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Chiral Condensate and Short-Time Evolution of QCD on the Light-Cone 1+1 Matthias Burkardt Department of Physics New Mexico State University Las Cruces, NM 88003-0001 U.S.A. Frieder Lenz and Michael Thies Institut fu¨r Theoretische Physik III Universität Erlangen-Nu¨rnberg Staudtstraße 7 D-91058 Erlangen Germany (Dated: February 1, 2008) 2 0 Chiral condensates in the trivial light-cone vacuum emerge if defined as short-time limits of 0 fermionpropagators. Ingaugetheories,thenecessaryinclusionofagaugestringincombinationwith 2 thecharacteristic light-coneinfrared singularities contain therelevant non-perturbativeingredients n responsible for formation of the condensate, as demonstrated for the’t Hooft model. a J I. INTRODUCTION the triviality of the light-cone vacuum on the one hand 9 2 and the non-perturbative nature of the short-time limit of light-cone correlationfunctions on the other. The triviality of the light-cone vacuum is the originof 1 most of the simple properties of field theories if quan- v tized onthe light-cone(for a recentreview, see Ref. [1]). 5 II. CONDENSATE AND SHORT-TIME 3 Forpurelykinematicalreasonsthereisnodistinctionbe- EVOLUTION 2 tween the ground state of free and interacting theories. 1 In view of the many physical consequences usually at- 0 tributedtonon-trivialvacua,thishasraisedconsiderable We define the condensate with respect to that of the 2 non-interacting theory, concern about the equivalence of light-cone and equal- 0 time quantization schemes. While for bosonic theories / ψ¯ψ = lim ψ¯ψ ψ¯ψ 0 , (2.1) h the common belief is that the key lies in the constrained h i ε 0 h iε−h iε -t zero-mode dynamics [2], in fermionic theories a way out → (cid:2) (cid:3) p of this apparent contradiction has been sought in pre- with he sscurcihptaiosntshefo“rpraergiutylarinizvaatriioannto”ftrhegeudlaivreizragteinotncroenldateinnsgatIeRs ψ¯ψ ε = 0ψ¯(ε)Peig 0εdxµAµψ(0)0 (2.2) : h i h | | i v and UV cutoffs [3]. It is doubtful that universal, kine- R Xi matical prescriptions exist which describe the formation regularized in a gauge invariant way. For evaluating the point-split condensate (2.2), we first consider a generic of condensates in theories such as gauge theories where r matrix element of the type a chiral symmetry breaking is not tantamount to fermion mdeafisnsecorerdateiropna.raInmtehteerspraessvenactuwuomrke,xtpheectpartoiopnosvaallu[4e]stoof M(ε)= 0A(ε)Peig 0sds′ddxsµ′ Aµ(x(s′))B(0)0 (2.3) h | | i products of Heisenberg operators,infinitesimally split in R wherex(s)isastraightpathwithx(0)=0andx(s)=ε. light-conetimedirection(inadditiontoaspacedirection) ByshiftingtheargumentofthegaugefieldA andusing willbe showntoyieldthecorrectcondensateinQCD µ 1+1 translationalinvariance of the vacuum, we can represent — the ’t Hooft model [5]. Unlike in standard quantiza- M(ε) as tion, the short-time limit of Heisenberg operators differs non-perturbatively from the corresponding Schrödinger M(ε)= 0A(0)W(s)B(0)0 (2.4) operators. Thisconnectionbetweencondensatesandthe h | | i non-trivialityofthesingularbehaviorofcorrelationfunc- with the point splitting now specified by the operator tions at short light-cone times will be the focus of this W(s) (momentum operator P ), µ work. ateTdhienc[o6n,d7]e.nTsahteesoefsttuhdei’etsHmoaokfetumseodoeflgheanserbaelernelaevtiaolnus- W(s)=e−iεµPµPeig 0sds′ddxsµ′Aµ(x(s′)) . (2.5) between properties of the excited states and the conden- R Differentiation with respect to s, sate,derivedinstandardquantization(Oakes-Rennerre- lation and sum rules). Here we will present a direct cal- dW(s) dxµ culation of the condensate which makes use explicitly of = i (Pµ gAµ(0))W(s) , (2.6) ds − ds − 2 and integration of this differential equation with the ini- with the self-energy given in principal value