LETTF, RE AL NUOVO CIMENTO VOL. 25, ~'. 17 25 Agosto 1979

Algorithm for Finite Regularization.

A. PATRASCIOIU (*)

Department o] Physics, University o/ Arizona . Tucson, Ariz . 85721

(ricevuto il 22 Maggie 1979)

We consider a local relativistic field theory described by some Lagrangian ~ . Knowing ~ one can derive Feynman rules (propagators and vertices) and start com- puting Feynman integrals. Very often those integrals are infinite and one must define a procedure for obtaining finite answers (regularize the theory). By now there are several ways of regularizing (Pauli-Villars (~), analytic (2) and dimensional (3)). All of these methods write the integral as an infinite plus a finite part. The infinite part is canceled by the introduction of a suitable counterterm in s The situation is hLrther complicated by the presence of divergent subgraphs. They are rendered finite also by the use of counterterms, which were already chosen when computing lower-order graphs.

In this paper we propose a method of regularization with which one can take any Fcynman integral and associate with it a finite integral (without actually computing the infinite parts and subtracting them with counterterms). The finite integrals so defined obey the sacred principles: Lorentz invariance, order-by-order unitarity, causality and gauge iuvariance. This regularization scheme can be used in both renormalizable and non renormalizable field theories, although, as we will show, there is an important difference between these two types of theories.

To simplify the algebra we will discuss first the case of scalar fields without derivative interaction. A Feynman diagram with /-loops and p-propagators in d-dimensions pro- duces an integral of the form

(1) I :-/~d-2 I-I ddk,,, ,

c o

(2) x: : / H -%.-" oxp i Z a,(q - - m ' § i e ) . J m - - 1 J r~ l ~' L r - 1

o

(*) Work supported by the National Science Foundation. (1) W . P a u l i a n d F. VILLARS: _/~V. Mod. Phys., 21, 434 (1948). (2) C. G. I~OLLINI, J . J . GIAMBIAOI a n d A. (~ONZALES DO]tIINIQUEZ" Nuovo Cimen~o, 31, 550 (1964); E. R . SPEER: Generalized Feynman Amplitudes ( P r i n c e t o n , N . J . , 1969) . (s) G. 'T IIOOFT a n d M. VELT~AN: NUCl. Phys., 44 B, 189 (1972).

527

5 2 8 A . P A T R A S C I O I U

T h e s t r a n g e f a c t o r ~td-2~ h a s b e e n i n t r o d u c e d so as t o m a k e a l l m o m e n t a a n d m a s s e s

dimensionless; its usefulness will become obvious shortly. Each qr is some linear com- binat ion of k~ and p~ (the external momenta). If all a 's are non zero, the integrations over the k's are convergent Gaussian ones. I t is convenient to lump kx, ..., kt into an /-dimensional vector k and use matrix notation (4). Equation (2) becomes

(3)

c o

/" ~ d a , P I = ptd-~.v | YI - - | dak exp [i(k "r" A k - - 2k T" Bp + pr . Fp - - a)] ,

J,-=x * J 0

where the matrices A, B and /" depend on a~ and

a - ( m ~ - i e ) Z ~ , .

Integrating over k one obtains

(4)

o

Final ly let us scale the a's:

(5)

r

[" ~ d~, [ n ~ ~1~" i = . e x p - IBp) + f " - -

~i = ;tfl i ,

~f l~ := 1 .

i = 1 . . . . , p ,

Remembering that det A and the exponent are homogeneous functions of s of degree l, respectively 1, one can write (4) as

(6)

1 c o

| J ~ l r j 0 0

There exist graphic rules for writing C and D (4). For now it suffices to say that C = C(fl) while D = D(fl, p, m). The integration over 2 converges at the upper limit because of the ie. At 0 though it diverges for 2n ~ l d - - 2 p >~ O. But this is precisely the overall degree of divergence of this Feynman graph. One regularizes this divergent integral as follows: for n~> 0

co

(7) f 2 - " -~ 0

is defined as

(8)

c o

f 0. 1 )] 1 d21n exp i2 + ie ~ ! ,Z ~-ff7"1

0

(~) R . J . EDEN, P. V. LA.N'DSHOFF, D. I. OLIVE and J. C. POLKINGHORNE: Th4~ Analytic S-3,falrix (Cambridge, Mass., 1966).

