ELSEVIER

2 June 1994

Physics Letters B 328 (1994) 411-419

PHYSICS LETTERS B

Large N QCD at high energies as two-dimensional field theory I.Ya. A r e f ' e v a 1

Steklov Mathematical Institute, Russian Academy of Sciences, Vavilov st.42, GSP-1,117966, Moscow, Russian Federation

Received 18 June 1993; revised manuscript received 6 April 1994 Editor: P.V. Landshoff

Abstract

Different aspects of the H. and E. Verlinde relation between high-energy effective scattering in QCD and a two- dimensional sigma-model are discussed. Starting from a lattice version of the truncated 4-dimensional Yang-Mills action we derive an effective theory with non-trivial longitudinal dynamics which has a form of the lattice two-dimensional chiral field model with non-trivial boundary conditions. To get quantum corrections coming from non-trivial longitudinal dynamics to transversal high-energy effective action one has to solve the two-dimensional chiral field model with non-trivial boundary conditions. We do this within an approximation scheme which takes into account one-dimensional excitations. Contributions of the one-dimensional excitations to quantum corrections for the high-energy effective action are calculated in the large N limit using the character expansion method.

1. Introduct ion

One of the striking results of intensive studies of high-energy scattering in QCD [ 1,2] is that the scattering amplitudes are related with two-dimensional field theory [3] . Recently H. Verlinde and E. Verlinde have formulated a simple model in which this two-dimensional nature of the interactions is manifest [4] . They have also shown that their formulation is in agreement with known results from standard perturbation theory [ 1,2].

There are some questions concerning the H. and E. Verlinde approach which we would like to address in this paper. First question is related with ultra-violet divergences and renormalizations. The H. and E. Verlinde (VV) effective action which describes the transversal dynamics has a form of the two-dimensional o--model. In spite of a non-convention form of this o--model it is reasonable to expect that it is asymptotic free. But the/3- functions corresponding to the 4-dimensional Yang-Mills theory and the two-dimensional o--model numerically have the different forms. Namely, the 4-dimensional Yang-Mills/3-function contains the factor or and/3-function o f the two-dimensional o--model does not. So one can expect that an additional renormalization comes from the longitudinal dynamics. This circumstance give us a raison to study the longitudinal dynamics more carefully. A treatment of longitudinal dynamics within the framework of the initial truncated Yang-Mills action is a main goal of this paper.

I E-mail: Arefeva@qft.mian.su.

Elsevier Science B.V. SSD10370-2693 ( 94 ) 00454-F

412 L Ya. Aref'eva /Physics Letters B 328 (1994) 411-419

To analyze quantum corrections to the longitudinal dynamics and the corresponding renormalization we have to introduce some regularization. As a regularization we can introduce a lattice in x-space. Recall that one can deal with 4-dimensional lattice only in the Euclidean space-time. However, the high-energy effective action has been obtained [4] in the Minkowski space-time. The Minkowski signature is essential, since it permits to use the property of the two-dimensional wave equation. We would like to know does a similar action arise in the Euclidean formulation and what are its quantum corrections. For this purpose we study the lattice version of the truncated action. It turns out that the lattice version of the truncated action looks like the usual lattice version of the two-dimensional chiral model with non-trivial boundary conditions. An interaction between fields living on the transversal planes appears due to the boundary effects in the two-dimensional chiral model. The lattice two-dimensional chiral model with non-trivial boundary conditions cannot be solved exactly. One can consider the one-dimensional chiral model and analyze the form of the corresponding transversal effective action. As in the two-dimensional case we can speculate that the one-dimensional answer describes the contribution coming from long one-dimensional excitations. In other words we will estimate long tubes contribution to the quantum version of VV effective action.

The paper is organized as follows. In Section 2 a lattice version of the VV truncated action is written down. In Section 3 we study the longitudinal dynamics for a model case then this dynamics is one-dimensional and calculate a leading term of the high-energy effective action for large N. We discuss the two-dimensional longitudinal dynamics in a concluding section.

