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The mass gap in the SU(N) × SU(N) chiral model by Green's function method
Volume 142, number 1,2
PHYSICS LETTERS
12 July 1984
THE MASS GAP IN THE SU(N) X SU(N) CHIRAL MODEL BY GREEN'S FUNCTION METHOD Shij ong RYANG
Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan Received 6 March 1984 Revised manuscript received 17 Ap'ril 1984 Using Green's function method we approximately obtain the mass gap of the SU(N) X SU(N) chiral model in a two-dimensional lattice for all values of the coupling constant. In the strong coupling region our results agree with those given by Shigemitsu and Kogut, while in the weak coupling limit the mass gap vanishes exponentially with the coupling constant.
Spin systems in two dimensions have received much attention as systems having considerable resemblances to the four-dimensional nonabelian gauge theories, with respect to asymptotic freedom, global topological structure and phase structure [ 1]. The two-dimensional O (N) vector spin model has been investigated by various theoretical techniques such as weak coupling perturbation [2], strong coupling expansion [3], Monte Carlo method [4] and Green's function method [5]. The continuous theory of SU(N) × SU(N) chiral model in two dimensions was proved to be asymptotically free by the weak coupling perturbative theory [6]. By applying the strong coupling expansion method to the lattice version of the two-dimensional chiral model, several authors investigated the mass gap [7] and the phase structure [8], and found that this model has a single phase. Studies of the specific heat by the Monte Carlo method also confirmed this phase structure [9]. Recently various types of the variational method to spin or gauge systems have been studied as an analytic approach applicable to the whole coupling region. In the magnetism theory the Green function method was developed to have validity in the entire region of temperature [ 10]. This approach has been applied to the O(N) vector spin model in 1+1 dimensions and the mass gap was calculated [5]. The SU(N) × SU(N) chiral model forms a more intimate analog of the SU(N) lattice gauge theory since their degrees of freedom are elements of a Lie group. The chiral model is a highly interacting theory and the physics should not be the same as in the vector model. There is another different point that the N + oo limit for the O(N) vector model is soluble by the well-defined 1IN expansion method, while no solution is found for the two-dimensional chiral model. We will investigate the SU(N) X SU(N) chiral model in 1+1 dimensions by the Green function method and evaluate via an approximation scheme, the mass gap for all values of the coupling constant. We begin with the hamiltonian of the SU(N) X SU(N) chiral model in 1+1 dimensions H - -_1 ~g
2 ~ lEa(x, t)]2 x
-~
1 ~
g2 x
[Tr U(x, t) U+(x + 1, t) + Tr U(x + 1, t) U+(x, t)],
(1)
where U(x) is an SU(N) matrix defined at site x (withx an integer 1 - L and the lattice constant is set to be unity), and E ~ is the ath generator of right SU(N) gauge transformation [left SU(N) gauge transformation could have been used equally well]. The equal-time commutation relations are given by [E~(x, t), U(x,t t)] = U(x, t) ~ X~ 8xx, ,
[E~(x, t), U+(x ', t)] = - 1 ~k¢~U+(X, t) ~XX"
[U(x, t), U(x', t)] = [U(x, t), U+(x ', t)] = [U+(x, t), U+(x ', t)] = 0.
