The status of lattice calculations of the nucleon structure functions

UCLEAR PHYSIC~

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 49 (1996) 250-255

ELSEVIER

T h e s t a t u s of l a t t i c e c a l c u l a t i o n s of t h e n u c l e o n s t r u c t u r e f u n c t i o n s * M. GSckelerL R. Horsley b, E.-M. Ilgenfritz b, H. Oelrich ~, H. Petit ¢, P. RakowL G. Schierholz ~'d and A. Schiller ¢ ~HSchstleistungsrechenzentrum HLRZ, c/o Forschungszentrum Jiilich, D-52425 Jiilich, Germany bInstitut fiir Physik, Humboldt-Universit£t, D-10115 Berlin, Germany CFak. f. Physik und Geowiss., Universit£t Leipzig, Augustusplatz 10-11, D-04109 Leipzig, G e r m a n y dDeutsches Elektronen-Synchrotron DESY, Notkestrafie 85, D-22603 Hamburg, Germany We review our progress on the lattice calculation of low moments of both the unpolarised and polarised nucleon structure functions.

1. I N T R O D U C T I O N Much of our knowledge about QCD and the structure of nucleons has been derived from charged current Deep Inelastic Scattering (DIS) experiments in which a lepton (usually an electron or muon) radiates a virtual photon at large spacelike m o m e n t u m _q2 _ Q2. This interacts with a nucleon (usually a proton) and measuring the total cross section yields information on 4 structure functions for the nucleon - F1, F2 when polarisations are summed over and gl, g2 when the incident lepton b e a m and the proton have definite polarisations. The structure functions are functions of the Bjorken variable z (0 < x < 1) and Q2. While the unpolarised case has been studied experimentally for m a n y years (since the pioneering discoveries at SLAC), only recently have experiments been reported with polarised beams and targets ([1,2]). Theoretically, deviations from the simplest parton model (in which the hadron may be considered as a collection of non-interacting quarks) is taken as strong evidence for the evidence of QCD, where we have interacting quarks and gluons. A direct theoretical calculation of the structure functions seems not to be possible; however using the Wilson Operator Product Expansion (OPE) we may relate moments of the structure functions to matrix elements of certain operators in a twist or Taylor expansion in 1/Q ~. For the unpolarised *Talk presented

structure functions we find 1,

2

/o

d~x,~-lFdx, Q2)

=

(I),n v(1)(#) + O(1/Q2), Z E Fx !

oldxx'~-"F2(~, & )

= ~E(n~,.¢n(,,.~. .._, + Off/@),

(1)

! where n is even starting at 2, f = u, d, s, g and

(g ~ o ~ " ' " ' d l g ~ = 2v~!)~

.... /°

- ~], (2)

with •

r~--I

Oq~...t~

(3) •

\2]

n--9.

~ °

For the polarised structure functions,

2

11

dxx'~ga(x,Q 2)

=

1 ~ ~ E(/)g,,na(/)(,~ ,,-, ~ q- o ( 1 / Q 2 ) , 1

1We use (if, s ~ , ~ ) = (2~r)a2Ep-~(ff-~)6~,~., with s2 = - m 2. The Wilson coefficients S =_ E(~2/Q2,g(,u)) are known perturbatively, for example for the charged current

by R. Horsley.

09_0-563_/96/$15.00 1996 Elsevier Science B.V. All rights reserved. PII: S0920-5632(96)00341-6

M. Gdckeler et al./Nuclear Physics B (Proc. Suppl.) 49 (1996) 250~255

Often we re-write the distribtions in terms of valence, v, and sea, S, contributions,

2 L t d z z ' ~ g 2 (z, Q2) 1 --

n

2 n ~

1E

E(/)9~,"'(d(l)Cu~n ,,-,

!

