Talk given at the Twelfth Workshop on Non-Perurbative Quantum Chromodynamics 10-13 June 2013
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QCD, Wilson loop and the interquark potentialMarco FrascaTwelfth Workshop on Non-Perturbative QuantumChromodynamics, Paris, June 10-13, 2013
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Plan of the TalkPlan of the talkClassical ﬁeld theoryScalar ﬁeld theoryYang-Mills theoryYang-Mills Green functionQuantum ﬁeld theoryScalar ﬁeld theoryYang-Mills theoryQCD in the infrared limitWilson loopInterquark potentialConclusions
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Classical ﬁeld theoryScalar ﬁeldA classical ﬁeld theory for a massless scalar ﬁeld is given byφ+λφ3=jThe homogeneous equation can be solved byExact solutionφ=µ( 2λ )14 sn(p·x+θ,i) p2=µ2 λ2being sn an elliptic Jacobi function and µ and θ two constants. Thissolution holds provided the given dispersion relation holds andrepresents a free massive solution notwithstanding we started from amassless theory.Mass arises from the nonlinearities when λ is taken to be ﬁnite.
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Classical ﬁeld theoryScalar ﬁeldWhen there is a current we ask for a solution in the limit λ→∞ as ouraim is to understand a strong coupling limit. So, we check a solutionφ=κ d4x G(x−x )j(x )+δφbeing δφ all higher order corrections.One can prove that this is indeed so providedNext-to-leading Order (NLO)δφ=κ2λ d4x d4x G(x−x )[G(x −x )]3j(x )+O(j(x)3)with the identiﬁcation κ=µ, the same of the exact solution, andG(x−x )+λ[G(x−x )]3=µ−1δ4(x−x ).The correction δφ is known in literature and yields a sunrise diagramin momenta. This needs a regularization.Our aim is to compute the propagator G(x−x ) to NLO.
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Classical ﬁeld theoryScalar ﬁeldIn order to solve the equationG(x−x )+λ[G(x−x )]3=µ−1δ4(x−x )we can start from the d=1+0 case ∂2t G0(t−t )+λ[G0(t−t )]3=µ2δ(t−t ).It is straightforwardly obtained the Fourier transformed solutionG0(ω)= ∞n=0(2n+1) π2K2(i)(−1)ne−(n+ 12 )π1+e−(2n+1)π1ω2−m2n+ibeing mn=(2n+1) π2K(i) (λ2 )14 µ and K(i)=1.311028777... an elliptic integral.We are able to recover the fully covariant propagator by boostingfrom the rest reference frame obtaining ﬁnallyG(p)= ∞n=0(2n+1) π2K2(i)(−1)ne−(n+ 12 )π1+e−(2n+1)π1p2−m2n+i.
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Classical ﬁeld theoryYang-Mills ﬁeldA classical ﬁeld theory for the Yang-Mills ﬁeld is given by∂µ∂µAaν −(1− 1ξ )∂ν (∂µAaµ)+gf abc Abµ(∂µAcν −∂ν Acµ)+gf abc ∂µ(AbµAcν )+g2f abc f cde AbµAdµAeν =−jaν .For the homogeneous equations, we want to study it in the formallimit g → ∞. We note that a class of exact solutions exists if we takethe potential Aaµ just depending on time, after a proper selection ofthe components [see Smilga (2001)]. These solutions are thesame of the scalar ﬁeld when spatial coordinates are set to zero(rest frame).Diﬀerently from the scalar ﬁeld, we cannot just boost away thesesolutions to get a general solution to Yang-Mills equations due togauge symmetry. But we can try to ﬁnd a set of similar solutionswith the proviso of a gauge choice.This kind of solutions will permit us to prove that a set of themexists supporting a trivial infrared ﬁxed point to build up aquantum ﬁeld theory.
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Classical ﬁeld theoryYang-Mills ﬁeldExactly as in the case of the scalar ﬁeld we assume the followingsolution to our ﬁeld equationsAaµ=κ d4x Dabµν (x−x )jbν (x )+δAaµAlso in this case, apart from a possible correction, this boils down toan expansion in powers of the currents as already guessed in the’80 [R. T. Cahill and C. D. Roberts, Phys. Rev. D 32, 2419 (1985)].This implies that the corresponding quantum theory, in a very strongcoupling limit, takes a Gaussian form and is trivial.The crucial point, as already pointed out in the eighties [T. Goldmanand R. W. Haymaker, Phys. Rev. D 24, 724 (1981), T. Cahill and C.D. Roberts (1985)], is the determination of the gluon propagatorin the low-energy limit.