(2)
Following Green's function method, we consider the following type of a double-time retarded (zeroth) Green's function 59
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((Tr (U(x, t); U+(x ', t')))) - O(t -- t') (Tr [U(x, t), U+(x', t')] ),
(3)
where U(x, t) is a Heisenberg operator and bracket ( ) shows the expectation value with respect to the ground state of the hamiltonian. Using the Heisenberg equation of motion for U(x, t),
Ot U(x, t) = I ig 2 {U(x, 0 1 xa Ea( x, t) + E%x, t) U(x, 0 1 Xa},
(4)
we get an equation of the zeroth Green function: ~ <(Tr(U(x); V+(x ')))> = i g2 <>+ i I g2 C2 ((Tr(U(x); U+(x'))}),
(5)
where times t and t' are suppressed and C 2 is the quadratic Casimir of the fundamental representation, i.e., C 2 = (N 2-1)/2N. Next we construct an equation of the "first-order" Green function ((Tr(U(x)I x~ E~(x); U+(x')) )) as ~t ((Tr (U(x) E(x); U+(x')))}
(t - t') (Tr [U(x) E(x), U+(x')]) + ig 2 ((Tr(U(x) E(x) E(x); U+(x'))))
+ i i g2 C2 <> +ig-2 g> ((Tr(U(x) I X~ [Tr(U(x + 1) + U(x - 1)) 1 ~,~ U+(x)] ; U+(x'))))},
(6)
where E(x) - 1 X~E%x). By using the formula
(g;k )ca = ~(Sad6bc - N-18abScd) the last term in eq (6) can be reexpressed in the form (i/2g 2) {((Tr(U(x) (U+(x+ l ) + U+(x-1)) U(x); U+(x'))))-((Tr(( U(x +l ) + U(x-1)); U+(x'))))}
- (i/2g2N)((Tr(U(x)[Tr(U(x) U+(x+l)+ U(x) U+(x-1) - U ( x + l ) U+(x) - U ( x - 1 ) U+(x))];U+(x')))). (7) The equal-time commutation relations (2) and U(x) U+(x) = 1 make the first term more compactified as
6(t - t') (Tr[U(x) E(x), U+(x')]) = - C2N bxx, 6 ( t - t'). Following the decoupling method in the magnetism theory, we now express the "higher-order" Green functions ((Tr(U(x) E(x)E(x); U+(x'))), ((Tr(U(x) [Tr U(x + 1) U+(x)] ; U+(x')))) etc. appearing in the above eqs. (6) and (7) in terms of the zeroth and first-order Green functions. The decoupling approximation for ((Tr(U(x)E(x)E(x); U+(x')))) is therefore to put it in the form ((Tr(U(x) E(x) E(x); U+(x')))) ~ A l((Tr(U(x) E(x); U+(x')))) + A 2 ((Tr(U(x); U+(x')))),
(8)
where A 1 and A 2 are supposed to depend on g and N, but not on space and time. Using the spectrum theorem [10] which describes the linear relation between Green's function and the correlation function, we obtain
(Tr[U(x)E(x)E(x), U+(x')])-~AI(Tr[U(x)E(x ), U+(x')])+A2(Tr[U(x ), U+(x')] ).
(9)
Setting t = t' and using commutation relations (2), we find A 1 to be - C 2. We regard A 2 as a parameter which depends on both g and N. Then A 2 will be determined self-consistently later. The other type of higher-order Green function ((Tr(U(x) [Tr U(x + 1) U+(x)] ; U+(x')))) may be decoupled as ((Tr(U(x) [Tr U(x+l) U+(x)]; U+(x'))))-~ BI(Tr(U(x+I) U+(x))) ((Tr(U(x); U+(x'))))+ B2((Tr(U(x+I ); U+(x')))).
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(Tr(U(x +1) U+(x))) in the first term is the nearest correlation function and its dependence on g and N will also be determined later self-consistently. The use of the spectrum theorem for eq. (10) leads to (Tr(U(x)[Tr U(x +1) U+(x)] ; U+(x'))) -~ BI(Tr(U(x+I) U+(x)))(Tr(U(x) U+(x'))} + B2(Tr(U(x+I) U+(x'))).
(11) Firstly, by setting x =x' at equal time in this equation, we obtain the relation
N=B1N+B2,
(12)
which can be regarded to hold in the whole coupling region. Secondly, for t = t' and x = x ' - 1, the eq. (11) becomes (Tr(U(x)[Tr U(x+l) U+(x)] U+(x+l))) = BI(Tr(U(x+!) U+(x))) (Tr(U(x) U+(x+l))) + B2N.
