~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 Adler- Bell-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 perturba- tion 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 expan- sion and that in many cases the chiral limit is sin- gular. Also, the coefficients in the chiral expan- sion (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 chi- ral 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 cor- rections 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 dif- ferent fi'om those in fidl QCD. Extrapolations of the quenched data from this range to the phys- ical light rr~,~, may prove to be sensible, and the
difference between the fldl and quenched coeffi- cients 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 ~1 p are already visible and statistically significant.
In Section 9 I switch gears and review the sta- tus 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 C t t I R A L P E R T U R B A - T I O N T H E O R Y
Morel [5] gave a Lagrangian description of the quenched theory by introducing ghost quark fields with Bose statistics. This Lagrangian ap- proach has been fltrther developed by Bernard- Golterman 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 contribu- tion to the ~l' propagator in full QCD which after summat ion of all diagrams has the form shown.
1
p2+ m 2
_ _ ) ( . _ _ 1 m 2 1
p2+ m 2 p2+ m 2
1
) ( ~ ) ( m 2 p2+ + m 2
quark mass matr ix, tt sets the scale of the mass term, and str is the supertrace over quarks and ghost quarks. The last two terms involve the field ~0 = ( r / - ~ ' ) /v/2, where ,)' is the ghost (coin- muting spin- i /2) field companion to the 7/'. These terms are t reated as interactions and give rise to "hairpin" vertices (see Fig. 1) in the rj' propa- gator. 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 ver- tex 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 s t rength of the vertex, m0 2, has been cap culated on the lattice by the Tsukuba Collabora- tion [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 pa- rameter that occurs repeatedly in the chiral ex- pansion 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; differ- ent lattice observables give varying estimates due to statistical and systematic errors.
Let me first give an intuitive picture of why the r I' propagator gives extra contributions. The enhanced logs due to the ~' are infrared diver- gent, so it suffices to consider the p2 = 0 limit in
the 7/ propagator. The single pole te rm is akin to the pion in the fllll theory, 1/rn~, while the double pole te rm (due to the hairpin vertex di-
1 Wt2 1 agram) is ~ "o~-2-- Thus any t ime there is a
nol"mai correction te rm like 2 2 rn,~Lnm,~ fl'om pion loops there will also be a singular t e rm of the
rrt~ ? - r 9 ~ 9 rnfiLnrn; aLnm 2. This is forln m~ mTrJ'nrn~r = "~
exactly what one finds in the c, lliral expansion for m, 2. Similarly, in the case of rn,~uczco~ the reg-
a 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 da ta for different observables, and extract 5 from them.
Further details on the formulation of the quenched ehiral lagrangian and on the cal- culation of I-loop corrections are given in Refs. [1] [31 [I61. The I-loop corrections in the fil l and quenched theories show tha t • the expaalsion coefficients are different, • there are enhanced chiral logs, • there are no kaon loops with strange sea quarks, • values for parameters like f , p,, .. are different in the quenched expressions. I will assmne that this difference is implicit in all subsequent dis- eussion even when the same symbols are used for the two theories. Before addressing the conse- quences of these differences for the various physi- cal quantities and their significance in the present data, I would like to mention the difference in the strategies, after l-loop corrections have been cal- culated, 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 parameters (or commonly accepted ones if these are unknown paramete rs in xPT) they require that the chiral corrections are small in both the fldl and quenched expressions, as well as in their difference. Observables satisfy- ing 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 87
expansion of such rat ios have terms propor t ional to the ex t ra pa rame te r 5. Since these terms can be singular in the chiral limit, it is necessary to a,ssume tha t there exists a window in m a where the l - loop result is reliable. Then 5 can be de- te rmined f rom fits to the quenched expression provided the fits to the quenched and hill theory are significantly different.
