Volume 167B, number 1 PHYSICS LETTERS 30 January 1986

EQUATION OF MOTION FOR A STRING OPERATOR IN T W O - D I M E N S I O N A L MASSLESS LATTICE QCD

Shijong R Y A N G

Department of Ph_)'sics, Osaka Universi(v, To_vonaka, Osaka 560, Japan

Received 23 July 1985

In two-dimensional massless lattice QCD we construct the equation of motion for a gauge invariant meson operator with a quark-antiquark pair connected by the path-ordered products of link variables. Using the large-N factorization property the meson wave equation is derived in a nearly identical form to the 't Hooft equation.

Lattice gauge theory is one of the promising approaches to describe the large distance behavior of QCD [1]. The descretization of space is designed to preserve the gauge symmetry. In lattice gauge theories quark con- finement can be understood in terms of stringlike structures mediating the forces between quark-antiquark pairs, where the gauge invariance plays a crucial role.

On the other hand collective field methods were applied to non-perturbative problems in QCD. It is of interest to express gauge theories in terms of gauge invariant Wilson loops and strings for large distance properties [2]. Two-dimensional continuum 0CD [3] was studied in terms of the gauge invariant path ordered operator in mas- sive fermion cases [4] and massless ferrnion cases [5].

In previous papers the equations of motion for the Green's functions were investigated with the large N factor- ization property in the lattice SU(N) × SU(N) chiral model, the chiral Gross-Neveu model and the lattice QCD, and the mass gaps or the pion mass were estimated [6]. In this paper we will be concerned with the path ordered operator formalism in two-dimensional massless lattice QCD. The equation of motion for a string operator, that is, a gauge invariant quark-antiquark operator connected by the path ordered operator will be formulated. A gauge invariant string operator will be shown to dissociate into two strings with time owing to the preservation of the gauge symmetry. Using the large N factorization property this equation will be transformed in a nearly iden- tical form to the 't Hooft equation in two-dimensional continuum QCD.

We begin with a hamlltonian for an SU(N) lattice QCD with a naive massless fermion in theA 0 = 0 gauge, n

" --~1 ~ [t~ +a(x)oeU+e'O(x)C't3(x + 1 ) - ~O+a(x + 1)aUC'O(x)~a(x)] , (1) x=--Ft,a ' X,Ot,/3

where E~ is the ath generator of the right SU(N) gauge transformation [the left SU(N) gauge transformation has been used equally well]. The equal-time commutators for E~ and the generator of the left SU(N) gauge transfor- marion, E~ are given by

[E~(x,t),U(x',t)] =U(x,t)~Xa6x,x , , [E[(x,t) ,U(x' , t)] = - ~ X U(x , t )Sx ,x ,+l . (2)

The gamma matrices are chosen in the chiral representation 70 = ol, 3 '1 = - i °2 and a = 75 = 03- It is convenient to write the massless quark field in terms of quark annihilation and creation operators

where L = 2n + 1 is the number of sites on the lattice. The canonical equal-time anticommutator for the quark

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Volume 167B, number 1 PHYSICS LETTERS 30 January 1986

field is given by

(~(x , t), ~+~(x', t)} = a~axx, . (4)

We will consider an equation of motion for a gauge invariant quark-antiquark operator (a meson operator)

Mji(v, x) = ~1 [~7%v)U~(y, x) ~(x) - ~ ( x ) U ~ ( v , x ) ~ j + ~ (y)],

UC~(y,x) = U , f o r y > x , UC~¢(y,x) = U+(z , f o r x > y , (5) " Z = X Z - - 1

where a quark and an antiquark are connected by the path ordered operator, that is, the path ordered products of link variables, and the quark fields have been symmetrized. Using the equation of motion for the quark field

aq,~(x)/at = ~ [ v + ~ ( x ) ¢ ~ ( x + 1) - u ~ ( x - 1)4,~(x -1)] - - ~ O ~ i k

we have an equation of the meson operator

aMh4y,x)/Ot = - ~V~k](Mki(Y + 1,x)--Mki(y - 1,x))--~(Mlk(Y,X + 1)--M]k(V,X- 1))C~ik

