N Masses from QCD Sum Rules

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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。

Gamma2

u ? d djN i = gA U (p) U (p) ; (1) where U (p) denotes the nucleon spinor. Assuming isospin symmetry, one nds (gA )p = ?(gA )n, and the value of (gA)p = 1:2573 0:0028, extracted from the neutron beta decay, has been quoted in the literature 1]. In nature, the isospin symmetryis broken by the current quark mass di erence as well as the electromagnetic interaction, and thus (gA )p 6= ?(gA )n . Previous studies of the nucleon isovector axial charge in the framework of external eld QCD sum-rule method have been made by various authors 2{4]. However, to our best knowledge, the isospin breaking e ects have been ignored in these studies. The goal of this Letter is to examine the di erence between (gA )p and (gA)n using the external eld QCD sum-rule approach, which has been used in studying various nucleon matrix elements of bilinear quark operators 2{13]. The isospin violation is re ected in mu 6= md and the isospin breaking in the vacuum condensates. Electromagnetic e ects will not be included. Invoking the Goldberger-Treiman relation, we also give an estimate of the isospin breaking in the pion-nucleon coupling constants, which is of interest in the investigation of charge symmetry breaking phenomena 14{17]. Let us start from the correlation function of the nucleon interpolating eld in the presence of a constant external isovector axial vector eld Z

Estimates of $m_d - m_u$ and $langlebar{d}drangle - langlebar{u}urangle$ from QCD sum rules

$Estimates of $m_d - m_u$ and $langlebar{d}drangle - langlebar{u}urangle$ from QCD sum rules$