2. Lattice effective action for QCD at high energies

Since this paper is inspired by the recent H. Verlinde and E. Verlinde paper [4], let us start from a brief presentation of their results. They are interested in the behaviour of scattering amplitudes in the kinematical regime where s is much larger than t, while t is also larger than the QCD scale AQCD. Introducing two light-cone coordinates y~ = (x +, x - ) with x :k = x :~ t and two transverse coordinates z i = (z 1, z2), authors assume that two fast moving particles have very large momenta in the x ± direction, while they remain at a relatively large distance in the z-direction. They also use the hypothesis, according to which the typical longitudinal momentum of the dynamical modes in this process grows proportional to the centre of mass energy, whereas the typical size of the transversal momenta is determined by the momentum transfer. A similar assumption was used also in other approaches to high energy QCD [ 1 ]. The distinguished feature of the H. and E. Verlinde approach is that the high-energy limit is taken directly at the level of the action.

From asymmetric dimensional analysis follows that the description of a scattering process with some s and t using the standard action is completely equivalent to that using the rescaled action

S~-M' - 4A2g e l tr ( F~F'~a ) + ~g2 t r (FaiF~i ) + 4g2-- tr ( F/i F a ) '

with rescaled s to s t = A2s, i.e.

( i)

A(Sr~, s) = A(S}~, a2s), (2)

where .A is a scattering amplitude describing the process. In (1) Fup is the non-abelian field strength, F~p = OuA~ - O, Au + [A u, A , ] , Au = A~r a where g is the coupling constant and r a are the generators of the Lie algebra of the gauge group G = S U ( N ) . One can use the correspondence (2) and reformulate the high-energy

1 limit s ~ cc in QCD as the A --~ 0 limit of the rescaled theory ( 1 ) with s r fixed, i.e. a ,-~ ~ --~ 0.

A subtle question related to the limit ,t --4 0 is the problem of divergences. To avoid ultra-violet divergences one can put the theory on the lattice. Let us consider a lattice version of the rescaled action (1)

L Ya. A re f ' eva /Phys ics Letters B 328 (1994) 411-419 413

4A2g 21 ~1 ~x ~(trU([~,i)_l)+_~g2~x Z( trU(Di , j )_ I ) SL~'M - Z Z ( t rU (D,,~) - 1) + x ct,[~ a,i i,j

(3)

Here x are the points of the 4-dimensional lattice, a , /3 are unit vectors in the longitudinal direction and i, j are unit vectors in the transversal direction; []~,~ is a single plaquette attached to the links (x, x + / x ) and (x,x + ~,), Ix, v = a , i . Performing the a ~ 0 limit in the rescaled lattice theory (3) we get a lattice truncated Lagrangian

S~yM = + ~x Z tr ( U ( D~,i ) - I ) , (4) o',i

with Ux,~ being a subject of the relation U([~,/3) - 1 = 0, i.e. Ux,~ is a zero-curvature lattice gauge field Ux,~ = VxVx+~,. Substituting this representation for Ux,~ in (4) and eliminating Vx by the gauge transformation Ux,i ---+ ~Jx,i = Vx+U+x,iVx+i w e get the lattice truncated Lagrangian

a D - 4

U y , z ; i U y + a , z ; i - 1). (5) Str= 4g z ~ - - ~ t r ( + i,z y,a

Here we restore for the moment a factor a D-4, which is omitted in the previous formula since we assumed D = 4. We also denote the points of 4-dimensional lattice as x = (y, z) , where y and z are the points of two two-dimensional lattices, say, y-lattice (longitudinal) and z-lattice (transversal). We see that the action (5) has a factorisation form. The indices z and i play a role of isotopic indices, Let (L) 2 is a number of points of the z-lattice. Then we can say that we have 2(L)2-copies of two-dimensional lattice models. Each model describes the chiral field Uv, attached to the sites of the two-dimensional y-lattice.

Note that there is some similarity between A rescaling and the high temperature limit of lattice gauge theory [7], in the first case one makes a decomposition (2, d - 2) and in the second (1, d - 1).

The classical longitudinal continuous limit of the action (5) gives (L)2(o-2Lcopies of the usual two- dimensional continuous chiral models

a Dag 2-4 / Str,cl = z ~ / d2ytr(OaUz;i(x)a,U+z.,i(x)). (6)

To remove the y-lattice in the quantum version of the two-dimensional chiral model one has to perform the renormalization. In the one-loop approximation this renormalization does not coincide with the Yang-Mills one-loop approximation renormalization [5]. This means that taking the limit ,~ ~ 0 we drop diagrams which contribute to the renormalization of the coupling constant. One can expect to compensate this deficient in the renormalization of the coupling constant in a renormalization accompanying the continuous limit of the transversal lattice.

