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Meson mass calculations in 3D U(1) lattice gauge theory
Volume 178, number 2,3
PHYSICS LETTERS B
2 October 1986
M E S O N M A S S C A L C U L A T I O N S IN 3D U(I) L A T F I C E G A U G E T H E O R Y Kostas F A R A K O S and George K O U T S O U M B A S Physics Department, National Technical University of Athens, 157 73 Zografou, Athens, Greece Received 25 June 1986
The meson masses are calculated in the quenched approximation to 3D lattice U(I) for several values of the coupling constant. The agreement with strong coupling estimates for the meson masses is fairly good. Evidence is presented for the dynamical breakdown of a global SU(2) symmetry.
Al=~73q~,
1. Introduction
Mass spectrum calculations in lattice gauge theories with fermions have so far been carried out in four dimensions for the SU(2) and SU(3) gauge groups [1]. There also exist similar investigations of two-dimensional theories on the lattice. In this work we present the study of a three-dimensional gauge theory with fermions, based on the group U(1) (calculations in 3D gauge groups without fermions can be found in ref. [2]). We performed Monte Carlo calculations of the meson spectrum in the so-called "quenched" approximation. The nonquenched calculations and the comparison to the quenched ones are the subject of work in progress. 2. 317 f e r m i o n s
The fermionic part of the lagrangian density is £f = ~ (i~ -- re)kit 2 = qs+70(u~2 i'yUDu - m ) ~ , D u = ~ - tea u .
(2.1)
I f m = 0, the theory has an SU(2) × U(1) global symmetry. The lagrangian density remains invariant under the transformation qJ ~ exp(i3F)q~, where 1-' is 1,73, 3'5 or A -- i7375 [3]. For m :~ 0 there will be a U(1) × U(1) symmetry; the matrix P may be 1 or A. The bound states will be generated by the operators
260
A2=C~2~5'I',
B l u = ~Tu73qJ, B3u = ~ T u q , ,
A3=C~q~,
A4=~JA~,
B2# = ~7u75 ~, B4u = CI,T u A ~ .
(2.2)
We consider two cases. (aJ m = 0. If the SU(2) × U(1) symmetry is realized in the states IA1), IA2) and IA3> form an SU(2) triplet. Similarly, IBlu>, IB2u) and IB3u> form a triplet. Thus, in this case, m(A1) = re(A2) = m(A3) , re(B1 u) = m(B2u ) = m(B3t
(2.3) But there is the alternative possibility that the global SU 2 symmetry is spontaneously broken, due to chiral symmetry breaking. I f ( ~ q o ~ 0, then 3,3 and 75 symmetry is broken. The two corresponding massless Goldstone particles would have the quantum numbers of the broken generators. These would be the states IA 1) and IA2). (b) m 4= O. In this case the global symmetry is U(1) X U(1), which does not imply relations between the masses of the bound states. 3. Lattice calculations o f meson masses and (Vp,~)
In the calculation we used Wilson fermions [1]. The quantities calculated are the correlation functions of the composite operators VPF~ representing meson states with quantum numbers determined by the Dirac matrix P. In the functional integral expres-
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-H011and Physics Publishing Division)
Volume 178, number 2,3
PHYSICS LETTERS B
2 October 1986
2.0
sion for G r ( x , y ) we integrate over the fermionic variables:
8~X16
,\
":
,xg.
