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A lattice calculation of the second moment of the pion's distribution amplitude
Volume 190, number 1,2
PHYSICS LETTERS B
21 May 1987
A LATTICE CALCULATION OF THE SECOND MOMENT OF THE PION'S DISTRIBUTION AMPLITUDE G. M A R T I N E L L I and C.T. S A C H R A J D A
CERN, CH-1211 Geneva 23, Switzerland Received 13 February 1987
We calculate <~2>, the second moment of the pion's distribution amplitude on a 103×20 lattice, with Wilson fermions in the quenched approximation and at fl = 6.0. We find <~2> = 0.26 + 0.13, in the lattice renormalisation scheme at a = (1.8 GeV) This is in disagreement with the previous lattice determination of this quantity. The reasons for this discrepancy are discussed.
The non-perturbative information required to predict hard exclusive processes in QCD, such as hadronic form factors or elastic scattering at large m o m e n t u m transfers, is contained in the quark distribution amplitudes q~(x~, Q2) [ 1-5]. Q2 is the square o f the m o m e n t u m transfer and the label i takes values from 1 to nv, where nv is the n u m b e r of valence partons. ~ 7L ~xi = 1 and in a physical gauge xi is interpreted as the fraction o f momentum carried by the valence parton i. In perturbation theory we are only able to predict the behaviour o f qb with Q2 (this, of course, is also the case for deep inelastic structure functions), hence for the calculation of ~b itself we have to go beyond perturbation theory. In this letter we shall present the results o f a lattice calculation of the second m o m e n t o f the quark distribution amplitude o f the pion. The moments o f the distribution amplitude can be expressed as matrix elements o f the lowest twist local operators between the hadron and the v a c u u m [ 6 ]. For the pion, defining
~=xq-x~,
(1)
we have < 0 [ O~o~,,..~,(0)I ~ (p) > =x/2f,~pu~Pu, ...p~,~ < ~ > + t e r m s containing factors ofp2gu,~,
(2)
wheref~ is the pion's decay constant = 94 MeV, O.ou, ...uo= ( - i ) " ~yuo ys Du, ...D~ ~u
(3)
symmetrized over the Lorentz indices, and 1
(~5_
_} d ; ~.,,~(¢, Q 2 ) ,
(4)
-1
with the normalisation chosen so that <~o> = 1. <~"> contains an explicit Q2 behaviour, where in eq. (2) Q2 is the renormalisation scale o f the operators Ouo...u,. Below we shall present our results for < ~2 > obtained from a lattice calculation. We start, however, with a brief review of previous calculations o f this quantity. The operator matrix elements (2) have been studied in considerable detail using Q C D sum rules [ 5,7-9 ]. The most complete study, using sum rules for two-point functions, is given in refs. [ 5,7 ], and the results for the On leave of absence from the Department of Physics, The University, Southampton SO9 5NH, UK. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
151
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21 May 1987
low moments are very close to those one would obtain from the distribution amplitude 0({, Q2 _~ (0.5 GeV) 2) = - ~ 2 ( 1 _~z) .
(5)
We also recall that as Q2 ~ , q ~ , s - - - ~ (1 _~2). Eq. (5) has the interesting feature that q~ is not peaked at ~= 0, but at ~ - + 0.7 where either the quark or the anti-quark carries about 85% of the pion's momentum. In contrast the asymptotic distribution amplitude ~as is peaked around ~ = 0. For the second moment the results are
(~2) = 0 . 4 0 +
(15--20)%
a t Q2 = 1.5 GeV 2
(6a)
at Q2 = 2.25 GeV 2
(6b)
at a2 =0.25 GeV 2
(6c)
(refs. [5,7]), ( ~ z ) = 0.39 (ref. [ 8 ] ) , ( ~ 2 ) = 0 . 3 8 + (10-15)% (ref. [ 9 ] ) , to be compared with (~2) = 0.2 at Q2= ~ . Thus QCD sum rules seem to indicate that the distribution amplitude of the pion is significantly broader than the asymptotic one. There is also a lattice calculation of ( ~2 ) [ 10 ], (on a 62 × 12 × 18 lattice, at fl = 5.7, using Wilson fermions in the quenched approximation), yielding the spectacular result (~2)=1.58_+0.23
a t Q = l GeV
(7)
