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Dyson-Schwinger Equations in QCD

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Dyson-Schwinger Equations in QCD Dyson-Schwinger Equations in QCD Craig D. Rob erts Physics Division, 203, Argonne National Lab oratory, Argonne, Illinois 60439-4843, USA Abstract. These three lectures describ e the nonp erturbative derivation of Dyson- Schwinger equations in quantum eld theory, their renormalisati on and, using the exp edient of a simple one-parameter mo del, their application to the calculation of hadronic observables and the prop erties of QCD at nite temp erature. 1 The Dyson-Schwinger Equations The Dyson-Schwinger equations [DSEs] provide a nonp erturbative, Poincar e in- variant approach to solving quantum eld theory, in which the fundamental elements are the Schwinger functions. The Schwinger functions are moments of the measure. For example, in a Euclidean quantum eld theory describing a self- interacting scalar eld, x, sp eci ed by a measure d[], which will involve the classical Euclidean action for the theory and, p erhaps, gauge xing terms, etc., then Z hx :::x i d[] x :::x ; 1 1 n 1 n R 1 where d[] represents a functional integral, is an n-p ointSchwinger function. In a manner analogous to that when a probability measure is involved, a quantum eld theory is completely sp eci ed if all of the moments of its measure are known. The fo cus of lattice eld theory is a numerical estimation of these Schwinger functions. The DSEs provide a continuum framework for their calculation. The DSEs are an in nite tower of coupled equations, with the equation for a given n-p oint function involving at least one m>n-p oint function. A tractable problem is only obtained if one truncates the system. Truncations that preserve the global symmetries of a theory; for example, chiral symmetry in QCD, are relatively simple to e ect [Bender, et al. 1996]. It is more dicult to preserve lo cal gauge symmetries, although much progress in this direction has b een made in Ab elian gauge theories Bashir and Pennington, 1994. One systematic means of truncating the system is a weak coupling expansion. In this way one readily nds that the DSEs contain p erturbation theory in the sense that, for a given theory, the weak coupling expansion of the equations generates all of the diagrams obtainable in p erturbation theory. In this way, at the very least, the DSEs can b e used as a generating-to ol for p erturbation 1 An intro duction to the functional integral formulation of quantum eld theory can b e found in Itzykson and Zub er 1980 and Rivers 1987. Dyson-Schwinger Equations in QCD 213 Σ D = γ S Γ Fig. 1. A diagrammatic representation of the DSE for a fermion self-energy, p 2 2 i p[Ap 1] + B p : S p=1=[i p + p] is the connected fermion 2-p oint function; D k is the connected gauge b oson 2-p oint function; and p; k is the 1-particle irreducible fermion{gauge-b oson 3-p oint function. Both D k and p; k satisfy their own DSEs and this illustrates the coupling of the equation for a given n-p oint function to those involving m>n-p oint functions. theory. This can also b e used as a constraint on alternative truncation schemes; i.e., they must b e such as to preserve the feature that p erturbative results are recovered in the weak coupling limit. There are many familiar examples of DSEs. For example, the gap equation that describ es Co op er pairing in ordinary sup erconductivity is simply a trun- cated DSE for a 2-p oint electron Schwinger function; Bethe-Salp eter equations [BSEs], which describ e relativistic two-b o dy b ound states, are DSEs for 4-p oint functions; and covariantFadde'ev equations, which describ e relativistic three- b o dy b ound states, are DSEs for 6-p oint functions. The statement that a theory is solved if all of its Schwinger functions are known can b e lo osely re-expressed as the statement that all observable S -matrix amplitudes can b e expressed in terms of the Schwinger functions of the elemen- tary elds in the theory. This entails that one can connect observables to the fundamental parameters of the theory via these Schwinger functions. The 2-p oint functions in a given theory contain imp ortant information. For example: in a gauge theory, the form and analytic prop erties of the gauge-b oson 2-p oint function can provide information ab out whether, due to interactions, the gauge b oson acquires a gauge