The Background-Field Method and Noninvariant

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:dubna:su :dubna:su t # RFL000 :dubna:su View metadata,citationandsimilarpapersatcore.ac.uk :j inr :j inr 1 1 :j inr 1 thsun y ISF gran thsun @ thsun HEP-TH-9411041@ @ av deev L kazakovD kalmykov Supp orted in part b E-mail: E-mail: E-mail: 1 2 3 4 o- es tial w ari- v 4 v 1 kground- o JINR-E2-94-388 yk ation TION provided byCERNDocumentServer der e u. Kalm ch, brought toyouby ar al Physics, kground- eld metho d giv ese M. Y ussian F etic hemes. By an example of a t or ,R ar R and ) kground- eld formalism when apply- 3 CORE v gion o e Abstract ow R atory of The or ab Mosc ( OUND-FIELD METHOD AND ov L , D. I. Kazak 2 GR ARIANT RENORMALIZA Joint Institute for Nucle Dubna goliub CK Bo mo del it is demonstrated that the bac 141 980 estigate the consistency of the bac v NONINV ein L. V. Avdeev arious regularizations and renormalization sc THE BA W t. In particular, it is found that the cut-o regularization and the di eren incorrect results when the regularization (and/or renormalization) is nonin an renormalization b elong to this eld class metho d and in are theories incompatible with with nonlinear the symmetries. bac ing v dimensional

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

L =

(@ n) ; n =1: (1)

It can b e treated as a sp ecial case of the generic b osonic mo del

j k

L = (@ ) g () @ ; (j; k =1;2; :::; n 1); (2)

where the metric is of the form

j k

g ()= + : (3)

jk jk

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

L =

m g () : (4)

This additional term serves only for eliminating infrared divergencies, naively present

in anytwo-dimensional theory with massless scalars. After the calculation of the

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

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.

Their contributions to Z are obtained by normalizing to the tree term ) g (@

. @

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

j k

(a) R (@ )(@ )

j k abc a

(b) R (@ )(@ ) R +3 R 2 R

j abck j ak

ab j k

? ?

+ (c) R R (@ )(@ )

j k ab

(ab)c j k

r r

(d) R R (@ )(@ )

j k abc

(ab)c

j k

r r

(e) R (@ )(@ ) R

k (cb)a

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. Lines of the diagrams refer to propagators

2 2

1=(p +m ), and arrows to p in numerators.

R = h (g g g g ) ; R =(n2) hg :

abj k aj bk ak bj jk jk

Besides, we should takeinto account the renormalization of the mass Z . In the

2 j k

rst lo op it is determined by the diagram of g. 2 (normalized to

m g ).

Although this op erator gives no direct contribution to the wave-function renormaliza-

tion, in all the diagrams that contribute to Z the mass ought to b e shifted by such

corrections. In the two-lo op approximation, only the g. 2 correction to g. 1 is essen-

tial.

2 j k

m R

Figure 2: The one-lo op mass correction to the e ective action.

The two-lo op renormalizations can b e carried out either directly | via a certain R

op eration, diagram by diagram, | or by means of re-expanding the one-lo op counter-

terms in the background eld and the quantum eld (provided wehaveanintermediate

regularization, and the counterterms can b e written down explicitly). The additional

diagrams that emerge in this way are shown in g. 3.

One can nd the function by requiring indep endence of the invariantcharge on

the normalization p oint. This leads to the following expression for the function

through the nite wave-function renormalization constant: 4

j k a

(a) (@ )(@ )(K R fig: 1a) R R

k aj

? ? ab j k 0

(b) R R (@ )(@ )(K R fig: 1a)

j abk

2 ab j k 0

j abk

Figure 3: The diagrams emerging from the background- eld expansion of the one-lo op

counterterms.

