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Composite operators in the hopping parameter expansion in the free quark model
Volume 131B, number 1,2,3
PHYSICS LEqq'ERS
10 November 1983
COMPOSITE OPERATORS IN THE HOPPING PARAMETER EXPANSION IN THE FREE QUARK MODEL Z. K U N S Z T 1
CERN, Geneva, Switzerland Received 2 June 1983 I have calculated hopping parameter series of meson and baryon propagators up to O(K32) in the Wilson formulation of the free quark model. The position of branch point singularities has been found with the help of Pad6 approximants. The values of the position of the singularities in K agreed with the exact values within 1-2% in case of mesons and 4-5% in case of baryons. It is argued that in QCD at the cross-over region the systematic errors of the method must be even smaller.
Considerable effort has been devoted recently to hadron spectrum calculations in lattice Q C D [1-9]. As experience has been accumulated it has become clear that longer series in the hopping [4] parameter expansion, larger volumes [7] and better statistics are needed in order to reach a quantitative understanding of the results of the Monte Carlo measurements. Fortunately, with improved computational technique, further development is expected in this direction [10]. If the inverse of the quark propagator is calculated by the Jacobi iterative method, we can generate very long O(K3°-K 4°) series in the hopping parameter expansion with manageable computer time [7]. The presence of a singularity in the hadron propagator in m o m e n t u m space implies a singularity in the hadron propagator in the hopping parameter as well [4]. With long series the Pad6 analysis is expected to become very reliable for the determination of the position and the strength of the lowest singularity. Therefore it is of some interest to obtain a quantitative understanding of the size of systematic errors given by the use of Pad6 approximants. xPart of this work has been done while the author was visiting the Rutherford and Appleton Laboratories, UK.
In Wilson's formulation of lattice Q C D the hadron propagators are very simple functions of the hopping parameter in the large Nc and strong coupling limit. They are ratios of low order polynomials of the hopping parameter [1,4]. Then of course the Pad6 analysis of even low order series gives exact result. In the weak coupling limit, however (g2 = 0), the meson propagators have a quark-antiquark, the baryon propagators have three-quark threshold singularities at K -1 = 6 + 2 exp(Ea/nq) where E is the external energy of the two-point function in the rest frame, nq is the number of quarks and a is the lattice distance. Therefore low order hopping parameter expansion cannot show the correct singularity structure of the amplitude [1]. Weaker singularities require longer series to reveal the radius of the convergence from the behaviour of the expansion coefficients. It is expected that at finite couplings these branch point singularities are converted at higher orders into a geometric series. Therefore, it appears to be justified to assume that if the series are long enough in the free quark model to find the correct singularity structure, then the method is b o u n d to work at finite values of g2 in the cross-over region (1/g 2~- 0.90). With this motivation, I studied long hopping parameter
0 031-9163/83/0000-0000/$03.00 © 1983 North-Holland
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Volume 131B, n u m b e r 1,2,3
PHYSICS L E T T E R S
series of composite operators in the free quark model. The action of the Wilson formulation of the free quark model is given as
s = - X clio rl
+ E K [ ~ . ( r + y ~ ) $ . ÷ ~ , +~+~,(r-%,)lk~]
(1)
n,bt
where n denotes the lattice sites, the ~/,,'s are the quark fields and the y~'s are the euclidean Dirac matrices, r may vary between 0 and 1. The positions of the branch point singularities of the hadron propagators at given K are at E -= nq 1 ln{[(1 - 6Kr)/2K][1 + (1 + 0t)1/2/(1 + r)]}, (2)
10 N o v e m b e r 1983
Pad6 approximant for mesons*Z and of 16/16 Pad6 approximants for baryons (r = 1.0) are given. The Pad6 tables of various approximants [L, M ] show high stability if rnqa > 0.2 for mesons and mqa >t 0.5 for baryons. In table 1 (last column), the exact values calculated with the help of eq. (1) (r = 1.0) are also given. The position of the lowest singularity in K has the same value for 7r, p, p and A. M o r e precisely, the approximate values agree with the exact ones within - 1 . 5 % in case of mesons and - 4 . 5 % in case of baryons. These error bars do not change significantly with the increase of mqa.