in the MS renormalisation scheme. Eq. (7) would imply that q~ not only has to have at least one maximum either side of ~ = 0, but that it also has to change sign. This result is clearly in conflict with those obtained using QCD sum rules. From the above discussion, we see that the calculation of (~2) requires the evaluation of the operator matrix element (2), with n = 2 . Since Lorentz (or euclidean) invariance is not a symmetry of the lattice, different choices of the indices/to,/tL,/t2 give rise not only to significant differences in the details of the calculations, but also raise important questions of principle. The operators O ~2) in general mix under renormalisation with operators of lower dimension, the mixing coefficients containing power divergences of the form 1/a 2 and 1/a (as can be seen even in first-order perturbation theory). Although for presently available lattices (fl ~- 5.7-6.0), the one-loop contribution to these mixing coefficients may be relatively small, the perturbative subtraction of these power divergences has been shown in several cases to give a wrong result, as for example in the evaluation of the critical value of the Wilson parameter K, and of (~t~u). By "perturbative subtraction" we mean the subtraction of the matrix elements of the operators of lower dimension (determined from lattice calculations) times the mixing coefficients calculated in perturbation theory. In order to avoid the problem of power divergences, we must take a combination of the operators O t2) which for symmetry reasons cannot mix with lowerdimensional ones. This excludes the possibility of taking Ouuu, i.e. an operator, with all three Lorentz indices equal, as was done in ref. [ 10 ]. The four operators Ouuu,/t = 1-4, transform like the ( ½, ½) irreducible representation of the hypercubic group (we are using the notation of ref. [ 11 ]) where the character table for this group can be found) and mix under renormalisation with ~t?4 ~'5~, O4 ~'~ 5~u and ~tr4~ ~ s D ~ ~, whxch transform in the same way under space-time symmetries [ 12 ]. In principle the choice Ou~v,with/t, v and p all different, is the most attractive. The four operators of this class transform like the ( ½, ½) irreducible representation of the hypercubic group (in the notation of ref. [ 11 ]). Since no lower-dimensional operator transforms in this way, there is no mixing with these operators under renormalisation. On the other hand, in order to get a non-zero matrix element at least two spatial components o f p must be non-zero. On currently available lattices, one would expect that in this case the pion's momentum is probably 152
Volume 190, number 1,2
PHYSICS LETTERSB
21 May 1987
tOO large to avoid lattice artefacts, an expectation which is borne out by our numerical studies. This leaves the final possibility Our, w i t h / t ~ v. The twelve operators in this set transform like the reducible ( ~, ½) + 8 representation of the hypercubic group. In order to eliminate mixing with lower-dimensional operators we take a linear combination of these operators which transforms like the eight-dimensional irreducible representation. A convenient choice is O411--0433. Thus one can eliminate the problem of power divergences, at the price of having to introduce just one non-zero spatial component of m o m e n t u m (Pl ¢ 0, P2 =P3 = 0 say). It seems to us that the third option is the optimal one on currently available lattices; indeed it is probably the only acceptable one. For the remainder of this letter we will consider matrix elements of the operators 0 ~ 0411--O433, with P, = n/5a, the smallest non-zero value on our 103× 20 lattice, and P2 =P3----0. We then have ( 0 1 0 ( 0 ) In ( P ) ) =x/2fxp4P 2 ( ~ 2 ) .
(8)
In order to "measure" ( ( 2 ) we consider the following two ratios of correlation functions ,l: Ex e i ' x (OlO(x)~/(O)ys~u(O)I0) C , ( t ) - E~ e ~ x (01 @¢(X)~475~/'/(X)~(0)~15~b¢(O)10>
(9a)
and Ex e ip'x (010(x)~(0)~14~)5 ~/J(0)10)
(9b)
C2(t) = ~x eipx ( 0 ] ~ / ( x ) y 4 y s ~ ( x ) ~ / ( 0 ) 7 4 y s ~ ( 0 ) 1 0 ) ' where the summations are over the lattice points in time slice t, both in the numerators and the denominators. If t is sufficiently large so that the correlation functions are dominated by the single-pion intermediate state, we have
C~ (t) = C2(t) = constant = p 2 ( ~ 2 ) .
