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Exploring hadron masses in lattice QCD with light quarks and an improved fermion action
Volume 200, number 1,2
PHYSICS LETTERS B
7 January 1988
EXPLORING H A D R O N M A S S E S IN LATTICE QCD WITH LIGHT QUARKS A N D AN IMPROVED F E R M I O N ACTION ~ Ph. DE F O R C R A N D CRAY Research, Minneapolis, MN, USA
R. G U P T A 1 Los Alamos National Laboratory. Los Alamos, NM 87545, USA
S. G O S K E N , K.-H. M O T T E R , A. PATEL, K. S C H I L L I N G and R. S O M M E R Physics Department, University of Wuppertal, D-5600 Wuppertal 1, Fed. Rep. Germany
Received 2 September 1987
Hadron masses are computed using quenched Wilson fermions on gauge configurations obtained after blocking 243 ×48 lattices twice with the v/3 block transformation. An improved Wilson fermion action, motivated by a truncated renormalisation group analysis within the block-diagonalisation scheme, is used. We present results obtained close to the chiral limit, i.e. in the range 0.4 < mJmo< 0.85. Moreover, we study finite size effects by combining results obtained with periodic and antiperiodic boundary conditions.
The realisation o f h a d r o n masses in lattice Q C D with light quark masses has p r o v e d to be an intriguing problem, even after several years o f M o n t e Carlo calculations. One needs large lattices to a v o i d finite size effects a n d a large d a t a sample to counter the large fluctuations o f the quark propagators in the vicinity o f the chiral limit. Since h a d r o n masses are extracted only from the long distance b e h a v i o u r o f the respective propagators, it is an attractive idea to reduce the n u m b e r o f degrees o f freedom. This can be achieved by performing a block t r a n s f o r m a t i o n to a coarser lattice a n d retaining only the light m o d e s that carry the infrared behaviour, before a t t e m p t i n g the numerical inversion o f the lattice Dirac operator. Such a m e t h o d ( n a m e d a p p r o x i m a t e block-diagonalisation) has been p u r s u e d by some o f us in a series o f papers, using two consecutive scale two blocking steps [ 1 - 3 ] . Let us a b b r e v i a t e it as SC2 for convenience. Work supported in part by the Deutsche Forschungsgemeinschaft grant Schi 257/2-1. J. Robert Oppenheimer fellow.
An alternative blocking geometry was suggested and a p p l i e d to the pure gauge theory case in ref. [4], where the links o f the blocked lattice are chosen to be the three-space diagonals o f the original hypercubic lattice. In this scheme ( S Q 3 ) , the scale factor between the coarse and fine lattices is xf3. The previous SC2 block-diagonalisation a p p r o a c h can readily be extended to the SQ3 blocking geometry, details o f which will be given in a forthcoming p a p e r [ 5]. In the present letter, we i n t e n d to show first results for the hadron masses using Wilson fermions blocked with the SQ3 scheme. These were o b t a i n e d on the CRAY X M P - 4 8 o f the newly established C o m p u t e r Center for Theoretical Physics at Jtilich. The effective gauge fields on the " c o a r s e " lattice are constructed from the link variables o f the " f i n e " lattice in precisely the same m a n n e r as in ref. [4]. After two blocking steps, the basis vectors o f the coarse lattice are parallel to those on the fine lattice. The block-diagonalised Dirac operator, truncated to keep only the leading order terms, has nearest neighbour as well as three-space diagonal interactions, the latter having been generated by the SQ3 blocking.
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
143
Volume 200, number 1,2
PHYSICSLETTERSB
The gluonic fields, U(x+e,, x), that carry the interaction across the three-space diagonals are constructed as the average [projected back onto SU (3) ] over the six possible three-link parallel transporters; i.e., the 0 are constructed exactly like the block links. The effective fermion action on the coarse lattice then is
M = 2mafi x ,.~ 4
+ Z U(X', x)[A, Cu(x', x) +A2?,S~,(x', x)] p= I 4
O ( x ' , x ) [ A 3 C- a ( x ,' x ) + A 4 ~ u ~ ( X ' ,
+Z
x)].
(1)
p=l
The second line contains the usual nearest neighbour couplings, while the third one adds new couplings along the three-space diagonals. Here
C,(x', x) = 2 - 6 x .x+~ - r x , x _ ~ , S,(x', x) =rx,.~+~-G,,x_~ , ~ , ( x ' , x) = 2 - 6 ~ . x + , , , - 6 ~ ' ......~,, , p
S,(x,x)=G,..,+e,,-G,
...... ,,,
2ma= l / K - 8 .
(2)
The three-space diagonal vectors, G~, read as follows: e l a = ( 0 , 1, 1, 1),
e2~=(1,0, 1, --1),
e3.=(1,-1,0,1),
e4.=(1,1,-1,0).
(3)
The corresponding rotated gamma matrices are 4
9,, = Z y , . e , , .