In order to see the i m p r o v e m e n t of the result if diagonal Pad6 approximants are used, in table 2 we summarized the K values obtained with
where a = (1 - r2)K2/(1 - 6Kr) 2 . E is the energy of the hadron in the rest frame, nq = 2 for mesons and nq = 3 for baryons. Since the threshold singularities come from the region of the loop m o m e n t u m integration where qi = 0,
q0 = I n [ ( 1 - 6 K ) / 2 K ]
(we use r = 1 for simplicity), the strength of the singularity is the same as in the continuum theory, i.e., we have square root singularities for mesons ,1 ( K ~ - K2) 1/2
(3a)
and weak logarithmic singularities for baryons ( K 2 - K2)2 ln(K 2 - K2).
(3b)
Both the position and strength of the leading (lowest singularity) are spin-independent. The hopping p a r a m e t e r series of hadron propagators have been generated by the Jacobi iterative method [7]. Although a cut is simulated by a multiple pole-zero structure in Pad6 approximants, the position of the lowest singularity gives the position of the branch point with good accuracy. In table 1, the results of 8/8 ,1 If E = 0 but mq ¥ 0 we have no branch cut for the m e s o n amplitudes, but we have a logarithmic singularity [1] in the pion propagator ~ m 21n(m2/A 2) given by the ultraviolet regularization of the loop integral.
174
Table 1 T h e position of the branch point singularities of 7r, p, proton and delta propagators as given by the diagonal Pad6 approximants 8/8 and 16/16 of the O ( K 32) series. T h e last column shows the exact value [see eq. (2) with r = 1]. mqa
K~
K~
Kp
K~
K~x~
0.25 0.50 1.00 2.00
0.119 0.1096 0.0889 0.0489
0.120 0.1094 0.0888 0.0489
0.138 0.1106 0.0924 0.0508
0.135 0.1104 0.0924 0.0507
0.1167 0.1076 0.0874 0.04813
Table 2 Position of the branch point singularities of rr and proton in K given by off-diagonal Pad6 approximants 1/7 and 1/15. At mqa <~0.5 there is no sign of convergence yet for proton. The last column shows the corrected values of Kc [see eq. (6)1. mqa
K'~
K~
Kcxact
K~corr
Kpcorr
0.25 0.50 1.00 2.00
0.121 0.1133 0.0926 0.0507
u. u. 0.0965 0.0544
0.1167 0.1076 0.0874 0.0481
0.115 0.108 0.0876 0.0483
u. u. 0.0875 0.0493
c
,2 It is amusing to remark that the positions of the series of poles and zeros are so close to each other for m e s o n s that the calculation had to be carried out in double precision using an I B M computer, in order to avoid the occurrence of instabilities due to rounding errors in t h e seventh digit of the coefficients.
Volume 131B, number 1,2,3
PHYSICS LETI'ERS
15/1 a n d 31/1 Pad6 a p p r o x i m a n t s for m e s o n s a n d b a r y o n s , respectively. It is also instructive to have a closer look at the coefficients of the series 16
DM(E)= ~
10 November 1983
Table 4 Ratios of the coefficients of the baryon series [see eqs. (4b) and (7b)] at mqa = 1. n
Cn-l/C, Ip
Cn-l/Cnla
W~
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
0.1194 0.1104 0.09458 0.1091 0.10072 0.09956 0.10315 0.09876 0.09972 0.09961 0.09804 0.09834 0.09779 0.09717 0.09714 0.09660
0.1125 0.1089 0.09456 0.10863 0.1001 0.09945 0.10269 0.09847 0.09951 0.09928 0.9783 0.09817 0.09755 0.09699 0.09694 0.09642
0.9246 0.8567 0.8669 1.0832 0.9232 0.9885 0.9574 1.0113 0.9637 0.9842 1.0031 0.9944 0.9937 0.9997 0.9945
32
C , K 2~,
DB(E)=~
h=0
C ' K ~.
