FROM SLAVNOV TAYLOR IDENTITIES TO THE ZJ EQUATION JEAN ZINN-JUSTIN

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1 FROM SLAVNOV TAYLOR IDENTITIES TO THE ZJ EQUATION JEAN ZINN-JUSTIN CEA, IRFU (irfu.cea.fr) Centre de Saclay, Gif-sur-Yvette Cedex, France ABSTRACT In their work devoted to the proof of the renormalizability of non-abelian gauge theories in the broken phase, Lee and Zinn-Justin have directly benefited from Slavnov s important contributions. Later generalized in the form of the BRST symmetry, the Slavnov Taylor identities eventually have led to a remarkable quadratic equation for the renormalized gauge action (sometimes called the Zinn-Justin (ZJ) equation), allowing a completely general proof of renormalizability of non-abelian gauge theories. Proceedings of the conference Gauge fields yesterday, today, tomorrow, Moscow 20 23/01/20010.

2 2 1 Slavnov Taylor identities: The origin We first explain the origin of Slavnov Taylor (ST) identities in a rather general context. In particular, we show that ST [1,2] identities in gauge theories owe less to gauge symmetry than to gauge fixing [3]. Note that below summation over repeated indices is always assumed. Let ϕ be a set of dynamical variables satisfying a system of equations, E (ϕ) = 0. (1.1) For notational simplicity, we write the variables and the equations with a discrete index, but the generalization to differential or functional equations is straightforward, summation over being replaced by integration and summation, and differentiation by functional differentiation. We assume that the functions E (ϕ) are smooth and that E = E (ϕ) is a one-to-one mapping in some neighbourhood of E = 0, which can be inverted in ϕ = ϕ (E). In particular, this implies that the equation has a unique solution ϕ s ϕ (0). In the neighbourhood of ϕ s, the determinant dete of the matrix E with elements E β β E does not vanish and thus one can choose E (ϕ) such that it is positive. For any function F(ϕ), one can derive a simple formal expression for F(ϕ s ), which does not involve solving the equation explicitly. One starts from the trivial identity { } F(ϕ s ) = de δ(e ) F ( ϕ(e) ), where δ(e) is Dirac s δ-function. One then changes variables E ϕ. This generates the Jacobian dete > 0. Thus, { F(ϕ s ) = } dϕ δ[e (ϕ)] dete(ϕ)f(ϕ). (1.2) In the context of non-abelian gauge theories, det E is the Faddeev Popov determinant [4,5]. 1.1 An invariant measure The integration measure de is an invariant measure for the group of translations E E +ν, ν constant. It follows that the measure dρ(ϕ) = dete(ϕ) dϕ (1.3)

3 3 is an invariant measure for the translation group realized non-linearly on the new coordinates ϕ (provided ν is small enough): ϕ ϕ with E (ϕ ) ν = E (ϕ). (1.4) The infinitesimal form of the transformation can be written more explicitly as δϕ = [E 1 (ϕ)] β ν β, (1.5) where one recognizes, in symbolic notation, the form introduced in [1]. This rather straightforward property of the measure and the corresponding infinitesimal transformations (1.5) have been used in the context of non-abelian gauge theories by Slavnov and, independently, Taylor to derive a set of important identities satisfied by Green s functions, thus called Slavnov Taylor identities [1,2,6,7]. These identities form the basis of the first proof by Lee and Zinn-Justin [8] of the renormalizability of non-abelian gauge theories in the broken phase, where the partial regularization proposed by Slavnov[9] was also used. This method has then been extended to various other field theories, for example, to the dynamics generated by a Langevin equation [10]. A number of papers devoted to non- Abelian gauge theories is collected in [11]. Reciprocal property. Conversely, one can characterize the general form of nonlinear representations of the translation group. One recovers the form of the previous measure. 2 From ST symmetry to BRST symmetry In quantum field theory, the non-linear and non-local character of the transformations (1.5) is the source of some technical complications. Remarkably enough, the invariance under the infinitesimal transformations (1.5) can be replaced by an invariance under linear anticommuting-type transformations at the expense of introducing additional Grassmann variables. One again starts from the identity (1.2) and first replaces the δ-function by its Fourier representation: δ[e (ϕ)] = d ϕ 2iπ e ϕ E (ϕ), where the ϕ integration runs along the imaginary axis. Moreover, the determinant can be written as an integral over Grassmann variables c and c [12]: dete = (d c dc )exp ( c E β c β).

