- Email: [email protected]
Chiral limit of QCD
Idl[lll=f'~,'|='-i'l¢l[llk~
PROCEEDINGS SUPPLEMENTS EISEVII-].~
Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
Chiral limit of QCD Rajah G u p t a ~ ~T-8 Group, MS B285, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 U. S. A. This talk contains an analysis of quenched chiral perturbation theory and its consequences. The chiral behavior of a number of quantities such as the pion mass ~r~, the Bernard-Golterman ratios R and X, the masses of nucleons, and the kaon B-parameter are examined to see if the singular terms induced by the additional Goldstone boson,,/, are visible ill present data. The overall conclusion (different from what I presented at the lattice meeting) of this analysis is that, with some caveats on the extra terms induced by '1' loops, the standard expressions break down when extrapolating the quenched data with mq < m,/2 to physical light quarks. I then show that due to the single and double poles in the quenched ~l', the axial charge of the proton cannot be calculated using the AdlerBell-Jackiw anomaly condition. I conclude with a review of the status of the calculation of light quark masses fi'om lattice QCD.
1. I N T R O D U C T I O N The main question this review attempts to answer is "should the ostrich care about the alarmists view of quenched QCD" ? The alarmists are two groups, Sharpe, Labrenz, and Zhang [3] [16] [18] and Bernard and Goltel'mall [1] [2]. They have calculated, using quenched chiral perturbation theory, a number of quantities to i-loop and find that in the quenched approximation r/ loops give rise to unphysical terms in the chiral expansion and that in many cases the chiral limit is singular. Also, the coefficients in the chiral expansion (including those of the normal ehiral logs) are different in the full and quenched theories. The ostrich is everyone who wishes to continue using the chiral expansions derived for the real world for extrapolating quenched data to the chiral linfit. The answer is, tmfortunately, YES they should care. The artifacts due to 7/ loops can potentially invalidate all the extrapolations to the chiral limit. The hope is that since these are loop corrections and potentially large only in the limit m,q --+ 0, therefore, there might exist a window in mq where the leading order chiral expansion is valid and sufficient, albeit with coefficients different fi'om those in fidl QCD. Extrapolations of the quenched data from this range to the physical light rr~,~, may prove to be sensible, and the 0920-5632/95/$09.50© 1995Elsevier Science B.V. All rights reserved. SSD! 0920-5632(95)00190-5
difference between the fldl and quenched coefficients taken as a measure of the goodness of the quenched approximation. With this goal in mind I a~lalyze the existing quenched data in the range ms~4 -- rn~ and show that terms induced by the ~1p are already visible and statistically significant. In Section 9 I switch gears and review the status of ~ and m,,. The quenched Wilson fermion data for ~, is almost a factor of two larger, even at [~ = 6.4, tha~l that for quenched staggered or n f = 2 staggered or Wilson fermion data. The estimates of m,~ depend on whether K or K ~ or ~b is used to set the strange scale. These systematic differences are nmch larger than statistical errors and need to be brought under control.
2. Q U E N C H E D CttIRAL TION THEORY
PERTURBA-
Morel [5] gave a Lagrangian description of the quenched theory by introducing ghost quark fields with Bose statistics. This Lagrangian approach has been fltrther developed by BernardGolterman into a calculational schelne. To the order we will be concerned with £1~a is
£BG = ]-g-~stl'[(O~EO,,Et) + 2/t(M~ + MEt) 1 +
- 0 0 , ~ 0 0 , e 0 -,,,.g~g
(1)
where f = f,~ ----- 131 MeV is the pion decay constant, ~; = exp(2iII/f), M is the hermitian
86
R. Gupta/Nuclear Physics B (Proc, Suppl.) 42 (1995) 85 95
Figure 1. The pseudoscalar propagator, (b) the hairpin vertex, and (c) the one bubble contribution to the ~l' propagator in full QCD which after s u m m a t i o n of all diagrams has the form shown. 1
p2+ m 2 _ _ )
(.__
1
m 2
p2+ m 2
1
p2+ m 2 1
) ( ~ ) (
p2+ m 2 + m2
quark mass matrix, tt sets the scale of the mass term, and str is the supertrace over quarks and ghost quarks. T h e last two terms involve the field ~0 = ( r / - ~')/v/2, where ,)' is the ghost (coinmuting spin-i/2) field companion to the 7/'. These terms are treated as interactions and give rise to "hairpin" vertices (see Fig. 1) in the rj' propagator. This introduces two new parameters, m 2 and a m o m e n t u m dependent coupling c~op 2, in the quenched analysis. In the filll theory this vertex and the tower generated by the insertion of bubble diagrams sum to give ~1' its large mass, m02/(1 - c~0), while in the quenched theory the ~1' remains a Goldstone boson and its propagator has a single and double pole. The strength of the vertex, m02, has been cap culated on the lattice by the T s u k u b a Collaboration [4] by taking the ratio of the disconnected to connected diagrams. It has also been determined using its relation to the topological susceptibility ,n 2 = 2 n y x t / f ~ = ,n2/ + rn27- 2,n %
(2)
measured on pure gauge configurations. These methods give 750 < m0 < 1150 M e V . The par a m e t e r that occurs repeatedly in the chiral expansion of quenched quantities is 6 - rn02/24~r2 f 2. Using f~ = 131 M e V and the above estimates for m0 gives 0.14 ~< d < 0.33. Current quenched data supports a value between 0.1 ~< 5 ~< 0.15; different lattice observables give varying estimates due to statistical and systematic errors. Let me first give an intuitive picture of why the rI' p r o p a g a t o r gives extra contributions. The enhanced logs due to the ~' are infrared divergent, so it suffices to consider the p2 = 0 limit in
the 7/ propagator. T h e single pole t e r m is akin to the pion in the fllll theory, 1/rn~, while the double pole t e r m (due to the hairpin vertex di1 agram) is ~1 Wt2"o~-2-Thus any time there is a nol"mai correction t e r m like rn,~Lnm,~ 2 2 fl'om pion loops there will also be a singular t e r m of the rrt~ ?-r 9 ~ 9 forln m~ mTrJ'nrn~r = rnfiLnrn; "~ a L n m 2. This is exactly what one finds in the c,lliral expansion for m,2. Similarly, in the case of rn,~uczco~ the rega and the ~' gives ular chiral correction is o( m,~, an extra term z( m2m,~. My goal is to expose these extra terms in the present lattice d a t a for different observables, and extract 5 from them. Further details on the formulation of the quenched ehiral lagrangian and on the calculation of I-loop corrections are given in Refs. [1] [31 [I61. The I-loop corrections in the fill and quenched theories show t h a t • the expaalsion coefficients are different, • there are enhanced chiral logs, • there are no kaon loops with strange sea quarks, • values for p a r a m e t e r s like f , p,, .. are different in the quenched expressions. I will assmne that this difference is implicit in all subsequent diseussion even when the same symbols are used for the two theories. Before addressing the consequences of these differences for the various physical quantities and their significance in the present data, I would like to mention the difference in the strategies, after l-loop corrections have been calculated, of the two groups of alarmists. I find that knowing their respective emphasis helps in reading their papers. Sharpe and collaborators focus on determining quantities that can be extracted reliably from quenched sinmlations. Using real world values to determine the chiral p a r a m e t e r s (or commonly accepted ones if these are unknown p a r a m e t e r s in x P T ) they require that the chiral corrections are small in both the fldl and quenched expressions, as well as in their difference. Observables satisfying these conditions are the "good" candidates. Bernard and Goltermaal concentrate on testing quenched x P T by forming ratios of quantities which are (a) free of O(p 4) terms in £~pt and (b) independent of the ultraviolet cutoff used to regularize loop integration. The quenched chiral
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
expansion of such ratios have terms proportional to the e x t r a p a r a m e t e r 5. Since these terms can be singular in the chiral limit, it is necessary to a,ssume t h a t there exists a window in m a where the l-loop result is reliable. T h e n 5 can be det e r m i n e d f r o m fits to the quenched expression provided the fits to the quenched and hill theory are significantly different.
87
Figure 2. Fit to staggered m ~ / m q data. At fl -6.0, an estimate of strange quark mass is m~a 0.025 1.85
ln(m
-
2/mq)
7T
mqa
-
KS(16 ~k
3. m~ V E R S U S mq
versus
+ KS(243)
. KS(U2 [] Stag(243)
1.8
Gasser a n d Leutwyler [6] [11] show t h a t in full
QCD m 2 = 2tt'm q
(1 + -~ 1 L(m~)-
6L(mn)+ O(mq) ) ( 3 ) 1.75
where L(rn) = m2Ln(m2/A2)/87r2f2.
Bernard
and aolterman [1] and Sharpe [3] show that these logs are M)sent in the quenched approximation. Instead, for c~0 -- 0, they get
(-,,;)Q "
=
--
0Ln(-~) -
+...).
