0 , k) where k is a three-vector with absolute value k = |k| and direction ˆ = k/k . We use the imaginary-time formalism, i.e., the k

O(alpha) QED Corrections to Polarized Elastic $mu e$ and Deep Inelastic l N Scattering

a r X i v :h e p -p h /9711228v 1 4 N o v 1997hep-ph/9711228October 1997O (α)QED Corrections to Polarized Elastic µe and Deep Inelastic lN ScatteringDima Bardin a,b,c ,Johannes Bl¨u mlein a ,Penka Christova a,d ,and Lida Kalinovskaya a,caDESY–Zeuthen,Platanenallee 6,D–15735Zeuthen,GermanybINFN,Sezione di Torino,Torino,ItalycJINR,ul.Joliot-Curie 6,RU–141980Dubna,RussiadBishop Konstantin Preslavsky University of Shoumen,9700Shoumen,BulgariaAbstractTwo computer codes relevant for the description of deep inelastic scattering oﬀpolarized targets are discussed.The code µe la deals with radiative corrections to elastic µe scattering,one method applied for muon beam polarimetry.The code HECTOR allows to calculate both the radiative corrections for unpolarized and polarized deep inelastic scattering,including higher order QED corrections.1IntroductionThe exact knowledge of QED,QCD,and electroweak (EW)radiative corrections (RC)to the deep inelastic scattering (DIS)processes is necessary for a precise determination of the nucleon structure functions.The present and forthcoming high statistics measurements of polarized structure functions in the SLAC experiments,by HERMES,and later by COMPASS require the knowledge of the RC to the DIS polarized cross-sections at the percent level.Several codes based on diﬀerent approaches for the calculation of the RC to DIS experiments,mainly for non-polarized DIS,were developped and thoroughly compared in the past,cf.[1].Later on the radiative corrections for a vast amount of experimentally relevant sets of kinematic variables were calculated [2],including also semi-inclusive situations as the RC’s in the case of tagged photons [3].Furthermore the radiative corrections to elastic µ-e scattering,a process to monitor (polarized)muon beams,were calculated [4].The corresponding codes are :•HECTOR 1.00,(1994-1995)[5],by the Dubna-Zeuthen Group.It calculates QED,QCD and EW corrections for variety of measuremets for unpolarized DIS.•µe la 1.00,(March 1996)[4],calculates O (α)QED correction for polarized µe elastic scattering.•HECTOR1.11,(1996)extends HECTOR1.00including the radiative corrections for polarized DIS[6],and for DIS with tagged photons[3].The beta-version of the code is available from http://www.ifh.de/.2The Programµe laMuon beams may be monitored using the processes ofµdecay andµe scattering in case of atomic targets.Both processes were used by the SMC experiment.Similar techniques will be used by the COMPASS experiment.For the cross section measurement the radiative corrections to these processes have to be known at high precision.For this purpose a renewed calculation of the radiative corrections toσ(µe→µe)was performed[4].The diﬀerential cross-section of polarized elasticµe scattering in the Born approximation reads,cf.[7],dσBORNm e Eµ (Y−y)2(1−P e Pµ) ,(1)where y=yµ=1−E′µ/Eµ=E′e/Eµ=y e,Y=(1+mµ/2/Eµ)−1=y max,mµ,m e–muon and electron masses,Eµ,E′µ,E′e the energies of the incoming and outgoing muon,and outgoing electron respectively,in the laboratory frame.Pµand P e denote the longitudinal polarizations of muon beam and electron target.At Born level yµand y e agree.However,both quantities are diﬀerent under inclusion of radiative corrections due to bremsstrahlung.The correction factors may be rather diﬀerent depending on which variables(yµor y e)are used.In the SMC analysis the yµ-distribution was used to measure the electron spin-ﬂip asymmetry A expµe.Since previous calculations,[8,9],referred to y e,and only ref.[9]took polarizations into account,a new calculation was performed,including the complete O(α)QED correction for the yµ-distribution,longitudinal polarizations for both leptons,theµ-mass eﬀects,and neglecting m e wherever possible.Furthermore the present calculation allows for cuts on the electron re-coil energy(35GeV),the energy balance(40GeV),and angular cuts for both outgoing leptons (1mrad).The default values are given in parentheses.Up to order O(α3),14Feynman graphs contribute to the cross-section forµ-e scattering, which may be subdivided into12=2×6pieces,which are separately gauge invariantdσQEDdyµ.(2) One may express(2)also asdσQEDdyµ+P e Pµdσpol kk=1−Born cross-section,k=b;2−RC for the muonic current:vertex+bremsstrahlung,k=µµ;3−amm contribution from muonic current,k=amm;4−RC for the electronic current:vertex+bremsstrahlung,k=ee;5−µe interference:two-photon exchange+muon-electron bremsstrahlung interference,k=µe;6−vacuum polarization correction,runningα,k=vp.The FORTRAN code for the scattering cross section(2)µe la was used in a recent analysis of the SMC collaboration.The RC,δA yµ,to the asymmetry A QEDµeshown inﬁgures1and2is deﬁned asδA yµ=A QEDµedσunpol.(4)The results may be summarized as follows.The O(α)QED RC to polarized elasticµe scattering were calculated for theﬁrst time using the variable yµ.A rather general FORTRAN codeµe la for this process was created allowing for the inclusion of kinematic cuts.Since under the conditions of the SMC experiment the corrections turn out to be small our calculation justiﬁes their neglection. 3Program HECTOR3.1Diﬀerent approaches to RC for DISThe radiative corrections to deep inelastic scattering are treated using two basic approaches. One possibility consists in generating events on the basis of matrix elements including the RC’s. This approach is suited for detector simulations,but requests a very hughe number of events to obtain the corrections at a high precision.Alternatively,semi-analytic codes allow a fast and very precise evaluation,even including a series of basic cuts andﬂexible adjustment to speciﬁc phase space requirements,which may be caused by the way kinematic variables are experimentally measured,cf.[2,5].Recently,a third approach,the so-called deterministic approach,was followed,cf.[10].It treats the RC’s completely exclusively combining features of fast computing with the possibility to apply any cuts.Some elements of this approach were used inµe la and in the branch of HECTOR1.11,in which DIS with tagged photons is calculated.Concerning the theoretical treatment three approaches are in use to calculate the radiative corrections:1)the model-independent approach(MI);2)the leading-log approximation(LLA); and3)an approach based on the quark-parton model(QPM)in evaluating the radiative correc-tions to the scattering cross-section.In the model-independent approach the QED corrections are only evaluated for the leptonic tensor.Strictly it applies only for neutral current processes.The hadronic tensor can be dealt with in its most general form on the Lorentz-level.Both lepton-hadron corrections as well as pure hadronic corrections are neglected.This is justiﬁed in a series of cases in which these corrections turn out to be very small.The leading logarithmic approximation is one of the semi-analytic treatments in which the diﬀerent collinear singularities of O((αln(Q2/m2l))n)are evaluated and other corrections are neglected.The QPM-approach deals with the full set of diagrams on the quark level.Within this method,any corrections(lepton-hadron interference, EW)can be included.However,it has limited precision too,now due to use of QPM-model itself. Details on the realization of these approaches within the code HECTOR are given in ref.[5,11].3.2O (α)QED Corrections for Polarized Deep Inelastic ScatteringTo introduce basic notation,we show the Born diagramr rr r j r r r r l ∓( k 1,m )l ∓( k 2,m )X ( p ′,M h )p ( p ,M )γ,Z ¨¨¨¨B ¨¨¨¨£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡£¢ ¡z r r r r r r r r r r r r r rr ¨¨¨¨B ¨¨¨¨r r r r j r r r r and the Born cross-section,which is presented as the product of the leptonic and hadronic tensordσBorn =2πα2p.k 1,x =Q 2q 2F 1(x,Q 2)+p µ p ν2p.qF 3(x,Q 2)+ie µνλσq λs σ(p.q )2G 2(x,Q 2)+p µ s ν+ s µ p νp.q1(p.q )2G 4(x,Q 2)+−g µν+q µq νp.qG 5(x,Q 2),(8)wherep µ=p µ−p.qq 2q µ,and s is the four vector of nucleon polarization,which is given by s =λp M (0, n )in the nucleonrest frame.The combined structure functions in eq.(8)F1,2(x,Q2)=Q2e Fγγ1,2(x,Q2)+2|Q e|(v l−p eλl a l)χ(Q2)FγZ1,2(x,Q2)+ v2l+a2l−2p eλl v l a l χ2(Q2)F ZZ1,2(x,Q2),F3(x,Q2)=2|Q e|(p e a l−λl v l)χ(Q2)FγZ3(x,Q2),+ 2p e v l a l−λl v2l+a2l χ2(Q2)F ZZ3(x,Q2),G1,2(x,Q2)=−Q2eλl gγγ1,2(x,Q2)+2|Q e|(p e a l−λl v l)χ(Q2)gγZ1,2(x,Q2),+ 2p e v l a l−λl v2l+a2l χ2(Q2)g ZZ1,2(x,Q2),G3,4,5(x,Q2)=2|Q e|(v l−p eλl a l)χ(Q2)gγZ3,4,5(x,Q2),+ v2l+a2l−2p eλl v l a l χ2(Q2)g ZZ3,4,5(x,Q2),(9) are expressed via the hadronic structure functions,the Z-boson-lepton couplings v l,a l,and the ratio of the propagators for the photon and Z-bosonχ(Q2)=Gµ2M2ZQ2+M2Z.(10)Furthermore we use the parameter p e for which p e=1for a scattered lepton and p e=−1for a scattered antilepton.The hadronic structure functions can be expressed in terms of parton densities accounting for the twist-2contributions only,see[12].Here,a series of relations between the diﬀerent structure functions are used in leading order QCD.The DIS cross-section on the Born-leveld2σBorndxdy +d2σpol Borndxdy =2πα2S ,S U3(y,Q2)=x 1−(1−y)2 ,(13) and the polarized partdσpol BornQ4λp N f p S5i=1S p gi(x,y)G i(x,Q2).(14)Here,S p gi(x,y)are functions,similar to(13),and may be found in[6].Furthermore we used the abbrevationsf L=1, n L=λp N k 12πSy 1−y−M2xy2π1−yThe O(α)DIS cross-section readsd2σQED,1πδVRd2σBorndx l dy l=d2σunpolQED,1dx l dy l.(16)All partial cross-sections have a form similar to the Born cross-section and are expressed in terms of kinematic functions and combinations of structure functions.In the O(α)approximation the measured cross-section,σrad,is deﬁne asd2σraddx l dy l +d2σQED,1dx l dy l+d2σpol radd2σBorn−1.(18)The radiative corrections calculated for leptonic variables grow towards high y and smaller values of x.Theﬁgures compare the results obtained in LLA,accounting for initial(i)andﬁnal state (f)radiation,as well as the Compton contribution(c2)with the result of the complete calculation of the leptonic corrections.In most of the phase space the LLA correction provides an excellent description,except of extreme kinematic ranges.A comparison of the radiative corrections for polarized deep inelastic scattering between the codes HECTOR and POLRAD[17]was carried out.It had to be performed under simpliﬁed conditions due to the restrictions of POLRAD.Corresponding results may be found in[11,13,14].3.3ConclusionsFor the evaluation of the QED radiative corrections to deep inelastic scattering of polarized targets two codes HECTOR and POLRAD exist.The code HECTOR allows a completely general study of the radiative corrections in the model independent approach in O(α)for neutral current reac-tions including Z-boson exchange.Furthermore,the LLA corrections are available in1st and2nd order,including soft-photon resummation and for charged current reactions.POLRAD contains a branch which may be used for some semi-inclusive DIS processes.The initial state radia-tive corrections(to2nd order in LLA+soft photon exponentiation)to these(and many more processes)can be calculated in detail with the code HECTOR,if the corresponding user-supplied routine USRBRN is used together with this package.This applies both for neutral and charged current processes as well as a large variety of diﬀerent measurements of kinematic variables. Aside the leptonic corrections,which were studied in detail already,further investigations may concern QED corrections to the hadronic tensor as well as the interference terms. References[1]Proceedings of the Workshop on Physics at HERA,1991Hamburg(DESY,Hamburg,1992),W.Buchm¨u ller and G.Ingelman(eds.).[2]J.Bl¨u mlein,Z.Phys.C65(1995)293.[3]D.Bardin,L.Kalinovskaya and T.Riemann,DESY96–213,Z.Phys.C in print.[4]D.Bardin and L.Kalinovskaya,µe la,version1.00,March1996.The source code is availablefrom http://www.ifh.de/~bardin.[5]A.Arbuzov,D.Bardin,J.Bl¨u mlein,L.Kalinovskaya and T.Riemann,Comput.Phys.Commun.94(1996)128,hep-ph/9510410[6]D.Bardin,J.Bl¨u mlein,P.Christova and L.Kalinovskaya,DESY96–189,hep-ph/9612435,Nucl.Phys.B in print.[7]SMC collaboration,D.Adams et al.,Phys.Lett.B396(1997)338;Phys.Rev.D56(1997)5330,and references therein.[8]A.I.Nikischov,Sov.J.Exp.Theor.Phys.Lett.9(1960)757;P.van Nieuwenhuizen,Nucl.Phys.B28(1971)429;D.Bardin and N.Shumeiko,Nucl.Phys.B127(1977)242.[9]T.V.Kukhto,N.M.Shumeiko and S.I.Timoshin,J.Phys.G13(1987)725.[10]G.Passarino,mun.97(1996)261.[11]D.Bardin,J.Bl¨u mlein,P.Christova,L.Kalinovskaya,and T.Riemann,Acta Phys.PolonicaB28(1997)511.[12]J.Bl¨u mlein and N.Kochelev,Phys.Lett.B381(1996)296;Nucl.Phys.B498(1997)285.[13]D.Bardin,J.Bl¨u mlein,P.Christova and L.Kalinovskaya,Preprint DESY96–198,hep-ph/9609399,in:Proceedings of the Workshop‘Future Physics at HERA’,G.Ingelman,A.De Roeck,R.Klanner(eds.),Vol.1,p.13;hep-ph/9609399.[14]D.Bardin,Contribution to the Proceedings of the International Conference on High EnergyPhysics,Warsaw,August1996.[15]M.Gl¨u ck,E.Reya,M.Stratmann and W.Vogelsang,Phys.Rev.D53(1996)4775.[16]S.Wandzura and F.Wilczek,Phys.Lett.B72(1977)195.[17]I.Akushevich,A.Il’ichev,N.Shumeiko,A.Soroko and A.Tolkachev,hep-ph/9706516.-20-18-16-14-12-10-8-6-4-200.10.20.30.40.50.60.70.80.91elaFigure 1:The QED radiative corrections to asymmetry without experimental cuts.-1-0.8-0.6-0.4-0.200.20.40.60.810.10.20.30.40.50.60.70.80.91elaFigure 2:The QED radiative corrections to asymmetry with experimental cuts.-50-40-30-20-100102030405000.10.20.30.40.50.60.70.80.91HectorFigure 3:A comparison of complete and LLA RC’s in the kinematic regime of HERMES for neutral current longitudinally polarized DIS in leptonic variables.The polarized parton densities [15]are used.The structure function g 2is calculated using the Wandzura–Wilczek relation.c 2stands for the Compton contribution,see [6]for details.-20-100102030405000.10.20.30.40.50.60.70.80.91HectorFigure 4:The same as in ﬁg.3,but for energies in the range of the SMC-experiment.-20-10010203040500.10.20.30.40.50.60.70.80.91HectorFigure 5:The same as in ﬁg.4for x =10−3.-200-150-100-5005010015020000.10.20.30.40.50.60.70.80.91HectorFigure 6:A comparison of complete and LLA RC’s at HERA collider kinematic regime for neutral current deep inelastic scattering oﬀa longitudinally polarized target measuring the kinematic variables at the leptonic vertex.。

