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Parity violating vacuum currents on the random lattice
Volume 213, number 2
PHYSICS LETTERS B
20 October 1988
PARITY V I O L A T I N G V A C U U M C U R R E N T S O N T H E R A N D O M L A T T I C E S.M. C A T T E R A L L and J.F. W H E A T E R Department of Theoretical Physics, University of O.~ford, 1 Keble Road, Oxford OX I 3NP, U K Received 19 July 1988
The Dirac operator is studied both with and without background gauge fields on a three-dimensional random lattice. It is shown that the correct continuum limit for free fermions is recovered in the zero field case and the doubling problem evaded at tree level, the doubles acquiring masses of order the cut-offand decoupling from long-distance physics. The vacuum current is calculated in the presence of a nontrivial U ( 1 ) field and shown to be in agreement with continuum calculations.
In this letter we describe the results of an investigation into the properties of Dirac fermions on random lattices in three dimensions. We study the fermion propagator and the vacuum charge in the presence of background gauge fields with nonzero winding number. Similar investigations have been done on regular lattices using Wilson fermions by Dagotto [ 1 ]. Previous work in ( 1 + 1 ) [2] and ( 3 + 1 ) [3] dimensions on random lattices, with background gauge fields, has shown that one obtains the correct continuum limit of a single light Dirac fermion species, the contribution of the doubled modes being suppressed, as a result of interaction with the randomness of the lattice. This is shown to be due to a spontaneous breaking of chiral symmetry (in the limit mayO), even for the Dirac theory without gauge fields. In this way the system gives masses O ( 1 / a ) (where a is the average lattice spacing) to the doubled sector; one also sees the corresponding (approximate) Goldstone boson in the pseudo-scalar channel with a mass O (,, m/-m~/a). This chiral symmetry breaking does not affect the primary fermions, at least at tree-graph level. If one considers radiative corrections, then the doubles contribute to primitively divergent graphs, even in the continuum limit. It is found that the form of these divergences differs from the continuum case and a careful fine-tuning is required to extract finite answers as the average lattice spacing tends to zero
[4]. It is of some interest to see whether the suppression 186
of the doubles carries over to ( 2 + 1) dimensions, where, in a minimal representation of the Dirac algebra, there is no 7 s operator, and hence no chiral symmetry. In three dimensions there is a discrete parity symmetry under which the continuum massless Dirac action is invariant. However the Dirac operator on a random lattice no longer possesses this symmetry even in the massless case, because the lattice itself is not symmetric. It is possible that breaking of this symmetry is manifested within the doubled sector and generates masses for the doubles of the order of the cut-off, leaving the primary fermions light. Long-distance physics would then be that of a single light Dirac fermion. The lattice itself is constructed from a random distribution of points in a cubical box of unit side length, with periodic boundary conditions imposed on all faces. If the circumsphere of four points contains no other sites, links are drawn between all members of such a cluster to form a tetrahedron. This procedure triangulates the box into a set o f non-overlapping tetrahedra which completely fill the system volume [ 5 ]. The length of a link is designated l u, the three-component vector along the link l~, and its unit vector /',!~. The dual lattice is found is by drawing all perpendicular bisecting planes to the links. Each link is then dual to an element of area a u, the union of these round a site enclosing the dual site cell volume o9,. We define the average lattice spacing a = N - ~/3, where N is the number of points. The fermion action is chosen to be
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 213, number 2
PHYSICS LETTERS B
_ It lt U,A&+ m ~ og,~,q/,, S = ~1 Y~ ~Ui20l~/~
t
20 October 1988
only the primary fermion contributes to the two-point function. Specifically we calculated the transverse fermion p r o p a g a t o r
where
2,/= 0",Jl v , = 0,
if i, j l i n k e d , otherwise.
