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The Background-Field Method and Noninvariant View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server JINR-E2-94-388 THE BACKGROUND-FIELD METHOD AND 1 NONINVARIANT RENORMALIZATION 2 3 4 L. V. Avdeev , D. I. Kazakov and M. Yu. Kalmykov Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna (Moscow Region),Russian Federation Abstract Weinvestigate the consistency of the background- eld formalism when apply- ing various regularizations and renormalization schemes. By an example of a two- dimensional mo del it is demonstrated that the background- eld metho d gives incorrect results when the regularization (and/or renormalization) is noninvari- ant. In particular, it is found that the cut-o regularization and the di erential renormalization b elong to this class and are incompatible with the background- eld metho d in theories with nonlinear symmetries. HEP-TH-9411041 1 Supp orted in part by ISF grant # RFL000 2 E-mail: av deev L@thsun1:j inr:dubna:su 3 E-mail: kazakovD@thsun1:j inr:dubna:su 4 E-mail: kalmykov@thsun1:j inr:dubna:su 1 Intro duction To obtain meaningful results in quantum eld theory, one has to remove ultraviolet and infrared divergencies. This goal can b e achieved by a renormalization pro cedure, that is, a prop er subtraction of singularities. In a general case, ultraviolet renormal- ization involves three steps: 1. Regularization of Feynman amplitudes byintro ducing some parameter which converts divergencies into singularities as this parameter tends to a particular limit value (say, zero or in nity). There should exist a smo oth limit of taking the regularization o : any amplitudes that were nite without it should not b e distorted. 2. Renormalization of the parameters of the theory (coupling constants, masses,etc) in order to absorb the singularities into a rede nition of these parameters, which is achieved byintro ducing some lo cal counterterms. 3. The choice of a renormalization scheme which xes the nite arbitrariness left after the regularization is taken o . In gauge theories, one has to b e very careful b ecause the renormalization pro cedure may violate the gauge invariance on the quantum level, thus destroying the renormal- izability of the theory. Therefore, when dealing with gauge theories, one is b ound to apply an invariant renormalization. By this we mean a renormalization that preserves all the relevant symmetries of the mo del on the quantum level, that is, preserves all the Ward identities [1] for the renormalized Green functions. On the one hand, this can b e achieved by applying an invariant regularization rst (resp ecting the symmetries in the regularized theory) and then using, for instance, the minimal subtraction scheme [2, 3] to x the nite arbitrariness. On the other hand, when the regularization is noninvariant or no explicit regularization is intro duced at all, there is no automatic preservation of the symmetries. Then one has to take care of that directly.For example, one can use some noninvariant regularization and consecutively cho ose certain nite counterterms to restore the invariance (the symmetry of the renormalized Green functions) order by order in p erturbation theory [4]. When treating theories with nonlinear realization of a symmetry, liketwo-dimen- sional mo dels or quantum gravity, one faces extraordinary complexity of p erturba- tive calculations. To simplify them, one usually applies the so-called background- eld metho d [5] which allows one to handle all the calculations in a strictly covariantway. This metho d was successfully applied to multilo op calculations in various gauge and scalar mo dels, b eing combined with the minimal subtraction scheme based on some invariant regularization. The most p opular and handy regularization used in these calculations was the dimensional regularization [6]. It has b een proved to b e an invariant regularization, preserving all the symmetries of the classical action that do not dep end explicitly on the space-time dimension [3, 7]. Moreover, any formal manipulations with the dimensionally regularized integrals are allowed. However, an obvious drawback of this 1 regularization is the violation of the axial invariance and of sup ersymmetry. That is why there are numerous attempts to nd some other regularization equally convenient and ecient. Among suchschemes the recently prop osed di erential renormalization [8] is discussed. In the present pap er weinvestigate the compatibility of the background- eld for- malism with various regularizations and renormalization prescriptions. Our conclusion is that the background- eld metho d necessarily