a rXiv:h ep-ph/932299v124Fe b1993TPI-MINN-92/69-T BUTP-93/2Estimates of m d −m u and ¯dd − ¯u u from QCD sum rules for D and D ∗isospin mass diﬀerences V.L.Eletsky †,Theoretical Physics Institute,University of Minnesota Minneapolis,MN 55455,USA and Institute for Theoretical Physics,Bern University Sidlerstrasse 5,CH-3012Bern,Switzerland and B.L.Ioﬀe Institute of Theoretical and Experimental Physics Moscow 117259,Russia Abstract The recent experimental data on D +−D 0and D ∗+−D ∗0mass diﬀerences are used as inputs in the QCD sum rules to obtain new estimates on the mass diﬀerence of light quarks and onthe diﬀerence of their condensates:m d −m u =3±1MeV , ¯dd − ¯u u =−(2.5±1)·10−3 ¯u u(at a standard normalization point,µ=0.5GeV ).The QCD sum rules invented more than a decade ago is now well known to be a very useful tool to study properties of hadrons at intermediate energies and to get information on the basic parameters of QCD,such as quark masses and non-perturbative condensates. One of the problems addressed already in the pioneering papers[1]was the relation between the isotopic symmetry violation on the level of hadrons and the diﬀerence between u-and d-quark masses and condensates.It was shown that the observedρ−ωmixing implies (m d−m u)/(m u+m d)∼0.3andγ∼−1.5·10−2,whereγ= ¯dd / ¯u u −1(1) ruling out a solution with m u=0,m d=0.The quark mass diﬀerence was estimated before the advent of QCD sum rules in refs.[2,3]with the result m d−m u≈3MeV and m d+m u≈11MeV.The diﬀerence of condensates was later estimated within the QCD sum rule framework in a number of papers[4]with the resultγ=−(3÷10)·10−3.The diﬀerence of condensates was also obtained in the framework of chiral perturbation theory[5],which givesγ≈−8·10−2,provided m d−m u=3MeV,1− ¯s s / ¯u u ≈0.2and m s≈150MeV. Recently isospin violation in QCD sum rules for the nucleon,ΣandΞwas considered[6]and the following results were obtained:m d−m u=3±1MeV,γ=−(2±1)·10−3.Thus, while most predictions for m d−m u agree and are grouped around3MeV,predictions for ¯dd − ¯u u are more diverse and range within an order of magnitude.Moreover,arguments were given in ref.[7]that m u may be equal to zero due to instanton contributions to the renormalization of the quark mass.Thus,additional independent estimates of diﬀerences between u−and d−quark masses and condensates are certainly welcome.In this paper we will consider QCD sum rules for isospin mass splittings of D∗and D mesons and make use of the recently reported[8]new results on these splittings,m D∗+−m D∗0=3.32±0.08±0.05MeVm D+−m D0=4.80±0.10±0.06MeV(2) to obtain such estimates.The sum rules are similar to those which were used in ref.[9]to succesfully predict the mass splittings m D∗s−m D∗=110±20MeV and m D s−m D= 120±20MeV.Let us start with the correlator of two pseudoscalar currents with quantum numbers of D,j5=¯cγ5q,where q is either u,or d,at Euclidean momentum−q2>1GeV2,∼C q=i d4xe iqx 0|T{j5(x),j+5(0)|0 (3)and consider its variationδC q as the light quark mass rises from zero to its actual value m q.To estimate C q it is possible to take into account only the unit operator and the quark condensate(Fig.1)in the operator product expansion,since the contribution of operators of higher dimension to the heavy-light correlators is negligible[10].Using the expansion of the quark condensate in the quark massq aα(x)¯q bβ(0) = −148m qˆxαβ ¯q qδab(4)where ¯q q itself also depends on m q,it is easy to see thatδC q is a function of m q and ¯q q − ¯q q 0where the subsript0denotes the chiral limit.On the other hand,saturating the correlators in the standard manner by the corresponding lowest mass resonanses D+and D0,subtracting the continuum from the contribution of the unit operator and applying the Borel transformation[1],(s+Q2)−1→M−2exp(−s/M2),we arrive at the following sum rule forδC d−δC u−βD+−βD02β2Dm D·{(m d−m u)[3m c2M2e−x(1+m2cM2m c e−x L4/9−(s D+−s D0)3−(βD∗+−βD∗0m2)M2+(m D∗+−m D∗0)hadr=D∗−m D∗e m2D∗/M24π2(1−3m4c s3)](7)D∗It is important to emphasize that since we do not take into account perturbative two-loop diagrams with one photon exchange in the”theoretical”part of the sum rules,only the hadronic parts of the isospin splittings,(∆m D)hadr and(∆m D∗)hadr,enter eqs.(5)and (7).To obtain them we use a quark model estimate[12]for the photon cloud part of the mass diﬀerence m D+−m D0=1.7±0.5MeV and also take into account the electromagnetic hyperﬁne splitting2πQ c Q q|Ψ(0)|2δm=−from non-perturbative power corrections and continuum in the chiral limit for u-and d-quarks are[10,11]1GeV2<∼M2<∼2GeV2for the pseudoscalar channel and1.5GeV2<∼M2<∼2.5GeV2for the vector channel.We will let the diﬀerence of thresholds vary around an estimate s D+−s D0∼s D∗+−s D∗0∼(m d−m u)√1For a discussion of possible reasons for this disagreement,see ref.[6].References[1]M.A.Shifman,A.I.Vainshtein and V.I.Zakharov,Nucl.Phys.B147,385,448,519(1979).[2]J.Gasser and H.Leutwyler,Nucl.Phys.B94,269(1975).[3]S.Weinberg,in:Festschrift for I.I.Rabi,ed.L.Motz,NY Academy of Sciences,NewYork,1978.[4]P.Pascual and R.Tarrach,Phys.Lett.B116,443(1982);E.Bagan,A.Bramon,S.Narison and N.Paver,Phys.Lett B135,463(1984);C.A.Dominguez and M.Loewe,Phys.Rev.D31,2930(1985);C.A.Dominguez and M.Loewe,Ann.Phys.(NY)174,372(1987);S.Narison,Rev.Nuovo Cim.10,1(1987);QCD Spectral Sum Rules,World Scientiﬁc Lecture Notes in Physics,v.26,Singapore,1989;E.G.Drukarev,St.-Petersburg Nuclear Physics Institute Report,PNPI-1780(1992).[5]J.Gasser and H.Leutwyler,Nucl.Phys.B250,465(1985).[6]C.Adami,E.G.Drukarev and B.L.Ioﬀe,Marmal Aid Preprint Series In TheoreticalNuclear Physics,MAP-151(1993).[7]Choi,Phys.Lett.B292,159(1992);Nucl.Phys.B383,58(1992).[8]D.Bortoletto et al,Phys.Rev.Lett.69,2046(1992).[9]B.Yu.Blok and V.L.Eletsky,Sov.J.Nucl.Phys.42,787(1985).[10]T.M.Aliev and V.L.Eletsky,Sov.J.Nucl.Phys.38,936(1983).[11]V.L.Eletsky and Ya.I.Kogan,Z.Phys.C28,155(1985).[12]J.Gasser and H.Leutwyler,Phys.Rep.87,78(1982).Figure Captions:•Fig.1:The diagrams taken into account in the sum rules.•Fig.2:Sum rule for D∗mass splitting:f V(M2)−M2d f V(M2)/dM2versus M2,wheref V(M2)is the r.h.s.of eq.(7)for m d−m u=1MeV(a),3MeV(b)and5MeV(c).Ineach of the three cases the lower and the upper curves correspond to s D∗+−s D∗0=0 and0.005GeV2,respectively.The dotted horizontal lines are the boundaries set by eq.(9).•Fig.3:Sum rule for D mass splitting:f P(M2)−M2d f P(M2)/dM2versus M2,wheref P(M2)is the r.h.s.of eq.(5)for m d−m u=3MeV and a)γ=−2.5·10−3;b)γ=−6·10−3.The numbers at the curves correspond to the value of s D+−s D0(in GeV2).The dotted horizontal lines are the boundaries set by eq.(9).。

格格格点点点QCD在在在K物物物理理理中中中的的的新新 ...