(13)
In the strong coupling limit where the nearest correlation is reduced to be zero, eq. (13) can be estimated as 1 = B2N. In the weak coupling limit the nearest correlation is so maximum that we get N 2 = B1 N2 +B2N , which is nothing but eq. (12). Therefore, the values B 1 = (N 2 - 1)/N2 and B 2 = 1IN can be used in the whole coupling region. For the other higher-order Green functions we take the similar decoupling procedure ((Tr(U(x) [Tr U(x) U+(x+l)l ; U+(x'))))m C(Tr(U(x) U+(x+i))} ((Tr(U(x); U+(x')))}, ((Tr(U(x) U+(x+i) U(x); U+(x'))))~D(Tr(U+(x+i) U(x)))((Tr(U(x); U+(x')))).
(14)
The spectrum theorem gives us (Tr(U(x) [Tr U(x) U+(x+l)] U+(x'))> ~ C(Tr(U(x) U+(x+l))) (Tr(U(x) U+(x'))), (Tr(U(x) U+(x+l) U(x) U+(x')))~-- D(Tr(U+(x+I ) U(x))) (Tr(U(x) U+(x'))).
(15)
C and D are uniquely determined b y setting x = x' at equal time in eq. (15) as C = 1 and D = 1IN. By the above decoupling approximation, eqs. (5) and (6) can be solved in closed form. The Fourier transformations of Green's functions are expressed as ((Tr(U(x); U+(x'))))=L -1 k ~ f d o o e ik(x-x')-k°(t-tS) Ou(k , co),
((Tr(U(x)E(x); U+(x')))) = L -1 E f d c o eik(x-x')-iw(t-t') OEu(k , ~). k
(16)
From eq. (6) we derive
GEu(k , oo) = [-C2N i/2rr + q/(k) Gu(k , co)]/(¢o - 2 1 g2 C2) ,
(17)
T(k) = _g2A2 + g - 2 ( 1 - N -2) (cos k - f/N),
(18)
where f is the nearest correlation function, f = (Tr(U(x) U+(x+l))) = (Tr(U(x) U+(x-1))> = (Tr(U+(x+l)
U(x))>= (Tr(U+(x-1) U(x)))
= (Tr(U(x+l) U+(x)))= ( T r ( U ( x - 1 ) U+(x))). From eqs. (5) and (17) we get the solution
Gu(k, 6o) = ig 2 (C2N/2rr)/(¢o 2 - E2(k)),
(19)
where 61
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E2(k) = (_~g2 C2)2 + g4A 2 - ( 1 - N -2) (cos k - f/N).
(20)
Substituting eq (19) into the formula of spectrum theorem (Tr(U(x, t) U(x', t'))) = ? deo [au(w + ie) - Gu(w - ie)] e -k° (t-t'), 0 where
(21)
1
GU(¢O) = ~ f d ( t - t') ((Tr(U(x, t); U+(x ', t')))) e ic° (t- t'), w e obtain the correlation function (Tr(U(x, t) U+(x ', t'))) = l g 2 C 2 N L - 1 ~
1 eik(x_x, )_iE(k)(t_t,)
(22)
By setting x =x' or x'=x +1 at equal time we get 1
l=~g
2
C2
L_I~
1
(23)
k E(k)'
f=± ((Tr(U(x) U+(x+l))) + (Tr(U(x) U+(x 1)))) =½g2C2NL-1 ~ 2
k
cosk
E(k)"
(24)
Here let us parametrize E2(k) in the form E2(k) = ( 1 - N -2) (1 - cos k) + m 2,
(25)
where rn corresponds to the mass gap. Substitution of eq. (25) into eq. (20) gives just the mass gap equation. The obtained mass gap together with eq. (25) will determine the nearest correlation function f i n eq. (24). Then, by combining eqs. (20) and (25) the parameter A 2 will be expressed in terms ofg and N in a self-consistent way. Taking a limit L ~ ~ in eqs. (23) and (24), in order to consider an infinite space, we derive
1 = [g2C2/Vr2rr (1 - N - 2 ) 1/2 ] X/raK(Vra), f= - [vr2g2C2Nfir(1-U-2) 1/2] (1/V~) E('Vt~-) +(2/~ - 1 ) N ,
(26)
a-~ [l+m2/2(1--N-2)] -1,
(27)
where K(Vta) and E(~c/a) are respectively the complete first-kind and second-kind elliptic integrals. Eq. (26) specifies the mass gap ovgr the whole coupling region. First we examine the gap equation (26) in the strong coupling region (g>> 1) which implies m >> 1 and ~ "¢ 1. Up to the third order (1/g 6) we get
1 g2 C2 _ 2/Ng2 + (4/C2N2)g-6. m = -~
(28)
The first and second results exactly agree with those of Kogut and Shigemitsu who used the strong-coupling expansion method in SU(N) groups (N t> 3), except for the SU(2) case [7]. Next taking a weak coupling limit (g ~ 1, m ~ 1, a ~ 1) in eq. (26), we derive m ~ 4V/2 (1 - N - 2 ) 1/2 exp [ - (2V~rc/g2N) (1 - N - 2 ) " 1/2].