(10)
Note that eq. (10) is valid even if we do not explicitly symmetrize on the Lorentz indices in O41~ and 0433, and below we shall not symmetrize. Out calculations were performed on a 103)<20 lattice at f l = 6 . 0 using Wilson fermions in the quenched approximation. We used 15 configurations at three different values of the Wilson parameter K (K=0.1515, 0.1530, 0.1545) and extrapolated to the physical value K = 0.1564. Further details of the simulation and of the generation of the quark propagators can be found in ref. [ 13 ]. For the discrete form of the derivatives we take the symmetric prescription
O,f(x) = (1/2a) [f(x + aft) - f i x - aft) ],
(I I a)
where # is a unit vector in the/z direction, and for the covariant derivative D;,f(x) = (1/2a)[ Uu(x)f(x+aft) - U~ ( x - a f t ) f ( x - a f t ) ] ,
(1 lb)
where Uu(x) is the gauge group matrix defined on the link from x to x+aft. Our first task is to demonstrate the existence of a plateau for C~ (t) and Ca (t), i.e. that for sufficiently large t and 2 0 - t (recall that our lattice has 20 points in the time direction) these quantities are independent of t. As will be seen below the errors in the measurements of Cl (t) and C2(t) are relatively large. For this reason we verify the dominance of the single-pion contribution to the unitarity sum by studying the correlation functions with O replaced by O4,~ or 0433. As an illustration of the general features of the results, we plot in fig. 1 the results for (22(t) with O replaced by O41~ and 0433 with K = 0.1530. This is a typical example. The results for K = 0.1515 have smaller errors and those for K = 0.1545 larger ones. The errors for C~ (t) (with O replaced by O4~ and 0433) are comparable to those for C2(t). The errors were estimated by using the procedure described :J f~ itself has also been calculated with the lattice used in this paper [ 13], with the resultf~ = 110 + 20 MeV. 153
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PHYSICS LETTERSB
0 replacedby
Cz(t) with
21 May 1987
Cz(t) with 0 replaced by
0/.II
K= 0.1530
0/*33
K= 0.1530
12
1.2 r •
1.0
1.0
M
O.B
0.8
.
0.6
0.6
0./,
0.Z~
0.2
0.2
(a) ,
I
I
3
5
,, I 7 13 Itt
,
i
15
,
I
t~
17
"÷,t+,+ ~ (b) ,
I
3
,
I
5
,
I][I
7 13
,
t
15
,
t
t
,
17
Fig. 1. Demonstration of the existence of a plateau for the correlator C2( t ) with O replacedby O4~, (a) and O433(b) at K= 0.1530. in ref. [ 13 ] of dividing the 15 configurations for each value of K into 3 sets of 5 configurations and treating the result from each set as independent. The two features of fig. 1 which we particularly wish to stress, are the following: (1) The plateaus for both O411 and 0433 are clearly present, and in all cases the plateau region can be taken to. be 5~
using Cl(t) and data from t = 5 and 15,
(12a)
(~2)L=0.27--+0.13
using Cl(t) a n d d a t a from t = 6 and 14,
(12b)
(~2)L =0.05+0.23
using Q ( t ) and data from t = 5 and 15,
(12c)
(~2)L =0.18+0.23
using C2(t) and data from t = 6 and 14.
(12d)
The results for t>~ 7 are consistent with eq. (12), but have very large errors. The subscript L is to remind us that these results are in the lattice renormalisation scheme at a = (1.8 GeV)-1; we will discuss the connection with 1.54
Volume 190, number 1,2
PHYSICS LETTERS B
21 May 1987
(F=2) obtained from El(t) at t= 5 and 15
K= 0.15¢5 0.2c 0.20
\
~
K
0"1530 = 0.1515
0.15 0.10 0.05 I
0.1
I
I
0.2
0.3
mz
Fig. 2. Example of the extrapolation to the physical limit of (~2). This figure is obtained using CI (t) at t = 5 and 15. The three points with their errors are correlated and the final error on (~2) is obtained using the clustering procedure discussed in the text.
c o n t i n u u m r e n o r m a l i s a t i o n schemes below. In c o m b i n i n g the results o f eq. (I 2) to o b t a i n a single number, it m u s t o f course be r e m e m b e r e d that they are correlated. Preferring to err on the side o f caution, we take as our final result, (~2)L =0.26+0.13.