用QCD因子化方法研究B→PV两体弱衰变过程

第26卷增刊高能物理与核物理V o1.26，Supp. 2002年12月HIGH ENERGY PHYSICS AND NUCLEAR PHYSICS Dec.，2002 Two-Body B Decays to Pseudoscalar and Vector Mesonsin QCD Factorization ApproachYANG Mao-Zhi1 YANG Ya-Dong21 (Institute of High Energy Physics, CAS, Beijing 100039, China)2 (Department of Physics, Technion, Haifa 32000, Israel)Abstract Motivated by recent CELO measurements and the progress of the theory of B decays,B→PV(P=π, K; V= K*, ρ, ω) decay modes are studied in the framework of QCD factorization.All the measured branching ratios are well accommodated in the reasonable parameter space andpredictions for other decay modes are well below the experimental upper limits.Key words factorization, weak decay, mesonB physics is one of the most important fields nowadays because it is of great help for testing the quark flavor mixing theory of the standard model and exploring the source of CP violation. Most of the theoretical studies of B decays to pseudocalar and vector final states are based on the popular Naive Factorization approach[1]. As it was ponited out years ago in Ref. [2], the dominant contribution in B decays comes from the so-called Feynman mechanism, where the energetic quark created in the weak decay picks up the soft spectator softly and carries nearly all of the final-state meson's momentum. It is also shown that Pion form factor in QCD at intermediate engery scale is dominated by Feynman mechanism[3—5]. From this point, we can understand why the naive factorization approach have worked well for B and D decays, and the many existing predictions for B decays based on naive factorization and spectator ansatz do have taken in the dominant physics effects although there are shortcommings. However, with the many new data available from CLEO and an abundance of data to arrive within few years from the B factories BaBar and Belle, it is demanded highly to go beyond the naive factorization approach.Recently, Beneke et al., have formed an interesting QCD factorization formula for B exclusive nonleptonic decays[6,7]. The factorization formula incorporates elements of the naive factorization approach (as leading contribution) and the hard-scattering approach (as subleading corrections), which allows us to calculate systematically radiative(subleading nonfactorizable) corrections to naive factorization for B exclusive nonleptonic decays. An important product of the formula is that the strong final-state interaction phases are calculable, which arise from the2 高能物理与核物理(HEP &NP) 第26卷hard-scattering kernel and hence process dependent. The strong phases are very important for studying CP violation in B decays.The amplitude of B decays to two light mesons, say M 1 and M 2, is obtained through the hadronic matrix element <M 1(p 1) M 2(p 2)⏐O i ⏐B (p )>, here M 1 denotes the final meson that picks up the light spectator quark in the B meson, and M 2 is the another meson which is composed of the quarks produced from the weak decay point of b quark. Since the quark pair, forming M 2, is ejected from the decay point of b quark carrying the large energy of order of m b , soft gluons with the momentum of order of ΛQCD decouple from it at leading order of ΛQCD /m b in the heavy quark limit. As a consequence any interaction between the quarks of M 2 and the quarks out of M 2 is hard at leading power in the heavy quark expansion. On the other hand, the light spectator quark carries the momentum of the order of ΛQCD , and is softly transferred into M 1 unless it undergoes a hard interaction.Any soft interaction between the spectator quark and other constituents in B and M 1 can be absorbed into the transition form factor of B →M 1. The non-factorizable contribution to B →M 1 M 2 can be calculated through the diagrams in Fig.1.Fig. 1. Order αs non-factorizable contributions in B →M 1M 2 decays.The O i 's incorporated in Fig.1 are the operators in the effective Hamiltonian for B decays [8], ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+++⎟⎟⎠⎞⎜⎜⎝⎛++=∑∑∑∑====21103g g ccqcb 21103g g uuq ub F eff2i i i i i i *i i i i i i *O C O C O C V V O C O C O C V V G H , (1)Where()()A V A V O －－ββααu 1b u u q ⋅=, ()()A V A V O －－αββαu 2b u u q ⋅=, ()()A V A V O －－ββααc 1b c c q ⋅=, ()()A V A V O －－αββαc 2b c c q ⋅=,()()AV A V O －－ββq αα3q q b q ′′⋅=∑′,()()AV A V O －－αβq βα4q q b q ′′⋅=∑′,()()AV A V O +′′′⋅=∑ββq αα5q q b q －, ()()A V A V O +′′′⋅=∑αβq βα6q q b q －, ()()A V A V e O +′′′′⋅=∑ββq q αα7q q b q 23－, ()()A V A V e O +′′′′⋅=∑αq q b q 23βq q βα8－,增刊杨茂志等：用QCD 因子化方法研究B →PV 两体弱衰变过程 3()()A V A V e O －－ββq q αα9q q b q 23′′⋅=∑′′, ()()A V A V e O －－αβq q βα10q q b q 23′′⋅=∑′′, ()()AA a g G b R m g O µνβαβµνλσ2/d π8/b 2s =. (2)Here q=d, s and (q'ε {u, d, s, c, b}), α and β are the SU (3) color indices and , A =1,...,8 are the Gell-Mann matrices, and denotes the gluonic field strength tensor. The Wilson coefficients evaluated at µ=m AαβλAG µνb scale are[8]C 1= 1.082, C 2=−0.185, C 3= 0.014, C 4=−0.035, C 5= 0.009, C 6=−0.041,C 7=−0.002/137, C 8=0.054/137, C 9=−1.292/137, C 10=0.262/137, C g =−0.143. (3) The non-factorizable contributions to B →M 1M 2 can be calculated through the diagrams in Fig.1. The details of the calculations can be found in Ref. [9]. In the numerical calculations we use[10]τ (B +) = 1.65×10－12s, τ (B 0) = 1.56×10－12s,M B = 5.2792GeV , m b = 4.8GeV , m c = 1.4GeV , f B= 0.180GeV , f π = 0.133GeV , f K = 0.158GeV , f K * = 0.214GeV , f ρ = 0.21GeV , f ω = 0.195GeV .For the chiral enhancement factors for the pseudoscalar mesons, we takeR π ±π= R K ±, 0 = －1.2 ,which are consistent with the values used in [6, 11, 12]. We should take care for R π0. As pointedout in Ref. [7], R π0 for π0 should be －2M /(m 2b (m u + m d )) and equal to R π± due to inclusion ofisospin breaking effects correctly.For the form factors, we take the results of light-cone sum rule[13,14]F B →π(0)=0.3, F B →K (0)=1.13F B →π(0), A =0.372, A =0.470,ρB 0→*K B 0→and assume (0)=1.2(0) since we find larger (0) is preferred by experimental data.ωB 0→A ρB 0→A ωB 0→A We take the leading-twist distribution amplitude (DA) φ(x ) and the twist-3 DA φ0(x ) of light pseudoscalar and vector mesons as the asymptotic form[15]φP,V (x ) =6x (1－x ), (x ) =1. (4) 0P φFor the B meson, the wave function is chosen as[16,17]()(),xM x x N x ⎥⎥⎦⎤⎢⎢⎣⎡=2B 22B 22B B 2exp 1ωφ－－ (5)with ωB =0.4GeV , and N B is the normalization constant to make(x ) =1. φ∫1Bd φx B (x ) is stronglypeaked around x =0.1, which is consistent with the observation of Heavy Quark Effective Theory that the wave function should be peaked around ΛQCD /M B .We have used the unitarity of the CKM matrix V *uq V ub +V *V cq cb +V *tq V tb =0 to decompose the4 高能物理与核物理(HEP &NP)第26卷amplitudes into terms containing , V *uq V ub and V *V cq cb , and⏐V us ⏐=λ=0.2196, ⏐V ub /V cb ⏐=0.085±0.02, ⏐V cb ⏐=0.0395±0.0017, ⏐V ud ⏐=1－λ2/2 . (6) We leave the CKM angle γ as a free parameter.The numerical results of the branching ratios B →PV are shown in Fig.2 as the function of CKM angle γ. We can see from Fig. 2(a), (b) and (c) that for the three detected channels the predicted branching ratios agree well with the CLEO experiment data [18]. Our predictions for other decay modes are well below their 90% C.L. upper limits.There are several works available with detailed analysis of the CLEO new data of the decays of B to charmless PV states[11,12,19]. It is worth to note that the shortcomings in the “generalizedfactorization” are resolved in the framework of QCD Factorization. Nonfactorizable effects are calculated in a rigorous way here instead of being parameterized by effective color number. Since the hard scattering kernals are convoluted with the light cone DAs of the mesons, gluon virtualityk 2=2b m x in the penguin diagram Fig. 1(e) has well defined meaning and leaves no ambiguity as tothe value of k 2, which has usually been treated as a free phenomenological parameter in the estimations of the strong phase generated though the BSS mechanism [20]. So that CP asymmetries are predicted soundly in this approach. We present the numerical result of the branching ratios of B →PV decays in Table 1 with the relevant strong phases shown explicitly. It shows that the strong phases are generally mode dependent.Table 1. Strong phases in the branching ratios (in units of 10－6) for thecharmless decays modes studied by CLEO. (γ =Arg V *u b )B (B －→π－ρ0)=6.65⏐0.11e －i86.5°+e －i γ⏐2B (0B →π+ρ－)=19.79⏐0.11e i9.02°+e －i γ⏐2B (0B →π－ρ+)=13.43⏐0.03e i172°+e －i γ⏐2B (B －→π－ω)=10.59⏐0.065e i26.01°+e －i γ⏐2B (0B →π0ρ0)=0.11⏐0.21e 2.90°+e －i γ⏐2B (B －→π0ρ－)=10.81⏐0.176e i7.20°+e －i γ⏐2B (0B →π－ω)=1.49×10－3⏐1.64e i148°+e －i γ⏐2B (B －→K －ρ0)=0.55⏐0.24e －i162°+e －i γ⏐2B (B －→π－⎯K *0)=0.0012⏐56.4e －i15.7°+e －i γ⏐2B (B －→K －K *0)=0.030⏐2.86e i164°+e －i γ⏐2B (B －→π0K *－)=0.59⏐2.80e －i169°+e －i γ⏐2B (B －→K －ω)=0.80⏐0.48e －i9.23°+e －i γ⏐2B (0B →K 0ω)=0.72⏐0.81e －i 11.8°+e －i γ⏐2B (⎯B 0→K －ρ+)=0.96⏐0.63e －i7.20°+e －i γ⏐2B (0B →π0⎯K *0)=0.004⏐12.89e i67.61°+e －i γ⏐2Hou, Smith and W ürthwein have performed a model dependent fit using the recent CLEOdata and found γ =114degree. Using SU (3) flavor symmetry, Gronau and Rosner have analyzedthe decays of B to charmless PV final states extensively and found several processes are consistent with cos γ < 0. In this paper we find cos γ < 0 is favored by the B 2521+－－→π－ρ0 and ⎯B 0→π－ρ++π+ρ－ if their experimental center values are taken seriously. To meet its center value with cos γ < 0 , B －→π增刊杨茂志等：用QCD 因子化方法研究B →PV 两体弱衰变过程 5－ω would indicate larger form factor i.e. A (0) > A (0). In our numerical calculation, wehave taken A (0) = 0.446 which is still consistent with the LCSR results 0.372 ± 0.074ω→B 0ρ→B 0ω→B 0[13]. It isalso interesting to note that ⎯B 0→π+ρ－ is suppressed by cos γ < 0 while ⎯B 0→π－ρ+ is enchanced. The defference between Br (⎯B 0→π+ρ－) and Br (⎯B 0→π－ρ+) is much more sensitive to γ than their sum.6 高能物理与核物理(HEP &NP) 第26卷Summarywe have calculated the branching ratios and CP asymmetries of the charmless decays B →PV(P = (π, K), V= (ρ,ω, K *)) in QCD factorization approach. We have used LCSR formfactors F B →π,K (0) and A (0) as inputs. The results of Br (B *K ,0ρ－→π－ρ0) and Br (⎯B 0→π±ρ) agree with CLEO m [18]very well and favor cos γ < 0 if their experimental center values are taken seriously. To meet its experimental center value and cos γ < 0, the decay B －→π－ω will prefer larger form factor (0). For the other decay modes, the branching ratios are predicted well below their 90% C.L. upper limits given in Ref. [18].ωB 0→A References1 Bauer M, Stech B, Wirbel M. Z. Phys., 1985, C29: 637; Z. Phys., 1987, C34: 1032 Chernyak V L, Zhitnitsky L R. Nucl. Phys., 1990, B345: 1373 Isgur N, Llewelyn-Smith C H. Phys. Rev. Lett., 1984, 52: 1080; Nucl. Phys., 1989, B317: 5264 Radyushkin A V . Acta Phys., 1984, Pol. 15: 4035 Stefanis N G . hep-ph/99113756 Beneke M, Buchalla G , Neubert M. Phys. Rev. Lett., 1999, 83: 19147 Beneke M, Buchalla G , Neubert M et al. hep-ph/00061248 Buchalla G , Buras A J, Lautenbacher M E. Rev. Mod. Phys., 1996, 68: 1125 9 YANG M Z, YANG Y Y . Phys. Rev., 2000, D62: 114019 10 Particle Data Group. Eur. Phys. J., 1998, C3: 1 11 CHENG H Y , YANG K C. hep-ph/991029112 HOU W S, Smith J G , W ürthwein F W. hep-ex/9910014 13 Ball P, Braun V M. Phys. Rev., 1998, D58: 094016 14 Ball P. JHEP09, 005(1998)15Lepage G P, Brodsky S J. Phys. Lett., 1979, B87: 359; Chernyak V L, Zhitinissky A R. Phys. Rep., 1983, 112: 173; Braun V M, Filyanov I E. Z. Phys., 1990, C48: 239 16 Keum Y Y , LI H -N, Sanda A I. Phys. Lett., 2001, B504: 2; Phys. Rev., 2001, D63: 054008 17 LÜ C D, Ukai D, YANG M Z. Phys. Rev., 2001, D63: 07400918 CLEO Collaboration. CLEO CONF 99-13; CLEO Collaboration. CLNS 99/1652 and CLEO 99-19 19 Gronau M, Rosner J L. Phys. Rev., 2000, D61: 073008 20Bander M, Silverman D, Soni A. Phys. Rev. Lett., 1979, 43: 242增刊杨茂志等：用QCD因子化方法研究B→PV两体弱衰变过程7 用QCD因子化方法研究B→PV两体弱衰变过程杨茂志1 杨亚东21 (中国科学院高能物理研究所北京 100039)2 (Department of Physics, Technion, Haifa 32000, Israel)摘要基于最近CLEO实验和B介子物理中理论研究的进展, 在QCD因子化方案下研究了B介子到一个赝标π, K和一个矢量介子ρ, ω的两体弱衰变过程.在合理的参数范围内, 理论计算与实验相符得很好.关键词因子化弱衰变介子。

利用迈克耳逊干涉仪测气体折射率实验报告

利用迈克耳逊干涉仪测气体折射率实验报告英文回答：Determination of Refractive Index of Gas Using Michelson Interferometer。

The Michelson interferometer is a versatile tool that can be used to measure various optical properties of materials, including the refractive index of gases. In this experiment, we employed the Michelson interferometer to determine the refractive index of an unknown gas sample.The Michelson interferometer consists of two mirrors, M1 and M2, mounted on a rigid frame in a configuration known as the "common path interferometer." A beam of light from a coherent source is split into two beams using asemi-transparent mirror, M3. One beam travels to mirror M1, while the other travels to mirror M2. The beams are then reflected back to the semi-transparent mirror, where theyrecombine to produce an interference pattern.The interference pattern is typically observed as a series of bright and dark fringes. The position of these fringes depends on the optical path length difference between the two interfering beams. By introducing a cell containing the gas sample into one of the arms of the interferometer, the optical path length is changed, resulting in a shift in the fringe pattern.The change in the fringe pattern can be used to determine the refractive index of the gas sample. The refractive index is a dimensionless quantity that describes the speed of light in a medium relative to its speed in vacuum. A higher refractive index indicates that the light travels slower in the medium.The refractive index of the gas sample can be calculated using the following formula:```。

RealtimePCRprotocol

2 colorless cap
H2O, PCRgrade
Storage and Stability
Store the kit at Ϫ15 to Ϫ25°C through the expiration date printed on the label.
N Keep the Reaction Mix (vial 1b, green cap) away from light.
Refer to the list below for additional reagents and equipment required to perform reactions with the LightCycler® FastStart DNA MasterPLUS SYBR Green I using the LightCycler® System: • LightCycler® System (LightCycler® 2.0 Instrument*, LightCycler® 1.5 Instrument*, or an instrument version below) • LightCycler® Capillaries* • Standard benchtop microcentrifuge containing a rotor for 2.0 ml reaction tubes L The LightCycler® System provides adapters that allow LightCycler® Capillaries to be centrifuged in a standard microcentifuge rotor. or • LightCycler® Carousel Centrifuge 2.0* for use with the LightCycler® 2.0 Carousel (optional) N If you use a LightCycler® Instrument version below 2.0, you need in addition the LightCycler® Carousel Centrifuge 2.0 Bucket 2.1*. To adapt the LightCycler® 2.0 Carousel to the former LightCycler® Carousel Centrifuge, you need the LightCycler® Carousel Centrifuge 2.0 Rotor Set*. • LightCycler® Uracil-DNA Glycosylase* (optional‡) • Nuclease-free, aerosol-resistant pipette tips • Pipettes with disposable, positive-displacement tips • Sterile reaction (Eppendorf) tubes for preparing master mixes and dilutions

