Pomeron and ®dderon in QCD and a Two Dimensional

Conformal Field Theory

L.N . Lipatov*

Deutsches Elektronen-Synchrotron - DESYHamburg, Fed . Rep. Germany

Abstract

Nuclear Physics B (Proc. Suppl.) 18C (1990) 6-12North-Holland

The problem of solving the Bethe-Salpeter equations in LLA for t-channel par-

tial waves corresponding to Feynman diagrams with many reggeized gluons issimplified significantly by using their conformal invariance in the impact para-meter representation and the separability property of their integral kernels . Inparticular, for the three gluon system with the odderon quantum numbers weobtain a one-dimensional integral equation .

It is known [1], that in the leading logarithmic approximation MLA) the gluonproduction amplitudes at large energies T have the multi-Regge form and areexpressed in terms of the reggeized gluon trajectory j = i+cj, w - g2 and theReggeon-Reggeon-particle vertex Y ti g, where g is the QCD coupling constant .High order- radiative corrections to these quantities and many Reggeon verticescan be calculated using a dispersive approach [2] .

The hadron scattering amplitudes in LLA are expressed through the solution ofthe Bethe-Salpeter equation for t-channel partial waves fw(k, k', q) describingthe pomeron built from two reggeized gluons [1, 3] . Further, the analogousequation for the three gluon compound state with the odderon quantum num-bers (Pi = C = -1) is constructed with the use of the integral kernels forpairlike gluon interactions which are proportional to the pomeron kernel [4].

It is convenient to perform the Fourier transform of the function f~(ki ,kl ~

depending on transverse components ki , k i . of virtual gluon momenta and topass to the impact parameter representation fjpl,pi) (here Pi and pi . are the

* Permanent address : Leningrad Nuclear Physics Institute, Gatchina, Leningrad,188350 USSR

0920-5632/91/$3.50 © Elsevier Science Publishers B .V . (North-llolland)

L.N. Lipatov/Pomeron and odderon in QCD

7

transverse coordinates of the initial and final gluons in the t-channel). TheBethe-Salpeter equations in this representation are conformal invariant andtheir solutions f(.~(p;,pi ") can be interpreted as the Green functions of a twodimensional Euclidean field theory [5] .

Here the fields ep(pi), q)(pi ") describe virtual gluons and belong to the adjointrepresentation of the SUM gauge group.

By inserting in eq. (I) the unit operator expanded In the sum of projectors intothe complete set of states corresponding some local operators O(PO) one canexpress f~ in terms of simpler matrix elements X pppo)

X(Pi' Po) =< 01 0(i)O(PO)Io > .

(L)i

The n-gluon wave function X satisfies In LLA the following homogenous equa-tion [4, 6]

WX(Pi, Po) =9

2 E (-T,. T,) K(Pr,Ps)X(Pi~ Po),87r r<s

The eigenvalues and the eigenfunctions of eq. (3) determine the position of thesingularities of the partial waves f., in LLA and their discontinuities on thesesingularities . In the complex coordinates pi, pig for impact parameters pi theoperator K in eq . (3) has the surprisingly simple separable structure [6]

K(Pr,Ps) _ %(MJ + k(Mr8 ),

(4)

2

°° (

21+1

2

1=0

rs

) .

where the colour group generators Tr . Ts and the integral operators K(pr ,ps )act at the variables of the r-th and s-th gluon.

Here

L.N. Lipatov/Pomeron and odderon in QCD

Ms = -(M° )2 + 2 (Mrs l~lrs + Mr"Mrs) -

Prsaras~

Prs = Pr

Psi CÎr

apr

is the Kasimir operator for the Möbius group of conformal transformationsand

are their generators .

M~ = Prar+ Ps

as, Mr, - Prar+ .02£9S

Mrs = ar + as '

The total momentum squared

M2 = (s Ms)2

commutes with all K. There-fore X belongs to a representation of the Möbius group:

M2X = rn(rn - 1)X, M*2X = in-(in- _ 1)X,

and the eigenvalues 6) are some functions of the conformal weights m and m' :

u; = w(m, fit), D = m -1- rn = 1 -}- 2iv, n = m - fin,

(9)

where ® and n are the dimension and the conformal spin of the operatorO(po) respectively .