!" #

(h)=h

1 h Z : (5)

@ @h

Using the dimensional regularization and the minimal subtraction scheme to cal-

culate the diagrams presented ab ove, one obtains the following well-known expression

for the two-lo op function of the n- eld mo del (1) [12 , 11]:

= (n 2)

1+2 : (6)

dim

4 4

As it has already b een mentioned, the dimensional regularization and the minimal

subtraction scheme provide us with an invariant renormalization pro cedure within the

background- eld metho d. Hence, the obtained expression for the function is correct,

and we can use it as a reference expression to compare with other approaches. Owing

to the presence of just one coupling constant in the mo del, the function should b e

renormalization-scheme indep endentuptotwo lo ops and should coincide with eq. (6).

Tocheck the validity of the background- eld metho d in conjunction with other reg-

ularizations and renormalization prescriptions, let us consider the calculation of the

function within twoschemes: the cut-o regularization and the di erential renormal-

ization.

3.1 The Cut-O Regularization

We start with the regularization that uses a cut-o in the momentum space. All

the integrals over the radial variable in the Euclidean space are cut at an upp er limit .

Strictly sp eaking, this is not a very promising regularization, since it explicitly breaks

the Lorentz, as well as gauge, invariance. However, we use it here to realize what may

happ en when a noninvariant regularization is applied.

We are going to use the minimal subtraction scheme which resp ects the invariance

prop erties of the applied intermediate regularization, keeps them intact, as they are.

When the regularization parameter has the dimension of a mass, the minimal sub-

traction pro cedure can b e de ned [13 ] so as just to convert the logarithms of the (in- 5

nite) cut-o into the logarithms of a nite renormalization p oint which app ears

in the theory after renormalizations:

n n n

2 2 2 n n

K ln ( )=ln ( ) ln ( ); K = ; (7)

so that

n n n

2 2 2 n

R ln ( )=(1K)ln( )=ln ( ); R =0: (8)

In case of overlapping divergencies, which generate p owers of the logarithms, one ought

to p erform the standard renormalization pro cedure prior to the subtractions. However,

if only the nal renormalized answers are of interest, one can simply drop all the con-

tributions of the minimally subtracted counterterms (7), since they will b e annihilated

by(1K ), eq. (8), irresp ectiveofanypowers of the logarithms from the residual graphs

with contracted subgraphs. The same will happ en to all the diagrams of g. 3, gener-

ated by re-expanding the counterterms.

Thus, it is sucient to calculate the regularized diagrams of gs. 1 and 2, up to

-p ower corrections and ultraviolet- nite two-lo op contributions, and then to replace

2 2 2 2

by and m in the one-lo op diagram of g. 1a by m Z , including the correction

from g. 2. The contributions of individual diagrams are

2 2

Z (fig: 1a) = (n 2) h=(4 ) ln( =m );

1 2 2 2 2 2

Z (fig: 1b) =

(n 2) (n +1) h =(4 ) ln ( =m );

h i

2 2 2 2 2 2 2

Z (fig: 1c) = (n 2) h =(4 ) ln ( =m ) ln( =m ) ;

2 2 2 2 2

Z (fig: 1d) = (n 2) h =(4 ) ln ( =m );

2 2 2 2

Z (fig: 1e) = (n 2) h =(4 ) ln ( =m );

2 2

Z (fig: 2) = (n 2) h=(4 ) ln( =m ):

The charge-renormalization constant proves then to b e

Z =1(n2) ln +0h ; (9)

cut

4 m

so that eq. (5) gives the function

(h)=(n2) (1 + 0 h) : (10)

cut

The di erence b etween this result and that obtained in dimensional renormalization

(6) is a direct manifestation of the noninvariance of the cut-o regularization, which

violates the translational invariance. However, one needs to explain the reason for the

failure to repro duce the correct function in the present case. Although the cut-o

regularization is noninvariant, still it has b een successfully used to p erform multilo op

calculations in scalar eld theories and in the quantum electro dynamics b oth within

the background- eld metho d and by the conventional diagram technique. 6

The p oint is that those theories were renormalizable in the ordinary sense, that is,

they had a nite number of typ es of divergent diagrams. In contrast, the n- eld mo del

is renormalizable only in the generalized sense. The total numb er of divergent struc-

tures here (with various external lines) is in nite, but they are related to each other by

general covariance of the renormalized theory (in case of an invariant renormalization).