夸克怎么批改英语作文

夸克怎么批改英语作文How Quark Corrects English Composition。

When it comes to correcting English compositions, Quark is a powerful tool that can help students improve their writing skills. With its advanced grammar and spelling checking features, Quark can quickly identify and correct errors in students' compositions, allowing them to focus on improving their writing style and content.To begin with, Quark's grammar checking feature is incredibly helpful for students who struggle with sentence structure and syntax. By highlighting grammatical errors and suggesting corrections, Quark helps students learn from their mistakes and develop a better understanding of English grammar rules. This not only improves the quality of their compositions but also helps them become more confident and proficient writers.In addition to grammar checking, Quark's spellingchecking feature is also essential for students who often make spelling mistakes. By flagging misspelled words and offering alternative spellings, Quark helps students avoid embarrassing typos and ensures that their compositions are error-free. This not only enhances the overall readability of their compositions but also helps them develop better spelling habits.Moreover, Quark's advanced editing tools allow students to revise and improve their compositions with ease. With features such as track changes and comments, Quark makes it easy for students to collaborate with their teachers and peers, receiving valuable feedback and making necessary revisions. This not only fosters a culture of continuous improvement but also helps students develop critical thinking and self-editing skills.Furthermore, Quark's plagiarism checking feature is crucial for ensuring the originality of students' compositions. By comparing their work against a vast database of academic sources, Quark helps students avoid unintentional plagiarism and maintain academic integrity.This not only reinforces the importance of ethical writing practices but also encourages students to develop their own unique voice and ideas.In conclusion, Quark is an invaluable tool for correcting English compositions. With its grammar and spelling checking features, advanced editing tools, and plagiarism checking capabilities, Quark helps students improve their writing skills and become more proficient and confident writers. By using Quark to correct their compositions, students can focus on honing their writing style and content, ultimately becoming better communicators and critical thinkers.。

圆二色谱Circular_CD

主要内容
CD原理
蛋白质CD谱
CD实验要点
CD原理
圆二色性(circular dichroism, CD)
当平面偏振光通过具有旋光活性的介质时，由于介质中同一种旋光活性分子存在手性不同的两种构型，故它们对平面偏振光所分解成的右旋和左旋圆偏振光吸收不同，从而产生圆二色性．
圆二色性的表示
椭圆度，摩尔椭圆度[] =2.303(AL – AR)/4 [] = 3298(L - R)3300 (L - R) 在蛋白质研究中，常用平均残基摩尔椭圆度
圆二色仪原理
蛋白质的CD谱
蛋白质的光学活性
The peptide bond is inherently asymmetric & is always optically active
构象
确定蛋白质构象最准确的方法是x-射线晶体衍射，但对结构复杂、柔性的生物大分子蛋白质来说，得到所需的晶体结构较为困难。二维、多维核磁共振技术能测出溶液状态下较小蛋白质的构象，可是对分子量较大的蛋白质的计算处理非常复杂。圆二色光谱：研究稀溶液中蛋白质构象，快速、简单、较准确
CD is very useful for looking at membrane proteins
蛋白的三级结构
1976年，Levitt和Chothia曾在Nature上报道，规则蛋白质的三级结构模型可分为4类 (1) 全α型，以仅α-螺旋结构为主，其分量大于40％，而 β-折叠的分量小于5％ (2) 全β型，以β-折叠这种结构为主，其分量大于40％，而仅一螺旋的分量小于5％； (3) α+β型，α螺旋及β-叠折分量都大于15％，这两种结构在空间上是分离的，且超过60％的折叠链是反平行排列； (4) α/β型， α-螺旋和B-折叠含量都大于15％，它们在空间上是相间的，且超过60％的折叠链平行排列。

Weak Interactions of Light Flavours

2Leabharlann Standard Model
LSM = LH (φ) + LG (W, Z, G) + Higgs Gauge ¯ / ψ+ ψiD
ψ=fermions ψ,ψ′ =fermions
The Standard Model Lagrangian has four parts: ¯ ′ gψψ′ ψφψ Yukawa
QCD and QED conserve C,P,T separately. Local Field theory by itself implies CPT. The fermion and Higgs2 part of the SM-lagrangian conserves CP and T as well. The only part that violates CP and as a consequence also T is the Yukawa part. The Higgs part is responsible for two parameters, the gauge part for three and the HiggsFermion part contains in principle 27 complex parameters, neglecting Yukawa couplings to neutrinos. Luckily most of the 54 real parameters in the Yukawa sector are unobservable. After diagonalizing the lepton sector there only the three charged lepton masses remain. The quark sector can be similarly diagonalized leading to 6 quark masses, but some parts remain in the diﬀerence between weak interaction eigenstates and mass-eigenstates. The latter is conventionally put in the couplings of the charged W -boson, which is given by Vud Vus Vub dα g α − uα cα t γ µ (1 − γ5 ) Vcd Vcs Vcb sα − √ Wµ 2 2 V V V b

Pentaquarks五夸克态.ppt

Pentaquarks: 五夸克态
Bo-Qiang Ma (马伯强)
PKU (北京大学物理学院)
?
August 14, 2004 talk at CCAST Wokshop on QCD and RHIC Physics
In Collaboration with B. Wu
Hep-ph/0312041, PRD69(2004)077501 Hep-ph/0312326, PLB586（2004）62 Hep-ph/0311331, Hep-ph/0402244 Hep-ph/0408121 all to appear in PRD
n K- (K+n)
an additional K+ is detected to reduce background for the missing mass spectrum and real photon is used instead of virtual photon.
12
What is a Pentaquark
• The corrections to the Gell-Mann-Okubo relations of baryons masses in SU(3) Skyrmen model are considered.
• The results could be regarded as a signal for the existence of the SU(3) rotation excitation states of baryons: 27-plet, 10*-let, and 35-let.
– MIT bag model predictions: MZ 1.7 GeV
• L(1405): molecular meson-baryon state uudsu? – interpretation problematic: could be uds

相差聚光镜_PCD2

BEFORE USEThis unit employs the UIS optical system and should be used only with UIS microscopes, eyepiecesand objectives. Less than optimum performance may result if inappropriate accessories are used.Caution:If the condenser is used in a manner not specified by this manual, the safety of the unit may be impaired. In addition, the unit may also be damaged. Always use the unit as outlined in this instruction manual.1.3 kinds of the light annuli are built into this turret condenser, allowing the following observations by changing the light annuli.Phase contrast observation:10X, 20X, 40X, 100XDarkfield observation:Up to 40XBrightfield observation2.This phase contrast condenser makes it possible to center each light annuli.3.Applicable objectives are as follows:UPlanFl10Xph, 20Xph, 40Xph, 100XphAch10Xph, 20Xph, 40Xph, 100XphPlan10Xph, 20Xph, 40Xph, 100XphPhase Contrast Turret Condenser Aperture iris diaphragm leverLight annulus centering knobPhase contrast turret Mounting dovetailAperture is wide-open ex-cept when this is set at thebrightfield position.Fig. 13ASSEMBLYMounting on the Microscope1.Rotate the coarse adjustment knob @ to fully raise the stage.#Be careful not to raise the stage too much to avoid acci-dental contact with the objective lens and specimen.2.Lower the condenser holder with the condenser height ad-justment knob ² until it stops at the lower limit.3.Loosen the condenser clamping knob ³.4.Insert the phase contrast turret condenser into the dovetailguideway of the condenser holder until the positioning pin ofthe condenser is engaged in the positioning groove of themounting dovetail guideway.5.Tighten the clamping knob of the condenser holder.6.Raise the condenser holder until it stops at the upper limit.U-PCD24OPERATIONCenter the Phase Contrast Condenser1.Rotate the phase contrast turret until the letter “0” can be seen in the central window, and open the aperture iris diaphragm fully.2.Place a specimen on the stage, bring the 10X objective into the light path and focus on the specimen.3.Reduce the field iris diaphragm of the microscope frame.4.While looking through the eyepieces, move the condenser up and down with the condenser height adjustment to focus on the image of the field iris diaphragm.5.Adjust the condenser centering knobs | to bring the diaphragm image into the center of the field of view.(Figs 1 & 2)·Widening the diameter of the field iris diaphragm progressively, repeat this operation until the reduced image of the diaphragm becomes inscribed in the circle which indicates the field of view.·When opening the field iris diaphragm, the image may be out of focus. Re-focus with the condenser height adjustment knob.Fig. 2Phase Contrast Observation1.Swing the objective of your choice into the light path.2.Rotate the phase contrast turret until the turret index dot indicates the Ph number seen through the central window which corresponds to the Ph number indicated on the side of objective lens.3.Focus roughly on the specimen.4.Remove one of the eyepieces, and insert the centering telescope U-CT30 into the eyepiece sleeve.5.Rotate the top lens assembly of the U-CT30 to bring the bright ring (light annulus in the condenser) and the dark ring (phase annulus in the objective) in focus.6.Rotate the two light annulus centering knobs @ of the condenser until the light annulus and phase annuls are concentric and superimposed. (Fig. 3)Before centration After centrationa =b =c = dFig. 3bac dU-PCD2#The other phase annuluses should be centered in the same manner.In case Ph1 is indicated on both 10X and 20X objectives, for observation purposes, once the phase annulus is aligned for 10X, it’s unnecessary to realign the phase annulus for 20X.However, for the purpose of photomicrography, it is recommended the phase annulus be aligned for each objective even though the same Ph number may be indicated on the objective lens.7.Remove the U-CT30, and insert the eyepiece back into the eyepiece sleeve.Now you can start your phase contrast observation with any phase contrast objective and corresponding light annulus as desired.Darkfield Observation1.Rotate the turret until “DF” can be seen in the central window.2.Swing the 40X or lower magnification objective into the light path.#It is possible to use objectives with NA0.75 (UPlanFl40X) or lower NA for darkfield observation.3.Place the specimen on the stage and move the stage up and down to bring the specimen into focus.4.Move the condenser up and down until uniform darkfield illumination is attained.5.Open the field iris diaphragm to the extent that even brightness is attained.#Keep eyes away from the eyepieces while changing the objective during darkfield observation, or changing from darkfield to another observation mode.2-43-2,Hatagaya, Shibuya-ku, Tokyo, JapanThis publication is printed on recycled paper.Printed in Japan 2000 03 M 005–@。