.: 1~x16
\ ~xx
•:EXTRAFO_AT, ON
G r ( x , y ) = <01T(~(x)r~I,(x)~(y)Pq,0,))10> 1.5
= ~ f D A DxpD* *(x)r~(x)~Cv)r*c~) X exp[-Sg(A)
1 fDA
_
ZA
M -- ~ Q ~ ] 1.0
[Tr(Q~y~(A)PQjxl(A)P)
-- Tr(Qxx1 (A) P) Tr(Qy1 (A) P) exp [ - S g ( A ) ] det Q(A)]. The second term has a significant contribution to the asymptotic behaviour only if the bound state is a singlet with respect to spin, flavor and color, i.e. only if the meson has the quantum numbers of the vacuum [4]. In the actual computation we did not take the second term into account, conforming to common usage, since that would cost too much in computer time. Thus the ~ , I ' and ~A,I, mesons will be degenerate and so will be the ~TuxI, and ~Tu A,I~ mesons. We now discuss the boundary conditions. In the time direction we supposed free ends. In the x and y directions we used antiperiodic boundary conditions. In fact, antiperiodic conditions have the disadvantage of giving, in the free-field case, a faster fall-off than what is implied by the explicit mass term in the action. These too large meson masses are brought down to the values of an infinite lattice by performing an appropriate scaling. Usually, the masses are supposed to be proportional to the Wilson parameter K:
m=aK +7. Our ansatz is that there will be corrections proportional to 1IN to the coefficients a and 7, where N is the number of lattice sites in the x ( o r y ) direction:
mN
=
(ct + 13/N)K + (7 + 6IN).
(3.1)
We checked this formula in the case of the freefermion theory. We used three lattices o f sizes 82 X 16, 122 × 16 and 162 X 16 and measured the mass associated with the Green's function of U~ysq~ for different K for an infinite lattice this mass can be obtained analytically. Measuring the mass o f each meson for the 82 X 16 and 122 X 16 lattices at vari-
0.5
\\
\ \
\'
\ \
I
O. 2
I
f
0.14 0.16 I KAPPA N
I
I
I
018
Fig. 1. Mass of ~-r5,l, as a function of K in the case of free fermion theory for the 82 X 16, 122 X 16 and 16 2 × 16 lattices and the infinite lattice extrapolation. ous values of K we determine the coefficients a,/3, 7, 6 and from eq. (3.1) we compute m16 and moo. The values of m 16 agree with the direct computation with the 162 X 16 lattice and m ~ agrees with the analytical calculation (fig. 1). Note that, for K near Kcritical the masses no longer lie on a straight line. We suppose, therefore, that this ansatz is applicable for the interacting theory as well, provided we stay way from K = KcriticalThe calculation was done as follows: Sixteen gauge field configurations were generated for some value of the coupling constant and the masses of the different mesons were calculated on two different lattices: 82 × 16 and 122 X 16. F o r each/3 we stopped increasing K, as soon as the masses declined from the straight line. Then from the above scaling method we obtained the masses that an infinite lattice would, presumably, predict. Fig. 2 illustrates the calculation just described for the ~3,5q~ propagator and/3 = 3. The masses measured on the lattices 82 X 16 and 122 × 16 are seen to lie on a straight line in the range of K investigated. 261
Volume 178, number 2,3
PHYSICS LETTERS B
2 October 1986
cosh mp = 1 + (1 - 12K2)/8K 2 ,
2.0
(4.1)
B=3 cosh m v = 1 + (1 - 8K2)2/4K2(1
• : 8"x16
-:12"x16 • : EXTRAPOLATION
1.5
,,~
.>-~10 7" 0 LO
-
4K2),
(4.2)
where rap, m v are the masses of the pseudoscalar (~3'5 q 0 and the vector (U~Tuq~) mesons respectively. In fig. 3 the result of our scaling m e t h o d for ~Tuq~ is plotted against the prediction of eq. (4.2). The agreement is fairly good. The pseudoscalar masses shown in fig. 4 are in good accord with eq. (4.1). In ref. [5] it is claimed that the ~ meson should have infinite mass in the strong coupling limit. In fig. 5 we give our results for the mass o f this state. We see
0.5 \
\ k
\ \
2.0 \
\ X
\
\ \
\
1"4 " 0.12 "
0.14
0.16
"~ ~..,,.
0.18
KAPPA
1.0 -
"-.. -.