-

q~(~,~) + s ( . , ~), ~(~,~,) ~ s(~,~),

a(/)(#))

+0(1/Q2),

(4)

and

~(s) :

,~ + ~ [ , ' F . . . . F °

-

n + l[(S'Pt'l -

d(j)

+...],

#,lp.)p~ . . . . p~"

+ . . . ] , (5)

with m ~hq~ * l " # " =

qT~TsD t~. . . . D~"q

(6)

q(~, ~)

~

s(., u)

~

a (q)

For gl we start with n : 0, while for g2 we begin with n = 2 (Osg, n = 2 is a special case). While the Wilson coefficients, E, can be calculated perturbatively, the matrix elements cannot: a non-perturbative method must be employed. In principle lattice gauge theories provide such a method to compute these matrix elements from first principles. In this talk we shall describe our progress towards this goal, [3,4], and try to discuss the problems that must be overcome. We will not dwell on lattice details and only give results in the form of Edinburgh (or APE) graphs which are plotted in terms of physical quantities. The moments, eq. (1), have a parton model interpretation, being powers of the fraction of the nucleon m o m e n t u m carried by the patton

V(nq)( I t ) =

<~n- 1) (q) (/.t) =

=

=

('-'~'-P.,°e"~...D"--,Fo=.°,

~,.

(7)

s(~,~),

(9)

(q = u, d). While in e(It)N DIS we can use only the charge conjugation positive m o m e n t s (ie for even n), in v N experiments we do not have this restriction and can use all moments. The lowest moment is particularly interesting, as due to conservation of the energy m o m e n t u m tensor we have the sum rule Zq
• n-1

~,~1

251

(xn- 1 )~q) (~t)

/01

(10)

e~--l[Aq(~, u)+ (-1)- A~(~, u)],

where A q ( z , # ) = qT(z,#) -- q l ( z , # ) (and similarly for Aq(z, , ) ) . The suitably modified eq. (9) is also taken to hold. Again the lowest moment is of particular interest because it can be related to the fraction of the spin carried by the quarks in the nucleon. Conventionally we define Aq by Aq =

dz[Aq(z, # ) + Aq(z, #)],

(11)

and then

2aq : a(: )

(12)

g2 contains not only a,~ (the so-called WandzuraWilczek contribution to g2) but also dn - a twist-3 contribution. It has been argued that the n = 0 moment of g2 should be 0, but it is not possible to check this sum rule on a lattice. Finally we note that g2 does not have a partonic interpretation.

dzz'~-t[q(z, #) + ( - 1 ) " ~ ( z , #)1,

2. T H E A P E P L O T where q(z, #) = qT(z, #) + q, (z, #) (and similarly for q(z, #)) with qi(z, #) being the quark distributions at some scale # and spin ~. For the gluon

¢:)(~)

: -----

<~--1V(~) Llda~a~n- lg(x, U).

(8)

We shall now motivate the idea of an Edinburgh (or APE) plot. In a lattice calculation, we must first Euclideanise (t --~ - i t ) and then discretise the QCD action and operators. This introduces a lattice spacing a as an ultraviolet cutoff in addition to the (bare) coupling constant

ML G6ckeler et al./Nuclear Physics B (Proc. Suppl.) 49 (1996)250-255

252

/3 = 6/g 2 and for the Wilson formulation of lattice fermions, l/R, which is related to the quark mass. (We shall only consider rnu = md here.) We first consider measuring the light hadron spectrum. Numerically we measure dimensionless numbers representing the different masses. In a region where scaling is taking place, [5], then mPhhys

mPhh,v,

mh

-- mh, quark mass--m,,/m~r,mK/mN .... g=~O ---* const. (13)

Thus if we plot, for example m N / m p as a function of m , / m p (here we will take m N / m p = .f((m,~/mp)2)), we must be in a region where we see universal scaring, ie for different g values the results tie on the same curve. The lattice spacing can be found from a = mh/mPh hv'. Measuring the various masses yields the plot shown in Fig. 1 The plot allows the whole range 1.7

'

'

I '

'

'

I

'

'

I '

'

'

I '

'

'

