Volume 51B, number 2 PHYSICS LETTERS 22 July 1974

T H E WILS ON A P P R O A C H T O T H E R E N O R M A L I Z A T I O N G R O U P

A N D T H E R E G G E O N C A L C U L U S

R. JENGO Istituto di Fisica Teorica dell'Universitd di Trieste, LN.F.N. Sezione de Trieste, Italy

Received 20 May 1974

Using a technique by Wilson which deals with the bare Lagrangian we investigate the problem of the critical be- haviour of the interacting Pomeron. We find, besides the strong coupling solution recently discussed, also the weak coupling solutions originally presented by Gribov and Migdal.

The problem of summing the multi-Pomeron cuts which all accumulate in the angular momentum plane J near J = 1 and determine the high energy behaviour of the total cross section and of the diffraction processes is viewed in the framework of the Reggeon calculus [1] as the search of the infrared behaviour of a non relativistic field theory for small Pomeron energy E(E = 1 - J ) and momenta k l (t = -k2 ) . Recently this investigation has received an important contribution [2, 3] by the results obtained with the use of the renormalization group techniqiaes. A strong coupling solution has been found in which the total cross section behaves at high squared c.m. energy s like (log sf/. At the first order in e, the dimensions of the transverse space being D --- 4 - e (the physical case is for e = 2), 7 = e/12. A second order calculation of the e expansion has also been performed [4]. In this note we present a study of the problem based on the Wilson approach [5] to the renormalization group for the statistical mechanics and we find besides the strong coupling solution also the weak coupling solutions originally proposed by Gribov and Migdal [6], in which the total cross section goes to a con- stant. The Wilson method [5] deals with the bare field theory, which we take to be specified by the action A

A

A = f [dp] {~o+*~O(ao + E - k 2 +a4p4q - DUQ) o

1 i(~o+,~o+,~o + h.c.) (go +g2 p2 + "'') --:2

+ ~ (~0+*~0+*~0"~0) (~"o +~'2 p2 + "")

+-~ (~0+*~o+*~0+*~o + h.c.) 0Co +f2p2 + ...)

+ { (~o+*~o) n/2) (h (n) +h(~)p2 + ...)).

(1)

We have introduced a symbolic notation: [dp] means dE i dDki, ~o(p) is the Pomeron field which depends on (p) = (E, k), p2,p4.., mean homogeneous polynomials of degree 1, 2, ... in E and k 2, { (~0+*~0) n/2} means the possible couplings o fn Pomerons. The slope ct' has been put equal to 1. A is a cut-off: it here represents our ignorance on what happens at high E and k. The problem is the evaluation of the Green functions G(~ ) (Pi) for E, k ~ 0. Following the Wilson method [4] we integrate over the high frequency modes ~o(p) for A/2< p < A and we redefine

~ ' ( p ) = ~ / 2 ) (2)

to obtain

G(~ ) (pi) = ~-m 2-2-D G(~)(2pi). (3)

A' is the new action in terms o f ¢ ' , specified by the parameters a~,g~, ~ , f~'., h(i n)'. ~ is fixed by the re- quirement that the coefficients of the kinetic terms E~0'+*~0 ' and k2~0'+*~0 ' inA' are the same as those of E~0+*~0 and k2~o+%0 inA. There are no invariance reasons why E~o+*~0 should have the same coefficient as k2~o +* ~0, therefore this would imply two conditions

for one quantity. So, we interpret the transformation in a more general way: p/2 will mean both El4 and lkl/21+8 ~" and 8 will then be fixed by the require- ment of the invariance of the kinetic term.

If, after transformations, we find a fixed point for which A' = A and evaluate the corresponding ~', eq. (3) gives the behaviour of G(~ n) (Pi) for Pi-* O. If at the fLxed point 8 4= 0 then this "critical" behaviour is ob- tained when E and k 7.'k/goes to zero at the fLxed

1+6 ratio kfk i[E . To write the relations among the parameters o fA ' and A let us define 143

Volume 51B, number 2 PHYSICS LETTERS 22 July 1974

(1/i)p(1,1) = a o + E - k 2 +a4p4 + ... + ~..

