evolution equation and relativistic bound state wave functions ... - SLAC

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EVOLUTION BOUND FIELD

STANLEY

EQUATION

STATE WAVE FUNCTIONS MODELS

J.

AND RELATIVISTIC

BRODSKY,

IN FOUR AND SIX DIMENSIONS*

CHUENG-RYONG

Stanford Stanford

FOR SCALAR

Linear

University,

JI AND

Accelerator

Stanford,

MIKOLAJ

SAWICKI+

Center

California,

94305

ABSTRACT _ We investigate framework

the evolution

equation

for distribution

amplitudes

of a scalar theory quantized on the light cone. We End general solu-

tions for the cases of 4 and 6 dimensions and use them to reconstruct relativistic

in the

bound state wave functions

at small distances. The relation between

the light-cone bound state equation and the Bethe-Salpeter

Submitted

two-body

equation is discussed.

to Physical Review D

* Work supported by the Department of Energy, contract DE - AC03 - 76SF00515. + On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.

1. Introductioti

-.

- -

A long-range goal in the study of quantum chromodynamics

is to actually

calculate the spectrum and wavefunctions of hadrons from first principles. portant theoretical

tool to this end is lattice gauge theory which may eventually The Bethe-Salpeter

provide accurate numerical values for hadronic properties.’ approach is primarily irreducible

An im-

suited to weak-binding

kernel corrections

bound state problems where higher

and non-perturbative

vacuum effects can be ne-

glected. An alternative approach2 to relativistic quantization

bound state problems is the light-cone

method, which provides a Hamiltonian

representation

formalism

of QCD at equal light-cone time r = t + z/c.

space bound-state

solutions to this system of relativistic

are functions of the light-cone variables z; = (kp + kf)/(p’ particle helicities X;. They are immediately observables, such as structure

functions,

anomalous moments, correlations

and Fock-state The momentum-

equations +(z,kl, +p”)

and zl;, and the

suitable for calculations distribution

X)

amplitudes,

of covariant form factors,

and other hadronic properties.

The first step in solving the full set of coupled Fock state equations on the light-cone

is to find a simple, analytically

lowest-particle-number for obtaining

tractable

equation for the valence,

sector, and to develop a systematic

perturbation

theory

higher particle number states and higher accuracy. These require-

ments are satisfied by the simplest approximation, order irreducible

corresponding

kernel; i.e. the light cone ladder approximation.

one can prove that higher Fock-state contributions mation in a renormalizable mentum momenta.

to the lowest Furthermore,

in light-cone ladder approxi-

theory are negligible at large relative transverse mo-

In gauge theories this statement 2

is true for physical gauges

for the vector fields, such as light-cone gauge. -Thus the covariant ladder approximation

is equivalent to light-cone ladder approximation

equivalence eliminates any possibility distribution

amplitude

at large $1. This

of cusp-like (non-analytic)

4(z;, Q) or light-cone wavefunction

behavior of the

of the type described

Since r$(z;, Q) satisfies the evolution equation which is derived

by Karmanov.3

by taking large cl limit of the light-cone projected Bethe-Salpeter equation, one can prove that +(zi, Q) is analytic (i.e. cusp free) in the whole xi region and can be expanded by Gegenbauer polynomials

in two-body

The cusp behavior is induced by the artificial

bound state problems.4

limit of taking the binding energy

to zero. The initial problem to be examined in this paper is the behavior of the twobody bound-state

wave function at large values of relative momentum.

To this

end we study first the properties of the corresponding valence-quark “distribution amplitudes” which control high-momentum distribution

amplitude

transfer exclusive reactions4

4(xi, Q) is the amplitude

The

for finding the I@) Fock-state

in the bound state collinear up to scale Q. Its variation with Q will be described by an evolution

equation.

use it to reconstruct Although

We find the solution of the evolution

a detailed form of the wave function

equation and

at short distances.

we will deal here with a simple spinless model, the methods are valid

independent of spin. The scalar models are also of interest to the extent that they give a first look at the nature of wave functions bound system.

We perform the analysis within

discuss two different 4-dimensions,

for relativistic,

the Wick-Cutkosky

cases with N = 4 and N = 6 dimensions.

we obtain the bound-state

matches the simplest approximation

wave function

model and Working

in

which asymptotically

to the light-cone wave function. 3

strongly

The case of

.I

I

6 dimensions is more interesting. has mathematical

Even though this case-is a nonphysical one, it

and graphical similarity

quark-gluon theories. In particular, renormalizable,

and furthermore

with the more physical non-Abelian

it has a fundamental

trilinear

it happens to be asymptotically

coupling, it is free. Because of

asymptotic freedom, the higher order kernels can be neglected at short distances. The asymptotic

behavior of the bound state Bethe-Salpeter

wave function

In this paper

such theory has also been investigated by Appelquist and Poggio.5

the analysis is based on the light-cone approach and the resulting wave function asymptotic

in

exhibits calculable anomalous dimension corrections

light-cone to a naive

behavior.

The paper is organized as follows: equation for the distribution

In Section 2 we present the evolution

amphtudes within

the Wick-Cutkosky

model.

In

section 3 we solve this equation for the case of N = 4 dimensions and use this solution to reconstruct the behavior of the two-body bound-state wave function at short distances. The QCD-like case of N = 6 dimensions is discussed in Section 5. The light-cone equation and its relation to the Bethe-Salpeter the distinction

between their respective ladder approximations

Appendices A and B.

equation and

are discussed in

I

2. The Evolution Eqtiatior.

- -

We shall consider the light-cone description of the relativistic

composite sys-

tem of two scalar particles interaction via the exchange of a massless scalar particle ( Wick-Cutkosky

model). “’ The interaction

Lagrangian is l = g$‘x,

where

$J is a “quark” field with mass m, and x is a massless “gluon” field. The bound state wave function can be described by means of the Fock-space components of the state vector 1~) (see Appendix A). The light-cone equation for the two-body

wave function

in N dimensions

reads (See A(6))

?+b(x;,&) = -J51x2

1 M2

_

mx

s2h)

2122

(2.1)

1

where x;, yi are the fractions of the total P+ momentum of bound state carried by the i-th valence quark (x1 +x2 = 1, [dy] = dwdw~(l-yl the N-2 dimensional perpendicular

momenta, (dNe2ti]

-n)>, = h

in, & are dNw2tl

and

M is the mass of the bound state.

The asymptotic

behavior of the coupling constant g2(Q) has the form

s2(Q) =

i.e.

there is a “running

Restricting

g2

for N = 4 ,

9: log(811113

forN=6

,

(2.2)

coupling constant” for -the theory in 6. dimensions.5

ourselves to the one-gluon-exchange only we obtain the light cone 5

ladder approximation

(LCLA)

O(Yl

-

to the kernel of Eq. (2.1);

-

1

Xl)

yl _ x1

The high-momentum

-

M2

_ m2z;k;

(&;f,)’

2) .

+ (1 *

m;tl

transfer exclusive reactions are controlled by “distribution

amplitudes” 4(x;, Q), which give the probability bound state with momentum

for finding the valence quarks in

fraction xi at relative perpendicular

distance not

i.e. collinear up to scale Q

smaller than bl - 0(1/Q),

IhI