The renormalization group equation for the Wilson loop

Volume 103B, number 4,5

PHYSICS LETTERS

30 July 1981

THE RENORMALIZATION GROUP EQUATION FOR THE WILSON LOOP Alan J. McKANE

Department of Physics, Universityof Pennsylvania,Philadelphia,PA 19104, USA and Michael STONE

Department of Physics, Universityof Illinois, Urbana,IL 61801, USA Received 28 February 1981

We formulate the renormalization group equation for Wilson loops in such a way as to make the analogy with spin systems more transparent. We thus obtain a quantity ~,(g2) analogous to the anomalous dimension 3,(g2) for spin systems. The value of f(ge) at the non-trivial fixed point in 4 + e dimensions gives the exponent r/defined by Peskin.

There are many analogies between the behavior of spin systems and gauge theories near two and four dimensions, respectively [ I - 3 ] . If the dimensionality of the system, d, is greater than the lower critical dimension, d c (viz. 2 or 4), then both have an ordered and disordered phase separated by a phase transition which is second order for sufficiently small values o f (d - d c ) . For d = d c they are disordered for all non-zero values of the coupling constant, with a crossover behavior between asymptotic freedom at short distances and symmetry restoration or confinement at long distance. One essential difference, however, is the absence of a local order parameter in the gauge theory case. In fact phase transitions in gauge theories were first studied [4] because o f this absence; it was only later that their utility in strong interactions was realized [5]. That one has a non-local (Wilson loop) order parameter not only hinders one from developing the intuitive picture that exists in the case of phase transitions in spin systems, but also complicates the application of renormalization group ideas to the behavior of the degree of disorder. One can write down renormalization group equations obeyed by the Wilson loop [6,7], but their similarity to the spin theory case is not obvious. In this note we aim to make the similarity a little more transparent. Most of what we shall do is cosmetic, in that we will just change the renormalization prescrip-

tion used previously by a finite multiplicative renormalization in order to make the behavior as close as possible to that seen in the spin system [3]. Nonetheless we believe that there is a genuine gain in understanding in doing this. The natural quantity to examine is the interquark potential:

lim 1 where

w(s)(r),

(1)

W(j)(I') is the Wilson loop,

W(j)(P)=(Pexp(- fr x~(j)A:dxU)),

(2)

and I" is a rectangular (R × T) loop, T>>R. The subscript J labels the representation under consideration. For physics we are usually interested in the fundamental representation, but knowing W(j)(I') for all J enables us to study the probability distribution for the group element associated with Y. One can obtain the renormalization group equation obeyed by V by extending the argument of ref. [7] to the case g2 @ g* 2, noting in passing that the key assumption of multiplicative renormalizability has been proved in ref. [8]. We find that, under a renormalization group transformation of finite scale X, V transforms as follows 365

Volume 103B, number 4,5

PHYSICS LETTERS lently

V(j)(R, g2) = g(j)0k ' g2) + )k-1 V(j)(R/~k, g2 0k)) , g2(x) lnX=-gf 2

(3) /3(g2dg2' ,,)

G (2) (j)~(R ,6,~2.~1 j,R~oo = ( °)2

Here/2(j) is the result of the multiplicative renormalization or alternatively the free energy in a shell ( A - l / X <~ r < A - 1 ) around the loop. Specializing to infinitesimal scale changes X = 1 + 6 we find, for consistency

= 6 ~(j)(g2)/eo,

(4)

and hence

(5) = f(j)(gZ)/R O. R 0 is a reference distance which we have extracted from ~'(j) in order to make it dimensionless. There are a few comments about this equation which should be made. Firstly, it is inhomogeneous. This is just a consequence of the fact that we are workhag with the logarithm of the correlation function. Indeed we can write the analogous renormalization group equation for the spin system ~,~ 2-) = 0 (6) [R O/OR +[j(g2)O/Og2 +7(j)(g2)] G(2) (j)t~,g in the same form:

[R O/OR +/3(g2) O/Og2 ] In [GI2] (R, g2)] - 1 = ,),(j)(g2).

