- Email: [email protected]
Monte Carlo study of long-range chiral structure in QCD
Volume 114B, number 6
PHYSICS LETTERS
12 August 1982
MONTE CARLO STUDY OF LONG-RANGE CHIRAL STRUCTURE IN QCD A. DUNCAN and R. ROSKIES Department o f Physics and Astronomy, University o f Pittsburgh, Pittsburgh, PA 15260, USA
and H. VAIDYA Bell Laboratories, Holmdel, NJ 07733, USA
Received 14 April 1982
Results of a Monte Carlo study of long-range chiral structure is massless4-dimensional QCD are preseted. The algorithm employed integrates fermion degrees of freedom exactly. The behavior of chiral correlations in both abelian and SU(2) color theories is compared. We find strong evidence for chiral breakdown, with a solution of the U(1) problem, in lattice QCD at large #.
1. Introduction
There has been considerable interest recently in the possibility of including fennionic degrees of freedom in Monte Carlo studies of lattice gauge theories, with encouraging results in both 2-dimensional gauge (and non-gauge) theories [1,2], and in 4-dimensional QCD in the approximation where internal fermion loops are neglected [3]. Indeed, the calculations of Hamber and Parisi [3] suggest that the latter approximation may be sufficient to give a very good fit to the observed hadron spectrum. Nevertheless, it is important to see exactly how important fermion determinantal effects are in determining the chiral structure of the theory. We shall present results of a Monte Carlo study of the chiral structure of massless 4dimensional QCD, using an approach in which the fermionic fields are exactly integrated, leaving only the gauge link variables to be integrated by standard Monte Carlo procedures. We work on quite small lattices (chiefly, 4 × 4 × 4 X 8 sites) where f'mite size effects are large and complicate the extraction of detailed quantitative results. As we shall see, however, the qualitative evidence for chiral symmetry breaking in massless QCD is very strong - in particular, the lattice theory appears to correctly resolve the U(1) problem, with the U(1) channel developing a massive ex0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
citation, while the SU(4) channels remain massless. The dependence of correlations on lattice spacing are qualitatively what one would expect from asymptotic freedom. We also fred [in an SU(2) color group, with 4 massless quark flavors] some quantitative discrepancy between our results and those in which effects of internal quark loops are neglected. The choice of action, and its relation to the continuum flavor fields is presented in section 2. The Monte Carlo technique is described in section 3, followed by results in section 4. We close with some discussion of work in progress. 2. The lattice model 2.A. Choice of a c t i o n . The primary limitation of the
Monte Carlo procedure we shall use lies in the necessity for storing all elements of the fennion propagator for any given gauge configuration. If there are Nf fermionic degrees of freedom, there are clearly N 2 such elements, and we must regard Nf as limited by the available core storage. Given this constraint, it is obvious that the modeling of the continuum gauge field will be improved by distributing the Nf fermion fields over as many lattice sites as possible. A method for doing this was first suggested by Susskind [4], and we have used a euclidean adaptation of the action he 439
Volume 114B, number 6
PHYSICS LETTERS
suggested. We have also generalized the action to allow for asymmetric lattices where the lattice spacing in the (euclidean) ~# direction is eua. In particular, we have worked with 4 X 4 X 4 X 8 lattices with e 4 _1 - ~, e 1 = e 2 = e 3 = 1. Such a lattice evidently models a hypercubic block of physical space-time, while by measuring correlations along the 4-direction, we effectively reap the advantage of an 8 4 lattice. It would not be advantageous to measure correlations over euclidean time separations greater than the spatial box size: such correlations (even for a massless excitation) are damped exponentially as the mass gap (due to the discrete m o m e n t u m eigenstates) makes itself felt. The action is
S = (2e/g 2) ~
{(l/e2e2)Retr(W~u(n) - 1)
rl,,u ~ v
#
v
+ 7uv(Re tr ( W ( n ) - 1)) 2}
12 August 1982
= 1, e 4 _-1 ~, we fred "Yi4 = 0.4795, "yq = 0 (i,i = 1,2, 3). We also obtain the following asymptotic freedom relation between 13= 4/g 2, the lattice spacing a, and the parameter ADR defining the l-loop subtractions in minimal dimensional regularization: a = (1/ADR)( ~ rr2/~)57/196 exp[---~ fr2~ -- 1.088)]. (2.3) The appearance of 7 (= 11 - 4) in the exponent in (2.3) confirms the presence of four flavors of massless fermions in this theory (we have reduced the 16fold corner degeneracy of the 4-dimensional Brillouin zone by going from 4 to 1 Dirac component per site). 2.B. Relation to flavor fields. The continuum fermion fields are constructed by combining the fermionic degrees of freedom within each 24 subblock of the lattiee. Labelling the fields as in fig. 1, one defines (i = 1, 2, 3,4) -1
--n~,# -~ fl.~nfbn( #ndPn+~# - U;n .~# dPn .~u )' -
g
_
_
(2.1) where e = I14=, e,f~n =fu(--l) "nppnp, where f : = (_l)nUuf#. u 1 We have chosen the representation fl =f3 = 1, f2 = - i , f 4 = - 1 , with
T/=
i00001 1
1
0
0
1
1
0
0 "
1
1
1
0
U~
'+
'
~2i Xi),
i-×i+
~( i + Xi
_
,_
,
~lli
Xi),
~i = 1 ~( 1]li - Xi __ ~it + Xti)"
(2.4)
~7uA u ~ + ia'~tuD75 if,
(2.5)
where ~ = (u, d, c, s), 2xu, D u are discrete first and sec(2.2)
Finally, to each site of the lattice are attached fermion color doublets en and ~'n" The second term of (2.1) vanishes relative to the first term in the continuum limit. Although the classical continuum limit of (2.1) yields a euclidean rotationally invariant continuum action, the quantum corrections in one-loop induce on the lattice asymmetric operators whose effect must be cancelled by a corresponding counterterm. The calculation (best performed by the elegant background field method developed for lattice applications by Dashen and Gross) [5] is lengthy and will be presented elsewhere. For the special case of interest here, with e I = e 2 = e 3 440
+
The conventional fields are then obtained b y a trivial rearrangement:~ for example, d 1 = d'2, d2 = - d l , d3 = - d 4 , d 4 = d 3 . With these definitions, the fermion part of (2.1), when all Unu = 1, is
The Uun are the usual SU(2) link variables, and U~
Ui- 2(~i +Xi
×5
q'3
....