Fig. 2. Predictions of the 8 2 X 16 and 12 2 X 16 lattices for the mass of ~'rs'I, at t3 = 3 and various K, the masses given by the extrapolation method and the error bars at K = 0.16. 018
The result of the scaling is also plotted. The statistical errors shown in the figure were obtained b y dividing the sixteen gauge field configurations in two blocks of eight configurations each and considering the variations in the masses over the two subsets. The error increases as we go to larger K or smaller/3. A typical value for the relative error (SM/M) is 0.05. We have also measured the condensate ( ~ ) . Chiral s y m m e t r y is explicitly b r o k e n in our model, since we used Wilson fermions. Nevertheless, we thought it interesting to display (Ckq0 from which we subtracted the value o f ( ~ ) for the free theory,
0.22
Q26 KA PPA
Q30
Q34
Fig. 3. The dashed line represents the prediction of eq. (5.2) for the vector meson mass. The full line is the result of our scaling method.
2-°r-x \
at K = Kcritical = ~.
4. Results ( a ) Strong coupling. We first present the results obtained for/3 = 0, which are to be compared to the strong coupling calculations of ref. [5]. The mass formulae of ref. [5] for three-dimensional s p a c e time are 262
LI
0.18
I
I
I
0.2 2
I
Q26
I
1
I
~*~
I
03 0
KAPPA Fig. 4. The dashed line represents the prediction of eq. (5.1) for the pseudoscalar meson mass. The full line is the result of our scaling method.
PHYSICS LETTERS B
Volume 178, number 2,3
2 October 1986
2.0
3.5
,,\
~--2
~3
• : 8"x16
\\
X\
•
~
:
'Pvsv'
-.--
+
-
• : 12~x16 ~.~EXTRAPOLATION
3.0
t.5
7 , . . . . . . : +~.y+,
):,, \
N
%,,, \<,',
2.5
~21.0
"%\, \\, \ '~?,,x\. lx
z 2.0
05
"~\ x,\. \\ ~ \
I
1.5 014
i
i
0.16
I
i
0.18
i
C~
0
i
J
• X
0.22
K APPA
i
i
r
i
o.13 Fig. 5. Mass of ~,I, at t3= 0 as a function of K for the 82 X 16 and 122 X 16 lattices. The infinite lattice extrapolation is plotted also.
i
o~5
i'~
o77
t
0 '79
K AP PA K~2) Fig. 6. Meson masses according to the infinite lattice extrapolation for ~3= 2. 2.0
that the 12 2 X 16 lattice predicts, at the same values of K, larger masses than the 8 2 X 16 lattice, something contrary to what happens in all other cases. The scaling method generates even greater masses, in qualitative agreement with the expectation o f ref. [5]. (b) Mass spectrum for 13= 2, 3 and 4. Figs. 6 - 8 give the meson masses according to the infinite lattice extrapolation. The states ~3,5q~ and ~3,uqz turn out to be lighter than the others and the difference between their masses is considerably smaller than any other mass difference. This is reminiscent o f the fact that, in the four-dimensional QCD calculations, the particles 7r and p come out with nearly the same mass. It is important that, at Kcrit., the masses o f ~3,5~I, and ~'),3 ~ are zero, while the mass o f ~ ' # gets a, rather large, value. According to what was said earlier, these three states belong to the triplet representation o f the global SU(2) symmetry, which exists in the massless limit. The difference found in the masses suggests that this symmetry is dynamically broken to a U(1). The two massless states ~')'3 qt and ~ 7 5 q~ are the resulting Goldstone bosons.
,
"-\
~\
B=3 \
: +~,
_
x\
..... :0+i+
:L.
1.5
1,9 03
\\ - , '~\ ",, ',.
1.0 Z O 03 kO
z
\ \ ,,',, \ \ ,,',.,
0.5
\ ' x ' , ', .
.
. . 0.13
.
. Q15
\',,',,',, 0.17 1
KAPPA
i
Q19
Kg3)
Fig. 7. Same as in fig. 6 for t\ = 3. 263
Volume 178, number 2,3
PHYSICS LETTERS B
2.0~
13=4 .
.
.
.
.
:
@vp~
:
~YsY~
0.101 0.08
2 October 1986
.
• : B--2
• : B=3 13=4 ": FREE
"
• :
"~ 0 . 0 6 [ 1.5
.
.
.
.
.
.
0"04 I U9 (.t)
,,cl: 1.0
0"02f
f !
