- Email: [email protected]
Fermion propagators in U (1) lattice gauge theory
Volume 243, number 4
PHYSICS LETTERS B
5 July 1990
Fermion propagators in U ( 1 ) lattice gauge theory A. Nakamura and R. Sinclair lnstitutfiir TheoretischePhysik, Freie UniversiRitBerlin.Arnimallee 14, D-IO00Berlin 33, Germany Received 8 March 1990
Fermion propagators are calculated for quenched compact U( 1) lattice gauge theory. These vanish in the standard Landau gauge due to gauge-equivalentcopies of a given field. We discuss how we remove them. Obtained fermion masses are analyzed as functions of the Wilson hopping parameter in confinement and Coulomb phases. The propagators are seen to change abruptly at re.
1. We need not fix a gauge in lattice gauge theories. If, however, we do fix a gauge in numerical simulations, Monte Carlo calculations become a much more powerful tool in studying quantum field theories. We can calculate gauge non-invariant objects such as hadron wavefunctions [ 1,2], propagators o f gauge fields [3,4] and offermions [5,6] even far from perturbative regions. Gauge fixing may help us to construct the smooth configurations required in the calculation of multigrid fermion propagators. We study a simple example in this paper: fermion propagators in compact U(1 ) gauge theory. This model is clear enough for one to see what is going on when one fixes a gauge, and is a fundamental prototype in lattice gauge theories. 2. Let us start by describing our Landau gauge fixing procedure and outlining a difficulty in the calculation of fermion propagators. We produce gauge configurations {Uu (x) -= exp [iAu (x) ] } using a Monte Carlo update without gauge fixing terms and then fix the gauge [ 7,3 ] through a gauge transformation
A ~ = A , +&(~+)Z,
(1)
such that
E &(~-)A'~(x)=0,
(2)
where &( + / - ) are lattice forward/backward difference operators. It should be understood that the gauge transformation is defined by 396
UAx)-~ r+(x) UAx) V(x+ ~), where V ( x ) = e x p [ i g ( x ) ] . We impose periodic boundary conditions on U and V. N o w we are faced with a difficulty: the Z, and therefore the A~ as well, are not uniquely determined. One can add any function o9 to Z, as long as to is a lattice harmonic (i.e. Ato = 0 ). There are many solutions of this condition due to the compact nature o f the theory. A simple example is to= E~, auxu+ b with a~,= 2zm/L~,, where n is an arbitrary integer and Lu is the length of the lattice in the direction #. In the continuum theory, one imposes the asymptotic boundary condition A~,(x)~O for x ~ _+oo, and obtains a unique solution for the abelian case. (For non-abelian continuum field theories there is the famous Gribov ambiguity.) Through a gauge transformation, fermion propagators transform as
<~,(x)~(y) > (exp{i [z(x) -x(y) ]} g/(x)~(y) ).
(3)
Hence if we impose only the Lorentz condition (2), the fermion propagator vanishes due to this averaging over an arbitrary phase. (A similar situation was discussed in ref. [ 8 ] with respect to the Higgs model.) On a lattice, the Lorentz condition alone is not enough to deliver a non-vanishing propagator. A gauge transformation satisfying the Lorentz condition for the U ( 1 ) case is easily found:
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 243, number 4
PHYSICS LETTERSB
Z(x) = Z a ( x - y ) ~ 6~u-)Au(Y), y
(4)
I.t
where a (x) is the lattice Green's function --1
a ( x , = 4"Volumel ~ ' e x p ( i p x ) , (~u sin2 (½Pu)) • (5) We omit the zero mode in the momentum sum. Hence Aa (x) = - Ox,o + 1/ Volume, hut the second term disappears in the calculation o f z ( x ) . Z can be evaluated fast enough numerically by working in Fourier space. 3. In order to fix the Landau gauge on a lattice for U ( 1 ) gauge theory, we propose the following procedure (which we refer to as the TL gauge in the following): (i) Transform configurations { Uu (x) } to satisfy a maximal tree temporal gauge condition [9]. See fig. 1. (ii) Then, calculating the gauge transformation by eqs. ( 1 ), (4) and (5), we transform again to satisfy the Lorentz condition. We use a maximal tree in the fixing of the temporal gauge since the temporal gauge alone (only A4(X ) = 0), as in continuum theories [ 10 ], does not completely remove gauge degrees of freedom. The configurations fixed by step (i) are therefore uniquely determined, although the transformation itself is not. Z(x, t ) + 0 also satisfies the maximal tree temporal gauge condition. We remove this ambiguity by using