So the numb er of indep endent structures remains nite. Expanding the lagrangian, we

get an in nite numb er of terms; however, the renormalization constants are not arbi-

trary but mutually related. Although the background- eld metho d formally preserves

the covariance of the mo del, the use of a noninvariant renormalization would break

the intrinsic connection b etween various diagrams (and b etween their renormalization

constants), thus leading to wrong results.

Therefore, we conclude that in generalized renormalizable (as well as nonrenormal-

izable) theories it is not allowed to use the cut-o regularization with the minimal

subtractions in the framework of the background- eld metho d.

Wenowwant in the same waytocheck the invariance prop erties of the di erential

renormalization metho d.

3.2 Di erential Renormalization

The idea of the di erential renormalization traces back to the foundations of the

renormalization pro cedure [14 ] as a rede nition of the pro duct of distributions at a

singular p oint. The metho d suggests to work in the co-ordinate space, where the

free Green functions are well de ned, although their pro duct at coinciding p oints suf-

fers from ultraviolet divergencies. The divergencies manifest themselves as singular

functions whichhavenowell-de ned Fourier transform. The recip e of the di erential

renormalization [8] consists in rewriting a singular pro duct in the form of a di erential

op erator applied to a nonsingular expression:

f (x ; :::; x )=D( ; :::; ) g (x ; :::; x ); (11)

j k j k

Eq. (11) should b e understo o d in the sense of distributions, that is, in the sense of

integration with a test function. Then one ignores any surface terms on rearranging the

derivatives via integration by parts. The nonsingular function g (x ; :::; x ) is obtained

j k

by solving a di erential equation, and hence, involves an obvious arbitrariness. The

latter can b e identi ed with the choice of a renormalization p oint and a renormalization

scheme. In this resp ect the di erential renormalization do es not di er from any other

renormalization prescription.

In the absence of a primary regularization this prescription might preserve all the

needed invariances and, what is imp ortant for applications, seems to renormalize ul-

traviolet singularities in the integer dimension. On the other hand, the absence of any

intermediate regularization prevents one from using the standard scheme: invariant

regularization + minimal subtractions. Therefore, to verify the invariance prop erties

of the di erential renormalization, one has to deal with renormalized amplitudes di-

rectly.

Two-lo op calculations of the renormalization constant in the two-dimensional 7

mo del in the framework of the di erential renormalization have b een p erformed in

ref. [15 ]. The authors have used the concept of the infrared R op eration to handle the

infrared divergencies. In the present case the infrared-renormalized free propagator in

the co-ordinate representation has the form

2 2

R (x)= ln(x N ); (12)

where N is an infrared renormalization scale.

An imp ortant role in the calculations plays the tadp ole diagram ( g. 1a). In four

dimensions, diagrams of this typ e diverge quadratically and can b e consistently renor-

malized to zero, as it was originally done in the metho d of the di erential renormal-

ization [8]. However, in two dimensions the leading one-lo op contribution to the

function comes from this very diagram. Hence, the tadp ole should b e di erent from

zero in any renormalization. This means that wehave to de ne the two-dimensional

tadp ole diagram in a self-consistentway in addition to the recip e of the di erential

renormalization. Such an extension has b een discussed in detail in ref. [15], where the

following expression for the massless tadp ole has b een suggested:

R (0) =

ln : (13)

4 N

The parameter M is an ultraviolet scale, and R denotes the complete infrared and

ultraviolet renormalization.

According to this mo di cation of the di erential renormalization rules, the expres-

sion for the function has b een found to b e

= (n 2)

+0h : (14)

dif f

Thus, the de nition of the tadp ole via eq. (13) in the massless case gives the correct

expression for the one-lo op function, but fails in two lo ops.