,2 ?D~q
d co - ~ ,
l-tO
~Lnrnq + Clmq + c2m~ (5)
to extract (~ fl'om a c o l n p e n d i u m of staggered fermioll d a t a at /~ = 6.0 [7][8][9]. Expressing all quantities in lattice units, the best fit gives 2 2 Ln(m~)c~ = 1.04-O.044Lnmq+l.2mq-2.8mq.(6) ~),,q
This implies t h a t 5 ~ 0.053, i.e. m u c h smaller t h a n the value ~ 0.3 based on hill Q C D p a r a m eters; however, the breaking of flavor s y m m e t r y in staggered fernfions has an interesting consequence for this analysis. T h e rf o p e r a t o r is a singlet u n d e r staggered flavor, and different f r o m the Goldstone pion which has flavor 75. TNts one should use the c o r r e s p o n d i n g n o n - G o l d s t o n e pion mass in terms t h a t come from the 7/. In Fig. 2 the fit uses the # (which has flavor 747s) mass in the log term as it is b e t t e r m e a s u r e d and consistent with the flavor singlet case. T h e result is Ln(m~)Q Y/tq
0.02
0.03
0.04
(4)
where A is some typical scale of x S B . This expression has been refined by Sharpe, who s m n m e d up the leading logs for the degenerate case rn~ = m.d = ms. We use his result [10] Ln (rl'r)Q
0.01
= 1.35_0.13Lnm2+l.5m2_2.4m~.(7 )
In this form the coefficient of the Lnmq t e r m is 5. Thus 5 ~ 0.13, a value consistent with the estimate 0.14 based on the calculation m0 -750 M e V . Also note t h a t since the mass of the flavor singlet state, #, does not vanish as mq --} O, therefore, there is no singularity at finite a due to the e n h a n c e d logs. T h e above analysis shows t h a t if one w a n t e d to extract the value of A~ in the expansion m 2 = A,m,,q + .... then the quenched d a t a worfld give a significantly different result depending on the kind of fit used. If one assumes t h a t the 5 d a t a points by the Staggered collaboration [7] represent a window in which x P T is valid and chiral corrections are negligible, i.e. the relation m 2 = A,m,,q is sufficient (as expected at smM1 e n o u g h mq in hill QCD), then one gets m,r2 --- 5.87mq [7], whereas Eq.7 gives A,~ ,-, 3.9, a significantly different value. T h e fit in Eq. 7 shows t h a t over a range of mq, the chiral log and higher order t e r m s can conspire to p r o d u c e a flat region. Finite size effects in r n , increase the value of (m~)Q/mq, so one might a t t r i b u t e the 4% deviation at mq = 0.0025 in Fig. 2 to this artifact. Fortunately, Kim and Sinclair [8] have o b t a i n e d high statistics d a t a for m o = 0.0025, 0.005, 0.01 on lattices of size L = 16, 24, and 32 as shown
R. G u p t a / N u c l e a r
88
P h y s i c s B (Proc. S u p p l . ) 4 2 ( 1 9 9 5 ) 8 5 95
in Fig. 2. There is clear indication of finite size effects on L = 16 lattices, but the near agreement between L = 24 and 32 d a t a confirnls that L = 32 is essentially infinite vohnne. To conclude, the d a t a show that the lowest order chiral expansion has broken down and the effects of r/ logs are manifest for mq < m , / 2 . A similar analysis with Wilson fermions is not yet useflfl because the lowest mq used in sinmlations is ,-, 0.4m~, i.e. the point where staggered fermions just start, to show significant deviations. 4. B E R N A R D - G O L T E R M A N AND f.
RATIO
Figure 3. The Bernard-Golterman ratio R versus the full QCD expression given in E q . l l .
1.04
,,.- 1.02
R 1
The ehiral behavior of f~ in full QCD has been analyzed by Gasser mid Leutwyler [11] to be f,
=
f[1-
L(m.) - }L(mK)
(8)
where L4 and L5 are two O(p 4) constants they introduce. In the quenched theory, with (t0 = 0, Bernard-Golte, rnlan and Sllarpe ge,t = f ( 1 + m,,Ls).
(9)
The absence of pion and kaon chiral logs ill the quenched expression is a 13 - 19% effect (corresponding to the range A = 0 . 7 7 - 1 G e V for the chiral synnnetry breaking scale in L ( m ) ) using flfll QCD parameters[ To reliably compare full and quenched theories without anlbiguities due to tile cutoff A and O (p4) terlns. Bernard-Golternlan construct, in a 4-flavor theory, the ratio ~, ----
f~ fil'f22'
(10)
and m2 = m,2,. The x P T expression for R in the full and quenched theories is where
~D, 1 ---- ~ ' 1 '
1 r 2 - / m , v L [n 7 5 - R e = 1 + -32~2f2 RQ = 1+ 5
mi2
0
0.02
0.04
0.06
X,.n/aan2fz
+
f ( m , , + m d + m.,)L4 + 'm,,Ls]
f,
× LANL
m,~v 2 m~2, ] ?~?'12 + zT~,22,Ln~--r~'12 [J
711i1~'
-
-
1 .
(11)
al.[14] collaborations are shown in Figs. 3 and 4 versus the full and quenched expressions given in E q . l l . Tim slope of the fit to R Q gives 5, while for R E the expected slope is unity. The d a t a favor the quenched expression and give 5 = 0.10(3). The caveat in this case is that the two points at largest X Q are obtained with m2 = 2m~, so one could argue that l-loop x P T is not reliable for these masses. Barring this technicality, I believe that this quantity provides the clemlest determination of 5. 5. B E R N A R D - G O L T E R M A N A N D (~¢)
X
Bernard-Golterman construct a second quantity that is independent of A and O(p 4) terms X - ('a'u)
~-~-2- - - - 7 ~ )
(12)
for which x P T gives Xtrec
XQ
_
n , , ~ - rod,
(13)
?Tts - - ??Lu
=
Xt~cc + 6[Ln m~' 71~,d
where the quantity witllin [] (called X) increases with the mass difference m2 - ml. The quenched Wilson d a t a for R obtained by the L A N L [12], UKQCD[13], and Bernard et
RATIO
XF
- -1
rod-- m,,Lnm~,], TD, s
[M~'+Ln~
I~l, u
7H, s
+
I'D, s - - ?It d ? M2o --M ko Ln--777-].
ms
mu
IVl~r
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
Figure 4. The Bernard-Golterlnan ratio R versus the quenched expression given in Eq.11. 1.06
'
'
'
'
I
'
Slop-
'
'
'
I
. . . .
1
i
1.04
~
Table 1 The Bernard-Goltermaal ratio X Staggered Wilson(GMOR) Wilson(WI)
0.549(30)
0.608(6)
0.614(5)
0.517(14) 0.509(15)
0.620(2) 0.616(2) 0.10(5)
0.620(2) 0.616(2) 0.05(4)
6. C H I R A L E X T R A P O L A T I O N NUCLEON MASS
OF THE
i
6 = 0.10(3)
1.02
~¢ XF
The behavior of baryon masses has been calculated in x P T and has the general form [6]
x LANL o BLS
[] U K Q C D i
i
i
i
I
0.1
i
i
l
i
]
i
i
89
i
0.2
MB
i
0.3
X = [ m ,22 / ( m ,2~ - m a2z ) ] l o g ( m 2n / m a 2a ) - i
To evaluate these expressions requires d a t a for the condensate at three values of ?l%q and pseudoscalar masses for the combinations rr = ?ti& K ° = sd, K + = su. At present only the staggered [7] and Wilson [12] fermion simulations at fl = 6.0 by the LANL collaboration have all the necessary data. Our results for 5 = ( X - X t , ' e e ) / Y , where Y is the factor nmltiplying 5 in the expression for XQ in Eq.14, are given in Table 1. The staggered d a t a have large errors and would give the wrong sign for 5. (I have not taken into account the difference between Goldstone and non-Goldstone mass in terms that come froin r/ loops.) With Wilson fermions the condensate in the chiral limit can be calculated in two ways, using the G M O R relation or the Ward Identity as explained in Ref. [15]. At finite mq there are lattice artifacts which we cannot control, nevertheless, the d a t a give reasonable vahle for a. This is probably fortuitous and I believe that much better d a t a is needed in order to extract 5 from the chiral condensate.
=
-M- + 2 _ ~~ q
(2),~vli2 + E
+ 0 (m4Lnm,)
c i(3)M/3 (14)
where Mi are 7:, K, 7? meson masses. The t e r m proportional to M/3 comes fi-om pion loops and is 25% - 50% of MB for the octet. For example, using the results of Bernard et al. [19] one finds MN = 0.97 + 0.24-- 0.27 respectively for the first three terms in Eq.14. Thus, the loop corrections in individual masses are large and one could question whether x P T is applicable at all to baryons. On the other hand x P T results for lnass differences and the Gellmann-Okubo formula work very well, just as in the quark model. So, it is possible that the loop effects somehow conspire to just shift the overall scale, in which case x P T is usefid and it is worthwhile examining the consequences of the quenched approximation. Labrenz and Sharpe [16] have extended the Lagrangian approach of B e r n a r d - G o l t e r m a n to baryons using the "heavy-quark" formalism of Jenkins mid Manohm" [17]. They show that along with a modification of the ci in Eq.14 one gets a rn02m~ term due to 7?' loops. The quenched expl"ession for degenerate quaxk masses is (assuming <~0 = 7 = 0, where 7 is a p a r a m e t e r in the baryon sector of £~pt and defined in [16])
M s = M + c(1)5M, + c (2) M~ + c(a) M~ + . . . (15) where c (1) ,-, - 2 . 5 , c (2) ,-, 3.4, and c (3) ,-, - 1 . 5 using full QCD values for the parameters. Fits to lattice d a t a using Eq. 15 are not very reliable because the number of light quark masses
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
90
Figure 5. Fit to the L A N L nucleon mass data. 2
. . . .
I . . . .
MN(GeV )
I . . . .
versus
I . . . .
I . . . .
m ~ ( G e V z)
1.8
1.6
mixing. It is one of the best m e a s u r e d lattice quantities. For details of the p h e n o m e n o l o g y and of the lattice m e t h o d o l o g y I refer you to Refs.[2111221123 I. Here, I present a smnmaa-y of just the chiral behavior. Zhang and S h a r p e [18] have calculated the chiral behavior of BK in b o t h the full a n d quenched theories. T h e fidl Q C D result is [23]
BK =-B 1-(3+:~)yLny+by+ce2y+O(y 2) (18) 1.4 where
y
=
0.2
and
=
-
ma)/(rn~ + rod) measures the degeneracy of s and
1.2
1
d quarks. B is the leading order value for BK, which is an input p a r a m e t e r in x P T , and b and c are u n k n o w n constants. T h e quenched result [18]
, , , I , , L , l , t , , I , , , , I , , , ~
0
0.2
0.4
0.6
0.8 BKQ
explored are typically 3 - 4 and only the point at the heaviest mass (typically rnq ~> 2rn~) shows any significant deviation from linearity. I find such 4 - p a r a m e t e r fits to 4 points very unstable. For example, in the case of L A N L d a t a [12], even the different J K smnples give completely different values of c(i). T h e best I could do was to fix one of the p a r a m e t e r s and lnake a 3-parameter fit and then vary the fixed p a r a m e t e r to minimize X2. T h e best fit (obtained by fixing any one of the less well d e t e r m i n e d coefficients, c (1), c (2) or c (3), as one gets the same final result on minimizing X2) to the L A N L d a t a expressed in units of GeV is shown in Fig.5 a n d gives MB = 1.16 -- 0.36M. + 1.6M 2 - 0.5M~.
