We report on a lattice computation of the second moment of the pion matrix
element of the twist-2 non-singlet operator corresponding to the average
momentum of parton densities. We apply a fully non-perturbatively evaluated
running renormalization constant as well as a careful extrapolation of our
results to the continuum limit. Thus the only limitation of our final result is
the quenched approximation.Comment: 6 pages, talk presented at the XXXth International Conference on High
  Energy Physics (ICHEP 2000), July 27-August 2, 2000, Osaka, Japa

Jansen, K.

English

arXiv

We report on a lattice computation of the second moment of the pion matrix element of the twist-2 non-singlet operator corresponding to the average momentum of parton densities. We apply a fully non-perturbatively evaluated running renormalization constant as well as a careful extrapolation of our results to the continuum limit. Thus the only limitation of our final result is the quenched approximation.We report on a lattice computation of the second moment of the pion matrix element of the twist-2 non-singlet operator corresponding to the average momentum of parton densities. We apply a fully non-perturbatively evaluated running renormalization constant as well as a careful extrapolation of our results to the continuum limit. Thus the only limitation of our final result is the quenched approximation

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For Publisher’s usev1.0 - 7th December 1998.Ω00 <000010038: submitted to World Scientific on October 26, 2000 1For Publisher’s useSTRUCTURE FUNCTIONS ON THE LATTICEK. JANSENCERN, TH Division, 1211 Geneva 23, SwitzerlandE-mail: Karl.Jansen@cern.chWe report on a lattice computation of the second moment of the pion matrix element of the twist-2non-singlet operator corresponding to the average momentum of parton densities. We apply a fullynon-perturbatively evaluated running renormalization constant as well as a careful extrapolation ofour results to the continuum limit. Thus the only limitation of our final result is the quenchedapproximation.1 IntroductionA reliable computation of parton distributionfunctions from rst principles would be veryimportant for future experiments plannede.g. at the LHC. The results of such cal-culations would oer a unique way to testwhether QCD is indeed the correct theory forthe strong interactions.In principle, the lattice regularization of-fers such a calculational scheme, where theonly starting point is the QCD lagrangian,and indeed is able to give results for mo-ments of parton density distributions in thecontinuum and fully non-perturbatively, as wewill show in this contribution. The work pre-sented here is a summary of a series of publi-cations 1;2;3;4;5 that provide the essential in-gredients to reach the above ambitious aim.The moments of parton density distri-butions are related to expectation values ofcertain local operators, which are renormal-ized multiplicatively by applying appropriaterenormalization factors Z(1=) that dependon the energy scale . This leads to con-sider renormalized matrix elements OrenSF ()to be computed in a certain, at this stage notspecied, renormalization scheme SF. If theenergy scale  is chosen large enough, it is tobe expected that the scale evolution is verywell described by perturbation theory, givingrise to the following denition of a renormal-ization group invariant matrix element:OrenINV = OrenSF ()  fSF(g2()) (1)with g() the running coupling andfSF(g2) = (g2())−γ0=2b0 (2) exp{−∫ g¯()0dg[γ(g)(g)− γ0b0g]}where (g) and γ(g) are the  andanomalous-dimension functions computed toa given order of perturbation theory in thespecied scheme, i.e. here the SF scheme.Once we know the value of OrenINV evaluatednon-perturbatively, the running matrix ele-ment in a preferred scheme can be computed,for example in the MS scheme:OrenMS() = OrenINV=fMS(g2()) (3)with now, of course, the  and γ functionscomputed in the MS scheme.A non-perturbatively obtained value ofthe renormalization group invariant matrixelement is hence of central importance. Itscalculation has to be performed in severalsteps. The reason is that we have to covera broad range of energy scales { from thedeep perturbative to