Therefore, wewould like to circumvent p ossible ambiguities of combining the in-

frared R op eration with the di erential renormalization. We are going to apply the

metho d to the massive mo del in which no infrared diculties ever app ear.

Intro ducing the mass term according to eq. (4), we obtain the free propagator of

the form

(x)=

K (mjxj); (15)

where K is the MacDonald function, ob eying the equation

2 2

m ) K (mjxj)= 2 (x): (16) (@

0 (2)

Bearing in mind the known expansion of the MacDonald function

1 1

X X

x 2k x 2k 1 (k +1)

(x)= ln K

+ ;

2 2

2 (k !) 2 (k!) 2

k =0 k =0 8

we come to the following natural generalization of eq. (13) to the massive case:

h i

1 M

R (x) (x) =(x) : (17) ln

4 m

In due course of the calculation we shall also need to de ne the pro duct of two

tadp oles. In the spirit of the consistent R op eration, the squared tadp ole ( g. 1b)

should b e de ned as the square of the renormalized value (17) for g. 1a, that is,

h i

R (x) (x) =(x)

: (18) ln

4 m

Nowwe are in a p osition to complete the calculation of the function. We present

it in more detail for the diagrams of g. 1c and g. 1d. The simple tadp ole subgraph

of g. 1c has already b een de ned via eq. (17). Consider another subgraph, with the

numerator,

d y

(x y) : (19)

Integrating by parts, ignoring the surface term, and then using eq. (16), we get

h i

2 2 2

d y m (x y ) (x y ) (xy) :

The integral of the rst term is nite and known (the normalization integral for

K ). On the other hand, the second term is just reduced to the basic tadp ole (17).

2 2

Thus, the result for eq. (19) is 1=(4 )[1ln(M =m )].

Generally sp eaking, the systematic di erential renormalization to all orders [16]

allows for intro ducing di erent ultraviolet renormalization scales in di erent diagrams

(all the scales varying prop ortionally to each other under renormalization-group trans-

formations). The ratio of these parameters can then b e sp ecially chosen [8] to satisfy

the Ward identities. Let us denote the scale that app ears in the tadp ole with the

numerator by M .

Pro ceed now to g. 1d. Its contribution to the e ective action is

Z Z

h ih i

2 2 abc j k 2

d x d yR (x) R (y ) @ (x) @ (y) (x y) @ @ (x y):

j abck m

Picking out the trace and traceless parts according to ref. [8], we get

Z Z

h ih i

j k 2 2 2 abc

(x) R (y ) @ (x) @ (y) (x y) d x d yR

j abck

h i

2 2 2 2

1 1 1

@ @ @ (x y )+ m (x y )+ @ m (x y ) :

m m m

2 2 2

The rst term, which is traceless, is nite and do es not generate any ultraviolet scale;

one can easily establish this fact in the momentum representation. The second term

vanishes as m !0. Thus, we are left with the last term. Via eq. (16) it is reduced to 9

eq. (18), that is, gives only the square of the logarithm. However, again the renormal-

ization scale M in the new diagram may di er from M .

Belowwe present the contributions of all the diagrams to the renormalization con-

stants:

2 2

Z (fig: 1a) = (n 2) h=(4 )ln(M=m );

2 2 2 2

Z (fig: 1b) =

(n 2) (n +1) h =(4 ) ln (M =m );

h i

2 2 2 2 2 2 2

Z (fig: 1c) = (n 2) h =(4 ) ln(M =m ) ln(M =m ) 1 ;

2 2 2 2

Z (fig: 1d) = (n 2) h =(4 ) ln (M =m );

2 2 2 2

Z (fig: 1e) = (n 2) h =(4 ) ln (M =m );

2 2

Z (fig: 2) = (n 2) h=(4 )ln(M=m ):

This gives the function

( " #)