_rx2
j/
/
~2 t//
1
. . . . .
i /
xx3
I 'l
i
I I
,
I '
II
1"4
I , "'x, I
x/,
~X 2
' ml I ---o
x,
Fig. 1. Labelling of fermi fields for x 4 odd (for x a even, all fields are primed).
Volume 114B, number 6
PHYSICS LETTERS
ond lattice derivatives respectively, and t u are four specific flavor matrices. The second term in (2.5) vanishes in the continuum limit, but it breaks the chiral symmetry on the lattice. This difficulty persists independently of the choice of flavor fields, Dirac representation, or of the specific form of the action (assuming nearest neighbor), as long as it has the correct continuum limit and one forms the continuum fields from combinations of the lattice degrees of freedom in a local 24 subblock. We must treat the second term in (2.5) as a chiral perturbation fixing the orientation of the vacuum after chiral symmetry breaking. This orientation can be determined if we assume that the perturbation is small and can be treated to first order, analogous to the quark mass term in PCAC. A single insertion of the chiral perturbing term in a graph for the effective potential (producing, for example, a vertex insertion iaEu q2 t~75 in a fermion loop carrying m o m e n t u m q) clearly has the same effect, after integration over q, as a momentum-independent insertion iaEu (q2)t~75. After a flavor and chiral redefinition of the fields, this is found to be equivalent to a flavor-independent mass term in the new fields • = (U, D, C, S). (2.5) then becomes
~7 u Au4 + 8m ¢,~.
(2.6)
The mass term still vanishes in the continuum limit, and does not appear to induce measurable masses for the Goldstone degrees of freedom (given the size of lattices available to us). However, its presence allows us to isolate the U(1) ("r/") degree of freedom ~75cb from the SU(N) ("rr") ones ~ 7 s t a ~ . The Susskind action does not have continuous 75 invariance, and strong coupling calculations [6] based on it have yielded large values for m,r. However, we see signs of massless pions emerging nevertheless.
12 August 1982
used as a positive definite factor multiplying exp Sgauge to determine the full Monte Carlo probability measure for the gauge field. The computation of the determinant, and of the various Wick contractions needed when measurements of correlation functions are performed, is facilitated by storing the inverse matrix (or propagator) D = M - 1(U/) and updating it as each U/ is updated. This can be done rapidly as the change in a single link changes only a small N c X N c (Nc = number of colors) block of the original matrix M. For example, for a U(1) gauge group, changing a single link U/ changes a single matrix element M; : by 6M~ ; , 0 z 010 ~0 1 0 resulting in the following exact changes in the det (/14) and D: det (M(U/0 + 6 Ulo)) = det(M(Uto)) (1 + D/oio 5MioJo),
Oi/(Ulo + 5Ulo) =Di/(UI o) -
DiiofUlo)D/oj(Ulo ) l+DjoioSMioJ ° 6Mt.oJo.
(3.2)
The generahzation to higher rank shifts, as occur in larger gauge groups, is trivial. On a 4 X 4 × 4 X 8 lattice for SU(2)c , D is a 256 X 256 matrix whose elements are themselves 2 X 2 complex matrices. The Monte Carlo procedure begins by setting all Ul = 1 and calculating D exactly. On an infinite lattice, the free part of the fermion action in (2.1) is readily seen to lead to a fermion propagator _
Omn
i~
d4k
2 a ~(k)
-. exp [lk (m - n)
--TT
X ~l f ~zrn sin (kueua), # e#
(3.3)
with
3. The Monte Carlo method [2] The fermionic part of the lattice action (2.1) can be written in the form Sf= ~
(~-liMi/.(U/)~2j- ~2iM]*(U/) fill) ,
(3.1)
31sin2(k e a), u e2
~
k(m- n)=a ~ k eu(m u - nu). la
where the ffl(ff2) fields occupy even (odd) sites of the lattice. The functional integral over the fermion fields can be performed. The fermion determinant is proportional to detM- d e t M t = IdetMI 2 and may be
On a finite lattice, the integrals in (3.3) go over to sums in an obvious way, and the resulting formula can be used to generate the free propagator at the beginning of the simulation. We have used free boundary 441
Volume 114B, number 6
PHYSICS LETTERS
12 August 1982
conditions on the long axis, and anti-periodic ones in the three "spatial" directions. The links are then updated, using both the gauge action and the fermion determinant in the measure, using the Metropolos procedure [7]. Whenever a new value for a link variable is accepted, the determinant and propagator are updated using (3.2). A single Monte Carlo interaction is defined to have occurred when as many updates have actually been performed as there are links on the lattice: this (depending on/3) may require anywhere from 4 to 20 sweeps of the lattice. The Metropolis method is superior to the heat bath when including determinantal effects as the allowed range of values of detM is enormous, whereas
[
I
-
'X
0.1 --
\8~.XN~ 'k ' ~ x ~
We have concentrated on measuring the two point chiral correlation functions Y~x(~3'5(b(x) ~ 75 q~(x + Ax)) (called the U(1) correlation function henceforth) and Zx(~75 Tcb(x)~75 T ~ ( x + 2xx)) (called the SU(4) correlation function) where T is a diagonal SU(4) flavor matrix, chosen so that T~75T~ corresponds in terms of the original fields ¢ in (2.1) to sums over ~i4)i at the same site. The chiral densities have a strongly lattice-dependent normalization and we shall only be concerned with the shape of the correlation function, not the absolute scale. Accordingly, all chiral correlations are normalized to unity at separation Ax 4 = a (corresponding to two lattice 442
Equilibrium U(1)Correlation
\\
\ • \
I
0.01
.