1.6

1.5

Q. "~ 1.4 z

15

mass limit to the physical point. In fact from the lattice m a n y quark mass worlds can be computed. (Unfortunately everything except the real world: this is because the u and d quark masses are very close to the chiral limit, which numerically is difficult to realise.) Our tightest quark mass ties at a b o u t the mass of the strange quark (in the middle of the plot, (m,~/rnp) 2 ,.~ 0.5). Do we have universal sealing? R o u g h l y the points seem to tie in a b a n d - a g o o d sign, although the smaller coupling values seem to tie slightly above the larger coupting value. Are there finite size effects? Looking at the tightest quark masses we see a slight dropping of the value when increasing the spatial lattice size. This might be interpreted as a sign of some finite volume effects. We note also the difficulty of linear extrapolations - using our results alone would give a rather different result to the physical result. Using lighter quark mass results as well gives a more satisfactory extrapolation, [4]. The curve from the heavy quark mass world to the physical world is certainly rather unpleasant - the curve first goes up before sinking down. We have also used the quenched a p p r o x i m a t i o n in which the fermion determinant is ignored (otherwise the c o m p u t a t i o n s are too costly). This introduces an uncontrolled a p p r o x i m a t i o n - for example it is thus not clear t h a t the ratio in eq. (13) need be the physical value. Finally as we have a choice of which h a d r o n mass to use to find a, if we are not in the n e i g h b o u r h o o d of the physical point, we will get ambiguities in the value. From [4] 2, we find a ..~ 2.0 - 2.5GeV -1 (using m N , rap). 3. L O W

MOMENTS

1.2

c o m p a r i s o n w e also s h o w the results of [6], (triangles,/3 -6.0,183 x 32 lattice), [7] (inverted triangles,/3 -- 6.0, 243 x 32 lattice), a n d [8] ( d i a m o n d s , / 3 = 5.7,163 × 20 lattice). T h e fiUed squares are the experimental result a n d h e a v y q u a r k limit (1,3/2).

We shall now present our results for the low m o m e n t s of the structure functions. The computational m e t h o d is to evaluate the nucleon two point functions with a bilinear quark or gluon operator insertion, leading to a numerical estimate of the matrix elements in eqs. (2,5). There are two basic types: T h e first is quark insertion in one of the nucleon quark lines (quark line connected). As this involves a valence quark, we shall take this as probing the valence distributions in eq. (9). In the second type the operator interacts via the

of quark mass to be shown, from the heavy quark

2ran

, i I J , , I 0.2 0.4

, ~ I , , , I , , i I 0.6 0.8 1.0

(~/p)**2

Figure 1 . T h e A P E p l o t our results (filled circles,/3

rnN/rn p

a g a i n s t (rn,drnp)2 f o r ---- 6 . 0 , 1 6 3 × 3 2 l a t t i c e ) , [4]. F o r

= 0.440(4),

r r % -- 0 . 3 2 1 ( 4 )

a t ~ c --- 0 . 1 5 6 9 9 ( 5 ) .

253

M. G6ckeler et al./Nuclear Physics B (Proc. Suppl.) 49 (1996) 25~255

exchange of gluons with the nucleon (for quark hne disconnected and gluon operators). These are taken as probes of the sea and gluon distributions. (Of course as we work in the quenched approximation there may be some distortions in the various distributions.) Due to technical difficulties, such as the operator growing too large in comparison with the lattice spatial size and the mixing of the lattice operators with lower dimensional operators, it has only been possible to measure the lowest three quark moments for the unpolarised valence structure functions, to give an estimate for the lowest gluon moment and for the polarised valence structure functions to compute a0, a2 and d2. (Some sea quark distributions also have been a t t e m p t e d in [8,9].) After computation the lattice matrix elements must be renormahsed. For this we use at present the one-loop perturbation results, [4]. (However improvements are being developed, [10].) Finally we shall quote all our results at the reference scale, 1 Q2 =/~2 = ~ ~ 4 - 6GeV 2

(14)

