QCD and the Nambu Jona-Lasinio Model

Transcription

1 Lecture 1 QCD and the Nambu Jona-Lasinio Model Ian Cloët The University of Adelaide & Argonne National Laboratory CSSM Summer School Non-perturbative Methods in Quantum Field Theory 11 th 15 th February 2013

2 Building Blocks of the Universe Fundamental constituents of the Standard Model (SM) of particle physics Quantum Chromodynamics (QCD) & Electroweak (EW) theories Local non-abelian gauge field theories a special type of relativistic quantum field theory SM Lagrangian has gauge symmetries: SU(3) c SU(2) L U Y (1) SM has 19 parameters which need to be determined by experiment however only 2 parameters are intrinsic to QCD: Λ QCD & θ QCD 10 9 frontpage table of contents appendices 2 / 31

3 Motivation of Lectures Explore non-perturbative structure of QCD, through the interplay of theory and experiment, as it relates to hadron and nuclear structure The tools available are: lattice QCD chiral perturbation theory QCD inspired models We will review the model of Nambu and Jona-Lasinio (NJL model) first proposed in 1961 as a theory of elementary nucleons with advent of QCD reinterpreted as a quark effective theory Some of the advantages of models over lattice and χpt are can explore a wider array of physics problems may provide better insight into important physics mechanisms facilitate a dynamic interplay between experiment and theory frontpage table of contents appendices 3 / 31

4 Plan of Lectures Lecture 1 Introduction to QCD and the non-perturbative framework provided by the Nambu Jona-Lasinio (NJL) model Lecture 2 Relativistic Faddeev (3-body) equation & electromagnetic form factors Lecture 3 Deep inelastic scattering and parton distribution functions Lecture 4 Quark degrees of freedom in nuclei and nuclear matter frontpage table of contents appendices 4 / 31

5 Recommended References Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity I, Phys. Rev. 122, 345 (1961). Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity II, Phys. Rev. 124, 246 (1961). U. Vogl and W. Weise, The Nambu and Jona Lasinio model: Its implications for hadrons and nuclei, Prog. Part. Nucl. Phys. 27, 195 (1991). S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64, 649 (1992). R. L. Jaffe, Deep Inelastic Scattering With Application To Nuclear Targets, MIT-CTP W. Bentz and A. W. Thomas, The Stability of nuclear matter in the Nambu-Jona-Lasinio model, Nucl. Phys. A 696, 138 (2001). I. C. Cloët, W. Bentz and A. W. Thomas, EMC and polarized EMC effects in nuclei, Phys. Lett. B 642, 210 (2006). I. C. Cloët, W. Bentz and A. W. Thomas, Isovector EMC effect explains the NuTeV anomaly, Phys. Rev. Lett. 102, (2009). frontpage table of contents appendices 5 / 31

6 Quantum Chromodynamics (QCD) QCD is the fundamental theory of the strong interaction, where the quarks and gluons are the basic degrees of freedom colour A = 1, 2, 3 { colour a = 1,...,8 (q α ) A f spin α =, A a µ spin ε ± µ flavour f = u, d, s, c, b, t QCD is a non-abelian gauge theory whose dynamics are governed by the Lagrangian L = q f ( i /D +m f ) qf 1 4 Fa µνf µν a ; i/d = γ µ( i µ +g s A a µt a) F a µν = µ A a ν ν A a µ +g s f abc A b µa c ν µ, a b, ν q a, µ b, ν β, B α, A a, µ p k c, σ c, σ d, λ Gluon self-interactions have many profound consequences frontpage table of contents appendices 6 / 31

7 Asymptotic Freedom At large Q 2 or short distances interaction strength becomes logarithmically small a striking features of QCD QED has opposite behaviour: α e α LO s (Q 2 ) = ( N f 4π ) ( ln Q 2 /Λ 2 QCD ) Asymptotic Freedom 2004 Nobel Prize Gross, Politzer and Wilczek Λ QCD most important parameter in QCD [dimenional transmutation of g s ] Λ QCD 200 MeV 1 fm 1 sets scale, QCDs standard kilogram Momentum-dependent coupling coupling depends on separation interaction strength between quarks and gluons grows with separation frontpage table of contents appendices 7 / 31