_
_
3a
2a
Ax 4 = a
4. Results
Starting Configurations (Ue= I)
Equilibrium Xx~ X~'N SU(4) correlation \x~x \-
detM(U/+ 8 U/)/detM(U/) is typically of order 1. For example, at/3 = 2.2 on a 4 × 4 × 4 × 8 lattice, ( f o r N c = 2), Idet(M)l 2 equilibrates at a value ~ 10 -68 that of the starting value (with all Ut = 1). At every stage, the Metropolis procedure uses only the ratio of successive values of the determinant, which from (3.2) is known to the accuracy of the elements of D. By explicit check, the norm o f the matrix DM-1 is found to be < 1 0 -12 even after 100 Monte Carlo iterations on the CRAY-1 (with a machine precision of 15 significant figures). The typical iteration time for a 4 × 4 × 4 × 8 lattice (updating 2K links) on the CRAY-1 was 140 s. Surprisingly, the limiting factor turns out to be the time required for measurements rather than for updating. As we shall see below, this becomes an especially serious constraint for exponentially falling correlations where statistical fluctuations are large.
I
Fig. 2. Monte Carlo simulation of chiral correlations in abelian theory. sites in the 4-direction). Measurements have been made at Ax 4 = a, 2a, 3a. The results for a Monte Carlo simulation at/3 = 2.2 of the abelian gauge theory (QED with four massless leptons) are shown in fig. 2. As a crude measure of the approach to equilibrium we have also given in table 1 the values o f - In [det MI 2 after each of the first 12 iterations (the starting value of the determinant is normalized to unity for convenience). Evidently, after 5 or so iterations equilibrium is reached. It should be noted that Idet MI 2 fluctuates by orders of magnitude from one iteration to the next. Nevertheless, our exact evaluation gives us control over its value at each stage to at least 12 significant digits. The U(1) and SU(4) correlations are shown in fig. 2 for the abelian theory. The dashed lines from A x 4 = 2a to 3a represent the starting values when all
Table 1 Iteration
-lnldetMI ~
Iteration
-lnldetMI 2
1 2 3 4 5 6
19.78 25.62 26.82 27.86 28.78 30.98
7 8 9 10 11 12
31.12 28.64 27.49 30.89 29.26 30.96
Volume 114B, number 6
PHYSICS LETTERS
links Ul = 1 (i.e. free lattice fermions). The rapid falloff is the lattice version of the continuum r -6 behavior. The starting difference between U(1) and SU(4) is a Finite-size lattice effect. After 12 iterations, the correlations are as shown with unbroken line *1. Both correlation functions fall off somewhat more slowly, with the U(1) correlation having increased rather more at Ax = 3a than the SU(4) (suggesting the additional contribution of multiphoton intermediate states in the interacting theory). Nevertheless, the fall off remains sharp - factors of 32, 74 for the U(1), SU(4) correlations between Ax 4 = a and Ax 4 = 3a. We should emphasize that the rapid decrease is due to a decrease in the magnitude of the matrix elementsDi/, not to cancellations between large contributions of fluctuating sign. Accordingly, the statistical fluctuations are quite small and measurements fairly stable (in fig. 1 the solid lines represent averages over 3 consecutive configurations, with error bars roughly twice the size of the data cicles). If we alter a few lines of program code to allow the selection of random SU(2)e rather than U(1)c link variables, these results change dramatically. We have performed measurements at/3 = 2.4 and/3 = 2.2 of the SU(4) correlation function at Ax 4 = a, 2a, 3a 4=1 The falloff from Ax 4 = a to AX4 = 2a is virtually unchanged after equilibration.
]
i
i
l
~
SU(4} ~=22 Fe mlon LOOPS
0. I
1 2.1_
/J: 2.2
0.01
AX 4 =
0
~
I SU(4) ~
I
I
20
30
' ~' : 2.4
Fig. 3. Monte Carlo simulation of chiral correlations in SU(2) color gauge theory.