+ ~ [ ~ [ a ( y ) ( a / a t ) V ~ ( y , x)O/~i(x ) - ¢~i(x)(a]at)ua~(v, x) ~[~(y)] . (6)

We will hereafter devote to the case y > x for definiteness, while the other case x > y will be similarly discussed. The equation of motion for a link variable U is given by

aV~O(x)/at = i ~_g2 [(V(x) ~Xa)a~E~(x) + E~(x)(U(x) ~-Xa) ~t~ ] (7)

= i ~g2 [(U(x) ~ X a ) ~ e ~ ( x ) + (U+(x)E~(x + 1) ½xau(x))~4JU(x)~'~ l , (8)

where we have made use of the relation between the generators, E~t and E~

U*(x) ~ XaE~ (x + 1) V(x) = ~XaE~(x) , (9)

which is derived from eq. (2). Using these generators we describe Gauss' law to be satisfied for the physical states as

[E~(x) - E[(x ) + ]~ (x)] Iphysical state) = 0 ,

]g(x) = ~ [~+~(x)(~-xa) a~, $/~(x)] . (10)

Taking account for eq. (9), we can find two equivalent solutions for eq. (10) x - 1

E~(x) = - ~ [2 Tr ~xaU(x - 1) ~X b [2 Tr ~xbU(x -- 2) ~X c l= --n

X [... [2 Tr ~XYU(1)~Xz]6(1)U+(1)I .--I U+(x-2)] U+(x- l ) ] - j~ (x) , (11)

?t

E~(x) = t=~x+112 Tr }VU+(x ) ~-xb[2 Tr ~xbU(x - 1) ~X c

× [... [2 Tr ~XYU+(I - 1) ~XZ]g(1)U(l - 1)l ...l U(x-1)] U(x)l +/g(x) , (12)

where we have assumed the boundary conditions E [ ( - n ) = E~(n) = 0. Since the constant color background fields cannot affect the gauge invariant color-singlet state, we can make them equal to zero. Thus each generator can be expressed in terms of the quark's color charge density operator and link variables only when it operates on physi- cal states. Later eq. (6) will be sandwiched with physical states. Substitution of eq. (8) together with eqs. (11) and (12) into the last two terms on the RHS of eq. (6) leads to

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Volume 167B, number 1 P H Y S I C S L E T T E R S 30 January 1986

2 Y...7-1 I

"~ l=x m=-n {3~k(y, m)Mki(m, x) + Mki(m , x)M/k(y , m)

+ (1/2N)IMkk(m , m)tp;°~(y)US#(y,x)~kai(x ) - tk3i(x)U°~#(y, x )~7~OP~kk (m, m)]}

• g 2 y - 1 n

- I - ~ - ~ ~ ( M / k ( Y , m ) M k i ( m , x ) + M k i ( m , x ) l ~ k ( Y , m ) l=x m=l+l

+ (1/2N) [~bfS(y) USa(y, x ) ~b~(X)Mkk(m , m) -- Mkk (m, m) $~i(x) USa(y, x ) $7s0,)] }, (13)

1 a 1 a 1 where the formula (~?~)~(~?t ) ~ = ~(~s,6~t~ - N - 1 f s a 6 ~ ) has been used. Introducing a step function care- fully we can express eq. (i 3) in a compact form

n l

i g2_ =~_( ~ [ { l ~ k ( Y , m ) , M k i ( m , x ) } + ( 1 / 2 N ) M k k ( m , m ) M / i O ' , x ) ] 4 l n m=-n

n

-- m=/+l ~ [ {M]k (Y' m ) ' M k i ( m ' x ) ) + (1/2N)Mkk(m' m ) ~ i ( Y ' x)] ) [O(l-x) --O(l-y)+ ~61¢ ¢ -- ~l,y]l , (14)

where a subtle cancellation has happened in arranging the O(1/N) terms into Mkk(m, m)M/i(y, x). The above meson operator equation has been formulated in the case y > x. Since the equation of motion for a

link variable U + is provided by

= i [ (V,(x) -xa)"a e f (x + 1) + (u(x)ef (x) (15)

where the relation (E~) 2 = (E~) 2 given by eq. (9) was used, we can show that the meson operator for x > y satis- fies the same equation of motion as the meson operator for y > x does.