8 Confinement Hadron structure & QCD is characterized by two emergent phenomena confinement and dynamical chiral symmetry breaking (DCSB) Both of these phenomena are not evident from the QCD Lagrangian All known hadrons are colour singlets, even though they are composed of coloured quarks and gluons: baryons (qqq) & mesons ( qq) Confinement conjecture: particles that carry the colour charge cannot be isolated and can therefore not be directly observed Related to $1 million Millennium Prize: for SU(3) c must prove that glueballs have a lower bound on their mass partial explanation as to why strong force is short ranged frontpage table of contents appendices 8 / 31

9 Chiral Symmetry Define left- and right-handed fields: ψ R,L = 1 2 (1±γ 5)ψ The QCD Lagrangian then takes the form [m = diag(m u,m d,m s,...)] L = ψ L i/dψ L + ψ R i/dψ R ψ R mψ L ψ L mψ R 1 4 Fa µνf µν a Therefore for m = 0 QCD Lagrangian is chirally symmetric SU(N f ) L SU(N f ) R = ψ L,R e iωa L,R Ta ψ L,R SU(N f ) L SU(N f ) R chiral symmetry is equivalent to SU(N f ) V SU(N f ) A = ψ e iωa V Ta ψ, ψ e iωa A Ta γ 5 ψ Global symmetries: Wigner-Weyl or Nambu-Goldstone modes Wigner-Weyl mode: vacuum is also invariant Nambu-Goldstone mode: vacuum breaks symmetry frontpage table of contents appendices 9 / 31

10 Dynamical Chiral Symmetry Breaking Recall for m = 0 QCD Lagrangian is invariant under SU(N f ) L SU(N f ) R SU(N f ) V SU(N f ) A N f = 2 corresponds to the isospin subgroup of SU(N f ) V transformations hadronic mass spectrum tells us nature largely respects isospin symmetry m π m π 0 m π +, m p m n, m Σ m Σ 0 m Σ + therefore SU(N f ) V is realized in the Wigner-Weyl mode SU(N f ) A transformations mix states of opposite parity expect hadronic mass spectrum to exhibit parity degeneracy m σ m π 300 MeV, m a1 m ρ 490 MeV, m N m N 600 MeV, etc recall: m u m d 5 MeV = cannot produce large mass splittings therefore SU(N f ) A must be realized in the Nambu-Goldstone mode Chiral symmetry broken dynamically: SU(N f ) L SU(N f ) R = SU(N f ) V frontpage table of contents appendices 10 / 31

11 Goldstone s Theorem Goldstone s theorem: if a continuous global symmetry is broken dynamically, then for each broken group generator there must appear in the theory a massless spinless particle (Goldstone boson) QCDs chiral symmetry is explicitly broken by small current quark masses m u = MeV & m d = MeV ( Λ QCD ) For N f = 2 expect N 2 f 1 = 3 Goldstone bosons: π+, π 0, π physical particle masses are not zero m π 140 MeV because of explicit chiral symmetry breaking: m u,d 0 Chiral symmetry and its dynamical breaking has profound consequences for the QCD mass spectrum and hadron structure this is not apparent from the QCD Lagrangian and is an innately non-perturbative phenomena Need non-perturbative methods to understand all consequences of QCD frontpage table of contents appendices 11 / 31

12 QCDs Dyson Schwinger Equations DSEs are the equations of motion for a quantum field theory must truncate infinite tower of coupled integral equations Truncation: gluon propagator becomes constant D µν (k) g µν Largely equivalent to the Nambu Jona Lasinio (NJL) model ETC! frontpage table of contents appendices 12 / 31