(29)
For finite N the mass gap vanishes exponentially in the weak coupling limit, from which g = 0 is considered as the critical coupling. It should be noted that the mass gap (29) has a finite value in the N-+ oo limit with g2N fixed at small values. To examine the behavior of the nearest correlation function f in the strong coupling region we substitute the solution (28) into eq. (27) and find f = 4 / N ( 1 - N - 2 ) g 4 + O(1/g8), 62
(30)
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where b o t h first 0 ( g 4) term and second O(1) terms exactly vanish and the third term 0 ( 1 / g 4) remains. Indeed the nearest correlation f u n c t i o n f reduces to small in the strong coupling region. In the weak coupling limit the correlat i o n f turns out to be N as expected. As a step to SU(N) lattice gauge theory in four dimensions, we investigated the mass gap o f SU(N) × SU(N) chiral model in 1+1 dimensions b y Green's function method. Our obtained mass gap agrees with strong coupling expansion series up to the second order and shows a non-perturbative behavior e - l / g 2 in the weak coupling limit. Over the whole coupling region there remains the finite mass gap, which is consistent with the Monte Carlo result that this model exists within a single phase. We hope that our techniques used in the chiral model will provide a useful analytic tool to study the string tension or the glueball mass over all ranges o f coupling constant in SU(N) lattice gauge theory. I would like to thank Professor T. Saito and Dr. K. Shigemoto for useful discussions.
References [1 ] A.A. Migdal, Soy. Phys. JETP 42 (1976) 413,743; A.A. Belavin and A.M. Polyakov, Soy. Phys. JETP Lett. 22 (1975) 245; E. Fradkin and L. Susskind, Phys. Rev. D17 (1979) 2637. [2] A.M. Polyakov, Phys. Lett. 59B (1975) 79; W.A. Bardeen, B.W. Lee and R.E. Shrock, Phys. Rev. D14 (1976) 985. [3] CJ. Hamer, J.B. Kogut and L. Susskind, Phys. Rev. D19 (1979) 3091. [4] G. Fox, R. Gupta, O. Martin and S. Otto, Nucl. Phys. B205 (1982) 188. [5] T. Matsui, Purdue preprint PURD-TH-83-18 (1983). [6] A. Mekane and M. Stone, Nucl. Phys. B163 (1980) 169. [7] J. Shigemitsu and J.B. Kogut, Nucl. Phys. B190 (1981) 365. [8] F. Green and S. Samuel, Nucl. Phys. B190 (1981) 113. [9] J. Kogut, M. Snow and M. Stone, Nucl. Phys. B215 (1983) 45. [10] V.L. Bonch-Bruevich and S.V. Tyablikov, in: The Green function method in statistical mechanics, ed. D. ter Haar (NorthHolland, Amsterdam, 1962); L. P. Kadanoff and G. Baym, in: Quantum statistical mechanics (Benjamin, New York, 1962); H.B. Callen, Phys. Rev. 130 (1963) 890; V. Mubayi and R.V. Lange, Phys. Rev. 178 (1969) 882.