A L G O R I T I t M FOB, F I N I T E R E G U L A R I Z A T I O N 529

for n integer and

(9) 1 1

nn--1 n--m ~ e x p i~ + i s ,

o

for n = m § m i n t e g c r , 0 < ~ < 1 .

Both of these formulae were obtained by integrat ing by parts and dropping the (infinite) surface term. Some tr ivial algcbra changes eqs. (8) and (9) to

(8') - - l ( iD) '~[y t ln ( - - iD)] ,

(9') F ( - - ~ 0 - - i .

In this way the divergent ).-integration in (6) has been made finite and (6) has become

(IO)

1

i oc ~,d-2rI f i dfl~ 5 [ ~ fl, _ l ] _l__ ~ D~ I~d/2)-r o

where i t is understood that if / d / 2 - - p is a nonnegative integer there is an addit ional factor of [~,-t- In (--i(D/C))].

The integrations over fl may also be divergent as a conscqucnce of C(fl) vanishing.

For any one-loop diagram C = ~ f l i = 1, but for more loops C is non tr ivial and it

will vanish whenever the fl's around any closed loop vanish. Since C appears as C -~/2 in (10), for sufficiently large d, thc fl-intcgrations will diverge. I t is easy to check tha t this happens precisely for tha t d for which that par t icular subintegration bccomes

divergence one scales the relevant fl's (fl~ = v~, ~ 'h = divergent. To handle the 1,

indcces)), then one integrates by parts in v throwing away the (infinite) i e (closed loop surface term at v = O. This operation must be pcrformed successively for all divergent subintegrals.

I t must be clear to the reader that this method of regularization can be used also in case of higher spin fields and derivative interactions. In such cases, in eq. (3) in the k-integration the exponential is mult ipl ied by a polynomial in k. This brings addit ional negative powers of Feynman parameters in (4), increasing the degree of divergence.

A final word about the arb i t ra ry mass scale ~ introduced in {1). I t makes the a's dimensionless; this is necessary in those cases where divergent integi als are transformed by integrat ion by parts into integrals involving logarithms. The logarithms are the only places where /~ survives when one reverts back to dimensioual nmmenta and masses. For instance if in (10) all subintegrations converge, then only if Id/2--p is a non negative integer is there any # dependence left; it comes from log (D/C) in (8').

If finite regularization produces physical ly acceptable amplitudes, then the finite par ts must be unique and obey the sacred principles s tated in the introduction. (Uni- queness means this : if the diagram has, say, an overall divergence plus certain subdi- vergences, the order of tackling them is irrelevant.)

~30 A. PATRASCIOIU

To investigate these problems, i t is convenient to compare this method of regular- ization with dimcnsionM regularization (a). Indeed the scheme described thru eq. (6) is amenable to dimensional regularization. For noninteger n = Id/2 - -p the dimensional regularization definition of (7) is precisely tha t of finite rcgularization (eq. (9')). For n approaching a non-negative integer value (n = m - - e, m integer, e --~ 0) dimensional regularization gives (7) the value

( (o) Normally the term of order 1/e is considered the pole par t and canceled against the counterterm. The finite par t in (11) differs from (8'7 by a polynomial in the external momenta

1 ( D \ m m l (12)

This term can be absorbed into the pole par t say by wri t ing

(13) e ~ e ' l + e ' .

Thus the finite parts obtained with finite and dimensional regularization can be made to coincide. Since dimensional regularization provides an unique answer obeying the sacred principles, so does finite regularization. The advantage of the la t ter method lies in the simple algori thm it uses (integration by parts), which el iminates the need for actual ly computing the infinite counterterms.

We have outlined a procedure for rcgularization. I t associates a finite integral with any Feynman integral. These finite answers will depend in general upon the parameters appearing in s (masses and coupling constants) and upon an arb i t ra ry mass scale/z. Fini te regularization can be used in either irenormalizablc or nourenormalizable field theories. The fundamental difference between these two types of theories is tha t in the former, through finite renormMization, the entire dependence upon/~ can be absorbed into a redefinition of the bare parameters.

In nonrenormalizable theories the dependence upon/z is genuine, so tha t the theory is characterized by one more parameter than those appearing in Lf. However, if the Green's functions so defined obey the sacred principles, they produce a self-consistent (and thus acceptable) theory for any /~.

The author is indebted to J. BRO~rZAN, D. ROSS and C. SACKRAJDA for useful discus- sions and to J. PRENTKY for invit ing him to visit CERN, where this work was begun.