(16)
A s s u m i n g c O) = - 2 . 5 , Eq.16 gives a ~ 0.14. T h e same m e t h o d applied to "012 sink" d a t a fi'om the G F l l collaboratim~[20] at fl = 5.93 gives ( upd a t e d version of the fit presented in Ref.[16]) MB = 1.18 - 1.0M~ + 3.0M~ - 1.3M~
(17)
which implies t h a t 5 ~ 0.4, and c (2) and c (a) have vahles close to those for full QCD. 7. T H E K A O N
B PARAMETER
T h e kaon B paralneter is a measure of the strong interaction corrections to the K ° - /~0
=
B Q[1 - (3 + /2
e2)yLny + bQ'y + cQc21] -e
2
i -
e
has exactly the same f o r m except for the additional term p r o p o r t i o n a l to & which is an artifact of quenching. T h e t e r m p r o p o r t i o n a l to 5 is singular in the limit e -+ 1, therefore e x t r a p o l a t i o n s of quenched results to the physical n o n - d e g e n e r a t e case are not reliable. For e = 0 this t e r m vanishes, so unless one works close to e ~ 1 (for which there is little incentive in the quenched a p p r o x i m a t i o n ) , it is unlikely t h a t we will, in the foreseeable filture, be able to e x t r a c t 5 using Eq.19. T h e constants B, b, c are different in the full and quenched theories and c a n n o t be fixed by x P T . Assuming B = B (?, the coefficient of the chiral log t e r m is the same for e = 0. This is the best agreement one can expect between the two theories. As a result S h a r p e [23] a d v o c a t e s t h a t BK with degenerate quarks is possibly a "good" quantity to calculate using the q u e n c h e d theory, t h o u g h systematic errors due to use of degenerate quarks are hard to estimate. Using full Q C D values, 3 y L n y ~ 1, so one can ask whether this n o r m a l chiral log is visible in the present d a t a and w h e t h e r it should be included in the extraction of BK? W i t h existing d a t a it is hard to distinguish this t e r m f r o m the one linear in y as the range of rnK is not large e n o u g h to
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95 Figure
6.
hanced
chiral 1
Evidence
I
logs '
'
of
finite
in
Bv.
'
'
size
effects
in
en-
8.
MATRIX
ELEMENT
AXIAL I
'
'
'
'
I
'
'
0.1
CURRENT
I
,
,
0.2
,
,
I
,
0.3
,
,
,
I
,
,
significantly affect the logarithm. Also, there exist data for mq ~ 7~,s/2, so for degenerate quarks (which, as explained above, is the best one can do with the quenched theory) there is no need for an extrapolation. For staggered fermions BK can be written as the sum of two terms, BK = B v + BA, each of which can be analyzed using xPT. These quantities are defined in Ref. [3] and are explicitly constructed such that they do not diverge as 1/m 2 in the chiral limit. Both B v and BA have enhanced logs (terms proportional to Lny and not suppressed by powers of y) that have nothing to do with quenching, i.e. are not due to the ~r. It is these logs, or more precisely the volume dependence of these logs, that has been seen in lattice data. Sharpe [3] has shown that this volume dependence is of the form
B y ( L ) - B~(~)
N I -~~s
=
= - ( B ~ ( L ) - BA(~)) (20) b ~6tt2e-mKL
SINGLET PROTON
lim ¢-4o ¢.
x
where ~ = iff-iffp and s is the proton's spin vector. The hope was that it would be easier to measure the M E of this purely gluonic operator. Since the r / p r o p a g a t o r contributes to this ME at tree level, the question arises whether Eq.21 is valid in the quenched approximation. The answer is NO [27]. Consider the Fourier transform of the anomaly relation
iqu (~', s[ Au(q)]fi, s) = 2mq (fi', s] P I~, s) + Ol s
NI~-~ (fi',sITr FFlfi, s). (22) Each of the three M E in Eq.22 can be parameterized in terms of form-factors as
(P', sl A~(q) Ifi, s) = f d % TsuG A - iq, ftTsuG A, (tiP, sl P Ig,
=
Ifi, s> =
(23)
In the quenched approximation the singularities in these form factors for on-shell M E with respect to q2 and due to the 77' propagators are
G~(q 2)
no 71' poles, a2
a
The constant b2 is not well determined, but the shape of the mK dependence is. The staggered fermion data at [~ = 6.0 on 16a and 243 lattices [24] are shown in Fig. 6 and qualitatively confirm the expected finite size effects in the chiral logs.
OF THE
(fi', s[TrF..P..(q")l~, s) (21)
0.4
mKa
IN
Ever since the measurement of the spin structure of protons using deep inelastic muon scattering from protons by the EMC collaboration[25], there has been much interest in the calculation of the forward matrix elements of the singlet axial current in the proton, (iff,sl(tiTu'~sq[~, s). There axe two possible Wick contractions that contribute to this matrix element (ME). These connected and disconnected diagrams are discussed in [27]. Since the disconnected diagram is hard to measure, Mandula [26] used the anomaly condition to derive the relation
(fi..~lA.lfi, s).~. 0.01
91
(q 2)
-
GP(q2)
_
(q2 _
P2
+
fl
+ F.
(q2 -
GF(q2)
_ (q:
-
al 2 ) + G2, _ m,
+
Pl + ~) (q: - m : . , ) (24)
92
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85 95
Equating the single and double pole terms gives two relations. Using these and taking the double limit, q2 __+ 0 and mq ~ 0, gives 2MpG~4(q 2 = O)
=
-a: + Nf~P
2mq(P2
)
(25
?
--
fi -- --5"- )
The term proportional to Nyc~s/2rr diverges in the chiral limit and there is 11o obvious way of extracting the physical answer from it alone. Thus the m e t h o d fails in the quenched theory. In the fifll theory, there are no (touble poles and all analogous analysis gives 2MpG1A(q 2 = O)
=
--al -[- N f ~ ] ~
= NI
(F- f'
(26)
which justifies the use of the anomaly :'elation. 9.
MASSES
OF
LIGHT
QUARKS
In order to extract light quark masses fl'om lattice simulations we use an ansatz for the chiral behavior of hadron masses. Theoretically, the best defined procedure is ) P T which :'elates the nlasses of p s e u d o s c a l a r mesons t o n~,u, rod, m s . The overall scale p, in the :::ass term of Eq.1 implies that only ratios of quark masses can be determined ::sing x P T . The predictions from x P T for the two independent ratios are [6] [321 (m.~ + md)/2m~ (')?'d -- ?D'u)/?T/'s
Lowest order Next order 1__ ± 215 44
31 29 "
In Lattice QCD it is traditional to make fits to the pseudoscalar spectrum assuming m,~2 = A,~(m: + rn2) and using either m o or f,~ to set the scale. (The expression in Eq.5 is not relevant for this discussion since most quenched simulations have ri?,q ~_ ?)~,s/2.) A consequence of using just the linear term is that the ratio m ~ / N = 25. i.e. these fits can be used to extract either ~t-Z, = (mu nt- rod)~2 or ?)t s by using the pbysical masses for m,. or inK, but not both. (One would get a different lmmber if O(m~) and chiral log terms are included in the :'elation.) Furthermore,
since lattice calculations are done in the isospin limit, m~, = rna, therefore x P T can be used to predict only one quark mass. T h e mass I prefer to extract, barring the complications of quenched x P T , is ~ as it avoids the question whether lowest order x P T is valid up to ms. Akira Ukawa reviewed the status of ~ at LATTICE92 [28] and I present an update on it. To convert lattice results to the continuum M S sche:ne I use 2
,nco,~t(q*) = m~att(o,) [ 1 - ~ ( l o g ( q X a ) - C ~ ) ]
(27)
where the renormalization scale p, is the same as q* (defined in [291) and chosen to be 7r/a., Cm = 2.159 for Wilson [30] and 6.536 for staggered fermions [31], and the rho mass is used to set the scale. (I have not used the tadpole improvement factor of U0 [29] in Cm and intact as this factor cancels in perturbation theory and is a small effect otherwise.) The value of boosted g 2 I use in Eq.27 is [29] 1 _ (plaq) + 0.025 g2 g2
(28)
latt
which is consistent with the continuum M S scheme vahle at Q = ~r/a g2(O)_ 167r2
1 ( ,b':Ln[Ln(x~-~)] ) fl0Ln( h~ ) 1 _Q2_ P02Ln(A~)
(29)
provided I use A = 245 M e V and 1 9 0 M e V for n / = 0 and 2 theories respectively. Note that the choice of A, q*, and the constant 0.025 in Eq.28 are interrelated and, at this order, one can trade changes between them. Finally, all the results are run down to Q = 2 G e V using
re(Q)
{ g2(O).~~ 1 + f ( Q ) - g:(q') ~-i
(
-[0fh)
-2-~0~
) . (30)
The status of calculations of ~ ( 2 G e V ) for quenched Wilson [351 [2O1 [361 [371 [121 [381, quenched staggered [391 [91 [401 [71 [81, dynamical (ni = 2) Wilson [41] [42], and dynaanical (n I = 2) staggered fer:nions [401 [431 [441 is show:,
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95 Figure 7. ,~, e x t r a c t e d using m~ d a t a with the scale set by m,p. ' I
.
.
.
.