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:\
0.5
,
X~, Q15
"(0.17
Kc(4)
018
•
Fig. 9. ~(~"I,)3, K is the difference (~'q'>3,K- <'t,'I')lfree, K = 1, The results of the measurements for 3 = 2, 3, 4 and for the free-fermion theory are given. Thepoints A, B, C, D are the Kcritical for the free-fermion theory and for 0 = 4, 3, 2, respectively.
"~'k "X. '~,, "',~
0.13
!
014 016 KAPPA
%.
0.19
KAPPA
We w o u l d like to t h a n k Professor G. Tiktopoulos for his help and for critically reading the manuscript.
References
Fig. 8. Same as in fig. 6 for fl = 4.
(c) C ondensate (UI,q,). In fig. 9 we give our results on the difference of <~q'> from the free field value o f (U~q,> at K = Kcr R •(3)13._,~ = g, 1 which • is • the value at which the pseudoscalar correlation f u n c t i o n of the free-fermion theory has n o exponential fall-off. We also indicate the value of Kcrit.(~) for 3 = 2, 3, 4. It would be interesting to study the deviation o f (Cpq,> _ <~xI'>lfree,K= 1/6 from zero at Kcrit.03 ) for each 3, because this would favour the assertion made previously, that the global SU(2) s y m m e t r y is dynamically broken. U n f o r t u n a t e l y , we c a n n o t draw del'mitive conclusions on this m a t t e r from fig. 9, since the largest K where we measured ( ~ I , ) is 0.16, whereas the values ofKcrit" range from 0.168 to 0.18.
264
[1] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792; F. Fucito et al., Nucl. Phys. B 210 [FS6] (1982) 407; A. Hasenfratz et al., Phys.Lett. B 110 (1982) 289; C. Bernard et al., Phys. Rev. D 27 (1983) 227; see also references therein. [2] E.D'Hoker, Nucl. Phys. B 180 [FS2] (1981) 341; G. Bhanot and M. Creutz, Phys. Rev. D 21 (1980) 2892; J. Ambjorn et al., Nucl. Phys. B 210 [FS6] (1982) 347. [3] J.F. Schonfeld, Nucl. Phys. B 185 (1981) 157; D. Boyanovsky, R. Blankenbecler and R. Yahalom, Physical origin of topological mass in 2 + 1 dimensions, preprint SLAC-PUB-3764 (September 1985); S. Deser et al., Phys. Rev. Lett. 48 (1982) 975; Arm. Phys. (NY) 140 (1982) 3372; I. Affleck et al., Nucl. Phys. B 206 (1982) 413; Th, Appelquist et al., Phys. Rev. Lett. 55 (1985) 1715. [4] R. Friedberg and T.D. Lee, Phys. Rev. D 18 (1978) 2623. [5] N. Kawamoto, Nucl. Phys. B 190 [FS3] (1981) 617.
PHYSICS LETTERS B
2 October 1986
M E S O N M A S S C A L C U L A T I O N S IN 3D U(I) L A T F I C E G A U G E T H E O R Y Kostas F A R A K O S and George K O U T S O U M B A S Physics Department, National Technical University of Athens, 157 73 Zografou, Athens, Greece Received 25 June 1986
The meson masses are calculated in the quenched approximation to 3D lattice U(I) for several values of the coupling constant. The agreement with strong coupling estimates for the meson masses is fairly good. Evidence is presented for the dynamical breakdown of a global SU(2) symmetry.
Al=~73q~,
1. Introduction
Mass spectrum calculations in lattice gauge theories with fermions have so far been carried out in four dimensions for the SU(2) and SU(3) gauge groups [1]. There also exist similar investigations of two-dimensional theories on the lattice. In this work we present the study of a three-dimensional gauge theory with fermions, based on the group U(1) (calculations in 3D gauge groups without fermions can be found in ref. [2]). We performed Monte Carlo calculations of the meson spectrum in the so-called "quenched" approximation. The nonquenched calculations and the comparison to the quenched ones are the subject of work in progress. 2. 317 f e r m i o n s
The fermionic part of the lagrangian density is £f = ~ (i~ -- re)kit 2 = qs+70(u~2 i'yUDu - m ) ~ , D u = ~ - tea u .
