Volume 112B, number 6 PHYSICS LETTERS 27 May 1982

THE UNEXPECTED INFRARED BEHAVIOR

OF THE ONE-INSTANTON CONTRIBUTION IN QCD

A. PATRASCIOIU t and A. ROUET 2

The Institute for Advanced Study, Princeton, NJ 08540, USA

Received 17 December 1981

We consider the one-instanton contribution to QCD defined on a sphere and investigate its dependence upon the radius R. We find it to be non-trivial. In particular the relevant scale for the running coupling constant turns out to be the infrared cut-off R,

The semi-classical approximation of the path inte- gral consists in finding classical solutions of finite ac- tion and computing quantum fluctuations in their background potential. I f there are zero modes due to certain degeneracies of the classical solutions, they are integrated by hand using the Faddeev-Popov technique. In the case of QCD4, the classical solu- tion of unit topologic charge [1] contains five ar- bitrary parameters: a u the position of the instan- ton and p its size. One is thus led to consider the following expression [2] :

(0 ) 1 ---- exp( Scl/g 2) f d4 a dp AF(a, , p)O(au, p)

DetMAo Det 'M¢ / D e t M 11/22)

Deft M A

where AF(au, p) is the Faddeev Popov determinant for the zero modes. M A,M~ and M s account for the fluctuations of the gauge, fermion and scalar field, respectively. Det ' is the determinant with all the zero modes removed. O(au, p) is some observable.

1 A.P. Sloan Foundation Fellow. Permanent address: Univer- sity of Arizona, Physics Department, Tucson, AZ 85721, USA.

2 Albert Einstein Professor. Permanent address: CNRS: Luminy-Case 907, Centre de Physique Theofique F-13288; Marseille Cedex 2, France.

This equation is ambiguous because the quantum determinants which enter it need both ultraviolet and infrared regularization. Although the need for infrared regularization was not appreciated at the beginning, work by Patrascioiu [3] and Seiler [4] made it clear that for massless fields in the presence of non-trivial topological charge these quantum deter- minants may be sensitive to the type of infrared cut- off used. More precisely, introducing a Pauli-Villars constant regulator mass in I:14 would leave an explicit infrared divergence the massive regularization term being infrared ffmite cannot compensate the in- frared divergence of the massless part. This is the reason why the usual method consists in formulating the theory on the sphere S 4 of radius R and studying the limit R -+ oo. Actually, a mass M 2 constant on S 4 means a space-time dependent mass M 2 R 4/(R 2 + x2) 2 on I:14 . Such a mass vanishes at Lx [ -+ oo, lead- ing to the same infrared behavior as the massless part, so that, in the regularized expression, the infrared divergence cancels between the massless and the regu- lator part. These computations appear in the litera- ture [5,6] and we will employ their results. In this letter we would like to point out the non-trivial de- pendence of eq. (1) upon the infrared cut-offR. Our method of analysis relies heavily upon the presence in the problem of a certain symmetry, which survives regularization.

We begin with the general results obtained by Berg

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Volume 112B, number 6 PHYSICS LETTERS 27 May 1982

and Ltischer [ref. [5], eq. (69)] t

ln(Det M~/Det M~ o) - Preg - 2Preg = 7 , (2)

which applied to the field configuration

Aau = (2/g)~Tauv x , / ( x2 + p2) . (3)

reads:

2 ' In N(p) + Y(p) (4) 7(p) = C - ~ lnM+ g l n p +

where: C is a numerical constant. R is the radius of the sphere,

7r2R 2 2~2R202 4~r2R4p 2 in p

N(p) - R 2 P 2 + + . -- (R 2 - 0 2 ) 2 (R 2 0 2 ) 3

y ( p ) _ 4 R 2 0 2 ÷ 404 ln~-

( R 2 _ 0 2 ) 2 (R 2 _ 0 2 ) 2

06 8 In p (5)