At first sight there are no raisons to get propagator modes in the z-plane, since there are not the corresponding kinetic terms. The H. and E. Verlinde have observed that a non-trivial dynamics in the transversal z-plane can be introduced due to non-trivial boundary conditions in the x:k-plane. They have arrived to this conclusion at the semiclassical level. In the next section we will examine this conclusion in the full quantum level for the case of the one-dimensional longitudinal dynamics.

3. One-dimensional longitudinal dynamics

It is rather instructive to discuss effects related with non-trivial boundary conditions in the action (5)in the case of the one-dimensional y-lattice. More precisely we mean the following boundary conditions

414 L Ya. Aref'eva /Physics Letters B 328 (1994) 411-419

H fl i" aD-4 g = Uy, z;iU;+l,z; i -~ Uy+I ~ + ,~;iUy,z;i 2)} Z,+2 /Jexp{ 4g 2 Z tr( ~ "+ - --

(z i) Y

X H dUy'z:idUo'z;idUL'z;idVo'zdWL'z) " ( 7 )

Oo.::,=Vo.~ Uo.:;,vg.: ~ , OL.:;,=v,..: v~,.~v~ +,

In (7) we have the product of the transition functions of the one-dimensional lattice chiral model. This correlation function can be calculated [6] using the characters expansion method,

L L t

lC(Uo, UL) =/exp{NB}--~ tr(UnU++~ +Un+IU+--2)} HdU,~= ~--~dimR(ZR)(L-1)xR(UoU+), (8) n=0 n=I R

where XR and dinar = XR (I ) are characters and dimensions of the R's irreducible representation respectively. Here L t = L - 1. ZR are given by the formula

1 / d U x R ( U + ) e x p { N / 3 t r ( U + U + - 2)}, (9) ZR - dimR

where ]3 = a°-4/4g 2. The representation R of U(N) can be labelled by N integers r = (nl,n2 . . . . . nu) with nx >_ n2 >_ . . . >_ nn , where n i corresponds to the number of boxes in the i-th row of a Young tableau. The explicit expression for the character XR(U) in terms of the eigenvalues e iqbk, 0 < q~k _< 277", of matrix U is given by the Weyl formula [ 8]. However, for our purpose we need the explicit dependence on the matrix U rather than on its eigenvalues. Such formulas are given by the well-known Frobenius relation [ 8 ] between characters and symmetric polynomials,

xR(U) = ~ XR(o) K,, n! H t r ( U ) kj, (10)

o'E S,, j=l

where n is a number of boxes in the Young tableau corresponding to the representation R of SU(N), Sn is the symmetric group on n objects. For a given permutation o- K~ is a total number of cycles and {kl . . . . kj . . . . kK~} are cycle lengths. In the right-hand side of (10) trU is trace in the fundamental representation. For the first characters one find explicit formulas in [9,10].

The explicit representations for the first ZR'S can be found from the explicit representations [9,10] and the generating functional

Z ( A , A +) = f dUe NBt~(vA++wA) ( I1)

The integral (1 l ) defines the well known object, this is the partition function of the Brezin-Gross model which describes one link gauge field in the external matrix source [ 12].

Z0(N/~, N) is nothing but the partition function of the Gross and Witten model [ l 1] and it exhibits the phase transition

{exp{-N 2(~ + ½1n2/3)} if/3 > 1/2, Z°(NB'N) N~oo e x p { - N 2 / 3 ( 2 - / 3 ) } i f / 3 < l /2; (12)

Z(l,o,0..0 (N/3, N) = ~ ( tr U ) , can be found from the relation

1 O ( t rU) = N + ~-~-:---~ log Zo(U/3, N ) , o H (13)

l, Ya. Aref'eva / Physics Letters B 328 (1994) 411-419 415

where

(f} = f dUf(U) exp{N/3 tr ( U + U + - 2) }

fdUexp{NBtr(U+ U + - 2 ) } '

We have

1 ~ ( t r U ) = f ( /3 ) ,

where

1-4--- ~ i f / 3 > 1/2,

f ( f l ) N ~ B if B_< 1/2.