(4)
o~=1
The relative strength of the Wilson term with respect to the naive term is chosen such that in the formal continuum limit the action acquires exactly the same form as the original Wilson fermion action with r = 1. The relative weights between nearest neighbour and the three-space diagonal couplings, A ~/A3 and Az/A4, are chosen to be the solutions of fixed point equations under iterative blocking, truncated to the fourdimensional coupling space spanned by {A~, A> A3, A4}. As a result, we find for {As} the values [5]
a,=l/3x/2, A4 _ 1
.
A2=¼, A 3 = ( x / 2 - 1 ) / 9 x / 2 , (5)
We briefly state the main features of the SQ3 block144
7 January 1988
diagonalisation, some of which are desirable advantages over the SC2 approach: (1) The block-diagonalisation process has a greater hypercubic rotational symmetry. (2) The separation between light and heavy modes is clean; there is one light mode with mass 2ma and four pairs of heavy modes with masses 2ma+ 9 +_3i. (3) There are no contact terms in the lowest order, leaving the mass term unrenormalised. This makes it easy to interpret the action in eqs. ( 1 ) - ( 5 ) , either as a block-diagonalised effective fermion action, or just as a new choice of action having the correct continuum limit. We have based our computation on 51 gluonic fields produced by one of us (PhdF) in the quenched approximation at fl = 6.3 on a 243X48 lattice. Starting from these fine lattice configurations, we produced the coarse lattice gluonic fields using two x/3 blocking steps. The numerical inversion of the effective fermion matrix, eq. (I), was then performed at two source points of the resulting 83X 16 lattice with the minimal residual method ~, after an even-odd partition of the problem. We have results for six x values, x=0.1345, 0.1351, 0.1355, 0.1358, 0.1360, 0.1362. The minimal residual algorithm was twice as fast as the commonly used conjugate gradient method on the five smaller values of to, and the two methods were comparable on the largest one. It helped to use the quark propagator for the previous smaller value of tc as a seed for starting the minimal residual algorithm. The error criterion for the convergence of the iterative inversion procedure for solving Mx=b, was chosen to be Ilb-Mxql/LIxll < 10 - 5. The number of iterations necessary to reach this accuracy ranged from 80 for the smallest ~c, to typically 400 for the largest one. For the largest K value though, we observed fluctuations between 250 and 1200 iterations. This whole calculation was performed with periodic (PBC) as well as anti-periodic (APBC) fermion boundary conditions in all the four directions. In the previous computation of hadron propagators with SC2 blocking geometry and PBC [ 3 ], based on the first 27 of the same "fine" gluonic configurations, large fluctuations were observed in the hadron propagators at small quark masses and large time "~ For instance as described in re£ [6],
Volume 200, number 1,2
PHYSICS LETTERS B
7 January 1988
45
a
4O
b
C
35
30 t 20 ,
! 10 5 0
0.
0.2
0.4
0.
0.2
0.4
0.2
O,
I I_ _ 1 0.4
L
Fig. 1. The histograms for the distribution of the pion propagators at Jc= 0.1360, t = 18. (a), (b), (c) correspond to PBC, APBC and FSR respectively. The data is not symmetrised and the two source points on the same configuration are plotted separately.
the nucleon p r o p a g a t o r far away from the source point. We have p e r f o r m e d the same analysis in the present SQ3 blocking scheme with both PBC a n d APBC. It turns out that configuration no. 3 is exceptional with PBC but not with APBC. On the other h a n d configuration no. 32 is exceptional with A P B C b u t not with PBC. Figs. 1 and 2 show the d i s t r i b u t i o n s o f the pion and nucleon propagators at t i m e sepa-
separations. They were i n t e r p r e t e d to be due to small eigenvalues o f the f e r m i o n m a t r i x which strongly fluctuate with the b a c k g r o u n d gauge field. These fluctuations were analysed by plotting the distribution o f h a d r o n propagators from the various background fields on fixed large time slices. The largest deviations from the m e a n were traced back to three " e x c e p t i o n a l " configurations, one o f which (no. 3 in the chronological o r d e r ) d o m i n a t e d the average o f
a
120
b
c
1 O0
8O
6O
4o
20
0 I ~ ~k -50 -20 -10
0
H~ = I 10 2 0
r 30
k i ~ 40-50-20-10
i . 0
~1 10
.i 20
i ~ ~ 30 40-,50-20-10
~ 0
10
20
50
I 40 -10-
6
Fig. 2. S a m e as fig. 1 but for the nucleon. T h e s h a r p n e s s o f the peak o b s e r v e d for A P B C is due to p r o j e c t i o n onto the lowest n o n - z e r o
momentum eigenstate. 145
Volume 200, number 1,2
PHYSICS LETTERSB