PhysRevD.79.114009

Relativistic correction to J=c production at hadron collidersYing Fan,*Yan-Qing Ma,†and Kuang-Ta Chao‡Department of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University,Beijing100871,China(Received20April2009;published10June2009)Relativistic corrections to the color-singlet J=c hadroproduction at the Tevatron and LHC arecalculated up to Oðv2Þin nonrelativistic QCD(NRQCD).The short-distance coefﬁcients are obtainedby matching full QCD with NRQCD results for the subprocess gþg!J=cþg.The long-distancematrix elements are extracted from observed J=c hadronic and leptonic decay widths up to Oðv2Þ.Usingthe CTEQ6parton distribution functions,we calculate the leading-order production cross sections andrelativistic corrections for the process pþ"pðpÞ!J=cþX at the Tevatron and LHC.Weﬁnd that theenhancement of Oðv2Þrelativistic corrections to the cross sections over a wide range of large transversemomentum p t is negligible,only at a level of about1%.This tiny effect is due to the smallness of thecorrection to short-distance coefﬁcients and the suppression from long-distance matrix elements.Theseresults indicate that relativistic corrections cannot help to resolve the large discrepancy between leading-order prediction and experimental data for J=c production at the Tevatron.DOI:10.1103/PhysRevD.79.114009PACS numbers:12.38.Bx,12.39.St,13.85.Ni,14.40.GxI.INTRODUCTION Nonrelativistic QCD(NRQCD)[1]is an effectiveﬁeld theory to describe production and decay of heavy quark-onium.In this formalism,inclusive production cross sec-tions and decay widths of charmonium and bottomonium can be factored into short-distance coefﬁcients,indicating the creation or annihilation of a heavy quark pair,and long-distance matrix elements,representing the evolvement of a free quark pair into a bound state.The short-distance part can be calculated perturbatively in powers of coupling constant s,while the nonperturbative matrix elements, which are scaled as v,the typical velocity of heavy quark or antiquark in the meson rest frame,can be estimated by nonperturbative methods or models,or extracted from experimental data.One important aspect of NRQCD is the introduction of the color-octet mechanism,which allows the intermediate heavy quark pair to exist in a color-octet state at short distances and evolve into the color-singlet bound state at long distances.This mechanism has been applied success-fully to absorb the infrared divergences in P-wave[1–3] and D-wave[4,5]decay widths of heavy quarkonia.In Ref.[6],the color-octet mechanism was introduced to account for the J=c production at the Tevatron,and the theoretical prediction of production rateﬁts well with experimental data.However,the color-octet gluon frag-mentation predicts that the J=c is transversely polarized at large transverse momentum p t,which is in contradiction with the experimental data[7].(For a review of these issues,one can refer to Refs.[8–10]).Moreover,in Refs.[11,12]it was pointed out that the color-octet long-distance matrix elements of J=c production may be much smaller than previously expected,and accordingly this mayreduce the color-octet contributions to J=c production at the Tevatron.In the past a couple of years,in order to resolve the largediscrepancy between the color-singlet leading-order(LO)predictions and experimental measurements[13–15]ofJ=c production at the Tevatron,the next-to-leading-order (NLO)QCD corrections to this process have been per-formed,and a large enhancement of an order of magnitudefor the cross section at large p t is found[16,17].But thisstill cannot make up the large discrepancy between thecolor-singlet contribution and data.Similarly,the observeddouble charmonium production cross sections in eþeÀannihilation at B factories[18,19]also signiﬁcantly differfrom LO theoretical predictions[20].Much work has beendone and it seems that those discrepancies could be re-solved by including NLO QCD corrections[21–24]andrelativistic corrections[25,26].One may wonder if therelativistic correction could also play a role to some extentin resolving the long standing puzzle of J=c production at the Tevatron.In this paper we will estimate the effect of relativisticcorrections to the color-singlet J=c production based on NRQCD.The relativistic effects are characterized by therelative velocity v with which the heavy quark or antiquarkmoves in the quarkonium rest frame.According to thevelocity scaling rules of NRQCD[27],the matrix elementsof operators can be organized into a hierarchy in the smallparameter v.We calculate the short-distance part pertur-batively up to Oðv2Þ.In order to avoid model dependence in determining the long-distance matrix elements,we ex-tract the matrix elements of up to dimension-8four fermionoperators from observed decay rates of J=c[28].Weﬁnd that the relativistic effect on the color-singlet J=c produc-tion at both the Tevatron and LHC is tiny and negligible,*ying.physics.fan@†@‡ktchao@PHYSICAL REVIEW D79,114009(2009)and relativistic corrections cannot offer much help to re-solve the puzzle associated with charmonium production at the Tevatron,and other mechanisms should be investigated to clarify the problem.The rest of the paper is organized as follows.In Sec.II, the NRQCD factorization formalism and matching condi-tion of full QCD and NRQCD effectiveﬁeld theory at long distances are described brieﬂy,and then detailed calcula-tions are given,including the perturbative calculation of the short-distance coefﬁcient,the long-distance matrix elements extracted from experimental data,and the parton-level differential cross section convolution with the parton distribution functions(PDF).In Sec.III,nu-merical results of differential cross sections over transverse momentum p t at the Tevatron and LHC are given and discussions are made for the enhancement effects of rela-tivistic corrections.Finally the summary of this work is presented.II.PRODUCTION CROSS SECTION IN NRQCDFACTORIZATIONAccording to NRQCD factorization[1],the inclusive cross section for the hadroproduction of J=c can be writ-ten asd dt ðgþg!J=cþgÞ¼XnF nm d nÀ4ch0j O J=c n j0i:(1)The short-distance coefﬁcients F n describe the production of a heavy quark pair Q"Q from the gluons,which come from the initial state hadrons,and are usually expressed in kinematic invariants.m c is the mass of charm quark.Thelong-distance matrix elements h0j O J=cn j0i with mass di-mension d n describe the evolution of Q"Q into J=c.The subscript n represents the conﬁguration in which the c"c pair can be for the J=c Fock state expansion,and it isusually denoted as n¼2Sþ1L½1;8J .Here,S,L,and J standfor spin,orbital,and total angular momentum of the heavy quarkonium,respectively.Superscript1or8means the color-singlet or color-octet state.For the color-singlet3S1c"c production,there are only two matrix elements contributing up to Oðv2Þ:the leading-order term h0j O J=cð3S½1 1Þj0i and the relativistic correction term h0j P J=cð3S½1 1Þj0i.Therefore the differential cross section takes the following form:d dt ðgþg!J=cþgÞ¼Fð3S½11Þm2ch0j O J=cð3S½1 1Þj0iþGð3S½11Þm4ch0j P J=cð3S½1 1Þj0iþOðv4Þ;(2)and the explicit expressions of the matrix elements are[1]h0j O J=cð3S½1 1Þj0i¼h0j y i cða y c a cÞc y i j0i;h0j P J=cð3S½1 1Þj0i¼12y i cða y c a cÞc y iÂÀi2D$2þH:c:;(3)where c annihilates a heavy quark, creates a heavy antiquark,a y c and a c are operators creating and annihilat-ing J=c in theﬁnal state,and D$¼~DÀD.In order to determine the short-distance coefﬁcients Fð3S½1 1Þand Gð3S½1 1Þ,the matching condition of full QCD and NRQCD is needed:ddtðgþg!J=cþgÞj pert QCD¼Fð3S½11Þm2ch0j O J=cð3S½1 1Þj0iþGð3S½11Þm4ch0j P J=cð3S½1 1Þj0ij pert NRQCD:(4)The differential cross section for the production of char-monium J=c on the left-hand side of Eq.(4)can be calculated in perturbative QCD.On the right-hand side the long-distance matrix elements are extracted from ex-perimental data.Quantities on both sides of the equation are expanded at leading order of s and next-to-leading order of v2.Then the short-distance coefﬁcients Fð3S½1 1Þand Gð3S½1 1Þcan be obtained by comparing the terms with powers of v2on both sides.A.Perturbative short-distance coefﬁcientsWe now present the calculation of relativistic correction to the process gþg!J=cþg.In order to determine the Oðv2Þcontribution in Eq.(2),the differential cross section on the left-hand side of Eq.(4)or equivalently the QCD amplitude should be expanded up to Oðv2Þ.We use FeynArts[29]to generate Feynman diagrams and am-plitudes,FeynCalc[30]to handle amplitudes,and FORTRAN to evaluate the phase space integrations.A typi-cal Feynman diagram for the process is shown in Fig.1.FIG.1.Typical Feynman diagram for3S½1 1c"c hadroproduction at LO.YING FAN,YAN-QING MA,AND KUANG-TA CHAO PHYSICAL REVIEW D79,114009(2009)The momenta of quark and antiquark in the lab frame are [26,31,32]:12Pþq ¼L ð12P rþq r Þ;12PÀq ¼L ð12P rÀq r Þ;(5)where L is the Lorenz boost matrix from the rest frame ofthe J=c to the frame in which it moves with four momen-tum P .P r ¼ð2E q ;0Þ,E q ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃm 2c þj ~qj 2p ,and 2q r ¼2ð0;~qÞis the relative momentum between heavy quark and antiquark in the J=c rest frame.The differential cross section on the left-hand side of Eq.(4)isddtðg þg !J=c þg Þj pert QCD ¼12"X j M ðg þg !J=c þg Þj 2h 0j O J=c ð3S ½1 1Þj 0i ;(6)where h 0j O J=c ð3S ½11Þj 0i is the matrix element evaluated at tree level,and the summation/average of the color and polarization degrees of freedom for the ﬁnal/initial statehas been implied by the symbol "P.The amplitude for the color-singlet process g ðp 1Þþg ðp 2Þ!J=c ðp 3¼P Þþg ðp 4ÞisM ðg þg !J=c þg Þ¼ﬃﬃﬃﬃﬃﬃ1E q s Tr ½C ½1 Åð1ÞM am ;(7)where M am denotes the parton-level amplitude amputatedof the heavy quark spinors.The factor ﬃﬃﬃﬃ1E q q comes from the normalization of the composite state j 3S ½11i[5].Here the covariant projection operator method [33,34]is adopted.For a color-singlet state,the color projector C ½1 ¼ ijﬃﬃﬃﬃﬃcp .The covariant spin-triplet projector Åð1Þin (7)is deﬁned byÅð1Þ¼X s "s v ðs Þ"u ð"s Þ 12;s ;12;"s j 1;S z ;(8)with its explicit formÅð1Þ¼1ﬃﬃﬃ2p ðE q þm c Þ P 2Àq Àm cÂP À2E q 4E qP 2þq þm c;(9)where the superscript (1)denotes the spin-triplet state andis the polarization vector of the spin 1meson.The Lorentz-invariant Mandelstam variables are deﬁned bys ¼ðp 1þp 2Þ2¼ðp 3þp 4Þ2;t ¼ðp 1Àp 3Þ2¼ðp 2Àp 4Þ2;u ¼ðp 1Àp 4Þ2¼ðp 2Àp 3Þ2;(10)and they satisfys þt þu ¼P 2¼4E 2q ¼4ðm 2c þj ~qj 2Þ:(11)Furthermore,the covariant spinors are normalized relativ-istically as "uu¼À"vv ¼2m c .Let M be short for the amplitude M ðg þg !J=c þg Þin Eq.(7),and it can be expanded in powers of v orequivalently j ~qj .That is M ¼ M¼ M ð0Þþ1q q @2M @q @q q ¼0þO ðq 4Þ;(12)where high order terms in four momentum q have been omitted.Terms of odd powers in q vanish because the heavy quark pair is in an S-wave conﬁguration.Note that the polarization vector also depends on q ,but it only has even powers of four momentum q ,and their expressions may be found e.g.in the appendix of Ref.[35].Therefore expansion on q 2of can be carried out after amplitude squaring.The following substitute is adopted:q q¼13j ~q j 2 Àg þP P P 213j ~q j 2Á :(13)This substitute should be understood to hold in the inte-gration over relative momentum ~qand in the S-wave case.Here,j ~qj 2can be identiﬁed as [33,36]j ~qj 2¼jh 0j y ðÀi2D $Þ2c j c ð3S ½1 1Þijjh 0j y c j c ð3S 1Þij ¼h 0j P J=c ð3S ½1 1Þj 0i h 0j O J=c ð3S ½1 1Þj 0i½1þO ðv 4Þ :(14)Then the amplitude squared deﬁned in Eq.(6)up to O ðv 2Þis Xj M j 2¼Mð0ÞM Ãð0ÞXÃþ16j ~q j 2 Á @2M @q @q q ¼0M Ãð0Þþ Á @2M Ã@q @q q ¼0M ð0Þ X Ã þO ðv 4Þ:(15)The heavy quark and antiquark are taken to be on shell,which means that P Áq ¼0,and then gauge invariance is maintained.The polarization sum in Eq.(15)isXÃ ¼Àg þP PP 2:(16)RELATIVISTIC CORRECTION TO J=c PRODUCTION ...PHYSICAL REVIEW D 79,114009(2009)It is clearly seen that the polarization sum above only contains even order powers of four momentum q,therefore it will make a contribution to the relativistic correction at Oðv2Þin theﬁrst term on the right-hand side of Eq.(15)when the contraction over indices and is carried out. However,since the second term on the right-hand side of Eq.(15)already has a term proportional to q2,i.e.j~q j2,the four momentum q can be set to zero throughout the index contraction.Then we haveXj M j2¼AþB j~q j2þOðv4Þ;(17)where A and B are independent of j~q j.By comparing Eqs.(4)and(6),we obtain the short-distance coefﬁcients shown explicitly below.The leading-order one isFð3S½1 1Þm2c ¼12111c1A¼12111c1ð4 sÞ35120m c½16ðs2þtsþt2Þm4cÀ4ð2s3þ3ts2þ3t2sþ2t3Þm2cþðs2þtsþt2Þ2 =½9ðsÀ4m2cÞ2ðtÀ4m2cÞ2ðsþtÞ2 ;(18)and the relativistic correction term isGð3S½1 1Þm4c ¼116 s21641412N c13B¼116 s21641412N c13ð4 sÞ3ðÀ2560Þ½2048ð3s2þ2tsþ3t2Þm10cÀ256ð5s3À2ts2À2t2sþ5t3Þm8cÀ320ð3s4þ10ts3þ10t2s2þ10t3sþ3t4Þm6cþ16ð21s5þ63ts4þ88t2s3þ88t3s2þ63t4sþ21t5Þm4cÀ4ð7s6þ18ts5þ23t2s4þ28t3s3þ23t4s2þ18t5sþ7t6Þm2cÀstðsþtÞðs2þtsþt2Þ2 =½27m cð4m2cÀsÞ3ð4m2cÀtÞ3ðsþtÞ3 :(19)Each of the factors has its own origin:1=16 s2isproportional to the inverse square of the Møller’s invariantﬂux factor,1=64and1=4are the color average andspin average of initial two gluons,respectively,1=2N ccomes from the color-singlet long-distance matrix elementdeﬁnition in Eq.(3)with N c¼3,1=3is the spin average for total spin J¼1states,andð4 sÞ3quantiﬁes the coupling in the QCD interaction vertices.Further-more the variable u has been expressed in terms of sand t through Eq.(11).To verify our results,weﬁndthat those in Ref.[31]discussed for J=c photoproduction are consistent with ours under replacementð4 Þe2c!ð4 sÞ,and the result in Ref.[37]agrees with ours at leading order after performing the polarization summation.B.Nonperturbative long-distance matrix elements The long-distance matrix elements may be determined by potential model[25,36]or lattice calculations[38],and by phenomenological extraction from experimental data. Here weﬁrst extract the decay matrix elements from experimental data.Up to NLO QCD and v2relativistic corrections,decay widths of the color-singlet J=c to light hadrons(LH)and eþeÀcan be expressed analytically as follows[33]:À½J=c!LH ¼F LHð3S½1 1Þm2ch H j O J=cð3S½1 1Þj H iþG LHð3S½1 1Þm4ch H j P J=cð3S½1 1Þj H i;À½J=c!eþeÀ ¼F eþeÀð3S½1 1Þm2ch H j O J=cð3S½1 1Þj H iþG eþeÀð3S½1 1Þm4ch H j P J=cð3S½1 1Þj H i;(20)where the short-distance coefﬁcients are[33]F LHð3S½1 1Þ¼ðN2cÀ1ÞðN2cÀ4ÞN3cð 2À9Þ3sð2m cÞÂ1þðÀ9:46C Fþ4:13C AÀ1:161N fÞ sþ2 e2QX Nfi¼1Q2i2e1À134C Fs;G LHð3S½1 1Þ¼À5ð19 2À132Þ7293sð2m cÞ;F eþeÀð3S½1 1Þ¼2 e2Q 2e31À4C F sð2m cÞ;G eþeÀð3S½1 1Þ¼À8 e2Q 2e9:(21)YING FAN,YAN-QING MA,AND KUANG-TA CHAO PHYSICAL REVIEW D79,114009(2009)Then,the production matrix elements can be related to the decay matrix elements through vacuum saturation approxi-mationh 0j O J=c ð3S ½1 1Þj 0i ¼ð2J þ1Þh H j OJ=c ð3S ½11Þj H i Â½1þO ðv 4Þ ¼3h H j O J=c ð3S ½11Þj H i½1þO ðv 4Þ :(22)Combining the above equations and the experimental data in [28],i.e.,À½J=c !LH ¼81:7KeV and À½J=c !e þe À ¼5:55KeV and excluding the NLO QCD radiative corrections in (21),we get the solutions accurate at leading order in sh 0j O J=c ð3S ½1 1Þj 0i ¼0:868GeV 3;h 0j P J=c ð3S ½1 1Þj 0i ¼0:190GeV 5;(23)and the enhanced matrix elements accurate up to NLO ins can be obtained by including NLO QCD radiative corrections in (21)h 0j O J=c ð3S ½1 1Þj 0i ¼1:64GeV 3;h 0j P J=c ð3S ½11Þj 0i¼0:320GeV 5:(24)The strong coupling constant evaluated at the charm quarkmass scale is s ð2m c Þ¼0:250for m c ¼1:5GeV .The other input parameters are chosen as follows:the QCD scale parameter ÃQCD ¼392MeV ,the number of quarks with mass less than the energy scale m c is N f ¼3,color factor C F ¼4=3and C A ¼3,the electric charge of the charm quark is e Q ¼2=3,Q i are the electric charges of the light quarks and ﬁne structure constant e ¼1=137.Our numerical values for the production matrix elementsh 0j O J=c ð3S ½1 1Þj 0i and h 0j PJ=c ð3S ½11Þj 0i are accurate up to NLO in v 2with uncertainties due to experimental errors and higher order corrections.C.Cross sections for p þ pðp Þ!J=c þX and phase space integration Based on the results obtained for the subprocess g þg !J=c þg we further calculate the production cross sections and relativistic corrections in the process p þ"pðp Þ!J=c þX ,which involves hadrons as the initial states.In order to get the cross sections at the hadron level,the partonic cross section deﬁned in Eq.(6)has to be convoluted with the parton distribution function (PDF)f g=p ðx ÆÞ,where x Ædenotes the fraction of the proton or antiproton beam energy carried by the gluons.We will work in the p "p center-of-mass (CM)frame and denote the p "penergy by ﬃﬃﬃS p ,the rapidity of J=c by y C ,and that of the gluon jet by y D .The differential crosssection of p þ"pðp Þ!J=c þX can be written as [39]d 3 ðp þ"pðp Þ!J=c þX Þdp 2t dy C dy D ¼x þf g=p ðx þÞx Àf g="p ðp Þðx ÀÞd ðg þg !J=c þg Þdt;(25)wherex Æ¼m C t exp ðÆy C Þþm Dt exp ðÆy D ÞﬃﬃﬃSp ;(26)with the transverse mass m C;D t ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃm 2C;D þp 2t q ,the mesonmass m C ¼2m c ,the gluon mass m D ¼0,and the trans-verse momentum p t .The Mandelstam variables can be expressed in terms of p t ,y C ,and y Ds ¼x þx ÀS;t ¼Àp 2t Àm C t m Dt exp ðy D Ày C Þ;u ¼Àp 2t Àm C t m D t exp ðy C Ày D Þ:(27)The accessible phase space puts kinetic constraints onvariables p t ,y C ,and y D for a ﬁxed value of two colliding hadron center-of-mass energy ﬃﬃﬃSp 0 p t12ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðS;m 2C ;m 2D ÞSs ;j y C j Arcosh S þm 2C Àm 2D2ﬃﬃﬃS p m C t;Àln ﬃﬃﬃS p Àm C t exp ðÀy CÞm D t y Dln ﬃﬃﬃS p Àm C t exp ðy C Þm D t ;(28)where ðx;y;z Þ¼x 2þy 2þz 2À2ðxy þyz þzx Þ.Thedistribution over p t of the differential cross section can be obtained after phase space integration.III.NUMERICAL RESULTS AND ANALYSIS The CTEQ6PDFs [40]are used in our numerical cal-culation.We present the distribution of J=c production differential cross section d =dp t over p t at the Tevatron with ﬃﬃﬃS p ¼1:96TeV and at the LHC with ﬃﬃﬃS p ¼14TeV in Figs.2–5.The solid line represents the distribution at leading order in O ðv 2Þ,and the dotted line describes the relativistic correction at next-to-leading order in O ðv 2Þ(excluding the leading-order result).The long-distance matrix elements are accurate up to leading order in s from Eq.(23)or next-to-leading order in s from Eq.(24).The variable p t is set to be from 5GeV to 30GeV (50GeV)for the Tevatron (LHC),and the distri-butions are depicted in logarithm unit along the vertical axis.All curves decrease rather rapidly as the transverse momentum p t increases,and the leading-order d =dp t behavior is not changed by the relativistic corrections.It can be seen that the ratio of relativistic correction to leading-order term is 1%or so,and less than 2%,which is insigniﬁcant and negligible.RELATIVISTIC CORRECTION TO J=c PRODUCTION ...PHYSICAL REVIEW D 79,114009(2009)The tiny effect of relativistic corrections is partly due to the smallness of the short-distance coefﬁcient correction.In fact,the ratio of the NLO short-distance coefﬁcient to the LO one from Eqs.(18)and (19)can be expanded as a series of the small quantity m c ,as compared with ﬃﬃﬃs p ,and this series reduces to a ﬁxed small number 16if only the leading-order term is kept,i.e.,G ð3S ½11ÞF ð3S 1Þ!16;as 2mc ﬃﬃﬃs p !0;2m cﬃﬃtp !0:(29)Together with the suppression from long-distance matrix elements,the tiny effect of relativistic corrections can be accounted for.Our results for relativistic corrections in theprocess p þ"pðp Þ!J=c þX are similar to that in the J=c photoproduction process discussed in Ref.[31].These results may indicate that the nonrelativistic approxi-mation in NRQCD is good for charmonium production at high energy collisions,and relativistic corrections are not important.This is in contrast to the case of double char-monium production in e þe Àannihilation at B factories,where relativistic corrections may be signiﬁcant.IV .SUMMARYIn this paper,relativistic corrections to the color-singlet J=c hadroproduction at the Tevatron and LHC are calcu-lated up to O ðv 2Þin the framework of the NRQCD facto-rization approach.The perturbative short-distance coefﬁcients are obtained by matching the full QCDdiffer-FIG.3.The p t distribution of d ðp þ"p !J=c þX Þ=dp t(with enhanced matrix elements)at the Tevatron with ﬃﬃﬃS p ¼1:96TeV .The O ðv 0Þand O ðv 2Þresults are represented by the solid and dotted lines,respectively.FIG.4.The p t distribution of d ðp þp !J=c þX Þ=dp t at the LHC with ﬃﬃﬃS p¼14TeV .The O ðv 0Þand O ðv 2Þresults are represented by the solid and dotted lines,respectively.FIG.5.The p t distribution of d ðp þp !J=c þX Þ=dp t (with enhanced matrix elements)at the LHC with ﬃﬃﬃS p ¼14TeV .The O ðv 0Þand O ðv 2Þresults are represented by the solid and dotted lines,respectively.FIG.2.The p t distribution of d ðp þ"p !J=c þX Þ=dp t at the Tevatron with ﬃﬃﬃS p ¼1:96TeV .The O ðv 0Þand O ðv 2Þresults are represented by the solid and dotted lines,respectively.YING FAN,YAN-QING MA,AND KUANG-TA CHAO PHYSICAL REVIEW D 79,114009(2009)ential cross section with the NRQCD effectiveﬁeld theorycalculation for the subprocess gþg!J=cþg.The nonperturbative long-distance matrix elements are ex-tracted from experimental data for J=c hadronic and leptonic decay widths up to Oðv2Þwith an approximate relation between the production matrix elements and decaymatrix ing the CTEQ6parton distributionfunctions,we then calculate the LO production cross sec-tions and relativistic corrections for the process pþ"pðpÞ!J=cþX at the Tevatron and LHC.Weﬁnd that the Oðv2Þrelativistic corrections to the differential cross sections over a wide range of large transverse momentum p t are tiny and negligible,only at a level of about1%.The tiny effect of relativistic corrections is due to the smallness of the short-distance coefﬁcient correction and the sup-pression from long-distance matrix elements.These results may indicate that the nonrelativistic approximation in NRQCD is good for charmonium production at high en-ergy hadron-hadron collisions,and relativistic corrections cannot offer much help to resolve the large discrepancy between the leading-order prediction and experimental data for J=c production at the Tevatron.Other mecha-nisms such as those suggested in[41–43]may need to be considered,aside from higher order QCD contributions.ACKNOWLEDGMENTSWe would like to thank Dr.Ce Meng for reading the manuscript and helpful discussions.This work was sup-ported by the National Natural Science Foundation of China(No.10675003,No.10721063)and the Ministry of Science and Technology of China(2009CB825200).[1]G.T.Bodwin,E.Braaten,and G.P.Lepage,Phys.Rev.D51,1125(1995);55,5853(E)(1997).[2]Han-Wen Huang and Kuang-Ta Chao,Phys.Rev.D54,3065(1996);56,7472(E)(1997);55,244(1997);54,6850 (1996);56,1821(E)(1997).[3] A.Petrelli,M.Cacciari,M.Greco,F.Maltoni,and 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无缝循环金色粒子光斑-透明通道

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The Higgs Boson Might Not Couple To B Quarks

to appear in Physics Letters B.