Using the Fourier transform of the step function in lattice space n

Ok -_ ~ e_ik n x + lxl _ i[cos(~(2n +m)k)- cos(~k)]/2 sin(~k) (16) x=-n 21xl

we obtain the equation of motion for the meson (string) operator in momentum space

i~M/i(Pl, p2)/3t = sin Pl ~k/Mki(Pl , P2) + sin P2 ~ k ( P l , P2)aik

~g2 L -1 ~ D ( q ) [ ~M/k(Pl, k), Mki(q - k, P2 - q)) - {MIk(Pl - q, k), Mki(q - k, P2))] q

_ g2 L-1 ~ D ( q ) [ M k k ( q _ k, k ) l~ i (P l , P2 - q) - Mkk(k, q - k)Mii(Pl - q, P2)] , (17) 4N q

where M/i(Pl,P2 ) is defined byMff~v,x)=L-2Zpl,p 2 exp(iPlY +iP2x)Mi/(pl,P2 ) and D(q)= {1 - exp[iq(n + 1)]}/ [2 sin(~-q)] 2. The third term on the RHS ofeq. (17) implies the dissociation of a stringM/i(p 1,P2) into two, and the fourth term is interpreted as the quark-antiquark pair creationeffect through the gluon emission, which is suppressed in the large Nlimit. The equation for the meson wave function M/z~o 1, p2) = (0 ~//t~o 1, P21 ) is obtained by sandwiching eq. (17) with the vacuum state and a meson state I ). In the large N limit we can adopt the factorization property such as

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Volume 167B, number 1 PHYSICS LETTERS 30 January 1986

(0 ~/]k (P 1 , k)Mki'(q-k,,p2-q)[ ) ~ (0[M]k (P 1 , k)lO)(O[Mki(q-k,P2-q)[ ),

where in the intermediate state the vacuum state 10) dominates and the contributions from the hadronic states will be suppressed. Moreover in the large N limit the normal ordering :M]i(Pl, P2): can be related to that of the quark field. Using the residual time independent gauge transformation, we can set U~O(x, t = O) = 0 so that the normal ordering of the meson operator at a certain time, t = 0 is defined by that of the quark field. The normal ordered Mji at t ¢ 0 is determined in such a way that it is consistent with the equation of motion. By this gauge independent normal ordering we can show

1 (18) M]i(Pl ,P2) = :M/i(Pl, P2): - 2NSpl +P2,0 751i,

where we have made use of the anticommutators (b~(Pl) , b+0(p2) ) = {d~(Pl), d+t~(p2)} = 5aOSpl ,p2 given from eqs. (3) and (4). In the large N limit we arrive at the meson wave equation in a closed form

~M/i(PJ, P2) = sin Pl ak/Mki(Pl , P2) + sin P2 M/k(Pl, P2)Ctik

--~g2NL-I ~ D(q)t'~5]k(Mki(Pl + q,P2 - q) --)~ki(Pl,P2)) -- 75ki(l~/k(Pl - q,P2 + q) - - / ~ / k ( P l , P 2 ) ) ] , q (19)

which can be further separated by a decomposition )1~ =)t~ 0 + i70/1~1 + i7075A~2 + 75J1~3 into two sets of coupled equations

¢°/~0(Pl, P2) = (sin PI + sin pE)/~ta(Pl, P2) + ~g 2NL -I ~ Da(q)[~3(p I + q, P2 - q) -/~ta(Pl - q ' P2 + q) ] , q