4 4 The expression (1.2) then becomes F(ϕ s ) = N (dϕ d ϕ d c dc )F(ϕ)exp[ S(ϕ, ϕ,c, c)], (2.1) in which N is a constant normalization factor, N 1 = (dϕ d ϕ d c dc )exp[ S(ϕ, ϕ,c, c)], and S(ϕ, ϕ,c, c) the function (and element of the Grassmann algebra) S(ϕ, ϕ,c, c) = ϕ E (ϕ) c E β (ϕ) c β. (2.2) Somewhat surprisingly, the function S has a new type of symmetry, which is an extension to Grassmann variables of the invariance of the measure det E(ϕ)dϕ under the infinitesimal transformations (1.5). In the context of non-abelian gauge theories, c and c are the Faddeev Popov ghosts. 2.1 BRST symmetry The BRST symmetry, first discovered in the context of quantized gauge theories by Becchi, Rouet, Stora, and Tyutin, [13,14] is a Grassmann symmetry in the sense that the parameter ε of the transformation is an anticommuting constant, an additional generator of the Grassmann algebra. The variations of the various dynamic variables are δϕ = ε c, δ c = 0, δc = ε ϕ, δ ϕ = 0 with ε 2 = 0, ε c + c ε = 0, εc +εc = 0. The transformation is obviously nilpotent of vanishing square: δ 2 = BRST symmetry and group manifolds When the variables ϕ parametrize an element g(ϕ) of a Lie group in some matrix representation, it is convenient to express BRST transformations on g(ϕ) directly and to parametrize the variation of g in terms of a Grassmann matrix C belonging to the Lie algebra of the group: δg = ε Cg. (2.3)

5 5 Thus, C = c g ϕ g 1. The calculation of the variation of C then yields δ C = c g ϕ g 1 ε c β g ϕ β g 1 = ε C 2. This is also the expression that appears in gauge theories, the group element being there associated with gauge transformations and the transformation (2.3) being an infinitesimal gauge transformation. 2.3 BRST transformations: Differential operator representation The BRST transformation, when acting on functions of {ϕ, ϕ,c, c}, can be represented by the Grassmann differential operator D c ϕ + ϕ c. (2.4) The nilpotency of the BRST transformation is then expressed by the identity D 2 = 0. (2.5) The differential operator D has the form of a cohomology operator, generalization of the exterior differentiation of differential forms. In particular, the first term c / ϕ in the BRST operator is identical to the differentiation of forms in a formalism in which the Grassmann variables c are introduced to represent forms. The equation D 2 = 0 implies that all quantities of the form DQ(ϕ, ϕ,c, c), quantities called BRST exact, are BRST invariant. One immediately verifies that the function S defined by equation (2.2) is BRST exact: It follows that S is BRST invariant, S = D[c E (ϕ)]. (2.6) DS = 0. (2.7) The reciprocal property, the meaning and implications of the BRST symmetry rely on simple considerations of BRST cohomology. These properties play an important role, in particular, in the discussion of the renormalization of non-abelian gauge theories, including gauge invariant operators [3,15,16].