K物理中的常规物理量: BK
Short distance 贡献占主导 ⇒ OPE ⇒ Wilson coeff. C(µ) × operator Q∆S=2(µ)
He∆ffS=2 占间接CP 破坏 K 贡献的绝大部份
K
=
exp(i φ
) sin(φ
)
⎡⎢⎢⎢⎢⎣
Im[⟨K
0 He∆ffS=2 ∆MK
格点QCD——超级计算机上的虚拟实验室
格点QCD：超级计算机上的虚拟实验室
格点QCD简介
格点QCD 40年
最早由诺贝奖获得者 Kenneth G. Wilson 在 1974 年提出计算机上的第一个数值计算由 M. Creutz 在 1979 年实现 QCD 超级计算机 1983 – 2011
未来几年内，QCD 计算机进入Eﬂops时代，每秒进行1018浮点运算
格子上的QCD
QCD 格点离散化夸克场位于格点上, ψ(x), xµ = nµa 胶子场由格点之间的链接来表示 Uµ(x ) = eiagAµ(x)
1 mπ
L a
计算机只能模拟有限的自由度 ⇒ 格距a不是无穷小，格子长度L不是无穷大欧氏时空路径积分: 闵氏时间被欧氏时间替代 x0 → −it ⇒ e−iHx0 → e−Ht = e−S[ψ,ψ¯,A] 但哈密顿量 H 在闵氏和欧氏时空是一致的
=++
FLAG average for = + RBC/UKQCD 15A RBC/UKQCD 13 FNAL/MILC 12I JLQCD 12 JLQCD 11 RBC/UKQCD 10 RBC/UKQCD 07
FLAG average for = ETM 10D (stat. err. only) ETM 09A QCDSF 07 (stat. err. only) RBC 06 JLQCD 05 JLQCD 05

Excited Heavy Mesons from QCD Sum Rules

arXiv:hep-ph/9802222v1 3 Feb 1998
EXCITED HEAVY MESONS FROM QCD SUM RULES
Chun Liu Center For Theoretical Physics, Seoul National University, Seoul 151-742, Korea
1 Masses
In HQET, the meson masses are expanded as
M
=
mQ
+
Λ¯
+
O(
1 mQ
)
,
(4)
where Λ¯ denotes the meson masses deﬁned in HQET. They are independent of heavy quark ﬂavor. For the sake of calculating them, the Green’s function is written as
The QCD sum rule 7 is a nonperturbative method rooted in QCD itself. The relevant interpolating current should be ﬁxed. For the excited heavy
1
mesons, we wrote the currents as follows 1,
ever, because the pion energy is about 500 MeV in this case. The decay widths
are given by, e.g.,
Γ(D0∗ → Dπ)
=

NORMA 4000 5000 Power Analyzer 用户说明手册说明书

Since some countries or states do not allow limitation of the term of an implied warranty, or exclusion or limitation of incidental or consequential damages, the limitations and exclusions of this warranty may not apply to every buyer. If any provision of this Warranty is held invalid or unenforceable by a court or other decision-maker of competent jurisdiction, such holding will not affect the validity or enforceability of any other provision.
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Nucleon sum rules in symmetric and asymmetric nuclear matter

(−) ¯(0)γµ d(0)|M = v u − v d vµ = M |u ¯(0)γµ u(0) − d µ µ i obtain nonzero values. In the rest frame of the matter vµ = v i δµ0 , vµ = vδµ0 , (−) vµ = v (−) δµ0 . We can present v i = n|q ¯iγ0 q i |n ρn + p|q ¯iγ0 q i |p ρp . The vali i ues N |q ¯ γ0 q |N are just the numbers of the valence quarks in the nucleons ¯ 0 d|p = 1, p|u ¯ 0 d|n = 2, and thus n|u ¯γ0 u|n = p|dγ ¯γ0 u|p = n|dγ
1
Introduction
We investigate the vector and scalar self-energies of nucleons in nuclear matter composed by the neutrons and protons, distributed with densities ρn and ρp . We calculate the dependence on the total density ρ = ρp + ρn and on the asymmetry parameter β = (ρn − ρp )/(ρp + ρn ). The QCD sum rules were invented in paper [1] to express the hadron parameters through the vacuum expectation values of QCD operators. Being initially used for the mesons, the method was expanded in [2] to the description of the baryons. The approach succeeded in describing the static characteristics as well as some of the dynamical characteristics of the hadrons in vacuum — see, e.g. the reviews [3]. Later the QCD sum rules were applied for investigation of modiﬁed nucleon parameters in the symmetric nuclear matter [4]. They were based on the Boreltransformed dispersion relation for the function Πm (q ) describing the propagation