(4a,b)
n=0
In table 3, we give the ratios un = (c,,_~/c,,) t/2 for the p i o n a n d r h o ( m q a = 0.5) a n d in t a b l e 4 the ratios u, = cn-t/c,, for the p r o t o n . A s n ~ ~ we have (ratio test) K~ = lim un. T h e c h a n g e in the values of the ratios u~ with the increase of n illustrates the rate of conv e r g e n c e as the o r d e r of the series is increased. If we k n o w the s t r e n g t h of the singularity, we m a y correct the v a l u e of K~n~= u~ for its a s y m p t o t i c value. I n case of (1 - z2) y we h a v e u,, = (C,,_I/C,,) = [1 + (1 + 3')/(n - y)]1/2,
(5a)
w h e r e y = ½for m e s o n s a n d if we c o n s i d e r (1 x) 2 ln(1 - x) we o b t a i n
A t n = 16 for m e s o n s a n d n = 32 for b a r y o n s , we h a v e larger c o r r e c t i o n s for b a r y o n s . T h e c o r r e c t e d values c a l c u l a t e d b y the f o r m u l a e
u~ = C n _ t / C , = 1 + 3 / ( n - 3)
K~
(5b)
which is the e x p e c t e d asymptotics for b a r y o n s . Table 3 Ratios of the coefficients of the meson series [see eqs. (4a) and (7a)] at mqa = 0.5. For comparison, the W, ratios are also given for (1 - x2)1/2. ,
(cn-~lC.yl~l. ( c . _ ~ / c . y % w g
w".
w~. '°°'
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.2346 0.1906 0.1664 0.1497 0.1376 0.1296 0.1245 0.1214 0.1193 0.1178 0.1167 0.1158 0.1150 0.1144 0.1139 0.1134
0.8547 0.8947 0.8204 0.7749 0.8815 0.9801 0.9929 1.0099 1.0052 1.0002 0.9991 0.9974 0.9974 0.9965 0.9969
0.8367 0.8944 0.9949 0.9661 0.9770 0.9834 0.9874 0.9910 0.9921 0.9935 0.9946 0.9954 0.9960 0.9965
0.2656 0.2270 0.2031 0.1666 0.1291 0.1138 0.1116 0.1124 0.1135 0.1141 0.1143 0.1142 0.1139 0.1136 0.1132 0.1129
0.8123 0.8734 0.8993 0.9195 0.9416 0.9610 0.9746 0.9828 0.9876 0.9905 0.9923 0.9935 0.9945 0.9953 0.9959
= (1 - 3 / n ) U 2 K ~ ) ,
(6a)
K fa~o~= (1 - 3 / n ) K ~ n) ,
(6b)
are given in t a b l e 2 (last two columns). W e see that with these c o r r e c t i o n factors we o b t a i n b e t t e r a g r e e m e n t with the exact results t h a n with the use of d i a g o n a l Pad6 a p p r o x i m a n t s . (The e r r o r is less t h a n 0.5% for m e s o n s a n d = 2 % for b a r y o n s . ) W e also c o n s i d e r e d the ratio W n = ( C 2 j C ~ _ t C n ) u2 , W~ = C 2 / C , _ I C n ,
formesons,
for b a r y o n s ,
(7a) (7b)
which give a good m e a s u r e to show b o t h the rate of c o n v e r g e n c e a n d the s t r e n g t h of the l e a d i n g singularity (see tables 3 a n d 4). I n case of 7r a n d p, the s t r e n g t h of the l e a d i n g singularity is c o n s i s t e n t with 3' = 0-6 -+ 0.15 [see eq. (5a)]. T h e s o m e w h a t slower a p p r o a c h to the a s y m p t o t i c v a l u e in case of b a r y o n s is e x p l a i n e d b y the w e a k e r singularity of the b a r y o n prop a g a t o r s a n d the p r e s e n c e of m o r e i m p o r t a n t s u b l e a d i n g c o n t r i b u t i o n s . E v e n for n > 20 we can o b s e r v e the r e m n a n t s of the structure of the 175
Volume 131B, number 1,2,3
PHYSICS LETTERS
Table 5 The coefficients of the hopping parameter series of the effective action at three values of r, r = 1.0, 0.5 and 0.25. Sell = En Self,nK 2n.