(14)

(15)

(16)

Let us examine the behaviour of ZR in the large N limit. Substituting the Frobenious relation (10) for the characters (10) in (9),

K~

z~,,, = ~ xR(o') f n! Htr(U)~Jexp{Nf l t r (U+U+-2)}dU' (17) o'E S,~ j= l

we see that different permutations o- give different powers of N in the integral over U. The main contribution comes from o- with the most number of cycles, i.e. ki -- 1 and K~ = n,

-- XRrn(( In ZRr, n! ) ) Z ° ( f ( ~ ) N ) n + O ( N n - l ) = - ~ (18)

dr.

where dr,, is the dimension of the representation of symmetry group given by the Young tableau Yn and the factorisation property of correlation functions in the large N limit is taken into account. Note, that we used the Frobenius relation for SU(N) group and Eq. (16) for U(N), since we expect that this difference can be neglected in the large N limit, direr being the character of the unit matrix also has the similar asymptotic behaviour for the large N

= dr, Nn dimR~ n! + (9(N n-1 ). (19)

The explicit formula for 1IN corrections to (19) can be written in terms of the lengths ni of the Young tableau [13].

Therefore for n fixed we get

ZRy. = Zo(f) n + O(N-]). (20)

Note that (20) for large /3, fixed n and large N is in agreement with well-known answer [ 14] ZR = 1 - C2(R)/2Nfl, where Cz(R) is a quadratic Casimir.

Representing the sum over all representations in (8) as

~(Uo, eL) = Z Z dimR( ZR)(L-1) xR(U°U+)' (21) n REY.

where the second sum in (21) is taken over all representations whose Young tableaux are in the set of Young tableaux with n boxes, we have

IC(Uo, UL) = (Zo) L' Z ( f ) nL' Z direr xR(UoU[). (22) n REY,,

416 I. Ya. Aref'eva / Physics Letters B 328 (1994) 411--419

Using the Frobenius relation once again we represent ]C(U0, UL) for large N as

fnL' K~ 1C(Uo, Uc) = (Z0)L' ~--~ ~ ~ dime Z X n ( o - ) I I t r (UoU+) k~, (23)

n RcY~ o'ES,, j= l

From this formula it is clear that the effective action describing an interaction of the field living on two infinite two-dimensional planes has all powers o n ~JO.z;i~J~,,z;i. If we are interested in the first term of the effective action

which is linear on /5'0,z;i we can keep in (23) the permutation with cycles ( l ~) and we have

f n L '

K;( U0, UL) = (Zo) c' Z ~ Z dime Xe( (1") ) ( tr [ UoU~] )~ + . . . . (24) n RCY,,

Dots denote the terms with higher power of UoU~[. Taking into account (19) at large N we have

fnL' )U(Uo, UL) = ( Zo)L' ~ ~ ~ d~. ( tr[ UoU~-])n + . . . . (25)

n Y,,

Recall that the sum in (25) is taken over all Young tableaux with n boxes. Due to the relation

~--'~ d~,, = n! (26) Y,

we have for the transition function

~(Uo, UL) = (Zo) L' exp{fL'Ntr( UoU +) + . . . } . (27)

Therefore for the large N and small g2 the transition function can be represented as

-L'/2N 2 r 3 _~ 1_) L, .. K2(U0, Uc) = (2/3) exp*t-~--~L + (1 - N t r ( UoU +) + .}. (28)

4f l -

In the strong coupling regime we have

1C(Uo, UL) = exp{-N2L' /3(2 - fl) + fl L'Ntr (UoU~[) + . . . } - (29)

Substituting the representations for the transition function in (7) we get

Z2+l ( 2 o ) # r 2 l e x p { f ( f l , L ' )N~ '~ tr[Vz + + = e°,z;iVz+i~z+iVL,u~i~z ] } 1-I dVzdnz ( H dUo,,z.idUL,z;i) d - -

i,z Z i

(30)

L ' for large D f(¢), L') = (1 - ~ ) and f ( / ) , L') = D L' for small/). The constant Z0 is different in the different regimes.

It seems reasonable to assume that in the week coupling regime U = log s. Performing in the lattice effective action the limit a ---, 0 one immediately recognizes an action similar to the W effective action

L, f S[V,I),Ai] = (1 - ~-~)N d2z tr [ V(z)D~V+(z)Y~(z)D+I~+(z)I, (31)

where

Di ~ = cgi + Ai ~, UL.z;i = e aA+(z) , Uo,z;i = e a z ' - ( z ) (32)

L Ya. A ref' eva / Physics Letters B 328 (1994) 411-419 417

In this example the analog of quark-quark scattering amplitude is given by the expectation value of two longitudinal Wilson lines