12 August 1982
and the U(1) correlation function at Ax 4 = a, 2a (the U(1) correlations at Ax 4 = 3a show enormous statistical fluctuations - on which more below). The SU(4) correlation function rises dramatically from its starting value and after only ~ 15 iterations stabilizes at the values indicated in fig. 3, obtained as an average over 25 consecutive configurations. We have also plotted the continuum r - 2 falloff for a massless pion propagator, shown as a dotted line. Between 13= 2.4 and/3 = 2.2 the physical lattice spacing roughly doubles [see eq. (2.3)]. If the U(1) correlation is dominated by a massive 1-particle 7?'-pole ,2 and the SU(4) correlation by a massless Goldstone pole, we expect to see the U(1) correlations decrease with decreasing /3, while the SU(4) correlations should increase from an initial r - 6 (short distance) falloff to the r -2 characteristic of a massless scalar excitation. This is indeed what appears to happen as we go from/3 = 2.4 to/3 = 2.2 [fig. (3)]. The value of/3 = 2.2 corresponds from (2.3) to a physical lattice spacing a = 0.02 AD 1, and the fact that the SU(4) correlation is already showing a behavior consistent with r -2 falloff suggests that the ADR parameter is considerably smaller in relation to physical mass scales (such as f~r) in SU(2) color gauge theory with 4 massless flavors than, say, in SU(3) with 2 massless flavors (where empirically, f,~ ~ ADR ). Unfortunately, the U(1) correlation function drops in virtue of delicate cancellations between numbers large compared to the Final average value. For the nonabelian theory, the typical matrix element Di/does not fall off rapidly (as is clear from the behavior of the SU(4) correlation) and the simulation must obtain exponential falloff by cancellation between quantities falling as a power. Thus, at separations Ax 4 = 2a, 3a one is forced to average over very many configurations. The results indicated in fig. 2 for the U(1) correlation at Ax 4 = 2a and at/3 = 2.2 and 2.4 represents an average over 160 configurations, representing several hours of CRAY-1 running time. This time is dominated by the actual correlation measurements (rather than link updating) which are complicated by the necessity for including gauge-covariantizing strings in each fermion bilinear. For the U(1) bilinears, the +2 We assume that the single particle rf-pole dominates the contribution of multi ( 9 4) pion states in the distance regime available on our lattice.
443
Volume 114B, number 6
PHYSICS LETTERS
fermi fields are located at third nearest neighbor sites, and an average is performed over all shortest string routes in each bilinear. In summary, though, we feel that the more rapid falloff of the U(1) correlation is strong qualitative eivdence that the lattice theory is properly modelling a massive 77'. If we assume that at/3 = 2.2 we are already measuring the asumptotic falloff of the r~' propagator, then a fit to the massive lattice scalar propagator (analytically computed) gives ADR ~ (0.01--0.02)mn,, again suggesting a rather small value in this theory for the A-parameter in terms of the hadron mass scale. However, a serious calculation of masses would require larger lattices and a careful study of the scaling over a range of values of 13. In fig. 3 we also show the results obtained for the SU(4) correlation when the determinantal factor is ignored in the Monte Carlo probability measure (i.e. ignoring fermion loops). The determinant now equilibrates at a value typically ~ 10 -86 (as compared to N 10-68 when determinantal effects are included in the update procedure). In other words, the absence of the restraining effect of the determinant factor allows the system to relax into a considerably more chaotic link configuration. On the other hand, the SU(4) falloff is considerably less rapid than in the full theory and is not a reasonable fit to a massless scalar propagator, although the gauge theory is known to be in the scaling region at/3 = 2.2. Of course, with four flavors of massless fermions, a considerable amount of asymptotic freedom is exhausted in SU(2)c (Nmax = 11) and perhaps in SU(3)c the effect of fermion loops may more safely be neglected.
5. Summary of ongoing work The results of the last section raise several interacing questions. On a simple level, is the one SU(4) correlation we studied typical of the other SU(4) correla-
444
12 August 1982
tion? Or are some of its characteristics related to being made up of same site ~)iqSi (as flavors and spatial symmetries are intimately related in the Susskind approach)? On a deeper level, it is clear that to study larger lattices and an SU(3) color group, one has to reduce the core requirements, by not storing the full fermion propagator at each stage. We have made substantial progress along these lines by exploiting the sparseness of the original matrix M(U) in (3.1). We can reduce the core requirement for a 4 × 4 X 4 × 4 lattice with SU(3) c to under 64 K, at some cost in running time. (The algorithm in this paper would have required over 300 K.) We have also considered adding a quark mass term to the original fermion action. We lose the nice property expressed in (3.1) that the action only couples even sites to odd ones, so storage requirements appear to increase. But this seems the only way of generating a realistic hadron spectrum.
References [ 1 ] F. Fucito, E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B180 (1981) 369; D. Weingarten and D. Petcher, Phys. Lett. 99B (1981) 333; D.J. Scalapino and R. Sugar, Phys. Rev. Lett. 46 (1981) 519; H. Hamber, Phys. Rev. D24 (1981) 951. [2] A. Duncan and M. Furman, Nucl. Phys. B190 (1981) 767. [3] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792: E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795. [4] L. Susskind, Phys. Rev. D16 (1977) 3031. [5] R. Dashen and D. Gross, Phys. Rev. D23 (1981) 2340. [6] T. Banks et al., Phys. Rev. D159 (1977) 1111-. [7] N. Metropolis, A.W. Rosenbluth, A.M. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087.