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

QUARK MASSES

{ 1{ QUARK MASSES
A. Introduction
(by A. Manohar, University of California, San Diego) This note discusses some of the theoretical issues involved in the determination of quark masses. Unlike the leptons, quarks are con ned inside hadrons and are not observed as physical particles. Quark masses cannot be measured directly, but must be determined indirectly through their in uence on hadron properties. As a result, the values of the quark masses depend on precisely how they are de ned; there is no one de nition that is the obvious choice. Though one often speaks loosely of quark masses as one would of the electron or muon mass, any careful statement of a quark mass value must make reference to a particular computational scheme that is used to extract the mass from observations. It is important to keep this scheme dependence in mind when using the quark mass values tabulated in the data listings. The simplest way to de ne the mass of a quark is by making a t of the hadron mass spectrum to a nonrelativistic quark model. The quark masses are de ned as the values obtained from the t. The resulting masses only make sense in the limited context of a particular quark model. They depend on the phenomenological potential used, and on how relativistic e ects are modelled. The quark masses used in potential models also cannot be connected with the quark mass parameters in the QCD Lagrangian. Fortunately, there exist other de nitions of the quark mass that have a more general signi cance, though they also depend on the method of calculation. The purpose of this review is to explain the most important such de nitions and their interrelations.

Lower Bound on the Pion Polarizability from QCD Sum Rules

ﬁnd interesting upper and/or lower bounds based on QCD sum rules. These bounds are a function QCD condensates and of the continuum (duality) thresholds. A suitable combination of these bounds results in a meaningful lower bound on the intrinsic polarizability α ˜E
present constraints on the QCD continuum (duality) threshold are accepted, this lower bound on the intrinsic polarizability α ˜ E is incompatible with some
models range between 3.6 and 13 [5]. On the experimental side there are three measurements of αE , ranging from 2.2 ± 1.6 [6] to 6.8 ± 1.4 ± 1.2 [7], and to 20 ± 12 [8]. In view of the large experimental and theoretical uncertainties it is worthwhile to search for theoretical bounds related to αE . This is done in this work using the relation between the polarizability and the axial current form factor in the decay π → l ν γ [9] and the above mentioned current algebra sum rule [3] which states that the intrinsic polarizability α ˜E is given by α α 2 α ˜ E = αE − < rπ >= 2 3Mπ Mπ fπ

Neutrino Mixing Sum Rules and Oscillation Experiments

Abstract
ν provides a possibility to The neutrino mixing sum rule θ12 − θ13 cos(δ) ≈ θ12 explore the structure of the neutrino mass matrix in the presence of charged lepton corrections, since it relates the 1-2 mixing angle from the neutrino mass ν , to observable parameters of the PMNS mixing matrix. The neutrino matrix, θ12 mixing sum rule holds if the charged lepton mixing angles are CKM-like, i.e., small and dominated by a 1-2 mixing, and for small 1-3 mixing in the neutrino mass matrix. These conditions hold in a wide class of well motivated ﬂavour models. We apply this sum rule to present oscillation data, and we investigate the prospects of future neutrino facilities for exploring the sum rule by simulating various setups for long-baseline reactor and accelerator experiments. As explicit examples, we use the sum rule to test the hypotheses of tri-bimaximal and bimaximal neutrino ν is predicted by sin2 (θ ν ) = 1/3 or 1/2, respectively, although mixing, where θ12 12 ν . the neutrino mixing sum rule can be used to test any prediction for θ12