the non-perturbative re-gion. With the scale-dependent renormaliza-tion factor Z(1=) we can write the renor-malized matrix element of eq. (1) asOrenINV =hjONSjiZSF(1=) fSF(g2()) ; (4)with hjONSji the expectation value of the(non-singlet) operator under consideration ingiven states, here the pion states.So far, all our discussions have been inthe continuum. However, if we think of the0010038: submitted to World Scientific on October 26, 2000 2For Publisher’s uselattice regularisation and eventual numeri-cal simulations to obtain non-perturbative re-sults, it would be convenient to compute therenormalized matrix element at only one con-venient (i.e. small hadronic) scale 0. Wetherefore rewrite the r.h.s. of eq. (4) as:hjONSjiZSF(1=0) ZSF(1=0)ZSF(1=)︸ ︷︷ ︸(=0;g¯())fSF(g2()) (5)where we introduce the step scaling function(=0; g()), which describes the evolutionof the renormalization factor from a scale 0to a scale . The advantage of concentratingon the step scaling function instead of therenormalization factor itself, is that the stepscaling function is well dened in the contin-uum and hence suitable for eventual contin-uum extrapolations of lattice results.We nally write the r.h.s. of eq. (4) asOrenSF (0)(=0; g())  fSF(g2())︸ ︷︷ ︸SUVINV(0)(6)with OrenSF (0) the renormalized matrix ele-ment, which is to be computed only onceat a scale 0 and the (ultraviolet) invariantstep scaling function SUVINV(0), which stilldepends on the infrared scale 0. The fol-lowing sections are devoted to a descriptionof how these two basic ingredients can bereliably computed on the lattice using non-perturbative methods, i.e. numerical simula-tions.2 The renormalization groupinvariant step scaling functionLet us start this section by disclosing what ishidden behind the ctitious SF scheme men-tioned in the introduction. SF stands forSchro¨dinger functional and denotes a nitephysical volume, V = L3  T , renormaliza-tion scheme where the energy scale  is iden-tied with the inverse spatial length of thebox itself, e.g.  = 1=L. The peculiarity ofthe Schro¨dinger functional set-up is that xedboundary conditions in time x0 are imposedwith classical elds at the time boundaries attime x0 = 0 and x0 = T . For a more detaileddiscussion we refer the reader to 7.To discuss the renormalization of oper-ators related to moments of parton distri-bution functions, we rst have to provide arenormalization condition. Denoting by jSF ia classical SF state, i.e. a classical quark eldat a time boundary with external momentump, the renormalization condition that we willuse readshSF jOren( =1L)jSF i = hSF jOtreejSF i :(7)The relation between the expectation value ofthe bare operator and the renormalized one isestablished through a scale-dependent renor-malization constant:OR() = Z−1(1=)Obare(1=L) : (8)In perturbation theory, on the 1-loop level,we have Z(1=) = 1− g2() [γ(0)ln() +B0]with γ(0) the anomalous dimension and B0the constant part. Up to this point, the dis-cussion is given solely in the continuum wherethe SF renormalization scheme is a perfectlyacceptable one. Dierent schemes such as theMS scheme can be related to the SF schemeas usual in perturbation theory.If we are interested, however, in a non-perturbative calculation, we have to detourfor a short time (which means, however, asubstantial computer time) to a nite latticewith non-zero lattice spacing a that allowsfor numerical simulations. A lattice repre-sentation of the twist-2 non-singlet operator,which is the only case we are considering here,is given byO12(x) = 14 (x)γf1$D2g32 (x) ; (9)where$D is the covariant derivative and thebracket around indices means symmetriza-tion. The operator is probed by boundary0010038: submitted to World Scientific on October 26, 2000 3For Publisher’s usequark elds  and , which reside at x0 = 0and a correlation function is constructedfO12(x0a) =∑x;y;zheip(y−z)O12(x)(y)γ23(z)i(10)with p the spatial 3-momentum.To take into account the eects of theboundary elds, we also consider the bound-ary operators dened at the time boundariesx0 = 0 and x0 = T :O0 = a6L3∑y;z(y)γ532(z);OT = a6L3∑y;z0(y)γ5320(z) (11)from which we construct the correlation func-tion:f1 = −hO0OT i : (12)The boundary wave-function contributioncan then