5 July 1990
a "punctured" gauge, in which Z(0, 0) =0. Our Landau gauge fixing technique (step (ii)) is unique by definition, and therefore does not produce any gaugeequivalent copies. As we saw before, the lattice Landau gauge does not have a one-to-one correspondence with that in the continuum theory. The lattice temporal gauge, on the contrary, does. In the continuum limit, the Landau and temporal gauges are related with each other in a well-defined way. It is therefore natural to make use of the TL gauge. Moreover, the temporal gauge ensures that the leading terms in the strong-coupling expansion of the fermion propagator do not average away to zero. This fact, and the uniqueness of step (ii), allows us to expect that our Landau gauge fixing delivers a non-zero propagator. 4. We define time-sliced fermion propagators as
G,~p(z, k=O )
For free Wilson fermions on an infinite lattice, this becomes [ 11 ] sgn(z) exp[-Ilog(l+m)'zl] Gfree (z, k = 0 ) =~'4 m-------) 2(1 + sgn(m) + - exp[-Ilog(l+m)-zl] 2(l+m) 1
+ ~ o 2( 1 + m ) ' I
I I
.....
I
+_+
----
- o - - -
- - - - 0
.
where m = l / 2 x - 4 . propagators as
V(z)=¼Try4G(z)
(7) We define vector and scalar (8)
and .
.
.
Fig. 1. Schematic diagram of maximal temporal gauge fixing in two dimensions.At,( x ) = 0 o n a tree (solid links).
S(r) = ~Tr G(z)
(9)
respectively, where the trace is taken over the Dirac indices. For the determination of xc it is of interest to know how these propagators behave as the lattice mass (m*) approaches zero. It is clearly related to the mass in the free particle propagator above by m * = l log( 1 + m) [. For masses close to zero we have m*-~ Iml, 397
Volume 243, number 4
Vf~(z)-
PHYSICS LETTERS B
½sgn (z) e x p ( - Imzl ),
combination of the conjugate-residual and conjugate-gradient methods with hopping parameter expansion preconditioning was used to invert the Wilson fermion matrix [ 12 ]. Typically, a hundred configurations, separated by one thousand sweeps in each case, were used in calculating a propagator. For each configuration used, we chose a representative point from each x4 = const, plane randomly, and then approximated the propagator by calculating from these points only and averaging over them. We imposed antiperiodic boundary conditions in the time direction for the fermionic fields, although the choice between periodic or antiperiodic boundary conditions is irrelevant in the case of a quenched U( 1 ) calculation (multiplying all the time-directed links at x4 = L4 by - 1e U ( 1 ) is a symmetry of the quenched action, but this also corresponds to interchanging periodic and antiperiodic boundary conditions). Defining xc as the value at which the fermion mass becomes zero, one would expect that tq= -~in the free case (A,,(x) = 0 ) . In the strong-coupling limit, Xc--[ 13 ]. The phase transition between the confined and Coulomb phases of the compact U ( 1 ) model occurs at r = 1 (see, for example, ref. [ 14] ). We calculated at r = 0.9 and r = 1.1 to examine differences between the confined and Coulomb phases respectively.
(10)
and Sfr~e(r)--,½ sgn(m) e x p ( - I m z l
)+½6~o.
(11)
One can see how the scalar propagator is extremely sensitive to the sign of m in the free field case (this behaviour persists in the case of a finite lattice for periodic or antiperiodic boundary conditions [ 11 ] ), and it is tempting to ask whether the sign o f S ( z > 0) could act as a signature of the transition at x~. Once the mass has reached zero (K---K¢), S ( r > 0 ) would change sign. Close to Kc one would expect that the propagators for individual gauge field configurations have differing signs, leading to bad statistics. The vector propagator, however, does not have this problem, as it does not (as is clear in the free case) change sign with the fermion mass (m). It would appear that the sign of the scalar propagator is a useful quantity to measure simply because it heralds the transition at x¢, but that the vector propagator is far more suited to an accurate determination of the lattice mass. 5. In order to see how our gauge fixing works, we performed Monte Carlo calculations on a small 63×12 lattice in the quenched approximation. A 10'
~
•
-
5 July 1990
~
,
.