X < u/(.,.,.,,,.:, ) ~(.,.,..,,~.-~) > ,
~,, ~,, are i n d e p e n d e n t (in euclidean space) twoc o m p o n e n t spinors, U~/is the gauge field link variable and m is the mass [5]. The lattice analogue o f the Dirac o p e r a t o r is
where V is the system volume. On a r a n d o m lattice this becomes
,~j~'"ll: u,/ where The discrete s y m m e t r y corresponding to parity is (in the c o n t i n u u m ) defined by the following transformations: XI ---+--XI ~
X2---+X2 ,
AI -->-AI ,
A2-+A2 ,
q/--~0"]///,
X3--+X3
~
A3--~A3,
~-+--~0"1 ,
AO, = O ( z i - z / -
(z-a~2)) -O(zi-Z/-
In a representation o f the g a m m a matrices such that ~z= 0"3 then G"/3(z) has zero off-diagonal elements by rotational invariance. The c o n t i n u u m G"~(z) in a cubical box o f side length L with periodic b o u n d a r y conditions has the following form z > 0: exp ( - mz) 1-exp(-mL)
which leaves the fermion kinetic t e r m ~0-. 0~u invariant. H o w e v e r any mass term changes sign u n d e r this symmetry. In a d d i t i o n one m a y a d d a new t e r m to the standard continuum action - the so-called C h e r n - Simons term [6,7]
and for z ~
1 m r d3x ~/'Vl'A/,Fvl,. I = 16~ Iml
GII (2) =
exp[ - m ( z + L ) ] 1-exp(-mL)
G22(2) =
exp[m(z+ L ) ]
J
This term is also o d d under parity and constitutes a topological mass term for the gauge field. It gives rise to a n o m a l o u s v a c u u m currents 1 rn ( J " ) = ( ~ 7 : ' ~ ' ) -- 8zt [m[ e/'"/'F" "
One m a y show [6,7 ], in the c o n t i n u u m , that in o r d e r to regulate the UV divergences o f the theory in a gauge invariant fashion one must necessarily introduce such a c o u n t e r t e r m into the effective action. Indeed oneloop v a c u u m polarisation effects with non-vanishing fermion masses also generate this term. As m--,0, a current survives with a sign ambiguity arising from the regularisation procedure. Firstly we e x a m i n e d the theory in the absence o f any background gauge field. The results confirm that
(z+a/2 ) ) .
G"(z)= G22(z) =
'
exp(mz) exp(mL) - 1
exp(mL) - 1 As L ~ o o we recover the expected behaviour, i.e, fermions propagate forward in time, and antifermions backwards in time. We use a conjugate gradient algorithm to obtain matrix elements o f the inverse Dirac o p e r a t o r needed in the calculation o f the lattice propagator. The propagator has the correct symmetries and the coefficients o f Z, and y,. are small. Fig. 1 shows G 11 calculated on a lattice o f 1 0 × 1 0 × 2 0 sites, f o r m e d from two identical 1000 site lattices displaced relative to one another by one lattice length in the z direction, c o m p a r e d with the c o n t i n u u m curve. The only deviations occur for separations less than about 187
Volume 213, number 2
PHYSICS
LETTERS
20 October 1988
B
Propagator 1.20
1.00
0.800 o
o
0600
0.400
0.200
e
0.00 0.00
5.00
o
100
o
I 20.0
15.0
z/o Fig. 1. Propagator on a 10X 10 × 20 lattice at ma = 0.2 with continuum curve.
three lattice spacings where the contribution of the doubles is evident. We also studied the fermion condensate ( ~ v ) . In the continuum with m o m e n t u m cut-offA this is ,1
f Tr
toroidal topology forces flux quantisation upon us and indeed quantises the possible vacuum charges [8]. The continuum charge is given by Q,, = j d 3 x ( ~ ' ~ , ~ ) .
A
d3p 1 (2z~)3i~+m-
2m
!
(2g) 3
d3p p2+m 2
0
= 4 m A / ( 2 re) 2 + finite terms.
With this field the only non-zero component o f the charge is the one associated with the z direction. Setting L to unity and the winding number to 1, the vacuum charge is one-half.