3. m~ V E R S U S mq
Gasser and Leutwyler [6] [11] show tha t in full QCD
m 2 = 2 t t ' m q (1 + 1 -~ L(m~) - 6L(mn) + O(mq) ) ( 3 )
where L(rn) = m2Ln(m2/A2)/87r2f2. Bernard and ao l t e rman [1] and Sharpe [3] show that these logs are M)sent in the quenched approximat ion. Instead, for c~0 -- 0, they get
(-,,;)Q - " = - - 0 L n ( - ~ ) + . . . ) . (4)
where A is some typical scale of x S B . This ex- pression has been refined by Sharpe, who smnmed up the leading logs for the degenerate case rn~ = m.d = ms. We use his result [10]
Ln (rl ' r)Q ,2 d
co - ~ , ~Lnrnq + Clmq + c2m~ (5) ?D~q l - t O
to ext rac t (~ fl 'om a co lnpendium of s taggered fermioll d a t a at /~ = 6.0 [7][8][9]. Expressing all quanti t ies in lat t ice units, the best fit gives
2 2 Ln(m~)c~ = 1.04-O.044Lnmq+l .2mq-2 .8mq.(6)
~),,q
This implies tha t 5 ~ 0.053, i.e. much smaller than the value ~ 0.3 based on hill QCD param- eters; however, the breaking of flavor s y m m e t r y in s taggered fernfions has an interesting conse- quence for this analysis. T he rf opera tor is a singlet under s taggered flavor, and different f rom the Golds tone pion which has flavor 75. TNts one should use the cor responding non-Golds tone pion mass in terms tha t come f rom the 7/. In Fig. 2 the fit uses the # (which has flavor 747s) mass in the log term as it is be t t e r measured and consistent with the flavor singlet case. The result is
Ln (m~)Q = 1 . 3 5 _ 0 . 1 3 L n m 2 + l . 5 m 2 _ 2 . 4 m ~ . ( 7 ) Y/tq
Figure 2. Fit to s taggered m ~ / m q data . At fl -- 6.0, an es t imate of s t range quark mass is m~a 0.025
l n ( m 2 / m q ) m q a - 1.85 - v e r s u s 7T
1.8
1.75
KS(16 ~ k + K S ( 2 4 3 )
. KS(U2 [] Stag(243)
0.01 0.02 0.03 0.04
In this form the coefficient of the Lnmq te rm is 5. Thus 5 ~ 0.13, a value consistent with the es t imate 0.14 based on the calculat ion m0 -- 750 M e V . Also note tha t since the mass of the flavor singlet state, #, does not vanish as mq --} O, therefore, there is no singulari ty at finite a due to the enhanced logs.
The above analysis shows tha t if one wanted to ext rac t 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 tha t the 5 d a t a points by the Staggered col laborat ion [7] repre- sent 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 enough
2 5.87mq [7], mq in hill QCD), then one gets m,r --- whereas Eq.7 gives A,~ ,-, 3.9, a significantly dif- ferent value. The fit in Eq. 7 shows tha t over a range of mq, the chiral log and higher order te rms can conspire to produce a flat region.
Finite size effects in rn , increase the value of (m~)Q/mq, so one might a t t r ibu te the 4% devi- at ion at mq = 0.0025 in Fig. 2 to this art ifact . Fortunately, Kim and Sinclair [8] have obta ined high statist ics d a t a for m o = 0.0025, 0.005, 0.01 on lattices of size L = 16, 24, and 32 as shown
88 R. G u p t a / N u c l e a r P h y s i c s B (Proc. Supp l . ) 42 (1995) 85 95
in Fig. 2. There is clear indication of finite size effects on L = 16 lattices, but the near agree- ment between L = 24 and 32 da ta confirnls that L = 32 is essentially infinite vohnne. To conclude, the da ta show that the lowest order chiral expan- sion 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 R A T I O R A N D f .
The ehiral behavior of f~ in full QCD has been analyzed by Gasser mid Leutwyler [11] to be
f , = f [ 1 - L ( m . ) - } L ( m K ) +
f (m, , + m d + m.,)L4 + 'm,,Ls] (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 , = f (1 + m,,Ls). (9)
The absence of pion and kaon chiral logs ill the quenched expression is a 13 - 19% effect (corre- sponding 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 ~ (10) f i l ' f22 '
w h e r e ~D, 1 ---- ~ ' 1 ' and m2 = m,2,. The x P T ex- pression for R in the full and quenched theories is
1 r 2 m,~v 2 m~2, ] = - - / m , v L n 7 5 - - + zT~,22,Ln~--- [ R e 1 + 32~2f2 [ ?~?'12 r~'12 J
R Q = 1 + 5 mi2 711i1~' - - 1 . (11)
where the quanti ty witllin [] (called X) increases with the mass difference m2 - ml .