(2.1)
I f m = 0, the theory has an SU(2) × U(1) global symmetry. The lagrangian density remains invariant under the transformation qJ ~ exp(i3F)q~, where 1-' is 1,73, 3'5 or A -- i7375 [3]. For m :~ 0 there will be a U(1) × U(1) symmetry; the matrix P may be 1 or A. The bound states will be generated by the operators
260
A2=C~2~5'I',
B l u = ~Tu73qJ, B3u = ~ T u q , ,
A3=C~q~,
A4=~JA~,
B2# = ~7u75 ~, B4u = CI,T u A ~ .
(2.2)
We consider two cases. (aJ m = 0. If the SU(2) × U(1) symmetry is realized in the states IA1), IA2) and IA3> form an SU(2) triplet. Similarly, IBlu>, IB2u) and IB3u> form a triplet. Thus, in this case, m(A1) = re(A2) = m(A3) , re(B1 u) = m(B2u ) = m(B3t
(2.3) But there is the alternative possibility that the global SU 2 symmetry is spontaneously broken, due to chiral symmetry breaking. I f ( ~ q o ~ 0, then 3,3 and 75 symmetry is broken. The two corresponding massless Goldstone particles would have the quantum numbers of the broken generators. These would be the states IA 1) and IA2). (b) m 4= O. In this case the global symmetry is U(1) X U(1), which does not imply relations between the masses of the bound states. 3. Lattice calculations o f meson masses and (Vp,~)
In the calculation we used Wilson fermions [1]. The quantities calculated are the correlation functions of the composite operators VPF~ representing meson states with quantum numbers determined by the Dirac matrix P. In the functional integral expres-
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-H011and Physics Publishing Division)
Volume 178, number 2,3
PHYSICS LETTERS B
2 October 1986
2.0
sion for G r ( x , y ) we integrate over the fermionic variables:
8~X16
,\
":
,xg.
.: 1~x16
\ ~xx
•:EXTRAFO_AT, ON
G r ( x , y ) = <01T(~(x)r~I,(x)~(y)Pq,0,))10> 1.5
= ~ f D A DxpD* *(x)r~(x)~Cv)r*c~) X exp[-Sg(A)
1 fDA
_
ZA
M -- ~ Q ~ ] 1.0
[Tr(Q~y~(A)PQjxl(A)P)
-- Tr(Qxx1 (A) P) Tr(Qy1 (A) P) exp [ - S g ( A ) ] det Q(A)]. The second term has a significant contribution to the asymptotic behaviour only if the bound state is a singlet with respect to spin, flavor and color, i.e. only if the meson has the quantum numbers of the vacuum [4]. In the actual computation we did not take the second term into account, conforming to common usage, since that would cost too much in computer time. Thus the ~ , I ' and ~A,I, mesons will be degenerate and so will be the ~TuxI, and ~Tu A,I~ mesons. We now discuss the boundary conditions. In the time direction we supposed free ends. In the x and y directions we used antiperiodic boundary conditions. In fact, antiperiodic conditions have the disadvantage of giving, in the free-field case, a faster fall-off than what is implied by the explicit mass term in the action. These too large meson masses are brought down to the values of an infinite lattice by performing an appropriate scaling. Usually, the masses are supposed to be proportional to the Wilson parameter K:
m=aK +7. Our ansatz is that there will be corrections proportional to 1IN to the coefficients a and 7, where N is the number of lattice sites in the x ( o r y ) direction:
mN
=
(ct + 13/N)K + (7 + 6IN).