.
.
.
.
~
•
v(:)
[]
s(;)
- - = -
,IB
10
1
I0
2
~'=o.9 10
-3
a:-O.14
10 -4 0
~ , _ , 1 _ , , _ , 1 , , 2 4
I, 6
i 8
10
Fig. 2. Typical measured vector and scalar fermion propagators (normalized to 1 at T= 1 ). Here r = 0.9 and r = 0.14.
398
Volume 243, number 4
PHYSICS LETTERS B
1.5
•
-
r
5 July 1990
:
!
'
'
I
(a) 8=0.9
!
//.
~
il
E
[]
05
0
Scalar (TL-Cauge)
•
Scalar
(Temporal)
Y~
Vector
(TL-Gauge)
•
Vector (Temporal)
I , , , , ' 1 ~ - - ,
5
I
6
!
,t
7
1 ,,/K
(b) ©
Pseudo-scalar
Vector 3
2
© /2=0.9 1
o 0
'
I
7
5
llx Fig. 3. Mass measurements in the confined phase (fl= 0.9 ). (a) Fermion masses from the two propagators and in different gauges against the naive strong-coupling result (solid curve). (b) Squared composite-state masses (statistical errors in the pseudo-scalar mass are smaller than the circular markers used).
In fig. 2 we show the vector and scalar fermion propagators at fl=0.9 and to=0.14. We plot the fermion masses obtained from such propagators in fig. 3a as a function of 1/x at fl= 0.9, and the corresponding vector and pseudo-scalar composite-state masses in fig. 3b for comparison. Some fermion masses obtained with the maximal temporal gauge fixing (i) alone are also shown in figs. 3a and 4a. They coincide with those obtained using the TL gauge within numerical error. This result is not trivial, even though
physical mass is a gauge-invariant quantity, since different gauges may define different cutoffs for a given lattice, resulting in different lattice masses. The noise was too great for us to see any propagators in an axial gauge. Through extrapolation of the pseudo-scalar composite-state mass at fl= 0.9 we get re--- 0.206 (2). The vector and pseudo-scalar composite-state masses split as x approaches Xo with the vector mass remaining finite as the pseudo-scalar mass goes to zero (like the 399
Volume 243, number 4
5 July 1990
PHYSICS LETTERS B
1
[] S c a l a r
A V e c h o r (TL G a u g e )
O5
•
o[
(TL-Gauge)
6
Vector (Temporal)
8 I'A
7
10
9
(b) ©
Pseudo
•
Vector
scalar
~E 2 g:t~ LI
6
7
8
9
10
1/r Fig. 4. Mass measurements in the Coulomb phase (fl= 1.1 ). (a) Fermion masses. (b) Squared composite-state masses. behaviour of p and n mesons in lattice Q C D ) . The fermion mass taken from the scalar and vector propagators follows roughly the naive strong-coupling expansion result m = - l o g 2x. This suggests that disordered states dominate in the confined region right up to the phase transition point (fl--- 1 ). In figs. 4a and 4b we see quite a different situation for fl= 1.1. The pseudo-scalar and vector compositestate masses remain equal (within numerical error). They go to zero at x = 0.142 (2). The fermion masses have a m i n i m u m at x = 0 . 1 4 5 ( 3 ) . We did not see the fermion mass completely vanish (which one could 400
understand as the result o f finite size effects). The scalar fermion propagator S(z) changes at ~c= 0.147 (2), where S - V= Tr ( 1 - 74) G/4 also jumps (for z > 0 ) . We plot the sign o f S ( 1 ) in fig. 5a. One would expect that these three values o f xc approach each other in the large lattice limit. As expected, the vector fermion propagator enabled accurate mass determinations even close to xc, while the sign of S( 1 ) proved to be a relatively simple and precise indicator of the transition. Aoki [ 15 ] has argued that parity is spontaneously broken at x~. We see that 62 (y7% ~ ) does indeed take finite values for x > 0.14 in fig. 5b.