I
m (MeV)
X
x
%
X xXXX XX X X
[] X
E]E]
0
i
i
o
,
[]
[]
r7
du
L
x nf=O Wilson [] nf=O Staggered x nf=2 Wilson n¢=2 Staggered I L , , I I , 0.5
J
1
a (CeV -t)
in Fig. 7. I have suppressed error bars as I want to first emphasize key qualitative features. T h e quenched staggered and the n I = 2 Wilson and staggered give ~ = 2 - 3 M e V and are roughly consistent; however, the quenched Wilson results seem to a p p r o a c h t h a t value from above and even at 3 = 6.4 are significantly higher. (The recent result m , = 128(18) M e V by Allton et al. [45] for b o t h Wilson and Sheikholeslami-Wohlert actions at/~ = 6.0 and 6.2 is consistent with the results in Fig. 7 once one notes t h a t ~ = m~/25.) I believe that, at this stage, it is i m p o r t a n t to u n d e r s t a n d why the quenched results with the Wilson action are so different f r o m the rest! A n alternative to using m,q = Z m a s s m ~ = Z ; - l m qL to calculate the quark masses with Wilson fermions is to use the Wal'd identity [34][15]
ZAm,~ ( A 4 ( r ) P ( 0 ) ) ""~ = zP
2
(31)
Using the p e r t u r b a t i v e values for ZA and Z p (with q* = 7r/a and b o o s t e d g2 defined in Eq.28) the L A N L Wilson d a t a [12] gives ~ = 3.53(10} M e V in c o n t r a s t to ~, = 5.15(15) M e V shown in Fig. 7. T h e statistical errors are calculated using a single elimination JK with a salnple
93
of 100 lattices of size 323 x 64, so the difference is significant. T h e R o m e collaboration [45] has found a similar discrepancy and argue t h a t it can be resolved if one uses the n o n - p e r t u r b a t i v e value for Z p , which they a d v o c a t e calculating using m a t r i x elmnents of the operators between q u a r k states in a fixed (Landau) gauge. Their results indicate t h a t p e r t u r b a t i o n theory (including tadpole improvement) fails for Zp. T h e two m e t h o d s for extracting r~--7give consistent results once the n o n - p e r t u r b a t i v e vahm of Zp is used. Having fixed 757 one Call extract re, s, mc, a n d mb using, for exalnple, N ~, D, and B meson masses provided it is a s s u m e d t h a t these masses are linear in the light quark mass and in the heavier quark lnass a r o u n d the physical value. Alternately, one can use m¢, d/g~, and T s p e c t r m n to get these quark masses directly w i t h o u t needing to extrapolate in the light quark mass. T h e results for rnc and m s have been reviewed by Sloan [33] at this conference so I will only analyze the d a t a for m,.~ and c o m p a r e these estimates to 25~,. Note t h a t the same d a t a used to compile Fig. 7 is used to calculate rn,e f r o m 7I~,K. and me. T h e p r o c e d u r e for translating the value to 2 G e V in the M S scheme is also the same. T h e results in Fig. 8 show t h a t t h a t the estimate of m~ from rn,¢ is systematically higher by 15 - 20% c o m p a r e d to 25~7,. I will use the L A N L data[12] to show t h a t the systematic errors due to choice of h a d r o n used to set the scale of the strallge qual'k are now a d o m i n a n t source of error. We find that, in the M S scheme at 2 G e V , rn, = 2 5 ~ = 129(4) M e V using M K , tn,~ = 151(15) M e V using M K . , a n d m~ = 157(13) M e V using Me. N o t e t h a t the latter two estimates give r n ~ / ~ ,,o 30, which is much closer to the "Next Order" prediction of x P T . T h e larger errors in these cases reflect the fact t h a t on the lattice masses of pseudoscalar mesons are m e a s u r e d with nmch b e t t e r statistical a c c u r a c y t h a n those of vector mesons. 10. C O N C L U S I O N S TRACTIONS
AND
COMING
AT-
T h e analysis of various quenched quantities show t h a t the p a r a m e t e r 5 characterizing the
R. Gupta/NuclearPhysics B (Proc. Suppl.) 42 (1995) 85 95
94
Figure 8. Comparison of rn,~ extracted using me and m~ = 25~,. The data are for quenched Wilson simulations.
200
~=m.(rn¢)
1:( }X
x
150 ;~
xx
X xXX X
X
x}~
x
x =25m
X
100 0
0.2
0.4
a
0.6
0.8
(GeV-t)
hairpin vertex in the ~/ propagator lies in the range 0.1 - 0 . 2 . As a result, for mq ~ m~/2 I find significant deviations from the lowest order chiral behavior in m~/mq. Therefore, I conclude that extrapolation of quenched data, obtained with mq < m~/2, to the ehiral limit cannot be done simply using flfll QCD fornnflae for quantities which have large contributions from enhanced logs. For quantities like the matrix elenlent of the singlet axial vector current using the Adler-Bell-Jackiw anomaly, the quenched approximation fails altogether. The alarmists are busy calculating l-loop corrections to other quantities to determine what can be extracted reliably from quenched sinmlations. Bernard and Golterman have extended the results presented at LATTICE93 [46] and calculated chiral corrections to the energy of two pions in a finite box as derived by Lfischer [47]. They find terms at O(1) and 0(1/L2), whose contribution could be substantial, in addition to modifications of the O( 1/L 3) term which is related to the ~r - 7r scattering amplitude [48]. Sharpe and Labrenz have extended the analysis of baryons to include the A decuplet [49]. Booth [50] and Zhang and Sharpe [18] have calculated corrections to heavylight meson properties like fB and BB. These new
results and more data should provide a clearer picture of what is possible with quenched QCD by L A T T I C E 95. In the calculations of light quark masses we need to understand the factor of two difference between the quenched Wilson and staggered data. On the other hand, the quenched staggered data is consistent with the n I = 2 Wilson and staggered data. The analysis presented here leaves open the question - - is the agreement between quenched Wilson (and O(a) improved SW action) data with the phenomenologically favored estimates of i~, (or equivalently rn~) fortuitous and an artifact of strong coupling? If so, then the n I = 0, 2 staggered and n I = 2 Wilson data give all estimate of ~, that is 2 - 3 times smaller than the commonly accepted phenomenological vahle. The systematic errors due to the choice of hadron nlass used in deternfining m~ are significant. Using 1:It,K. or me to extract ms gives a ~ 20% larger value than that obtained from m,g. Even though the statistical errors are larger when extracting m~ from vector nmsons, these estimates provide information beyond the lowest order x P T result rn,~ = 25~-7. Phenomenological estimates involving extrapolation to strange quark mass need to take this systematic difference into account.
11.
ACKNOWLEDGEMENTS
I thank Claude Bernard, Tanmoy Bhattacharya, Maarten Golterlnan, and especially Steve Sharpe for many enlightening discussions. I thank Martin Liischer for reminding me that the 7/ is a staggered flavor singlet. I am grateful to Claude Bernard, Richard Kenway and Don Sinclair for providing unpublished data and to Akira Ukawa for the quark ma~ss data he presented at LATTICE92. The staggered and Wilson fermion calculations by the LANL group have been done as part of the DOE H P C C Grand Challenges program. The recent Wilson fermion sinmlations on 323 x 64 lattices have been done on the CM5 and we gratefully acknowledge the tremendous support provided by the ACL at Los Alamos and by NCSA at Urbana-Champaign.
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
REFERENCES
1. C. Bernard and M. Golterman, Phys. Rev. D46 (1992) 853; LATTICE92, Pg. 217; LATTICE93, Pg. 334; 2. M. Golterman, hep-lat/9405002 and heplat/9411005. 3. S. Sharpe, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 146, Plays. Rev. D41 (1990) 1990, Phys. Rev. D46 (1992) 3146, and "Standard Model hadron Phenomenology and weak decays on the lattice", Ed. G. Martinelli, World Scientific, 1994. 4. Y. Kuramashi, et aJ., Phys. Rev. Lett. 72 (1994) 3448. 5. A. Morel, J. Phys. (Paris) 48 (1987) 1111. 6. J. Gasser, H. Leutwyler, Phys. Rep. C87 (1982) 77. 7. R. Gupta et al., Phys. Rev. D43 (1992) 2003. 8. S. Kim, D. Sinclair, Private conmmnications. 9. Ape Collaboration Phys. Left. 258B (1991) 202. 10. S. Sharpe, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 213. 11. J. Gasser, H. Leutwyler, Nucl. Phys. B250 (1985) 465. 12. T. Bhattacharya and R. Gupta, these proceedings. 13. R. Kenway, Private comnmnications. 14. C. Bernard, Private communications. 15. R. Gupta e t al., Phys. Rev. D46 (1992) 3130. 16. J. Labrenz, S. Sharpe, LATTICE93, Pg335. 17. A. Manohar, E. Jenkins, Phys. Lett. 255B (1991) 558. 18. Y. Zhang and S. Sharpe, ill preparation. 19. V. Bernard et al., Z. Phys.C60 (1993) 111. 20. GF11 collaboration, hep-lat/9405003. 21. S. Sharpe, LATTICE 93, Pg.403. 22. R. Gupta et al., Phys. Rev. D47 (1993) 5113. 23. S. Sharpe, Lectures at 1994 TASI, Boulder, hep-ph/9412243. 24. G. Kilcup, et al., Phys. Rev. Lett. 64 (1990) 25. 25. J. Ashman, et al., Phys. Lett. 2068 (1988) 364; Nucl. Phys. B328 (1989) 1. 26. J.E. Mandula, in Proceedings of ARC Conference on Polarization Dynamics, Trieste (1992) (hep-lat/9206027).
95
27. R. Gupta, J.E. Mandula, Phys. Rev. D50 (1994) 6931. 28. A. Ukawa, LATTICE92, Nucl. Phys. B (Proc. Suppl.) 30 (1992) 3. 29. P. Lepage and P. Mackenzie, Phys. Rev. D48 (1993) 2250. 30. G. Martinelli, Z. Yi-Cheng, Phys. Lett. 123B (1983) 433. 31. M. Golterman, J. Smit, Phys. Lett. 140B (1984) 392. 32. J. Donoglme, B. Holstein, D. Wyler, Phys. Rev. Left. 69 (1992) 3444. 33. J. Sloan, these proceedings. 34. L. Maiaini, G. Martinelli, Phys. Left. 178B (1986) 265. 35. APE collaboration, Phys. Lett. 258B (1991) 195; Nucl. Phys. B376 (1992) 172; Nucl. Phys. 8378 (1992) 616. 36. QCDPAX collaboration, Phys. Lett. 216B (1989) 387. 37. HEMCGC collaboration, Phys. Rev. D46 (1992) 2169. 38. UKQCD collaboration, Nuel. Phys. B407 (1993) 331. 39. APE collaboration, Nucl. Phys. B343 (1990) 228. 40. HEMCGC collaboration, Phys. Rev. D49 (1994) 6026. 41. A. Ukawa, Unpublished. 42. R. Gupta, et al., Phys. Rev. D44 (1991) 3272. 43. Tsukuba Collaboration, Phys. Rev. D50 (1994) 486. 44. Columbia Collaboration, Phys. Rev. Lett. 67 (1991) 1062. 45. C. R. Allton, et al., hep-ph/9406233. 46. C. Bernard et al., Nucl. Phys. B (Proc. Suppl.) 34 (1994) 334. 47. M. Lfischer, Nucl. Phys. B354 (1991) 531. 48. C. Bernard and M. Golterman, in preparation. 49. J. Labrenz and S. Sharpe, in preparation. 50. M. Booth, hep-ph/9411433; hep-ph/9412228.