(13)
It is clear that o u r result disagrees with that o f the previous lattice calculation, eq. ( 7 ) . (In ref. [ 10 ] the result in the lattice r e n o r m a l i s a t i o n scheme is ( ~ 2 ) L ~ 1.37 _+0.20 at an a o f 1 G e V - 1 . ) It should be n o t e d that we work on a different lattice a n d at fl = 6.0 rather than 5.7. In addition, as stressed above, our m a i n theoretical objection to the calculation o f ref. [ 10 ], is that there is no way o f checking directly whether the one-loop perturbative subtraction o f p o w e r divergences was correct or not. In our case the n o n - p e r t u r b a t i v e subtraction o f power divergences p r o d u c e d a significant numerical cancellation, which then was responsible for the large errors in eqs. (12) a n d (13) ~2. In ref. [ 10] neither the subtraction o f the traces, n o r the p e r t u r b a t i v e r e m o v a l o f the r e m a i n i n g p o w e r divergences seems to spoil the numerical accuracy o f the result. Before c o m p a r i n g o u r results with those o b t a i n e d using Q C D sum rules, we have to state ours in some continu u m r e n o r m a l i s a t i o n scheme (MS, say), using
( ~2 )~.g ~. ( Zo/ZA)
(~2)L
,
(14)
where Zo is the r e n o r m a l i s a t i o n constant for the o p e r a t o r O (or m o r e precisely the ratio o f r e n o r m a l i s a t i o n constants in the MS a n d lattice schemes) a n d ZA is the r e n o r m a l i s a t i o n constant o f the axial current q/Y4)'5 ~. In p e r t u r b a t i o n theory ZA----0.87 a n d although we have not calculated Zo explicitly, the t a d p o l e d o m i n a n c e o f lattice p e r t u r b a t i o n theory (see, e.g., refs. [ 12,14 ] ) leads us to expect confidently that Zo--- 1.13 (with an estim a t e d uncertainty o f 0.05) ~3. Thus z2 Even the one-loop perturbative contribution to the power divergences of O4~ is much larger than the matrix element of the operator O. :3 Notice that since the renormalisation constants are at most logarithmically divergent, they can he computed in perturbation theory for sufficiently large fl [ 15 ]. 155
Volume 190, number 1,2
PHYSICS LETTERS B
(~2)~_g ,~ 1.3 ( ~ 2 ) L -
21 May 1987 (15)
The r e n o r m a l i z a t i o n factor o f 1.3 brings the central value o f our result very close to those f o u n d using Q C D sum rules (eq. ( 13 ) now b e c o m e s ( ~2 ) ~g = 0.34 + 0.17 ) although o f course our error is very large. W h e t h e r p e r t u r b a t i o n theory is enough to calculate Zo/ZA sufficiently accurately at fl = 6.0 is one o f the systematic uncertainties o f this calculation. A n o n - p e r t u r b a t i v e calculation o f ZA on the same lattice gave a result o f ZA--~0.7 [ 13 ] as c o m p a r e d to the p e r t u r b a t i v e calculation o f 0.87, b u t there is no equivalent calculation o f Zo. We end with a b r i e f recapitulation o f our calculations. The m a i n result o f this p a p e r is eq. (13). The relatively large statistical error ( c o r r e s p o n d i n g to 15 configurations) is due to the significant cancellation which occurred when the power divergences were subtracted non-perturbatively. O u r result is in disagreement with that found in the previous lattice calculation [ 10 ], however, we do not accept that the t r e a t m e n t o f power divergences in ref. [ 10] was valid. O u r value o f ( ~ 2 ) is consistent with those f o u n d using Q C D sum rules. Clearly it would be desirable to reduce the error, which can be done either b y increasing the statistics or by working on a large lattice with the o p e r a t o r O412, which is free o f power divergences. We acknowledge helpful discussions with B. Gavela, A. K r o n f e l d a n d L. Maiani, and the support for computing t i m e by I N F N a n d C N R .
References [ 1] S.J. Brodsky and G.P. Lepage, Phys. Lett. B 87 (1979) 359; Phys. Rev. D 22 (1980) 2157. [2] A.V. Efremov and A.V. Radyushkin, Phys. Lett. B 94 (1980) 245; Riv. Nuovo Cimento 3 (1980) 1. [3] A. Duncan and A. Mueller, Phys. Rev. D 21 (1980) 1636; Phys. Lett. B 98 (1980) 159. [4] G.R. Farrar and D.R. Jackson, Phys. Rev. Lett. 43 (1979) 246. [5] V.L. Chernyak and A.R. Zhitnitsky, Phys. Rep. 112 (1984) 173. [6] S.J. Brodsky, Y. Frishman, G.P. Lepage and C.T. Sachrajda, Phys. Lett. B 91 (1980) 239. [7] V.L. Chernyak and A.R. Zhitnitsky, Nucl. Phys. B 201 (1982) 492. [8] M.J. Lavelle, Z. Phys. C 29 (1985) 203. [9 ] D.A. Johnston and H.F. Jones, Imperial College Preprint TP/85-86/28 (1986). [ 10] S. Gottlieb and A.S. Kronfeld, Phys. Rev. D 33 (1985) 227. [ 11 ] J. Mandula, G. Zweig and J. Goevarts, Nucl. Phys. B 228 (1983) 91. [ 12] A.S. Kronfeld and D.M. Photiadis, Phys. Rev. D 31 (1985) 2939. [ 13] L. Maiani and G. Martinelli, Phys. Lett. B 178 (1986) 265. [ 14] G. Martinelli and Y.G. Zhang, Phys. Lett. B 123 (1983) 433; Phys. Lett. B 125 (1983) 77. [ 15 ] M. Bochicchio et al., Nucl. Phys. B 262 (1985) 331.