¢°M3(Pl, P2) = (sin Pl + sin p2)/~t0(Pl, P2) + ~g 2NL-I ~ Da(q)[Mo(Pl + q, P2 - q) -/~0(Pl - q ' P2 + q)l , q

(20)

w/l~l (Pl , P2) = (--sin Pl + sin p2) /~Z(p l , P2) -- ~g 2NL -1 ~ Ds(q ) [/l~rZ(pl + q ,P2 - q) - / ~ 2 ( P l , P2)] , q

w/I~2(Pl ,P2) = ( - s i n P l + sinP2)M-l(Pl,P2)--½g 2NL-1 ~ Ds(q)[MI(Pl +q,P2 - q ) - M I ( P l , P2)] , (21) q

where D a (q)..= - i sin [q(n + 1)] / [2 sin(X~-q)] 2 and Ds(q) = ( I - cos [q(n + 1)] ) / [2 sin(½q)] 2. The combination M+ = M 1 + M 2 rearranges eq. (21) into

~ + ( P l , P2) = ( - sin Pl + sin P2)M+(Pl , P2) -- ~g 2NL- 1 ~ Ds(q ) [M+(Pl + q, P2 - q) - )I~+(Pl, P2)] , (22) q

w/l~t-(Pl, P2) = (sin Pl - sin p 2 ) / ~ _ ( p l , P2) + }g 2NL-1 ~ Ds(q)[M-(Pl + q, P2 - q) - / k t - ( P l , P2)] • (23) q

/~±(Pl , P2) is just the Fourier transform of - ~ i [~ (y ) (1 + "Y5)U(y, x), ¢/(x)]. Since the vacuum state (01 is char- acterized by ( 0 ~ + a ~ ) = (01d+:'(p) = 0 and I ) is the quark and antiquark bound state, only the ~r+ does not vanish among M 0, M 3 and M e. In the center-of-mass frame P2 = - P l = P eq. (22) becomes

Ucb(p) = 2 sin p cb(p) _ ~g2NL-1 ~ Ds(q)[cb(p _ q) _ qbfp)] , ep(p)-- ~I+(--p, p) , (24) q

which is the eigenvalue equation for the meson mass ta. Eq. (24) corresponds to the ' t Hooft equation in con- t inuum two-dimensional QCD. It is noted that the resulting q summation is deaf ly free of infrared singularity. Even when the lattice size L is taken infinite this property is retained, as is shown by

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Volume 167B, number 1 PttYSICS LETTERS 30 January 1986

~r dq q 1

- ~g2Nf __ 27r [2 sin(~q)] 2 p q [~b(p - q) _ o(P)] • (25)

It is o f interest that the principal value prescription appears. We cannot further express eq. (22) in terms o f mo- mentum rescale variables in contrast with the continuum theory.

In this paper we showed ~hat the path ordered operator formulation can be naturally incorporated in two- dimensional massless latt ice QCD. The equation of mot ion for the meson operator was so constructed that the string picture might be visualized. In this construction Gauss' law had an essential role, which was solved with link variables. Gauss' law makes the gauge symmetry preserved so that a physical state is identified with a gauge in- variant state and a gauge invariant string operator dissociates into two gauge invafiant string operators with time. Using the large N factorization proper ty we presented the wave equation of the meson in massless lattice QCD, which has structures analogous to the ' t Hooft equation. We hope that our formulation in the latt ice space will be useful in the four-dimensional extension of two-dimensional QCD. The numerical analysis for the solution o f our eigenvalue equation and the investigation for the Wilson fermion case are left as future tasks.

R e F e r e n c e s

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[2] G. 't Hooft, Nucl. Phys. B138 (1978) 1; Y. Nambu, Phys. Lett. 80B (1979) 372; Yu.M. Makeenko and A.A. Migdal, Phys. Lett. 88B (1979) 135; A. Suura, Phys. Rev. D20 (1979) 1412; A.M. Polyakov, Nucl. Phys. B164 (1980) 171 ; B. Sakita, Phys. Rev. D21 (1980) 1067; A. Migdal, Phys. Rep. 102 (1983) 199.

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