rations t = 18,2, which is the most important one for the subsequent mass determinations. The entries from the two exceptional configurations are marked in black. For the largest x value the exceptional configurations stick out a lot more. In ref. [ 3 ], it was argued that the exceptional configurations might be a disease of the quenched approximation which - due to the absence of the fermion determinant - is not protected against the occurrence of small eigenvalue fluctuations. Our present experience shows that in a finite volume these fluctuations are very sensitive to the boundary conditions. When working with Wilson fermions one is bound to encounter such fluctuations, as the effective quark mass depends on the background gauge field through the r-term. The presence of the fermion determinant may reduce the spread of the fluctuations, but it cannot eliminate the contribution of the small eigenvalues which is needed in order to obtain a massless pion at some xcri,. As a matter of fact for any ~C~[~,~Ccr,] there is a non-zero probability of finding configurations which will become critical at that particular •. The path integral is supposed to sum over all such configurations. This is precisely what is analytically done in strong coupling expansions, and one obtains a sensible result at the end. But the Monte Carlo procedure is not efficient in integrating over singular regions of phase space, and one singular configuration is enough to throw the whole process of path integration into total disarray. Not having a clear way out of the dilemma, we just exclude by hand from our spectrum analysis configuration nos. 1, 2, 3 and 32, the first two to guard against the possibility of incomplete thermalisation ~3. Finally we point out that the remarks above do not apply to staggered fermions since there the chiral symmetries fix the critical point to be at m = 0. In order to reduce the effects of a finite lattice volume, which are particularly harmful in the region of
7 January 1988
small pion masses, one can use the trick of varying the fermionic boundary conditions [7,8,5]. Consider an infinite periodic gauge field built up from finite cells each carrying the given background field configuration. The quark propagator computed on this system senses just one source term, while the quark propagator computed on the finite cell with PBC or APBC feels periodic mirror sources in each cell, coupled with different signs as dictated by the boundary conditions. This sign change can be exploited to eliminate the source terms in every other cell: let Qp and Qa respectively denote the quark propagators computed with PBC and APBC. Let H+ and H_ respectively be the zero momentum hadron propagators calculated using ½(Qp+ Qa) and ½( Q p - Q a ) as the quark propagator. The combination H+ + H then will show reduced finite size ef-
=
.1562,
FSR
o
o
o
10
10 -2
10 5
A A /
10 4
10 - 5
-
-
•
PION
o
RHO
•
NUCLEON
z~
DELTA
10 - 6
10 -7
~2 T h o u g h the actual c o m p u t a t i o n was done on the coarse 83 × 16 lattice, all the n u m b e r s q u o t e d are scaled by a factor of three, as if they were o b t a i n e d on the fine 243 X 48 lattice. T h i s is j u s t to facilitate a c o m p a r i s o n with the p r e v i o u s SC2 calculations, ~3 510 t h e r m a l i s a t i o n sweeps were discarded after c o n s t r u c t i n g a 24~X48 lattice by periodically stacking up a 2 4 3 X 6 one. Thereafter configurations were stored every 100 sweeps,
146
n_
l lO
u 20
I 50
I 40
l t
Fig. 3. The FSR hadron propagators at ~: = 0.1362, symmetrized in forward-backward directions and averaged over 47 back-
ground fields.
Volume 200, number 1,2
PHYSICS LETTERS B
7 January 1988
PBC a n d F S R masses as a function o f 1/x are shown in fig. 4. The errors displayed were calculated with the Jack-knife m e t h o d [ 9 ]. We notice that m ~ falls on a straight line. On the other hand m o shows a slight positive curvature a n d mN a slightly negative one. The splitting between N a n d A increases with increasing x. The c o m p a r i s o n o f PBC and F S R results d e m o n s t r a t e s the i m p a c t o f finite size effects, a n d the gain o f using F S R results is obvious in the error bars. It can be seen, from the systematic difference between PBC a n d F S R results, that enlarging the physical v o l u m e increases mp and decreases mN, though the tiny effects are well within our present error bars. Fitting m o to its experimental value provides an estimate o f the scale on the fine lattice, 1 / a = 4 . 0 ( 3 ) GeV. The above features o f the d a t a are better expressed
fects. We used this c o m b i n a t i o n (called F S R ) at the three largest values o f ~c. The fluctuations in the propagators with p e r i o d i c a n d a n t i - p e r i o d i c b o u n d ary conditions are anti-correlated [7], and consequently the F S R results show reduced spread (figs. 1 a n d 2). This effect is m o r e d r a m a t i c for the baryons, where in some cases the errors reduce by a factor o f ~ 2, than for the mesons. This is as a n t i c i p a t e d since the baryons are physically bigger than the mesons. In fig. 3 we show our F S R results for the n, p, N a n d A propagators at the smallest quark mass. The d a t a are s y m m e t r i z e d in the f o r w a r d - b a c k w a r d directions (for the baryons this means c o m b i n i n g upper a n d lower spinor c o m p o n e n t s ) , and the errors are statistical. All the propagators were fit with single mass exponentials from t = 15 to t = 24. The resulting
HADRON MASS RATLOS HADRON MASSES PBC 1.8 0.5
++++
0.4
+
1.6
u
FSR
I
experiment
u
infinite querk mass
--
quark model
1
t
~
~
1.z
t{ +* +
0.5
•
m~ 2
•
rap,
PBC
0.2 PBC
1.
m N , PBC •
m~,
PBC
o m 2 , FSR 0.1
z~ m~ , FSR ~' m . •
O.