So we expect that ( ~)
Psibar Psi
~ma/a2 ,
1.46
unless there are extra heavy fermions present. Fig. 2 shows a plot of a2(~v~v) versus m a for different lattices. We see that a 2 ( ~ v ) is fixed for fixed ma, but does not vary like ma. This confirms that, as expected there are extra particles in the spectrum with mass ~ 1/a. The figure suggests that this conclusion survives in the limit rna-oO and indicates that the doubles acquire masses of order the cut-off. We have examined the electromagnetic current in the background gauge field A:=O,
A.,.=-2nny&(x)/L,
A , , = 2 1 r n x / L 2,
1.44 \ 1.42
v
188
\
1.40
1.58
136
134 0.00
which has winding number n and yields a constant field strength everywhere, E,.,,= 2 ~ n / L ~-. Note that the
c,
I 5.00x10-2
I 0.100
; 0350
] 0200
] 0250
rno
Fig. 2. a-~(~vq/) versus ma forN= 1000 (~); N=2000 (o) sites.
Volume 213, number 2
PHYSICS LETTERS B
20 October 1988
Log Vacuum Charge
On a random lattice we must first determine the form of the gauge invariant conserved current. This is done by the following trick. In the continuum replace A , as follows
0.00
-0.500
-1.00
A,(x) ~A,(x) +a.,
Q .
\
-1.50
then
÷
O'
+
'-.~ Q\
x*
+
-2.00
Q, = i ~
\
In Z . . . . o"
+
x \*
x
-2.50 x N
On the random lattice
-3.00
_ it S= ~1 Z ~/i6d[d7 Uiiexp(la'l~)~j+m ~i ¢.oi~iq/i,
-3.50
,
-4.00 0.00
x
o
\
o
\
o
I 5 00x10"2
I 0.100
I 0.150
I 0.250
I 0200
I 0300
mQ
SO
Fig. 3. Log vacuum charge versus ma for N= 500 ( + ), N= 1000 (X), N=2000 ( ~ ) , and N=4000 (o) sites. The continuum prediction In ~ is also shown. The dashed line through the ( o ) points is to guide the eye.
as
--
1 2 ~
a,J,~l;Uu x/e~,~
[TrT"Op+m)-'].
We experimented with both periodic and antiperiodic boundary conditions on the fermion field. The results for periodic boundary conditions show strong finite-size effects, due to the contribution of the (approximate) zero-mode, rendering extrapolation to the limit m a - - , O impossible on the relatively small lattices we employed. Those employing antiperiodic boundary conditions however show much improved behaviour and in this limit we do indeed see a vacuum charge close to one-half. This is shown in fig. 3, where we plot in Q : against m a for four lattice sizes. In the figure the continuum result is shown and we see that the data is consistent with a charge of onehalf in the continuum limit m a - , O . N o t e that at small enough m a the curves for small lattice size turn over to give vanishing charge at m a = 0, so our extrapolations must be done at points before these finite-size effects b e c o m e important. In order to further test the model, we also calculated the charge on 1000 site lattices for winding number 2 fields. Fig. 4 compares this with the winding number 1 case. The results confirm that the vacuum charge doubles. We also checked that Q : does indeed change sign when the sign o f m is changed. In conclusion it appears that the random lattice provides a suitable regulator for fermionic theories
in odd dimension, the doubled modes do not contribute to correlation functions beyond a few lattice spacings and appear to acquire masses of the order of the cut-off. That this is so signals a breaking of parity in the system. This is in agreement with other continuum methods of regularisation which are forced to induce parity anomalies in order to maintain gauge invariance. In particular, our results are very similar