13 The Nambu Jona-Lasinio Model The Nambu Jona-Lasinio (NJL) Model was invented in 1961 by Yoichiro Nambu and Giovanni Jona-Lasinio while at The University of Chicago was inspired by the BCS theory of superconductivity was originally a theory of elementary nucleons rediscovered in the 80s as an effective quark theory It is a relativistic quantum field theory, that is relatively easy to work with, and is very successful in the description of hadrons, nuclear matter, etc Nambu won half the 2008 Nobel prize in physics in part for the NJL model: for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics [Nobel Committee] frontpage table of contents appendices 13 / 31

14 NJL Model NJL model is interpreted as low energy chiral effective theory of QCD G Θ(k2 Λ2) Z(k 2 ) 50 ω = 0.4 Can be motivated by infrared enhancement of quark gluon interaction e.g. DSEs and Lattice QCD αeff(k 2 )/k 2 (GeV 2 ) ω = 0.5 α s (k 2 )/k 2 A. Holl, et al, Phys. Rev. C 71, (2005) Investigate the role of quark degrees of freedom NJL has same flavour symmetries as QCD frontpage table of contents appendices 14 / k 2 (GeV 2 ) NJL is non-renormalizable = cannot remove regularization parameter

15 NJL Lagrangian In general the NJL Lagrangian has the form L NJL = L 0 +L I = ψ ( i/ m ) ψ + α G α ( ψγα ψ ) 2 Γ α represents a product of Dirac, colour and flavour matrices What about L I? effective theories should maintain symmetries of QCD In chiral limit QCD Lagrangian has symmetries S QCD = SU(3) c SU(N f ) V SU(N f ) A U(1) V U(1) A C P T SU(N f ) A is broken dynamically DCSB U(1) A is broken in the anomalous mode U(1) problem massive η NJL interaction Lagrangian must respect the symmetries S NJL = SU(3) c SU(N f ) V SU(N f ) A U(1) V C P T in NJL SU(3) c will be considered a global gauge symmetry U(1) A is often broken explicitly = m η 0 frontpage table of contents appendices 15 / 31

16 NJL Lagrangian (2) S NJL = SU(3) c SU(N f ) V SU(N f ) A U(1) V C P T The NJL Lagrangian should be symmetric under the transformations SU(N f ) V : ψ e it θ V ψ SU(N f ) A : ψ e iγ 5t θ A ψ U(1) V : ψ e iθ ψ U(1) A : ψ e iγ5θ ψ ψ it θ ψe V ψ iγ ψe 5 t θ A ψ iθ ψe iγ ψ ψe 5 θ Nambu and Jona-Lasinio choose the Lagrangian L = ψ ( i/ m ) ψ +G π [ ( ψψ ) 2 ( ψγ5 τ ψ ) 2 ] Can choose any combination of these 4 fermion interactions ( ψψ ) 2, ( ψγ5 ψ ) 2, ( ψγ µ ψ ) 2 ( ψγ µ γ 5 ψ ) 2, ( ψiσ µν ψ ) 2, ( ψtψ ) 2, ( ψγ5 tψ ) 2, ( ψγ µ tψ ) 2, ( ψγ µ γ 5 tψ ) 2, ( ψiσ µν tψ ) 2. frontpage table of contents appendices 16 / 31

