The 3-loop calculation of critical exponents in the Reggeon calculus

Volume 62B, number 4

PHYSICS LETTERS

21 June 1976

T H E 3 - L O O P C A L C U L A T I O N O F C R I T I C A L E X P O N E N T S IN T H E R E G G E O N C A L C U L U S ~' S.J. HARRINGTON University of Washington, USA Received 21 April 1976 Critical exponents are calculated in the 34oop approximation for the Reggeoncalculus. The values 3' = -0.13 and z = 1.07 are obtained. This work is concerned with the asymptotic properties of diffraction scattering. The scattering is described by Gribov's Reggeon calculus [I ]. If the Pomeron intercept is assumed to be 1, then the asymptotic behavior of the elastic amplitude was shown to be of the form

[2]: T(s, t) = son s)-~ f [ t ( l n s) z] ,

(1)

provided the Gell-Mann-Low function has a zero (this is due to the coincidence of an infinite number of branch points at t =0). The critical exponents 7 and z do not depend upon the magnitude of the Reggeon coupling, and have been extimated using a variety of approximation schemes which are also used for calculating critical behavior in solid state physics. Calculations have been carried out using the epsilon expansion [3], the N-loop expansion [4], and the high-temperature expansion [5]. The results of these calculations obtained to date, have been somewhat unsatisfactory. The epsilon expansion has been carried out to second order, but unfortunately the second order term is the same magnitude as the linear term, creating some doubt as to the convergence of the series. The 2-loop calculation has also been performed; however, in 2-loops the GellMann-Low function does not have a zero. The hightemperature expansion has so far only been carried out to low order, and has given a wide range of possible values. Higher order calculations should aid in dealing with these problems by giving further insight into the rate of convergence, as well as new values for the exponents. Supported by the U.S. Energy Research and Development Administration.

Since for each loop order, the number of calculations necessary increases on the order of (2n)!, while the complexity of each calculation increases exponentially, considerable effort is required to extend the loop order. This makes calculation beyond 2-loops at least tedious and error-prone. The N-loop expansion, however, lends itself to automation, and the 3-loop calculation can be carried out entirely by computer [6]. Following the notation of ref. [5] the following definitions are made: r 0 is the bare coupling,/x is the mass. The dimensionless bare coupling is

(2)

g~o2 0, A, -ro/a Z -1 = ~--~i F(1,1)(E, K 2 , r 0, a o , A)

(3) = l + a 2 g 2 +a4g 4 +a6g6o, t~'Z - I t

ao =

m

--

f 1 ~ iF(1,1)(E,K 2 , r o , a o,A) oto aK 2 e

(4)

1+c2g02 + C494o + c6g6o ,

r,~la)(Ei,Ki, ro, a,o,A)=_~_~,a~(1 ro +cl3~ . +dsgo+d7go) 4 6 (5)

'

Ref. [5] gives the values: (4r0a2=-l/8,

(4r0c2=-1/16,

(4~r)d3---1/2,

(41r)2a4 = 0.0579, (4rr)2c4=0.0253 , (47r)2d 5 = 0.3893. This calculation yields the values: (4rr)3a6=-0.0387, (47r)3c6=-0.0146, (47r)3d7=-0.387 , where numerical errors are estimated at less than 10%. The renormalized coupling is 433

03

'

I

'

I

21 June 1976

PHYSICS LETTERS

Volume 62B, number 4 '

I

'

I

'

[

'

I

' I

02--

3-

It gives a critical couphng o f g 2-- 1.4 (47r). Finally the two critical exponents are

I /

LOOP//

--

aA-AaZ a2g2 + ( 2 a 4 _ 2 w a 2 _ a 2 ) gc 2 4

7-Z

/ /

01--

+ (7w2a 2 - 2w4a 2 - 8 wa 4 + 4 wa 2 + 3 a 6 - 3 a2a 4

OP

0 + a 23) g 6c = - 0 . 1 3 ,

(11)

-01 and

-02

2 4 z = 1 - 7 + c2g2 + (2c4 - 2 w c 2 - c2)gc

-03

+(7w2c 2-2w4c2-8wc4+4wc

-04--0.~

--

,

I

02

~

I

04

,

I

06

,

I

08

~

I

I0

J

I \,

1.2

=

Z3/2 l"(1,2)(0,0,ro , O~'o,A) (t~'A)l/2

I

14

(6)

2+ 4 g o ( l + w g o w 4 g o+ w6 g6) .

where w = d 3 - a 2 - c2/2 = - 0 . 3 4 4 / ( 4 7 r ) ,

w 4 = d 5 - d3(a2 + c2/2) + a 2 + (3/8)c 2 + a2c2/2

(12)

- 3c2c 4 + c 3 ) g 6 = 1.07.

Fig. 1. The Gell-Mann-Low function for the 1-, 2-, and 3loop cases.

g = (2~')3/2

2 +3c 6

(7)

The values obtained here, 7 = - 0 . 1 3 and z = 1.07, are somewhat smaller than values obtained in lower order calculations. The first and second order epsilon expansion values for 7 are - 0 . 1 7 , and - 0 . 3 2 respectively. For z, they are 1.08, and 1.15. The l-loop expansion yields - 0 . 1 8 for 7 and 1.08 for z. The effect of the 3-loop corrections on the Gell-Mann Low function is to cancel the 2-loop contribution, which inhibited the zero, giving a critical coupling very close to that o f the l-loop case (see fig. 1). The 3-loop corrections to the coefficient o f the critical exponent expressions are small. These are encouraging signs. They generate some confidence in the expansion, although the/3 function may lose its zero for even orders. It would be exciting if a third order epsllon expansion were to confirm these findings.

(8)

- a 4 - c4/2 = 0.262/(47r)2,

w 6 = d 7 - ( 1 / 2 ) c 6 + ( 3 / 4 ) c 2 c 4 - (5/16)c 3 + (3/8)d3 c2 - ( 1 / 2 ) d 3 c 4 - ( 1 / 2 ) c 2 d 5 - a 6 + 2a2a 4 - a32 + a2d3 _ (1/2)a2c2 - a4d3 + (1/2)a4c 2 - (3/8)a2c2

(9)

+ (1/2)a2c 4 - a2d 5 + (1/2)azd3c 2 = -0.266/(4rr) 3 . The Gell-Mann-Low function is: /3= - g / 2 (1 +2 w g 2 + ( 4 w 4 - 6w2)g 4 (10) + (24w 3 - 2 6 w w 4 + 6w6)g6 ) .

434

References

[1] V.N. Gnbov, Soy. Phys. JETP 26 (1968) 414; A. Migdal, A. Polyakov and K. Ter Martirosyan, Phys. Lett. 48B (1974) 239. [2] H.D.I. Abarbanel and J.B. Bronzan, Phys. Lett. 48B (1974) 345. [3] M. Baker, Phys. Lett. 51B (1974) 158; J.B. Bronzan and J.W. Dash, Phys. Lett. 51B (1974) 496. [4] J.W. Dash, S.J. Harrington, Phys. Lett. 59B (1975) 249. [5] J. Ellis, R. Savlt, Nucl. Phys. B94 (1975) 477. [6] S.J. Harrington, Masters thesis, University of Washington.