156
PHYSICS LETTERS B
21 May 1987
A LATTICE CALCULATION OF THE SECOND MOMENT OF THE PION'S DISTRIBUTION AMPLITUDE G. M A R T I N E L L I and C.T. S A C H R A J D A
CERN, CH-1211 Geneva 23, Switzerland Received 13 February 1987
We calculate <~2>, the second moment of the pion's distribution amplitude on a 103×20 lattice, with Wilson fermions in the quenched approximation and at fl = 6.0. We find <~2> = 0.26 + 0.13, in the lattice renormalisation scheme at a = (1.8 GeV) This is in disagreement with the previous lattice determination of this quantity. The reasons for this discrepancy are discussed.
The non-perturbative information required to predict hard exclusive processes in QCD, such as hadronic form factors or elastic scattering at large m o m e n t u m transfers, is contained in the quark distribution amplitudes q~(x~, Q2) [ 1-5]. Q2 is the square o f the m o m e n t u m transfer and the label i takes values from 1 to nv, where nv is the n u m b e r of valence partons. ~ 7L ~xi = 1 and in a physical gauge xi is interpreted as the fraction o f momentum carried by the valence parton i. In perturbation theory we are only able to predict the behaviour o f qb with Q2 (this, of course, is also the case for deep inelastic structure functions), hence for the calculation of ~b itself we have to go beyond perturbation theory. In this letter we shall present the results o f a lattice calculation of the second m o m e n t o f the quark distribution amplitude o f the pion. The moments o f the distribution amplitude can be expressed as matrix elements o f the lowest twist local operators between the hadron and the v a c u u m [ 6 ]. For the pion, defining
~=xq-x~,
(1)
we have < 0 [ O~o~,,..~,(0)I ~ (p) > =x/2f,~pu~Pu, ...p~,~ < ~ > + t e r m s containing factors ofp2gu,~,
(2)
wheref~ is the pion's decay constant = 94 MeV, O.ou, ...uo= ( - i ) " ~yuo ys Du, ...D~ ~u
(3)
symmetrized over the Lorentz indices, and 1
(~5_
_} d ; ~.,,~(¢, Q 2 ) ,
(4)
-1
with the normalisation chosen so that <~o> = 1. <~"> contains an explicit Q2 behaviour, where in eq. (2) Q2 is the renormalisation scale o f the operators Ouo...u,. Below we shall present our results for < ~2 > obtained from a lattice calculation. We start, however, with a brief review of previous calculations o f this quantity. The operator matrix elements (2) have been studied in considerable detail using Q C D sum rules [ 5,7-9 ]. The most complete study, using sum rules for two-point functions, is given in refs. [ 5,7 ], and the results for the On leave of absence from the Department of Physics, The University, Southampton SO9 5NH, UK. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
151
Volume 190,number 1,2
PHYSICSLETTERSB
21 May 1987
low moments are very close to those one would obtain from the distribution amplitude 0({, Q2 _~ (0.5 GeV) 2) = - ~ 2 ( 1 _~z) .
(5)
We also recall that as Q2 ~ , q ~ , s - - - ~ (1 _~2). Eq. (5) has the interesting feature that q~ is not peaked at ~= 0, but at ~ - + 0.7 where either the quark or the anti-quark carries about 85% of the pion's momentum. In contrast the asymptotic distribution amplitude ~as is peaked around ~ = 0. For the second moment the results are
(~2) = 0 . 4 0 +
(15--20)%
a t Q2 = 1.5 GeV 2
(6a)
at Q2 = 2.25 GeV 2
(6b)
at a2 =0.25 GeV 2
(6c)
(refs. [5,7]), ( ~ z ) = 0.39 (ref. [ 8 ] ) , ( ~ 2 ) = 0 . 3 8 + (10-15)% (ref. [ 9 ] ) , to be compared with (~2) = 0.2 at Q2= ~ . Thus QCD sum rules seem to indicate that the distribution amplitude of the pion is significantly broader than the asymptotic one. There is also a lattice calculation of ( ~2 ) [ 10 ], (on a 62 × 12 × 18 lattice, at fl = 5.7, using Wilson fermions in the quenched approximation), yielding the spectacular result (~2)=1.58_+0.23
a t Q = l GeV
(7)
