Slavnov-Taylor identities

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Author: Dr. Andrei A. Slavnov, Steklov Mathematical Institute, Moscow, Russia

Slavnov-Taylor identities- relations between Green functions, which provide the gauge invariance of the quantum theory including non-Abelian gauge fields.

Contents

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Introduction

Principle of relativity in the internal space, according to which two field configurations related by a gauge transformation describe the same physical situation, determines uniquely the form of a minimal classical Lagrangian in gauge theories. To quantize a gauge invariant theory one has to choose a representative in a class of gauge equivalent configurations, for example by imposing on the fields the Coulomb gauge condition \partial_iA_i=0 (i=1,2,3). Although a gauge condition breaks explicit gauge invariance of the classical theory, in accordance with the relativity principle in internal space all observables are independent on the particular choice of a gauge.

In quantum theory requirement of independence of observables on the choice of a gauge leads to the relations between the Green functions that is the vacuum expectation values of chronologically ordered field products <T\bar{\psi}(x_1) \ldots \psi(x_m)A_{\mu}(x_{m+1}) \ldots A_{\nu}(x_n)>. In Abelian theories like Quantum Electrodynamics such relations were obtained by J.C.Ward and later in a more general form by E.S.Fradkin and Y.Takahashi. In Quantum Electrodynamics imposing a gauge condition does not break the conservation of the electromagnetic current, which is a source of the gauge field, and the Ward identities follow directly from the conservation of this current. In non-Abelian models, like the Yang-Mills theory, fixing a gauge breaks the conservation of the current which is the source of the gauge field. Moreover, fixing a gauge in non-Abelian theories must be accompanied by a further modification of the classical Lagrangian. This modification may be presented as a new interaction of the gauge fields with fictitious anticommuting scalar particles, Faddeev-Popov ghosts. This additional interaction also breaks gauge invariance. As a result in non-Abelian models simple Ward identities do not hold.

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Necessary and sufficient conditions for gauge independence of observables in non-Abelian gauge theories.

The relations between the Green functions in the quantum Yang-Mills theory, which are necessary and sufficient conditions of the gauge independence of observables were obtained by A.A.Slavnov and J.C.Taylor. They may be written in the form

\int \exp\{i\int[L_{ef}(x)+J_{\mu}^a(x)A_{\mu}^a(x)]dx\}\{- \frac{1}{\alpha}\partial_{\mu}A_{\mu}^a(y) +\int \bar{c}^a(y)J_{\mu}^b(z)[D_{\mu}c(z)]^b dz\}\prod_x dA_{\mu}(x)d \bar{c}(x)dc(x)=0

where the effective Lagrangian is given by the equation

L_{ef}= - \frac{1}{4}F_{\mu\nu}^aF_{\mu\nu}^a+ \frac{1}{2\alpha}(\partial_{\mu}A_{\mu})^2-\partial_{\mu} \bar{c}^a(D_{\mu}c)^a

Here A_{\mu}^a is the Yang-Mills field, \bar{c}^a, c^b are Faddeev-Popov ghosts , F_{\mu\nu}^a=\partial_{\mu}A_{\nu}^a-\partial_{\nu}A_{\mu}^a+gt^{abc}A_{\mu}^bA_{\nu}^c is the Yang-Mills stress tensor and (D_{\mu}c)^a=\partial_{\mu}c^a-gt^{abd}A_{\mu}^bc^d is the covariant derivative of the ghost field. J_{\mu} is the classical external source.

Differentiating the identity (1) with respect to external source J_{\mu} and putting J_{\mu}=0 one obtains the relations between Green functions with different number of external lines, which guarantee independence of the observables on the choice of the gauge, and hence physical equivalence of different gauge conditions.

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Slavnov-Taylor identities and renormalization

A quantum theory includes a new element: renormalization. Calculation of radiative corrections in any four dimensional theory leads to appearance of divergent integrals. A consistent treatment of these integrals is provided by the renormalization procedure which may be considered as modification of the Lagrangian by introducing the new terms (counterterms) and is equivalent to redefinition of the parameters (masses, charges) of the classical Lagrangian. To guarantee the gauge independence of observables this procedure must respect the principle of relativity in the internal space, and hence the Slavnov-Taylor identities must impose certain conditions on possible counterterms. In particular it follows from Slavnov-Taylor identities that only the transversal part of the two point Green function of the Yang-Mills field is modified by the radiative corrections.