As far as the mass generation within the framework of the standard electroweak model is concerned, one must diﬀerentiate between the mass generation for the electroweak bosons W , Z , the mass generation for the heavy t quark, and the generation of mass for the leptons and the ﬁve remaining, relatively light quarks. While there exists no freedom in the choice of the interaction strengths of the weak bosons with the scalar ﬁeld, which is dictated by the gauge invariance [1], there is such a freedom with respect to the fermions. The masses of the fermions are given by the various Yukawa coupling constants, which parametrize the interactions of the leptons and quarks with the scalar ﬁeld. The Yukawa coupling constant of the t quark ﬁeld is of the same order as the gauge coupling constant, while the other fermions couple much weakly (0.018 for the b quark, 0.005 for the c quark, etc.). The origin of the light fermion masses is still mysterious, and alternative views or slight variations of the standard electroweak theory might indeed give a diﬀerent view. Taking into account the observed ﬂavor mixing phenomenon, one could speculate, for example, that the masses of the light quarks and of the leptons are due to the mixing. In the absence of the mixing the mass matrix of the quarks in the u-sector would simply be proportional to a diagonal matrix with the entries (0, 0, 1), and there would only be a coupling of the scalar ﬁeld to the t quark. Once the ﬂavor mixing is switched on, the mass eigenstates for the light quarks are not necessarily coupled to the scalar ﬁeld, with a strength given by the mass eigenvalues. In particular these couplings could remain zero. It is well-known that the renormalizability of the theory requires a coupling of the fermion to the scalar ﬁeld [1]. Otherwise the unitarity in the s + − ¯ channel is violated at high energies for the reaction fL f L → W W . However, for all fermions except the t quark these problems appear only at extremely high energies. Modiﬁcations of the electroweak theory, which involve an energy scale not orders of magnitude above the typical electroweak scale of about 0.3 TeV, e.g. theories which do not rely on the Higgs mechanism, can take care of this problem. Recently we have discussed such a modiﬁcation [2], or rather an alternative description of the standard model, based on the complementarity between conﬁnement and Higgs phase [3]. We suppose that the electroweak interactions are described by the conﬁnement phase, and not by the Higgs phase, as usually assumed. This provides an alternative view of the electroweak bosons, which are not the basic gauge bosons of the underlying gauge theory, but “bound states” of an underlying scalar ﬁeld, which in the Higgs phase plays the role of the Higgs doublet. Both the charged W bosons and the neutral Z boson are J = 1 bound systems of the type hh, (hh)† or ¯ respectively. There is a corresponding J = 0, hh ¯ system, which is to be hh 1

H--ggg(gqqbar) at Two Loops in the Large-M_t Limit

a r X i v :h e p -p h /9707448v 1 24 J u l 1997MSUHEP-70723July 1997hep-ph/9707448H →ggg (gq ¯q )at Two Loops in the Large-M t LimitCarl R.SchmidtDepartment of Physics and AstronomyMichigan State University East Lansing,MI 48824,USAAbstractWe present a calculation of the two-loop helicity amplitudes for the processes H →ggg and H →gq ¯q in the large-M t limit.In this limit the calculation can be performed in terms of one-loop diagrams containing an eﬀective Hgg operator.These amplitudes are required for the next-to-leading order (NLO)corrections to the Higgs transverse momentum distribution and the next-to-next-to-leading order (NNLO)corrections to the Higgs production cross section via the gluon fusion mechanism.1IntroductionThe Higgs Boson H is the only particle of the Standard Model remaining to be discovered.Its role is to provide a simple mechanism to break the electroweak gauge symmetry and to give mass to the weak gauge bosons and the fermions. Of course,it is possible that nature uses more than a single scalar boson for this purpose,but still the Standard Model and its supersymmetric extension are the primary examples of the class of symmetry-breaking models which interact weakly. Therefore,the search for the Higgs boson is of the highest priority for the Large Hadron Collider(LHC)at CERN.The detection of the Higgs Boson above background at the LHC will be a chal-lenging task.In particular,if the mass of the Higgs is below∼140GeV,near the threshold for decay into W boson pairs,the detection of the Higgs is quite sub-tle.Although the largest production mechanism by far is gluon-gluon fusion,the equally large backgrounds require the use of the H→γγdecay channel,which has a branching ratio of O(10−3).To prepare for the search,we need the best theoret-ical predictions possible,and this means the inclusion of quantum chromodynamic (QCD)corrections to Higgs production and decay.Recent relevant reviews are given in reference[1].The Higgs production via gluon-gluon fusion proceeds at lowest order(LO) through a quark loop.This loop is dominated by the top quark,because the Higgs coupling is proportional to the quark mass.The two-loop next-to-leading order (NLO)QCD corrections have also been calculated[2],and they are quite large:1∼50−100%.An interesting feature of this NLO calculation is that it becomes much simpler in the limit of large top quark mass(M t→∞).In this limit,one can integrate out the heavy top quark loop,leaving behind an eﬀective gauge-invariant Hgg coupling.Thus,the number of loops at each order is reduced byone.It has been shown that the NLO corrections in this large-M t limit give agood approximation to the complete two-loop result over a large range of Higgsmasses[3].The large NLO correction suggests that even higher orders still maybe important.A soft-gluon resummation in the large-M t limit has recently beenperformed,which gives an estimate of the next-to-next-to-leading order(NNLO)corrections[4].Meanwhile,other groups have considered less inclusive quantities,such as thetransverse momentum spectrum of the Higgs boson.This observable has beenconsidered at the one-loop Born level,both in the large-top-mass limit and withfull M t dependence[5].In addition,the eﬀects of soft gluons have also been studied,which modify the spectrum at small Higgs p⊥[6].However,a NLO calculation hasnot been done.The realﬁve-point H→gggg amplitudes which are needed havebeen calculated by Dawson and Kauﬀman[7]in the large-M t limit,and recentlyKauﬀman et al.[8]have calculated theﬁve-point amplitudes with light externalquarks.In this paper we present the virtual corrections to the four-point H→ggg(gq¯q)amplitudes in the large-M t limit,which completes the set of amplitudes neededto study the Higgs p⊥spectrum at NLO.In this limit,the two-loop results canbe computed from eﬀective one-loop diagrams.The large-M t approximation to2the Higgs p⊥spectrum will be good for some range of M H and Higgs p⊥,and furthermore this calculation oﬀers a check of the complete M t-dependent result, when it should become available.Moreover,these amplitudes are necessary for a full NNLO calculation of the cross section in the large-M t limit[9].In addition to using the eﬀective Higgs-gluon operator in the large-M t limit,we have also used several other techniques that have been found convenient in QCD loop calculations[10].These include the use of helicity spinors,color ordering,and background-ﬁeld gauge.In section2we discuss the details of the calculation,while in section3we present the amplitudes and discuss various cross-checks.In section 4we summarize our conclusions.2Calculational DetailsIn the large-M t limit the top quark can be removed from the full theory,leaving a residual Higgs-gluon coupling term in the lagrangian of the eﬀective theory:L eﬀ=−13πH.(2)4πFollowing Mangano and Parke[11],we use the unconventional normalization for√the SU(3)generator matrices Tr(T a T b)=δab and[T a,T b]=i2from the helicity amplitudes below.3As suggested by string theory methods [10],we use the background-ﬁeld gauge to calculate the one-particle irreducible parts of the Feynman diagrams.The gluon ﬁeld G µis split into a background component B µand a quantum component Q µ,i.e,G µ=B µ+Q µ.Two reasonable choices for the gauge-ﬁxing term areL (1)gf=−121−αsv(1+∆)(D B µQ µ)2(3)whereD B µQµ=∂µQ µ−(ig/√3πvg sbe written[12].(5)ǫ±(p)µ=± p±|γµ|q±2 q∓|p±The arbitrary reference vector q satisﬁes q2=0and q·p=0.A change in the reference vector just shiftsǫ(p)µby a term proportional to pµ,which drops out of the gauge-invariant helicity amplitude.There are two independent Hggg helicity subamplitudes,which at tree level arem0(1+,2+,3+)=−M4H.(6)[12][23][31]Here we have used the notation ij = p i−|p j+ and[ij]= p i+|p j− .These spinor products are antisymmetric and satisfy ij [ji]=2p i·p j≡S ij.All other subamplitudes can be obtained by invariance under cyclic permutations,charge conjugation,and parity.We also consider the process0→Hgq¯q with p H+p+q+¯q=0.The amplitude for a gluon and quarks with helicitiesλ,h,¯h and colors a,i,¯ıcan be writtenM=−αs2T a i¯ım(p,λ;q,h;¯q,¯h).(7)At tree level we have[p¯q]2m0(g+,q−,¯q+)=3The Helicity AmplitudesFigures 1and 2show representative box diagrams for the Hggg and Hgq ¯q ampli-tudes,respectively.The Feynman diagrams have been evaluated with the aid of the symbolic manipulation program MAPLE using the straightforward Passarino-Veltman reduction.For the sake of generality,we have regularized the loop in-tegrals by continuing the loop momenta to (4−2ǫ)dimensions,while taking the number of helicity states of the internal gluons to be (4−2δR ǫ).Thus,δR =1corresponds to the t’Hooft-Veltman scheme [13]and δR =0corresponds to the four-dimensional-helicity scheme [14].For the Hggg amplitudes we obtain m 1(1+,2+,3+)=m 0(1+,2+,3+)αs −M 2HǫN c U+1M 4Hm 1(1−,2+,3+)=m 0(1−,2+,3+)αs −M 2HǫN c U(9)+1S 223,where the prefactor isr Γ=Γ(1+ǫ)Γ2(1−ǫ)ǫ2−−M 2H−S 23ǫ−−M 2H2−ln−S 12−M 2H−ln−S 12−M 2H−ln−S 23−M 2H6−2Li2 1−S12M2H −2Li2 1−S314πrΓ 4πµ2N c V2+n f V3 ,(12) withV1=1−S gqǫ− −M2H6ǫ −M2H−M2Hln −S q¯q−M2Hln −S q¯qM2H −Li21−S gq M2H (13) +836+π22S q¯qǫ2+3−S q¯qǫ+ln −S gq−M2H+Li2 1−S gq M2H (14) +72−π22S q¯q3ǫ −M2H9.(15)In these expressions,N c=3is the number of colors,n f is the number of light fermions,and Li2is the dilogarithm function.The amplitudes are written for S ij<0and M2H<0,but they can be analytically continued to the physical region by letting−S ij=−S ij−iηand−M2H=−M2H−iηforη→0+.Note that these amplitudes are ultraviolet-unrenormalized amplitudes.Including the7renormalization gives the modiﬁcationm1→m1+(∆+3δg)m0,(16) where∆is theﬁnite renormalization of the eﬀective Hgg operator,given in eq.(2) andδg is the gauge-coupling ing theǫαs6−n fcalculation of the Higgs p⊥spectrum in the large-M t limit,which is currently in progress[18].In addition,they are part of the set needed for a complete NNLO calculation of the total cross-section in this limit.AcknowledgementsWe would like to thank Vittorio Del Duca,Lance Dixon,Chris Glosser,and Jim Amundson for useful discussions.References[1]M.Spira,preprint CERN-TH/97-68,hep-ph/9705337(1997);Z.Kunszt,S.Moretti,and W.J.Stirling,Z.Phys.C74,479(1997);S.Dawson,preprint BNL-HET-SD-97-004,hep-ph/9703387(1997).[2]D.Graudenz,M.Spira,and P.Zerwas,Phys.Rev.Lett.70,1372(1993);M.Spira,A.Djouadi,D.Graudenz,and P.Zerwas,Nucl.Phys.B453,17 (1995).[3]S.Dawson,Nucl.Phys.B359,283(1991);A.Djouadi,M.Spira,and P.Zerwas,Phys.Lett.B264,440(1991).[4]M.Kr¨a mer, enen,and M.Spira,preprint CERN-TH/96-231,hep-ph/9611272(1997).9[5]R.K.Ellis,I.Hinchliﬀe,M.Soldate,and J.J.Van Der Bij,Nucl.Phys.B297,221(1988);U.Baur and E.Glover,Nucl.Phys.B339,38(1990).[6]I.Hinchliﬀe and S.Novaes,Phys.Rev.D38,3475(1988);R.Kauﬀman,Phys.Rev.D44,1415(1991);Phys.Rev.D45,1512(1992);C.-P.Yuan,Phys.Lett.B283,395(1992).[7]S.Dawson and R.Kauﬀman,Phys.Rev.Lett.68,2273(1992).[8]R.Kauﬀman,S.Desai,and D.Risal,Phys.Rev.D55,4005(1997).[9]A recent NNLO calculation in the large-M t limit of the crossed process,H→gg,has been done by K.Chetyrkin,B.Kniehl,and M.Steinhauser, Phys.Rev.Lett.79,353(1997).[10]For a review,see Z.Bern,L.Dixon,and D.Kosower,Ann.Rev.Nucl.Part.Sci.46,109(1996).[11]M.Mangano and S.Parke,Phys.Rep.200,301(1991).[12]Z.Xu,D.Zhang,and L.Chang,Nucl.Phys.B291,392(1987);R.Kleiss and W.Stirling,Nucl.Phys.B262,235(1985).[13]G.t’Hooft and M.Veltman,Nucl.Phys.B63,277(1973).[14]Z.Bern and D.Kosower,Phys.Rev.Lett.66,1669(1991);Z.Bern and D.Kosower,Nucl.Phys.B379,451(1992).10[15]S.Catani and M.Seymour,Nucl.Phys.B485,291(1997).[16]Z.Kunszt,A.Signer,and Z.Tr´o cs´a nyi,Nucl.Phys.B411,397(1994);S.Catani,M.Seymour,and Z.Tr´o cs´a nyi,Phys.Rev.D55,6819(1997).[17]Z.Bern,L.Dixon,D.Dunbar,and D.Kosower,Nucl.Phys.B425,217(1994).[18]C.Glosser and C.Schmidt,(to be published).11Fig.1:Box diagram for the Hggg amplitude.The dot represents the eﬀective Hgg vertex in the large-M t limit.Fig.2:Box diagrams for the Hgq¯q amplitude.The dot represents the eﬀective Hgg vertex in the large-M t limit.12。

