Desy 04194 Edinburgh 2004/24 Lth 638 LuItp 2004/039 October 2004 A lattice determination of moments of unpolarised nucleon structure functions using improved Wilson fermions
Abstract
Within the framework of quenched lattice QCD and using improved Wilson fermions and nonperturbative renormalisation, a high statistics computation of low moments of the unpolarised nucleon structure functions is given. Particular attention is paid to the chiral and continuum extrapolations.
1 Introduction
The results of a lattice simulation of Quantum Chromodynamics (QCD) give in principle a direct probe of certain low energy aspects of the theory, such as hadronic masses and matrix elements. This is at present the only way of getting these quantities from QCD, without additional modeldependent assumptions. A useful theoretical tool in conjunction with QCD and deep inelastic scattering (or DIS) experiments is the operator product expansion, OPE. At leading twist the OPE relates moments of an experimentally measured structure function, generically denoted by , to certain matrix elements where
(1) 
is a function of two variables – , the spacelike momentum transfer to the nucleon and , the Bjorken variable ( is a normalisation factor). are the nucleon matrix elements of certain operators and are the associated Wilson coefficients. These are perturbatively known at high energies where the coupling constant becomes small and are found in a specified scheme at scale . Usually, of course, we take at scale few GeV. We also assume that is large enough, so that higher twist terms ie terms are negligible.
As will be discussed later (section 4.3), lattice computations are presently restricted to determining nonsinglet, NS, nucleon structure functions
(2) 
ie the difference between proton, , and neutron, , results. Note in particular that this means that nucleon matrix elements of gluonic operators have cancelled.
In this article we shall only be concerned with unpolarised structure functions. The same matrix elements contribute to the scattering of charged leptons and of neutrinos, but the weights are different in the two cases. Thus for charged lepton–nucleon DIS, , which is mediated by a photon, we have with and . For neutrino–nucleon charged weak current interactions for example , () or , () which are mediated by , respectively, then we have and , (neglecting the CKM mixing matrix) with . (Alternatively setting in all cases one has the same matrix elements and s as for in eq. (1), but different Wilson coefficients. The additional structure functions, occuring because of parity nonconservation also obey eq. (1) with and again with .) Similar expressions hold for the neutral currents, but with more complicated expressions for the s.
In all cases the relevant matrix elements are given by first defining the sequence of quark bilinear forms
(3) 
where . Symmetrising the indices and removing traces, gives the Lorentz decomposition of the proton (ie nucleon, N) matrix element of^{1}^{1}1The nucleon states are normalised with the convention .
(4) 
For example we have for ,
(5)  
and more complicated expressions hold for higher moments. Finally, the nonsinglet, NS, matrix element is defined as
(6) 
In this article, we shall compute , and in the quenched approximation (), by finding the appropriate matrix elements in eq. (4). As will be seen most effort will be spent on , as this is technically less complicated than the higher moments, and also numerically the lattice results are more precise. The lattice approach discretises Euclidean spacetime, with lattice spacing , in the path integral and simulates the resulting highdimensional integral for the partition function using Monte Carlo techniques. Matrix elements can then be obtained from suitable ratios of correlation functions, [1, 2]. Note that the lattice programme is rather like an experiment: careful account must be taken of error estimations and extrapolations. There are three limits to consider:

The spatial box size must be large enough so that finite size effects are small. Currently sizes of seem large enough (the nucleon diameter is about )^{2}^{2}2Additionally all our current lattices have where is the pseudoscalar mass.. This situation is probably more favourable for quenched QCD (where one drops the fermion determinant in the path integral, see section 4), as due to the suppression of the pion cloud, we would expect the radius of the nucleon to be somewhat smaller. This indeed seems to be the case, see eg [3].

The continuum limit, . We use improved Wilson fermions (where the discretisation effects of the action and matrix elements have been arranged to be ). For unimproved Wilson fermions, or where one has not succeeded in entirely improving the matrix element we should extrapolate in rather than .

The chiral limit, when the quark mass approaches zero. There has recently been much activity on deriving formulae for this limit, [4, 5, 6, 7, 8, 9]. However while most of these results are valid around the physical pion mass on the lattice, it is difficult to calculate quark propagators at quark masses much below the strange quark mass. Thus the use of these formulae is not straightforward.
In addition the lattice matrix element must also be renormalised.
Previous lattice studies gave discrepancies to the phenomenological results, especially for . In this work we want to try to narrow down the sources for this difference. In particular we shall present here nonperturbative, NP, results for the renormalisation constants (and as many previous studies used results based on perturbation theory compare with these other results). We also consider improvement and operator mixing to enable a reliable continuum extrapolation to be performed.
Compared to our previous work [2] we have improved our techniques in several respects:

We employ nonperturbatively improved Wilson fermions instead of unimproved Wilson fermions. This should reduce cutoff effects.