+ 3 (R 2 _ 02)3

(please note that 7(P) is regular at p = R). We notice that Under the change

p ~ R2/p ,

the quantities N(0) and Y(p) become

W(p) ~ N(R 2/p) = Co 2/t2 2)W(p) ,

y(p) __> y(R2/p) = y(p) _ 4 log(p/R), (6)

so that

7(P) -~ 7(R2/p) = 7(P). (7)

Eq. (7) is the basis of the results we would like to present henceforth. It arises as a natural consequence of the fact that the theory is defined on a sphere of radius R. Indeed, following Berg and Lfischer, we have been using sterographic coordinates to describe the classical solution. One can easily verify that the change p -+R2/p corresponds to a change of the pro- jection point from the north to the south pole of the sphere, followed by a gauge transformation. It is clear that the calculations of the functional deter- minants must be left invariant under those changes of the parameters of the classical solution (a u and p) which correspond only to reparametrizations of the surface of the sphere S 4 and to gauge transforma-

tions. This is the symmetry to which we alluded be- fore. Let us emphasize that this symmetry has noth- ing to do with the conformal invariance of the classi- cal Yang-Mills action and thus it is not affected by regularization, as we have explicitly demonstrated above.

It is commonly argued [2,6] that, as R goes to infinity eq. (4) can be approximated by its asym- ptotic behavior as o/R -+ O, that is

7(P) o/R-*O "> --ln M + ~ In Mp. (8)

Making this approximation is incorrect, since, how- ever large R may be, configurations with arbitrary value of p/R are a priori as important as the ones with p/R -+ O. Moreover, as it can be seen from eq. (7) and follows from our discussion above, p/R -~ gives an identical contribution as p/R -+ O. Therefore,

• even at large R (MR >> 1), the correct expression for 7(p) is

2 .7(P) = In R - g In MR + F (p /R ) , (9)

where

F ( p l R ) = C + l n N ( p ) + Y(p)+~ln(p /R) (10)

is a dimensionless function satisfying

F(p/n) = F(R /p ) . (11)

The same comment can be made with respect to • the position a: however, large R may be, au/R must be integrated over all I:14, so that eq. (1) actually reads:

(0) = (MR) 22/3

X e x p ( - S c . ) f d 4 ( ~ - ) d ( ~ ) ( : I ) ( ~ , ~ -) (12) \ g2

Here a > 0 depends upon the matter fields involved {see eq. (12.5) of ref. [2]}. It is also clear that the same problem will occur for higher topological charges.

This expression is similar to the one obtained in the CP ± case [7]. In the same way, the relevant scale in the running coupling constant is the radius of the sphere. Finally let is notice that if the integral in eq. (12) converges at p/R = 0, it also does at p/R = oo and similarly forau/R. Let us point out that this dif-

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Volume 112B, number 6 PHYSICS LETTERS 27 May 1982

ficulty cannot be escaped if we are to regard the semi-classical approximation defined as a saddle- point approximation, which is the point of view we are discussing here. Of course, if, for phenomenolog- ical reasons, a cut off on the size and position of the instantons is introduced the problem we raise here disappears.

We thank H. Neuberger for useful discussions and the Institute for Advanced Study for its hospitality. One of us (A.R.) acknowledges the support of a grant from the Federal Republic of Germany.

References

[1] A.A. Belavin et al., Phys. Lett. 59B (1975) 85. [2] G.'t Hooft, Phys. Rev. D14 (1976) 3432. [3] A. Patrascioiu, Phys. Rev. D17 (1978) 2764. [4] E. Seiler, Phys. Rev. D22 (1980) 2412. [5] B. Berg and M. Ltischer, Nucl. Phys. B160 (1979) 281. [6.] E.Corrigan, P. Goddard, H. Osborn and S. Templeton,

Nucl. Phys. B159 (1979) 469. [7] A. Patrascioiu and A. Rouet, The non-linear a-model;

a critical analysis of semi-classical results, I.A.S. prepfint (November 1981).

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