prescription tial condition W(0) = 1 allows one to simplify this ex- by pression, yielding Ng2 W(s)=e−iεµ(Pµ−gAµ(0)). (2.7) m2r ≡m2− 2π . (2.15) With this form of the operator W the chiral condensate The right hand side of (2.14) combines with the remain- of Eq. (2.2) is seen to be given by the space-time evolu- derofthematrixelementin(2.13)toyieldagainC(p,t). tionofa systemoflight quarkscoupledto the currentof ThetermproportionaltoA+(0)inHeff canbeexpressed an infinitely heavy quark. Thus, on the light-cone with via the Poisson equation in terms of the fermion color its kinematical vacuum, the dynamics of a heavy-light charge density, quark system determines the chiral condensate. So far, everything is rather general and applies equally well to [A (0)] = g dp′dp′′ϕ†j(p′)ϕi(p′′) . (2.16) QCD in 4 dimensions. We now specialize to the ’t Hooft + ij −2 −2π 2π (p p )2 Z ′− ′′ model [5] and write Eq. (2.2) as Withthisresult,Eq. (2.13)canbesimplifiedbyreplacing ψ¯ψ ε = 0ψ¯i(0)(e−iε+Heff)ijψj(0)0 (2.8) in the large N limit the operator h i h | | i w0)i,ththeeffectiveHamiltonianinlight-conegauge(A− = ϕ†i(p)ϕi(p′)→2πδ(p−p′)Nθ(−p), (2.17) by its expectation value. Here the triviality of the light- (H ) =(P +λP )δ g[A (0)] . (2.9) conevacuumisexplicitlyused. Choosingunits suchthat eff ij + ij + ij − − We have denoted the slope of the path, ε /ε+, by λ Ng2 − =1 , (2.18) and will take ε+ and λ as independent parameters from 2π now on; due to the subtraction of the free value, the factoring out a step function θ( p) from C(p,t) and condensatewillturnoutto be independentofλ. P+ and − changing p into p, the time evolution of C can finally P aretheHamiltonianandmomentumoperatorforthe − ’t−Hooft model, respectively. be cast into the form of a typical light-cone Schrödinger equation, For evaluation of the chiral condensate, we represent the spinor ψ in terms of the unconstrained right-handed m2 1 1 component ϕ, iC˙(p,t) = − +λp C(p,t)+ C(0,t) 2p 2p (cid:18) (cid:19) 1 1 1 ∞ 1 ∂C(p′,t) ψ(x)= m ϕ , (2.10) dp′ . (2.19) 21/4 (cid:18) i√2∂− (cid:19) − 2Z−0 p′−p ∂p′ and find (after Fourier transforming the fermion fields) This evolution equation for C(p,t) at short times to- gether with the initial condition (cf. Eq. (2.12)) dpm ψ¯ψ = C(p,ε+) (2.11) h iε 2π p C(p,0)=N (2.20) Z with determines the condensate. We note that dq m2 C(p,t)= 2πh0|ϕ†i(p)(e−iHefft)ijϕj(q)|0i . (2.12) iC˙(p,t=0)=N 2p +λp . Z (cid:18) (cid:19) In order to compute this correlation function we derive For non-interacting fermions, its equation of motion. In m2 dq C0(p,t)=Nexp it +λp (2.21) iC˙(p,t)= 2πh0|ϕ†i(p) Heffe−iHefft ijϕj(q)|0i , (cid:26)− (cid:18)2p (cid:19)(cid:27) Z (cid:0) (cid:1) (2.13) solvestheevolutionequationwiththecorrectinitialcon- we treat the two terms of H (cf. Eq. (2.9)) separately. dition. Due to the presence of singularities, C(p,t) devi- eff Inthefirstterm,theproductoftheoperators(P +λP ) ates significantly for arbitrarily small times from its ini- + and ϕ†i(p) can be replaced by their commutator. In t−he tialvalue N. Characteristicforthe short-timelight-cone large N limit, this commutator generates a combination dynamics is the infrared singularity which implies ofthequarkself-energyandmomentumwhere,following limC(0,t)=0= limC(p,0) . (2.22) ’t Hooft’s original paper [5], t 0 6 p 0 → → m2 The short-time behavior for large momenta is the same h0|[ϕ†i(p),P++λP−]= 2pr +λp h0|ϕ†i(p) , (2.14) in the interacting and in the free theory. It is therefore (cid:18) (cid:19) 3 irrelevant for the evaluation of the condensate and we valuefor the condensate(see below),i.e., the condensate drop in the following the ultraviolet regulator, is directly connected to the changein the infraredsingu- larity of the quark propagator. λ=0 . (2.23) As a consequence of the light-cone singularity in the infrared, determination of the short-time