The critical exponents of reggeon field theory

Volume 67B, number 1

PHYSICS LETTERS

14 March 1977

THE CRITICAL EXPONENTS OF REGGEON FIELD THEORY John L. CARDY

CERN, Geneva,Switzerland Received 23 December 1976 By using the method of Lipatov to determine the behaviour of perturbation theory at large order, we derive a convergent numerical procedure for the calculation of the critical exponents of Reggeon field theory, based on the loop expansion and the e expansion. We find the results 7 = -0.26 ± 0.02, z = 1.13 ± 0.01, and h = 0.49 -+ 0.01.

Recently, Lipatov [1 ] has developed a simple method for determining the behaviour of perturbation theory at large order in quantum field theory. The technique has been extended and applied to the calculation of critical exponents by Br4zin et al. [2]. The critical exponents of Reggeon field theory [3] determine the asymptotic behaviour of elastic scattering when a(0) = 1 according to the following laws: Otot o= (In S)-'r [1 + O((ln s)-X)],

(1)

oe I cc On s) - 2 7 - z [1 + O ( ( l n s ) - ~')].

(2)

Previous calculations of these exponents, based on the loop expansion, the e expansion, or other methods [4], have been unreliable because of their lack of convergence or of any estimate of the error involved. We shall show that the addition of the knowledge of the large order behaviour of the perturbation expansion enables one to convert the loop expansion and the e expansion into convergent procedures. The current knowledge of the loop expansion [5, 6] (out to three loops), and the e expansion [7, 8] (out to O(e2)), then gives the exponents to an accuracy of better than 10%. The Green functions of Reggeon field theory are defined by the generating functional

Z[J,J] = f c ' o t ~

exp(-A [~b, ~ , J , Y ] ) ,

(3)

where

A [~, ~,J,J]

=fdy dDb{½~ayt~ +a'OV~'Vtk + A ~ (4)

+ ~ i r 0 ff ff(~b + ~k) +J~k +)-~ + counterterms). In the loop expansion [5], the renormalization

group functions are defined in the theory with A > 0. All quantities of interest will be expanded in the dimensionless coupling g 0 = ro/(a'oA) 1/2 in D = 2. Thus t we set a 0 = A = 1 and r 0 = go for convenience. The method of Lipatov [1 ] consists of using the Cauchy formula for the coefficient ofg~0 in the functional integral, and estimating this using the method of steepest descent as n ~ oo. The calculation for Reggeon field theory follows closely that of Br~zin et al. [2] for ~b2N theory, and will be given in detail elsewhere. The results are as follows; for a Green function G(r) with r external lines,

G(r)

(...+ ~_1 c(r'F(n+~+P)(-g20/m) n ={

nlarge

[...+g-~

0nlarg e

\

z

(r e v e n )

z/

c(r)r(n+~-+e+L~(-g2_l~) n ~' 2 2 2] 0 (r odd),

(5)

where C (r) depends on the external momenta, etc., but not on n. A is the action of a solution of the equations of motion with an unphysical value of the coupling constant (in this case, a real triple-Pomeron coupling) .Y[~,~] = fdydOb { ~ 1- ay~ + v ~ - v ~ + ~ - ½ ~ ( ~ + ~ ) )

(6) The integer p is the number of arbitrary constants appearing in such a solution. In the physical case D = 2, we have p = 3, corresponding to translational invarlance in rapidity and impact parameter. For this case, we have been unable to find a solution of finite action analytically, nor prove that it must exist. However, it is intuitively clear that such solutions should exist, and, since only the action is required, one can use variation97

Volume 67B, number 1

PHYSICS LETTERS

al techniques to estimate it. The error should be small even if the trial functions are rather crude. We use trial functions of the form

¢~y, b) : f(y)u(b),

~2(y,b) : f ( y ) u ( b ) .

(7)

The functional iT[C, ~] then reduces to the form

14 March 1977

tion expansions, but indicate that they are Borel summable. The Borel transform of 3(g), defined by dividing the coefficient o f g 2n+l by n!, has radius of convergence A, with a leading singularity of the form [t + iT] -4.5. If instead we define a modified Borel transform by oo

3(g) =g ~ tt

+'~ i F o f f ( f + f ) ) ,

f:

(8)

3.

n=0 r(n + ~_)g2n,

where X 0 = f t ( g u ) 2 + u21 dDb/fu 2 dDb,

(9)

--ir0 : f u 3 dDb/ f u 2 dDb.

(10)

Apart from the factor fu2dDb, eq. (8) has the form of the action for the D = 0 problem, for which the solutions of the classical equations have been classified [9]. The only solutions with finite action at arbitrarily high rapidity have the action -2~x2/r 2 (S O> 0). The problem is thus reduced to finding the lowest extremum of the functional

=

fu 3 dDb

2 dDb .

(11)

Variational estimates with combinations of rational and exponential functions yield .~/4rr ~, 4.47 in D = 2. The renormalization group functions ~g), 7(g), z(g) are defined in terms of derivatives with respect to A of the two and three-point functions evaluated at zero external energy and momentum. This differentiation introduces an extra factor of n. One must then substitute the renormalized couplingg for the bare coupling go" This does not change the large n dependence. Finally, we have, in D = 2

3(g)=-~g+...+g

~

n large

"},(g) = - - ~ ( g 2 / 4 ~ ' ) + . . . +

CP(n +~)(_g2/~)n,

~ C'['(n +.~)(_g2/~)n, n large (13)

(14)

(15)

then fl(t) will have only a weak singularity at t = - A . 3(g) can be recovered from 3(t) by 3(g) = g f e - t t 11/2 ~(tg2) dt. 0

(16)