From eqs . (4) - (6) one can obtain the normalization condition for X

IIXII 2 = J

d2P,X* Il (ar)2 X.

(10)i

r

k,k _

rh

pil

ki)

R-T (

Pi-

)ki

Efkil F-jk;j Cfki)fk;j

( pjO,, O

t

4040

E ki = k, E k, = %,

6

It is remarkable that the function co (9) has simple analytic properties in thevariables m and m : it contains poles at integer points k = l, 2, . . . and V = 1,2, . . . accordingly . To find the residues in these poles let us notice firstly thatin integer m,-M' the operator O(po) can be constructed as a conformally covari-ant polynomial in the derivatives of free fields 9 . Therefore the matrix ele-ment (2) for m = k, m =

equals

where the coefficients C(ki )(ki ) can be calculated from a secular equation de-rived from eq. (3). However, in the pole approximation for w(m,en) It turns outthat the corresponding anomalous dimension matrix is degenerate.

To show this in general it is necessary to consider a trial function X which isnonsingular at pil = 0 for all i and j, has the correct dimension and coincideswith (11) in the limit m -3~ k, m -1 . But when in r.h.s . of eq. (3) we applyone of the operators K(pr,ps) to this function it is possible to restrict oursel-ves to the simpler expression:

m-kX -

Pra

Prs

Xk,k~

(12)PrOPso PrOPâo

because K(pr,ps) does not strengthen its singularity . Moreover, X(12) Is an ei-genfunction of K(pr,ps) . Then using the same procedure for each kernal (5) Inr.h.s . of eq. (3) we obtain its eigenvalue in the pole approximation

w(m ---+ k, in- -)

L.N. Lipatov/Pomemn and odderon in QCD

g

k)_ 9

2 nN

1ti87r2 2

(k-m

where we used the identity rs (-TraTsa ) _

N. The result (13) is valid for ar-bitrary linear combinations of eigenfunctions (11) with ki,kI Z 1, which corresponds to the matrix elements of the gauge invariant operators.

e simpleanalytic structure of (,) (13) and the large degeneracy of the anomalous dimen-sion matrix in the pole approximation are presumably important for finding anexact solution of eq . (3).

Let us return now to the separability relation (4) for the kernel K. In caseswhere the colour structure of X is factorized (it takes place for n = 2,3 andN = oo) we can write down it due to eq. (4) in the form

X(P,~ Po) = E X 2(P=~ Po)X2(P~ ~ Pô)~r

k-im}

where Xr and Xr satisfy the simpler equations in which K is substituted by k.The functions yr . Xr are analogous to "conformal blocks" of the conformalfield theory [7].

To construct k(Mrs) in the form of an integral operator one can use its fol-lowing representation :

°°

exp ( -~TÎ)k(M,) = 2 f dT (c(-r)tr exp(--rm.)

exP (- ITl )00

(13)

(14)

(15)

L.N. Lipatov/Pomeron and odderon in QCD

where "tr" in front of the subsequent operator means the sum over its matrixelements between the states inside the irreducible representation with thehighest weight m-1 .

Expression (15) can be rewritten using the operation of the invariant integrationas follows

3

k(M;) _

dacth(

-z,2)E(zo)9(-z~) exp (ZMMN,) - C,27r f zu

e_zC = 4 f

dT

T, T =

-z~, z~ = -zô -I- zi -{- z2 .

(16)0

1 - e-

fiere Zp and MV are vectors in the three-dimensional Minkowski space .

The factor e ZpM carries out the arbitrary Möbius transformation of the varia-bles pr , Ps with the group parameters Zp . Therefore eq . (16) can be presentedin the form of an integral operator acting at the function depending on thetransformed variables pr, ps .