Log Vacuum Charge 0.00
-0.500
-I.00
-1.50
c
-2.00
-2.50
-3.00
-3.50
-4.00 0.00
I 5.OOxlO-2
I 0.100
I 0.150
I 0.200
I 0250
m(]
Fig. 4. Log vacuum charge versus ma for N= 1000 sites and winding number 1 ( ~ ) , winding number 2 ( o ) fields. 189
Volume 213, number 2
PHYSICS LETTERS B
to those o b t a i n e d by D a g o t t o using W i l s o n f e r m i o n s [1]. We are able to d e m o n s t r a t e the existence o f a topological t e r m in the e f f e c t i v e action for the gauge fields, which yields p a r i t y - v i o l a t i n g currents in the v a c u u m , w h o s e m a g n i t u d e d e p e n d s only o n the sign o f the bare f e r m i o n mass t e r m . T h e m e a s u r e d charge agrees well with the p r e d i c t e d fractional value. T h e fact that a n o n - z e r o v a l u e is f o u n d for the charge is a n o t h e r i n d i c a t i o n that in the c o n t i n u u m l i m i t the doubles d e c o u p l e f r o m the theory, since any e v e n n u m b e r o f f e r m i o n species ( s u c h as are o b t a i n e d in a t h e o r y e x h i b i t i n g the d o u b l i n g p h e n o m e n o n ) p a i r up with e q u a l a n d o p p o s i t e mass signs in o d d d i m e n sion, their c o n t r i b u t i o n s to the current cancelling [ 9 ]. We h a v e not c o n s i d e r e d the s i t u a t i o n w i t h d y n a m i c a l gauge fields. T h e calculations were d o n e o n the C R A Y - X M P / 48 at the R u t h e r f o r d L a b o r a t o r y u n d e r S E R C grant GR/E/3209.0.
190
20 October 1988
References [ 1] E. Dagono, Phys. Rev. D 34 (1986) 2457. [2 ] D. Espriu, M. Gross, P.E.L. Rackow and J.F. Wheater, Nucl. Phys. B 275 (1986) 39. [ 3 ] Y. Pang and H-C. Ren, Colombia University preprint (1987) CU-TP-369. [4] S.J. Perantonis and J.F. Wheater Nucl. Phys. B 295 [FS21 ] (1988) 443. [5] N.H. Christ, R. Friedberg and T.D. Lee, Nucl. Phys. B 202 (1982) 89; Nucl. Phys. B 210 [FS6] (1982) 310. [6] A. Niemi and G. Semenoff, Phys. Rev. Lett. 51 ( 1983 ) 2077; A.N. Redlich, Phys. Rev. D 29 (1984) 2366; R. Jackiw, Phys. Rev. D 29 (1985) 2375. J.F. Schonfeld, Nucl. Phys. B 185 (1981) 157; S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Len. 48 (1982) 975. [ 7 ] L. Alvarez-Gaum6, S. Della-Pietra and G. Moore, Ann. Phys. (NY) 163 (1985) 288. [8] R. Blankenbecler and D. Boyanovsky, Phys. Rev. D 34 (1986) 2612. [9] T.A. Appelquist, M.J. Bowick, D. Karakali and L.C.R. Wijewardna Phys. Rev. D 33 (1986) 3774; G. Semenoff, Phys. Rev. Len. 53 (1984) 2449.
PHYSICS LETTERS B
20 October 1988
PARITY V I O L A T I N G V A C U U M C U R R E N T S O N T H E R A N D O M L A T T I C E S.M. C A T T E R A L L and J.F. W H E A T E R Department of Theoretical Physics, University of O.~ford, 1 Keble Road, Oxford OX I 3NP, U K Received 19 July 1988
The Dirac operator is studied both with and without background gauge fields on a three-dimensional random lattice. It is shown that the correct continuum limit for free fermions is recovered in the zero field case and the doubling problem evaded at tree level, the doubles acquiring masses of order the cut-offand decoupling from long-distance physics. The vacuum current is calculated in the presence of a nontrivial U ( 1 ) field and shown to be in agreement with continuum calculations.