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Classical ﬁeld theoryYang-Mills ﬁeldThe question to ask is: Does a set of classical solutions exist forYang-Mills equations supporting a trivial infrared ﬁxed point for thecorresponding quantum theory?The answer is yes! These solutions are instantons in the formAaµ = ηaµφ with ηaµ a set of constants and φ a scalar ﬁeld.By direct substitution into Yang-Mills equations one recovers theequation for φ that is∂µ∂µφ− 1N2−11− 1ξ(ηa·∂)2φ+Ng2φ3=−jφbeing jφ=ηaµjµa and use has been made of the formula ηνaηaν =N2−1.In the Landau gauge (Lorenz gauge classically) this equation isexactly that of the scalar ﬁeld given before and we get again a currentexpansion.So, a set of solutions of the Yang-Mills equations existssupporting a trivial infrared ﬁxed point. Our aim is to study thetheory in this case.
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Classical ﬁeld theoryYang-Mills-Green functionThe instanton solutions given above permit us to write downimmediately the propagator for the Yang-Mills equations in theLandau gauge for SU(N) being exactly the same given for the scalarﬁeld:Gluon propagator in the infrared limit∆abµν (p)=δab ηµν −pµpνp2∞n=0Bnp2−m2n+i+O 1√NgbeingBn=(2n+1) π2K2(i)(−1)n+1e−(n+ 12 )π1+e−(2n+1)πandmn=(2n+1) π2K(i)Ng2214ΛΛ is an integration constant as µ for the scalar ﬁeld.This is the propagator of a massive ﬁeld theory
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Quantum ﬁeld theoryScalar ﬁeldWe can formulate a quantum ﬁeld theory for the scalar ﬁeld startingfrom the generating functionalZ[j]= [dφ] exp[i d4x(12(∂φ)2−λ4φ4+jφ)].We can rescale the space-time variable as x →√λx and rewrite thefunctional asZ[j]= [dφ] exp[i 1λd4x(12(∂φ)2−14φ4+ 1λjφ)].Then we can seek for a solution series as φ= ∞n=0 λ−nφn and rescale thecurrent j → j/λ being this arbitrary.The leading order correction can be computed solving the classicalequationφ0+φ30=jthat we already know how to manage. This is completely consistentwith our preceding formulation [M. Frasca, Phys. Rev. D 73, 027701(2006)] but now all is fully covariant.
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Quantum ﬁeld theoryScalar ﬁeldUsing the approximation holding at strong couplingφ0=µ d4xG(x−x )j(x )+...it is not diﬃcult to write the generating functional at the leadingorder in a Gaussian formZ0[j]=exp[ i2d4x d4x j(x )G(x −x )j(x )].This conclusion is really important: It says that the scalar ﬁeld theoryin d=3+1 is trivial in the infrared limit!This functional describes a set of free particles with a mass spectrummn=(2n+1) π2K(i) (λ2 )14 µthat are the poles of the propagator, the same of the classical theory.
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Quantum ﬁeld theoryYang-Mills ﬁeldWe can now take the form of the propagator given above, e.g. in theLandau gauge, asDabµν (p)=δab ηµν −pµpνp2∞n=0Bnp2−m2n+i+O 1√Ngto do a formulation of Yang-Mills theory in the infrared limit.Then, the next step is to use the approximation that holds in a strongcoupling limitAaµ=Λ d4x Dabµν (x−x )jbν (x )+O 1√Ng+O(j3)and we note that, in this approximation, the ghost ﬁeld just decouplesand becomes free and one ﬁnally has at the leading orderZ0[j]=N exp[ i2d4x d4x jaµ(x )Dabµν (x −x )jbν (x )].Yang-Mills theory has an infrared trivial ﬁxed point in the limit of thecoupling going to inﬁnity and we expect the running coupling to go tozero lowering energies. So, the leading order propagator cannotconﬁne.