PROCEEDINGS SUPPLEMENTS EISEVII-].~
Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
Chiral limit of QCD Rajah G u p t a ~ ~T-8 Group, MS B285, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 U. S. A. This talk contains an analysis of quenched chiral perturbation theory and its consequences. The chiral behavior of a number of quantities such as the pion mass ~r~, the Bernard-Golterman ratios R and X, the masses of nucleons, and the kaon B-parameter are examined to see if the singular terms induced by the additional Goldstone boson,,/, are visible ill present data. The overall conclusion (different from what I presented at the lattice meeting) of this analysis is that, with some caveats on the extra terms induced by '1' loops, the standard expressions break down when extrapolating the quenched data with mq < m,/2 to physical light quarks. I then show that due to the single and double poles in the quenched ~l', the axial charge of the proton cannot be calculated using the AdlerBell-Jackiw anomaly condition. I conclude with a review of the status of the calculation of light quark masses fi'om lattice QCD.
1. I N T R O D U C T I O N The main question this review attempts to answer is "should the ostrich care about the alarmists view of quenched QCD" ? The alarmists are two groups, Sharpe, Labrenz, and Zhang [3] [16] [18] and Bernard and Goltel'mall [1] [2]. They have calculated, using quenched chiral perturbation theory, a number of quantities to i-loop and find that in the quenched approximation r/ loops give rise to unphysical terms in the chiral expansion and that in many cases the chiral limit is singular. Also, the coefficients in the chiral expansion (including those of the normal ehiral logs) are different in the full and quenched theories. The ostrich is everyone who wishes to continue using the chiral expansions derived for the real world for extrapolating quenched data to the chiral linfit. The answer is, tmfortunately, YES they should care. The artifacts due to 7/ loops can potentially invalidate all the extrapolations to the chiral limit. The hope is that since these are loop corrections and potentially large only in the limit m,q --+ 0, therefore, there might exist a window in mq where the leading order chiral expansion is valid and sufficient, albeit with coefficients different fi'om those in fidl QCD. Extrapolations of the quenched data from this range to the physical light rr~,~, may prove to be sensible, and the 0920-5632/95/$09.50© 1995Elsevier Science B.V. All rights reserved. SSD! 0920-5632(95)00190-5
difference between the fldl and quenched coefficients taken as a measure of the goodness of the quenched approximation. With this goal in mind I a~lalyze the existing quenched data in the range ms~4 -- rn~ and show that terms induced by the ~1p are already visible and statistically significant. In Section 9 I switch gears and review the status of ~ and m,,. The quenched Wilson fermion data for ~, is almost a factor of two larger, even at [~ = 6.4, tha~l that for quenched staggered or n f = 2 staggered or Wilson fermion data. The estimates of m,~ depend on whether K or K ~ or ~b is used to set the strange scale. These systematic differences are nmch larger than statistical errors and need to be brought under control.
2. Q U E N C H E D CttIRAL TION THEORY
PERTURBA-
Morel [5] gave a Lagrangian description of the quenched theory by introducing ghost quark fields with Bose statistics. This Lagrangian approach has been fltrther developed by BernardGolterman into a calculational schelne. To the order we will be concerned with £1~a is
£BG = ]-g-~stl'[(O~EO,,Et) + 2/t(M~ + MEt) 1 +
- 0 0 , ~ 0 0 , e 0 -,,,.g~g
(1)
where f = f,~ ----- 131 MeV is the pion decay constant, ~; = exp(2iII/f), M is the hermitian
86
R. Gupta/Nuclear Physics B (Proc, Suppl.) 42 (1995) 85 95
Figure 1. The pseudoscalar propagator, (b) the hairpin vertex, and (c) the one bubble contribution to the ~l' propagator in full QCD which after s u m m a t i o n of all diagrams has the form shown. 1
p2+ m 2 _ _ )
(.__
1
m 2
p2+ m 2
1
p2+ m 2 1
) ( ~ ) (
p2+ m 2 + m2
quark mass matrix, tt sets the scale of the mass term, and str is the supertrace over quarks and ghost quarks. T h e last two terms involve the field ~0 = ( r / - ~')/v/2, where ,)' is the ghost (coinmuting spin-i/2) field companion to the 7/'. These terms are treated as interactions and give rise to "hairpin" vertices (see Fig. 1) in the rj' propagator. This introduces two new parameters, m 2 and a m o m e n t u m dependent coupling c~op 2, in the quenched analysis. In the filll theory this vertex and the tower generated by the insertion of bubble diagrams sum to give ~1' its large mass, m02/(1 - c~0), while in the quenched theory the ~1' remains a Goldstone boson and its propagator has a single and double pole. The strength of the vertex, m02, has been cap culated on the lattice by the T s u k u b a Collaboration [4] by taking the ratio of the disconnected to connected diagrams. It has also been determined using its relation to the topological susceptibility ,n 2 = 2 n y x t / f ~ = ,n2/ + rn27- 2,n %
(2)
measured on pure gauge configurations. These methods give 750 < m0 < 1150 M e V . The par a m e t e r that occurs repeatedly in the chiral expansion of quenched quantities is 6 - rn02/24~r2 f 2. Using f~ = 131 M e V and the above estimates for m0 gives 0.14 ~< d < 0.33. Current quenched data supports a value between 0.1 ~< 5 ~< 0.15; different lattice observables give varying estimates due to statistical and systematic errors. Let me first give an intuitive picture of why the rI' p r o p a g a t o r gives extra contributions. The enhanced logs due to the ~' are infrared divergent, so it suffices to consider the p2 = 0 limit in
the 7/ propagator. T h e single pole t e r m is akin to the pion in the fllll theory, 1/rn~, while the double pole t e r m (due to the hairpin vertex di1 agram) is ~1 Wt2"o~-2-Thus any time there is a nol"mai correction t e r m like rn,~Lnm,~ 2 2 fl'om pion loops there will also be a singular t e r m of the rrt~ ?-r 9 ~ 9 forln m~ mTrJ'nrn~r = rnfiLnrn; "~ a L n m 2. This is exactly what one finds in the c,lliral expansion for m,2. Similarly, in the case of rn,~uczco~ the rega and the ~' gives ular chiral correction is o( m,~, an extra term z( m2m,~. My goal is to expose these extra terms in the present lattice d a t a for different observables, and extract 5 from them. Further details on the formulation of the quenched ehiral lagrangian and on the calculation of I-loop corrections are given in Refs. [1] [31 [I61. The I-loop corrections in the fill and quenched theories show t h a t • the expaalsion coefficients are different, • there are enhanced chiral logs, • there are no kaon loops with strange sea quarks, • values for p a r a m e t e r s like f , p,, .. are different in the quenched expressions. I will assmne that this difference is implicit in all subsequent diseussion even when the same symbols are used for the two theories. Before addressing the consequences of these differences for the various physical quantities and their significance in the present data, I would like to mention the difference in the strategies, after l-loop corrections have been calculated, of the two groups of alarmists. I find that knowing their respective emphasis helps in reading their papers. Sharpe and collaborators focus on determining quantities that can be extracted reliably from quenched sinmlations. Using real world values to determine the chiral p a r a m e t e r s (or commonly accepted ones if these are unknown p a r a m e t e r s in x P T ) they require that the chiral corrections are small in both the fldl and quenched expressions, as well as in their difference. Observables satisfying these conditions are the "good" candidates. Bernard and Goltermaal concentrate on testing quenched x P T by forming ratios of quantities which are (a) free of O(p 4) terms in £~pt and (b) independent of the ultraviolet cutoff used to regularize loop integration. The quenched chiral
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
expansion of such ratios have terms proportional to the e x t r a p a r a m e t e r 5. Since these terms can be singular in the chiral limit, it is necessary to a,ssume t h a t there exists a window in m a where the l-loop result is reliable. T h e n 5 can be det e r m i n e d f r o m fits to the quenched expression provided the fits to the quenched and hill theory are significantly different.
87
Figure 2. Fit to staggered m ~ / m q data. At fl -6.0, an estimate of strange quark mass is m~a 0.025 1.85
ln(m
-
2/mq)
7T
mqa
-
KS(16 ~k
3. m~ V E R S U S mq
versus
+ KS(243)
. KS(U2 [] Stag(243)
1.8
Gasser a n d Leutwyler [6] [11] show t h a t in full
QCD m 2 = 2tt'm q
(1 + -~ 1 L(m~)-
6L(mn)+ O(mq) ) ( 3 ) 1.75
where L(rn) = m2Ln(m2/A2)/87r2f2.
Bernard
and aolterman [1] and Sharpe [3] show that these logs are M)sent in the quenched approximation. Instead, for c~0 -- 0, they get
(-,,;)Q "
=
--
0Ln(-~) -
+...).