n
S~(r = 1.0),
Sell(r= 0.5),
S~(r = 0.25),
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
0.0 0.96 × 102 0.2186 x 104 0.8198 x 105 0.2458 × 10 7 0.7508 × 108 0.2302 × 101° 0.6948 x 10ll 0.2001 x 1013 0.5147 x 1014 0.9456 x 1015 -0.8001 × 1016 -0.2296 × 1019 -0.1804 x 1021 -0.1130 × 1023 -0.6456 x 1024
0.12 × 102 0.15 × 101 -0.64 × 102 -0.2576 × 105 0.2152 x 102 0.6186 x 104 0.4964 x 105 0.1558 × 106 -0.2312 x 107 -0.4066 × 108 -0.2287 × 109 0.1106 x 101° 0.2779 × 1011 0.1325 × 1012 -0.1688 × 1013 -0.3406 × 10TM
0.15 × 102 -0.5166× 102 0.2285 x 103 -0.1134 x l& 0.6128 × 10 4 -0.3580 x 105 0.2257 x 106 -0.1537 × 107 0.1131 x 108 -0.8988 x 108 0.7719 × 109 -0.7132 x 101°
strong coupling limit where only the coefficients C~3×k) (k = 0, 1, 2 . . . . ) a r e n o n - v a n i s h i n g . W e e x p e c t t h a t at i n t e r m e d i a t e v a l u e s of 1/g 2, w h e r e b o t h t h e m e s o n s a n d b a r y o n s h a v e a p o l e as l e a d i n g s i n g u l a r i t y , t h e m e t h o d will b e e v e n m o r e r e l i a b l e . If t h e statistical e r r o r s c a n b e k e p t u n d e r c o n t r o l ,3, t h e c a l c u l a t i o n o f l o n g h o p p i n g p a r a m e t e r s e r i e s will p r o v i d e us w i t h a p o w e r f u l t o o l to o b t a i n t h e v a l u e s of t h e l o w lying h a d r o n m a s s e s in l a t t i c e Q C D in t h e crosso v e r r e g i o n . H e r e I w o u l d l i k e to e m p h a s i z e that the analysis was reliable only above certain m a s s v a l u e s (mqa >10.25 f o r m e s o n s a n d mqa >1 0.5 f o r b a r y o n s ) . It is n o t c l e a r w h a t a r e t h e a l l o w e d m a s s v a l u e s at finite g2 f o r s e r i e s of O(K32).
I also c a r r i e d o u t t h e a n a l y s i s at s m a l l e r v a l u e s of r (r = 0.5, 0.25) a n d it w a s f o u n d t h a t ,3 In QCD at finite coupling y = -1. It is expected that the size of the statistical errors in 3' with 220 configuration will not be larger than Ay ~ 0.3--0.5, at least at larger quark mass values.
176
10 November 1983
subleading effects increase with the decrease of r. T h i s s u g g e s t s t h a t t h e c h o i c e r = 1.0 is p r e f e r r e d if t h e h o p p i n g p a r a m e t e r e x p a n s i o n is u s e d . W e l i s t e d t h e e x p a n s i o n c o e f f i c i e n t s o f Se~ at r = 1.0, 0.5 a n d 0.25 in t a b l e 5 u p to O(K32). It is r e m a r k a b l e t h a t o n e n e e d s o n l y - 3 m i n I B M 3081 C P U c o m p u t e r t i m e to g e n e r a t e t h e e x a c t c o e f f i c i e n t s of t a b l e 5. I a m i n d e b t e d to P. H a s e n f r a t z a n d I. M o n t vay for very useful discussions and correspond e n c e . I also t h a n k t h e T h e o r y G r o u p of t h e Rutherford Laboratory and the Theoretical P h y s i c s D i v i s i o n of C E R N f o r t h e i r k i n d h o s p i tality.