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THE MASS GAP IN THE SU(N) X SU(N) CHIRAL MODEL BY GREEN'S FUNCTION METHOD Shij ong RYANG
Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan Received 6 March 1984 Revised manuscript received 17 Ap'ril 1984 Using Green's function method we approximately obtain the mass gap of the SU(N) X SU(N) chiral model in a two-dimensional lattice for all values of the coupling constant. In the strong coupling region our results agree with those given by Shigemitsu and Kogut, while in the weak coupling limit the mass gap vanishes exponentially with the coupling constant.
Spin systems in two dimensions have received much attention as systems having considerable resemblances to the four-dimensional nonabelian gauge theories, with respect to asymptotic freedom, global topological structure and phase structure [ 1]. The two-dimensional O (N) vector spin model has been investigated by various theoretical techniques such as weak coupling perturbation [2], strong coupling expansion [3], Monte Carlo method [4] and Green's function method [5]. The continuous theory of SU(N) × SU(N) chiral model in two dimensions was proved to be asymptotically free by the weak coupling perturbative theory [6]. By applying the strong coupling expansion method to the lattice version of the two-dimensional chiral model, several authors investigated the mass gap [7] and the phase structure [8], and found that this model has a single phase. Studies of the specific heat by the Monte Carlo method also confirmed this phase structure [9]. Recently various types of the variational method to spin or gauge systems have been studied as an analytic approach applicable to the whole coupling region. In the magnetism theory the Green function method was developed to have validity in the entire region of temperature [ 10]. This approach has been applied to the O(N) vector spin model in 1+1 dimensions and the mass gap was calculated [5]. The SU(N) × SU(N) chiral model forms a more intimate analog of the SU(N) lattice gauge theory since their degrees of freedom are elements of a Lie group. The chiral model is a highly interacting theory and the physics should not be the same as in the vector model. There is another different point that the N + oo limit for the O(N) vector model is soluble by the well-defined 1IN expansion method, while no solution is found for the two-dimensional chiral model. We will investigate the SU(N) X SU(N) chiral model in 1+1 dimensions by the Green function method and evaluate via an approximation scheme, the mass gap for all values of the coupling constant. We begin with the hamiltonian of the SU(N) X SU(N) chiral model in 1+1 dimensions H - -_1 ~g
2 ~ lEa(x, t)]2 x
-~
1 ~
g2 x
[Tr U(x, t) U+(x + 1, t) + Tr U(x + 1, t) U+(x, t)],
(1)
where U(x) is an SU(N) matrix defined at site x (withx an integer 1 - L and the lattice constant is set to be unity), and E ~ is the ath generator of right SU(N) gauge transformation [left SU(N) gauge transformation could have been used equally well]. The equal-time commutation relations are given by [E~(x, t), U(x,t t)] = U(x, t) ~ X~ 8xx, ,
[E~(x, t), U+(x ', t)] = - 1 ~k¢~U+(X, t) ~XX"
[U(x, t), U(x', t)] = [U(x, t), U+(x ', t)] = [U+(x, t), U+(x ', t)] = 0.