invariant mass that screens the interaction Schwinger-mass generation or a strong enhancement at small momenta that can b e a signal of con nement. Either of these prop erties will in uence the propagation characteristics of other mo des in the theory and hence physical observables. In all gauge theories the fermions act as a source of the gauge eld and their propagation characteristics are strongly a ected by their interaction with their self-generated gauge eld. This is describ ed by the DSE for the fermion 2-p oint function, illustrated in Fig. 1, in which the gauge-b oson 2-p oint function app ears as a driving term. For example, in QCD, whether the gauge-b oson 2-p oint function is nite or strongly enhanced at small momenta determines whether chiral symmetry is 214 Craig D. Rob erts dynamically broken and/or whether quarks are con ned. Euclidean Metric. Herein I employ a Euclidean metric formulation of eld theory; i.e., a non-negative metric for real vectors: 4 X a b a b a b 2 i i i=1 2 where is the Kronecker-delta. In this case, Q is a spacelikevector if Q > 0. My Dirac matrices satisfy y 3 [ ] = ; f ; g =2 and . One realisation of this algebra is provided by 5 4 1 2 3 E 0 E j ; i ;j=1;2;3 4 4 i j where are the usual contravariant Dirac matrices in Minkowski space. I adopt the p oint of view that the Euclidean formulation is primary; i.e., that a eld theory should b e de ned in Euclidean space, which is the p ersp ective employed in constructive eld theory and, usually as a pragmatic arti ce, in the lattice formulation and numerical simulation of eld theories. The Schwinger functions can then b e calculated and the question of the existence of the Wight- man functions, and hence the Minkowski space propagators, addressed subse- quently. [A fuller discussion of these p oints can b e found in Sec. 2.3 of Rob erts and Williams 1994.] This is imp ortant b ecause the analytic structure of a nonp erturbatively dressed Schwinger function is not necessarily the same as that of its p erturba- tive seed. Given this, one cannot know a priori the singularities in the integrand of the integral equation. Hence the true consequences of rotating the momen- tum space integration contour, as one do es in a Wick rotation, are unclear; i.e., the correct form of the \Wick rotated" equation mayinvolve contributions from p oles, branch cuts, etc., that cannot b e anticipated based on the p ertur- bative form of the Schwinger functions involved. To elucidate this, the following Euclidean , Minkowski Transcription Rules are valid at each order in p erturba- tion theory: Momentum Space Con guration Space Z Z Z Z M E M E 4 M 4 E 4 M 4 E d k ! i d k 1. d x !i dx 1. E E E E 2. =k !i k 2. =@ ! i @ E E E E 3. k q !k q 3. A= !i A E E E E 4. k x !k x , 4. A B !A B by which I mean that the correct Minkowski space integral for a given diagram in p erturbation theory is obtained by applying these transcription rules to the Euclidean integral. However, for skeleton diagrams; i.e., those in which each line and vertex represents a fully dressed n-p oint function, this cannot b e guaranteed. Dyson-Schwinger Equations in QCD 215 1.1 Quantum Electro dynamics in d-Dimensions, QED d I will use QED to illustrate the nonp erturbative derivation and form of the d DSEs. The generating functional or partition function for QED is d E E E E Z [ ; ;J ]= 5 Z Z E E E E d fE fE fE fE E E d ; ;A exp ; d x + + A J E E E E fE where there is an implicit normalisation Z [ =0; =0;J =0]=1; , fE E , J are auxiliary source elds; E E E E d ; ;A 6 Y Y Y fE fE E E E E D x D x DA x exp S [ ; ;A ] x f indicates a functional integration and sp eci es the measure; and the action is 2 Z 1 1 2 E E E E E E E E d E 4 F F @ + A 7 S [ ; ;A ]= d x 4 2 0 3 N f X f f fE E E E E fE 5 + @ + m + ie A 0 0 f =1 E E E E E with F = @ A @ A the eld strength tensor and f lab elling the fermion \ avour". This generating functional contains all the information ab out the eld theory; the b ound state sp ectrum, reaction rates, etc. - one must only enquire of it in the appropriate manner. The nonp erturbative derivation of the DSEs follows from the simple obser- vation that the integral of a total derivative is zero, assuming appropriate and sensible b oundary conditions; e.g., that the elds vanish on the b oundary of the compacti cation of Euclidean space used in rigorously de ning Eq.
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