We note that (16) involves a continuation of 3(t) outside the circle of convergence, where the perturbation expansion in t diverges. This difficulty can be overcome by mapping the t plane, cut from -"~ to - A , into the unit circle Iwl < 1, for example by the transformation w=

(t + ~z~)l]2 _ ~T112

(t + ~)t/2 +iyl/2 '

(17)

which also maps t = 0 into w = 0. If3"(t) is then analytic in the cut plane (which assumes that all extrema of (6) have real action >~,4) then, as a function of w, it is analytic in Iwl < 1, and therefore has a convergent perturbation expansion in w for all t in the interval (0, 0% From the work of Dash and Harrington [5] and Harrington [6], we know the renormalization group functions through three loops. We can therefore calculate the perturbation expansion of 3", etc., in w through O(w3). For examp!e, from ref. [6]

3(g)/g = - 2 +1 0.344 h - 0.165 h 2 + 0.124 h 3 + O(h4), (18)

(12)

z(g) has a similar character to 7(g), while k(g) = 3'(g) gains an extra factor of n on 3(g). These results imply the divergence of the perturba98

3ng 2n,

=0

where h = g2/4n. The perturbation expansion of 3(t) in w is as follows: 3(t) = -~3-13 {--~ + 0.946 w + 0.811 w 2 1 i.~-1

+ 0.224 w 3 + O(w4)).

(19)

Volume 67B, number 1

PHYSICS LETTERS

Table 1 Exponents calculated in the loop expansion, to two decimal places. Errors corresponding to a 10% uncertainty in A are shown, where significant

Table 2 Exponents calculated in the ~-expansion, to two decimal places. Errors corresponding to a 10% uncertainty in R are shown, where significant.

Number of loops

g21/4r¢ -3' z h

2

3

3.15 +- 0.3 0.24 1.12 0.68 -+0.01

3.00 -+ 0.3 0.26 1.13 0.58 -+ 0.01

Although (19) is apparently no more convergent than (18), we know that in fact the higher order terms in (19) must go to a constant (since the radius o f convergence is unity), while in (18) they will start to diverge rapidly (around O(h6)). We then reconstruct/3(g) via (16) and look for a zero g = g l at which/3'(gl) ~> 0. The same method is then applied to the other renormalization group functions to evaluate them at g = g l ' This yields the exponents 3', z and ~. The results obtained by truncating the series at the two and three loop levels are shown in table 1. The results at the single loop level are not sensible. The closeness of the two and three loop resuits indicates the rapid convergence of our procedure. We conclude with several r.emarks. (1) One can test for the sensitivity of the results to the exact value of A . We find that a 10% error in~T resuits in a similar error in g l , but only a small variation (~1%) in the exponents. (2) One can further m o d i f y the Borel transform (15) by dividing by F(n + b') instead o f F(n + ~ ) . The resuits are insensitive to such a change if Ib' - ~ 1 ~ 1. (3) The small change between the two and three loop results then indicates (since the procedure is convergent) that the error at the three loop level is less than 10%, at least for 3' and z. (4) These results agree with the Padd-Borel analysis by Bronzan et al. [10] o f the three loop expansion, but improve the error involved. We have applied the same techniques to the zero of/3(g) in D -- 0, where it is known [10] t h a t g 1 = 2. At the three loop level we find g l ~ 2.07, which is an improvement o f a factor of more than two in the error over the Padd-Borel method [10], which f i n d s g I ~ 1.84 at three loops. (5) We have also determined the behaviour o f the e expansion o f the exponents at large orders. The de-

14 March 1977

-~, z h

O(e)

O(e 2)

0.08 -+0.01 1.04 0.49 -+ 0.01

0.18 -+0.01 1.09 -+0.01 0.49 -+0.01

tails of this calculation will be given elsewhere. We find 3,=-(e/12)+...+

~ n large

C"F(n+6)(-e/R) n,

(20)

where R ~ 6.91. The exponents z and ~ have a similar character. The techniques described above can then be applied to calculate at c = 2, using the O(e 2) results of Bronzan and Dash [7]. These results are shown in table 2. For 7 and z they do not converge, nor agree with the loop expansion. This is because (20) predicts that the e expansion be an alternating series for large n, but, to O(e2), 7 and z do not alternate. However, the series for ~ does alternate, and diverges rapidly, and the method is very successful, giving a better result than the loop expansion. The author acknowledges useful discussions with J. Zinn-Justin on the equivalent problem in statistical mechanics.

References [1 ] L.N. Lipatov, JETP Lett. 24 (1976) 179, and Leningrad Preprints. [2] E. Brdzin, 1.C. Le Guillou and J. Zinn-Justin, Saclay Preprint D.Ph.T 76/102 (1976). [3] H.D.I. Abarbanel et al., Phys. Reports 21C (1975) 119. [4] M. Moshe, Lectures at Canadian Institute of Particle Physics Summer School (1976), Preprint UCSD-10P10166 (1976). [5] J.W. Dash and SJ. Harrington, Phys. Lett. 59B (1975) 269. [6] S.J. Harrington, University of Washington Preprint RLO1388-709 (1976). [7] J.B. Bronzan and J.W. Dash, Phys. Lett. 51B (1974) 436; Phys. Rev. D10 (1974) 4203; Phys. Rev. D12 (1975) 1850. [8] M. Baker, Phys. Lett. 51B (1974) 158; Nucl. Phys. B80 (1974) 62. [91 D. Amati, L. Caneschi and R. lengo, Nucl. Phys. 13101 (1975) 397. [10] J.B. Bronzan, LA. Shapiro and R.L. Sugar, Phys. Rev. D14 (1976) 618. 99