in the MS renormalisation scheme. Eq. (7) would imply that q~ not only has to have at least one maximum either side of ~ = 0, but that it also has to change sign. This result is clearly in conflict with those obtained using QCD sum rules. From the above discussion, we see that the calculation of (~2) requires the evaluation of the operator matrix element (2), with n = 2 . Since Lorentz (or euclidean) invariance is not a symmetry of the lattice, different choices of the indices/to,/tL,/t2 give rise not only to significant differences in the details of the calculations, but also raise important questions of principle. The operators O ~2) in general mix under renormalisation with operators of lower dimension, the mixing coefficients containing power divergences of the form 1/a 2 and 1/a (as can be seen even in first-order perturbation theory). Although for presently available lattices (fl ~- 5.7-6.0), the one-loop contribution to these mixing coefficients may be relatively small, the perturbative subtraction of these power divergences has been shown in several cases to give a wrong result, as for example in the evaluation of the critical value of the Wilson parameter K, and of (~t~u). By "perturbative subtraction" we mean the subtraction of the matrix elements of the operators of lower dimension (determined from lattice calculations) times the mixing coefficients calculated in perturbation theory. In order to avoid the problem of power divergences, we must take a combination of the operators O t2) which for symmetry reasons cannot mix with lowerdimensional ones. This excludes the possibility of taking Ouuu, i.e. an operator, with all three Lorentz indices equal, as was done in ref. [ 10 ]. The four operators Ouuu,/t = 1-4, transform like the ( ½, ½) irreducible representation of the hypercubic group (we are using the notation of ref. [ 11 ]) where the character table for this group can be found) and mix under renormalisation with ~t?4 ~'5~, O4 ~'~ 5~u and ~tr4~ ~ s D ~ ~, whxch transform in the same way under space-time symmetries [ 12 ]. In principle the choice Ou~v,with/t, v and p all different, is the most attractive. The four operators of this class transform like the ( ½, ½) irreducible representation of the hypercubic group (in the notation of ref. [ 11 ]). Since no lower-dimensional operator transforms in this way, there is no mixing with these operators under renormalisation. On the other hand, in order to get a non-zero matrix element at least two spatial components o f p must be non-zero. On currently available lattices, one would expect that in this case the pion's momentum is probably 152
Volume 190, number 1,2
PHYSICS LETTERSB
21 May 1987
tOO large to avoid lattice artefacts, an expectation which is borne out by our numerical studies. This leaves the final possibility Our, w i t h / t ~ v. The twelve operators in this set transform like the reducible ( ~, ½) + 8 representation of the hypercubic group. In order to eliminate mixing with lower-dimensional operators we take a linear combination of these operators which transforms like the eight-dimensional irreducible representation. A convenient choice is O411--0433. Thus one can eliminate the problem of power divergences, at the price of having to introduce just one non-zero spatial component of m o m e n t u m (Pl ¢ 0, P2 =P3 = 0 say). It seems to us that the third option is the optimal one on currently available lattices; indeed it is probably the only acceptable one. For the remainder of this letter we will consider matrix elements of the operators 0 ~ 0411--O433, with P, = n/5a, the smallest non-zero value on our 103× 20 lattice, and P2 =P3----0. We then have ( 0 1 0 ( 0 ) In ( P ) ) =x/2fxp4P 2 ( ~ 2 ) .
(8)
In order to "measure" ( ( 2 ) we consider the following two ratios of correlation functions ,l: Ex e i ' x (OlO(x)~/(O)ys~u(O)I0) C , ( t ) - E~ e ~ x (01 @¢(X)~475~/'/(X)~(0)~15~b¢(O)10>
(9a)
and Ex e ip'x (010(x)~(0)~14~)5 ~/J(0)10)
(9b)
C2(t) = ~x eipx ( 0 ] ~ / ( x ) y 4 y s ~ ( x ) ~ / ( 0 ) 7 4 y s ~ ( 0 ) 1 0 ) ' where the summations are over the lattice points in time slice t, both in the numerators and the denominators. If t is sufficiently large so that the correlation functions are dominated by the single-pion intermediate state, we have
C~ (t) = C2(t) = constant = p 2 ( ~ 2 ) .