(3.1)
We checked this formula in the case of the freefermion theory. We used three lattices o f sizes 82 X 16, 122 × 16 and 162 X 16 and measured the mass associated with the Green's function of U~ysq~ for different K for an infinite lattice this mass can be obtained analytically. Measuring the mass o f each meson for the 82 X 16 and 122 X 16 lattices at vari-
0.5
\\
\ \
\'
\ \
I
O. 2
I
f
0.14 0.16 I KAPPA N
I
I
I
018
Fig. 1. Mass of ~-r5,l, as a function of K in the case of free fermion theory for the 82 X 16, 122 X 16 and 16 2 × 16 lattices and the infinite lattice extrapolation. ous values of K we determine the coefficients a,/3, 7, 6 and from eq. (3.1) we compute m16 and moo. The values of m 16 agree with the direct computation with the 162 X 16 lattice and m ~ agrees with the analytical calculation (fig. 1). Note that, for K near Kcritical the masses no longer lie on a straight line. We suppose, therefore, that this ansatz is applicable for the interacting theory as well, provided we stay way from K = KcriticalThe calculation was done as follows: Sixteen gauge field configurations were generated for some value of the coupling constant and the masses of the different mesons were calculated on two different lattices: 82 × 16 and 122 X 16. F o r each/3 we stopped increasing K, as soon as the masses declined from the straight line. Then from the above scaling method we obtained the masses that an infinite lattice would, presumably, predict. Fig. 2 illustrates the calculation just described for the ~3,5q~ propagator and/3 = 3. The masses measured on the lattices 82 X 16 and 122 × 16 are seen to lie on a straight line in the range of K investigated. 261
Volume 178, number 2,3
PHYSICS LETTERS B
2 October 1986
cosh mp = 1 + (1 - 12K2)/8K 2 ,
2.0
(4.1)
B=3 cosh m v = 1 + (1 - 8K2)2/4K2(1
• : 8"x16
-:12"x16 • : EXTRAPOLATION
1.5
,,~
.>-~10 7" 0 LO
-
4K2),
(4.2)
where rap, m v are the masses of the pseudoscalar (~3'5 q 0 and the vector (U~Tuq~) mesons respectively. In fig. 3 the result of our scaling m e t h o d for ~Tuq~ is plotted against the prediction of eq. (4.2). The agreement is fairly good. The pseudoscalar masses shown in fig. 4 are in good accord with eq. (4.1). In ref. [5] it is claimed that the ~ meson should have infinite mass in the strong coupling limit. In fig. 5 we give our results for the mass o f this state. We see
0.5 \
\ k
\ \
2.0 \
\ X
\
\ \
\
1"4 " 0.12 "
0.14
0.16
"~ ~..,,.
0.18
KAPPA
1.0 -
"-.. -.
Fig. 2. Predictions of the 8 2 X 16 and 12 2 X 16 lattices for the mass of ~'rs'I, at t3 = 3 and various K, the masses given by the extrapolation method and the error bars at K = 0.16. 018
The result of the scaling is also plotted. The statistical errors shown in the figure were obtained b y dividing the sixteen gauge field configurations in two blocks of eight configurations each and considering the variations in the masses over the two subsets. The error increases as we go to larger K or smaller/3. A typical value for the relative error (SM/M) is 0.05. We have also measured the condensate ( ~ ) . Chiral s y m m e t r y is explicitly b r o k e n in our model, since we used Wilson fermions. Nevertheless, we thought it interesting to display (Ckq0 from which we subtracted the value o f ( ~ ) for the free theory,
0.22
Q26 KA PPA
Q30
Q34
Fig. 3. The dashed line represents the prediction of eq. (5.2) for the vector meson mass. The full line is the result of our scaling method.
2-°r-x \
at K = Kcritical = ~.
4. Results ( a ) Strong coupling. We first present the results obtained for/3 = 0, which are to be compared to the strong coupling calculations of ref. [5]. The mass formulae of ref. [5] for three-dimensional s p a c e time are 262
LI
0.18
I
I
I
0.2 2
I
Q26
I
1
I
~*~
I
03 0
KAPPA Fig. 4. The dashed line represents the prediction of eq. (5.1) for the pseudoscalar meson mass. The full line is the result of our scaling method.