Modified operators are used for , , which improves the numerical signal.

In [2] we had simulations for a single lattice spacing only. Here we shall present results for three different values of the lattice spacing so that we can monitor lattice artefacts.

As mentioned before, the 1loop perturbative renormalisation factors of [2] have been replaced by nonperturbatively calculated renormalisation constants. In addition we shall pay close attention to possible mixing problems of the operators involved.

We have increased the number of quark masses at each value of the lattice spacing in order to improve the chiral extrapolation.
The organisation of this paper is as follows. In section 2 various continuum results for the and functions and for the Wilson coefficients in the scheme are collated and renormalisation group invariant quantities are introduced, while in section 3 some NS phenomenological results are discussed to compare later with the lattice results. Section 4 describes our lattice techniques: choice of operators, operator mixing problems, improvement and gives the bare, ie unrenormalised results. Section 5 discusses and compares various renormalisation results: oneloop perturbation theory and a tadpole improvement, together with the nonperturbative method. In section 6 we discuss the results and give continuum and chiral extrapolations. Finally in section 7 we present our conclusions.
2 Continuum QCD results
In this section we shall consider the RHS of eq. (1). Much of the functional form is already known: the lattice input is reduced to the computation of a single number (for each moment).
The running of the coupling constant as the scale is changed is controlled by the function. This is defined as
(7)  
with the bare parameters being held constant. This function is given perturbatively as a power series expansion in the coupling constant. The first two coefficients in the expansion are universal (ie scheme independent). In the scheme where , the expansion is now known to four loops [10, 11]. The threeloop result for quenched QCD is
(8) 
We may immediately integrate eq. (7) to obtain the solution,
(9) 
where is an integration constant.
While eq. (1) is the conventional definition of the moment of a structure function, for us it is more convenient to rewrite it using renormalisation group invariant (or RGI) functions. The bare operator (or matrix element) must first be renormalised
(10) 
giving , the anomalous dimension of the operator,
(11)  
(The first coefficient is again scheme independent.) One may also change the scale and/or scheme for the operator by
(12) 
This leads to two anomalous dimension equations, obtained by either differentiating with respect to or . Integrating these equations gives
(13) 
where we have defined
(14) 
From eqs. (12) and (13), we see that we can define a RGI operator by
(15) 
Then obviously is independent of the scale and scheme. The function thus controls how the matrix element changes as the scale is varied. Note also that the normalisation of depends on the convention chosen for , here given in eq. (14).
As the LHS of eq. (1) is a physical quantity, the RGI form for the Wilson coefficient is given by
(16) 
It is convenient to choose , as then
(17) 
and has no large numbers in it, so that a perturbative power series in becomes tenable. In two schemes and from eq. (17) we have
(18) 
because in this limit . Hence is independent of the scheme. With our convention for this is , so that
(19) 
Practically we shall here only consider the , and moments. For these moments we have, for quenched QCD (ie ) [12, 13, 14, 15],
(20)  
for , and respectively (). The operator has anomalous dimensions given by, [16, 17, 18],
(21)  
again for , and , respectively. Solving first eq. (9) for and then using eq. (14) gives the results for shown in Fig. 1.
Note that by loop expansion, we mean using the and function result to the appropriate order; we do not expand eqs. (9), (14) any further, but solve them numerically.
To determine a result in GeV, we shall use the scale here, [19]. From [20, 21] we take for quenched QCD and together with the scale choice this gives for an energy of for example, . The Wilson coefficient, can also be found and is shown in Fig. 2 for , . To obtain from eq. (17) we must simply multiply the results from Fig. 1 with those of Fig. 2. From this latter
figure we see that the change from the tree level result for the moment in the Wilson coefficient is at most for and is practically negligible. This is not so for the higher moments, when the Wilson coefficient deviates significantly from one. Useful values for (relevant for the forthcoming lattice results) are given in Table 6 in Appendix A.
3 Phenomenology and experimental data
Ideally we would like to make a direct comparison between the theoretical and experimental result, by rewriting eq. (1) as,
(22) 
The RHS of this equation has a clean separation between a number , which can only be obtained using a nonperturbative method (eg the lattice approach) and a function, , which describes all the momentum behaviour of the moment.
More conventional (and practical) however is to use parton densities. Usually phenomenological fits using parton densities are obtained from global fits (such as MRST, [22] and CTEQ, [23]) to the data. In this section we shall compare whether taking moments of the structure function gives the same answer as taking moments of the parton density. This could also help in estimating the error in the phenomenological fit. Parton densities , are implicitly defined by
(23) 
We may relate the structure function to the parton density via a convolution. Defining similar but separate Mellin transformations for even and odd by
(24) 
(where in the inverse transformation, in is analytically continued from even/odd integer values to complex numbers and is chosen so that all singularities lie to the left of the line ) then gives,
(25)  
where for even (ie for ) and for odd (ie for ). To lowest order in the coupling constant we get from eq. (19), and so^{3}^{3}3There are many references to the relationship between structure functions and parton densities. See for example [24].
(26) 
The parton densities are usually determined from global fits to the data, with an assumed functional form, typically for MRST results like
(27) 
with parameters , , , and at some given reference scale . For the MRST results given here we use the fit MRST200E, [25] together with the error analysis of [22] at a scale of . As a comparison we also consider the CTEQ fit CTEQ61M, with errors calculated from [23]. (Practically, in both cases, we use the parton distribution calculator [26] to compute the moments in eq. (23).)
Let us now briefly consider some lepton–nucleon DIS experimental results. While is well known, experiments with deuterium to find are much more difficult, due to target nuclear effects. We shall use here the results from [27] which employ both proton and neutron (deuterium) targets in the same experiment, which thus minimizes systematic errors. In [28] this has been combined with the world data, [29], and is shown in Fig. 3
in the form of a series of bins at different values. (Naively, if there were no QCD interactions, the parton model would give a deltafunction distribution at . This distribution has been considerably washed out here though.) There is a paucity of data for larger . However is dropping rapidly to zero, so any error here will not affect the low moments. As shown in the figure, we have simply made a linear extrapolation to , to estimate this region. To find the moments for eq. (1), we simply need to find the area under the bins weighted with the appropriate moment, ie
(28)  
where the first error is from the data and the second error is the effect of dropping the estimated last bin. (As expected, we see that higher moments are more sensitive to this bin.)
In Table 1 we give estimates of
‘World’  MRST  CTEQ  