6 6 3 From BRST symmetry to the ZJ equation In non-abelian gauge theories, the BRST transformations take the explicit form (we omit matter fields for simplicity, but the extension is simple) [13,14,17] { δaµ (x) = εd µ C(x), δ C(x) = ε C2 (x), δc(x) = ελ(x), δλ(x) = 0, (3.1) where the transformation of the gauge field follows from the gauge transformation (2.3). The BRST symmetry of the action can then be expressed by the functional equation [ ] d 4 δs x tr δa µ (x) δa µ (x) +δ C(x) δs δ C(x) +λ(x) δs = 0. (3.2) δc(x) From the viewpoint of renormalization, the important remark is that the BRST variations of A µ and C are quadratic in the fields and are, therefore, renormalized: the explicit form (3.2) of the BRST transformations is not stable under renormalization. Stable equations can be derived by first adding to the quantized action S sources K µ and L for the non-trivial BRST transformations: S(A µ,c, C,λ) S(A µ,c, C,λ,K µ,l) with S(A µ,c, C,λ,K µ,l) = S(A µ,c, C,λ) d 4 xtr ( K µ (x)d µ C(x)+L(x) C2 (x) ). (3.3) The sources K µ and L for BRST transformations have been later named antifields. No other terms are required because the two composite operators in the BRST transformations (3.1) are BRST invariant since D 2 = 0. It can be easily verified[17,3,18,19] that equation(3.2) implies for the complete quantized action (3.3) with these added sources a quadratic relation, the ZJ equation, which does not involve the explicit form of the BRST transformations (3.1). In the component form, the equation reads d 4 x ( δs δs δa µ(x) δkµ(x) + δs ) δs δs δ C (x) δl (x) +λ (x) δc = 0. (3.4) (x) Remarkably enough, this equation is stable under renormalization and the general solution of this equation, taking into account locality, power counting and ghost number conservation, takes the form of a BRST invariant gauge action.

7 7 References [1] A.A. Slavnov, Ward Identities in Gauge Theories, Theor. Math. Phys. 10 (1972) [2] J.C. Taylor, Ward identities and charge renormalization of the Yang-Mills field, Nucl. Phys. B33 (1971) [3] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, (Oxford University Press, Oxford 1989), chap. 16 Fourth Edition, International Series of Monographs on Physics 113, 1054 pp. (2002). [4] L.D. Faddeev and V.N. Popov, Feynman diagrams for the Yang-Mills field, Phys. Lett. 25B (1967) [5] L.D. Faddeev, Faddeev Popov ghosts, (2009) Scholarpedia, 4(4):7389. [6] L.D. Faddeev and A.A. Slavnov, Gauge Fields: Introduction to Quantum Field Theory (Benjamin, Reading, MA 1980). [7] A.A. Slavnov, Slavnov Taylor identities, (2008) Scholarpedia, 3(10): [8] B.W. Lee and J. Zinn-Justin, Spontaneously Broken Gauge Symmetries I,II,III,IV, Phys. Rev. D5 (1972) , , ; ibidem D7 (1973) [9] A.A. Slavnov, Invariant regularization of gauge theories, Theor. Mat. Phys. 13 (1972) [10] J. Zinn-Justin, Renormalization and stochastic quantization, Nucl. Phys. B275 (1986) [11] C.H. Lai (ed.) Gauge Theory of Weak and Electromagnetic Interactions, (World Scientific, Singapore 1981). [12] F.A. Berezin, The Method of Second Quantization, translated from Russian Akad Nauk Moscow (1965) by N. Mugibayashi and A. Jeffrey, Academic Press, New York (1966). [13] C. Becchi, A. Rouet and R. Stora, Renormalization of the abelian Higgs- Kibble model, Commun. Math. Phys. 42 (1975) ; ibidem Renormalization of gauge theories, Ann. Phys. 98 (1976) 28. [14] I.V. Tyutin, Lebedev preprint FIAN 39 (1975), unpublished. [15] H. Kluberg-Stern and J.B. Zuber, Ward identities and some clues to the renormalization of gauge-invariant operators, Phys. Rev. D 12 (1975) [16] G. Barnich, and M. Henneaux, Renormalization of gauge invariant operators and anomalies in Yang-Mills theory, Phys. Rev. Lett. 72 (1994) ;

8 8 G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in the antifield formalism: I,II, Comm. Math. Phys. 174 (1995) 57-91, [17] J. Zinn-Justin in Trends in Elementary Particle Physics. Lectures Notes in Physics 37, H. Rollnik and K. Dietz eds. (Springer-Verlag, Berlin 1975). [18] J. Zinn-Justin, Renormalization of gauge theories and master equation, Modern Physics Letters A14 (1999) [19] Jean Zinn-Justin, Zinn-Justin equation, (2009) Scholarpedia, 4(1): 7120.