I• ° 7.34
iiio 00 ~ 7.36
0.~
FSR
[] m~ , FSR __
I 7..38
I 7.4
I_ _ J 7.42 7.44
I 746
'
1 // : Fig. 4. The hadron masses as a function of 1/K. The FSR results are horizontally shifted so as to match them to the PBC scorn.
O.
/ 0.2
~ 0.4
I 0.6
I 0.8
__
I 1. r]q~ J
__ mp
Fig. 5. T h e h a d r o n m a s s r a t i o s a s a f u n c t i o n o f m ~ l m w T h e u p p e r
set of points are for raN~rap, while the lower ones correspond to mN/m•.
147
Volume 200, number 1,2
PHYSICS LETTERS B
in a mass-ratio plot as in fig. 5. The errors again were calculated with the Jack-knife method. Note that we are able to go down to rn,~/mp= 0.46, while previous calculations [10] (except ref. [3]) stopped a little above mJrnp=0.7. In the additional range we observe a b e n d i n g down ~4 of mN/m o towards its experimental value, as rn,~/rn~, decreases. The ratio mN/ma points towards its physical value as well. We are aware, however, that all our results for raN~r%lie systematically above the predictions of a quark model including hyperfine splitting [ 1 1 ]. One of the possible reasons for this discrepancy is the neglect of dyn a m i c a l fermions. The first 27 configurations of our data sample have been used before in ref. [ 12] to compute hadron masses with staggered fermions and APBC, over a comparable range of rn,~/rnp. There the value of r% extrapolated to the chiral limit was 0.40(4) in lattice units, to be compared with the value 0.193 (I 4) obtained in the present work. This implies that on our coarse lattices the scale has a sizeable dependence on the form of the action we choose to work with. We are currently extending the SC2 analysis to the full data sample a n d a detailed comparison of the SQ3 a n d SC2 results will be presented in a forthcoming paper. We are grateful to the staff of the C o m p u t e r Center for Theoretical Physics at J~ilich for their generous help.
t~4This feature becomes far more pronounced if the exceptional configurationsare included in the analysis.
148
7 January 1988
References [1] K.-H. Miitter and K. Schilling,Nucl. Phys. B 230 (FS10) (1984) 275; A. Kt~nig,K.-H. Mfitter and K. Schilling,Phys. Left. B 147 (1984) 145;Nucl. Phys. B259 (1985) 33; A. K6nig,K.-H. Miitter, K. Schillingand J. Smit, Phys. Lett. B 157 (1985) 421; K.-H. M/itter and K. Schilling,in: Advances in lattice gauge theory, eds. D.W. Duke and J.F. Owens (World Scientific, Singapore, 1985); Ph. de Forcrand, A. K6nig, K.-H. Miitter, K. Schillingand R. Sommer, in: Lattice gauge theory - A challenge in large scale computing, Proc. 1985 Workshop (Wuppertal), eds. B. Bunk,K.-H. Mfitterand K. Schilling(Plenum, New York, 1986). [2] O. Haan, E. Schnepf, E. Laermann, K.-H. Miitter, K. Schillingand R. Sommer, Phys. Lett. B 190 (1987) 147. [3] Ph. de Forcrand, A. K/Snig,K.H. MiJtter, K. Schilling and R. Sommer, in: Proc. Intern. Symp. on Lattice gauge theory (Brookhaven, 1986), (Plenum, New York, 1987). [4] R. Cordery, R. Gupta and M. Novotny, Phys. Lett. B 128 (1983) 425; R. Gupta, G. Guralnik, A. Patel, T. Warnock and C. Zemach, Phys. Rev. Len. 53 (1984) 1721; A. Patel and R. Gupta, Nucl. Phys. B 251 (1985) 789. [ 5] K.-H. Miitter et al., in preparation. [6] Y. Oyanagi, Comput. Phys. Commun. 42 (1986) 333. [ 71 R. Gupta and A. Patel, Nucl. Phys. B 226 (1983) 152. [8 ] P. Gibbs, University of Glasgow preprint ( 1985). [9] See e.g.S. Gottlieb, P.B. Mackanzie, H.B. Thacker and D. Weingarten,Nucl. Phys. B 263 (1986) 704. [10] H. Hamber, Phys. Lett. B 178 (1986) 277; S. Itoh, Y. Iwasaki and T. Yoshie, Phys. Lett. B 183 (1987) 351; R. Gupta, G. Guralnik, G. Kilcup,A. Patel and S.R. Sharpc, Cornell preprint, CLNS-87-79 (1987). [ 11 ] S. Ono, Phys. Rev. D 17 (1978) 888. [12] S.R. Sharpe, A. Patel, R. Gupta, G. Guralnik and G.W. Kilcup, Nucl. Phys. B 286 (1987) 253.