= / d2ze iqz (V(0)'I)+ (z)), (33) A

V(Z) = tr [ P exp ldyA(y , z) 1. (34)

The lattice version of (34) is simply

Vz = Vo,~VL+z. (35)

Note that in the model example of the one-dimensional transversal plane one can perform the integration over fields dVz d~z explicitly. Indeed, taking into account the form (8) for the transition function/C (U0, UL) and the orthogonality condition for characters we have

ZI+, [ f i l ~ ( V z + + = Uo,z ~ +i, fZz ULz 12z +i) dVz d[~ z dUo,z ;idUL,z ;i J Z | =

[Z( ,+II o = ZR)(L-I)(M-1)XR(I~+VI ,z;i MP- L,z;i)dUO,z;idUL,z;i. (36) R Z Z

Here P+ and P_ are ordering and antiordering products. This answer was obvious from the beginning since in this case we deal with the two-dimensional QCD [ 15,16].

4. Concluding remarks and discussion

Let us make some comments about the 4-dimensional case. In this case the y-lattice is two-dimensional and we have to deal with the following lattice integrals

J f aN (J .~+ - 1 c.c.)}d/*, Z2+2 = H exP{4e z ~ tr ( y,z;,Uy+~,e;i + (37) (z,i) u y,a

where the integration measure includes

d l-t = d[ f y,z ;idUyl ,O,z ;idUyl ,Lz,z ;idULl ,y2,z ;idUo,y2,z ;idVy 1,0,z dVvl,L~ ,z dVL1 ,y2,z dVo,yz,z ,

and the following boundaries conditions are assumed:

/.7/yl,O,z;i ~-- v y l , O , z U y l , O , z ; , W ; ~ , o , z _ ~ i , (7/yl,t2,z;/= V v l , t 2 , z U y i , L 2 , z ; i W y ~ l , L 2 , z q _ i ,

Uo,y2,z;i = Wo,y2,zfO,y2,z;iWo+,y2,z+i , [fLl,y2,z;i: WL,,yz,zULi,y2,z;iW~,yz,z+i" (38)

In (37) we have the product of the transition functions of the two-dimensional lattice chiral model. If we assume the periodic boundary conditions in one of two longitudinal direction, say y2-direction,one can expect that for L1 >> L2 the main contribution comes from the one dimensional excitations, i.e.

~ = (Co) f e x p { ( 1 - l ogs y2 4B ) N . ~ tr [ Vy2,zao,y2,z;iVy+2,z +i~r~yz,z +iUL,y2,z;i~;=,z ] } l,z, y2,

x 1-I(dVy>edfZy>z (1-I dUo,y2,z;idUL,y2,z;i)). (39) z i

418 L Ya. Aref'eva/Physics Letters B 328 (1994) 411--419

the contribution of long one-dimensional tubes apparently is still described by the In the continuous version action (31), in spite of the summation over Y2 in (39), i.e.

Z2+z ~ (2~0) v°l' exp{(1 - d2z tr [ V(z)D~-V+(z)I2(z)D+II+(z)}

× 1-I d v ( z ) d a ( z ) ( 1-[ d A + ( z ) d A ; ( z ) ) " (40) z i=1,2

Eq. (39) describes a rough approximation, however it gives a rather acceptable physical picture. This picture is in agreement with the H. and E. Verlinde answer which is in accordance with perturbation calculations [2] and has the shock-wave [ 17-19] semiclassical interpretation [4] .

Let us stress that the question about dominate contributions in (39) or in others words the question about the behaviour o f non-linear o" model with non-trivial boundary conditions is rather complicated and without doubt it is worthy more attention.

The Lagrangians (5) and (39) show a relation between the 4-dimensional Yang-Mills theory and the two- dimensional chiral models. A relation was expected long time ego and was motivated by the fact that the both theories have dimensionless coupling constant and both are asymptotically free. Note also that some similarity was expected between the usual local formulation of the two-dimensional chiral model and the loop formulation, i.e. dynamics o f long tubes excitations in high-dimensional Yang-Mills theory [20,21]. The H. and E. Verlinde truncated action [4] together with the assumption (39) make this expected relation more tangible.

In concluding, it has been argued that consideration of quantum fluctuations of one-dimensional excitations confirms the two-dimensional picture of high-energy scattering in QCD,

Acknowledgment

The author is grateful to G. Arutyunov and K. Zarembo for useful discussions. This work is supported in part by RFFR under grant N93-011-147.

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