2 2 2 2

h h 1 M 4 M 2 M

1 2 3

= (n 2) 1+ (n2) ln + ln + ln : (20)

dif f

2 2 2

4 4 3 M 3 M 3 M

We see that the result explicitly dep ends on the ratio of the renormalization scale pa-

rameters in di erent diagrams. Such a dep endence on the details of the renormaliza-

tion prescription is b eyond the usual scheme arbitrariness. It would never o ccur to two

lo ops in the conventional p erturbation theory for ordinary renormalizable one-charge

mo dels. There the arbitrariness would b e completely absorb ed into a nite number of

counterterms which are of the op erator typ es present in the tree lagrangian. Hence, we

should try to x the parameters of the di erential renormalization by imp osing some

additional requirements. In the quantum electro dynamics the gauge Ward identities

could b e used to this end [8]. For the mo del in the background- eld formalism the

situation is not so clear.

The parameter M that app ears in the one-lo op tadp ole subgraph of g. 1c with the

numerator can b e xed as follows. In the momentum representation we can easily see

that the sum of this diagram and the simple tadp ole ( g. 1a) is just an ultraviolet- nite

integral which equals 1=(4 ). The value will b e correctly repro duced by the di erential

renormalization if wecho ose the same scale for b oth tadp ole graphs: M =M .Thus,

for these diagrams the renormalization seems to b e automatically invariant.

Let us p oint out that this simple checkisby no means trivial. For example, the

straightforward Feynman regularization of the quantum- eld propagator in the mo-

2 2 2 2 2 2

mentum space 1=(p + m ) ! 1=(p + m ) 1=(p + M )would not stand the test.

As a result, the co ecient of the lower-order logarithm generated by g. 1c would b e

incorrect, and a contribution prop ortional to (n 2) would b e left in the two-lo op

function [as for M 6=M in eq. (20)]. The Feynman cut-o is therefore a noninvariant

regularization and cannot b e freely combined with the background- eld metho d.

Exp ecting that the di erential renormalization is automatically invariant, wewould

=M =M as well. However, then eq. (20) would again give us the wrong result set M

2 3

(14) obtained under the assumption of the automatic invariance in the massless theory 10

via the infrared R op eration. Hence, the ratio of the renormalization parameters ought

to b e somehow tuned in order to restore the invariance.

The identical situation was encountered in a nonrenormalizable chiral theory al-

ready at the one-lo op level for physical observables [17]. Inside the di erential renor-

malization, one nds no a priori internal criterion for cho osing the ratios of auxiliary

masses, to get reliable results. Of course, comparing eq. (20) to eq. (6) in the di-

mensional renormalization [or the results for g. 1(d,e) individually], we can infer the

2 2 2 2

values that would ensure the invariance: ln(M =M )=ln(M =M ) = 1. But by it-

2 3

self the di erential renormalization remains ambiguous if we apply it to a theory that

is not renormalizable in the ordinary sense, and is not directly compatible with the

background eld metho d.

4 Conclusion

Our examples show that the background- eld formalism requires one to use an

invariant renormalization pro cedure in order to obtain valid results in a generalized-

renormalizable theory. A noninvariant regularization or renormalization may break

an implicit correlation b etween di erent diagrams, which is essential as one formally

expands the action in the background and quantum elds.

Wehave demonstrated by direct two-lo op calculations that the regularization via

a cut-o in the momentum space is noninvariant and gives a wrong result for the

function of the n- eld mo del within the background- eld formalism.

Wehave also found that the di erential renormalization is not automatically in-

variant. The result dep ends on the ratio of the auxiliary scale parameters b eyond the

allowed scheme arbitrariness in the second order of p erturbation theory.We can par-

tially x the ambiguityby imp osing a condition on divergent one-lo op tadp ole-typ e

diagrams a combination of which should b e nite. But this is not enough, and there

seems to b e no algorithm of generalizing such conditions to more complicated graphs.

Wewould like to stress once more that the calculations in nonlinear mo dels like the

mo del or sup ergravity are hardly p ossible without the background- eld formalism.

Thus, the need in the regularization that preserves the underline symmetries and is

practically usefull at the same time is of vital imp ortance. The example considered

ab ove clearly demonstrate the problems arising when using a non-invariant pro cedure. 11

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