QCD Sum Rules Intercrossed Relations for the Sigma^0-Lambda Mass Splitting

a rXiv:h ep-ph/31345v23Mar24QCD Sum Rules:Intercrossed Relations for the Σ0−ΛMass Splitting A.¨Ozpineci ∗International Centre for Theoretical Physics,Strada Costiera 11,I34100,Trieste,Italy S.B.Yakovlev,and V.S.Zamiralov †Institute of Nuclear Physics,M.V.Lomonosov Moscow State University,Vorobjovy Gory,Moscow,Russia.February 1,2008Abstract New relations between QCD Borel sum rules for masses of Σ0and Λhyperons are constructed.It is shown that starting from the sum rule for the Σ0hyperon mass it is straightforward to obtain the cor-responding sum rule for the Λhyperon mass and vice versa .PACS:11.30.Hv,11.55.HxKeywords :Octet Baryons,Mass sum rules,SU (3)f Symmetry1IntroductionRecently a series of papers were dedicated to study hadron properties of the Σ,Σc baryons as well as of theΛ,Λc ones in the framework of various QCD sum rules[1,2,3,4,5,6]which have their origin in the works[7,8,9].In [7,9],the nucleons were studied using the QCD sum rules approach and in [10,11],the study was extended to the whole baryon octet.In[8],the whole baryon octet was studied using QCD sum rules together with Gell-Mann-Okubo relation to obtain the mass of theΛ.Many interesting results were obtained.But full expressions for mass or magnetic moment sum rules often become too long and tedious to achieve and prove.Is it possible to relate all these results among themselves and derive,say,Λhyperon properties from that ofΣones and vice versa or just to check them mutually?We propose here nonlinear intercrossed relations which relate matrix ele-ments ofΣ-like baryons with those ofΛ-like ones and vice versa.Their origin lies in the relation between isotopic,U-and V-spin quantities and is quasi obvious in the framework of the quark model.These relations prove to be valid for any QCD sum rules and seem to be useful while obtaining hadron properties of theΛ-like baryons from those of theΣ-like baryons(and vice versa)or checking expressions for them reciprocally.The latter proves to be important asﬁnal QCD SR’s comes to be rather long and cumbersome so it becomes a diﬃcult work to compare or prove them term by term.2Relation between magnetic moments of hy-peronsΣ0andΛin the NRQMWe begin with a simple example.Let us write magnetic moments of hyperons Σ0andΛof the baryon octet in the NRQM:µ(Σ0(ud,s))=23µd−13µu−1But magnetic moment of theΛhyperon can be also obtained from that of the Σ0one,as well as magnetic moment of theΣ0can be obtained from that of theΛone.For that purpose let us formally perform in Eq.(1)the exchange d↔s to getµ(˜Σ0d↔s)=23µs−13µd+23µu;µ(˜Λu↔s)=µu.(3)The following relations are valid:2(µ(˜Σ0d↔s)+µ(˜Σ0u↔s))−µ(Σ0)=3µ(Λ);(4)2(µ(˜Λd↔s)+µ(˜Λu↔s))−µ(Λ)=3µ(Σ0).The origin of these relations lies in the structure of baryon wave functions in the NRQM with isospin I=1,0and I3=0:2√3|Λ>,−2|˜Λd↔s>=−√3|Λ>,2|˜Λu↔s>=√3Relation between QCD correlators forΣ0 andΛhyperonsNow we demonstrate how similar considerations work for QCD sum rules on the example of QCD Borel mass sum rules.The starting point would be two-point Green’s function for hyperonsΣ0 andΛof the baryon octet:ΠΣ0,Λ=i d4xe ipx<0|T{ηΣ0,Λ(x),ηΣ0,Λ(0)}|0>,(5)where isovector(with I3=0)and isocalarﬁeld operators could be chosen as [3]ηΣ0=12ǫabc[ u aT Cs b γ5d c+ d aT Cs b γ5u c−u aT Cγ5s b d c− d aT Cγ5s b u c],ηΛ=16ǫabc[−2 u aT Cd b γ5s c− u aT Cs b γ5d c+ d aT Cs b γ5u c+2 u aT Cγ5d b s c(6) + u aT Cγ5s b d c− d aT Cγ5s b u c],where a,b,c are the color indices and u,d,s are quark wave functions,C is charge conjugation matrix,We show now that one can operate withΣhyperon and obtain the results for theΛhyperon.The reasoning would be valid also for charm and beaty Σ-like andΛ-like baryons.In order to arrive at the desired relations we write not only isospin quan-tities but also U-spin and V-spin ones.Let us introduce U-vector(with U3=0)and U-scalarﬁeld operators just formally changing(d↔s)in the Eq.(7):˜ηΣ0(d↔s)=12ǫabc[ u aT Cd b γ5s c+ s aT Cd b γ5u c − u aT Cγ5d b s c− s aT Cγ5d b u c],˜ηΛ(d↔s)=16ǫabc[−2 u aT Cs b γ5d c−3u aT Cd b γ5s c+ s aT Cd b γ5u c+2 u aT Cγ5s b d c+ u aT Cγ5s b d c− s aT Cγ5d b u c],(7) Similarly we introduce V-vector(with V3=0)and V-scalarﬁeld operators just changing(u↔s)in the Eq.(7):˜ηΣ0(u↔s)=12ǫabc[ s aT Cu b γ5d c+ d aT Cu b γ5s c − s aT Cγ5u b d c− d aT Cγ5u b s c],˜ηΛ(u↔s)=16ǫabc[−2 s aT Cd b γ5u c−s aT Cu b γ5d c+ d aT Cu b γ5s c+2 s aT Cγ5d b u c+ s aT Cγ5u b d c− d aT Cγ5u b s c],(8) Field operators of the Eq.