(2)
Following Green's function method, we consider the following type of a double-time retarded (zeroth) Green's function 59
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((Tr (U(x, t); U+(x ', t')))) - O(t -- t') (Tr [U(x, t), U+(x', t')] ),
(3)
where U(x, t) is a Heisenberg operator and bracket ( ) shows the expectation value with respect to the ground state of the hamiltonian. Using the Heisenberg equation of motion for U(x, t),
Ot U(x, t) = I ig 2 {U(x, 0 1 xa Ea( x, t) + E%x, t) U(x, 0 1 Xa},
(4)
we get an equation of the zeroth Green function: ~ <(Tr(U(x); V+(x ')))> = i g2 <
(5)
where times t and t' are suppressed and C 2 is the quadratic Casimir of the fundamental representation, i.e., C 2 = (N 2-1)/2N. Next we construct an equation of the "first-order" Green function ((Tr(U(x)I x~ E~(x); U+(x')) )) as ~t ((Tr (U(x) E(x); U+(x')))}
(t - t') (Tr [U(x) E(x), U+(x')]) + ig 2 ((Tr(U(x) E(x) E(x); U+(x'))))
+ i i g2 C2 <
(6)
where E(x) - 1 X~E%x). By using the formula
(g;k )ca = ~(Sad6bc - N-18abScd) the last term in eq (6) can be reexpressed in the form (i/2g 2) {((Tr(U(x) (U+(x+ l ) + U+(x-1)) U(x); U+(x'))))-((Tr(( U(x +l ) + U(x-1)); U+(x'))))}
- (i/2g2N)((Tr(U(x)[Tr(U(x) U+(x+l)+ U(x) U+(x-1) - U ( x + l ) U+(x) - U ( x - 1 ) U+(x))];U+(x')))). (7) The equal-time commutation relations (2) and U(x) U+(x) = 1 make the first term more compactified as
6(t - t') (Tr[U(x) E(x), U+(x')]) = - C2N bxx, 6 ( t - t'). Following the decoupling method in the magnetism theory, we now express the "higher-order" Green functions ((Tr(U(x) E(x)E(x); U+(x'))), ((Tr(U(x) [Tr U(x + 1) U+(x)] ; U+(x')))) etc. appearing in the above eqs. (6) and (7) in terms of the zeroth and first-order Green functions. The decoupling approximation for ((Tr(U(x)E(x)E(x); U+(x')))) is therefore to put it in the form ((Tr(U(x) E(x) E(x); U+(x')))) ~ A l((Tr(U(x) E(x); U+(x')))) + A 2 ((Tr(U(x); U+(x')))),
(8)
where A 1 and A 2 are supposed to depend on g and N, but not on space and time. Using the spectrum theorem [10] which describes the linear relation between Green's function and the correlation function, we obtain
(Tr[U(x)E(x)E(x), U+(x')])-~AI(Tr[U(x)E(x ), U+(x')])+A2(Tr[U(x ), U+(x')] ).
(9)
Setting t = t' and using commutation relations (2), we find A 1 to be - C 2. We regard A 2 as a parameter which depends on both g and N. Then A 2 will be determined self-consistently later. The other type of higher-order Green function ((Tr(U(x) [Tr U(x + 1) U+(x)] ; U+(x')))) may be decoupled as ((Tr(U(x) [Tr U(x+l) U+(x)]; U+(x'))))-~ BI(Tr(U(x+I) U+(x))) ((Tr(U(x); U+(x'))))+ B2((Tr(U(x+I ); U+(x')))).
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(Tr(U(x +1) U+(x))) in the first term is the nearest correlation function and its dependence on g and N will also be determined later self-consistently. The use of the spectrum theorem for eq. (10) leads to (Tr(U(x)[Tr U(x +1) U+(x)] ; U+(x'))) -~ BI(Tr(U(x+I) U+(x)))(Tr(U(x) U+(x'))} + B2(Tr(U(x+I) U+(x'))).
(11) Firstly, by setting x =x' at equal time in this equation, we obtain the relation
N=B1N+B2,
(12)
which can be regarded to hold in the whole coupling region. Secondly, for t = t' and x = x ' - 1, the eq. (11) becomes (Tr(U(x)[Tr U(x+l) U+(x)] U+(x+l))) = BI(Tr(U(x+!) U+(x))) (Tr(U(x) U+(x+l))) + B2N.