The quenched Wilson da ta for R obtained by the LANL [12], UKQCD[13], and Bernard et
Figure 3. The Bernard-Gol terman ratio R versus the full QCD expression given in Eq . l l .
1.04
,,.- 1 .02
1 × LANL
0 0 . 0 2 0 . 0 4 0 . 0 6 X,.n/aan2f z
al.[14] collaborations are shown in Figs. 3 and 4 versus the full and quenched expressions given in Eq . l l . Tim slope of the fit to R Q gives 5, while for R E the expected slope is unity. The da ta 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 quanti ty provides the clemlest determination of 5.
5. B E R N A R D - G O L T E R M A N R A T I O X A N D (~¢)
Bernard-Gol terman construct a second quan- tity that is independent of A and O(p 4) terms
X - ('a'u) ~-~-2 - - - - 7 ~ ) (12)
for which x P T gives
X t r e c
XQ
XF
_ n , , ~ - rod, (13) ?Tts - - ??Lu
= Xt~cc + 6[Ln m~' rod-- m, ,Lnm~,] , 71~, d TD, s I~l, u 7H, s
- - 1 [ M ~ ' + L n ~ +
I'D, s - - ?It d ? M 2 o - - - - M k o L n - - 7 7 7 - ] . ms mu IVl~r
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95 89
Figure 4. The Bernard-Golter lnan ratio R versus the quenched expression given in Eq.11.
1.06
1.04
' ' ' ' I
Slop- ' ' ' ' I . . . . i
6 = 0.10(3) 1
i
x LANL
o BLS
[] UKQCD
i i l i ] i i i i
0.2 0.3
~ 1.02
i i i i I
0.1 2 2 2 2 2 X = [m,2 / (m,~-maz)] log(mn/maa)- i
To evaluate these expressions requires da ta for the condensate at three values of ?l%q and pseu- doscalar masses for the combinations rr = ?ti&
K ° = sd, K + = su. At present only the stag- gered [7] and Wilson [12] fermion simulations at fl = 6.0 by the LANL collaboration have all the necessary data. Our results for 5 = (X-Xt , ' ee ) /Y , where Y is the factor nmltiplying 5 in the expres- sion for XQ in Eq.14, are given in Table 1.
The staggered da ta 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 Identi ty as explained in Ref. [15]. At finite mq there are lattice artifacts which we cannot control, never- theless, the da ta give reasonable vahle for a. This is probably fortuitous and I believe that much bet ter da ta is needed in order to extract 5 from the chiral condensate.
6. C H I R A L E X T R A P O L A T I O N O F T H E
N U C L E O N M A S S
The behavior of baryon masses has been calcu- lated in x P T and has the general form [6]
- - ~ (2), 2 + E (3)M/3 MB = M + 2 _ ~ q ~vli c i
+ 0 ( m 4 L n m , ) (14)
where Mi are 7:, K, 7? meson masses. The te rm proport ional to M/3 comes fi-om pion loops and is 25% - 50% of MB for the octet. For exam- ple, 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 cor- rections 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 Gel lmann-Okubo for- mula 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 Bernard-Gol terman 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 ex- pl"ession for degenerate quaxk masses is (assuming <~0 = 7 = 0, where 7 is a paramete 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 da ta using Eq. 15 are not very reliable because the number of light quark masses
9 0 R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95
Figure 5. Fit to the L A N L nucleon mass data .
2 . . . . I . . . . I . . . . I . . . . I . . . .