behavior of Inthis case the evolutionequationtogether with the ini- C(p,t) requires a non-perturbative calculation of the tial condition implies that C(p,t) (as well as C (p,t)) 0 function K(q). In terms of K (q), the condensate is (0) depends only on the ratio of the variables p and t, written (cf. Eqs. (2.1),(2.11)) as C(0)(p,−iτ)=NK(0)(p/τ) , (2.24) ψ¯ψ = N m ∞ dq(K(q) K (q)) . (2.30) 0 h i − 2π q − where we have switched to imaginary time. With Eq. Z0 (2.24) the time evolution is converted into the integro- Due to the scaling property (2.24) the dependence on differential equation the regulator ε+ has disappeared entirely from the expression of the condensate. dK(q) m2 1 1 ∞ 1 dK(q′) q = − K(q) dq′ (2.25) dq 2q − 2 − q q dq Z0 ′− ′ and the asymptotic behavior of K(q) is determined by III. CALCULATION OF THE CONDENSATE the initial condition for C(p,t), Wenowbrieflysketchtheevaluationofthecondensate lim K(q)=1 . (2.26) q by Mellin transformation of the equation for K. The →∞ techniques developed in [9] will be used. We define We can also determine the infrared behavior of K(q). Using results for Hilbert transforms of powers (cf. [8]) it γ(κ)= ∞dqqκ 1K(q) . (3.1) is seen that for small q − Z0 K(q) qβ0 (2.27) The Mellin transform is only well defined if ∼ where βn denote the solutions of the ’t Hooft boundary −β0 <κ<0 . (3.2) condition For non-negative κ the integral (3.1) diverges at large πqcotπq =1 m2 (2.28) values of q, while the infraredbehavior(2.27)entails the − lower limit. The Mellin transform converts Eq. (2.25) ordered according to into the recursion relation (cf. [10]) 1 q = βn , n=0,1,2... with βn [n,n+1]. (2.29) γ(κ)= m2 1+π(κ 1)cotπκ γ(κ 1) . (3.3) ± ∈ −2κ − − − Thus the (confining) interaction in the ’t Hooft model (cid:2) (cid:3) As can be easily verified, changes the essential singularity of K (q) of the non- 0 interacting theory κ u+1sin2πu exp 2π du 2 π κ − 0 sin2πu γ(κ)= K0(q)=e−m2/2q N 2 nβ0cosRπ(κ+(cid:16)β0+1/2) (cid:17)o (cid:16) (cid:17) toabranchpoint. Thisremarkablephenomenonisdueto ∞ 1+ (m2π−β1n)−ta1nπκ πβ0−(m2−1)tanπκ tthioenpCre(spe,ntc)e. oFfotrhecogmapuagreissotnrinwgeinretmhaerckortrhealtattihone fsuanmce- · nY=11+ (m2π−(κ1)+tnan)πκ πκ+(m2−1)tanπκ   (3.4) calculation in the Gross-Neveu model, involving a sim- ilar large N approximation, yields the non-interacting solves the recursion relation (3.3). Condition (2.26) de- form of the correlation function with the fermion mass termines the normalization . The solution has the ex- modified by the interactions. N pected poles at the endpoints of the regularity interval For the general case, we have not been able to ex- (3.2) and is free of singularities within this interval. press the solution of Eq. (2.25) via the Mellin trans- According to Eq. (2.30) the condensate is directly form (see below) in simple terms. In the chiral limit given by the Mellin transform (β √3m/π 0), 0 ≈ → m ψ¯ψ = N lim(γ(κ) γ (κ)) (3.5) 0 C(p,−iτ)≈N(p/τ)β0θ(1−(p/τ)). h i − 2π κ→0 − where It can be verified that Eq. (2.25) is satisfied up to terms of O(β ). This expression displays the subtleties of the m2 κ 0 γ (κ)= Γ( κ) (3.6) p,t 0 limit. It reproduces in the chiral limit the exact 0 2 − → (cid:18) (cid:19) 4 is the Mellin transform of K (q) for non-interacting Heisenberg operators [4]. In this way, the condensate is 0 fermions. Expansion of the Mellin transforms (3.4, 3.6) obtained as the short light-cone time limit of an appro- around κ = 0, with the normalization factor chosen priate correlation function. In light-cone quantization, N such that the singular pieces ( 1/κ) cancel, yields non-vanishing order parameters reflect the non-triviality ∼ of the short-time limit of the relevant Heisenberg opera- ψ¯ψ = N m 1+γ+ln m2 (3.7) tors. h i − 2π π We have carried out a study of the chiral condensate of (cid:26) (cid:18) (cid:19) two