PHYSICS LETTERS
12 August 1982
MONTE CARLO STUDY OF LONG-RANGE CHIRAL STRUCTURE IN QCD A. DUNCAN and R. ROSKIES Department o f Physics and Astronomy, University o f Pittsburgh, Pittsburgh, PA 15260, USA
and H. VAIDYA Bell Laboratories, Holmdel, NJ 07733, USA
Received 14 April 1982
Results of a Monte Carlo study of long-range chiral structure is massless4-dimensional QCD are preseted. The algorithm employed integrates fermion degrees of freedom exactly. The behavior of chiral correlations in both abelian and SU(2) color theories is compared. We find strong evidence for chiral breakdown, with a solution of the U(1) problem, in lattice QCD at large #.
1. Introduction
There has been considerable interest recently in the possibility of including fennionic degrees of freedom in Monte Carlo studies of lattice gauge theories, with encouraging results in both 2-dimensional gauge (and non-gauge) theories [1,2], and in 4-dimensional QCD in the approximation where internal fermion loops are neglected [3]. Indeed, the calculations of Hamber and Parisi [3] suggest that the latter approximation may be sufficient to give a very good fit to the observed hadron spectrum. Nevertheless, it is important to see exactly how important fermion determinantal effects are in determining the chiral structure of the theory. We shall present results of a Monte Carlo study of the chiral structure of massless 4dimensional QCD, using an approach in which the fermionic fields are exactly integrated, leaving only the gauge link variables to be integrated by standard Monte Carlo procedures. We work on quite small lattices (chiefly, 4 × 4 × 4 X 8 sites) where f'mite size effects are large and complicate the extraction of detailed quantitative results. As we shall see, however, the qualitative evidence for chiral symmetry breaking in massless QCD is very strong - in particular, the lattice theory appears to correctly resolve the U(1) problem, with the U(1) channel developing a massive ex0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
citation, while the SU(4) channels remain massless. The dependence of correlations on lattice spacing are qualitatively what one would expect from asymptotic freedom. We also fred [in an SU(2) color group, with 4 massless quark flavors] some quantitative discrepancy between our results and those in which effects of internal quark loops are neglected. The choice of action, and its relation to the continuum flavor fields is presented in section 2. The Monte Carlo technique is described in section 3, followed by results in section 4. We close with some discussion of work in progress. 2. The lattice model 2.A. Choice of a c t i o n . The primary limitation of the
Monte Carlo procedure we shall use lies in the necessity for storing all elements of the fennion propagator for any given gauge configuration. If there are Nf fermionic degrees of freedom, there are clearly N 2 such elements, and we must regard Nf as limited by the available core storage. Given this constraint, it is obvious that the modeling of the continuum gauge field will be improved by distributing the Nf fermion fields over as many lattice sites as possible. A method for doing this was first suggested by Susskind [4], and we have used a euclidean adaptation of the action he 439
Volume 114B, number 6
PHYSICS LETTERS
suggested. We have also generalized the action to allow for asymmetric lattices where the lattice spacing in the (euclidean) ~# direction is eua. In particular, we have worked with 4 X 4 X 4 X 8 lattices with e 4 _1 - ~, e 1 = e 2 = e 3 = 1. Such a lattice evidently models a hypercubic block of physical space-time, while by measuring correlations along the 4-direction, we effectively reap the advantage of an 8 4 lattice. It would not be advantageous to measure correlations over euclidean time separations greater than the spatial box size: such correlations (even for a massless excitation) are damped exponentially as the mass gap (due to the discrete m o m e n t u m eigenstates) makes itself felt. The action is
S = (2e/g 2) ~
{(l/e2e2)Retr(W~u(n) - 1)
rl,,u ~ v
#
v
+ 7uv(Re tr ( W ( n ) - 1)) 2}
12 August 1982
= 1, e 4 _-1 ~, we fred "Yi4 = 0.4795, "yq = 0 (i,i = 1,2, 3). We also obtain the following asymptotic freedom relation between 13= 4/g 2, the lattice spacing a, and the parameter ADR defining the l-loop subtractions in minimal dimensional regularization: a = (1/ADR)( ~ rr2/~)57/196 exp[---~ fr2~ -- 1.088)]. (2.3) The appearance of 7 (= 11 - 4) in the exponent in (2.3) confirms the presence of four flavors of massless fermions in this theory (we have reduced the 16fold corner degeneracy of the 4-dimensional Brillouin zone by going from 4 to 1 Dirac component per site). 2.B. Relation to flavor fields. The continuum fermion fields are constructed by combining the fermionic degrees of freedom within each 24 subblock of the lattiee. Labelling the fields as in fig. 1, one defines (i = 1, 2, 3,4) -1
--n~,# -~ fl.~nfbn( #ndPn+~# - U;n .~# dPn .~u )' -
g
_
_
(2.1) where e = I14=, e,f~n =fu(--l) "nppnp, where f : = (_l)nUuf#. u 1 We have chosen the representation fl =f3 = 1, f2 = - i , f 4 = - 1 , with
T/=
i00001 1
1
0
0
1
1
0
0 "
1
1
1
0
U~
'+
'
~2i Xi),
i-×i+
~( i + Xi
_
,_
,
~lli
Xi),
~i = 1 ~( 1]li - Xi __ ~it + Xti)"
(2.4)
~7uA u ~ + ia'~tuD75 if,
(2.5)
where ~ = (u, d, c, s), 2xu, D u are discrete first and sec(2.2)
Finally, to each site of the lattice are attached fermion color doublets en and ~'n" The second term of (2.1) vanishes relative to the first term in the continuum limit. Although the classical continuum limit of (2.1) yields a euclidean rotationally invariant continuum action, the quantum corrections in one-loop induce on the lattice asymmetric operators whose effect must be cancelled by a corresponding counterterm. The calculation (best performed by the elegant background field method developed for lattice applications by Dashen and Gross) [5] is lengthy and will be presented elsewhere. For the special case of interest here, with e I = e 2 = e 3 440
+
The conventional fields are then obtained b y a trivial rearrangement:~ for example, d 1 = d'2, d2 = - d l , d3 = - d 4 , d 4 = d 3 . With these definitions, the fermion part of (2.1), when all Unu = 1, is
The Uun are the usual SU(2) link variables, and U~
Ui- 2(~i +Xi
×5
q'3
....