I-'(1'2) = go +g2 p2 + "'" +R(l '2) (4)

( l / i ) 1 -'(nl2'nlz) = h(o n) + h(~)p 2 + ... +R (nl2'n/2) ,

where F are the bare amputated irreducible Green functions, and, as before p2r is a polynomial of de- gree r in the external variables Ei, k i'k/. ~ and R are in general functions of the parameters of A, of the cut-off A, and of the external variables. We call A ~ ( A R ) the difference AZ = Z(A) - Z(A/2) (AR = R ( A ) - R ( A / 2 ) ) . A~ and AR can be Taylor expanded in the external variables Pi near Pi = 0, since the range of the internal loop variables is from A/2 to A. We need in particular the quantities

a Q = A ~ p = o, C = ~ - ~ A ~ p = o, B = - - ~ - ~ A 2 ] I p = o,

G=AR(1,2)tpi=O, S = A R (2,2) pi=0 , T = A R (1'3) pi=O

At the lowest order in the interaction we have

5 = (1/2 log 2) ( B - C )

~ =41+ D/4+(1/21og2)((D/4)B - C(1/2+ D/4)~ ( 6 )

and the transformation which brings A into A' is t

a ° = 4(a ° + Q - a o C ) (7) t go = 2(2-O/2) [go + G-go{(D/4)B + C(3/2 -D/4)) ] (8)

3`0 2(2-D) [Xo +S-Xo{(D/Z)B +C(2-D/2))] (9)

fo = 2(2-D)[fo+ T-fo{(D/Z)B+ C(2-D/2))] (10)

The other couplings, like a4, g2, 3"2, f2, h(o n) -'" are called "irrelevant" in the Wilson method [ 5]. They have dimensions [p] - ~ with t~ positive (at least for D 1> 2), for instance [a 4 ] = p - 2, [g2 ] = P-D~2, [3'2] = p-D , and their zero order transformation rules are

, , = 2 - D x 2 + . . . a 4 = ¼ a 4 + . . . , X 2

g2 = 2-D/2g2 +''' ' h(n) '= 2(2-(n-2)D/2)h(n) + .... ( l 1) O

I f D = 4 - e with e small even 3`o andfo are irrelevant. All the irrelevant coupling are pushed to zero (or to higher orders in e for the strong coupling solution)

after many transformations. This is clear from eq. (11). We can now see two possible critical behaviours. First, the strong coupling solution: go goes to a fixed point different from zero. In such a case G, B, C and Q are dominated in the lowest order by the couplingg o (G is o f o r d e r g 3 and so on) and eq. (8) gives the fLxed point for go = go. It can be shown that the fixed point is indeed attractive for the imaginary triple Pomeron coupling, see eq. (1), and it is of the order.v/e. Eq. (7) gives then the fixed point for a o, of the order e. That one is however repulsive and in order to reach the fixed point in the (go, ao) plane we have to stay from the beginning on a critical line, or on a critical surface in the general space of the para- meters, like in the problem of the critical temperature in statistical mechanics [4]. The critical line corres- ponds, in the approach which deals with the renorm- alized Green functions, to the requirement that the renormalized Pomeron singularity is at J = 1. Why the Pomeron is at J = 1 it is not explained by the renormalization groups techniques. Once the fixed point has been reached, eqs. (6) and (3) give the be- haviour of the Green functions and it is easy to check ~ that the result is the same found in refs. [2] and [3].

Second, the weak coupling solutions. Those solu- tions correspond to more particular critical surfaces in the space of parameters. Along those surfaces asymptotically go < < g2 and the triple Pomeron inter- action is dominated by the termg2p2(~0+* ~0,~o+ h.c.) which is the linear triple Pomeron zero of Gribov and Migdal [6]. Those solutions given an asymptotically free theory, in particular the Pomeron propagator behaves like the free propagator and 8 = 0. Case a). Asymptotically go << g2 and X o < < X2, fo <<f2 : in this case also the four Pomeron vertex has a linear zero. Now G is dominated by the coupling g2, it is of the order g~, S and T are of the order X 2, f2 or X2.f2 (B and C contribute to higher order cor- rections). In the physical case D = 2 we have from eq. (11), after l transformations,

g2( l ) "~ 2 - 1 ' X2(I) "~ f 2 ( l ) ~ 4 -1"

The surfaces along which the solution is asymptotic-

* That is at the lowest order in e. We do not presently know if the second order would give in our approach the same result of ref. [4].