17 NJL Lagrangian (3) The most general N f = 2 NJL Lagrangian that respects the symmetries is L = ψ ( i/ m ) [ ) 2 ψ +G π ( ψψ ( ψγ5 τ ψ ) ] 2 +G ω ( ψγ µ ψ ) [ 2 +Gρ ( ψγ µ τ ψ ) 2 + ( ψγ µ γ 5 τ ψ ) ] 2 +G h ( ψγ µ γ 5 ψ ) [ 2 +Gη ( ψγ5 ψ ) 2 ) ] [ 2 ( ψτ ψ +G T ( ψiσ µν ψ ) 2 ( ψiσ µν τ ψ ) ] 2 L I is U(1) A invariant if: G π = G η & G T = 0 ψψ σ ψγ 5 τ ψ π ψγ µ ψ ω ψγ µ τ ψ ρ ψγ µ γ 5 ψ f 1,h 1 ψγ µ γ 5 τ ψ a 1 ψτψ a 0 ψγ 5 ψ η,η ( J P,T ) = ( 0 +,0 ) ( J P,T ) = ( 0,1 ) ( J P,T ) = ( 1,0 ) ( J P,T ) = ( 1,1 ) ( J P,T ) = ( 1 +,0 ) ( J P,T ) = ( 1 +,1 ) ( J P,T ) = ( 0 +,1 ) ( J P,T ) = ( 0,0 ) frontpage table of contents appendices 17 / 31

18 NJL Lagrangian (4) The most general N f = 2 NJL Lagrangian that respects the symmetries is [ L I = 1 2 G ) 2 π ( ψψ ( ψγ5 τ ψ ) ] G ω ( ψγ µ ψ ) [ G ρ ( ψγ µ τ ψ ) 2 ( ψγ µ γ 5 τ ψ ) ] G f ( ψγ µ γ 5 ψ ) [ G η ( ψγ5 ψ ) 2 ) ] [ 2 ( ψτ ψ 1 2 G T ( ψiσ µν ψ ) 2 ( ψiσ µν τ ψ ) ] 2 L I is U(1) A invariant if: G π = G η & G T = 0 The most general N f = 3 NJL Lagrangian that respects the symmetries is L I = G π [ 1 6 ( ψψ ) 2 + ( ψtψ ) ( ψγ5 ψ ) 2 ( ψγ5 tψ ) 2 ] 1 2 G ρ[ ( ψγ µ tψ ) 2 + ( ψγ µ γ 5 tψ ) 2 ] 1 2 G ω ( ψγ µ ψ ) G f ( ψγ µ γ 5 ψ ) 2 Enlarging the SU(N f ) V SU(N f ) A chiral group from N f = 2 to N f = 3 reduces the number of coupling from six to four The N f = 3 Lagrangian is automatically U(1) A invariant U(1) A is then often broken by the t Hooft term a 6-quark interaction L (6) I = K [ det ( ψ(1+γ5 )ψ ) +det ( ψ(1 γ5 )ψ )] frontpage table of contents appendices 18 / 31

19 NJL Interaction Kernel Using Wick s theorem and the NJL Lagrangian their are 2 diagrams for the interaction between a quark and an anti-quark γ α δ α 2iG [ ] Ω i αβ Ωi γδ Ω i αδ Ωi γβ δ Direct β Time γ Exchange β Using Fierz transformations can express each exchange term as a sum of direct terms The SU(2) NJL interaction kernel then takes the form K αβ,γδ = 2iG π [ (½) αβ (½) γδ (γ 5 τ) αβ (γ 5 τ) γδ ] 2iG ω (γ µ ) αβ (γ µ ) γδ 2iG ρ [ (γ µ τ) αβ (γ µ τ) γδ +(γ µ γ 5 τ) αβ (γ µ γ 5 τ) γδ ]+... This kernel enters the NJL gap and meson Bethe-Salpeter equations frontpage table of contents appendices 19 / 31

20 Regularization Schemes The NJL model is non-renormalizable = cannot remove regularization regularization parameter(s) play a dynamical role Popular choices are: 3-momentum cutoff: p 2 < Λ 2 4-momentum cutoff p 2 E < Λ2 Pauli-Villars We will use the proper-time regularization scheme 1 X n = 1 (n 1)! 0 dτ τ n 1 e τ X 1 (n 1)! 1/Λ 2 IR 1/Λ 2 UV dτ τ n 1 e τ X only Λ UV is need to render the theory finite however, as we shall see, Λ IR plays a very important role; it prevents quarks going on their mass shell and hence simulates confinement frontpage table of contents appendices 20 / 31