In this letter we describe the results of an investigation into the properties of Dirac fermions on random lattices in three dimensions. We study the fermion propagator and the vacuum charge in the presence of background gauge fields with nonzero winding number. Similar investigations have been done on regular lattices using Wilson fermions by Dagotto [ 1 ]. Previous work in ( 1 + 1 ) [2] and ( 3 + 1 ) [3] dimensions on random lattices, with background gauge fields, has shown that one obtains the correct continuum limit of a single light Dirac fermion species, the contribution of the doubled modes being suppressed, as a result of interaction with the randomness of the lattice. This is shown to be due to a spontaneous breaking of chiral symmetry (in the limit mayO), even for the Dirac theory without gauge fields. In this way the system gives masses O ( 1 / a ) (where a is the average lattice spacing) to the doubled sector; one also sees the corresponding (approximate) Goldstone boson in the pseudo-scalar channel with a mass O (,, m/-m~/a). This chiral symmetry breaking does not affect the primary fermions, at least at tree-graph level. If one considers radiative corrections, then the doubles contribute to primitively divergent graphs, even in the continuum limit. It is found that the form of these divergences differs from the continuum case and a careful fine-tuning is required to extract finite answers as the average lattice spacing tends to zero
[4]. It is of some interest to see whether the suppression 186
of the doubles carries over to ( 2 + 1) dimensions, where, in a minimal representation of the Dirac algebra, there is no 7 s operator, and hence no chiral symmetry. In three dimensions there is a discrete parity symmetry under which the continuum massless Dirac action is invariant. However the Dirac operator on a random lattice no longer possesses this symmetry even in the massless case, because the lattice itself is not symmetric. It is possible that breaking of this symmetry is manifested within the doubled sector and generates masses for the doubles of the order of the cut-off, leaving the primary fermions light. Long-distance physics would then be that of a single light Dirac fermion. The lattice itself is constructed from a random distribution of points in a cubical box of unit side length, with periodic boundary conditions imposed on all faces. If the circumsphere of four points contains no other sites, links are drawn between all members of such a cluster to form a tetrahedron. This procedure triangulates the box into a set o f non-overlapping tetrahedra which completely fill the system volume [ 5 ]. The length of a link is designated l u, the three-component vector along the link l~, and its unit vector /',!~. The dual lattice is found is by drawing all perpendicular bisecting planes to the links. Each link is then dual to an element of area a u, the union of these round a site enclosing the dual site cell volume o9,. We define the average lattice spacing a = N - ~/3, where N is the number of points. The fermion action is chosen to be
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 213, number 2
PHYSICS LETTERS B
_ It lt U,A&+ m ~ og,~,q/,, S = ~1 Y~ ~Ui20l~/~
t
20 October 1988
only the primary fermion contributes to the two-point function. Specifically we calculated the transverse fermion p r o p a g a t o r
where
2,/= 0",Jl v , = 0,
if i, j l i n k e d , otherwise.
X < u/(.,.,.,,,.:, ) ~(.,.,..,,~.-~) > ,
~,, ~,, are i n d e p e n d e n t (in euclidean space) twoc o m p o n e n t spinors, U~/is the gauge field link variable and m is the mass [5]. The lattice analogue o f the Dirac o p e r a t o r is
where V is the system volume. On a r a n d o m lattice this becomes
,~j~'"ll: u,/ where The discrete s y m m e t r y corresponding to parity is (in the c o n t i n u u m ) defined by the following transformations: XI ---+--XI ~
X2---+X2 ,
AI -->-AI ,
A2-+A2 ,
q/--~0"]///,
X3--+X3
~
A3--~A3,
~-+--~0"1 ,
AO, = O ( z i - z / -
(z-a~2)) -O(zi-Z/-
In a representation o f the g a m m a matrices such that ~z= 0"3 then G"/3(z) has zero off-diagonal elements by rotational invariance. The c o n t i n u u m G"~(z) in a cubical box o f side length L with periodic b o u n d a r y conditions has the following form z > 0: exp ( - mz) 1-exp(-mL)
which leaves the fermion kinetic t e r m ~0-. 0~u invariant. H o w e v e r any mass term changes sign u n d e r this symmetry. In a d d i t i o n one m a y a d d a new t e r m to the standard continuum action - the so-called C h e r n - Simons term [6,7]
and for z ~
1 m r d3x ~/'Vl'A/,Fvl,. I = 16~ Iml
GII (2) =
exp[ - m ( z + L ) ] 1-exp(-mL)
G22(2) =
exp[m(z+ L ) ]
J
This term is also o d d under parity and constitutes a topological mass term for the gauge field. It gives rise to a n o m a l o u s v a c u u m currents 1 rn ( J " ) = ( ~ 7 : ' ~ ' ) -- 8zt [m[ e/'"/'F" "
One m a y show [6,7 ], in the c o n t i n u u m , that in o r d e r to regulate the UV divergences o f the theory in a gauge invariant fashion one must necessarily introduce such a c o u n t e r t e r m into the effective action. Indeed oneloop v a c u u m polarisation effects with non-vanishing fermion masses also generate this term. As m--,0, a current survives with a sign ambiguity arising from the regularisation procedure. Firstly we e x a m i n e d the theory in the absence o f any background gauge field. The results confirm that
(z+a/2 ) ) .