(13)
In the strong coupling limit where the nearest correlation is reduced to be zero, eq. (13) can be estimated as 1 = B2N. In the weak coupling limit the nearest correlation is so maximum that we get N 2 = B1 N2 +B2N , which is nothing but eq. (12). Therefore, the values B 1 = (N 2 - 1)/N2 and B 2 = 1IN can be used in the whole coupling region. For the other higher-order Green functions we take the similar decoupling procedure ((Tr(U(x) [Tr U(x) U+(x+l)l ; U+(x'))))m C(Tr(U(x) U+(x+i))} ((Tr(U(x); U+(x')))}, ((Tr(U(x) U+(x+i) U(x); U+(x'))))~D(Tr(U+(x+i) U(x)))((Tr(U(x); U+(x')))).
(14)
The spectrum theorem gives us (Tr(U(x) [Tr U(x) U+(x+l)] U+(x'))> ~ C(Tr(U(x) U+(x+l))) (Tr(U(x) U+(x'))), (Tr(U(x) U+(x+l) U(x) U+(x')))~-- D(Tr(U+(x+I ) U(x))) (Tr(U(x) U+(x'))).
(15)
C and D are uniquely determined b y setting x = x' at equal time in eq. (15) as C = 1 and D = 1IN. By the above decoupling approximation, eqs. (5) and (6) can be solved in closed form. The Fourier transformations of Green's functions are expressed as ((Tr(U(x); U+(x'))))=L -1 k ~ f d o o e ik(x-x')-k°(t-tS) Ou(k , co),
((Tr(U(x)E(x); U+(x')))) = L -1 E f d c o eik(x-x')-iw(t-t') OEu(k , ~). k
(16)
From eq. (6) we derive
GEu(k , oo) = [-C2N i/2rr + q/(k) Gu(k , co)]/(¢o - 2 1 g2 C2) ,
(17)
T(k) = _g2A2 + g - 2 ( 1 - N -2) (cos k - f/N),
(18)
where f is the nearest correlation function, f = (Tr(U(x) U+(x+l))) = (Tr(U(x) U+(x-1))> = (Tr(U+(x+l)
U(x))>= (Tr(U+(x-1) U(x)))
= (Tr(U(x+l) U+(x)))= ( T r ( U ( x - 1 ) U+(x))). From eqs. (5) and (17) we get the solution
Gu(k, 6o) = ig 2 (C2N/2rr)/(¢o 2 - E2(k)),
(19)
where 61
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E2(k) = (_~g2 C2)2 + g4A 2 - ( 1 - N -2) (cos k - f/N).
(20)
Substituting eq (19) into the formula of spectrum theorem (Tr(U(x, t) U(x', t'))) = ? deo [au(w + ie) - Gu(w - ie)] e -k° (t-t'), 0 where
(21)
1
GU(¢O) = ~ f d ( t - t') ((Tr(U(x, t); U+(x ', t')))) e ic° (t- t'), w e obtain the correlation function (Tr(U(x, t) U+(x ', t'))) = l g 2 C 2 N L - 1 ~
1 eik(x_x, )_iE(k)(t_t,)
(22)
By setting x =x' or x'=x +1 at equal time we get 1
l=~g
2
C2
L_I~
1
(23)
k E(k)'
f=± ((Tr(U(x) U+(x+l))) + (Tr(U(x) U+(x 1)))) =½g2C2NL-1 ~ 2
k
cosk
E(k)"
(24)
Here let us parametrize E2(k) in the form E2(k) = ( 1 - N -2) (1 - cos k) + m 2,
(25)
where rn corresponds to the mass gap. Substitution of eq. (25) into eq. (20) gives just the mass gap equation. The obtained mass gap together with eq. (25) will determine the nearest correlation function f i n eq. (24). Then, by combining eqs. (20) and (25) the parameter A 2 will be expressed in terms ofg and N in a self-consistent way. Taking a limit L ~ ~ in eqs. (23) and (24), in order to consider an infinite space, we derive
1 = [g2C2/Vr2rr (1 - N - 2 ) 1/2 ] X/raK(Vra), f= - [vr2g2C2Nfir(1-U-2) 1/2] (1/V~) E('Vt~-) +(2/~ - 1 ) N ,
(26)
a-~ [l+m2/2(1--N-2)] -1,
(27)
where K(Vta) and E(~c/a) are respectively the complete first-kind and second-kind elliptic integrals. Eq. (26) specifies the mass gap ovgr the whole coupling region. First we examine the gap equation (26) in the strong coupling region (g>> 1) which implies m >> 1 and ~ "¢ 1. Up to the third order (1/g 6) we get
1 g2 C2 _ 2/Ng2 + (4/C2N2)g-6. m = -~
(28)
The first and second results exactly agree with those of Kogut and Shigemitsu who used the strong-coupling expansion method in SU(N) groups (N t> 3), except for the SU(2) case [7]. Next taking a weak coupling limit (g ~ 1, m ~ 1, a ~ 1) in eq. (26), we derive m ~ 4V/2 (1 - N - 2 ) 1/2 exp [ - (2V~rc/g2N) (1 - N - 2 ) " 1/2].