21 NJL Quark Propagator Complete expression for the quark propagator cannot be obtained need a truncation Do not in include diagrams like: would give a momentum dependent mass function Include all diagrams of the form: = All these diagrams can be summed via an integral equation 1 = 1 + The most general quark propagator has the form S(p) = 1 /p m Σ(p) = Z(p2 ) /p M(p 2 ) frontpage table of contents appendices 21 / 31

22 NJL Gap Equation The NJL gap equation has the form 1 S 1 (k) = S 1 0 (k) Σ(k) = [ /k m] j = 1 + d 4 l [ (2π) 4Tr S(l)Ω j] Ω j The only piece of the interaction kernel that contributes is: K σ αβ,γδ = 2iG π(½) γδ (½) αβ Solving this equation give a quark propagator of the form S 1 (k) = /k M +iε The constituent quark mass satisfies the equation d 4 l 1 M = m+48ig π M (2π) 4 l 2 M 2 +iε = m+m 3G π π 2 dτ e τ M2 τ 2 frontpage table of contents appendices 22 / 31

23 NJL Gap Equation (2) M = m+m 3G π π 2 dτ e τ M2 τ 2 For the case m = 0 the gap equation has two solutions: trivial solution: M = 0 & non-trivial solution: M 0 Which solution does nature choose, that is, which solution minimizes the energy. Compare vacuum energy density, E, for each case E(M) E(M = 0) = 3 4π 2 dτ 1 ( τ 3 e τm2 1 )+ M2 4G π E [GeV 4 ] G π = 5 GeV 2 G π = G π,crit G π = 20 GeV 2 G π = 30 GeV M [GeV] For G π > G π,crit the lowest energy solution has M 0 Therefore for G π > G π,crit NJL has DCSB DCSB generates mass from nothing frontpage table of contents appendices 23 / 31

24 NJL & DSE gap equations Dynamical Quark Mass (MeV) m q = 0MeV m q = 5MeV m q = 50MeV G/G crit NJL constituent mass is given by: M = m 2G π ψψ Chiral condensate is defined by ψψ lim x y Tr[ is(x y)] = d 4 k (2π) 4 Tr[iS(k)] Mass is generated via interaction with vacuum Dynamically generated quark masses ψψ 0 ψψ ūu+ = dd is an order parameter for DSCB frontpage table of contents appendices 24 / 31

25 Confinement in NJL model In general the NJL model is not confining; quark propagator is simply S(k) = 1 /k M +iε = /k +M k 2 M 2 +iε quark propagator has a pole = quarks are part of physical spectrum However the proper-time scheme is unique S(k) = 0 [ dτ (/k +M)e τ (k 2 M 2 e ) (k2 M 2 )/Λ 2 UV e (k2 M 2 )/Λ 2 ] IR [/k +M] k 2 M }{{ 2 } Z(k 2 ) quark propagator does not have a pole: Z(k 2 ) k2 M 2 = 1 Λ 2 IR 1 Λ 2 UV Are confinement and DCSB related? NJL model is proof that DCSB can exist without confinement however commonly believed cannot have confinement without DCSB frontpage table of contents appendices 25 / 31

26 Hadron Spectrum In QFT physical states appear as poles in n-point Green Functions For example, the full quark antiquark scattering matrix or T -matrix, contains poles for all qq bound states, that is, the physical mesons The quark antiquark T -matrix is obtained by solving the Bethe-Salpeter equation T = K + T K In principle kernel, K, contains all possible 2PI diagrams K = NJL = Kernel of gap and BSEs are intimately related q µ Γ µ,i 5 (p,p) = S 1 (p )γ τ i τ iγ 5 S 1 (p)+2mγ i π(p,p) frontpage table of contents appendices 26 / 31