G"(z)= G22(z) =
'
exp(mz) exp(mL) - 1
exp(mL) - 1 As L ~ o o we recover the expected behaviour, i.e, fermions propagate forward in time, and antifermions backwards in time. We use a conjugate gradient algorithm to obtain matrix elements o f the inverse Dirac o p e r a t o r needed in the calculation o f the lattice propagator. The propagator has the correct symmetries and the coefficients o f Z, and y,. are small. Fig. 1 shows G 11 calculated on a lattice o f 1 0 × 1 0 × 2 0 sites, f o r m e d from two identical 1000 site lattices displaced relative to one another by one lattice length in the z direction, c o m p a r e d with the c o n t i n u u m curve. The only deviations occur for separations less than about 187
Volume 213, number 2
PHYSICS
LETTERS
20 October 1988
B
Propagator 1.20
1.00
0.800 o
o
0600
0.400
0.200
e
0.00 0.00
5.00
o
100
o
I 20.0
15.0
z/o Fig. 1. Propagator on a 10X 10 × 20 lattice at ma = 0.2 with continuum curve.
three lattice spacings where the contribution of the doubles is evident. We also studied the fermion condensate ( ~ v ) . In the continuum with m o m e n t u m cut-offA this is ,1
f Tr
toroidal topology forces flux quantisation upon us and indeed quantises the possible vacuum charges [8]. The continuum charge is given by Q,, = j d 3 x ( ~ ' ~ , ~ ) .
A
d3p 1 (2z~)3i~+m-
2m
!
(2g) 3
d3p p2+m 2
0
= 4 m A / ( 2 re) 2 + finite terms.
With this field the only non-zero component o f the charge is the one associated with the z direction. Setting L to unity and the winding number to 1, the vacuum charge is one-half.
So we expect that ( ~)
Psibar Psi
~ma/a2 ,
1.46
unless there are extra heavy fermions present. Fig. 2 shows a plot of a2(~v~v) versus m a for different lattices. We see that a 2 ( ~ v ) is fixed for fixed ma, but does not vary like ma. This confirms that, as expected there are extra particles in the spectrum with mass ~ 1/a. The figure suggests that this conclusion survives in the limit rna-oO and indicates that the doubles acquire masses of order the cut-off. We have examined the electromagnetic current in the background gauge field A:=O,
A.,.=-2nny&(x)/L,
A , , = 2 1 r n x / L 2,
1.44 \ 1.42
v
188
\
1.40
1.58
136
134 0.00
which has winding number n and yields a constant field strength everywhere, E,.,,= 2 ~ n / L ~-. Note that the
c,
I 5.00x10-2
I 0.100
; 0350
] 0200
] 0250
rno
Fig. 2. a-~(~vq/) versus ma forN= 1000 (~); N=2000 (o) sites.
Volume 213, number 2
PHYSICS LETTERS B
20 October 1988
Log Vacuum Charge
On a random lattice we must first determine the form of the gauge invariant conserved current. This is done by the following trick. In the continuum replace A , as follows
0.00
-0.500
-1.00
A,(x) ~A,(x) +a.,
Q .