dimensional QCD within light-cone quantization. A 1 ∞ 1 1 +(1 m2) + . direct and explicit calculation of the condensate within − "β0 n=1(cid:18)βn − n(cid:19)#) the ’t Hooft model has been presented. The kinematical X structure of the light-cone vacuum has been an essential This agrees with the result obtained in [7]. In particular ingredientinthiscalculation. Ourcalculationshowsthat in the chiral limit where the 1/β0 term dominates, the the short-time limit of the quark propagator is afflicted result by non-perturbative physics. On the light-cone this cor- relationfunction is singularin the infrared;the singular- N ψ¯ψ = , (3.8) ity depends on the dynamics. The essential singularity h i −√12 of the non-interacting theory is converted by the interactions in QCD to a branch point — a phenomenon first derived in [6] is reproduced. The calculations [6, 7] 1+1 whichdefiesaperturbativedescriptionandisresponsible are based on low energy theorems and connect the con- for generation of the condensate in the chiral limit. The densate with properties of the mesons of the ’t Hooft resultingfermioncorrelationfunctiondifferssignificantly model (in particular the “Goldstone boson”in the chiral atshortlight-conetimes fromthe correlationfunctionof limit). Inourapproachwiththequarkpropagatorasthe the Gross-Neveu model. In this two dimensional model essentialingredient,the condensateis determinedby the a chiral condensate emerges in the process of mass gen- short-time behavior of a heavy-light quark system. The eration [4]. Here the essential singularity in the infrared equivalenceofthesequitedifferentapproachessuggestsa persists,itsparametersaremodifiedbyinteractions. The relationin light-conequantizationsimilar to the relation successfuldescriptionofthecondensatesinthesedynam- in ordinary coordinates based on chiral Ward identities ically very different models strongly supports the idea of whichconnectsquarkpropagatorsandtheGoldstonebo- reconciling the non-trivial vacuum properties with the son Bethe-Salpeter wave function (cf. [11]). kinematical nature of the light-cone vacuum by the dy- namical, non-perturbative short light-cone time limit of IV. CONCLUSIONS Heisenberg operators. In quantum field theory, condensates are calculated in Acknowledgments M.B. was supported by a grant generalasexpectationvaluesofSchrödingeroperatorsin from DOE (FG03-95ER40965) and through Jefferson the corresponding vacua. Non-vanishing order parame- Lab by contract DE-AC05-84ER40150 under which the tersreflectthenon-trivialityofthevacuum. Inlight-cone Southeastern Universities Research Association (SURA) quantization, the kinematical structure of the light-cone operates the Thomas Jefferson National Accelerator Fa- vacuum gives rise to trivial vacuum expectation values cility. ofSchrödingeroperatorswhich thereforecannotserveas order parameters. On the light-cone, condensates have to be defined as vacuum expectation values of limits of [1] S.J. Brodsky, H.-C. Pauli, and S.S. Pinsky, Phys. Rep. [8] A.Erdélyi,W.Magnus,F.OberhettingerandF.Tricomi, 301, 299 (1998). Tables of Integral Transforms, Vols. 1 and 2, McGraw- [2] K. Yamawaki, in: Seoul 1997, QCD, light-cone physics Hill, New York (1954). and hadron phenomenology 116-199, hep-th/9802037, [9] R.C. Brower, W.L. Spence and J.H. Weis, Phys. Rev.D and references therein. 19, 3024 (1979). , [3] K. Itakura and S. Maedan, Progr. Theor. Phys. 97, 635 [10] L.Debnath,IntegralTransformsandTheirApplications, (1997). CRC Press, Boca Raton, New York, London, Tokyo [4] F. Lenz, M. Thies, and K. Yazaki, Phys. Rev. D 63, (1995). 045018 (2001). [11] V.A.Miransky,DynamicalSymmetryBreakinginQuan- [5] G. ’t Hooft, Nucl. Phys. B 75, 461 (1974). tum Field Theories, World Scientific, Singapore (1993). [6] A.R.Zhitnitsky,Phys.Lett. B 165, 405 (1985). [7] M. Burkardt,Phys. Rev.D 53, 933 (1996).