MN(GeV ) v e r s u s m ~ ( G e V z)
1.8
1.6
1.4
1.2
1 , , , I , , L , l , t , , I , , , , I , , , ~
0 0.2 0.4 0.6 0.8
explored are typically 3 - 4 and only the point at the heaviest mass (typically rnq ~> 2rn~) shows any significant deviat ion f rom linearity. I find such 4-paramete 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 JK smnples give completely different values of c (i). The best I could do was to fix one of the pa ramete r s and lnake a 3-parameter fit and then vary the fixed pa rame te r to minimize X 2. The best fit (obta ined by fixing any one of the less well de te rmined coefficients, c (1), c (2) or c (3), as one gets the same final result on minimizing X 2) to the L A N L d a t a expressed in units of GeV is shown in Fig.5 and gives
MB = 1.16 -- 0.36M. + 1.6M 2 - 0.5M~. (16)
Assuming c O) = - 2 . 5 , Eq.16 gives a ~ 0.14. The same m e t h o d applied to "012 sink" da t a fi 'om the G F l l collaboratim~[20] at fl = 5.93 gives ( up- da ted version of the fit presented in Ref.[16])
MB = 1.18 - 1.0M~ + 3.0M~ - 1.3M~ (17)
which implies tha 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 P A R A M E T E R
The kaon B para lne ter is a measure of the s t rong interact ion corrections to the K ° - /~0
mixing. It is one of the best measured lat t ice quantities. For details of the phenomeno logy and of the latt ice me thodo logy I refer you to Refs.[2111221123 I. Here, I present a smnmaa-y of just the chiral behavior.
Zhang and Sharpe [18] have calculated the chi- ral behavior of BK in b o t h the full and quenched theor ies . The fidl QCD result is [23]
BK =-B 1-(3+:~)yLny+by+ce2y+O(y 2) (18)
w h e r e y = 0 . 2 a n d = -
ma)/(rn~ + rod) measures the degeneracy of s and d quarks. B is the leading order value for BK, which is an input pa rame te r in x P T , and b and c are unknown constants . The quenched result [18]
= B Q[1 - (3 + e2)yLny + bQ'y + BK Q cQc21]
/ 2 - e 2 i - e
has exactly the same form except for the addi- tional te rm propor t iona l to & which is an ar t i fact of quenching. The te rm propor t iona l to 5 is singu- lar in the limit e -+ 1, therefore ex t rapola t ions of quenched results to the physical non-degenera te case are not reliable. For e = 0 this t e rm vanishes, so unless one works close to e ~ 1 (for which there is little incentive in the quenched approximat ion) , it is unlikely tha t we will, in the foreseeable fil- ture, be able to ex t rac t 5 using Eq.19.
The constants B, b, c are different in the full and quenched theories and cannot be fixed by x P T . Assuming B = B (?, the coefficient of the chiral log te rm is the same for e = 0. This is the best agreement one can expect between the two theories. As a result Sharpe [23] advoca tes tha t BK with degenerate quarks is possibly a "good" quant i ty to calculate using the quenched theory, though sys temat ic errors due to use of degenera te quarks are hard to est imate.
Using full QCD values, 3yLny ~ 1, so one can ask whether this normal chiral log is visible in the present da t a and whether it should be included in the extract ion of BK? W i t h existing d a t a it is hard to distinguish this t e rm f rom the one linear in y as the range of rnK is not large enough to
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95 91
F i g u r e 6 . E v i d e n c e o f f i n i t e s i z e e f f e c t s i n e n -
h a n c e d c h i r a l l o g s i n B v .
1 I ' ' ' ' I ' ' ' ' I ' '
0 . 1
0 . 0 1 I , , , , I , , , , I , ,
0 . 2 0 . 3 0 . 4
m K a
significantly affect the logarithm. Also, there ex- ist 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 quanti- ties are defined in Ref. [3] and are explicitly con- structed such that they do not diverge as 1/m 2 in the chiral limit. Both B v and BA have en- hanced 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 depen- dence of these logs, that has been seen in lattice data. Sharpe [3] has shown that this volume de- pendence is of the form
B y ( L ) - B~(~) = - ( B ~ ( L ) - BA(~)) (20) b ~ 6 t t 2 e - m K L
The constant b2 is not well determined, but the shape of the mK dependence is. The staggered fermion data at [~ = 6.0 on 16 a and 243 lattices [24] are shown in Fig. 6 and qualitatively confirm the expected finite size effects in the chiral logs.