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Quantum ﬁeld theoryYang-Mills ﬁeldNow, we can take a look at the ghost part of the action. We justnote that, for this particular form of the propagator, inserting ourapproximation into the action produces an action for a free ghost ﬁeld.Indeed, we will haveSg =− d4x ¯ca∂µ∂µca+O 1√Ng+O(j3)A ghost propagator can be written down asGab(p)=−δabp2+i+O 1√Ng.Our conclusion is that, in a strong coupling expansion 1/√Ng, we getthe so called decoupling solution.
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Quantum ﬁeld theoryYang-Mills ﬁeldA direct comparison of our results with numerical Dyson-Schwingerequations gives the following:that is strikingly good (ref. A. Aguilar, A. Natale, JHEP 0408, 057(2004)).
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Quantum ﬁeld theoryQCD at infrared limitWhen use is made of the trivial infrared ﬁxed point, QCD action canbe written down quite easily.Indeed, we will have for the gluon ﬁeldSgf =12d4x d4x jµa(x )Dabµν (x −x )jνb(x )+O 1√Ng+O(j3)and for the quark ﬁeldsSq= q d4x¯q(x) i /∂−mq−gγµ λa2d4x Dabµν (x−x )jνb(x )−g2γµ λa2d4x Dabµν (x−x ) q ¯q (x ) λb2γν q (x )+O 1√Ng+O(j3) q(x)We recognize here an explicit Yukawa interaction and aNambu-Jona-Lasinio non-local term. Already at this stage we are ableto recognize that NJL is the proper low-energy limit for QCD.
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Quantum ﬁeld theoryQCD at infrared limitNow we operate the Smilga’s choice ηaµηbν = δab(ηµν − pµpν/p2) forthe Landau gauge.We are left with the infrared limit QCD using conservation of currentsSgf =12d4x d4x jaµ(x )∆(x −x )jµa(x )+O 1√Ng+O(j3)and for the quark ﬁeldsSq= q d4x¯q(x) i /∂−mq−gγµ λa2d4x ∆(x−x )jaµ(x )−g2γµ λa2d4x ∆(x−x ) q ¯q (x ) λa2γµq (x )+O 1√Ng+O(j3) q(x)We want to give explicitly the contributions from gluon resonances.In order to do this, we introduce the bosonic currents jaµ(x) = ηaµj(x)with the current j(x) that of the gluonic excitations.
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Quantum ﬁeld theoryQCD at infrared limitUsing the relation ηaµηµa=3(N2c −1) we get in the endSgf =32(N2c −1) d4x d4x j(x )∆(x −x )j(x )+O 1√Ng+O(j3)and for the quark ﬁeldsSq= q d4x¯q(x) i /∂−mq−gηaµγµ λa2d4x ∆(x−x )j(x )−g2γµ λa2d4x ∆(x−x ) q ¯q (x ) λa2γµq (x )+O 1√Ng+O(j3) q(x)Now, we recognize that the propagator is just a sum of Yukawapropagators weighted by exponential damping terms. So, we introducethe σ ﬁeld and truncate at the ﬁrst excitation. This is a somewhatrough approximation but will be helpful in the following analysis.This means the we can write the bosonic currents contribution ascoming from a boson ﬁeld and written down asσ(x)=√3(N2c −1)/B0 d4x ∆(x−x )j(x ).
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Quantum ﬁeld theoryQCD at infrared limitSo, low-energy QCD yields a NJL model as given in [M. Frasca, PRC84, 055208 (2011)]Sσ= d4x[12(∂σ)2−12m20σ2]and for the quark ﬁeldsSq= q d4x¯q(x) i /∂−mq−gB03(N2c −1)ηaµγµ λa2σ(x)−g2γµ λa2d4x ∆(x−x ) q ¯q (x ) λa2γµq (x )+O 1√Ng+O(j3) q(x)Now, we obtain directly from QCD (2G(0) = G is the standard NJLcoupling)G(p)=−12g2 ∞n=0Bnp2−(2n+1)2(π/2K(i))2σ+i=G2C(p)with C(0) = 1 ﬁxing in this way the value of G using the gluonpropagator. This yields an almost perfect agreement with the case ofan instanton liquid (see Ref. in this page).