Mesons with a Light Quark-Antiquark Pair and the Bethe-Salpeter Equation

−1 i i i i i i
(2.6) (2.7)
(2.8) (2.9)
˜ Gg = f ˜ G′ g f
˜ ∈S ˜, g ∈S ∀f
By the Bethe-Salpeter relation derived above,
˜ G−1φ = φ ˜ ( K −1 − I )φ = φ ˜φ φ ˜φ φ ˜φ λ i −1 = φ i i i i i i i i i i G′ ≡ ˜ Iφ ) φ φ ˜ ˜K φ − φ ∑ (φ
As any matrix and its transpose share the same characteristic equation, they share identical eigenvalues. Let M be the subspace spanned by the nondegenerate eigenfunctions of G. In this subspace, the indexing of both sets of eigenfunctions can thus be aligned by matching eigenvalues.
HUTP-94/A030
Mesons with a light quark-antiquark pair and the Bethe-Salpeter equation
Dean Lee1
Physics Department, Harvard University, Cambridge MA 02138
3
˜G = λφ ˜ index i: Gφi = λ iφi , φ i i i ˜φ = φ ˜ Gφ = λ φ ˜φ ⇒ φ ˜ φ = cδ . Since λ iφ i k i k k i k i k i ik ˜ ) be the subspace of M spanned by φ ( φ ˜ ) for c nonzero. Using orthogonality relations, the Let S( S i i i eigenfunctions can be used to write a bilinear form, G′ , equivalent to G on these restricted subspaces. G′ ≡ ˜φ ) φφ ˜ ∑ λ (φ

N=4 Super-Yang-Mills Theory, QCD and Collider Physics

a rXiv:h ep-th/04121v12O ct24SLAC–PUB–10739,IPPP/04/59,DCPT/04/118,UCLA/04/TEP/40Saclay/SPhT–T04/116,hep-th/0410021October,2004N =4Super-Yang-Mills Theory,QCD and Collider Physics Z.Bern a L.J.Dixon b 1 D.A.Kosower c a Department of Physics &Astronomy,UCLA,Los Angeles,CA 90095-1547,USA b SLAC,Stanford University,Stanford,CA 94309,USA,and IPPP,University of Durham,Durham DH13LE,England c Service de Physique Th´e orique,CEA–Saclay,F-91191Gif-sur-Yvette cedex,France1Introduction and Collider Physics MotivationMaximally supersymmetric (N =4)Yang-Mills theory (MSYM)is unique in many ways.Its properties are uniquely speciﬁed by the gauge group,say SU(N c ),and the value of the gauge coupling g .It is conformally invariant for any value of g .Although gravity is not present in its usual formulation,MSYMis connected to gravity and string theory through the AdS/CFT correspon-dence[1].Because this correspondence is a weak-strong coupling duality,it is diﬃcult to verify quantitatively for general observables.On the other hand, such checks are possible and have been remarkably successful for quantities protected by supersymmetry such as BPS operators[2],or when an additional expansion parameter is available,such as the number ofﬁelds in sequences of composite,large R-charge operators[3,4,5,6,7,8].It is interesting to study even more observables in perturbative MSYM,in order to see how the simplicity of the strong coupling limit is reﬂected in the structure of the weak coupling expansion.The strong coupling limit should be even simpler when the large-N c limit is taken simultaneously,as it corresponds to a weakly-coupled supergravity theory in a background with a large radius of curvature.There are diﬀerent ways to study perturbative MSYM.One approach is via computation of the anomalous dimensions of composite,gauge invariant operators[1,3,4,5,6,7,8].Another possibility[9],discussed here,is to study the scattering amplitudes for(regulated)plane-wave elementaryﬁeld excitations such as gluons and gluinos.One of the virtues of the latter approach is that perturbative MSYM scat-tering amplitudes share many qualitative properties with QCD amplitudes in the regime probed at high-energy colliders.Yet the results and the computa-tions(when organized in the right way)are typically signiﬁcantly simpler.In this way,MSYM serves as a testing ground for many aspects of perturbative QCD.MSYM loop amplitudes can be considered as components of QCD loop amplitudes.Depending on one’s point of view,they can be considered either “the simplest pieces”(in terms of the rank of the loop momentum tensors in the numerator of the amplitude)[10,11],or“the most complicated pieces”in terms of the degree of transcendentality(see section6)of the special functions entering theﬁnal results[12].As discussed in section6,the latter interpreta-tion links recent three-loop anomalous dimension results in QCD[13]to those in the spin-chain approach to MSYM[5].The most direct experimental probes of short-distance physics are collider experiments at the energy frontier.For the next decade,that frontier is at hadron colliders—Run II of the Fermilab Tevatron now,followed by startup of the CERN Large Hadron Collider in2007.New physics at colliders always contends with Standard Model backgrounds.At hadron colliders,all physics processes—signals and backgrounds—are inherently QCD processes.Hence it is important to be able to predict them theoretically as precisely as possi-ble.The cross section for a“hard,”or short-distance-dominated processes,can be factorized[14]into a partonic cross section,which can be computed order by order in perturbative QCD,convoluted with nonperturbative but measur-able parton distribution functions(pdfs).For example,the cross section for producing a pair of jets(plus anything else)in a p¯p collision is given byσp¯p→jjX(s)= a,b1 0dx1dx2f a(x1;µF)¯f b(x2;µF)×ˆσab→jjX(sx1x2;µF,µR;αs(µR)),(1)where s is the squared center-of-mass energy,x1,2are the longitudinal(light-cone)fractions of the p,¯p momentum carried by partons a,b,which may be quarks,anti-quarks or gluons.The experimental deﬁnition of a jet is an in-volved one which need not concern us here.The pdf f a(x,µF)gives the prob-ability forﬁnding parton a with momentum fraction x inside the proton; similarly¯f b is the probability forﬁnding parton b in the antiproton.The pdfs depend logarithmically on the factorization scaleµF,or transverse resolution with which the proton is examined.The Mellin moments of f a(x,µF)are for-ward matrix elements of leading-twist operators in the proton,renormalized at the scaleµF.The quark distribution function q(x,µ),for example,obeys 10dx x j q(x,µ)= p|[¯qγ+∂j+q](µ)|p .2Ingredients for a NNLO CalculationMany hadron collider measurements can beneﬁt from predictions that are accurate to next-to-next-to-leading order(NNLO)in QCD.Three separate ingredients enter such an NNLO computation;only the third depends on the process:(1)The experimental value of the QCD couplingαs(µR)must be determinedat one value of the renormalization scaleµR(for example m Z),and its evolution inµR computed using the3-loopβ-function,which has been known since1980[15].(2)The experimental values for the pdfs f a(x,µF)must be determined,ide-ally using predictions at the NNLO level,as are available for deep-inelastic scattering[16]and more recently Drell-Yan production[17].The evolu-tion of pdfs inµF to NNLO accuracy has very recently been completed, after a multi-year eﬀort by Moch,Vermaseren and Vogt[13](previously, approximations to the NNLO kernel were available[18]).(3)The NNLO terms in the expansion of the partonic cross sections must becomputed for the hadronic process in question.For example,the parton cross sections for jet production has the expansion,ˆσab→jjX=α2s(A+αs B+α2s C+...).(2)The quantities A and B have been known for over a decade[19],but C has not yet been computed.Figure 1.LHC Z production [22].•real ×real:×•virtual ×real:×•virtual ×virtual:×•doubly-virtual ×real:×Figure 2.Purely gluonic contributionsto ˆσgg →jjX at NNLO.Indeed,the NNLO terms are unknown for all but a handful of collider puting a wide range of processes at NNLO is the goal of a large amount of recent eﬀort in perturbative QCD [20].As an example of the im-proved precision that could result from this program,consider the production of a virtual photon,W or Z boson via the Drell-Yan process at the Tevatron or LHC.The total cross section for this process was ﬁrst computed at NNLO in 1991[21].Last year,the rapidity distribution of the vector boson also be-came available at this order [17,22],as shown in ﬁg.1.The rapidity is deﬁned in terms of the energy E and longitudinal momentum p z of the vector boson in the center-of-mass frame,Y ≡1E −p z .It determines where the vector boson decays within the detector,or outside its acceptance.The rapidity is sensitive to the x values of the incoming partons.At leading order in QCD,x 1=e Y m V /√s ,where m V is the vector boson mass.The LHC will produce roughly 100million W s and 10million Z s per year in detectable (leptonic)decay modes.LHC experiments will be able to map out the curve in ﬁg.1with exquisite precision,and use it to constrain the parton distributions —in the same detectors that are being used to search for new physics in other channels,often with similar q ¯q initial states.By taking ratios of the other processes to the “calibration”processes of single W and Z production,many experimental uncertainties,including those associated with the initial state parton distributions,drop out.Thus ﬁg.1plays a role as a “partonic luminosity monitor”[23].To get the full beneﬁt of the remarkable experimental precision,though,the theory uncertainty must approach the 1%level.As seen from the uncertainty bands in the ﬁgure,this precision is only achievable at NNLO.The bands are estimated by varying the arbitrary renormalization and factorization scales µR and µF (set to a common value µ)from m V /2to 2m V .A computation to all orders in αs would have no dependence on µ.Hence the µ-dependence of a ﬁxed order computation is related to the size of the missing higher-order terms in the series.Althoughsub-1%uncertainties may be special to W and Z production at the LHC, similar qualitative improvements in precision will be achieved for many other processes,such as di-jet production,as the NNLO terms are completed.Even within the NNLO terms in the partonic cross section,there are several types of ingredients.This feature is illustrated inﬁg.2for the purely gluonic contributions to di-jet production,ˆσgg→jjX.In theﬁgure,individual Feynman graphs stand for full amplitudes interfered(×)with other amplitudes,in order to produce contributions to a cross section.There may be2,3,or4partons in theﬁnal state.Just as in QED it is impossible to deﬁne an outgoing electron with no accompanying cloud of soft photons,also in QCD sensible observables require sums overﬁnal states with diﬀerent numbers of partons.Jets,for example,are deﬁned by a certain amount of energy into a certain conical region.At leading order,that energy typically comes from a single parton, but at NLO there may be two partons,and at NNLO three partons,within the jet cone.Each line inﬁg.2results in a cross-section contribution containing severe infrared divergences,which are traditionally regulated by dimensional regula-tion with D=4−2ǫ.Note that this regulation breaks the classical conformal invariance of QCD,and the classical and quantum conformal invariance of N=4super-Yang-Mills theory.Each contribution contains poles inǫranging from1/ǫ4to1/ǫ.The poles in the real contributions come from regions ofphase-space where the emitted gluons are soft and/or collinear.The poles in the virtual contributions come from similar regions of virtual loop integra-tion.The virtual×real contribution obviously has a mixture of the two.The Kinoshita-Lee-Nauenberg theorem[24]guarantees that the poles all cancel in the sum,for properly-deﬁned,short-distance observables,after renormal-izing the coupling constant and removing initial-state collinear singularities associated with renormalization of the pdfs.A critical ingredient in any NNLO prediction is the set of two-loop ampli-tudes,which enter the doubly-virtual×real interference inﬁg.2.Such ampli-tudes require dimensionally-regulated all-massless two-loop integrals depend-ing on at least one dimensionless ratio,which were only computed beginning in 1999[25,26,27].They also receive contributions from many Feynman diagrams, with lots of gauge-dependent cancellations between them.It is of interest to develop more eﬃcient,manifestly gauge-invariant methods for combining di-agrams,such as the unitarity or cut-based method successfully applied at one loop[10]and in the initial two-loop computations[28].i,ij+ i iFigure3.Illustration of soft-collinear(left)and pure-collinear(right)one-loop di-vergences.3N=4Super-Yang-Mills Theory as a Testing Ground for QCDN=4super-Yang-Mills theory serves an excellent testing ground for pertur-bative QCD methods.For n-gluon scattering at tree level,the two theories in fact give identical predictions.(The extra fermions and scalars of MSYM can only be produced in pairs;hence they only appear in an n-gluon ampli-tude at loop level.)Therefore any consequence of N=4supersymmetry,such as Ward identities among scattering amplitudes[29],automatically applies to tree-level gluonic scattering in QCD[30].Similarly,at tree level Witten’s topological string[31]produces MSYM,but implies twistor-space localization properties for QCD tree amplitudes.(Amplitudes with quarks can be related to supersymmetric amplitudes with gluinos using simple color manipulations.)3.1Pole Structure at One and Two LoopsAt the loop-level,MSYM becomes progressively more removed from QCD. However,it can still illuminate general properties of scattering amplitudes,in a calculationally simpler arena.Consider the infrared singularities of one-loop massless gauge theory amplitudes.In dimensional regularization,the leading singularity is1/ǫ2,arising from virtual gluons which are both soft and collinear with respect to a second gluon or another massless particle.It can be char-acterized by attaching a gluon to any pair of external legs of the tree-level amplitude,as in the left graph inﬁg.3.Up to color factors,this leading diver-gence is the same for MSYM and QCD.There are also purely collinear terms associated with individual external lines,as shown in the right graph inﬁg.3. The pure-collinear terms have a simpler form than the soft terms,because there is less tangling of color indices,but they do diﬀer from theory to theory.The full result for one-loop divergences can be expressed as an operator I(1)(ǫ) which acts on the color indices of the tree amplitude[32].Treating the L-loop amplitude as a vector in color space,|A(L)n ,the one-loop result is|A(1)n =I(1)(ǫ)|A(0)n +|A(1),ﬁnn ,(3)where |A (1),ﬁn nis ﬁnite as ǫ→0,and I (1)(ǫ)=1Γ(1−ǫ)n i =1n j =i T i ·T j 1T 2i 1−s ij ǫ,(4)where γis Euler’s constant and s ij =(k i +k j )2is a Mandelstam invariant.The color operator T i ·T j =T a i T a j and factor of (µ2R /(−s ij ))ǫarise from softgluons exchanged between legs i and j ,as in the left graph in ﬁg.3.The pure 1/ǫpoles terms proportional to γi have been written in a symmetric fashion,which slightly obscures the fact that the color structure is actually simpler.We can use the equation which represents color conservation in the color-space notation, n j =1T j =0,to simplify the result.At order 1/ǫwe may neglect the (µ2R /(−s ij ))ǫfactor in the γi terms,and we have n j =i T i ·T j γi /T 2i =−γi .So the color structure of the pure 1/ǫterm is actually trivial.For an n -gluon amplitude,the factor γi is set equal to its value for gluons,which turns out to be γg =b 0,the one-loop coeﬃcient in the β-function.Hence the pure-collinear contribution vanishes for MSYM,but not for QCD.The divergences of two-loop amplitudes can be described in the same for-malism [32].The relation to soft-collinear factorization has been made more transparent by Sterman and Tejeda-Yeomans,who also predicted the three-loop behavior [33].Decompose the two-loop amplitude |A (2)n as|A (2)n =I (2)(ǫ)|A (0)n +I (1)(ǫ)|A (1)n +|A (2),ﬁn n,(5)where |A (2),ﬁn n is ﬁnite as ǫ→0and I (2)(ǫ)=−1ǫ+e −ǫγΓ(1−2ǫ)ǫ+K I (1)(2ǫ)+e ǫγT 2i µ22C 2A ,(8)where C A =N c is the adjoint Casimir value.The quantity ˆH(2)has non-trivial,but purely subleading-in-N c ,color structure.It is associated with soft,rather than collinear,momenta [37,33],so it is theory-independent,up to color factors.An ansatz for it for general n has been presented recently [38].3.2Recycling Cuts in MSYMAn eﬃcient way to compute loop amplitudes,particularly in theories with a great deal of supersymmetry,is to use unitarity and reconstruct the am-plitude from its cuts [10,38].For the four-gluon amplitude in MSYM,the two-loop structure,and much of the higher-loop structure,follows from a sim-ple property of the one-loop two-particle cut in this theory.For simplicity,we strip the color indices oﬀof the four-point amplitude A (0)4,by decomposing it into color-ordered amplitudes A (0)4,whose coeﬃcients are traces of SU(N c )generator matrices (Chan-Paton factors),A (0)4(k 1,a 1;k 2,a 2;k 3,a 3;k 4,a 4)=g 2 ρ∈S 4/Z 4Tr(T a ρ(1)T a ρ(2)T a ρ(3)T a ρ(4))×A (0)4(k ρ(1),k ρ(2),k ρ(3),k ρ(4)).