(7) Secondly, the " 1 " on the left-hand side of eq. (5) is

not an anomalous dimension; it is in fact the canonical dimension of the source. This is zero for the point source in a spin system [eq. (7)] and one for the line source in the gauge theory. Thirdly, eq. (6) seems to differ from the equation for V derived in ref. [6] and quoted in ref. [7] .1 : (8)

The reason for this is simply the renormalization convention. Even in spin systems one can choose a renorrealization which makes the anomalous dimension ~,(g2) vanish to all orders! One simply renormalizes so .that the magnetization (o) = constant, say 1, or equiva4:1 There is also a sign error in ref. [6] which has crept into ref. [7]. 366

(9)

O/Og2 + 1,),(g2)] (o) = 0,

(10)

we see that 3,(g 2) = 0 for all g2. There is nothing wrong with this as long as one stays in the magnetized phase but it makes studying the way (o) -~ 0 as g2 -+ g2* rather difficult. However, one can easily obtain a more sensible scheme. Define G(j)(R, ~(2) g 2 ) by performing a -

finite renormalization on ~(2)(R (j), , g 2 ) =

[R O/OR +/3(g2) O/Og2 + 1] V(j)(R, g2)

[R O/OR +(3(g2)O/Og2 + 1] V(j)(R,g 2) = 0.

= 1.

Since (o) satisfies

g2=g2(1)"

[fi(g2)

/J(j)(1 + 6, g2)

30 July 1981

G(2)(R (j),, , o~2~. J"

-1 2 )G(j)(R,g (2) 2 ), Z(j)(g

(I1)

with the renormalization condition G(2)(R g2)] R = 1, (12) (j)~ , :R o where R 0 is a finite reference point rather than the infinite one in eq. (9). Thus eq. (t 1) at R = R 0 reads

Z(J) (g2) : G(2)(R(J)"O' g2),

(13)

It is then easy to verify that the new anomalous dimension is given by "~(j)(g2) = 7(j)(g2) + [j(g2)(O/Og2 ) In

Z(j)(g 2) (14)

= 13(gZ)(O/Og2) in G(j)(R (2) O, g 2 ), using eq. (13) and .),(j)(g2) = 0. This is just what has happened in Fischler's calculation of V(R, g2) [6]. By leaving out the graphs which do not connect the two sides of the loop he has implicitly arranged that ~-(g2) -z 0. Using the analogy displayed in eqs. (5) and (7) we see that a corresponding finite renormalization to (11) and (13) is

"V(j)(R,g2) = V(j)(R,g 2) - V(j)(Ro, g2), thus if V(j)(R, g2) satisfies eq. (8), isfy eq. (5) with

(15)

"V(j)(R, g2) will sat-

~(j)(g2)/R 0 = - [/3(g2) O/Og2 + 1] V(j)(Ro, g2). (16) Therefore we may simply take Fischler's result for V(j)(R, g2) and evaluate ¢(j)(g2) using (16). ff we work with the gauge theory in d = 4 + e dimensions, the renormalization group/3-function takes the form [9]

Volume 103B, number 4,5

PHYSICS LETTERS

where 12 is an arbitrary momentum scale and where

68/-,2 g6/(161r2)2 ¢1(g2) = eg2 - ~ C A g4/16rr2 -- ~-~A

(17)

+ O(g 8), where facdfbcd = CA 6ab and fabc are the structure constants of the group. The non-trivial fixed point is found from (17) to be g2* = (16rrZ/CA)[3e _ z-g-~e 153 _2 + O(e3)l,

(18)

indicating a second-order phase transition for sufficiently small values of e. At this fixed point the potential has the form [7]

V(j)(R, g2*) =(RI/R)[V(j)(RI,g 2" )-~(j)(g 2* )/R 0] 2"

(19)

+ ~(j)(g )/R O,

and so from Peskin's definition of the exponent r~(j) [7]

2rr(d- 4

xa(j)xa(j)= Cj1. Thus if we take Ro =12-1, for exam pie, we easily find from eqs. (16) and (17) that

~(j)(g2)/R 0 = (12/8n){2g2Cj -g2eCj(3" +In 4n - 2) +g4(CACj/161r2)[~(3" + In 4~) + ~ -- 441 -5-

(26)

+ O(g 6 , g4 G g2e2)). A more elegant form can be obtained if we choose R 0 such that In 12R0 = 1 - 5-(3' + In 41r). This corresponds to a minimally subtracted scheme in which there is no dependence on e in ~-(g2): ~-(j)(g2) = (1/87r)

(27)

X [2Cjg 2 + ~(CACj/16n2)g4 + O(g6)].

as may be verified by solving eq. (5) at the fixed point. Here R 1 is an arbitrary reference distance. In Fischler's scheme the corresponding equation is simply V(j)(R,g 2* ) = R -1 [R 1 V(j)(R,g2*)], (20)

V(j)(R,g 2* ) =

30 July 1981

+'O(j))/R,

(21)

one finds

r~(j) + d - 4 = -(1/2~r)RIV(j)(RI,g 2* ).