27 Bethe-Salpeter Equation for the Pion How does the pion become (almost) massless when it is composed of two massive constituents The pion is realized as the lowest lying pole in the quark anti-quark T -matrix in the pseudo-scalar channel In the NJL model this T -matrix is given by d 4 k T (q) αβ,γδ = K αβ,γδ + (2π) 4 K αβ,λǫs(q +k) εε S(k) λ λt (q) ε λ,γδ, γ δ q α β = γ δ α β + γ δ ε λ ε λ α β K π = 2iG π (γ 5 τ) αβ (γ 5 τ) λǫ The NJL pion t-matrix is T (q) i αβ,γδ = (γ 5τ i ) αβ 2iG π 1+2G π Π π (q 2 ) (γ 5τ i ) γδ The pion mass is then given by: 1+2G π Π π (q 2 = m 2 π) = 0 frontpage table of contents appendices 27 / 31

28 The Pion as a Goldstone Boson Recall the pion pole condition 1+2G π Π π (q 2 = m 2 π) = 0 where Π π (q 2 ) = m 2G π M 1 q 2 I(q 2 ) 2G π have used the gap equation to obtain this expression The pion pole condition therefore implies m 2 π = m 2G π M I(m 2 π) Therefore in the chiral limit m 0 (M 0) pion is massless Can show other chiral symmetry relations are also satisfied: f π g πqq = M g Aqq Goldberger Treiman (GT) relation f 2 π m 2 π = 1 2 (m u +m d ) ūu+ dd Gell-Mann Oakes Renner (GMOR) α α µ 0 A µ q β β q a π b (p) = if π p µ δ ab frontpage table of contents appendices 28 / 31

29 Chiral Partners If chiral symmetry was NOT dynamically broken in nature expect mass degenerate chiral partners, e.g., m σ m π & m a1 m ρ The ρ and a 1 are the lowest lying vector (J P = 1 ) and axial-vector (J P = 1 + ) qq bound states: m exp t ρ 770 MeV & m exp t a MeV Masses given by T -matrix poles in the vector and axial-vector qq channels q = + q K = 2iG ρ [ (γµ τ)(γ µ τ) +(γ µ γ 5 τ)(γ µ γ 5 τ) ] Pole conditions: 1+2G ρ Π ρ (m 2 ρ) = 0 & 1+2G ρ Π a1 (m 2 a 1 ) = 0 Π a1 (q 2 ) = M 2 I(q 2 )+Π ρ (q 2 ) DCSB splits masses; NJL gives: m ρ 770 MeV & m a MeV good agreement with the Weinberg sum rule result: m a1 2m ρ Pion conditions for π and σ = m 2 σ m 2 π +4M 2 frontpage table of contents appendices 29 / 31

30 Homogeneous Bethe-Salpeter vertex functions T = K + T K = Γ = Γ K Near a bound state pole of mass m a two-body t-matrix behaves as T (p,k) Γ(p,k) Γ(p,k) p 2 m 2 where p = p 1 +p 2, k = p 1 p 2 Γ(p,k) is the homogeneous Bethe-Salpeter vertex and describes the relative motion of the quark and anti-quark while they form the bound state Expanding the pion T -matrix about the pole gives T = γ 5 τ i 2iG π 1+2G π Π π (q 2 ) γ 5τ i ig2 πqq q 2 m 2 π(γ 5 τ i )(γ 5 τ i ) = Γ π = g π γ 5 τ i g πqq is effective pion-quark coupling constant Bethe-Salpeter vertex needed for calculations e.g. f π if π q µ δ ij = d 4 k Tr [ 1 (2π) 4 2 γµ γ 5 τ j S(k)Γ i πs(k q) ] α α µ q β β q frontpage table of contents appendices 30 / 31

31 Table of Contents building blocks motivation plan recommended references QCD asymptotic freedom QCD & hadron structure chiral symmetry DCSB goldstone s theorem DSEs introduction njl model njl Lagrangian njl interaction regularization gap equation njl & dse gap equations njl confinement hadron spectrum pion goldstone boson chiral partners BS vertex functions frontpage table of contents appendices 31 / 31