_rx2
j/
/
~2 t//
1
. . . . .
i /
xx3
I 'l
i
I I
,
I '
II
1"4
I , "'x, I
x/,
~X 2
' ml I ---o
x,
Fig. 1. Labelling of fermi fields for x 4 odd (for x a even, all fields are primed).
Volume 114B, number 6
PHYSICS LETTERS
ond lattice derivatives respectively, and t u are four specific flavor matrices. The second term in (2.5) vanishes in the continuum limit, but it breaks the chiral symmetry on the lattice. This difficulty persists independently of the choice of flavor fields, Dirac representation, or of the specific form of the action (assuming nearest neighbor), as long as it has the correct continuum limit and one forms the continuum fields from combinations of the lattice degrees of freedom in a local 24 subblock. We must treat the second term in (2.5) as a chiral perturbation fixing the orientation of the vacuum after chiral symmetry breaking. This orientation can be determined if we assume that the perturbation is small and can be treated to first order, analogous to the quark mass term in PCAC. A single insertion of the chiral perturbing term in a graph for the effective potential (producing, for example, a vertex insertion iaEu q2 t~75 in a fermion loop carrying m o m e n t u m q) clearly has the same effect, after integration over q, as a momentum-independent insertion iaEu (q2)t~75. After a flavor and chiral redefinition of the fields, this is found to be equivalent to a flavor-independent mass term in the new fields • = (U, D, C, S). (2.5) then becomes
~7 u Au4 + 8m ¢,~.
(2.6)
The mass term still vanishes in the continuum limit, and does not appear to induce measurable masses for the Goldstone degrees of freedom (given the size of lattices available to us). However, its presence allows us to isolate the U(1) ("r/") degree of freedom ~75cb from the SU(N) ("rr") ones ~ 7 s t a ~ . The Susskind action does not have continuous 75 invariance, and strong coupling calculations [6] based on it have yielded large values for m,r. However, we see signs of massless pions emerging nevertheless.
12 August 1982
used as a positive definite factor multiplying exp Sgauge to determine the full Monte Carlo probability measure for the gauge field. The computation of the determinant, and of the various Wick contractions needed when measurements of correlation functions are performed, is facilitated by storing the inverse matrix (or propagator) D = M - 1(U/) and updating it as each U/ is updated. This can be done rapidly as the change in a single link changes only a small N c X N c (Nc = number of colors) block of the original matrix M. For example, for a U(1) gauge group, changing a single link U/ changes a single matrix element M; : by 6M~ ; , 0 z 010 ~0 1 0 resulting in the following exact changes in the det (/14) and D: det (M(U/0 + 6 Ulo)) = det(M(Uto)) (1 + D/oio 5MioJo),
Oi/(Ulo + 5Ulo) =Di/(UI o) -
DiiofUlo)D/oj(Ulo ) l+DjoioSMioJ ° 6Mt.oJo.
(3.2)
The generahzation to higher rank shifts, as occur in larger gauge groups, is trivial. On a 4 X 4 × 4 X 8 lattice for SU(2)c , D is a 256 X 256 matrix whose elements are themselves 2 X 2 complex matrices. The Monte Carlo procedure begins by setting all Ul = 1 and calculating D exactly. On an infinite lattice, the free part of the fermion action in (2.1) is readily seen to lead to a fermion propagator _
Omn
i~
d4k
2 a ~(k)
-. exp [lk (m - n)
--TT
X ~l f ~zrn sin (kueua), # e#
(3.3)
with
3. The Monte Carlo method [2] The fermionic part of the lattice action (2.1) can be written in the form Sf= ~
(~-liMi/.(U/)~2j- ~2iM]*(U/) fill) ,
(3.1)
31sin2(k e a), u e2
~
k(m- n)=a ~ k eu(m u - nu). la
where the ffl(ff2) fields occupy even (odd) sites of the lattice. The functional integral over the fermion fields can be performed. The fermion determinant is proportional to detM- d e t M t = IdetMI 2 and may be
On a finite lattice, the integrals in (3.3) go over to sums in an obvious way, and the resulting formula can be used to generate the free propagator at the beginning of the simulation. We have used free boundary 441
Volume 114B, number 6
PHYSICS LETTERS
12 August 1982
conditions on the long axis, and anti-periodic ones in the three "spatial" directions. The links are then updated, using both the gauge action and the fermion determinant in the measure, using the Metropolos procedure [7]. Whenever a new value for a link variable is accepted, the determinant and propagator are updated using (3.2). A single Monte Carlo interaction is defined to have occurred when as many updates have actually been performed as there are links on the lattice: this (depending on/3) may require anywhere from 4 to 20 sweeps of the lattice. The Metropolis method is superior to the heat bath when including determinantal effects as the allowed range of values of detM is enormous, whereas
[
I
-
'X
0.1 --
\8~.XN~ 'k ' ~ x ~
We have concentrated on measuring the two point chiral correlation functions Y~x(~3'5(b(x) ~ 75 q~(x + Ax)) (called the U(1) correlation function henceforth) and Zx(~75 Tcb(x)~75 T ~ ( x + 2xx)) (called the SU(4) correlation function) where T is a diagonal SU(4) flavor matrix, chosen so that T~75T~ corresponds in terms of the original fields ¢ in (2.1) to sums over ~i4)i at the same site. The chiral densities have a strongly lattice-dependent normalization and we shall only be concerned with the shape of the correlation function, not the absolute scale. Accordingly, all chiral correlations are normalized to unity at separation Ax 4 = a (corresponding to two lattice 442
Equilibrium U(1)Correlation
\\
\ • \
I
0.01
.