PHYSICS LETTERS B
7 January 1988
EXPLORING H A D R O N M A S S E S IN LATTICE QCD WITH LIGHT QUARKS A N D AN IMPROVED F E R M I O N ACTION ~ Ph. DE F O R C R A N D CRAY Research, Minneapolis, MN, USA
R. G U P T A 1 Los Alamos National Laboratory. Los Alamos, NM 87545, USA
S. G O S K E N , K.-H. M O T T E R , A. PATEL, K. S C H I L L I N G and R. S O M M E R Physics Department, University of Wuppertal, D-5600 Wuppertal 1, Fed. Rep. Germany
Received 2 September 1987
Hadron masses are computed using quenched Wilson fermions on gauge configurations obtained after blocking 243 ×48 lattices twice with the v/3 block transformation. An improved Wilson fermion action, motivated by a truncated renormalisation group analysis within the block-diagonalisation scheme, is used. We present results obtained close to the chiral limit, i.e. in the range 0.4 < mJmo< 0.85. Moreover, we study finite size effects by combining results obtained with periodic and antiperiodic boundary conditions.
The realisation o f h a d r o n masses in lattice Q C D with light quark masses has p r o v e d to be an intriguing problem, even after several years o f M o n t e Carlo calculations. One needs large lattices to a v o i d finite size effects a n d a large d a t a sample to counter the large fluctuations o f the quark propagators in the vicinity o f the chiral limit. Since h a d r o n masses are extracted only from the long distance b e h a v i o u r o f the respective propagators, it is an attractive idea to reduce the n u m b e r o f degrees o f freedom. This can be achieved by performing a block t r a n s f o r m a t i o n to a coarser lattice a n d retaining only the light m o d e s that carry the infrared behaviour, before a t t e m p t i n g the numerical inversion o f the lattice Dirac operator. Such a m e t h o d ( n a m e d a p p r o x i m a t e block-diagonalisation) has been p u r s u e d by some o f us in a series o f papers, using two consecutive scale two blocking steps [ 1 - 3 ] . Let us a b b r e v i a t e it as SC2 for convenience. Work supported in part by the Deutsche Forschungsgemeinschaft grant Schi 257/2-1. J. Robert Oppenheimer fellow.
An alternative blocking geometry was suggested and a p p l i e d to the pure gauge theory case in ref. [4], where the links o f the blocked lattice are chosen to be the three-space diagonals o f the original hypercubic lattice. In this scheme ( S Q 3 ) , the scale factor between the coarse and fine lattices is xf3. The previous SC2 block-diagonalisation a p p r o a c h can readily be extended to the SQ3 blocking geometry, details o f which will be given in a forthcoming p a p e r [ 5]. In the present letter, we i n t e n d to show first results for the hadron masses using Wilson fermions blocked with the SQ3 scheme. These were o b t a i n e d on the CRAY X M P - 4 8 o f the newly established C o m p u t e r Center for Theoretical Physics at Jtilich. The effective gauge fields on the " c o a r s e " lattice are constructed from the link variables o f the " f i n e " lattice in precisely the same m a n n e r as in ref. [4]. After two blocking steps, the basis vectors o f the coarse lattice are parallel to those on the fine lattice. The block-diagonalised Dirac operator, truncated to keep only the leading order terms, has nearest neighbour as well as three-space diagonal interactions, the latter having been generated by the SQ3 blocking.
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
143
Volume 200, number 1,2
PHYSICSLETTERSB
The gluonic fields, U(x+e,, x), that carry the interaction across the three-space diagonals are constructed as the average [projected back onto SU (3) ] over the six possible three-link parallel transporters; i.e., the 0 are constructed exactly like the block links. The effective fermion action on the coarse lattice then is
M = 2mafi x ,.~ 4
+ Z U(X', x)[A, Cu(x', x) +A2?,S~,(x', x)] p= I 4
O ( x ' , x ) [ A 3 C- a ( x ,' x ) + A 4 ~ u ~ ( X ' ,
+Z
x)].
(1)
p=l
The second line contains the usual nearest neighbour couplings, while the third one adds new couplings along the three-space diagonals. Here
C,(x', x) = 2 - 6 x .x+~ - r x , x _ ~ , S,(x', x) =rx,.~+~-G,,x_~ , ~ , ( x ' , x) = 2 - 6 ~ . x + , , , - 6 ~ ' ......~,, , p
S,(x,x)=G,..,+e,,-G,
...... ,,,
2ma= l / K - 8 .
(2)
The three-space diagonal vectors, G~, read as follows: e l a = ( 0 , 1, 1, 1),
e2~=(1,0, 1, --1),
e3.=(1,-1,0,1),
e4.=(1,1,-1,0).
(3)
The corresponding rotated gamma matrices are 4
9,, = Z y , . e , , .