(7)and Eq.(7)can be related through−2˜ηΛ(d↔s)=ηΛ−√3ηΛ+ηΣ0,2˜ηΛ(u↔s)=ηΛ+√3ηΛ−ηΣ0,Upon using Eqs.(7,10)two-point Green functions of the Eq.(5)for hyper-onsΣ0andΛof the baryon octet can be related as2[˜ΠΣ0(d↔s)+˜ΠΣ0(u↔s)]−ΠΣ0=3ΠΛ,(10)2[˜ΠΛ(d↔s)+˜ΠΛ(u↔s)]−ΠΛ=3ΠΣ0.(11) These are essentially nonlinear relations.It is seen that starting calculations,e.g.,fromΣ-like quantities one arrives at the correspondong quantities forΛ-like baryons and vice versa.It should be noted that since the overall normalizations of the currents depend on the convention,in Eqs.(10)and(11),there is an ambiguity in these relations in the ratio of the coeﬃcients of the correlators obtained from Σ0correlator and lambda correlator.This ambiguity results in the freedom to multiply the LHS or the RHS on only one of the Eqs.(10)and(11)by an arbitrary constant.Once this is done on one of the relations,the coeﬃcients in the other relation areﬁxed.In Eq.(7),the normalization is chosen so that the obtained relations for the correlators resemble the relations obtained for the magnetic moments in NRQM,Eq.(4)44Intercrossed relations for the QCD Borel sum rulesIn order to see how it works,we prefered not to use the results of one of us with coautors in[3,4],which also satisfy our relations,but we have repeated calculations of theﬁrst of the QCD mass sum rules for theΣ0hyperon following[1].conserving non-degenerated quantities for u and d quarks, namelyM632L−4/9E0+a u a d48M2L2/27−m s a s m20(s)4L4/9[a s m s−(a u−a d)(m d−m u)]−1Now changing(d↔s)and(u↔s)in the LHS(Σ0)of the Eq.(12) to obtain LHS(¯Σ0(d↔s))and LHS(¯Σ0(u↔s)),respectively,and using Eq.(10)we obtain for theΛ-mass SR:M632L4/9E0+2a s(a u+a d)−a u a d144M2[2(a u+a d)a s m20(s)+2(m20(u)a u+m20(d)a d)−a u a d(m20(u)+m20(d))]−M2E0m s12L4/9E0[3(m u a u+m d a d)+m d a u+m u a d−2(m u+m d)a s]−124(m u+m d−m s)a s m20(s)L−26/27=β2Λe−(M2Λ/M2)+e.s.c.With m0(u)=m0(d)=m0in Eqs.(15)one returns to the expressions given by Eq.(23)in[1].If also a0(u)=a0(d)=a,m20=m20(s)and m u=m d=0,one returnsto mass sum rules of[7]in the form given by Eq.(3)in[12]upon neglecting factors L−2/27and L−26/27in two terms at the LHS:M632L−4/9+a224M2L2/27−(15)a s m s M224L26/27=β2Σ0e−(M2Σ0/M2)+e.s.c.,M632L4/9−a2s m2012L4/9−m s a s m2012(2a u+2a d−a s)E1−bαs243M 2[108a u a d a s +a s (a 2u +a 2d )−2(a u a d +a 2s )(a u +a d )](M 6E 296L 8/9)(2m u +2m d −m s )+136[12a s (m u a d +m d a u )+a s (m u a u +m d a d )−2(m u +m d )a u a d ]=β2ΛM Λe −(M 2Λ/M 2)+e.s.c.Performing changes s ↔d (u ↔d )we arrive at the corresponding Borel sum rules for ˜Λd ↔s (˜Λu ↔s ).Putting these expressions into Eq.(11)it is straightforward to obtaina s M 472+αs 81M 2[−(a 2u +a 2d )+36a u a d ]a s +M 632L 8/9m s E 0+(18)13m s a u a d =β2Σ0M Σ0e −(M 2Σ0/M 2)e.s.c.,which is just the relation given by the Eq.(22)in [1].5ConclusionWe have shown that starting from the QDC Borel mass sum rules for the Σhyperon it is straightforward to obtain the corresponding quantities for the Λhyperon and vice versa upon using intercrossed relations of the type given by Eqs.(2,3)and Eqs.(10,11).More generally these relations can be used not only to obtain properties of the the Σ-like baryons from those of Λ-like ones and vice versa but also to check reciprocally many-terms relations for the Σ-like and Λ-like baryons.7AcknowledgmentsWe are grateful to T.Aliev,V.Dubovik,F.Hussain,N.Paver,G.Thompson for discussions.We are also grateful to B.L.Ioﬀe for illuminating discussions and comments on theﬁnal version of our work.One of us(V.Z.)is grateful to Prof.S.Randjbar-Daemi for the hospitality extended to him at HE section of ICTP(Trieste,Italy).8References[1]Shi-lin Zhu,W-Y.P.Hwang and Ze-sen Yang,Phys.Rev.D56,7273(1997).[2]W-Y.P.Hwang and K.-C.Yang,Phys.Rev.D49,460(1994).[3]T.M.Aliev,A.¨Ozpineci,M.Savci,Phys.Rev.D62,053012(2000).[4]T.M.Aliev,A.¨Ozpineci,M.Savci,Nucl.Phys.A678,443(2000).[5]K.-C.Yang,W-Y.Hwang,E.M.Henley,and L.S.Kisslinger,Phys.Rev.D47,3001(1993).[6]Frank X.Lee and Xinyu Liu,Phys.Rev.D6*******(2002)[7]V.M.Belyaev, B.L.Ioﬀe,JETP56,493(1982); B.L.Ioﬀe,Smilga,Nucl.Phys.B232,109(1984).[8]V.M.Belyaev,B.L.Ioﬀe,Sov.Phys.JETP57(1983)716[Zh.Eksp.Teor.Fiz.84(1983)1236][9]I.I.Balitsky and A.V.Yung,Phys.Lett.B129,328(1983).[10]Ch.B.Chiu,J.Pasupathy,S.L.Wilson,Phys.Rev.D33,1961(1986).[11]J.P.Singh,J.Pasupathy,S.L.Wilson,Ch.B.Chiu,Phys.Rev.D36,1442(1986).[12]Ch.B.Chiu,S.L.Wilson,J.Pasupathy,J.P.Singh,Phys.Rev.D36,1553(1987).9。