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Volume 51B, number 2 PHYSICS LETTERS 22 July 1974

ally reached are ~, from eqs. (8), (9) and (10), 2

go(t) x2(t) •

The Green functions F (n,m) for n + m > 4 are dominated by the couplings h(n+m). Taking into account the ~ factors (from eq. (6) ~" = 43/2) one find that they go to a constant, i.e. they are more regular than expected from the canonical dimen- sions. Of course, also here, like before and in the following, a o must stay on a critical surface, which in this case and in the case b gives asymptotically

a o = 0. Case b). Asymptotically go < g2 but ~0, f0 > ~2,f2. The theory is again asymptotically free because there is an attractive fixed point at the origin for ~0 and f0, it is only reached more slowly for D = 2, i.e. like an inverse power o f logarithm (like in the relativistic case of tfl 4 in 4 space-time dimensions). In eqs. (9) and (10) S and T are dominated by ~0, f0, they are o f order 2 2 ~0, f0, k0fo- By looking closely at the relevant graphs and at their relative numerical weight one finds that after l iterations *

~o(t) - 1/log (2 l) fo(l) ~ 1/(log (2l)) 3

Again, there is a critical surface along which go(l) ~ g2(l) ~0(l) <~ g2(l) the triple Pomeron is dominated by g2 and therefore it goes to zero linearly in E, k 2, except for extra factors 1/log E, 1/log k 2 which are easily calculated. The Green functions F (n,m) for (n + m ) even and bigger than 4 have the behaviour given by the canonical dimen- sions, except for extra 1/log p, for (n + m) odd and bigger than 4 they are more regular than the canonical expectation. It is as the theory became asymptotically ~4, which is infrared free.

In conclusion, starting from a quite general form for the action A, we have found various critical surfaces in the space of the parameters. The strong coupling solution is the solution for which the minimum number of conditions are necessary, then

For instance, if G = g3k, then asymptotically the line in the (go, g2) space isgo(/) = - (16/15)kg3(l).

~+ There are also more particular solutions, not essentiadly different from the one presented.

we find also the weak coupling solutions b) and a), which correspond to a higher number of constraints. Why nature should select any of those critical sur- faces is left unexplained by the renormalization group, in our as well as in other approaches.

It could be that the critical surfaces for the weak coupling solutions intersect the axis g2 = f2 = ~'2 = 0 = a 4 = h(~) = ... and therefore that this solution is obtained formally starting from an action in which only go, ~0, fo are :/: 0 #. They must however have some particular value (e.g., for dimensional reasons, go must be some specific fraction of the cut-off A). Formally, the weak coupling solution disappears when A ~ oo in the limit of a local renormalizable field theory with usual propagators. Since we do not see any special reason why the interacting Pomeron should be described by such a local renormalizable field theory, we think that a priori the more partic- ular weak coupling solution should be considered on the same footing as the strong coupling one.

When this work was being written I was informed by J. Ellis that he and R. Brower had also found very similar results, considering a theory with de- rivate couplings and non linear terms in the propa- gator. I thank J. Ellis for a useful conservation on that argument. I thank also G. Calucci for much valuable help he gave me during the present in- vestigation.

# This happens in a sort of solvable example.

References

[1] V.N. Gribov, Zh. Eksp. Teor Fiz. 53 (1967) 654; (JEPT 26 (1968) 414).

[2] A.A. Migdal, A.M. Polyakov, K.A. Ter-Martirosyan Phys. Lett. 48B (1974) 239.

[3] H.D.I. Abarbanel, J.B. Bronzan, Phys. Lett. 48B (1974) 345.

[4] M. Baker, The e-expansion of Pomeron amplitudes, Moscow preprint (1974).

[5] K.G. Wilson and K. Kogut, The renormalization group and the e-expansion, Lectures given at Princeton University (1972).

[6] V.N. Gribov and A.A. Migdal, Yad. Fiz. 8, (1968) 1002; (Soviet J. of N. Phys. 8 (1969) 583).

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