References [1] K.G. Wilson, in: New phenomena in subnuclear physics, ed. A. Zichichi (Plenum, New York, 1977). [2] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792; E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795. [3] D.H. Weingarten, Phys. Lett. B109 (1982) 57, Nucl. Phys. B215 [FS7] (1983) 1. [4] A. Hasenfratz, P. Hasenfratz, Z. Kunszt and C.B. Lang, Phys. Lett. B l l 0 (1982) 282; Bl17 (1982) 81. [5] H. Hamber, E. Marinari, G. Parisi and C. Rebbi, Phys. Lett. B108 (1982) 314. [6] F. Fucito, G. Martinelli, C. Omero, G. Parisi, R. Petronzio and F. Rapuano, Nucl. Phys. B120 [FS6] (1982) 407. [7] P. Hasenfratz and I. Montvay, Phys. Rev. Lett. 50 (1983) 309. [8] H. Hamber and G. Parisi, Brookhaven report BNL 31322 (1982). [9] C. Bernard, T. Draper, K. Olynyk and M. Rushton, Phys. Rev. Lett. 49 (1982) 1076; C. Bernard, T. Draper and K. Olynyk, UCLA preprint (1982); R. Gupta and A. Patel, Caltech preprint (1982); K.C. Bowler, G.S. Pawley, D. Wallace, E. Marinari and F. Rapuano, Edinburgh preprint 82/236 (1982); M. Fukugita, T. Kaneko and A. Ukawa, KEK preprint (1982). [10] H. Lipps, G. Martinelli, R. Petronzio and F. Rapuano, Phys. Lett. 126B (1983) 250.
PHYSICS LEqq'ERS
10 November 1983
COMPOSITE OPERATORS IN THE HOPPING PARAMETER EXPANSION IN THE FREE QUARK MODEL Z. K U N S Z T 1
CERN, Geneva, Switzerland Received 2 June 1983 I have calculated hopping parameter series of meson and baryon propagators up to O(K32) in the Wilson formulation of the free quark model. The position of branch point singularities has been found with the help of Pad6 approximants. The values of the position of the singularities in K agreed with the exact values within 1-2% in case of mesons and 4-5% in case of baryons. It is argued that in QCD at the cross-over region the systematic errors of the method must be even smaller.
Considerable effort has been devoted recently to hadron spectrum calculations in lattice Q C D [1-9]. As experience has been accumulated it has become clear that longer series in the hopping [4] parameter expansion, larger volumes [7] and better statistics are needed in order to reach a quantitative understanding of the results of the Monte Carlo measurements. Fortunately, with improved computational technique, further development is expected in this direction [10]. If the inverse of the quark propagator is calculated by the Jacobi iterative method, we can generate very long O(K3°-K 4°) series in the hopping parameter expansion with manageable computer time [7]. The presence of a singularity in the hadron propagator in m o m e n t u m space implies a singularity in the hadron propagator in the hopping parameter as well [4]. With long series the Pad6 analysis is expected to become very reliable for the determination of the position and the strength of the lowest singularity. Therefore it is of some interest to obtain a quantitative understanding of the size of systematic errors given by the use of Pad6 approximants. xPart of this work has been done while the author was visiting the Rutherford and Appleton Laboratories, UK.
In Wilson's formulation of lattice Q C D the hadron propagators are very simple functions of the hopping parameter in the large Nc and strong coupling limit. They are ratios of low order polynomials of the hopping parameter [1,4]. Then of course the Pad6 analysis of even low order series gives exact result. In the weak coupling limit, however (g2 = 0), the meson propagators have a quark-antiquark, the baryon propagators have three-quark threshold singularities at K -1 = 6 + 2 exp(Ea/nq) where E is the external energy of the two-point function in the rest frame, nq is the number of quarks and a is the lattice distance. Therefore low order hopping parameter expansion cannot show the correct singularity structure of the amplitude [1]. Weaker singularities require longer series to reveal the radius of the convergence from the behaviour of the expansion coefficients. It is expected that at finite couplings these branch point singularities are converted at higher orders into a geometric series. Therefore, it appears to be justified to assume that if the series are long enough in the free quark model to find the correct singularity structure, then the method is b o u n d to work at finite values of g2 in the cross-over region (1/g 2~- 0.90). With this motivation, I studied long hopping parameter
0 031-9163/83/0000-0000/$03.00 © 1983 North-Holland
173
Volume 131B, n u m b e r 1,2,3
PHYSICS L E T T E R S
series of composite operators in the free quark model. The action of the Wilson formulation of the free quark model is given as
s = - X clio rl
+ E K [ ~ . ( r + y ~ ) $ . ÷ ~ , +~+~,(r-%,)lk~]
(1)
n,bt
where n denotes the lattice sites, the ~/,,'s are the quark fields and the y~'s are the euclidean Dirac matrices, r may vary between 0 and 1. The positions of the branch point singularities of the hadron propagators at given K are at E -= nq 1 ln{[(1 - 6Kr)/2K][1 + (1 + 0t)1/2/(1 + r)]}, (2)
10 N o v e m b e r 1983
Pad6 approximant for mesons*Z and of 16/16 Pad6 approximants for baryons (r = 1.0) are given. The Pad6 tables of various approximants [L, M ] show high stability if rnqa > 0.2 for mesons and mqa >t 0.5 for baryons. In table 1 (last column), the exact values calculated with the help of eq. (1) (r = 1.0) are also given. The position of the lowest singularity in K has the same value for 7r, p, p and A. M o r e precisely, the approximate values agree with the exact ones within - 1 . 5 % in case of mesons and - 4 . 5 % in case of baryons. These error bars do not change significantly with the increase of mqa.