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Quantum ﬁeld theoryWilson loopLow-energy QCD, being at infrared ﬁxed point, is not conﬁning (NJLmodel is not conﬁning). This agrees with the analysis given in [P.Gonz´alez, V. Mathieu, and V. Vento, PRD 84, 114008 (2011)] for thedecoupling solution of the propagators in the Landau gauge. Indeed,one hasW [C]=exp −g22C2(R) d4p(2π)4 ∆(p2) ηµν −pµpνp2 C dxµC dyν e−ip(x−y) .For the decoupling solution (at infrared ﬁxed point) one hasW [C]≈exp −T g22C2(R) d3p(2π)3 ∆(p,0)e−ip·x =exp[−TVYM (r)]The potential is (assuming a ﬁxed point value for g in QCD)VYM (r)=−C2(R)g22∞n=0(2n+1) π2K2(i)(−1)ne−(n+ 12 )π1+e−(2n+1)πe−mnrrand due to massive excitations one gets a screened but not conﬁningpotential. This agrees very well with Gonz´alez&al.
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Quantum ﬁeld theoryWilson loopThe leading order of the gluon propagator, as also emerging fromlattice computations, is insuﬃcient to give reason for conﬁnement.We need to compute the sunrise diagram going to NLO:∆R (p2)−∆(p2)=λ 1µ2 ∆(p2)d4p1(2π)4d4p2(2π)4 n1,n2,n3Bn1p21−m2n1Bn2p22−m2n2Bn3(p−p1−p2)2−m2n3.This integral is well-known [Caﬀo&al. Nuovo Cim. A 111, 365(1998)] At small momenta will yieldField renormalization factorZφ(p2)=1− 1λ1227π8 + 1λ3.3·48π8 1+ 316p2µ2 +O λ− 32 .This implies for the gluon propagator (λ=C2(R)g2, Z0=Zφ(0))Dabµν (p2)=δab ηµν −pµpνp2∞n=0Z−10Bnp2+ 1λ3.3·9π8p4µ2 +m2n(p2)+O λ− 32
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Quantum ﬁeld theoryWilson loopWe note thatm2n(p2)=m2n(0) Z0+ 1λ3.3·9π8p2µ2 +O λ− 32that provides very good agreement with the scenario by Dudal&al.obtained by postulating condensates. Here we have an existenceproof. Masses run with momenta.This correction provides the needed p4 Gribov contribution to thepropagator to get a linear term in the potential.Now, from Wilson loop, we have to evaluateVYM (r)=−g22C2(R) d3p(2π)3 ∆R (p,0)e−ip·x.beingD abµν (p2)=δab ηµν −pµpνp2 ∆R (p2)the renormalized propagator.
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Quantum ﬁeld theoryInterquark potentialSo,VYM (r)=− g28πrC2(R)Z−10∞n=0 Bn∞−∞ dpp sin(pr)p2+ 1λ3.3·9π8p4µ2 +m2n(p2).We rewrite it asVYM (r)≈− g28πrC2(R)Z−10π8λµ23.3·9∞−∞ dpp sin(pr)(p2+κ2)2−κ4being κ2=π8λµ23.3·9, neglecting running masses that go like√λ.Finally, for κr 1, this yields the well-known linear contribution:VYM (r)≈−g28rC2(R)e− κ√2rsinh κ√2r ≈− g28πC2(R) π√2κ−π2κ2r+O((κr)2) .
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Quantum ﬁeld theoryInterquark potentialFrom the given potential it is not diﬃcult to evaluate the stringtension, similarly to what is done in d=2+1 for pure Yang-Millstheory.The linear rising term givesσ=π4g24πC2(R)κ2.Remembering that λ=d(R)g2,String tension for SU(N) in d=3+1:√σ ≈π9211g2 C2(R)d(R)4πµthat compares really well with the case in d=2+1 [D. Karabali,V. P. Nair and A. Yelnikov, Nucl. Phys. B 824, 387 (2010)] being√σd=2+1≈g2 C2(R)d(R)4π.
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ConclusionsConclusionsWe provided a strong coupling expansion both for classical andquantum ﬁeld theory for scalar ﬁeld and QCD.A low-energy limit of QCD is so obtained that reduces to anon-local Nambu-Jona-Lasinio model with all the parameters andthe form factor properly ﬁxed.We showed how the leading order for the gluon propagator is notconﬁning and we need to compute Next-to-Leading Orderapproximation given by a sunrise diagram.Next-to-Leading Order correction provides the p4 Gribovcontribution granting a conﬁning potential.String tension can be computed and appears to be consistent withexpectations from d=2+1 case.Helpful discussions with Marco Ruggieri are gratefully acknowl-edged.
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