V o l u m e 243, n u m b e r 4
PHYSICS LETTERS B
5 J u l y 1990
rr r
i
o
~gn s(1) (a)
-I b
4.
i
I
[
I
I
I
i
I-I
I
'
~
-
l
i 0 -s
2 . 1 0 -5
! 6
(b) 9
7
t
10
1/x Fig. 5. Indicators of re in the Coulomb phase (fl= 1.1 ). (a) The sign of the scalar fermion propagator at z= 1. (sgn S(z)-~ (S(z))/ I (S(z)) I. ) (b) The parity-breaking order parameter c~2( ff9'5¥). The curve is to be understood as only a guide to the eye.
6. In conclusion, we have presented a gauge-fixing procedure for U ( 1 ) lattice gauge m o d e l s which allows fermion propagators to be m e a s u r e d in the T L gauge. The maximal tree temporal gauge ( i ) may also be used alone. M e a s u r e m e n t o f f e r m i o n propagators is useful: we can see the relation between fermion a n d compositestate masses, a n d the d i s a p p e a r a n c e o f f e r m i o n mass m a y be seen directly through either mass measurements using the vector f e r m i o n propagator, or watching the sign o f S( z > 0) as a function o f x. The fact that we can see the fermion propagators so clearly encourages one to consider studying the full theory including dynamlt.el fermions. In the T L a n d m a x i m a l tree t e m p o r a l gauges, we can measure other gauge n o n - i n v a r i a n t quantities such as vertex functions. Calculations o f fermion propagators in the nonabelian case is, needless to say, highly challenging. We would like to thank C. Borgs, V. Linke a n d F. Nill for m a n y helpful discussions. We are i n d e b t e d to Ph. de F o r c r a n d and W. Theis for their critical reading o f the m a n u s c r i p t a n d valuable comments. R. Sinclair wishes to t h a n k the Deutscher Akade-
mischer Austauschdienst a n d F a m i l i e Prager for supporting h i m in Berlin.
Note added. It has come to our attention, after the submission o f this letter, that Bernard, Murphy, Soni a n d Yee have studied lattice fermion propagators for SU ( 3 ) [ 16 ], a n d also found that the fermion mass remains finite at the value o f x where the pseudoscalar mass vanishes in the confined phase. We thank A. Soni for correspondence.
References
[ 1 ] B. Velikson and D. Weingarten, Nucl. Phys. B 249 ( 1985 ) 433. [2 ] Ph. de Forcrand and K.S. Liu, to be published. [3] J.E. Mandula and M. Ogilvie, Phys. Lett. B 185 (1987) 127. [4] P. Coddington, A. Hey, J. Mandula and M. Ogilvie, Phys. Lett. B 197 (1987) 191. [ 5 ] R. Roskies and J.C. Wu, Phys. Rev. D 33 (1986) 2469. [6] A. Nakamura and H.C. Hege, in: New trends in strong coupling gauge theories, eds. M. Bando, T. Muta and K. Yamawaki (World Scientific, Singapore, 1989 ) p. 227. [7] K.G. Wilson, in: Recent developments in gauge theories, ed. G. 't Hooft (Plenum, New York, 1980) p. 363.
401
Volume 243, number 4
PHYSICS LETTERS B
[8] F. Nill, Phys. Lett. B 195 (1987) 443; C. Borgs and F. NiU, Nucl. Phys. B 270 (1986) 92. [ 9 ] M. Creutz, Phys. Rev. D 15 ( 1977 ) 1128. [ 10 ] L.D. Faddeev and A.A. Slavnov, Gauge fields (Benjamin/ Cummings, New York, 1980). [ 11 ] D.B. Carpenter and C.F. Baillie, Nucl. Phys. B 260 ( 1985 ) 103. [12]A. Nakamura, G. Feuer, H.C. Hege, V. Linke and M. Haraguchi, Comput. Phys. Commun. 51 ( 1988 ) 301.
402
5 July 1990
[ 13] N. Kawamoto, Nucl. Phys. B 190 [FS3] ( 1981 ) 617. [ 14 ] W. Janke and H. Kleinert, in: Frontiers in nonperturbative field theory, eds. Z. Horvath, L. Palla and A. Patkos (World Scientific, Singapore, 1988 ) p. 279. [15] S. Aoki, Ph.D. thesis, University of Tokyo (1987); Phys. Rev. D 30 (1984) 2653; Phys. Rev. Lett. 57 (1986) 3136; Phys. Lett. B 190 (1987) 140. [ 16] C. Bernard, D. Murphy, A. Soni and K. Yee, in: Proc. Intern. Workshop Lattice '89, Nucl. Phys. B (Proc. Suppl. ), to be published.