0.161(4)  0.157(9)  0.155(17)  
0.226(14)  0.220(18)  0.217(27)  
  0.0565(26)  0.0551(51)  
  0.0972(95)  0.095(12)  
0.0241(13)  0.0231(11)  0.0230(23)  
0.0480(58)  0.0460(54)  0.0458(67) 
, , using estimates of the Wilson coefficients and given in the figure caption. (These numbers are similar to the quenched results, as can be seen from the caption of Fig. 2 and Table 6.) We find that there is good agreement between the two methods, with the lowest moment from MRST or CTEQ being slightly smaller than the experimental result. Thus these global fits describe the (low) moment data well^{4}^{4}4Note that a recent analysis, [30], gives similar results of and .. In future for definiteness we use the MRST results.
4 The Lattice Approach
Euclidean lattice operators^{5}^{5}5Our Euclidean conventions are described in [31]. are defined by
(29) 
where is taken to be either a or quark and is an arbitrary product of Dirac gamma matrices. (The index will be usually suppressed.) We have used the lattice definitions
(30) 
and . For the operators corresponding to eq. (3), we shall only need . However for the discussion on mixing under renormalisation, we shall also use and .
4.1 Choice of lattice operators
We now take the simplifying choices of two momenta or with being the lowest nonzero momentum possible on a periodic lattice ie where the number of sites in each spatial direction is . We take our lattice operators as^{6}^{6}6, are the same operators as we used previously in [2], while there are some modifications to and . For we have effectively made the proper rotation (ie preserving ) of (and ) of the operator in [2]. This means that the measured ratio in eq. (39) is rather than and hence the signal is better by a factor . For we have symmetrised over the , indices of the operator given in [2], which makes the measurement of the ratio for the new operator numerically a little less noisy.,
(31) 
Of course, there are other possibilities. However these will all require nonzero momenta in two directions or suffer from more severe mixing problems. As we shall see, even a nonzero momentum in one direction leads to a strong degradation of the signal and with two nonzero momenta very little signal is observed, [32].
Operator  

Note in particular that the ‘offdiagonal’ () and ‘diagonal’ operators () for belong to different representations, in distinction to the continuum operator. Thus we expect that although these operators should have different lattice artefacts and renormalisation factors, in the continuum limit both should lead to the same result – potentially a useful check.
4.2 Mixing of lattice operators
A given operator of engineering dimension can mix with operators (with the same quantum numbers) of lower dimension, the same dimension or higher dimension. improvement involves mixing with one dimension higher operators (irrelevant operators) where the choice of improvement coefficients is conventionally treated separately to mixing with operators of dimension (relevant operators). We shall follow this practice here.