In order to see the i m p r o v e m e n t of the result if diagonal Pad6 approximants are used, in table 2 we summarized the K values obtained with
where a = (1 - r2)K2/(1 - 6Kr) 2 . E is the energy of the hadron in the rest frame, nq = 2 for mesons and nq = 3 for baryons. Since the threshold singularities come from the region of the loop m o m e n t u m integration where qi = 0,
q0 = I n [ ( 1 - 6 K ) / 2 K ]
(we use r = 1 for simplicity), the strength of the singularity is the same as in the continuum theory, i.e., we have square root singularities for mesons ,1 ( K ~ - K2) 1/2
(3a)
and weak logarithmic singularities for baryons ( K 2 - K2)2 ln(K 2 - K2).
(3b)
Both the position and strength of the leading (lowest singularity) are spin-independent. The hopping p a r a m e t e r series of hadron propagators have been generated by the Jacobi iterative method [7]. Although a cut is simulated by a multiple pole-zero structure in Pad6 approximants, the position of the lowest singularity gives the position of the branch point with good accuracy. In table 1, the results of 8/8 ,1 If E = 0 but mq ¥ 0 we have no branch cut for the m e s o n amplitudes, but we have a logarithmic singularity [1] in the pion propagator ~ m 21n(m2/A 2) given by the ultraviolet regularization of the loop integral.
174
Table 1 T h e position of the branch point singularities of 7r, p, proton and delta propagators as given by the diagonal Pad6 approximants 8/8 and 16/16 of the O ( K 32) series. T h e last column shows the exact value [see eq. (2) with r = 1]. mqa
K~
K~
Kp
K~
K~x~
0.25 0.50 1.00 2.00
0.119 0.1096 0.0889 0.0489
0.120 0.1094 0.0888 0.0489
0.138 0.1106 0.0924 0.0508
0.135 0.1104 0.0924 0.0507
0.1167 0.1076 0.0874 0.04813
Table 2 Position of the branch point singularities of rr and proton in K given by off-diagonal Pad6 approximants 1/7 and 1/15. At mqa <~0.5 there is no sign of convergence yet for proton. The last column shows the corrected values of Kc [see eq. (6)1. mqa
K'~
K~
Kcxact
K~corr
Kpcorr
0.25 0.50 1.00 2.00
0.121 0.1133 0.0926 0.0507
u. u. 0.0965 0.0544
0.1167 0.1076 0.0874 0.0481
0.115 0.108 0.0876 0.0483
u. u. 0.0875 0.0493
c
,2 It is amusing to remark that the positions of the series of poles and zeros are so close to each other for m e s o n s that the calculation had to be carried out in double precision using an I B M computer, in order to avoid the occurrence of instabilities due to rounding errors in t h e seventh digit of the coefficients.
Volume 131B, number 1,2,3
PHYSICS LETI'ERS
15/1 a n d 31/1 Pad6 a p p r o x i m a n t s for m e s o n s a n d b a r y o n s , respectively. It is also instructive to have a closer look at the coefficients of the series 16
DM(E)= ~
10 November 1983
Table 4 Ratios of the coefficients of the baryon series [see eqs. (4b) and (7b)] at mqa = 1. n
Cn-l/C, Ip
Cn-l/Cnla
W~
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
0.1194 0.1104 0.09458 0.1091 0.10072 0.09956 0.10315 0.09876 0.09972 0.09961 0.09804 0.09834 0.09779 0.09717 0.09714 0.09660
0.1125 0.1089 0.09456 0.10863 0.1001 0.09945 0.10269 0.09847 0.09951 0.09928 0.9783 0.09817 0.09755 0.09699 0.09694 0.09642
0.9246 0.8567 0.8669 1.0832 0.9232 0.9885 0.9574 1.0113 0.9637 0.9842 1.0031 0.9944 0.9937 0.9997 0.9945
32
C , K 2~,
DB(E)=~
h=0
C ' K ~.