(to avoid having to re-sum large logarithms in Z or the Wilson coefficients). Unpolarised In Fig. 2 we show the A P E plot for the 3 lowest valence moments. Also shown are the physical values (taken from [11]) and the heavy quark mass hmit (u~(z) ---* 2/36(z - 1/3), d~(z) -+ 1/36(z - 1/3), S(z) --, 0). We see that the results roughly seem to he on a smooth extrapolation from the heavy quark mass hmit to the physical result. (Although the hghtest quark mass seems to be levelling off, this should be checked on a larger lattice to see that there are no finite volume effects present.) It is to be hoped that smaller quark masses will continue to drift downwards. Certainly httle sign of overshooting is seen (as happened in the A P E mass plot). In Fig. 3 we show the ratios of the u~ to d~ moments. There seem indications of a very smooth (linear?) behaviour with the quark mass. In Fig. 4 we show the lowest gluon moment. The experimental result comes from [12]. (In the heavy quark mass hmit we would expect all the nucleon m o m e n t u m to reside in the quarks and none in the gluons.)

. . . . .

~

'

'

'

I

'

'

'

I

'

'

'

I

.6

A

.5

I c

× V

.5

.2

.I

0

0.2

0,4

m

0.6

al

0.8

Q

1.0

(nlp)**2 Figure 2. The A P E plot for valence results: (~n-1)(~) with n ----2,3,4, top to bottom, in turn first for the u quarks (squares) and then for the d quarks (circles).(For (x), we have evaluated the matrix elements in two distinct ways, [4],methods a,b.) The experimental numbers and heavy quark limit are the filledsymbols. At present the signal is not so good. This is numerically a difficult calculation however a n d requires a large statistic. N o statement at present can be m a d e on the m o m e n t u m s u m rule. Polarlsed W e n o w turn to a consideration of the polarised structure function m o m e n t s . W e first consider the non-singlet results for a0. T h e octet a n d triplet currents are given by 3FA -- D A = A u + A d F A + D A = gA = A u -

2As ~ Auv + Ad, Ad'~

Au,, - Ad,,

(15)

and thus we need only look at connected diagrams. Space prevents us from showing results for gA, 3 F A -- D A or F A / D A , we refer the interested reader to the nice review [13]. (Comparing the results there with the experimental and the heavy quark mass limit we again see for gA an overshoot effect: we seem to have to climb up to reach the experimental value.) For the singlet results we have Au + Ad + As -- A E and

254

M. GSckeler et al./Nuclear Physics B (Proc. Suppl.) 49 (1996) 250-255

1.0 .3.2

I i

,

,

i

,

,

,

i

,

,

,

i

,

,

,

i

,

,

,

i

L 1

0.9 0.8

5.0

0.7 2.8 0.6

2'

o o

×

2.6

0.5

V

-I 0.4

2.40.3 2.2

L

0,2 0,1

2.0 ,

,

,

J

0.2

,

,

,

I

,

,

h

I

0.4 0.6 (~/p)**2

,

,

,

I,

,

,

0,0 0.2

0.8

Figure 3. The APE plot for the ratio of the valence moments: (xn-x)(~)/(~r'~-l)(a). Method a (n = 2) are triangles. Method b (inverted triangles) and n = 3 (squares) are slightly displaced for clarity. so we expect the disconnected terms to be more important. This is computationally a very difficult problem, [8,9]. Recent numerical estimates however show an encouraging trend to the experimental result, [13]. Of the higher a moments we have been able to look at a2. Changing to p -- proton, n = neutron (rather than u, d), we plot a2 in Fig. 5. (The numbers for a2 and d2 are conveniently collated in [14].) Again we see a rather smooth behaviour between the heavy quark mass limit and the experimental result. Indeed, perhaps due to the present rather large errors there, we could claim that we have reasonable agreement with experiment. Finally we turn to an estimation of d2 - the first non-leading twist operator in the O P E expansion. In Fig. 6 we plot our results. While one might say that for n we are compatible with the experimental result, for p we have crass disagreement. At present we have no explanation for this result. (Indeed as we tend towards the chiral limit, the discrepancy grows.) In Fig. 7 we compare our results with other theoretical estimates. At present the lattice result comes out a

0.4 0.6 ( u / p )**2

0.8

1.0

Figure 4. The APE plot for (x)(9). poor last. (But one should note that the most compatible result to the experiment appears to be the heavy quark mass limit.) Conclusions Although progress has been made, it is clear that much more remains to be done. In particular for reliable extrapolations to the chiral limit, smaller quark mass simulations are necessary.