\
-1.50
then
÷
O'
+
'-.~ Q\
x*
+
-2.00
Q, = i ~
\
In Z . . . . o"
+
x \*
x
-2.50 x N
On the random lattice
-3.00
_ it S= ~1 Z ~/i6d[d7 Uiiexp(la'l~)~j+m ~i ¢.oi~iq/i,
-3.50
,
-4.00 0.00
x
o
\
o
\
o
I 5 00x10"2
I 0.100
I 0.150
I 0.250
I 0200
I 0300
mQ
SO
Fig. 3. Log vacuum charge versus ma for N= 500 ( + ), N= 1000 (X), N=2000 ( ~ ) , and N=4000 (o) sites. The continuum prediction In ~ is also shown. The dashed line through the ( o ) points is to guide the eye.
as
--
1 2 ~
a,J,~l;Uu x/e~,~
[TrT"Op+m)-'].
We experimented with both periodic and antiperiodic boundary conditions on the fermion field. The results for periodic boundary conditions show strong finite-size effects, due to the contribution of the (approximate) zero-mode, rendering extrapolation to the limit m a - - , O impossible on the relatively small lattices we employed. Those employing antiperiodic boundary conditions however show much improved behaviour and in this limit we do indeed see a vacuum charge close to one-half. This is shown in fig. 3, where we plot in Q : against m a for four lattice sizes. In the figure the continuum result is shown and we see that the data is consistent with a charge of onehalf in the continuum limit m a - , O . N o t e that at small enough m a the curves for small lattice size turn over to give vanishing charge at m a = 0, so our extrapolations must be done at points before these finite-size effects b e c o m e important. In order to further test the model, we also calculated the charge on 1000 site lattices for winding number 2 fields. Fig. 4 compares this with the winding number 1 case. The results confirm that the vacuum charge doubles. We also checked that Q : does indeed change sign when the sign o f m is changed. In conclusion it appears that the random lattice provides a suitable regulator for fermionic theories
in odd dimension, the doubled modes do not contribute to correlation functions beyond a few lattice spacings and appear to acquire masses of the order of the cut-off. That this is so signals a breaking of parity in the system. This is in agreement with other continuum methods of regularisation which are forced to induce parity anomalies in order to maintain gauge invariance. In particular, our results are very similar
Log Vacuum Charge 0.00
-0.500
-I.00
-1.50
c
-2.00
-2.50
-3.00
-3.50
-4.00 0.00
I 5.OOxlO-2
I 0.100
I 0.150
I 0.200
I 0250
m(]
Fig. 4. Log vacuum charge versus ma for N= 1000 sites and winding number 1 ( ~ ) , winding number 2 ( o ) fields. 189
Volume 213, number 2
PHYSICS LETTERS B
to those o b t a i n e d by D a g o t t o using W i l s o n f e r m i o n s [1]. We are able to d e m o n s t r a t e the existence o f a topological t e r m in the e f f e c t i v e action for the gauge fields, which yields p a r i t y - v i o l a t i n g currents in the v a c u u m , w h o s e m a g n i t u d e d e p e n d s only o n the sign o f the bare f e r m i o n mass t e r m . T h e m e a s u r e d charge agrees well with the p r e d i c t e d fractional value. T h e fact that a n o n - z e r o v a l u e is f o u n d for the charge is a n o t h e r i n d i c a t i o n that in the c o n t i n u u m l i m i t the doubles d e c o u p l e f r o m the theory, since any e v e n n u m b e r o f f e r m i o n species ( s u c h as are o b t a i n e d in a t h e o r y e x h i b i t i n g the d o u b l i n g p h e n o m e n o n ) p a i r up with e q u a l a n d o p p o s i t e mass signs in o d d d i m e n sion, their c o n t r i b u t i o n s to the current cancelling [ 9 ]. We h a v e not c o n s i d e r e d the s i t u a t i o n w i t h d y n a m i c a l gauge fields. T h e calculations were d o n e o n the C R A Y - X M P / 48 at the R u t h e r f o r d L a b o r a t o r y u n d e r S E R C grant GR/E/3209.0.
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