(4)
o~=1
The relative strength of the Wilson term with respect to the naive term is chosen such that in the formal continuum limit the action acquires exactly the same form as the original Wilson fermion action with r = 1. The relative weights between nearest neighbour and the three-space diagonal couplings, A ~/A3 and Az/A4, are chosen to be the solutions of fixed point equations under iterative blocking, truncated to the fourdimensional coupling space spanned by {A~, A> A3, A4}. As a result, we find for {As} the values [5]
a,=l/3x/2, A4 _ 1
.
A2=¼, A 3 = ( x / 2 - 1 ) / 9 x / 2 , (5)
We briefly state the main features of the SQ3 block144
7 January 1988
diagonalisation, some of which are desirable advantages over the SC2 approach: (1) The block-diagonalisation process has a greater hypercubic rotational symmetry. (2) The separation between light and heavy modes is clean; there is one light mode with mass 2ma and four pairs of heavy modes with masses 2ma+ 9 +_3i. (3) There are no contact terms in the lowest order, leaving the mass term unrenormalised. This makes it easy to interpret the action in eqs. ( 1 ) - ( 5 ) , either as a block-diagonalised effective fermion action, or just as a new choice of action having the correct continuum limit. We have based our computation on 51 gluonic fields produced by one of us (PhdF) in the quenched approximation at fl = 6.3 on a 243X48 lattice. Starting from these fine lattice configurations, we produced the coarse lattice gluonic fields using two x/3 blocking steps. The numerical inversion of the effective fermion matrix, eq. (I), was then performed at two source points of the resulting 83X 16 lattice with the minimal residual method ~, after an even-odd partition of the problem. We have results for six x values, x=0.1345, 0.1351, 0.1355, 0.1358, 0.1360, 0.1362. The minimal residual algorithm was twice as fast as the commonly used conjugate gradient method on the five smaller values of to, and the two methods were comparable on the largest one. It helped to use the quark propagator for the previous smaller value of tc as a seed for starting the minimal residual algorithm. The error criterion for the convergence of the iterative inversion procedure for solving Mx=b, was chosen to be Ilb-Mxql/LIxll < 10 - 5. The number of iterations necessary to reach this accuracy ranged from 80 for the smallest ~c, to typically 400 for the largest one. For the largest K value though, we observed fluctuations between 250 and 1200 iterations. This whole calculation was performed with periodic (PBC) as well as anti-periodic (APBC) fermion boundary conditions in all the four directions. In the previous computation of hadron propagators with SC2 blocking geometry and PBC [ 3 ], based on the first 27 of the same "fine" gluonic configurations, large fluctuations were observed in the hadron propagators at small quark masses and large time "~ For instance as described in re£ [6],
Volume 200, number 1,2
PHYSICS LETTERS B
7 January 1988
45
a
4O
b
C
35
30 t 20 ,
! 10 5 0
0.
0.2
0.4
0.
0.2
0.4
0.2
O,
I I_ _ 1 0.4
L
Fig. 1. The histograms for the distribution of the pion propagators at Jc= 0.1360, t = 18. (a), (b), (c) correspond to PBC, APBC and FSR respectively. The data is not symmetrised and the two source points on the same configuration are plotted separately.
the nucleon p r o p a g a t o r far away from the source point. We have p e r f o r m e d the same analysis in the present SQ3 blocking scheme with both PBC a n d APBC. It turns out that configuration no. 3 is exceptional with PBC but not with APBC. On the other h a n d configuration no. 32 is exceptional with A P B C b u t not with PBC. Figs. 1 and 2 show the d i s t r i b u t i o n s o f the pion and nucleon propagators at t i m e sepa-
separations. They were i n t e r p r e t e d to be due to small eigenvalues o f the f e r m i o n m a t r i x which strongly fluctuate with the b a c k g r o u n d gauge field. These fluctuations were analysed by plotting the distribution o f h a d r o n propagators from the various background fields on fixed large time slices. The largest deviations from the m e a n were traced back to three " e x c e p t i o n a l " configurations, one o f which (no. 3 in the chronological o r d e r ) d o m i n a t e d the average o f
a
120
b
c
1 O0
8O
6O
4o
20
0 I ~ ~k -50 -20 -10
0
H~ = I 10 2 0
r 30
k i ~ 40-50-20-10
i . 0
~1 10
.i 20
i ~ ~ 30 40-,50-20-10
~ 0
10
20
50
I 40 -10-
6
Fig. 2. S a m e as fig. 1 but for the nucleon. T h e s h a r p n e s s o f the peak o b s e r v e d for A P B C is due to p r o j e c t i o n onto the lowest n o n - z e r o
momentum eigenstate. 145