PHYSICS LETTERS B
5 July 1990
Fermion propagators in U ( 1 ) lattice gauge theory A. Nakamura and R. Sinclair lnstitutfiir TheoretischePhysik, Freie UniversiRitBerlin.Arnimallee 14, D-IO00Berlin 33, Germany Received 8 March 1990
Fermion propagators are calculated for quenched compact U( 1) lattice gauge theory. These vanish in the standard Landau gauge due to gauge-equivalentcopies of a given field. We discuss how we remove them. Obtained fermion masses are analyzed as functions of the Wilson hopping parameter in confinement and Coulomb phases. The propagators are seen to change abruptly at re.
1. We need not fix a gauge in lattice gauge theories. If, however, we do fix a gauge in numerical simulations, Monte Carlo calculations become a much more powerful tool in studying quantum field theories. We can calculate gauge non-invariant objects such as hadron wavefunctions [ 1,2], propagators o f gauge fields [3,4] and offermions [5,6] even far from perturbative regions. Gauge fixing may help us to construct the smooth configurations required in the calculation of multigrid fermion propagators. We study a simple example in this paper: fermion propagators in compact U(1 ) gauge theory. This model is clear enough for one to see what is going on when one fixes a gauge, and is a fundamental prototype in lattice gauge theories. 2. Let us start by describing our Landau gauge fixing procedure and outlining a difficulty in the calculation of fermion propagators. We produce gauge configurations {Uu (x) -= exp [iAu (x) ] } using a Monte Carlo update without gauge fixing terms and then fix the gauge [ 7,3 ] through a gauge transformation
A ~ = A , +&(~+)Z,
(1)
such that
E &(~-)A'~(x)=0,
(2)
where &( + / - ) are lattice forward/backward difference operators. It should be understood that the gauge transformation is defined by 396
UAx)-~ r+(x) UAx) V(x+ ~), where V ( x ) = e x p [ i g ( x ) ] . We impose periodic boundary conditions on U and V. N o w we are faced with a difficulty: the Z, and therefore the A~ as well, are not uniquely determined. One can add any function o9 to Z, as long as to is a lattice harmonic (i.e. Ato = 0 ). There are many solutions of this condition due to the compact nature o f the theory. A simple example is to= E~, auxu+ b with a~,= 2zm/L~,, where n is an arbitrary integer and Lu is the length of the lattice in the direction #. In the continuum theory, one imposes the asymptotic boundary condition A~,(x)~O for x ~ _+oo, and obtains a unique solution for the abelian case. (For non-abelian continuum field theories there is the famous Gribov ambiguity.) Through a gauge transformation, fermion propagators transform as
<~,(x)~(y) > (exp{i [z(x) -x(y) ]} g/(x)~(y) ).
(3)
Hence if we impose only the Lorentz condition (2), the fermion propagator vanishes due to this averaging over an arbitrary phase. (A similar situation was discussed in ref. [ 8 ] with respect to the Higgs model.) On a lattice, the Lorentz condition alone is not enough to deliver a non-vanishing propagator. A gauge transformation satisfying the Lorentz condition for the U ( 1 ) case is easily found:
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
Volume 243, number 4
PHYSICS LETTERSB
Z(x) = Z a ( x - y ) ~ 6~u-)Au(Y), y
(4)
I.t
where a (x) is the lattice Green's function --1
a ( x , = 4"Volumel ~ ' e x p ( i p x ) , (~u sin2 (½Pu)) • (5) We omit the zero mode in the momentum sum. Hence Aa (x) = - Ox,o + 1/ Volume, hut the second term disappears in the calculation o f z ( x ) . Z can be evaluated fast enough numerically by working in Fourier space. 3. In order to fix the Landau gauge on a lattice for U ( 1 ) gauge theory, we propose the following procedure (which we refer to as the TL gauge in the following): (i) Transform configurations { Uu (x) } to satisfy a maximal tree temporal gauge condition [9]. See fig. 1. (ii) Then, calculating the gauge transformation by eqs. ( 1 ), (4) and (5), we transform again to satisfy the Lorentz condition. We use a maximal tree in the fixing of the temporal gauge since the temporal gauge alone (only A4(X ) = 0), as in continuum theories [ 10 ], does not completely remove gauge degrees of freedom. The configurations fixed by step (i) are therefore uniquely determined, although the transformation itself is not. Z(x, t ) + 0 also satisfies the maximal tree temporal gauge condition. We remove this ambiguity by using