(10)
Note that eq. (10) is valid even if we do not explicitly symmetrize on the Lorentz indices in O41~ and 0433, and below we shall not symmetrize. Out calculations were performed on a 103)<20 lattice at f l = 6 . 0 using Wilson fermions in the quenched approximation. We used 15 configurations at three different values of the Wilson parameter K (K=0.1515, 0.1530, 0.1545) and extrapolated to the physical value K = 0.1564. Further details of the simulation and of the generation of the quark propagators can be found in ref. [ 13 ]. For the discrete form of the derivatives we take the symmetric prescription
O,f(x) = (1/2a) [f(x + aft) - f i x - aft) ],
(I I a)
where # is a unit vector in the/z direction, and for the covariant derivative D;,f(x) = (1/2a)[ Uu(x)f(x+aft) - U~ ( x - a f t ) f ( x - a f t ) ] ,
(1 lb)
where Uu(x) is the gauge group matrix defined on the link from x to x+aft. Our first task is to demonstrate the existence of a plateau for C~ (t) and Ca (t), i.e. that for sufficiently large t and 2 0 - t (recall that our lattice has 20 points in the time direction) these quantities are independent of t. As will be seen below the errors in the measurements of Cl (t) and C2(t) are relatively large. For this reason we verify the dominance of the single-pion contribution to the unitarity sum by studying the correlation functions with O replaced by O4,~ or 0433. As an illustration of the general features of the results, we plot in fig. 1 the results for (22(t) with O replaced by O41~ and 0433 with K = 0.1530. This is a typical example. The results for K = 0.1515 have smaller errors and those for K = 0.1545 larger ones. The errors for C~ (t) (with O replaced by O4~ and 0433) are comparable to those for C2(t). The errors were estimated by using the procedure described :J f~ itself has also been calculated with the lattice used in this paper [ 13], with the resultf~ = 110 + 20 MeV. 153
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0 replacedby
Cz(t) with
21 May 1987
Cz(t) with 0 replaced by
0/.II
K= 0.1530
0/*33
K= 0.1530
12
1.2 r •
1.0
1.0
M
O.B
0.8
.
0.6
0.6
0./,
0.Z~
0.2
0.2
(a) ,
I
I
3
5
,, I 7 13 Itt
,
i
15
,
I
t~
17
"÷,t+,+ ~ (b) ,
I
3
,
I
5
,
I][I
7 13
,
t
15
,
t
t
,
17
Fig. 1. Demonstration of the existence of a plateau for the correlator C2( t ) with O replacedby O4~, (a) and O433(b) at K= 0.1530. in ref. [ 13 ] of dividing the 15 configurations for each value of K into 3 sets of 5 configurations and treating the result from each set as independent. The two features of fig. 1 which we particularly wish to stress, are the following: (1) The plateaus for both O411 and 0433 are clearly present, and in all cases the plateau region can be taken to. be 5~
using Cl(t) and data from t = 5 and 15,
(12a)
(~2)L=0.27--+0.13
using Cl(t) a n d d a t a from t = 6 and 14,
(12b)
(~2)L =0.05+0.23
using Q ( t ) and data from t = 5 and 15,
(12c)
(~2)L =0.18+0.23
using C2(t) and data from t = 6 and 14.
(12d)
The results for t>~ 7 are consistent with eq. (12), but have very large errors. The subscript L is to remind us that these results are in the lattice renormalisation scheme at a = (1.8 GeV)-1; we will discuss the connection with 1.54
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21 May 1987
(F=2) obtained from El(t) at t= 5 and 15
K= 0.15¢5 0.2c 0.20
\
~
K
0"1530 = 0.1515
0.15 0.10 0.05 I
0.1
I
I
0.2
0.3
mz
Fig. 2. Example of the extrapolation to the physical limit of (~2). This figure is obtained using CI (t) at t = 5 and 15. The three points with their errors are correlated and the final error on (~2) is obtained using the clustering procedure discussed in the text.
c o n t i n u u m r e n o r m a l i s a t i o n schemes below. In c o m b i n i n g the results o f eq. (I 2) to o b t a i n a single number, it m u s t o f course be r e m e m b e r e d that they are correlated. Preferring to err on the side o f caution, we take as our final result, (~2)L =0.26+0.13.
