Slavnov-Taylor identities

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Andrei A. Slavnov (2008), Scholarpedia, 3(10):7119. doi:10.4249/scholarpedia.7119 revision #137427 [link to/cite this article]
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Slavnov-Taylor identities are relations between Green functions, which provide the gauge invariance of quantum theory, including non-Abelian gauge fields.



The principle of relativity in the internal space (the principle of gauge invariance), according to which two field configurations are related by a gauge transformation describe the same physical situation, determines uniquely the form of a minimal classical Lagrangian in gauge theories and constraints the structure of composite operators corresponding to physical observables, e.g. the energy-momentum tensor.

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, for example scattering aplitudes, are independent on the particular choice of a gauge.

Quantum theory cannot be constructed only in terms of observable quantities. To formulate some basic principles, like causality, it is necessary to introduce objects which are not directly observable and therefore may depend on a gauge. The most important objects of this kind are so called Green functions that is the vacuum expectation values of chronologically ordered field products \(<T\bar{\psi}(x_1) \ldots \psi(x_m)A_{\mu}(x_{m+1}) \ldots A_{\nu}(x_n)>\) where the symbol \(T\) means chronological ordering of the field arguments. Although these functions cannot be measured directly in any experiment, they are related to scattering amplitudes and to provide the gauge independence of observables they must satisfy certain relations. These relations are Slavnov-Taylor identities.

In Abelian theories like quantum electrodynamics the relations between Green functions providing the gauge independence of observables 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 the simple Ward identities do not hold.

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

where the effective Lagrangian is given by the equation \[\tag{2} L_{ef}= - \frac{1}{4}F_{\mu\nu}^aF_{\mu\nu}^a+ \frac{1}{2\alpha}(\partial_{\mu}A_{\mu})^2-\partial_{\mu} \bar{c}^a(D_{\mu}c)^a \]

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, \((D_{\mu}c)^a=\partial_{\mu}c^a+gt^{abd}A_{\mu}^bc^d\) is the covariant derivative of the ghost field, and \(t^{abc}\) are the structure constants of the gauge group. \(J_{\mu}\) is the classical external source.

To understand the meaning of this equation it is instructive to consider the scattering matrix in the Yang-Mills theory. In terms of the path integral the scattering matrix may be written as follows \[\tag{3} S= \int \exp \{i \int L_{eff}(x)dx \} \Pi_xdA_{\mu}(x)d \bar{c}(x)dc(x) \]

Here the effective Lagrangian is given by the eq.(2) and depends explicitly on the gauge parameter \( \alpha\ .\) Nevertheless the value of the integral (3) which determines the scattering amplitudes for the Yang-Mills field is gauge independent. This is a consequence of the relativity principle in the internal space according to which the fields related by a gauge transformation describe the same physical situation, and therefore in the path integral for the scattering matrix one should integrate over the classes of the gauge equivalent fields. In practice that means that one has to select a single representative in the gauge equivalent class by imposing some gauge condition and normalising correctly the integration measure by adding the integration over Faddeev-Popov ghost fields. The result does not depend on the particular choice of the gauge because the transformation from one gauge to another may be accomplished by a simple change of variables in the path integral.

Contrary to the scattering matrix the Green functions are not directly observable and do depend on the gauge. Any Green function may be presented as a variational derivative of the Green function generating functional given by the following path integral \[\tag{4} Z(\eta)= \int \exp \{i \int[L_{eff}(x)+J_{\mu}^a(x)A_{\mu}^a(x)]dx \}dA_{\mu}d \bar{c}dc \]

\[\tag{5} <TA_{\mu}^a(x_1) \ldots A_{\nu}^c(x_n)>= \frac{\delta^nZ}{\delta J_{\mu}^a(x_1) \ldots J_{\nu}^c(x_n)}_{J=0} \]

One sees that the integrand in the formula for the Green functions generating functional differs of the corresponding integrand for the scattering matrix by the presence of the gauge non invariant term \( \int J_{\mu}^a(x)A_{\mu}^a(x)dx\) in the exponent. When one passes from one gauge to another by the change of variables this functional is changed. Its variation is given by the left hand side of the eq.(1). As a change of variables does not influence the value of integral this variation is equal to zero. Hence the Slavnov-Taylor identities (1) describe the response of the Green function generating functional to the change of gauge conditions.

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. The simplest identity is \[\tag{6} \alpha^{-1}<T \partial_{\mu}A_{\mu}^a(x)A_{\nu}^b(y)>=-i<T \bar{c}^a(x)(D_{\nu}c)^b(y)> \]

Differentiating this equation with respect to \( \nu\) and taking into account the equation of motion for the ghost field we conclude that this equation implies the absence of radiative corrections to the longitudinal part of the two point Green function \[\tag{7} \partial_{\mu} \partial_{\nu}G_{\mu \nu}^{ab}(x,y)= \alpha \delta^{ab} \delta (x-y) \ .\]

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 and multiplicative renormalization of the fields \[\tag{8} A_{\mu}^a \rightarrow \sqrt{z_2}\,A_{\mu}^a, \quad \bar{c}^a \rightarrow \sqrt{\tilde{z}_2} \,\bar{c}^a, \quad c^a \rightarrow \sqrt{\tilde{z}_2}\,c^a \]

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.

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 \[\tag{9} \delta m=0; \quad z_1z_2^{-1}= \tilde{z}_1 \tilde{z}_2^{-1}: \quad z_4=z_1^2z_2^{-1} \]

where \(\delta m\) is the mass renormalization of the Yang-Mills field, \(z_1\) 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.

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 to the 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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Internal references

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. 4th edition. Oxford University Press, USA. ISBN 0198509235.

See also

Gauge invariance, Gauge theories, Becchi-Rouet-Stora-Tyutin symmetry, Zinn-Justin equation

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