8 . M A T R I X E L E M E N T O F S I N G L E T
A X I A L C U R R E N T I N T H E P R O T O N
Ever since the measurement of the spin struc- ture of protons using deep inelastic muon scatter- ing 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 con- tribute to this matrix element (ME) . These con- nected and disconnected diagrams are discussed in [27]. Since the disconnected diagram is hard to measure, Mandula [26] used the anomaly con- dition to derive the relation
~ s l i m x (fi..~lA.lfi, s).~. = N I - ~ ¢-4o ¢.
(fi', s[TrF..P..(q")l~, s) (21)
where ~ = iff-iff p 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 ropaga to 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 parame- terized in terms of form-factors as
(P', sl A~(q) Ifi, s) = f d% TsuG A - iq, ftTsuG A,
( t iP , sl P Ig , =
<fi', 1TrF, 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 7 7' propagators are
G~(q 2) no 71' poles, a2
al 2 ) + G2, a (q 2) - (q2 _ + _ m ,
GP(q2) _ P2 + Pl + ~) (q2 - (q : - m : . , )
GF(q2) _ f l + F. ( 2 4 ) ( q : -
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 : + N f ~ P (25
2 m q ( P 2 ) ? f i -- -- --5"- )
The term proportional to Nyc~s/2rr diverges in the chiral limit and there is 11o obvious way of ex- tracting the physical answer from it alone. Thus the method fails in the quenched theory.
In the fifll theory, there are no (touble poles and all analogous analysis gives
= --al -[- N f ~ ] ~
= NI (F- f'
2MpG1A(q 2 = O)
(26)
which justifies the use of the anomaly :'elation.
9 . M A S S E S O F L I G H T Q U A R K S
In order to extract light quark masses fl'om lat- tice simulations we use an ansatz for the chiral behavior of hadron masses. Theoretically, the best defined procedure is ) P T which :'elates the n l a s s e s of p s e u d o s c a l a r m e s o n s t o n~,u, rod, m s .
The overall scale p, in the :::ass term of Eq.1 im- plies that only ratios of quark masses can be de- termined ::sing xPT. The predictions from x P T for the two independent ratios are [6] [321
Lowest order Next order (m.~ + md)/2m~ 1__ ± ( ' ) ? ' d - - ? D ' u ) / ? T / ' s 215 31
4 4 29 "
In Latt ice 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 simula- tions 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 n t- 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. The mass I prefer to extract, barring the complications of quenched x P T , is ~ as it avoids the question whether low- est 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 stag- gered fermions [31], and the rho mass is used to set the scale. (I have not used the tadpole im- provement factor of U0 [29] in Cm and intact as this factor cancels in perturbat ion theory and is a small effect otherwise.) The value of boosted g 2 I use in Eq.27 is [29]
1 _ (plaq) + 0.025 (28) g2 g 2 latt
which is consistent with the continuum M S scheme vahle at Q = ~r/a
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') - [ 0 f h ) ~ - i ( -2-~0 ~ ) . (30)
The status of calculations of ~ ( 2 GeV) for quenched Wilson [351 [2O 1 [361 [371 [121 [381, quenched staggered [391 [91 [401 [71 [81, dynam- ical (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 93
Figure 7. scale set by m,p.
0 i i
o
,~, ex t rac ted using m~ da ta with the
' I . . . . I
m (MeV)
x % X
X x X X X
XX X X [ ]
X
E]E] [] [] r7 du
x nf=O Wilson [] nf=O Staggered x nf=2 Wilson
n¢=2 Staggered , L I L , , I I , J
0.5 1 a (CeV -t)
in Fig. 7. I have suppressed error bars as I want to first emphasize key quali tat ive features. The quenched s taggered and the n I = 2 Wilson and s taggered give ~ = 2 - 3 M e V and are roughly consistent; however, the quenched Wilson results seem to approach tha t value f rom above and even at 3 = 6.4 are significantly higher. (The recent result m , = 128(18) M e V by All ton 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 ha t ~ = m~/25.) I believe that , at this stage, it is impor t an t to unders tand why the quenched results with the Wilson act ion are so different f rom the rest!