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rations t = 18,2, which is the most important one for the subsequent mass determinations. The entries from the two exceptional configurations are marked in black. For the largest x value the exceptional configurations stick out a lot more. In ref. [ 3 ], it was argued that the exceptional configurations might be a disease of the quenched approximation which - due to the absence of the fermion determinant - is not protected against the occurrence of small eigenvalue fluctuations. Our present experience shows that in a finite volume these fluctuations are very sensitive to the boundary conditions. When working with Wilson fermions one is bound to encounter such fluctuations, as the effective quark mass depends on the background gauge field through the r-term. The presence of the fermion determinant may reduce the spread of the fluctuations, but it cannot eliminate the contribution of the small eigenvalues which is needed in order to obtain a massless pion at some xcri,. As a matter of fact for any ~C~[~,~Ccr,] there is a non-zero probability of finding configurations which will become critical at that particular •. The path integral is supposed to sum over all such configurations. This is precisely what is analytically done in strong coupling expansions, and one obtains a sensible result at the end. But the Monte Carlo procedure is not efficient in integrating over singular regions of phase space, and one singular configuration is enough to throw the whole process of path integration into total disarray. Not having a clear way out of the dilemma, we just exclude by hand from our spectrum analysis configuration nos. 1, 2, 3 and 32, the first two to guard against the possibility of incomplete thermalisation ~3. Finally we point out that the remarks above do not apply to staggered fermions since there the chiral symmetries fix the critical point to be at m = 0. In order to reduce the effects of a finite lattice volume, which are particularly harmful in the region of
7 January 1988
small pion masses, one can use the trick of varying the fermionic boundary conditions [7,8,5]. Consider an infinite periodic gauge field built up from finite cells each carrying the given background field configuration. The quark propagator computed on this system senses just one source term, while the quark propagator computed on the finite cell with PBC or APBC feels periodic mirror sources in each cell, coupled with different signs as dictated by the boundary conditions. This sign change can be exploited to eliminate the source terms in every other cell: let Qp and Qa respectively denote the quark propagators computed with PBC and APBC. Let H+ and H_ respectively be the zero momentum hadron propagators calculated using ½(Qp+ Qa) and ½( Q p - Q a ) as the quark propagator. The combination H+ + H then will show reduced finite size ef-
=
.1562,
FSR
o
o
o
10
10 -2
10 5
A A /
10 4
10 - 5
-
-
•
PION
o
RHO
•
NUCLEON
z~
DELTA
10 - 6
10 -7
~2 T h o u g h the actual c o m p u t a t i o n was done on the coarse 83 × 16 lattice, all the n u m b e r s q u o t e d are scaled by a factor of three, as if they were o b t a i n e d on the fine 243 X 48 lattice. T h i s is j u s t to facilitate a c o m p a r i s o n with the p r e v i o u s SC2 calculations, ~3 510 t h e r m a l i s a t i o n sweeps were discarded after c o n s t r u c t i n g a 24~X48 lattice by periodically stacking up a 2 4 3 X 6 one. Thereafter configurations were stored every 100 sweeps,
146
n_
l lO
u 20
I 50
I 40
l t
Fig. 3. The FSR hadron propagators at ~: = 0.1362, symmetrized in forward-backward directions and averaged over 47 back-
ground fields.
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7 January 1988
PBC a n d F S R masses as a function o f 1/x are shown in fig. 4. The errors displayed were calculated with the Jack-knife m e t h o d [ 9 ]. We notice that m ~ falls on a straight line. On the other hand m o shows a slight positive curvature a n d mN a slightly negative one. The splitting between N a n d A increases with increasing x. The c o m p a r i s o n o f PBC and F S R results d e m o n s t r a t e s the i m p a c t o f finite size effects, a n d the gain o f using F S R results is obvious in the error bars. It can be seen, from the systematic difference between PBC a n d F S R results, that enlarging the physical v o l u m e increases mp and decreases mN, though the tiny effects are well within our present error bars. Fitting m o to its experimental value provides an estimate o f the scale on the fine lattice, 1 / a = 4 . 0 ( 3 ) GeV. The above features o f the d a t a are better expressed
fects. We used this c o m b i n a t i o n (called F S R ) at the three largest values o f ~c. The fluctuations in the propagators with p e r i o d i c a n d a n t i - p e r i o d i c b o u n d ary conditions are anti-correlated [7], and consequently the F S R results show reduced spread (figs. 1 a n d 2). This effect is m o r e d r a m a t i c for the baryons, where in some cases the errors reduce by a factor o f ~ 2, than for the mesons. This is as a n t i c i p a t e d since the baryons are physically bigger than the mesons. In fig. 3 we show our F S R results for the n, p, N a n d A propagators at the smallest quark mass. The d a t a are s y m m e t r i z e d in the f o r w a r d - b a c k w a r d directions (for the baryons this means c o m b i n i n g upper a n d lower spinor c o m p o n e n t s ) , and the errors are statistical. All the propagators were fit with single mass exponentials from t = 15 to t = 24. The resulting
HADRON MASS RATLOS HADRON MASSES PBC 1.8 0.5
++++
0.4
+
1.6
u
FSR
I
experiment
u
infinite querk mass
--
quark model
1
t
~
~
1.z
t{ +* +
0.5
•
m~ 2
•
rap,
PBC
0.2 PBC
1.
m N , PBC •
m~,
PBC
o m 2 , FSR 0.1
z~ m~ , FSR ~' m . •
O.