Let us consider now eq. (3) in the simple case of the three gluon system (n=3)with odderon quantum numbers (PI = C = -1) . Here the colour structure of Xis factorized and we , can substitute (- TaTsa) by N = 2 . After that we obtainthe following equation for Xr in eq . (14)

rvw2 =92

3

2

k(M_)x2, w = w' -F- w't .1<T<s<3

By putting the conformally covariant ansatz In this equation

(17)

X2 =

P12P23P23

f(

'x)'

- P12P3o

P10P20P30

)

P13P2o

(18)

we can construct the one-dimensional equation for the function f(x) . The nu-merical solution of this equation will be published later [8] .

An approximate method of solving eqs . (3), (17) was suggested In ref . [5] . It

consists in using a "diffusion approximation" for K(4) :

Acknowledgement:

L.N. Lipatov/Pomeron and odderon in QCD

K(P, ps) ti 81n2 + 14C(3)(.~t; + M;; + ?

2

,(19)

which corresponds to expanding MM 2) (S) in a series near the point MZ = - VIn this approximation eq . (3) is simplified:

_wX = HX

H - s2 (--[81n 2 + 7( (3)]2 + 14«3)[(T,MT )z + (TaMT~)2~)

(20)

Here the summation over a, m and r is implied . H commutes with the opera-tors

,r=,Trb Mrv,

rE Trb, rE! R4rv which are generating elements of an algebra.Therefore (,) can be calculated in terms of the highest weight for each repre-sentation of the algebra . These representations are infinitely degenerate becauseH commute with the operator ( T~b Mw)2 ( T~b , ( Mw)2 for a!1 subsetsti ,- r r r r r rr of the set r = 1, 2 . . . n .

As is seen from eq . (20), in this model the position w of the leading singula-rity of the partial waves grows linearly with the number n of reggeized gluonsin accordance with the pole approximation (13), which is apparently a generalphenomenon . Nevertheless, the "diffusion approximation" is bad from the theo-retical point of view because the operator in r.h .s . of eq. (19) in contrast withthe exact one (4) is not restricted from above for states with a large confor-mal spin n,_.-q. Therefore, the odderon intercept obtained in ref. [S] with theuse of this approximation is not reliable . One of the possible ways to correctthis defect is to put some constraints an the solution of eq. (20) to supressthe contribution of the states with nonzero values of nrs-

In conclusion it is necessary to stress that contributions of diagrams withmany reggeized gluons are very important for constructing the QCD scatteringamplitudes at high energies satisfying general requirements of the s- and t-channel unitarity.

1 thank the Institute of the Nuclear Physics (Orsay) and the DESY Theory Di-vision for their kind hospitality during my stay . I express my gratitude to B .Nicolescu, P. Gouron and J . Bartels for helpful discussions.

12

L.N. Lipatov/Pomeron and odderon in QCD

References

[1]

E.A. Kuraev, L.N . Lipatov, V .S . Fadin, Sov . Phys. JETP 44 (1976) 433;45 (1977) 199

[2]

V.S. Fadin, L.N . Lipatov, preprint 89-13, Institute of Nuclear Physics,Novosibirsk, 1989

[3]

Ya.Ya.

Balitskii, L.N .

Lipatov,

Sov. J . Nucl . Phys . 28 (1978) 822

[4]

L. Kwiecinslki, M. Praszalowicz, Phys . Lett . 94B (1980) 413;1'. Jaroszewicz, J. Kwiecinski, Z. Phys. C12 (1982) 167

[S]

L.N. Lipatov, Sov. Phys . JETP 63 (5) (1986) 904

[6]

L.N. Lipatov, in "Perturbative QCD", edited by A.H. Mueller - AdvancedSeries in Directions in Iiigh Energy Physics - World Scientific (1989)

[7]

A.A . Belavin, A.M . Polyakov, and A.B . Zamolodchikov, Nucl . Phys . B241(1984) 333

[8]

P. Gouron, L. Lipatov, B. Nicolesku, work in progress .