5 July 1990
a "punctured" gauge, in which Z(0, 0) =0. Our Landau gauge fixing technique (step (ii)) is unique by definition, and therefore does not produce any gaugeequivalent copies. As we saw before, the lattice Landau gauge does not have a one-to-one correspondence with that in the continuum theory. The lattice temporal gauge, on the contrary, does. In the continuum limit, the Landau and temporal gauges are related with each other in a well-defined way. It is therefore natural to make use of the TL gauge. Moreover, the temporal gauge ensures that the leading terms in the strong-coupling expansion of the fermion propagator do not average away to zero. This fact, and the uniqueness of step (ii), allows us to expect that our Landau gauge fixing delivers a non-zero propagator. 4. We define time-sliced fermion propagators as
G,~p(z, k=O )
For free Wilson fermions on an infinite lattice, this becomes [ 11 ] sgn(z) exp[-Ilog(l+m)'zl] Gfree (z, k = 0 ) =~'4 m-------) 2(1 + sgn(m) + - exp[-Ilog(l+m)-zl] 2(l+m) 1
+ ~ o 2( 1 + m ) ' I
I I
.....
I
+_+
----
- o - - -
- - - - 0
.
where m = l / 2 x - 4 . propagators as
V(z)=¼Try4G(z)
(7) We define vector and scalar (8)
and .
.
.
Fig. 1. Schematic diagram of maximal temporal gauge fixing in two dimensions.At,( x ) = 0 o n a tree (solid links).
S(r) = ~Tr G(z)
(9)
respectively, where the trace is taken over the Dirac indices. For the determination of xc it is of interest to know how these propagators behave as the lattice mass (m*) approaches zero. It is clearly related to the mass in the free particle propagator above by m * = l log( 1 + m) [. For masses close to zero we have m*-~ Iml, 397
Volume 243, number 4
Vf~(z)-
PHYSICS LETTERS B
½sgn (z) e x p ( - Imzl ),
combination of the conjugate-residual and conjugate-gradient methods with hopping parameter expansion preconditioning was used to invert the Wilson fermion matrix [ 12 ]. Typically, a hundred configurations, separated by one thousand sweeps in each case, were used in calculating a propagator. For each configuration used, we chose a representative point from each x4 = const, plane randomly, and then approximated the propagator by calculating from these points only and averaging over them. We imposed antiperiodic boundary conditions in the time direction for the fermionic fields, although the choice between periodic or antiperiodic boundary conditions is irrelevant in the case of a quenched U( 1 ) calculation (multiplying all the time-directed links at x4 = L4 by - 1e U ( 1 ) is a symmetry of the quenched action, but this also corresponds to interchanging periodic and antiperiodic boundary conditions). Defining xc as the value at which the fermion mass becomes zero, one would expect that tq= -~in the free case (A,,(x) = 0 ) . In the strong-coupling limit, Xc--[ 13 ]. The phase transition between the confined and Coulomb phases of the compact U ( 1 ) model occurs at r = 1 (see, for example, ref. [ 14] ). We calculated at r = 0.9 and r = 1.1 to examine differences between the confined and Coulomb phases respectively.
(10)
and Sfr~e(r)--,½ sgn(m) e x p ( - I m z l
)+½6~o.
(11)
One can see how the scalar propagator is extremely sensitive to the sign of m in the free field case (this behaviour persists in the case of a finite lattice for periodic or antiperiodic boundary conditions [ 11 ] ), and it is tempting to ask whether the sign o f S ( z > 0) could act as a signature of the transition at x~. Once the mass has reached zero (K---K¢), S ( r > 0 ) would change sign. Close to Kc one would expect that the propagators for individual gauge field configurations have differing signs, leading to bad statistics. The vector propagator, however, does not have this problem, as it does not (as is clear in the free case) change sign with the fermion mass (m). It would appear that the sign of the scalar propagator is a useful quantity to measure simply because it heralds the transition at x¢, but that the vector propagator is far more suited to an accurate determination of the lattice mass. 5. In order to see how our gauge fixing works, we performed Monte Carlo calculations on a small 63×12 lattice in the quenched approximation. A 10'
~
•
-
5 July 1990
~
,
.