(29)
For finite N the mass gap vanishes exponentially in the weak coupling limit, from which g = 0 is considered as the critical coupling. It should be noted that the mass gap (29) has a finite value in the N-+ oo limit with g2N fixed at small values. To examine the behavior of the nearest correlation function f in the strong coupling region we substitute the solution (28) into eq. (27) and find f = 4 / N ( 1 - N - 2 ) g 4 + O(1/g8), 62
(30)
Volume 142, number 1,2
PHYSICS LETTERS
12 July 1984
where b o t h first 0 ( g 4) term and second O(1) terms exactly vanish and the third term 0 ( 1 / g 4) remains. Indeed the nearest correlation f u n c t i o n f reduces to small in the strong coupling region. In the weak coupling limit the correlat i o n f turns out to be N as expected. As a step to SU(N) lattice gauge theory in four dimensions, we investigated the mass gap o f SU(N) × SU(N) chiral model in 1+1 dimensions b y Green's function method. Our obtained mass gap agrees with strong coupling expansion series up to the second order and shows a non-perturbative behavior e - l / g 2 in the weak coupling limit. Over the whole coupling region there remains the finite mass gap, which is consistent with the Monte Carlo result that this model exists within a single phase. We hope that our techniques used in the chiral model will provide a useful analytic tool to study the string tension or the glueball mass over all ranges o f coupling constant in SU(N) lattice gauge theory. I would like to thank Professor T. Saito and Dr. K. Shigemoto for useful discussions.
References [1 ] A.A. Migdal, Soy. Phys. JETP 42 (1976) 413,743; A.A. Belavin and A.M. Polyakov, Soy. Phys. JETP Lett. 22 (1975) 245; E. Fradkin and L. Susskind, Phys. Rev. D17 (1979) 2637. [2] A.M. Polyakov, Phys. Lett. 59B (1975) 79; W.A. Bardeen, B.W. Lee and R.E. Shrock, Phys. Rev. D14 (1976) 985. [3] CJ. Hamer, J.B. Kogut and L. Susskind, Phys. Rev. D19 (1979) 3091. [4] G. Fox, R. Gupta, O. Martin and S. Otto, Nucl. Phys. B205 (1982) 188. [5] T. Matsui, Purdue preprint PURD-TH-83-18 (1983). [6] A. Mekane and M. Stone, Nucl. Phys. B163 (1980) 169. [7] J. Shigemitsu and J.B. Kogut, Nucl. Phys. B190 (1981) 365. [8] F. Green and S. Samuel, Nucl. Phys. B190 (1981) 113. [9] J. Kogut, M. Snow and M. Stone, Nucl. Phys. B215 (1983) 45. [10] V.L. Bonch-Bruevich and S.V. Tyablikov, in: The Green function method in statistical mechanics, ed. D. ter Haar (NorthHolland, Amsterdam, 1962); L. P. Kadanoff and G. Baym, in: Quantum statistical mechanics (Benjamin, New York, 1962); H.B. Callen, Phys. Rev. 130 (1963) 890; V. Mubayi and R.V. Lange, Phys. Rev. 178 (1969) 882.
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