requires one to use an invariant renor- malization pro cedure. As the invariance do es not hold, the metho d gives incorrect results. We demonstrate this by an example of the two-dimensional O (n) -mo del, comparing the dimensional regularization, the cut-o regularization within the mini- mal subtraction scheme, and the di erential renormalization metho d. 2 Invariant Renormalization in the Background-Field Formalism To preserve all the symmetries on the quantum level, one has to apply an invariant renormalization pro cedure. The simplest way to construct such a pro cedure is to use an invariant regularization and the minimal subtraction scheme. An invariant regularization should p ermit any formal manipulations with the functional integral that are needed to ensure the Ward identities. Let us list the prop erties of an invariant regularization [9]: R R D D 1. translational invariance d xf(x+y)= d xf(x); R R R R D D D D 2. unambiguity of the order of integrations d x d yf(x; y )= d y d xf(x; y ); R R P P D D 3. linearity d x a f (x)= a d xf (x); j j j j j j 4. Lorentz covariance; 5. integration by parts, neglecting the surface terms; 6. p ossibility of canceling the numerator with the denominator; 7. commutativity of the space-time or momentum integration and di erentiation with resp ect to an external parameter. The only known regularization ob eying all these requirements is the dimensional regularization. Combined with the minimal subtraction scheme, it makes an invariant renormalization for which the action principle is valid [7]. Using the dimensional renor- malization in conjunction with the background- eld metho d leads to covariant results of multilo op calculations in any theory unless its symmetry prop erties dep end on the particular numb er of dimensions. On the other hand, if one applies some noninvariant regularization, the initial sym- metry may b e violated, and one has to explore the p ossibility of using such a regular- ization in the framework of the background- eld metho d. 2 In multilo op calculations by this metho d an invariant regularization automatically provides some implicit correlations b etween di erent diagrams, whichmay b e essential as the formal background- eld expansion of the action is p erformed. Any violation of these ne correlations by a noninvariant regularization or by an improp er choice of nite counterterms may result in wrong answers, although covariant in form. 3 Two-dimensional nonlinear O (n) mo del Let us consider the two-dimensional mo del of the O (n) principal chiral eld (n eld) and calculate the two-lo op function, using various approaches. The mo del is describ ed by the lagrangian 1 2 2 L = (@ n) ; n =1: (1) 2h It can b e treated as a sp ecial case of the generic b osonic mo del j k 1 L = (@ ) g () @ ; (j; k =1;2; :::; n 1); (2) jk 2 where the metric is of the form h j k g ()= + : (3) jk jk 2 1h The background- eld expansion of the action can b e done in a strictly covariant fashion [10]. To separate the ultraviolet and infrared divergencies, we add an auxiliary mass term to the initial lagrangian (2) 2 j k 1 L = m g () : (4) m jk 2 This additional term serves only for eliminating infrared divergencies, naively present in anytwo-dimensional theory with massless scalars. After the calculation of the 2 ultraviolet logarithms, one should set m =0. The mo del (2) with the particular choice of the metric (3) b ecomes renormalizable. All the covariant structures that may app ear as counterterms are reducible to the metric, so that the only thing that happ ens is a renormalization of the kinetic term. By rescaling the elds the renormalization can b e absorb ed into the charge. The 1 invariantcharge h=Z h is de ned through the eld renormalization constant Z .To calculate Z within the background- eld metho d, one has to consider the one-particle- irreducible diagrams with two external lines of the background eld, and quantum elds inside the lo ops. Up to two lo ops the relevant diagrams [11 ] are shown in g. 1. h 1 j Their contributions to Z are obtained by normalizing to the tree term ) g (@ jk 2 i k . @ The Riemann and Ricci tensors are the functionals of the background eld. In our mo del with the metric given by eq. (3) they are evaluated to 3 1 j k (a) R (@ )(@ ) jk r 2 @ @ 1 j k abc a r (b) R (@ )(@ ) R +3 R 2 R j abck j ak 12 r 1 ab j k q ? ? + (c) R R (@ )(@ ) j k ab r 6 @ @ 4 (ab)c j k - r r (d) R R (@ )(@ ) j k abc 9 8 (ab)c j k r r (e) R (@ )(@ ) R j k (cb)a 9 Figure 1: The one- and two-lo op corrections to the e ective action of the two-dimen- sional b osonic mo del without torsion.
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