(9)The two-particle cut can be written as a product of two four-point color-ordered amplitudes,summed over the pair of intermediate N =4states S,S ′crossing the cut,which evaluates toS,S ′∈N =4A (0)4(k 1,k 2,ℓS ,−ℓ′S ′)×A (0)4(ℓ′S ′,−ℓS ,k 3,k 4)=is 12s 23A (0)4(k 1,k 2,k 3,k 4)×1(ℓ−k 3)2,(10)where ℓ′=ℓ−k 1−k 2.This equation is also shown in ﬁg.4.The scalar propagator factors in eq.(10)are depicted as solid vertical lines in the ﬁgure.The dashed line indicates the cut.Thus the cut reduces to the cut of a scalar box integral,deﬁned byI D =4−2ǫ4≡ d 4−2ǫℓℓ2(ℓ−k 1)2(ℓ−k 1−k 2)2(ℓ+k 4)2.(11)One of the virtues of eq.(10)is that it is valid for arbitrary external states in the N =4multiplet,although only external gluons are shown in ﬁg.4.Therefore it can be re-used at higher loop order,for example by attaching yet another tree to the left.N =41234=i s 12s 231234Figure 4.The one-loop two-particle cuts for the four-point amplitude in MSYM reduce to the tree amplitude multiplied by a cut scalar box integral (for any set of four external states).i 2s 12s121234+s 121234+perms Figure 5.The two-loop gg →gg amplitude in MSYM [11,39].The blob on theright represents the color-ordered tree amplitude A (0)4.(The quantity s 12s 23A (0)4transforms symmetrically under gluon interchange.)In the the brackets,black linesare kinematic 1/p 2propagators,with scalar (φ3)vertices.Green lines are color δab propagators,with structure constant (f abc )vertices.The permutation sum is over the three cyclic permutations of legs 2,3,4,and makes the amplitude Bose symmetric.At two loops,the simplicity of eq.(10)made it possible to compute the two-loop gg →gg scattering amplitude in that theory (in terms of speciﬁc loop integrals)in 1997[11],four years before the analogous computations in QCD [36,37].All of the loop momenta in the numerators of the Feynman di-agrams can be factored out,and only two independent loop integrals appear,the planar and nonplanar scalar double box integrals.The result can be writ-ten in an appealing diagrammatic form,ﬁg.5,where the color algebra has the same form as the kinematics of the loop integrals [39].At higher loops,eq.(10)leads to a “rung rule”[11]for generating a class of (L +1)-loop contributions from L -loop contributions.The rule states that one can insert into a L -loop contribution a rung,i.e.a scalar propagator,transverse to two parallel lines carrying momentum ℓ1+ℓ2,along with a factor of i (ℓ1+ℓ2)2in the numerator,as shown in ﬁing this rule,one can construct recursively the external and loop-momentum-containing numerators factors associated with every φ3-type diagram that can be reduced to trees by a sequence of two-particle cuts,such as the diagram in ﬁg.7a .Such diagrams can be termed “iterated 2-particle cut-constructible,”although a more compact notation might be ‘Mondrian’diagrams,given their resemblance to Mondrian’s paintings.Not all diagrams can be computed in this way.The diagram in ﬁg.7b is not in the ‘Mondrian’class,so it cannot be determined from two-particle cuts.Instead,evaluation of the three-particle cuts shows that it appears with a non-vanishing coeﬃcient in the subleading-color contributions to the three-loop MSYM amplitude.ℓ1ℓ2−→i (ℓ1+ℓ2)2ℓ1ℓ2Figure 6.The rung rule for MSYM.(a)(b)Figure 7.(a)Example of a ‘Mondrian’diagram which can be determined re-cursively from the rung rule.(b)Theﬁrst non-vanishing,non-Mondrian dia-grams appear at three loops in nonplanar,subleading-color contributions.4Iterative Relation in N =4Super-Yang-Mills TheoryAlthough the two-loop gg →gg amplitude in MSYM was expressed in terms of scalar integrals in 1997[11],and the integrals themselves were computed as a Laurent expansion about D =4in 1999[25,26],the expansion of the N =4amplitude was not inspected until last fall [9],considerably after similar investigations for QCD and N =1super-Yang-Mills theory [36,37].It was found to have a quite interesting “iterative”relation,when expressed in terms of the one-loop amplitude and its square.At leading color,the L -loop gg →gg amplitude has the same single-trace color decomposition as the tree amplitude,eq.(9).Let M (L )4be the ratio of this leading-color,color-ordered amplitude to the corresponding tree amplitude,omitting also several conventional factors,A (L ),N =4planar 4= 2e −ǫγg 2N c2 M (1)4(ǫ) 2+f (ǫ)M (1)4(2ǫ)−12(ζ2)2is replaced by approximately sixpages of formulas (!),including a plethora of polylogarithms,logarithms and=+f(ǫ)−12(ζ2)2+O(ǫ)f(ǫ)=−(ζ2+ǫζ3+ǫ2ζ4+...)Figure8.Schematic depiction of the iterative relation(13)between two-loop and one-loop MSYM amplitudes.polynomials in ratios of invariants s/t,s/u and t/u[37].The polylogarithm is deﬁned byLi m(x)=∞i=1x i t Li m−1(t),Li1(x)=−ln(1−x).(14)It appears with degree m up to4at theﬁnite,orderǫ0,level;and up to degree4−i in the O(ǫ−i)terms.In the case of MSYM,identities relating these polylogarithms are needed to establish eq.(13).Although the O(ǫ0)term in eq.(13)is miraculously simple,as noted above the behavior of the pole terms is not a miracle.It is dictated in general terms by the cancellation of infrared divergences between virtual corrections and real emission[24].Roughly speaking,for this cancellation to take place,the virtual terms must resemble lower-loop amplitudes,and the real terms must resemble lower-point amplitudes,in the soft and collinear regions of loop or phase-space integration.At the level of theﬁnite terms,the iterative relation(13)can be understood in the Regge/BFKL limit where s≫t,because it then corresponds to expo-nentiation of large logarithms of s/t[40].For general values of s/t,however, there is no such argument.The relation is special to D=4,where the theory is conformally invariant. That is,the O(ǫ1)remainder terms cannot be simpliﬁed signiﬁcantly.For ex-ample,the two-loop amplitude M(2)4(ǫ)contains at O(ǫ1)all three independent Li5functions,Li5(−s/u),Li5(−t/u)and Li5(−s/t),yet[M(1)4(ǫ)]2has only the ﬁrst two of these[9].The relation is also special to the planar,leading-color limit.The subleading color-components of theﬁnite remainder|A(2),ﬁnn deﬁned by eq.(5)show no signiﬁcant simpliﬁcation at all.For planar amplitudes in the D→4limit,however,there is evidence that an identical relation also holds for an arbitrary number n of external legs, at least for certain“maximally helicity-violating”(MHV)helicity amplitudes. This evidence comes from studying the limits of two-loop amplitudes as two of the n gluon momenta become collinear[9,38,41].(Indeed,it was by analyzing these limits that the relation for n=4wasﬁrst uncovered.)The collinear limits turn out to be consistent with the same eq.(13)with M4replaced by M n everywhere[9],i.e.M(2)n(ǫ)=12(ζ2)2+O(ǫ).(15)The collinear consistency does not constitute a proof of eq.(15),but in light of the remarkable properties of MSYM,it would be surprising if it were not true in the MHV case.Because the direct computation of two-loop amplitudes for n>4seems rather diﬃcult,it would be quite interesting to try to examine the twistor-space properties of eq.(15),along the lines of refs.[31,42].(The right-hand-side of eq.(15)is not completely speciﬁed at order1/ǫandǫ0for n>4.The reason is that the orderǫandǫ2terms in M(1)n(ǫ),which contribute to theﬁrst term in eq.(15)at order1/ǫandǫ0,contain the D=6−2ǫpentagon integral[43],which is not known in closed form.On the other hand, the diﬀerential equations this integral satisﬁes may suﬃce to test the twistor-space behavior.Or one may examine just theﬁnite remainder M(L),ﬁnn deﬁnedvia eq.(5).)It may soon be possible to test whether an iterative relation for planar MSYM amplitudes extends to three loops.An ansatz for the three-loop planar gg→gg amplitude,shown inﬁg.9,was provided at the same time as the two-loop re-sult,in1997[11].The ansatz is based on the“rung-rule”evaluation of the iterated2-particle cuts,plus the3-particle cuts with intermediate states in D=4;the4-particle cuts have not yet been veriﬁed.Two integrals,each be-ginning at O(ǫ−6),are required to evaluate the ansatz in a Laurent expansion about D=4.(The other two integrals are related by s↔t.)The triple ladder integral on the top line ofﬁg.9was evaluated last year by Smirnov,all the way through O(ǫ0)[44].Evaluation of the remaining integral,which contains a factor of(ℓ+k4)2in the numerator,is in progress[45];all the terms through O(ǫ−2)agree with predictions[33],up to a couple of minor corrections.5Signiﬁcance of Iterative Behavior?It is not yet entirely clear why the two-loop four-point amplitude,and prob-ably also the n-point amplitudes,have the iterative structure(15).However, one can speculate that it is from the need for the perturbative series to=i3s12s212+s223+2s12(ℓ+k4)+2s23(ℓ+k1)21Figure9.Graphical representation of the three-loop amplitude for MSYM in the planar limit.be summable into something which becomes“simple”in the planar strong-coupling limit,since that corresponds,via AdS/CFT,to a weakly-coupled supergravity theory.The fact that the relation is special to the conformal limit D→4,and to the planar limit,backs up this speculation.Obviously it would be nice to have some more information at three loops.There have been other hints of an iterative structure in the four-point correlation func-tions of chiral primary(BPS)composite operators[46],but here also the exact structure is not yet clear.Integrability has played a key role in recent higher-loop computations of non-BPS spin-chain anomalous dimensions[4,5,6,8].By imposing regularity of the BMN‘continuum’limit[3],a piece of the anoma-lous dimension matrix has even been summed to all orders in g2N c in terms of hypergeometric functions[7].The quantities we considered here—gauge-invariant,but dimensionally regularized,scattering amplitudes of color non-singlet states—are quite diﬀerent from the composite color-singlet operators usually treated.Yet there should be some underlying connection between the diﬀerent perturbative series.6Aside:Anomalous Dimensions in QCD and MSYMAs mentioned previously,the set of anomalous dimensions for leading-twist operators was recently computed at NNLO in QCD,as the culmination of a multi-year eﬀort[13]which is central to performing precise computations of hadron collider cross sections.Shortly after the Moch,Vermaseren and Vogt computation,the anomalous dimensions in MSYM were extracted from this result by Kotikov,Lipatov,Onishchenko and Velizhanin[12].(The MSYM anomalous dimensions are universal;supersymmetry implies that there is only one independent one for each Mellin moment j.)This extraction was non-trivial,because MSYM contains scalars,interacting through both gauge and Yukawa interactions,whereas QCD does not.However,Kotikov et al.noticed, from comparing NLO computations in both leading-twist anomalous dimen-sions and BFKL evolution,that the“most complicated terms”in the QCDcomputation always coincide with the MSYM result,once the gauge group representation of the fermions is shifted from the fundamental to the adjoint representation.One can deﬁne the“most complicated terms”in the x-space representation of the anomalous dimensions—i.e.the splitting kernels—as follows:Assign a logarithm or factor ofπa transcendentality of1,and a polylogarithm Li m or factor ofζm=Li m(1)a transcendentality of m.Then the most complicated terms are those with leading transcendentality.For the NNLO anomalous dimensions,this turns out to be transcendentality4.(This rule for extracting the MSYM terms from QCD has also been found to hold directly at NNLO,for the doubly-virtual contributions[38].)Strikingly,the NNLO MSYM anomalous dimension obtained for j=4by this procedure agrees with a previous result derived by assuming an integrable structure for the planar three-loop contribution to the dilatation operator[5].7Conclusions and OutlookN=4super-Yang-Mills theory is an excellent testing ground for techniques for computing,and understanding the structure of,QCD scattering amplitudes which are needed for precise theoretical predictions at high-energy colliders. One can even learn something about the structure of N=4super-Yang-Mills theory in the process,although clearly there is much more to be understood. Some open questions include:Is there any AdS/CFT“dictionary”for color non-singlet states,like plane-wave gluons?Can one recover composite operator correlation functions from any limits of multi-point scattering amplitudes?Is there a better way to infrared regulate N=4supersymmetric scattering amplitudes,that might be more convenient for approaching the AdS/CFT correspondence,such as compactiﬁcation on a three-sphere,use of twistor-space,or use of coherent external states?Further investigations of this arena will surely be fruitful.AcknowledgementsWe are grateful to the organizers of Strings04for putting together such a stim-ulating meeting.This research was supported by the US Department of En-ergy under contracts DE-FG03-91ER40662(Z.B.)and DE-AC02-76SF00515 (L.J.D.),and by the Direction des Sciences de la Mati`e re of the Commissariat `a l’Energie Atomique of France(D.A.K.).。