(13)
It is clear that o u r result disagrees with that o f the previous lattice calculation, eq. ( 7 ) . (In ref. [ 10 ] the result in the lattice r e n o r m a l i s a t i o n scheme is ( ~ 2 ) L ~ 1.37 _+0.20 at an a o f 1 G e V - 1 . ) It should be n o t e d that we work on a different lattice a n d at fl = 6.0 rather than 5.7. In addition, as stressed above, our m a i n theoretical objection to the calculation o f ref. [ 10 ], is that there is no way o f checking directly whether the one-loop perturbative subtraction o f p o w e r divergences was correct or not. In our case the n o n - p e r t u r b a t i v e subtraction o f power divergences p r o d u c e d a significant numerical cancellation, which then was responsible for the large errors in eqs. (12) a n d (13) ~2. In ref. [ 10] neither the subtraction o f the traces, n o r the p e r t u r b a t i v e r e m o v a l o f the r e m a i n i n g p o w e r divergences seems to spoil the numerical accuracy o f the result. Before c o m p a r i n g o u r results with those o b t a i n e d using Q C D sum rules, we have to state ours in some continu u m r e n o r m a l i s a t i o n scheme (MS, say), using
( ~2 )~.g ~. ( Zo/ZA)
(~2)L
,
(14)
where Zo is the r e n o r m a l i s a t i o n constant for the o p e r a t o r O (or m o r e precisely the ratio o f r e n o r m a l i s a t i o n constants in the MS a n d lattice schemes) a n d ZA is the r e n o r m a l i s a t i o n constant o f the axial current q/Y4)'5 ~. In p e r t u r b a t i o n theory ZA----0.87 a n d although we have not calculated Zo explicitly, the t a d p o l e d o m i n a n c e o f lattice p e r t u r b a t i o n theory (see, e.g., refs. [ 12,14 ] ) leads us to expect confidently that Zo--- 1.13 (with an estim a t e d uncertainty o f 0.05) ~3. Thus z2 Even the one-loop perturbative contribution to the power divergences of O4~ is much larger than the matrix element of the operator O. :3 Notice that since the renormalisation constants are at most logarithmically divergent, they can he computed in perturbation theory for sufficiently large fl [ 15 ]. 155
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(~2)~_g ,~ 1.3 ( ~ 2 ) L -
21 May 1987 (15)
The r e n o r m a l i z a t i o n factor o f 1.3 brings the central value o f our result very close to those f o u n d using Q C D sum rules (eq. ( 13 ) now b e c o m e s ( ~2 ) ~g = 0.34 + 0.17 ) although o f course our error is very large. W h e t h e r p e r t u r b a t i o n theory is enough to calculate Zo/ZA sufficiently accurately at fl = 6.0 is one o f the systematic uncertainties o f this calculation. A n o n - p e r t u r b a t i v e calculation o f ZA on the same lattice gave a result o f ZA--~0.7 [ 13 ] as c o m p a r e d to the p e r t u r b a t i v e calculation o f 0.87, b u t there is no equivalent calculation o f Zo. We end with a b r i e f recapitulation o f our calculations. The m a i n result o f this p a p e r is eq. (13). The relatively large statistical error ( c o r r e s p o n d i n g to 15 configurations) is due to the significant cancellation which occurred when the power divergences were subtracted non-perturbatively. O u r result is in disagreement with that found in the previous lattice calculation [ 10 ], however, we do not accept that the t r e a t m e n t o f power divergences in ref. [ 10] was valid. O u r value o f ( ~ 2 ) is consistent with those f o u n d using Q C D sum rules. Clearly it would be desirable to reduce the error, which can be done either b y increasing the statistics or by working on a large lattice with the o p e r a t o r O412, which is free o f power divergences. We acknowledge helpful discussions with B. Gavela, A. K r o n f e l d a n d L. Maiani, and the support for computing t i m e by I N F N a n d C N R .
References [ 1] S.J. Brodsky and G.P. Lepage, Phys. Lett. B 87 (1979) 359; Phys. Rev. D 22 (1980) 2157. [2] A.V. Efremov and A.V. Radyushkin, Phys. Lett. B 94 (1980) 245; Riv. Nuovo Cimento 3 (1980) 1. [3] A. Duncan and A. Mueller, Phys. Rev. D 21 (1980) 1636; Phys. Lett. B 98 (1980) 159. [4] G.R. Farrar and D.R. Jackson, Phys. Rev. Lett. 43 (1979) 246. [5] V.L. Chernyak and A.R. Zhitnitsky, Phys. Rep. 112 (1984) 173. [6] S.J. Brodsky, Y. Frishman, G.P. Lepage and C.T. Sachrajda, Phys. Lett. B 91 (1980) 239. [7] V.L. Chernyak and A.R. Zhitnitsky, Nucl. Phys. B 201 (1982) 492. [8] M.J. Lavelle, Z. Phys. C 29 (1985) 203. [9 ] D.A. Johnston and H.F. Jones, Imperial College Preprint TP/85-86/28 (1986). [ 10] S. Gottlieb and A.S. Kronfeld, Phys. Rev. D 33 (1985) 227. [ 11 ] J. Mandula, G. Zweig and J. Goevarts, Nucl. Phys. B 228 (1983) 91. [ 12] A.S. Kronfeld and D.M. Photiadis, Phys. Rev. D 31 (1985) 2939. [ 13] L. Maiani and G. Martinelli, Phys. Lett. B 178 (1986) 265. [ 14] G. Martinelli and Y.G. Zhang, Phys. Lett. B 123 (1983) 433; Phys. Lett. B 125 (1983) 77. [ 15 ] M. Bochicchio et al., Nucl. Phys. B 262 (1985) 331.
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