_
_
3a
2a
Ax 4 = a
4. Results
Starting Configurations (Ue= I)
Equilibrium Xx~ X~'N SU(4) correlation \x~x \-
detM(U/+ 8 U/)/detM(U/) is typically of order 1. For example, at/3 = 2.2 on a 4 × 4 × 4 × 8 lattice, ( f o r N c = 2), Idet(M)l 2 equilibrates at a value ~ 10 -68 that of the starting value (with all Ut = 1). At every stage, the Metropolis procedure uses only the ratio of successive values of the determinant, which from (3.2) is known to the accuracy of the elements of D. By explicit check, the norm o f the matrix DM-1 is found to be < 1 0 -12 even after 100 Monte Carlo iterations on the CRAY-1 (with a machine precision of 15 significant figures). The typical iteration time for a 4 × 4 × 4 × 8 lattice (updating 2K links) on the CRAY-1 was 140 s. Surprisingly, the limiting factor turns out to be the time required for measurements rather than for updating. As we shall see below, this becomes an especially serious constraint for exponentially falling correlations where statistical fluctuations are large.
I
Fig. 2. Monte Carlo simulation of chiral correlations in abelian theory. sites in the 4-direction). Measurements have been made at Ax 4 = a, 2a, 3a. The results for a Monte Carlo simulation at/3 = 2.2 of the abelian gauge theory (QED with four massless leptons) are shown in fig. 2. As a crude measure of the approach to equilibrium we have also given in table 1 the values o f - In [det MI 2 after each of the first 12 iterations (the starting value of the determinant is normalized to unity for convenience). Evidently, after 5 or so iterations equilibrium is reached. It should be noted that Idet MI 2 fluctuates by orders of magnitude from one iteration to the next. Nevertheless, our exact evaluation gives us control over its value at each stage to at least 12 significant digits. The U(1) and SU(4) correlations are shown in fig. 2 for the abelian theory. The dashed lines from A x 4 = 2a to 3a represent the starting values when all
Table 1 Iteration
-lnldetMI ~
Iteration
-lnldetMI 2
1 2 3 4 5 6
19.78 25.62 26.82 27.86 28.78 30.98
7 8 9 10 11 12
31.12 28.64 27.49 30.89 29.26 30.96
Volume 114B, number 6
PHYSICS LETTERS
links Ul = 1 (i.e. free lattice fermions). The rapid falloff is the lattice version of the continuum r -6 behavior. The starting difference between U(1) and SU(4) is a Finite-size lattice effect. After 12 iterations, the correlations are as shown with unbroken line *1. Both correlation functions fall off somewhat more slowly, with the U(1) correlation having increased rather more at Ax = 3a than the SU(4) (suggesting the additional contribution of multiphoton intermediate states in the interacting theory). Nevertheless, the fall off remains sharp - factors of 32, 74 for the U(1), SU(4) correlations between Ax 4 = a and Ax 4 = 3a. We should emphasize that the rapid decrease is due to a decrease in the magnitude of the matrix elementsDi/, not to cancellations between large contributions of fluctuating sign. Accordingly, the statistical fluctuations are quite small and measurements fairly stable (in fig. 1 the solid lines represent averages over 3 consecutive configurations, with error bars roughly twice the size of the data cicles). If we alter a few lines of program code to allow the selection of random SU(2)e rather than U(1)c link variables, these results change dramatically. We have performed measurements at/3 = 2.4 and/3 = 2.2 of the SU(4) correlation function at Ax 4 = a, 2a, 3a 4=1 The falloff from Ax 4 = a to AX4 = 2a is virtually unchanged after equilibration.
]
i
i
l
~
SU(4} ~=22 Fe mlon LOOPS
0. I
1 2.1_
/J: 2.2
0.01
AX 4 =
0
~
I SU(4) ~
I
I
20
30
' ~' : 2.4
Fig. 3. Monte Carlo simulation of chiral correlations in SU(2) color gauge theory.