Contrary to the Ward identities in Quantum Electrodynamics the Slavnov-Taylor identities include composite operators, such as (D_{\mu}c)^a. This leads to considerable differences in renormalization of Quantum Electrodynamics and non-Abelian gauge theories.In particular in the Yang-Mills theory the Slavnov-Taylor identities impose the following conditions on the possible counterterms

\delta m=0; \quad z_1z_2^{-1}= \tilde{z}_1 \tilde{z}_2^{-1}: \quad z_4=z_1^2z_2^{-1}

where \delta m is the mass renormalization of the Yang-Mills field , z_2 and \tilde{z}_2 are the renormalization constants of the Yang-Mills field and the ghost field correspondingly, z_1 and z_4 are the renormalization constants of the Yang-Mills Green functions with three and four external lines, and \tilde{z}_1 is the renormalization constant of the vertex function describing the interaction of the Yang-Mills field with the ghost fields. These relations provide the gauge invariance of the renormalized Lagrangian, however the parameters of the gauge transformation depend on the renormalization constants.

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Alternatives and generalizations.

It was noted by C.Becchi,A.Rouet,R.Stora and by I.Tyutin that the Slavnov-Taylor Identities may be interpreted as consequence of a charge conservation. However this charge is not related tothe source of the gauge field, as it happens in Quantum Electrodynamics, but includes also the contribution from Faddeev-Popov ghosts. Its conservation follows from the invariance of the effective action with respect to some supersymmetry transformations (Becchi-Rouet-Stora-Tyutin symmetry). Similar identities may be also written for the strongly connected Green functions (B.W.Lee; J.Zinn-Justin). See also Zinn-Justin equation. Slavnov-Taylor identities play the key role in the proof of the gauge invariant renormalizability of non-Abelian gauge theories.

As was shown by G t'Hooft the quantization of the gauge invariant models with spontaneously broken symmetry may be performed along the same lines as the quantization of the Yang-Mills theory leading to renormalizability of such models, in particular Weinberg-Salam model. The Slavnov-Taylor identities also may be generalized in a straightforward way to spontaneously broken models (B.W.Lee, J.Zinn-Justin), and their fulfillment guarantees the gauge invariance and hence unitarity of renormalized gauge models with spontaneously broken symmetries.

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References

  • Becchi, C M; Rouet, O and Stora, R (1974). Phys.Lett. B52: 344.
  • Becchi, C M; Rouet, O and Stora, R (1975). Comm.Math.Phys. 42: 127.
  • Faddeev, L D and Popov, V N (1967). Phys.Lett. B25: 30.
  • Fradkin, E S (1955). JETP 29: 288.
  • 't Hooft, G (1971). Nucl.Phys. B35: 167.
  • Lee, B W (1974). Phys.Rev. D9: 933.
  • Lee, B W and Zinn-Justin, J (1972). Phys.Rev. D5: 3137.
  • Salam, A (1968). Elementary Particle Theory. Svartholm, N editor. Almquist Ferlag AB, Stockholm.
  • Slavnov, A A (1972). Theor.Math.Phys. 10: 99.
  • Takahashi, Y (1957). Nuovo Cimento. 6: 370.
  • Taylor, J C (1971). Nucl.Phys. B33: 436.
  • Tyutin, I; 1975. Preprint of Lebedev Physical Institute, 39, 1975.
  • Zinn-Justin, J (1974). Lecture Notes in Physics, 37. Springer Verlag, Berlin.
  • Ward, J C (1950). Phys.Rev. 77: 2931.
  • Weinberg, S (1967). Phys.Rev.Lett. 19: 1264.
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Further reading

  • Faddeev, L D and Slavnov, A A (1991). Gauge Fields. Introduction to quantum theory. Second edition. Addison-Wesley Publishing Company, T. ISBN 0201524724.
  • Zinn-Justin, J (2002). Quantum Field Theory and Critical Phenomena. 4-th edition. Oxford University Press, USA. ISBN 0198509235.
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See also

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