PHYSICS LETTERS B
Volume 178, number 2,3
2 October 1986
2.0
3.5
,,\
~--2
~3
• : 8"x16
\\
X\
•
~
:
'Pvsv'
-.--
+
-
• : 12~x16 ~.~EXTRAPOLATION
3.0
t.5
7 , . . . . . . : +~.y+,
):,, \
N
%,,, \<,',
2.5
~21.0
"%\, \\, \ '~?,,x\. lx
z 2.0
05
"~\ x,\. \\ ~ \
I
1.5 014
i
i
0.16
I
i
0.18
i
C~
0
i
J
• X
0.22
K APPA
i
i
r
i
o.13 Fig. 5. Mass of ~,I, at t3= 0 as a function of K for the 82 X 16 and 122 X 16 lattices. The infinite lattice extrapolation is plotted also.
i
o~5
i'~
o77
t
0 '79
K AP PA K~2) Fig. 6. Meson masses according to the infinite lattice extrapolation for ~3= 2. 2.0
that the 12 2 X 16 lattice predicts, at the same values of K, larger masses than the 8 2 X 16 lattice, something contrary to what happens in all other cases. The scaling method generates even greater masses, in qualitative agreement with the expectation o f ref. [5]. (b) Mass spectrum for 13= 2, 3 and 4. Figs. 6 - 8 give the meson masses according to the infinite lattice extrapolation. The states ~3,5q~ and ~3,uqz turn out to be lighter than the others and the difference between their masses is considerably smaller than any other mass difference. This is reminiscent o f the fact that, in the four-dimensional QCD calculations, the particles 7r and p come out with nearly the same mass. It is important that, at Kcrit., the masses o f ~3,5~I, and ~'),3 ~ are zero, while the mass o f ~ ' # gets a, rather large, value. According to what was said earlier, these three states belong to the triplet representation o f the global SU(2) symmetry, which exists in the massless limit. The difference found in the masses suggests that this symmetry is dynamically broken to a U(1). The two massless states ~')'3 qt and ~ 7 5 q~ are the resulting Goldstone bosons.
,
"-\
~\
B=3 \
: +~,
_
x\
..... :0+i+
:L.
1.5
1,9 03
\\ - , '~\ ",, ',.
1.0 Z O 03 kO
z
\ \ ,,',, \ \ ,,',.,
0.5
\ ' x ' , ', .
.
. . 0.13
.
. Q15
\',,',,',, 0.17 1
KAPPA
i
Q19
Kg3)
Fig. 7. Same as in fig. 6 for t\ = 3. 263
Volume 178, number 2,3
PHYSICS LETTERS B
2.0~
13=4 .
.
.
.
.
:
@vp~
:
~YsY~
0.101 0.08
2 October 1986
.
• : B--2
• : B=3 13=4 ": FREE
"
• :
"~ 0 . 0 6 [ 1.5
.
.
.
.
.
.
0"04 I U9 (.t)
,,cl: 1.0
0"02f
f !
Z 0
:\
0.5
,
X~, Q15
"(0.17
Kc(4)
018
•
Fig. 9. ~(~"I,)3, K is the difference (~'q'>3,K- <'t,'I')lfree, K = 1, The results of the measurements for 3 = 2, 3, 4 and for the free-fermion theory are given. Thepoints A, B, C, D are the Kcritical for the free-fermion theory and for 0 = 4, 3, 2, respectively.
"~'k "X. '~,, "',~
0.13
!
014 016 KAPPA
%.
0.19
KAPPA
We w o u l d like to t h a n k Professor G. Tiktopoulos for his help and for critically reading the manuscript.
References
Fig. 8. Same as in fig. 6 for fl = 4.
(c) C ondensate (UI,q,). In fig. 9 we give our results on the difference of <~q'> from the free field value o f (U~q,> at K = Kcr R •(3)13._,~ = g, 1 which • is • the value at which the pseudoscalar correlation f u n c t i o n of the free-fermion theory has n o exponential fall-off. We also indicate the value of Kcrit.(~) for 3 = 2, 3, 4. It would be interesting to study the deviation o f (Cpq,> _ <~xI'>lfree,K= 1/6 from zero at Kcrit.03 ) for each 3, because this would favour the assertion made previously, that the global SU(2) s y m m e t r y is dynamically broken. U n f o r t u n a t e l y , we c a n n o t draw del'mitive conclusions on this m a t t e r from fig. 9, since the largest K where we measured ( ~ I , ) is 0.16, whereas the values ofKcrit" range from 0.168 to 0.18.
264
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