12 August 1982
and the U(1) correlation function at Ax 4 = a, 2a (the U(1) correlations at Ax 4 = 3a show enormous statistical fluctuations - on which more below). The SU(4) correlation function rises dramatically from its starting value and after only ~ 15 iterations stabilizes at the values indicated in fig. 3, obtained as an average over 25 consecutive configurations. We have also plotted the continuum r - 2 falloff for a massless pion propagator, shown as a dotted line. Between 13= 2.4 and/3 = 2.2 the physical lattice spacing roughly doubles [see eq. (2.3)]. If the U(1) correlation is dominated by a massive 1-particle 7?'-pole ,2 and the SU(4) correlation by a massless Goldstone pole, we expect to see the U(1) correlations decrease with decreasing /3, while the SU(4) correlations should increase from an initial r - 6 (short distance) falloff to the r -2 characteristic of a massless scalar excitation. This is indeed what appears to happen as we go from/3 = 2.4 to/3 = 2.2 [fig. (3)]. The value of/3 = 2.2 corresponds from (2.3) to a physical lattice spacing a = 0.02 AD 1, and the fact that the SU(4) correlation is already showing a behavior consistent with r -2 falloff suggests that the ADR parameter is considerably smaller in relation to physical mass scales (such as f~r) in SU(2) color gauge theory with 4 massless flavors than, say, in SU(3) with 2 massless flavors (where empirically, f,~ ~ ADR ). Unfortunately, the U(1) correlation function drops in virtue of delicate cancellations between numbers large compared to the Final average value. For the nonabelian theory, the typical matrix element Di/does not fall off rapidly (as is clear from the behavior of the SU(4) correlation) and the simulation must obtain exponential falloff by cancellation between quantities falling as a power. Thus, at separations Ax 4 = 2a, 3a one is forced to average over very many configurations. The results indicated in fig. 2 for the U(1) correlation at Ax 4 = 2a and at/3 = 2.2 and 2.4 represents an average over 160 configurations, representing several hours of CRAY-1 running time. This time is dominated by the actual correlation measurements (rather than link updating) which are complicated by the necessity for including gauge-covariantizing strings in each fermion bilinear. For the U(1) bilinears, the +2 We assume that the single particle rf-pole dominates the contribution of multi ( 9 4) pion states in the distance regime available on our lattice.
443
Volume 114B, number 6
PHYSICS LETTERS
fermi fields are located at third nearest neighbor sites, and an average is performed over all shortest string routes in each bilinear. In summary, though, we feel that the more rapid falloff of the U(1) correlation is strong qualitative eivdence that the lattice theory is properly modelling a massive 77'. If we assume that at/3 = 2.2 we are already measuring the asumptotic falloff of the r~' propagator, then a fit to the massive lattice scalar propagator (analytically computed) gives ADR ~ (0.01--0.02)mn,, again suggesting a rather small value in this theory for the A-parameter in terms of the hadron mass scale. However, a serious calculation of masses would require larger lattices and a careful study of the scaling over a range of values of 13. In fig. 3 we also show the results obtained for the SU(4) correlation when the determinantal factor is ignored in the Monte Carlo probability measure (i.e. ignoring fermion loops). The determinant now equilibrates at a value typically ~ 10 -86 (as compared to N 10-68 when determinantal effects are included in the update procedure). In other words, the absence of the restraining effect of the determinant factor allows the system to relax into a considerably more chaotic link configuration. On the other hand, the SU(4) falloff is considerably less rapid than in the full theory and is not a reasonable fit to a massless scalar propagator, although the gauge theory is known to be in the scaling region at/3 = 2.2. Of course, with four flavors of massless fermions, a considerable amount of asymptotic freedom is exhausted in SU(2)c (Nmax = 11) and perhaps in SU(3)c the effect of fermion loops may more safely be neglected.
5. Summary of ongoing work The results of the last section raise several interacing questions. On a simple level, is the one SU(4) correlation we studied typical of the other SU(4) correla-
444
12 August 1982
tion? Or are some of its characteristics related to being made up of same site ~)iqSi (as flavors and spatial symmetries are intimately related in the Susskind approach)? On a deeper level, it is clear that to study larger lattices and an SU(3) color group, one has to reduce the core requirements, by not storing the full fermion propagator at each stage. We have made substantial progress along these lines by exploiting the sparseness of the original matrix M(U) in (3.1). We can reduce the core requirement for a 4 × 4 X 4 × 4 lattice with SU(3) c to under 64 K, at some cost in running time. (The algorithm in this paper would have required over 300 K.) We have also considered adding a quark mass term to the original fermion action. We lose the nice property expressed in (3.1) that the action only couples even sites to odd ones, so storage requirements appear to increase. But this seems the only way of generating a realistic hadron spectrum.
References [ 1 ] F. Fucito, E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B180 (1981) 369; D. Weingarten and D. Petcher, Phys. Lett. 99B (1981) 333; D.J. Scalapino and R. Sugar, Phys. Rev. Lett. 46 (1981) 519; H. Hamber, Phys. Rev. D24 (1981) 951. [2] A. Duncan and M. Furman, Nucl. Phys. B190 (1981) 767. [3] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792: E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795. [4] L. Susskind, Phys. Rev. D16 (1977) 3031. [5] R. Dashen and D. Gross, Phys. Rev. D23 (1981) 2340. [6] T. Banks et al., Phys. Rev. D159 (1977) 1111-. [7] N. Metropolis, A.W. Rosenbluth, A.M. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087.