ACKNOWLEDGMENTS The numerical calculations were performed on the QUADRICS (Q16 and QH2) at DESY (Zeuthen) and at Bielefeld University. We wish to thank both institutions for their support and in particular the system managers H. Simma, W. Friebel and M. Plagge for their help.

REFERENCES 1. 2. 3.

J. Ashman et al., Nucl. Phys. B238 (1990) 1. K. Abe et al., Phys. Rev. Lett 75 (1995) 346, ibid 75 (1995) 25; hep-ex/9511013. M. G6ckeler et al., Nucl. Phys. B(Proc. Suppl.) 42 (1995) 337, (hep-lat/9412055); Yarnagata Workshop (hep-lat/9509079); Lat95

fvL Gdckeler et al./Nuclear Physics B (Proc. Suppl.) 49 (1996) 250 255

0

.12

.12

.10

.10

.08

.08

.06

.06

.04

.04

.02

*~

E

' ' I ' '

' I ' '

255

' I '

'

' I ' ' '

I

.02

C

c"

0

c

0

0

0

o -.02

8

(:l

--o2

CL

-.04

- . 0 4

o -.06

-.06

-.08

-.08

-.10

-.10

-.12 ,

,

,

I

0.2

,

¢

J

J

0.4

~

,

,

I

0.6

,

,

,

I

0.8

~

,

J

m ¢

-.12

I

1.0

0.2

Figure proton

4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

0.8

1.0

(~/p).,2 Figure 6.

5 . T h e A P E p l o t f o r a 2~'n. T h e s q u a r e s are the r e s u l t s , w h i l e t h e circles are t h e n e u t r o n r e s u l t s .

(hep-lat/9510017); Brussels HEP Conference (hep-lat/9511013); Zeuthen Spin Workshop (hep-lat/9511025). M. G6ckelcr, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. Rakow, G. Schierholz and A. Schiller, hep-lat/9508004 (accepted for PRD). R. D. Kenway, QCD 20 years later, Aachen, 1992. S. Cabasino et al. Phys. Lett. B214 (1988) 115. S. Cabasino et al. Phys. Lett. B258 (1991) 195. M. Fukugita et al., Phys. Rev. Left. 75 (1995) 2092, (hep-lat/9501010). S. 3. Dong et al., Phys. Rev. Lett. 75 (1995) 2096, (hep-lat/9502334). G. Martinelli et al., Nucl. Phys. B445(1995) 81, (hep-lat/9411010). A. D. Martin et al., Phys. Rev. D47 (1993) 867. A. D. Martin et al., Phys. Left. B354 (1995) 155, (hep-ph/9502336). M. Okawa, Lat95 (hep-lat/9510047). L. Mankiewicz et al., Zeuthen Spin Workshop (hep-ph/9510418).

0.6

0.4

( ~ / p ) * * 2

T h e A P E p l o t f o r d~ 'n. T h e s a m e n o t a t i o n

as

f o r F i g . 5.

.06

_'

'

'

'

r

'

'

'

I

'

'

'

[

'

'

,

J

,

I

J

,

I

'

I

,

.04 .02

t 0

"~" -.02 0 0

-.04 ÷

-.06

+

- . 0 8

-.10 --.12

J

--.04

J

,

I

,

k

,

I

--.02 0 .02 d2 (neutron)

.04

Figure 7. d(2P)versus d(2=) . The lattice results as squares,

sum rule as (inverted)

triangles,

are shown and the bag

e s t i m a t e as a diamond. The experimental value is a filled s q u a r e a n d t h e h e a v y q u a r k m a s s l i m i t is a f i l l e d c i r c l e .