arXiv0712.0473v1[hep-ph]4Dec2007

a r X i v :0712.0473v 1 [h e p -p h ] 4 D e c 2007Frascati Physics Series Vol.XL VI (2007),pp.000-000HADRON07:XII Int.Conf.on Hadron Spectroscopy –Frascati,October 8-13,2007Plenary/Parallel Session pick one THE EFFECT OF ISOSPIN VIOLATION ON SCALAR MESON PRODUCTION C.Hanhart 1,B.Kubis 2,and J.R.Pel´a ez 31Instit¨u t f¨u r Kernphysik (Theorie),Forschungzentrum J¨u lich,D-52425J¨u lich,Germany 2HISKP (Theorie),Universit¨a t Bonn,Nussallee 14-16,D-53115Bonn,Germany.3Departamento de F´ısica Te´o rica II,Universidad Complutense,E-28040Madrid.Spain Abstract We investigate the isospin-violating mixing of the light scalar mesons a 0(980)and f 0(980)within the unitarized chiral approach.Isospin-violating eﬀects are considered to leading order in the quark mass diﬀerence and electromag-netism.In this approach both resonances are generated through meson-meson dynamics.Our results provide a description of the mixing phenomenon within a framework consistent with chiral symmetry and unitarity,where these res-onances are not predominantly q ¯q states.We discuss in detail the reactionsJ/Ψ→φS ,where S denotes a suitable pair of pseudo–scalar mesons in the scalar channel,namely π0η,K +K −,and K 0¯K0.In this work predictions for the cross section in the kaon channels are given for the ﬁrst time with isospin violating eﬀects included.1IntroductionAlthough the light scalar mesons a0(980)and f0(980)have been established as resonances long ago,there is still a heated debate going on in the literature regarding the very nature of these states.Naively one might assign them a conventional q¯q structure,however,at present no quark model is capable of describing both states simultaneously as q¯q states—see,e.g.,Ref.1).On the other hand,as early as1977it was stressed that especially in the scalar channel the interaction of four-quark systems(two quarks,two antiquarks)is attractive2).Some authors have found indications for the existence of compact four-quark states3−5).However,the same short-ranged interaction can also be the kernel to the scattering of pseudoscalars,giving rise to extended four-quark states that one might call hadronic molecules or extraordinary hadrons6−9). Independently,a similar conclusion was found in diﬀerent approaches10−13).In Refs.14,15)it is stressed that the eﬀective coupling constants of the scalar mesons to the K¯K channel contain the essential structure information. Especially,the larger the molecular component,the larger the residue at the resonance pole,which acquires its maximum value in case of a pure molecule. It should be stressed,however,that this connection can be made rigorous only for stable bound states and if the bound state pole is on theﬁrst sheet very close to the elastic threshold14).However,if the state of interest is narrow and the inelastic threshold is suﬃciently far away,the argument should still hold15).Note that both conditions apply for a0(980)and f0(980).Therefore one should aim at observables that are very sensitive to the eﬀective coupling constants.The resonance signal,as seen in production experiments for both states,is not very useful to determine those couplings,for it turned out to be mainly sensitive to ratios of couplings16).It is therefore important to investigate other observables.When this formalism was applied to the scalar mesons a0(980)and f0(980) it was found from an analysis of a series of reactions17)that the latter is indeed predominantly a K¯K molecule,in line with the results of Refs.7,10,11,13), while the results for the former did not lead to an unambiguous interpretation. This might either point at a prominent non–molecular contribution to the a0 structure,or the a0is not a bound state,but a virtual state.To decide on this issue it is important to collect more information especially on the a0.In Ref.18)it was stressed that the amplitude for the isospin violating a0−f0Figure1:Graphical illustration of the leading contribution to the a0−f0mixing matrix element,driven by the kaon mass diﬀerences.transition should be very sensitive to the product of the eﬀective coupling to the two–kaon channels of the f0and the a0.If we assume that the f0is a molecule,its coupling to the K¯K channel isﬁxed,as discussed above.Then the value for the a0−f0mixing amplitude should be very sensitive to the structure of the a0.The reason why the mixing amplitude of a0and f0is sensitive to the eﬀective couplings is that it gets a prominent contribution from the kaon–loops (see Fig.1).Their isospin violating part is driven by the kaon mass diﬀerences giving rise to an eﬀect that can be shown to scale as1This decay was identiﬁed as very useful to study isospin violation for scalar mesons in Refs.21,22).Figure2:Graphical illustration of the subleading contribution to the a0−f0 mixing matrix element,driven by isospin-violating vertices,denoted by the crosses.coupling of the f0to kaons and should therefore provide an independent cross check for the size of corrections of order(m d−m u)/m s.In the next section the formalism is brieﬂy reviewed and the results are given.We close with a short summary.2Formalism and ResultsDetails on the formalism are given in Ref.19).Thus,here we will only repeat the essential physics that went into the calculations.In the Standard Model there are two sources of isospin violation present,namely the up–down quark mass diﬀerence as well as electromagnetism.Both appear in a well deﬁned form in the corresponding Lagrangian density formulated in terms of the fundamen-tal degrees of freedom,here photons,gluons,and quarks.Chiral perturbation theory23,24)allows for a consistent representation of those terms for a theory describing the interaction of the(pseudo)Goldstone bosons with each other. At leading order in the chiral Lagrangian the only parameter that appears, in addition to the various particle masses,is the pion decay constant.Thus, also the leading eﬀect of isospin violation is predicted.It should be stressed that already at leading order both isospin violating mass diﬀerences as well as isospin violating interactions are present.Already in isospin symmetric calculations of meson–meson scattering am-plitudes,when unitarizing the leading chiral Lagrangian,an a priori unknown constant needs to be adjusted to the data.In addition,a few more parameters need to beﬁtted to the production data—in our case they wereﬁxed from a ﬁt to the data on J/Ψ→φπ+π−,K+K−as well as J/Ψ→ωπ+π−25).Once this is done,the isospin violating signals emerge as predictions.Our resultsFigure3:Left:predicted signal for J/Ψ→φπ0η.The solid line shows the result without isospin violation in the production operator(IVPO),while the band shows the eﬀect of its inclusion.Right:predicted signal for J/Ψ→φK¯K. For both kaon channels the solid band,which includes the uncertainties,is the full result,while for the dashed line the same decay amplitude was used in both channels(see text).for the various channels are shown in Fig.3.The predicted signal in theπ0ηchannel was already published in Ref.19).Note that in an isospin symmetric world this reaction would not be allowed,thus any signal is directly propor-tional to the square of an isospin violating amplitude.Based on very general scale arguments one can show that in addition this amplitude is dominated by the isospin violation that emerges from a0−f0mixing26).This was conﬁrmed on the grounds of the current model:theﬁlled band in the left panel of Fig.3 shows the eﬀect of possible isospin violation in the production operator.The right panel of the sameﬁgure shows the J/Ψdecay with a pair of kaons in the ﬁnal state.Please note that for these two channels there would be a decay rate also in an isospin symmetric world,however,then both signals would agree. The solid lines show our results for the two channels including the eﬀects of isospin violation as described in Ref.19).In addition,the presence of charged particles in theﬁnal state called for an additional treatment of soft photons. Here we followed Ref.27).The decay spectrum for J/Ψ→φK+K−,based on an isospin symmetric calculation,is shown in Ref.25).To see how much of the diﬀerence in the two kaon channels originates from the diﬀerent phase space thresholds alone(2m K+−2m K0=8MeV),in theﬁgure we also show the signals that emerge when the same decay amplitude is used for both channels(dashed lines).For this calculation we took the average of the charged and the neutral kaon amplitudes,corresponding to a formal isospin0combination.Obviously the by far most important diﬀerence between the channels is driven by the displacement of the thresholds,the eﬀect of which is further enhanced by the strongly varying amplitudes precisely in this threshold region.The original idea was to extract information on the charge dependence of the couplings of the f0to kaons from a comparison of J/Ψ→φK+K−and J/Ψ→φK0¯K0.However,as can be seen from the ﬁgure,the diﬀerences between the solid and the dashed lines are too small to be accessible experimentally.Therefore we do not expect data for the kaon channels to be suﬃciently sensitive to extract isospin violating eﬀects in the decay amplitudes.3SummaryIn this work we calculated the reactions J/Ψ→φS,where S denotes a suitable pair of pseudo–scalar mesons in the scalar channel,namelyπ0η,K+K−,and K0¯K0.The goal of this study was to gain a better quantitative understanding of the phenomenon of a0−f0mixing,which should eventually reveal important information on the structure of the a0(980).In addition to theπ0ηchannel,the kaon channels including isospin violation were discussed here for theﬁrst time. We found that,at least within the current model,the impact of the charge dependence of the coupling of the f0to kaons is too small to be deduced from a comparison of the rates for K+K−,and K0¯K0.On the other hand theπ0ηdecay channel appears to be not only very sensitive to the eﬀective coupling constants that encode the structure information but also theoretically under control19).For a discussion of additional background eﬀects see Ref.22).The corresponding measurements can be performed,once BES III is in operation.4AcknowledgmentsThe authors acknowledge partialﬁnancial support by the EU integrated in-frastructure initiative HADRONPHYSICS PROJECT,under contract RII3-CT-2004-506078. B.K.is supported by the DFG(SFB/TR16).The re-search of J.R.P.is partially funded by Banco Santander/Complutense con-tract PR27/05-13955-BSCH and Spanish CICYT contracts FPA2007-29115-E, FPA2005-02327,BFM2003-00856.References1.C.R.M¨u nz,J.Resag,B.C.Metsch and H.R.Petry,Nucl.Phys.A578,418(1994)[arXiv:nucl-th/9307027].2.R.L.Jaﬀe,Phys.Rev.D15,267(1977).3.N.N.Achasov,Phys.Atom.Nucl.67,1529(2004)[Yad.Fiz.67,1552(2004)][arXiv:hep-ph/0309118].4.F.Giacosa,Phys.Rev.D74,014028(2006)[arXiv:hep-ph/0605191].5.J.Vijande,F.Fern´a ndez,A.Valcarce and B.Silvestre-Brac,arXiv:hep-ph/0206263.6.R.L.Jaﬀe and F.E.Low,Phys.Rev.D19,2105(1979).7.J.R.Pel´a ez,Phys.Rev.Lett.92,102001(2004)[arXiv:hep-ph/0309292].8.J.R.Pel´a ez and G.R´ıos,Phys.Rev.Lett.97,242002(2006).[arXiv:hep-ph/0610397].9.R.L.Jaﬀe,arXiv:hep-ph/0701038.10.J.D.Weinstein and N.Isgur,Phys.Rev.Lett.48,659(1982);Phys.Rev.D27,588(1983);Phys.Rev.D41,2236(1990).11.G.Janssen,B.C.Pearce,K.Holinde and J.Speth,Phys.Rev.D52,2690(1995)[arXiv:nucl-th/9411021].12.E.van Beveren and G.Rupp,Eur.Phys.J.C22,493(2001)[arXiv:hep-ex/0106077].13.J.A.Oller and E.Oset,Nucl.Phys.A620,438(1997)[Erratum-ibid.A652,407(1999)][arXiv:hep-ph/9702314].14.S.Weinberg,Phys.Rev.130,776(1963);Phys.Rev.131,440(1963);Phys.Rev.137,B672(1965).15.V.Baru,J.Haidenbauer,C.Hanhart,Y.Kalashnikova and A.Kudryavt-sev,Phys.Lett.B586,53(2004)[arXiv:hep-ph/0308129].16.V.Baru,J.Haidenbauer, C.Hanhart, A. E.Kudryavtsev and U.-G.Meißner,Eur.Phys.J.A23,523(2005)[arXiv:nucl-th/0410099].17.C.Hanhart,Yu.S.Kalashnikova,A.E.Kudryavtsev and A.V.Nefediev,Phys.Rev.D75,074015(2007)[arXiv:hep-ph/0701214];Yu.Kalashnikova,A.E.Kudryavtsev,A.V.Nefediev,J.Haidenbauer and C.Hanhart,Phys.Rev.C73,045203(2006)[arXiv:nucl-th/0512028];Yu.S.Kalashnikova,A.E.Kudryavtsev,A.V.Nefediev,C.Hanhart and J.Haidenbauer,Eur.Phys.J.A24,437(2005)[arXiv:hep-ph/0412340].18.N.N.Achasov,S.A.Devyanin and G.N.Shestakov,Phys.Lett.B88,367(1979).19.C.Hanhart,B.Kubis and J.R.Pel´a ez,Phys.Rev.D76,074028(2007)[arXiv:0707.0262[hep-ph]].20.J.R.Pel´a ez,Mod.Phys.Lett.A19,2879(2004)[arXiv:hep-ph/0411107].21.F. E.Close and A.Kirk,Phys.Lett.B489,24(2000)[arXiv:hep-ph/0008066];22.J.J.Wu,Q.Zhao and B.S.Zou,Phys.Rev.D75,114012(2007)[arXiv:0704.3652[hep-ph]].23.S.Weinberg,Physica A96,327(1979);J.Gasser and H.Leutwyler,AnnalsPhys.158,142(1984);Nucl.Phys.B250,465(1985).24.R.Urech,Nucl.Phys.B433,234(1995)[arXiv:hep-ph/9405341].25.T. A.L¨a hde and U.-G.Meißner,Phys.Rev.D74,034021(2006)[arXiv:hep-ph/0606133].26.C.Hanhart,AIP Conf.Proc.688,61(2004)[arXiv:nucl-th/0306073];C.Hanhart,Phys.Rept.397,155(2004)[arXiv:hep-ph/0311341].27.G.Isidori,arXiv:0709.2439[hep-ph].。

夸克禁闭和渐近自由 ICTS_0909_THuang

Spectrum from quenched LQCD
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一个普遍的看法是认为夸克囚禁是由量子色动力学物理真空性质造成的。微扰量子色动力学理论是建立在微扰真空的基础上，而量子色动力学物理真空完全不同于微扰真空。在物理真空中真空不空，它充满着夸克、反夸克对以及胶子，物质与真空中的夸克、反夸克对和胶子不断发生相互作用造成新的强子结构图像。因此揭示真空本质将导致找到夸克囚禁疑难解。
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Light hadron spectrum on the lattice
For the first time, the lattice results appear on Sciences
LQCD verifies naï prediction that glueballs should exist: ve
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高精度
• 美国、日本B工厂Super-B • 高亮度Tau-Charm (Tau-Charm factory)
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北京正负电子对撞机（BEPC）鸟瞰图
BEPC --->BEPCII
τ粲物理研究高精度测量前沿：高统计性和小系统误差的
测量，精确检验标准模型，探索超越标准模型的新物理现象。 BEPC 对该能区物理有独特优势:J／和′共振峰截面大，阈值区域本底小，是B工厂无法替代的，对QCD的发展，包括微扰QCD和非微扰QCD及其过渡区域十分重要，同时能探索新物理现象。
19
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物质结构的层次原子→原子核物理→粒子物理(高能物理)
粒子物理就是研究物质结构的最小组成成份 (夸克、轻子)及其基本相互作用(电磁、弱、强相互作用)规律的科学。
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二.渐近自由--量子色动力学理论 (Quantum Chromodynamics (QCD))

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a r X i v :h e p -p h /9503444v 1 27 M a r 1995hep-ph/9503444TTP95-151March 1995TWO-LOOP LIGHT QUARK CORRECTIONSTO THE TOP WIDTH 2ANDRZEJ CZARNECKIInstitut f¨u r Theoretische Teilchenphysik,Universit¨a t KarlsruheD-76128Karlsruhe,GermanyAbstractA method of computing two-loop fermionic contributions to the width of a heavy quark is described.An analytical formula for this eﬀect in the limit of mass of the quark much larger than the decay products is obtained for the ﬁrst time.The result conﬁrms previous numerical studies.1IntroductionThe recently discovered top quark attracts attention of many physicists who regard its extraordinarily large mass as a hint of its connection with “new physics”.One consequence of the unusually big mass is the large width of the top quark Γt .This quantity is certainly sensitive to possible exotic particles with which top can interact and will be subject of future precision experimental studies.It is important to know the standard model prediction for Γt as precisely as possible and a lot of eﬀort has been invested in studies of quantum eﬀects which modify it.In the framework of the standard model the transition of the top quark into a bottom quark and a W boson is by far the dominant decay mode and it has been the focus of the recent studies.In particular,one-loop QCD corrections have been found to reduceΓ(t→bW)by about 10%[1].Recently theﬁrst step has been made towards numerical calculation of two-loop strong corrections to this reaction,namely the subset of diagrams containing a fermion loop or an additional fermion pair in theﬁnal state[2]. The relevant diagrams are shown inﬁg.1.(a)(b)(c)Figure1:Diagrams with virtual and real fermions contributing to top’s width.Calculation of the fermionic contributions is particularly important be-cause it helps to establish the correct mass scale at which the strong coupling constant should be taken for the calculation of one-loop corrections.We note that the relevant expansion parameter for the calculations of the decay widthΓ(t→bW)is m2W/m2t∼<0.2.Therefore taking m W=0,as well as neglecting masses of all fermions except the decaying top quark as in ref.2is a very good approximation.The purpose of this paper is to show a method in which the fermionic subset of two-loop corrections can be calculated analyti-cally.2Real and virtual contributionsThe tree level decay rate of a top quark into massless W and b is(we take the relevant Kobayashi-Maskawa matrix element equal1)Γ0=G F m3t2(2π)2−D R2(1)where m t is the pole mass of the top quark(we shall use it as the unit of mass throughout this calculation)and R2denotes the volume of the two-body phase spaceR2≡π1−εΓ(1−ε)In order to regularize infrared and ultraviolet divergences we perform the cal-culations in D=4−2εdimensions.In particular we need to know the one-loop gluonic corrections toΓ(t→bW)including terms O(ε).Weﬁnd(neglecting terms containing the Euler constantγE and ln4πwhich vanish in theﬁnal result)Γ1=−G F R2g2s C F222D−3π32+24+5(4π)D/22−2ε√3(1−2ε)2(1−3ε)(2−3ε)(3−2ε)×Γ2(−ε)Γ(ε)Γ(2ε)Γ(−4ε)√23D−4π2D−2× 136ε2+ 15772 1648+5518ζ(3) (5)Similarly,the correction to the external quark leg givesΓZ=−G F R22g4s C F3(1−3ε)(2−3ε)Γ(ε)Γ2(−ε)Γ(2ε)Γ(−4ε)√23D−4π2D−2 18ε+5924 (6)It is more diﬃcult to calculate the eﬀect of real quarks in theﬁnal state shown inﬁg.1b,c.Especially the square of the amplitude of the gluon emission oﬀthe decaying quark and the interference between amplitudes with emissions from both quarks are troublesome.This is because the integration over the four particle phase space in the presence of the massive propagator in the diagrams leads in D dimensions to hypergeometric functions.However,for our aims it is suﬃcient to expand the result in a Laurent series inε.WeﬁndΓR=−G F R22g4s C F12ε3+5216−11ε+1998527π2−733Results and summaryIt is interesting to look closer at the cancellation of divergences among the diagrams calculated in the previous section.The most singular,1/ε3terms, which have a purely infrared origin,vanish after summing the vertex and the real radiation diagrams(in fact only the interference of the two real radiation amplitudes is relevant at this level).The sum of the two formulasΓV+ΓR contains1/ε2poles which are cancelled after adding the external leg correction ΓZ.The singularity of the resulting formula is proportional to theﬁnite part of the one-loop gluonic correctionΓ1and is removed by expressing the one-loop result in terms of the unrenormalized coupling constant.Of course at this stage we have the freedom of choosing theﬁnite normalization ofαs.Various choices have been discussed recently[2,4].In order to better understand the terminology involved we look again at the gluon vacuum polarization in eq.(4). Expanding the numerical factor in this formula inεwe getΠ(q2)=−4ig2s6ε+5MS scheme corresponds to taking only the divergent part of this expansion for the renormalization of the coupling constant.In this scheme our result readsΓferm(t→W b)= αs3Γ0 −89ζ(2)+2ζ(3) (9)whereζ(2)=π2/6andζ(3)≈1.2020569,and N f is the number of lightﬂavors of quarks.The authors of ref.2recommend using the so-called V scheme[5] which in our case amounts to taking−q2=1in eq.8(we use m t as the unit of mass).Ourﬁnal result then becomesΓferm(t→W b)= αs3Γ0 13+2ζ(3) (10)Evaluation of the square bracket gives3.452425...which conﬁrms the numer-ical result given in ref.22.54 2π26 =3.45 (11)The analytical result given in eq.(10)describes the two-loop O(N fα2s)cor-rection to the decay of the top quark into massless b quark and W boson.It is possible to extend this calculation to include the eﬀects of both masses.In order to avoid diﬃculties connected with integrals over a four particle massive phase space I would recommend employing the method of asymptotic expan-sions(see e.g.[6]).This would reduce the problem to calculating derivatives of the amplitudes with respect to m b and m W in the massless limit.Therefore, one would obtain the same integral structures with possible higher powers of propagators.4AcknowledgementsI gratefully acknowledge helpful discussions with M.Buza,M.Je˙z abek,and Q.P.Xu.This research was supported by a grant BMFT056KA93P.References[1]M.Je˙z abek and J.H.K¨u hn.Nucl.Phys.B314(1989)1.[2]B.H.Smith and M.B.Voloshin.Phys.Lett.B340(1994)176.[3]T.Muta.Foundations of Quantum Chromodynamics.World Scientiﬁc,Singapore,1987.[4]M.Shifman and N.G.Uraltsev.A closer look at perturbative correctionsin the b→c semileptonic transitions,hep-ph/9412398.[5]S.J.Brodsky,G.P.Lepage,and P.B.Mackenzie.Phys.Rev.D28(1983)228.[6]V.A.Smirnov.Renormalization and asymptotic expansions.Birkh¨a user,1991.5。