(4a,b)
n=0
In table 3, we give the ratios un = (c,,_~/c,,) t/2 for the p i o n a n d r h o ( m q a = 0.5) a n d in t a b l e 4 the ratios u, = cn-t/c,, for the p r o t o n . A s n ~ ~ we have (ratio test) K~ = lim un. T h e c h a n g e in the values of the ratios u~ with the increase of n illustrates the rate of conv e r g e n c e as the o r d e r of the series is increased. If we k n o w the s t r e n g t h of the singularity, we m a y correct the v a l u e of K~n~= u~ for its a s y m p t o t i c value. I n case of (1 - z2) y we h a v e u,, = (C,,_I/C,,) = [1 + (1 + 3')/(n - y)]1/2,
(5a)
w h e r e y = ½for m e s o n s a n d if we c o n s i d e r (1 x) 2 ln(1 - x) we o b t a i n
A t n = 16 for m e s o n s a n d n = 32 for b a r y o n s , we h a v e larger c o r r e c t i o n s for b a r y o n s . T h e c o r r e c t e d values c a l c u l a t e d b y the f o r m u l a e
u~ = C n _ t / C , = 1 + 3 / ( n - 3)
K~
(5b)
which is the e x p e c t e d asymptotics for b a r y o n s . Table 3 Ratios of the coefficients of the meson series [see eqs. (4a) and (7a)] at mqa = 0.5. For comparison, the W, ratios are also given for (1 - x2)1/2. ,
(cn-~lC.yl~l. ( c . _ ~ / c . y % w g
w".
w~. '°°'
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.2346 0.1906 0.1664 0.1497 0.1376 0.1296 0.1245 0.1214 0.1193 0.1178 0.1167 0.1158 0.1150 0.1144 0.1139 0.1134
0.8547 0.8947 0.8204 0.7749 0.8815 0.9801 0.9929 1.0099 1.0052 1.0002 0.9991 0.9974 0.9974 0.9965 0.9969
0.8367 0.8944 0.9949 0.9661 0.9770 0.9834 0.9874 0.9910 0.9921 0.9935 0.9946 0.9954 0.9960 0.9965
0.2656 0.2270 0.2031 0.1666 0.1291 0.1138 0.1116 0.1124 0.1135 0.1141 0.1143 0.1142 0.1139 0.1136 0.1132 0.1129
0.8123 0.8734 0.8993 0.9195 0.9416 0.9610 0.9746 0.9828 0.9876 0.9905 0.9923 0.9935 0.9945 0.9953 0.9959
= (1 - 3 / n ) U 2 K ~ ) ,
(6a)
K fa~o~= (1 - 3 / n ) K ~ n) ,
(6b)
are given in t a b l e 2 (last two columns). W e see that with these c o r r e c t i o n factors we o b t a i n b e t t e r a g r e e m e n t with the exact results t h a n with the use of d i a g o n a l Pad6 a p p r o x i m a n t s . (The e r r o r is less t h a n 0.5% for m e s o n s a n d = 2 % for b a r y o n s . ) W e also c o n s i d e r e d the ratio W n = ( C 2 j C ~ _ t C n ) u2 , W~ = C 2 / C , _ I C n ,
formesons,
for b a r y o n s ,
(7a) (7b)
which give a good m e a s u r e to show b o t h the rate of c o n v e r g e n c e a n d the s t r e n g t h of the l e a d i n g singularity (see tables 3 a n d 4). I n case of 7r a n d p, the s t r e n g t h of the l e a d i n g singularity is c o n s i s t e n t with 3' = 0-6 -+ 0.15 [see eq. (5a)]. T h e s o m e w h a t slower a p p r o a c h to the a s y m p t o t i c v a l u e in case of b a r y o n s is e x p l a i n e d b y the w e a k e r singularity of the b a r y o n prop a g a t o r s a n d the p r e s e n c e of m o r e i m p o r t a n t s u b l e a d i n g c o n t r i b u t i o n s . E v e n for n > 20 we can o b s e r v e the r e m n a n t s of the structure of the 175
Volume 131B, number 1,2,3
PHYSICS LETTERS
Table 5 The coefficients of the hopping parameter series of the effective action at three values of r, r = 1.0, 0.5 and 0.25. Sell = En Self,nK 2n.