《QCD SPECTRAL SUM RULES》(《量子色动力学谱的求和

《QCD SPECTRAL SUM RULES》（《量子色动力学谱的求和规则》）简介
一、作者简介
Stephan Narison 是一个基本粒子物理方面的研究员，他在Montpellier 数学物理实验室工作，同时他还是CERN和ICTP的联合研究员。S．Narison 在研究领域的第一个工作是强子对μ子g-2的贡献。他在电弱方面做了若干工作之后（例如：轻子数守恒、τ衰变过程和超出标准模型的检验。）于1978年将兴趣转向了量子色动力学（QCD）和ＱＣＤ谱的求和规则。S．Narison应Ａ．Salam教授邀请，在ICTP做了两年博士后，应M. K. Gaillard 教授的邀请在LAPP访问一年。他在982-84被授予CERN奖学金，并于1983 年获得Montpellier终身教职。
除了对QCD拉氏量中夸克质量的重要预言之外，本书还介绍了一下若干问题：100-150MeV能标的的QCD、0.04-0.08GeV4 的胶子凝聚、2-3倍于真空饱和中得到的夸克凝聚、夸克-胶子混合凝聚、K介子PCAC 50%的SU（3）破坏和奇异夸克的凝聚。
本书中计算了若干介子的衰变常数，如fD和fB 。此外还计算了，若干弱作用过程中的矩阵元、ρ介子谱在有限温度的行为、若干强子耦合常数和介子波函数，并且预言了奇异介子（胶球，混杂子和四夸克态）的质量、宽度和混合。特别地，当我们将QSSR的结果和QCD低能理论的结果联合起来的时候，可以得到一个在\bar{q}q-胶球混合机制范围内标量介子的复杂谱的合理解释。
二、内容简介
本书第一部分介绍了QCD谱的求和规则（QSSR）的理论基础和计算方法。这些对于初学者、学生和有志于投身于该领域的人来讲是非常有帮助的。
本书第二部分讨论了强子物理中利用QSSR的方法得到的大量的唯象的预言和若干基于电弱相互作用的模型的限制。列举了QSSR方法中的两个重要标准：1，求和规则变量的稳定性；2，QCD连续态阈值的确定。

D_s(2317) and D_s(2457) from HQET Sum Rules

(γtµgtν
ρ
+
γtν gtµρ)
−
1 3
gtµν
γtρ
for Ds∗0 , for Ds∗1 , for Ds1 ,
for Ds∗2
(4)
where γtµ =
γtµ ≡ (0, γ))
γµ − vµ and gtµν
v (In the ≡ gµν −
rest vµvν
frame . The
of the meson, related decay
II. HQET DESCRIPTION
Let us ﬁrst describe the p-wave Ds states. There are four states: Ds∗0(0+), Ds∗1(1+), Ds1(1+) and Ds∗2(2+). These kinds of heavy mesons can be systematically
bDepartment of Physics, Peking University, Beijing 100871, China
Within the framework of the heavy quark eﬀective theory, QCD sum rules are used to calculate the masses of p-wave c¯s states. The results for 0+ and 1+ states with the angular momentum of the light component jl = 1/2 are consistent with the experimental values for Ds(2317) and Ds(2457), respectively.