I• ° 7.34
iiio 00 ~ 7.36
0.~
FSR
[] m~ , FSR __
I 7..38
I 7.4
I_ _ J 7.42 7.44
I 746
'
1 // : Fig. 4. The hadron masses as a function of 1/K. The FSR results are horizontally shifted so as to match them to the PBC scorn.
O.
/ 0.2
~ 0.4
I 0.6
I 0.8
__
I 1. r]q~ J
__ mp
Fig. 5. T h e h a d r o n m a s s r a t i o s a s a f u n c t i o n o f m ~ l m w T h e u p p e r
set of points are for raN~rap, while the lower ones correspond to mN/m•.
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in a mass-ratio plot as in fig. 5. The errors again were calculated with the Jack-knife method. Note that we are able to go down to rn,~/mp= 0.46, while previous calculations [10] (except ref. [3]) stopped a little above mJrnp=0.7. In the additional range we observe a b e n d i n g down ~4 of mN/m o towards its experimental value, as rn,~/rn~, decreases. The ratio mN/ma points towards its physical value as well. We are aware, however, that all our results for raN~r%lie systematically above the predictions of a quark model including hyperfine splitting [ 1 1 ]. One of the possible reasons for this discrepancy is the neglect of dyn a m i c a l fermions. The first 27 configurations of our data sample have been used before in ref. [ 12] to compute hadron masses with staggered fermions and APBC, over a comparable range of rn,~/rnp. There the value of r% extrapolated to the chiral limit was 0.40(4) in lattice units, to be compared with the value 0.193 (I 4) obtained in the present work. This implies that on our coarse lattices the scale has a sizeable dependence on the form of the action we choose to work with. We are currently extending the SC2 analysis to the full data sample a n d a detailed comparison of the SQ3 a n d SC2 results will be presented in a forthcoming paper. We are grateful to the staff of the C o m p u t e r Center for Theoretical Physics at J~ilich for their generous help.
t~4This feature becomes far more pronounced if the exceptional configurationsare included in the analysis.
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7 January 1988
References [1] K.-H. Miitter and K. Schilling,Nucl. Phys. B 230 (FS10) (1984) 275; A. Kt~nig,K.-H. Mfitter and K. Schilling,Phys. Left. B 147 (1984) 145;Nucl. Phys. B259 (1985) 33; A. K6nig,K.-H. Miitter, K. Schillingand J. Smit, Phys. Lett. B 157 (1985) 421; K.-H. M/itter and K. Schilling,in: Advances in lattice gauge theory, eds. D.W. Duke and J.F. Owens (World Scientific, Singapore, 1985); Ph. de Forcrand, A. K6nig, K.-H. Miitter, K. Schillingand R. Sommer, in: Lattice gauge theory - A challenge in large scale computing, Proc. 1985 Workshop (Wuppertal), eds. B. Bunk,K.-H. Mfitterand K. Schilling(Plenum, New York, 1986). [2] O. Haan, E. Schnepf, E. Laermann, K.-H. Miitter, K. Schillingand R. Sommer, Phys. Lett. B 190 (1987) 147. [3] Ph. de Forcrand, A. K/Snig,K.H. MiJtter, K. Schilling and R. Sommer, in: Proc. Intern. Symp. on Lattice gauge theory (Brookhaven, 1986), (Plenum, New York, 1987). [4] R. Cordery, R. Gupta and M. Novotny, Phys. Lett. B 128 (1983) 425; R. Gupta, G. Guralnik, A. Patel, T. Warnock and C. Zemach, Phys. Rev. Len. 53 (1984) 1721; A. Patel and R. Gupta, Nucl. Phys. B 251 (1985) 789. [ 5] K.-H. Miitter et al., in preparation. [6] Y. Oyanagi, Comput. Phys. Commun. 42 (1986) 333. [ 71 R. Gupta and A. Patel, Nucl. Phys. B 226 (1983) 152. [8 ] P. Gibbs, University of Glasgow preprint ( 1985). [9] See e.g.S. Gottlieb, P.B. Mackanzie, H.B. Thacker and D. Weingarten,Nucl. Phys. B 263 (1986) 704. [10] H. Hamber, Phys. Lett. B 178 (1986) 277; S. Itoh, Y. Iwasaki and T. Yoshie, Phys. Lett. B 183 (1987) 351; R. Gupta, G. Guralnik, G. Kilcup,A. Patel and S.R. Sharpc, Cornell preprint, CLNS-87-79 (1987). [ 11 ] S. Ono, Phys. Rev. D 17 (1978) 888. [12] S.R. Sharpe, A. Patel, R. Gupta, G. Guralnik and G.W. Kilcup, Nucl. Phys. B 286 (1987) 253.