,2 ?D~q
d co - ~ ,
l-tO
~Lnrnq + Clmq + c2m~ (5)
to extract (~ fl'om a c o l n p e n d i u m of staggered fermioll d a t a at /~ = 6.0 [7][8][9]. Expressing all quantities in lattice units, the best fit gives 2 2 Ln(m~)c~ = 1.04-O.044Lnmq+l.2mq-2.8mq.(6) ~),,q
This implies t h a t 5 ~ 0.053, i.e. m u c h smaller t h a n the value ~ 0.3 based on hill Q C D p a r a m eters; however, the breaking of flavor s y m m e t r y in staggered fernfions has an interesting consequence for this analysis. T h e rf o p e r a t o r is a singlet u n d e r staggered flavor, and different f r o m the Goldstone pion which has flavor 75. TNts one should use the c o r r e s p o n d i n g n o n - G o l d s t o n e pion mass in terms t h a t come from the 7/. In Fig. 2 the fit uses the # (which has flavor 747s) mass in the log term as it is b e t t e r m e a s u r e d and consistent with the flavor singlet case. T h e result is Ln(m~)Q Y/tq
0.02
0.03
0.04
(4)
where A is some typical scale of x S B . This expression has been refined by Sharpe, who s m n m e d up the leading logs for the degenerate case rn~ = m.d = ms. We use his result [10] Ln (rl'r)Q
0.01
= 1.35_0.13Lnm2+l.5m2_2.4m~.(7 )
In this form the coefficient of the Lnmq t e r m is 5. Thus 5 ~ 0.13, a value consistent with the estimate 0.14 based on the calculation m0 -750 M e V . Also note t h a t since the mass of the flavor singlet state, #, does not vanish as mq --} O, therefore, there is no singularity at finite a due to the e n h a n c e d logs. T h e above analysis shows t h a t if one w a n t e d to extract the value of A~ in the expansion m 2 = A,m,,q + .... then the quenched d a t a worfld give a significantly different result depending on the kind of fit used. If one assumes t h a t the 5 d a t a points by the Staggered collaboration [7] represent a window in which x P T is valid and chiral corrections are negligible, i.e. the relation m 2 = A,m,,q is sufficient (as expected at smM1 e n o u g h mq in hill QCD), then one gets m,r2 --- 5.87mq [7], whereas Eq.7 gives A,~ ,-, 3.9, a significantly different value. T h e fit in Eq. 7 shows t h a t over a range of mq, the chiral log and higher order t e r m s can conspire to p r o d u c e a flat region. Finite size effects in r n , increase the value of (m~)Q/mq, so one might a t t r i b u t e the 4% deviation at mq = 0.0025 in Fig. 2 to this artifact. Fortunately, Kim and Sinclair [8] have o b t a i n e d high statistics d a t a for m o = 0.0025, 0.005, 0.01 on lattices of size L = 16, 24, and 32 as shown
R. G u p t a / N u c l e a r
88
P h y s i c s B (Proc. S u p p l . ) 4 2 ( 1 9 9 5 ) 8 5 95
in Fig. 2. There is clear indication of finite size effects on L = 16 lattices, but the near agreement between L = 24 and 32 d a t a confirnls that L = 32 is essentially infinite vohnne. To conclude, the d a t a show that the lowest order chiral expansion has broken down and the effects of r/ logs are manifest for mq < m , / 2 . A similar analysis with Wilson fermions is not yet useflfl because the lowest mq used in sinmlations is ,-, 0.4m~, i.e. the point where staggered fermions just start, to show significant deviations. 4. B E R N A R D - G O L T E R M A N AND f.
RATIO
Figure 3. The Bernard-Golterman ratio R versus the full QCD expression given in E q . l l .
1.04
,,.- 1.02
R 1
The ehiral behavior of f~ in full QCD has been analyzed by Gasser mid Leutwyler [11] to be f,
=
f[1-
L(m.) - }L(mK)
(8)
where L4 and L5 are two O(p 4) constants they introduce. In the quenched theory, with (t0 = 0, Bernard-Golte, rnlan and Sllarpe ge,t = f ( 1 + m,,Ls).
(9)
The absence of pion and kaon chiral logs ill the quenched expression is a 13 - 19% effect (corresponding to the range A = 0 . 7 7 - 1 G e V for the chiral synnnetry breaking scale in L ( m ) ) using flfll QCD parameters[ To reliably compare full and quenched theories without anlbiguities due to tile cutoff A and O (p4) terlns. Bernard-Golternlan construct, in a 4-flavor theory, the ratio ~, ----
f~ fil'f22'
(10)
and m2 = m,2,. The x P T expression for R in the full and quenched theories is where
~D, 1 ---- ~ ' 1 '
1 r 2 - / m , v L [n 7 5 - R e = 1 + -32~2f2 RQ = 1+ 5
mi2
0
0.02
0.04
0.06
X,.n/aan2fz
+
f ( m , , + m d + m.,)L4 + 'm,,Ls]
f,
× LANL
m,~v 2 m~2, ] ?~?'12 + zT~,22,Ln~--r~'12 [J
711i1~'
-
-
1 .
(11)
al.[14] collaborations are shown in Figs. 3 and 4 versus the full and quenched expressions given in E q . l l . Tim slope of the fit to R Q gives 5, while for R E the expected slope is unity. The d a t a favor the quenched expression and give 5 = 0.10(3). The caveat in this case is that the two points at largest X Q are obtained with m2 = 2m~, so one could argue that l-loop x P T is not reliable for these masses. Barring this technicality, I believe that this quantity provides the clemlest determination of 5. 5. B E R N A R D - G O L T E R M A N A N D (~¢)
X
Bernard-Golterman construct a second quantity that is independent of A and O(p 4) terms X - ('a'u)
~-~-2- - - - 7 ~ )
(12)
for which x P T gives Xtrec
XQ
_
n , , ~ - rod,
(13)
?Tts - - ??Lu
=
Xt~cc + 6[Ln m~' 71~,d
where the quantity witllin [] (called X) increases with the mass difference m2 - ml. The quenched Wilson d a t a for R obtained by the L A N L [12], UKQCD[13], and Bernard et
RATIO
XF
- -1
rod-- m,,Lnm~,], TD, s
[M~'+Ln~
I~l, u
7H, s
+
I'D, s - - ?It d ? M2o --M ko Ln--777-].
ms
mu
IVl~r
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
Figure 4. The Bernard-Golterlnan ratio R versus the quenched expression given in Eq.11. 1.06
'
'
'
'
I
'
Slop-
'
'
'
I
. . . .
1
i
1.04
~
Table 1 The Bernard-Goltermaal ratio X Staggered Wilson(GMOR) Wilson(WI)
0.549(30)
0.608(6)
0.614(5)
0.517(14) 0.509(15)
0.620(2) 0.616(2) 0.10(5)
0.620(2) 0.616(2) 0.05(4)
6. C H I R A L E X T R A P O L A T I O N NUCLEON MASS
OF THE
i
6 = 0.10(3)
1.02
~¢ XF
The behavior of baryon masses has been calculated in x P T and has the general form [6]
x LANL o BLS
[] U K Q C D i
i
i
i
I
0.1
i
i
l
i
]
i
i
89
i
0.2
MB
i
0.3
X = [ m ,22 / ( m ,2~ - m a2z ) ] l o g ( m 2n / m a 2a ) - i
To evaluate these expressions requires d a t a for the condensate at three values of ?l%q and pseudoscalar masses for the combinations rr = ?ti& K ° = sd, K + = su. At present only the staggered [7] and Wilson [12] fermion simulations at fl = 6.0 by the LANL collaboration have all the necessary data. Our results for 5 = ( X - X t , ' e e ) / Y , where Y is the factor nmltiplying 5 in the expression for XQ in Eq.14, are given in Table 1. The staggered d a t a have large errors and would give the wrong sign for 5. (I have not taken into account the difference between Goldstone and non-Goldstone mass in terms that come froin r/ loops.) With Wilson fermions the condensate in the chiral limit can be calculated in two ways, using the G M O R relation or the Ward Identity as explained in Ref. [15]. At finite mq there are lattice artifacts which we cannot control, nevertheless, the d a t a give reasonable vahle for a. This is probably fortuitous and I believe that much better d a t a is needed in order to extract 5 from the chiral condensate.
=
-M- + 2 _ ~~ q
(2),~vli2 + E
+ 0 (m4Lnm,)
c i(3)M/3 (14)
where Mi are 7:, K, 7? meson masses. The t e r m proportional to M/3 comes fi-om pion loops and is 25% - 50% of MB for the octet. For example, using the results of Bernard et al. [19] one finds MN = 0.97 + 0.24-- 0.27 respectively for the first three terms in Eq.14. Thus, the loop corrections in individual masses are large and one could question whether x P T is applicable at all to baryons. On the other hand x P T results for lnass differences and the Gellmann-Okubo formula work very well, just as in the quark model. So, it is possible that the loop effects somehow conspire to just shift the overall scale, in which case x P T is usefid and it is worthwhile examining the consequences of the quenched approximation. Labrenz and Sharpe [16] have extended the Lagrangian approach of B e r n a r d - G o l t e r m a n to baryons using the "heavy-quark" formalism of Jenkins mid Manohm" [17]. They show that along with a modification of the ci in Eq.14 one gets a rn02m~ term due to 7?' loops. The quenched expl"ession for degenerate quaxk masses is (assuming <~0 = 7 = 0, where 7 is a p a r a m e t e r in the baryon sector of £~pt and defined in [16])
M s = M + c(1)5M, + c (2) M~ + c(a) M~ + . . . (15) where c (1) ,-, - 2 . 5 , c (2) ,-, 3.4, and c (3) ,-, - 1 . 5 using full QCD values for the parameters. Fits to lattice d a t a using Eq. 15 are not very reliable because the number of light quark masses
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
90
Figure 5. Fit to the L A N L nucleon mass data. 2
. . . .
I . . . .
MN(GeV )
I . . . .
versus
I . . . .
I . . . .
m ~ ( G e V z)
1.8
1.6
mixing. It is one of the best m e a s u r e d lattice quantities. For details of the p h e n o m e n o l o g y and of the lattice m e t h o d o l o g y I refer you to Refs.[2111221123 I. Here, I present a smnmaa-y of just the chiral behavior. Zhang and S h a r p e [18] have calculated the chiral behavior of BK in b o t h the full a n d quenched theories. T h e fidl Q C D result is [23]
BK =-B 1-(3+:~)yLny+by+ce2y+O(y 2) (18) 1.4 where
y
=
0.2
and
=
-
ma)/(rn~ + rod) measures the degeneracy of s and
1.2
1
d quarks. B is the leading order value for BK, which is an input p a r a m e t e r in x P T , and b and c are u n k n o w n constants. T h e quenched result [18]
, , , I , , L , l , t , , I , , , , I , , , ~
0
0.2
0.4
0.6
0.8 BKQ
explored are typically 3 - 4 and only the point at the heaviest mass (typically rnq ~> 2rn~) shows any significant deviation from linearity. I find such 4 - p a r a m e t e r fits to 4 points very unstable. For example, in the case of L A N L d a t a [12], even the different J K smnples give completely different values of c(i). T h e best I could do was to fix one of the p a r a m e t e r s and lnake a 3-parameter fit and then vary the fixed p a r a m e t e r to minimize X2. T h e best fit (obtained by fixing any one of the less well d e t e r m i n e d coefficients, c (1), c (2) or c (3), as one gets the same final result on minimizing X2) to the L A N L d a t a expressed in units of GeV is shown in Fig.5 a n d gives MB = 1.16 -- 0.36M. + 1.6M 2 - 0.5M~.