n
S~(r = 1.0),
Sell(r= 0.5),
S~(r = 0.25),
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
0.0 0.96 × 102 0.2186 x 104 0.8198 x 105 0.2458 × 10 7 0.7508 × 108 0.2302 × 101° 0.6948 x 10ll 0.2001 x 1013 0.5147 x 1014 0.9456 x 1015 -0.8001 × 1016 -0.2296 × 1019 -0.1804 x 1021 -0.1130 × 1023 -0.6456 x 1024
0.12 × 102 0.15 × 101 -0.64 × 102 -0.2576 × 105 0.2152 x 102 0.6186 x 104 0.4964 x 105 0.1558 × 106 -0.2312 x 107 -0.4066 × 108 -0.2287 × 109 0.1106 x 101° 0.2779 × 1011 0.1325 × 1012 -0.1688 × 1013 -0.3406 × 10TM
0.15 × 102 -0.5166× 102 0.2285 x 103 -0.1134 x l& 0.6128 × 10 4 -0.3580 x 105 0.2257 x 106 -0.1537 × 107 0.1131 x 108 -0.8988 x 108 0.7719 × 109 -0.7132 x 101°
strong coupling limit where only the coefficients C~3×k) (k = 0, 1, 2 . . . . ) a r e n o n - v a n i s h i n g . W e e x p e c t t h a t at i n t e r m e d i a t e v a l u e s of 1/g 2, w h e r e b o t h t h e m e s o n s a n d b a r y o n s h a v e a p o l e as l e a d i n g s i n g u l a r i t y , t h e m e t h o d will b e e v e n m o r e r e l i a b l e . If t h e statistical e r r o r s c a n b e k e p t u n d e r c o n t r o l ,3, t h e c a l c u l a t i o n o f l o n g h o p p i n g p a r a m e t e r s e r i e s will p r o v i d e us w i t h a p o w e r f u l t o o l to o b t a i n t h e v a l u e s of t h e l o w lying h a d r o n m a s s e s in l a t t i c e Q C D in t h e crosso v e r r e g i o n . H e r e I w o u l d l i k e to e m p h a s i z e that the analysis was reliable only above certain m a s s v a l u e s (mqa >10.25 f o r m e s o n s a n d mqa >1 0.5 f o r b a r y o n s ) . It is n o t c l e a r w h a t a r e t h e a l l o w e d m a s s v a l u e s at finite g2 f o r s e r i e s of O(K32).
I also c a r r i e d o u t t h e a n a l y s i s at s m a l l e r v a l u e s of r (r = 0.5, 0.25) a n d it w a s f o u n d t h a t ,3 In QCD at finite coupling y = -1. It is expected that the size of the statistical errors in 3' with 220 configuration will not be larger than Ay ~ 0.3--0.5, at least at larger quark mass values.
176
10 November 1983
subleading effects increase with the decrease of r. T h i s s u g g e s t s t h a t t h e c h o i c e r = 1.0 is p r e f e r r e d if t h e h o p p i n g p a r a m e t e r e x p a n s i o n is u s e d . W e l i s t e d t h e e x p a n s i o n c o e f f i c i e n t s o f Se~ at r = 1.0, 0.5 a n d 0.25 in t a b l e 5 u p to O(K32). It is r e m a r k a b l e t h a t o n e n e e d s o n l y - 3 m i n I B M 3081 C P U c o m p u t e r t i m e to g e n e r a t e t h e e x a c t c o e f f i c i e n t s of t a b l e 5. I a m i n d e b t e d to P. H a s e n f r a t z a n d I. M o n t vay for very useful discussions and correspond e n c e . I also t h a n k t h e T h e o r y G r o u p of t h e Rutherford Laboratory and the Theoretical P h y s i c s D i v i s i o n of C E R N f o r t h e i r k i n d h o s p i tality.
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