用势模型求π介子的波函数

用势模型求π介子的波函数作者：闫红星来源：《读与写·上旬刊》2013年第12期摘要：本文从夸克模型理论出发，考虑到QCD渐近自由和色禁闭，把夸克势取为线性禁闭势和色库仑势之和。

并利用微扰论对夸克势模型进行处理，从而求解了介子空间波函数。

再根据介子的色、味、自旋等特征，构造出由色波函、味波函、自旋波函和空间波函数的乘积表示的介子的波函数。

关键词：π介子；夸克势模型；波函数中图分类号：G633.6 文献标识码：B 文章编号：1672-1578（2013）12-0286-01为了揭示核力的本质及新的物理机制，人们做了很多工作，然而在这些工作中波函数无疑都是至关重要的。

同样，在对π介子的研究过程中，它的波函数也是不可或缺的，所以求解π介子的波函数是很有必要的。

下面我们来构造介子的波函数。

1.空间波函数π介子的总波函可以表示成色波函、味波函、自旋波函和空间波函数的乘积形式，下面先来求解空间波函数。

π介子是由夸克q和反夸克q组成的，在介子的夸克势模型中，夸克作非相对论运动，服从薛定谔方程。

1.1 介子的哈密顿量的确定。

夸克间的相互作用用势描述，为满足QCD渐进自由和色禁闭这两个要求，在雅可比坐标中，取介子的哈密顿量为：（1）为禁闭势，可取为线性势（Ar）；为单胶子交换势，取。

A和都为常数，采用自然单位计算时，可取A=1、。

设（看做微扰），其余为H0（看做非微扰）。

（1）式可写成1.2针对非微扰部分在球坐标中求解径向方程。

因为我们所考虑的体系具有球对称性，我们用球坐标。

角动量在中心力场中守恒，选择为力学量完全集，在球坐标中：H0的本征函数和能级记为，能量本征方程为（2）再将共同本征态写为（3）将（3）式代入（2）式可得径向方程能量与磁量子数m无关，但与角动量量子数l有关。

令，令，则约化径向方程为（4）显然p=0，∞是方程的两个奇点。

取径向波函数为：代入（4）式得：（5）设则（5）化为合流超几何方程，方程在p=0领域有界的解为合流超几何函数，无穷级数解，不能满足无穷远处的束缚态边界条件。

QCD Sum Rules - a Working Tool for Hadronic Physics

QCD sum rules are overviewed with an emphasize on the practical applications of this method to the physics of light and heavy hadrons.
talk given at Continuous Advances in QCD 2002/ ARKADYFEST (honoring the 60th birthday of Arkady Vainshtein), Minneapolis, Minnesota, 17-23 May 2002.
TTP02-23
arXiv:hep-ph/0209166v1 15 Sep 2002
QCD SUM RULES - A WORKING TOOL FOR HADRONIC PHYSICS
ALEXANDER KHODJAMIRIAN
∗)
Institut f¨ ur Theoretische Teilchenphysik, Universit¨ at Karlsruhe, D-76128 Karlsruhe, Germany
1. Introduction Imagine a big birthday cake for Arkady Vainshtein, each candle on that cake corresponding to one of his outstanding contributions to the modern particle theory. I think, a very bright and illuminating candle should then mark QCD sum rules. The renown papers introducing QCD sum rules [ 1] have been published by Shifman, Vainshtein and Zakharov in 1979. The method, known also under the nickname of SVZ or ITEP sum rules, very soon became quite popular in the particle theory community, especially in Russia. Not only experienced theorists, but also many students of that time contributed to the development of this ﬁeld with important results. It was indeed a lot of fun to start with an explicit QCD calculation in terms of quark-gluon Feynman diagrams and end up estimating dynamical characteristics of real hadrons. The ﬂexibility and universality of the sum rule method allowed one to go from one interesting problem to another, describing, in the same framework, very diﬀerent hadronic objects, from pions and nucleons to charmonium and B mesons. Nowadays, QCD sum rules are still being