.
.
.
.
~
•
v(:)
[]
s(;)
- - = -
,IB
10
1
I0
2
~'=o.9 10
-3
a:-O.14
10 -4 0
~ , _ , 1 _ , , _ , 1 , , 2 4
I, 6
i 8
10
Fig. 2. Typical measured vector and scalar fermion propagators (normalized to 1 at T= 1 ). Here r = 0.9 and r = 0.14.
398
Volume 243, number 4
PHYSICS LETTERS B
1.5
•
-
r
5 July 1990
:
!
'
'
I
(a) 8=0.9
!
//.
~
il
E
[]
05
0
Scalar (TL-Cauge)
•
Scalar
(Temporal)
Y~
Vector
(TL-Gauge)
•
Vector (Temporal)
I , , , , ' 1 ~ - - ,
5
I
6
!
,t
7
1 ,,/K
(b) ©
Pseudo-scalar
Vector 3
2
© /2=0.9 1
o 0
'
I
7
5
llx Fig. 3. Mass measurements in the confined phase (fl= 0.9 ). (a) Fermion masses from the two propagators and in different gauges against the naive strong-coupling result (solid curve). (b) Squared composite-state masses (statistical errors in the pseudo-scalar mass are smaller than the circular markers used).
In fig. 2 we show the vector and scalar fermion propagators at fl=0.9 and to=0.14. We plot the fermion masses obtained from such propagators in fig. 3a as a function of 1/x at fl= 0.9, and the corresponding vector and pseudo-scalar composite-state masses in fig. 3b for comparison. Some fermion masses obtained with the maximal temporal gauge fixing (i) alone are also shown in figs. 3a and 4a. They coincide with those obtained using the TL gauge within numerical error. This result is not trivial, even though
physical mass is a gauge-invariant quantity, since different gauges may define different cutoffs for a given lattice, resulting in different lattice masses. The noise was too great for us to see any propagators in an axial gauge. Through extrapolation of the pseudo-scalar composite-state mass at fl= 0.9 we get re--- 0.206 (2). The vector and pseudo-scalar composite-state masses split as x approaches Xo with the vector mass remaining finite as the pseudo-scalar mass goes to zero (like the 399
Volume 243, number 4
5 July 1990
PHYSICS LETTERS B
1
[] S c a l a r
A V e c h o r (TL G a u g e )
O5
•
o[
(TL-Gauge)
6
Vector (Temporal)
8 I'A
7
10
9
(b) ©
Pseudo
•
Vector
scalar
~E 2 g:t~ LI
6
7
8
9
10
1/r Fig. 4. Mass measurements in the Coulomb phase (fl= 1.1 ). (a) Fermion masses. (b) Squared composite-state masses. behaviour of p and n mesons in lattice Q C D ) . The fermion mass taken from the scalar and vector propagators follows roughly the naive strong-coupling expansion result m = - l o g 2x. This suggests that disordered states dominate in the confined region right up to the phase transition point (fl--- 1 ). In figs. 4a and 4b we see quite a different situation for fl= 1.1. The pseudo-scalar and vector compositestate masses remain equal (within numerical error). They go to zero at x = 0.142 (2). The fermion masses have a m i n i m u m at x = 0 . 1 4 5 ( 3 ) . We did not see the fermion mass completely vanish (which one could 400
understand as the result o f finite size effects). The scalar fermion propagator S(z) changes at ~c= 0.147 (2), where S - V= Tr ( 1 - 74) G/4 also jumps (for z > 0 ) . We plot the sign o f S ( 1 ) in fig. 5a. One would expect that these three values o f xc approach each other in the large lattice limit. As expected, the vector fermion propagator enabled accurate mass determinations even close to xc, while the sign of S( 1 ) proved to be a relatively simple and precise indicator of the transition. Aoki [ 15 ] has argued that parity is spontaneously broken at x~. We see that 62 (y7% ~ ) does indeed take finite values for x > 0.14 in fig. 5b.
V o l u m e 243, n u m b e r 4
PHYSICS LETTERS B
5 J u l y 1990
rr r
i
o
~gn s(1) (a)
-I b
4.
i
I
[
I
I
I
i
I-I
I
'
~
-
l
i 0 -s
2 . 1 0 -5
! 6
(b) 9
7
t
10
1/x Fig. 5. Indicators of re in the Coulomb phase (fl= 1.1 ). (a) The sign of the scalar fermion propagator at z= 1. (sgn S(z)-~ (S(z))/ I (S(z)) I. ) (b) The parity-breaking order parameter c~2( ff9'5¥). The curve is to be understood as only a guide to the eye.