(16)
A s s u m i n g c O) = - 2 . 5 , Eq.16 gives a ~ 0.14. T h e same m e t h o d applied to "012 sink" d a t a fi'om the G F l l collaboratim~[20] at fl = 5.93 gives ( upd a t e d version of the fit presented in Ref.[16]) MB = 1.18 - 1.0M~ + 3.0M~ - 1.3M~
(17)
which implies t h a t 5 ~ 0.4, and c (2) and c (a) have vahles close to those for full QCD. 7. T H E K A O N
B PARAMETER
T h e kaon B paralneter is a measure of the strong interaction corrections to the K ° - /~0
=
B Q[1 - (3 + /2
e2)yLny + bQ'y + cQc21] -e
2
i -
e
has exactly the same f o r m except for the additional term p r o p o r t i o n a l to & which is an artifact of quenching. T h e t e r m p r o p o r t i o n a l to 5 is singular in the limit e -+ 1, therefore e x t r a p o l a t i o n s of quenched results to the physical n o n - d e g e n e r a t e case are not reliable. For e = 0 this t e r m vanishes, so unless one works close to e ~ 1 (for which there is little incentive in the quenched a p p r o x i m a t i o n ) , it is unlikely t h a t we will, in the foreseeable filture, be able to e x t r a c t 5 using Eq.19. T h e constants B, b, c are different in the full and quenched theories and c a n n o t be fixed by x P T . Assuming B = B (?, the coefficient of the chiral log t e r m is the same for e = 0. This is the best agreement one can expect between the two theories. As a result S h a r p e [23] a d v o c a t e s t h a t BK with degenerate quarks is possibly a "good" quantity to calculate using the q u e n c h e d theory, t h o u g h systematic errors due to use of degenerate quarks are hard to estimate. Using full Q C D values, 3 y L n y ~ 1, so one can ask whether this n o r m a l chiral log is visible in the present d a t a and w h e t h e r it should be included in the extraction of BK? W i t h existing d a t a it is hard to distinguish this t e r m f r o m the one linear in y as the range of rnK is not large e n o u g h to
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95 Figure
6.
hanced
chiral 1
Evidence
I
logs '
'
of
finite
in
Bv.
'
'
size
effects
in
en-
8.
MATRIX
ELEMENT
AXIAL I
'
'
'
'
I
'
'
0.1
CURRENT
I
,
,
0.2
,
,
I
,
0.3
,
,
,
I
,
,
significantly affect the logarithm. Also, there exist data for mq ~ 7~,s/2, so for degenerate quarks (which, as explained above, is the best one can do with the quenched theory) there is no need for an extrapolation. For staggered fermions BK can be written as the sum of two terms, BK = B v + BA, each of which can be analyzed using xPT. These quantities are defined in Ref. [3] and are explicitly constructed such that they do not diverge as 1/m 2 in the chiral limit. Both B v and BA have enhanced logs (terms proportional to Lny and not suppressed by powers of y) that have nothing to do with quenching, i.e. are not due to the ~r. It is these logs, or more precisely the volume dependence of these logs, that has been seen in lattice data. Sharpe [3] has shown that this volume dependence is of the form
B y ( L ) - B~(~)
N I -~~s
=
= - ( B ~ ( L ) - BA(~)) (20) b ~6tt2e-mKL
SINGLET PROTON
lim ¢-4o ¢.
x
where ~ = iff-iffp and s is the proton's spin vector. The hope was that it would be easier to measure the M E of this purely gluonic operator. Since the r / p r o p a g a t o r contributes to this ME at tree level, the question arises whether Eq.21 is valid in the quenched approximation. The answer is NO [27]. Consider the Fourier transform of the anomaly relation
iqu (~', s[ Au(q)]fi, s) = 2mq (fi', s] P I~, s) + Ol s
NI~-~ (fi',sITr FFlfi, s). (22) Each of the three M E in Eq.22 can be parameterized in terms of form-factors as
(P', sl A~(q) Ifi, s) = f d % TsuG A - iq, ftTsuG A, (tiP, sl P Ig,
=
Ifi, s> =
(23)
In the quenched approximation the singularities in these form factors for on-shell M E with respect to q2 and due to the 77' propagators are
G~(q 2)
no 71' poles, a2
a
The constant b2 is not well determined, but the shape of the mK dependence is. The staggered fermion data at [~ = 6.0 on 16a and 243 lattices [24] are shown in Fig. 6 and qualitatively confirm the expected finite size effects in the chiral logs.
OF THE
(fi', s[TrF..P..(q")l~, s) (21)
0.4
mKa
IN
Ever since the measurement of the spin structure of protons using deep inelastic muon scattering from protons by the EMC collaboration[25], there has been much interest in the calculation of the forward matrix elements of the singlet axial current in the proton, (iff,sl(tiTu'~sq[~, s). There axe two possible Wick contractions that contribute to this matrix element (ME). These connected and disconnected diagrams are discussed in [27]. Since the disconnected diagram is hard to measure, Mandula [26] used the anomaly condition to derive the relation
(fi..~lA.lfi, s).~. 0.01
91
(q 2)
-
GP(q2)
_
(q2 _
P2
+
fl
+ F.
(q2 -
GF(q2)
_ (q:
-
al 2 ) + G2, _ m,
+
Pl + ~) (q: - m : . , ) (24)
92
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85 95
Equating the single and double pole terms gives two relations. Using these and taking the double limit, q2 __+ 0 and mq ~ 0, gives 2MpG~4(q 2 = O)
=
-a: + Nf~P
2mq(P2
)
(25
?
--
fi -- --5"- )
The term proportional to Nyc~s/2rr diverges in the chiral limit and there is 11o obvious way of extracting the physical answer from it alone. Thus the m e t h o d fails in the quenched theory. In the fifll theory, there are no (touble poles and all analogous analysis gives 2MpG1A(q 2 = O)
=
--al -[- N f ~ ] ~
= NI
(F- f'
(26)
which justifies the use of the anomaly :'elation. 9.
MASSES
OF
LIGHT
QUARKS
In order to extract light quark masses fl'om lattice simulations we use an ansatz for the chiral behavior of hadron masses. Theoretically, the best defined procedure is ) P T which :'elates the nlasses of p s e u d o s c a l a r mesons t o n~,u, rod, m s . The overall scale p, in the :::ass term of Eq.1 implies that only ratios of quark masses can be determined ::sing x P T . The predictions from x P T for the two independent ratios are [6] [321 (m.~ + md)/2m~ (')?'d -- ?D'u)/?T/'s
Lowest order Next order 1__ ± 215 44
31 29 "
In Lattice QCD it is traditional to make fits to the pseudoscalar spectrum assuming m,~2 = A,~(m: + rn2) and using either m o or f,~ to set the scale. (The expression in Eq.5 is not relevant for this discussion since most quenched simulations have ri?,q ~_ ?)~,s/2.) A consequence of using just the linear term is that the ratio m ~ / N = 25. i.e. these fits can be used to extract either ~t-Z, = (mu nt- rod)~2 or ?)t s by using the pbysical masses for m,. or inK, but not both. (One would get a different lmmber if O(m~) and chiral log terms are included in the :'elation.) Furthermore,
since lattice calculations are done in the isospin limit, m~, = rna, therefore x P T can be used to predict only one quark mass. T h e mass I prefer to extract, barring the complications of quenched x P T , is ~ as it avoids the question whether lowest order x P T is valid up to ms. Akira Ukawa reviewed the status of ~ at LATTICE92 [28] and I present an update on it. To convert lattice results to the continuum M S sche:ne I use 2
,nco,~t(q*) = m~att(o,) [ 1 - ~ ( l o g ( q X a ) - C ~ ) ]
(27)
where the renormalization scale p, is the same as q* (defined in [291) and chosen to be 7r/a., Cm = 2.159 for Wilson [30] and 6.536 for staggered fermions [31], and the rho mass is used to set the scale. (I have not used the tadpole improvement factor of U0 [29] in Cm and intact as this factor cancels in perturbation theory and is a small effect otherwise.) The value of boosted g 2 I use in Eq.27 is [29] 1 _ (plaq) + 0.025 g2 g2
(28)
latt
which is consistent with the continuum M S scheme vahle at Q = ~r/a g2(O)_ 167r2
1 ( ,b':Ln[Ln(x~-~)] ) fl0Ln( h~ ) 1 _Q2_ P02Ln(A~)
(29)
provided I use A = 245 M e V and 1 9 0 M e V for n / = 0 and 2 theories respectively. Note that the choice of A, q*, and the constant 0.025 in Eq.28 are interrelated and, at this order, one can trade changes between them. Finally, all the results are run down to Q = 2 G e V using
re(Q)
{ g2(O).~~ 1 + f ( Q ) - g:(q') ~-i
(
-[0fh)
-2-~0~
) . (30)
The status of calculations of ~ ( 2 G e V ) for quenched Wilson [351 [2O1 [361 [371 [121 [381, quenched staggered [391 [91 [401 [71 [81, dynamical (ni = 2) Wilson [41] [42], and dynaanical (n I = 2) staggered fer:nions [401 [431 [441 is show:,
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95 Figure 7. ,~, e x t r a c t e d using m~ d a t a with the scale set by m,p. ' I
.
.
.
.