An al ternat ive to using m,q = Zmassm~ =
Z; - lmq L to calculate the quark masses with Wil- son fermions is to use the Wal'd identi ty [34][15]
ZA m,~ (A 4( r )P (0 ) ) (31) ""~ = zP 2 <p(~)p(0)>
Using the pe r tu rba t ive values for ZA and Zp (with q* = 7r/a and boos ted g2 defined in Eq.28) the L A N L Wilson da t a [12] gives ~ = 3.53(10} M e V in cont ras t to ~, = 5.15(15) M e V shown in Fig. 7. The statist ical errors are calcu- lated using a single el imination JK with a salnple
of 100 lattices of size 323 x 64, so the difference is significant. The R o m e col laborat ion [45] has found a similar discrepancy and argue tha t it can be resolved if one uses the non-per tu rba t ive value for Zp , which they advoca te calculat ing using matr ix elmnents of the opera tors between quark states in a fixed (Landau) gauge. Their results indicate tha t pe r tu rba t ion theory (including tad- pole improvement) fails for Zp. The two me thods for extract ing r~--7 give consistent results once the non-per turba t ive vahm of Zp is used.
Having fixed 757 one Call ext rac t re, s, mc, and mb using, for exalnple, N ~, D, and B meson masses provided it is assumed tha t these masses are lin- ear in the light quark mass and in the heavier quark lnass a round the physical value. Alter- nately, one can use m¢, d/g~, and T s p e c t r m n to get these quark masses directly wi thout needing to ext rapola te in the light quark mass. The re- sults for rnc and ms have been reviewed by Sloan [33] at this conference so I will only analyze the da t a for m,.~ and compare these es t imates to 25~,. Note tha t the same d a t a used to compile Fig. 7 is used to calculate rn, e f rom 7I~,K. and me. The procedure for t ransla t ing the value to 2 G e V in the M S scheme is also the same. The results in Fig. 8 show tha t tha t the es t imate of m~ f rom rn,¢ is systematical ly higher by 15 - 20% compared to 25~7,.
I will use the L A N L data[12] to show tha t the sys temat ic errors due to choice of had ron used to set the scale of the strallge qual 'k are now a dominan 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 MK, tn,~ = 151(15) M e V using M K . , and m~ = 157(13) M e V using Me. Note tha t the lat ter two est imates give r n ~ / ~ ,,o 30, which is much closer to the "Next Order" predic t ion of x P T . The larger errors in these cases reflect the fact tha t on the latt ice masses of pseudoscalar mesons are measured with nmch be t t e r s tat is t ical accuracy than those of vector mesons.
10. C O N C L U S I O N S A N D C O M I N G A T - TRACTIONS
The analysis of various quenched quanti t ies show tha t the pa ramete r 5 character iz ing the
94 R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85 95
Figure 8. Comparison of rn,~ extracted using me and m~ = 25~,. The data are for quenched Wil- son simulations.
200 ~=m.(rn¢)
150
100
0
1:(
}X
x x x ;~ X
x X X X X
x}~ x x =25m
X
0 .2 0 . 4 0 .6 0 . 8 a (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 or- der chiral behavior in m~/mq. Therefore, I con- clude that extrapolation of quenched data, ob- tained with mq < m~/2, to the ehiral limit can- not be done simply using flfll QCD fornnflae for quantities which have large contributions from enhanced logs. For quantities like the matrix ele- nlent of the singlet axial vector current using the Adler-Bell-Jackiw anomaly, the quenched approx- imation fails altogether.
The alarmists are busy calculating l-loop cor- rections 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 chi- ral corrections to the energy of two pions in a fi- nite 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 heavy- light 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 LATTICE 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 stag- gered 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 esti- mates 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 signif- icant. 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. Phenomenologi- cal estimates involving extrapolation to strange quark mass need to take this systematic differ- ence into account.
1 1 . A C K N O W L E D G E M E N T S
I thank Claude Bernard, Tanmoy Bhat- tacharya, 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 Sin- clair 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 HPCC Grand Challenges pro- gram. The recent Wilson fermion sinmlations on 323 x 64 lattices have been done on the CM5 and we gratefully acknowledge the tremendous sup- port provided by the ACL at Los Alamos and by NCSA at Urbana-Champaign.
R. Gupta/Nuclear Physics B (Proc. Suppl.) 42 (1995) 85-95 95
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