6. In conclusion, we have presented a gauge-fixing procedure for U ( 1 ) lattice gauge m o d e l s which allows fermion propagators to be m e a s u r e d in the T L gauge. The maximal tree temporal gauge ( i ) may also be used alone. M e a s u r e m e n t o f f e r m i o n propagators is useful: we can see the relation between fermion a n d compositestate masses, a n d the d i s a p p e a r a n c e o f f e r m i o n mass m a y be seen directly through either mass measurements using the vector f e r m i o n propagator, or watching the sign o f S( z > 0) as a function o f x. The fact that we can see the fermion propagators so clearly encourages one to consider studying the full theory including dynamlt.el fermions. In the T L a n d m a x i m a l tree t e m p o r a l gauges, we can measure other gauge n o n - i n v a r i a n t quantities such as vertex functions. Calculations o f fermion propagators in the nonabelian case is, needless to say, highly challenging. We would like to thank C. Borgs, V. Linke a n d F. Nill for m a n y helpful discussions. We are i n d e b t e d to Ph. de F o r c r a n d and W. Theis for their critical reading o f the m a n u s c r i p t a n d valuable comments. R. Sinclair wishes to t h a n k the Deutscher Akade-
mischer Austauschdienst a n d F a m i l i e Prager for supporting h i m in Berlin.
Note added. It has come to our attention, after the submission o f this letter, that Bernard, Murphy, Soni a n d Yee have studied lattice fermion propagators for SU ( 3 ) [ 16 ], a n d also found that the fermion mass remains finite at the value o f x where the pseudoscalar mass vanishes in the confined phase. We thank A. Soni for correspondence.
References
[ 1 ] B. Velikson and D. Weingarten, Nucl. Phys. B 249 ( 1985 ) 433. [2 ] Ph. de Forcrand and K.S. Liu, to be published. [3] J.E. Mandula and M. Ogilvie, Phys. Lett. B 185 (1987) 127. [4] P. Coddington, A. Hey, J. Mandula and M. Ogilvie, Phys. Lett. B 197 (1987) 191. [ 5 ] R. Roskies and J.C. Wu, Phys. Rev. D 33 (1986) 2469. [6] A. Nakamura and H.C. Hege, in: New trends in strong coupling gauge theories, eds. M. Bando, T. Muta and K. Yamawaki (World Scientific, Singapore, 1989 ) p. 227. [7] K.G. Wilson, in: Recent developments in gauge theories, ed. G. 't Hooft (Plenum, New York, 1980) p. 363.
401
Volume 243, number 4
PHYSICS LETTERS B
[8] F. Nill, Phys. Lett. B 195 (1987) 443; C. Borgs and F. NiU, Nucl. Phys. B 270 (1986) 92. [ 9 ] M. Creutz, Phys. Rev. D 15 ( 1977 ) 1128. [ 10 ] L.D. Faddeev and A.A. Slavnov, Gauge fields (Benjamin/ Cummings, New York, 1980). [ 11 ] D.B. Carpenter and C.F. Baillie, Nucl. Phys. B 260 ( 1985 ) 103. [12]A. Nakamura, G. Feuer, H.C. Hege, V. Linke and M. Haraguchi, Comput. Phys. Commun. 51 ( 1988 ) 301.
402
5 July 1990
[ 13] N. Kawamoto, Nucl. Phys. B 190 [FS3] ( 1981 ) 617. [ 14 ] W. Janke and H. Kleinert, in: Frontiers in nonperturbative field theory, eds. Z. Horvath, L. Palla and A. Patkos (World Scientific, Singapore, 1988 ) p. 279. [15] S. Aoki, Ph.D. thesis, University of Tokyo (1987); Phys. Rev. D 30 (1984) 2653; Phys. Rev. Lett. 57 (1986) 3136; Phys. Lett. B 190 (1987) 140. [ 16] C. Bernard, D. Murphy, A. Soni and K. Yee, in: Proc. Intern. Workshop Lattice '89, Nucl. Phys. B (Proc. Suppl. ), to be published.