I
m (MeV)
X
x
%
X xXXX XX X X
[] X
E]E]
0
i
i
o
,
[]
[]
r7
du
L
x nf=O Wilson [] nf=O Staggered x nf=2 Wilson n¢=2 Staggered I L , , I I , 0.5
J
1
a (CeV -t)
in Fig. 7. I have suppressed error bars as I want to first emphasize key qualitative features. T h e quenched staggered and the n I = 2 Wilson and staggered give ~ = 2 - 3 M e V and are roughly consistent; however, the quenched Wilson results seem to a p p r o a c h t h a t value from above and even at 3 = 6.4 are significantly higher. (The recent result m , = 128(18) M e V by Allton et al. [45] for b o t h Wilson and Sheikholeslami-Wohlert actions at/~ = 6.0 and 6.2 is consistent with the results in Fig. 7 once one notes t h a t ~ = m~/25.) I believe that, at this stage, it is i m p o r t a n t to u n d e r s t a n d why the quenched results with the Wilson action are so different f r o m the rest! A n alternative to using m,q = Z m a s s m ~ = Z ; - l m qL to calculate the quark masses with Wilson fermions is to use the Wal'd identity [34][15]
ZAm,~ ( A 4 ( r ) P ( 0 ) ) ""~ = zP
2
(31)
Using the p e r t u r b a t i v e values for ZA and Z p (with q* = 7r/a and b o o s t e d g2 defined in Eq.28) the L A N L Wilson d a t a [12] gives ~ = 3.53(10} M e V in c o n t r a s t to ~, = 5.15(15) M e V shown in Fig. 7. T h e statistical errors are calculated using a single elimination JK with a salnple
93
of 100 lattices of size 323 x 64, so the difference is significant. T h e R o m e collaboration [45] has found a similar discrepancy and argue t h a t it can be resolved if one uses the n o n - p e r t u r b a t i v e value for Z p , which they a d v o c a t e calculating using m a t r i x elmnents of the operators between q u a r k states in a fixed (Landau) gauge. Their results indicate t h a t p e r t u r b a t i o n theory (including tadpole improvement) fails for Zp. T h e two m e t h o d s for extracting r~--7give consistent results once the n o n - p e r t u r b a t i v e vahm of Zp is used. Having fixed 757 one Call extract re, s, mc, a n d mb using, for exalnple, N ~, D, and B meson masses provided it is a s s u m e d t h a t these masses are linear in the light quark mass and in the heavier quark lnass a r o u n d the physical value. Alternately, one can use m¢, d/g~, and T s p e c t r m n to get these quark masses directly w i t h o u t needing to extrapolate in the light quark mass. T h e results for rnc and m s have been reviewed by Sloan [33] at this conference so I will only analyze the d a t a for m,.~ and c o m p a r e these estimates to 25~,. Note t h a t the same d a t a used to compile Fig. 7 is used to calculate rn,e f r o m 7I~,K. and me. T h e p r o c e d u r e for translating the value to 2 G e V in the M S scheme is also the same. T h e results in Fig. 8 show t h a t t h a t the estimate of m~ from rn,¢ is systematically higher by 15 - 20% c o m p a r e d to 25~7,. I will use the L A N L data[12] to show t h a t the systematic errors due to choice of h a d r o n used to set the scale of the strallge qual'k are now a d o m i n a n t source of error. We find that, in the M S scheme at 2 G e V , rn, = 2 5 ~ = 129(4) M e V using M K , tn,~ = 151(15) M e V using M K . , a n d m~ = 157(13) M e V using Me. N o t e t h a t the latter two estimates give r n ~ / ~ ,,o 30, which is much closer to the "Next Order" prediction of x P T . T h e larger errors in these cases reflect the fact t h a t on the lattice masses of pseudoscalar mesons are m e a s u r e d with nmch b e t t e r statistical a c c u r a c y t h a n those of vector mesons. 10. C O N C L U S I O N S TRACTIONS
AND
COMING
AT-
T h e analysis of various quenched quantities show t h a t the p a r a m e t e r 5 characterizing the
R. Gupta/NuclearPhysics B (Proc. Suppl.) 42 (1995) 85 95
94
Figure 8. Comparison of rn,~ extracted using me and m~ = 25~,. The data are for quenched Wilson simulations.
200
~=m.(rn¢)
1:( }X
x
150 ;~
xx
X xXX X
X
x}~
x
x =25m
X
100 0
0.2
0.4
a
0.6
0.8
(GeV-t)
hairpin vertex in the ~/ propagator lies in the range 0.1 - 0 . 2 . As a result, for mq ~ m~/2 I find significant deviations from the lowest order chiral behavior in m~/mq. Therefore, I conclude that extrapolation of quenched data, obtained with mq < m~/2, to the ehiral limit cannot be done simply using flfll QCD fornnflae for quantities which have large contributions from enhanced logs. For quantities like the matrix elenlent of the singlet axial vector current using the Adler-Bell-Jackiw anomaly, the quenched approximation fails altogether. The alarmists are busy calculating l-loop corrections to other quantities to determine what can be extracted reliably from quenched sinmlations. Bernard and Golterman have extended the results presented at LATTICE93 [46] and calculated chiral corrections to the energy of two pions in a finite box as derived by Lfischer [47]. They find terms at O(1) and 0(1/L2), whose contribution could be substantial, in addition to modifications of the O( 1/L 3) term which is related to the ~r - 7r scattering amplitude [48]. Sharpe and Labrenz have extended the analysis of baryons to include the A decuplet [49]. Booth [50] and Zhang and Sharpe [18] have calculated corrections to heavylight meson properties like fB and BB. These new
results and more data should provide a clearer picture of what is possible with quenched QCD by L A T T I C E 95. In the calculations of light quark masses we need to understand the factor of two difference between the quenched Wilson and staggered data. On the other hand, the quenched staggered data is consistent with the n I = 2 Wilson and staggered data. The analysis presented here leaves open the question - - is the agreement between quenched Wilson (and O(a) improved SW action) data with the phenomenologically favored estimates of i~, (or equivalently rn~) fortuitous and an artifact of strong coupling? If so, then the n I = 0, 2 staggered and n I = 2 Wilson data give all estimate of ~, that is 2 - 3 times smaller than the commonly accepted phenomenological vahle. The systematic errors due to the choice of hadron nlass used in deternfining m~ are significant. Using 1:It,K. or me to extract ms gives a ~ 20% larger value than that obtained from m,g. Even though the statistical errors are larger when extracting m~ from vector nmsons, these estimates provide information beyond the lowest order x P T result rn,~ = 25~-7. Phenomenological estimates involving extrapolation to strange quark mass need to take this systematic difference into account.
11.
ACKNOWLEDGEMENTS
I thank Claude Bernard, Tanmoy Bhattacharya, Maarten Golterlnan, and especially Steve Sharpe for many enlightening discussions. I thank Martin Liischer for reminding me that the 7/ is a staggered flavor singlet. I am grateful to Claude Bernard, Richard Kenway and Don Sinclair for providing unpublished data and to Akira Ukawa for the quark ma~ss data he presented at LATTICE92. The staggered and Wilson fermion calculations by the LANL group have been done as part of the DOE H P C C Grand Challenges program. The recent Wilson fermion sinmlations on 323 x 64 lattices have been done on the CM5 and we gratefully acknowledge the tremendous support provided by the ACL at Los Alamos and by NCSA at Urbana-Champaign.
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
REFERENCES
1. C. Bernard and M. Golterman, Phys. Rev. D46 (1992) 853; LATTICE92, Pg. 217; LATTICE93, Pg. 334; 2. M. Golterman, hep-lat/9405002 and heplat/9411005. 3. S. Sharpe, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 146, Plays. Rev. D41 (1990) 1990, Phys. Rev. D46 (1992) 3146, and "Standard Model hadron Phenomenology and weak decays on the lattice", Ed. G. Martinelli, World Scientific, 1994. 4. Y. Kuramashi, et aJ., Phys. Rev. Lett. 72 (1994) 3448. 5. A. Morel, J. Phys. (Paris) 48 (1987) 1111. 6. J. Gasser, H. Leutwyler, Phys. Rep. C87 (1982) 77. 7. R. Gupta et al., Phys. Rev. D43 (1992) 2003. 8. S. Kim, D. Sinclair, Private conmmnications. 9. Ape Collaboration Phys. Left. 258B (1991) 202. 10. S. Sharpe, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 213. 11. J. Gasser, H. Leutwyler, Nucl. Phys. B250 (1985) 465. 12. T. Bhattacharya and R. Gupta, these proceedings. 13. R. Kenway, Private comnmnications. 14. C. Bernard, Private communications. 15. R. Gupta e t al., Phys. Rev. D46 (1992) 3130. 16. J. Labrenz, S. Sharpe, LATTICE93, Pg335. 17. A. Manohar, E. Jenkins, Phys. Lett. 255B (1991) 558. 18. Y. Zhang and S. Sharpe, ill preparation. 19. V. Bernard et al., Z. Phys.C60 (1993) 111. 20. GF11 collaboration, hep-lat/9405003. 21. S. Sharpe, LATTICE 93, Pg.403. 22. R. Gupta et al., Phys. Rev. D47 (1993) 5113. 23. S. Sharpe, Lectures at 1994 TASI, Boulder, hep-ph/9412243. 24. G. Kilcup, et al., Phys. Rev. Lett. 64 (1990) 25. 25. J. Ashman, et al., Phys. Lett. 2068 (1988) 364; Nucl. Phys. B328 (1989) 1. 26. J.E. Mandula, in Proceedings of ARC Conference on Polarization Dynamics, Trieste (1992) (hep-lat/9206027).
95
27. R. Gupta, J.E. Mandula, Phys. Rev. D50 (1994) 6931. 28. A. Ukawa, LATTICE92, Nucl. Phys. B (Proc. Suppl.) 30 (1992) 3. 29. P. Lepage and P. Mackenzie, Phys. Rev. D48 (1993) 2250. 30. G. Martinelli, Z. Yi-Cheng, Phys. Lett. 123B (1983) 433. 31. M. Golterman, J. Smit, Phys. Lett. 140B (1984) 392. 32. J. Donoglme, B. Holstein, D. Wyler, Phys. Rev. Left. 69 (1992) 3444. 33. J. Sloan, these proceedings. 34. L. Maiaini, G. Martinelli, Phys. Left. 178B (1986) 265. 35. APE collaboration, Phys. Lett. 258B (1991) 195; Nucl. Phys. B376 (1992) 172; Nucl. Phys. 8378 (1992) 616. 36. QCDPAX collaboration, Phys. Lett. 216B (1989) 387. 37. HEMCGC collaboration, Phys. Rev. D46 (1992) 2169. 38. UKQCD collaboration, Nuel. Phys. B407 (1993) 331. 39. APE collaboration, Nucl. Phys. B343 (1990) 228. 40. HEMCGC collaboration, Phys. Rev. D49 (1994) 6026. 41. A. Ukawa, Unpublished. 42. R. Gupta, et al., Phys. Rev. D44 (1991) 3272. 43. Tsukuba Collaboration, Phys. Rev. D50 (1994) 486. 44. Columbia Collaboration, Phys. Rev. Lett. 67 (1991) 1062. 45. C. R. Allton, et al., hep-ph/9406233. 46. C. Bernard et al., Nucl. Phys. B (Proc. Suppl.) 34 (1994) 334. 47. M. Lfischer, Nucl. Phys. B354 (1991) 531. 48. C. Bernard and M. Golterman, in preparation. 49. J. Labrenz and S. Sharpe, in preparation. 50. M. Booth, hep-ph/9411433; hep-ph/9412228.