(22)

We will work instead with the potential defined in eq. (15) and so, taking the arbitrary distance scale R 1 to be R0, we find eq. (19) becomes V(j)(R,g 2* ) = ~(j)(g-9" )/R + ~(j)(g 2* )/R O, (23) and thus the exponent r/(j) is now given by

rl(j) + d - 4 = (1/2~r)~(y)(g 2* ),

(24)

which is analogous to the spin theory result [ 10] r / + d - 2 = T(g2*).

The calculation of ~-(j)(g2) in 4 + e dimensions is straightforward. The perturbative calculation of the potential carried out by Fischler yields [6,7]

Using the value o f g 2. given in eq. (18) and the definition (24) of r/(j) we find from eq. (26) or (27) that +

=

+

35

2

+ o(d)l.

(28)

This exponent was calculated in ref. [7] from eq. (22) and agrees with the result above, except for a slight numerical error in the O(e 2) term. One of the things we can do with the expression for ~(j)(g2) is to look for an exponent analogous to the quantity "fl" in a spin system which tells us how the magnetization vanishes [ 10] (a) cc (g2* _ g2)t3

as g2 _+g2*,

(29)

fi = _3,(g2*)/2fl'(g2*). One might naively expect the radiative corrections to the quark self-mass ½ V(R ~ 0%g2) __1 = 5- Voo to blow up as one enters the confined phase, driving the Wilson loop to zero at large R. Let us see what actually happens. From eq. (15) we see that Voo = -V(Ro,g2) and thus eq. (16) gives

~(g2)/Ro = [fi(g2)O/Og2 + 1] V .

(30)

Solving this equation in the neighborhood o f g .2, where fl(g2) = - u - 1 (g2 _ g*2) yields

V(j) (R, 12,g2) = _g2 Cj/ 4rrR + (g2eCj/4rrR)[ln gR + ½(3" + In 4~')]

V(u)--[ul"(A+t,?~(u')lu'l-~-ldu

-- (g4CA Cj/16~'2)(1/4rcR)

(31)

X [ ~ in 12R + -5-(7 + In 4n) + + O(g 6, g4e, g2e2),

')

astt -+0

,

(25) 367

Volume 103B, number 4,5

PHYSICS LETTERS

where u = g2 _ g2*. The first term tends to zero as the inverse correlation length (which is typical for any quantity with dimensions o f mass at a second-order transition), while the second contribution

(32)

I.t v f ¢(.')1.'1- u-1 du',

30 July 1981

We wish to thank E. Fradkin and J. Stack for useful discussions. AJM wishes to thank Professor J. Kogut and the Physics Department, University of Illinois for their hospitality during the period this work was being carried out. This work was supported in part by NSF grant PHY79-O0272.

References

is analytic at u = 0 if ~'(u) is, that is, U

=

mm-V

lul m, (33)

provided v is not a positive integer (if v is a positive integer there are logarithmic corrections). The reason for the lack of any really singular behavior is not hard to find. The interquark potential at g2 =g2* goes like I/R [eq. (19)] which is n o t confining. One concludes therefore that there is no/3-like exponent for a gauge theory. A final comment. At higher orders in g2, the A p p l e q u i s t - D i n e - M u z i n i c h effect [11 ] implies that the interquark potential is not analytic in g2. Will this be reflected in if(g2)? If so, this will be an unusual effect for a renormalization group function.

368

[1] A.A. Migdal, JETP 42 (1976) 413, 743. [2] L.P. Kadanoff, Ann. Phys. 100 (1976) 359; Rev. Mod. Phys. 49 (1977) 267. [3] A.J. McKane and M. Stone, Nucl. Phys. B163 (1980) 169. [4] F. Wegner, J. Math. Phys. 12 (1971) 2259. [5] K.G. Wilson, Phys. Rev. D10 (1974) 2445. [6] W. Fischler, Nucl. Phys. B129 (1977) 157. [7] M. Peskin, Phys. Lett. 94B (1980) 161. [8] V.S. Dotsenko and S.N. Vergeles, Nucl. Phys. B169 (1980) 527. [9] W.E. Caswell, Phys. Rev. Lett. 33 (1974) 244; D.R.T. Jones, NucL Phys. B75 (1974) 531. [10] E. Br~zin and J. Zinn-Justin, Phys. Rev. B14 (1976) 3110. [11 ] T. Applequist, M. Dine and I.J. Muzinich, Phys. Rev. DI7 (1978) 2074.