be taken out by considering the ra-tio fO12(x0=a)=pf1.There are several physical scales in ourproblem which all have to be given in unitsof L, which is the only scale that is to bechanged. Therefore, in order to arrive at adenition of the renormalization constant, wehave to specify the spatial momentum p, thequark mass mq and the time x0 where weread o the expectation value of the operatorbetween SF states from the correlation func-tion. The nal physical result does, of course,not depend on our choice of these quantities,but we choose them solely for convenience.In particular we selectmq = 0 ; x0 = T=4 ; (13)p  (p1; p2; p3) = (p1 = 2=L; 0; 0) :The choice of a zero quark mass results inusing a massless renormalization scheme andthe choice of the smallest available momen-tum on the lattice minimizes lattice artefacts.With the above choice, it is indeed only thephysical box length L (assuming T = L) thatwe identify with the inverse scale, which isvaried in the problem.We are now in a position to give theprecise denition of the renormalization con-stantZ(L) = Z(L)=Z1(L) ; (14)withZ(L) = fO12(T=4)=ftreeO12 (T=4) ;Z1(L) =√f1(L)=√f tree1 (L) ; (15)where we divide by the corresponding treelevel expression as required by the renormal-ization condition, eq. (7). Instead of comput-ing the Z-factors, we concentrate on the stepscaling functionsZ¯ =Z(2L)Z(L); f1 =Z1(2L)Z1(L); Z =Z(2L)Z(L);(16)because, in contrast to the Z-factors, the stepscaling functions have a well-dened contin-uum limit. The strategy is now to computethe step scaling functions at various valuesof the lattice spacing while keeping xed theconditions in eqs. (13) and the physical scale = L−1 (determined by the running cou-pling g()) and to extrapolate the resultsthus obtained to a = 0.It is one of the basic ingredients and char-acteristics of our work that almost all simula-tion results at non-zero lattice spacings havebeen obtained by employing the standardWilson action and the non-perturbatively im-proved clover action. Since these two formu-lations lead to dierent lattice artefacts, it isa very crucial test of our results that theircontinuum extrapolations give consistent re-sults. That this is indeed the case is demon-strated in g. 1. It shows that for the two stepscaling functions f1 and Z¯ the continuumlimit of both discretizations agree within theerror bars. We note that in the case of Z¯ aquadratic extrapolation in the lattice spacinga is necessary while for f1 a linear extrap-olation is sucient. After checking that asimilar behaviour is found at all values of thecoupling we have simulated, we performed0010038: submitted to World Scientific on October 26, 2000 4For Publisher’s useFigure 1. Continuum extrapolation of the step scal-ing functions Z¯ and f1 performed separately forthe Wilson action (circles and dotted lines) and thenon-perturbatively improved action (squares and fulllines) at a fixed value of the running coupling g¯2 =2:45.constraint ts, demanding that the contin-uum value of the step scaling functions bethe same from both actions. A summary ofour results for Z¯ is shown in g. 2.At this point, having performed all nec-essary continuum extrapolations, we end ourdetour on a lattice with non-zero lattice spac-ing and come back to the discussion in thecontinuum. With the results on the step scal-ing function, extrapolated to the continuumlimit, which were obtained at 9 values of therunning coupling constant, we can now com-pute the (ultraviolet) invariant step scalingfunctionSUVINV(0) = (=0; g2(0))  f(g2()) ;(17)which still depends on the infrared scale 0.This scale dependence will only be cancelledwhen multiplying with the matrix element,renormalized at the scale 0. The functionFigure 2. Constraint continuum extrapolation of Z¯(notation as in fig. 1).0010038: submitted to World Scientific on October 26, 2000 5For Publisher’s useFigure 3. The values of SUVINV(0) for two choices ofthe running coupling.f(g2()) is the same as in eq. (3) and the and γ functions are taken up to 3 loops in theSF schemea.In g. 3 we show SUVINV(0) as a func-tion of =0. For large enough energy scales,=0 > 100, SUVINV(0) does not changewithin the errors and we can determine avalue for it by tting the last, say, 4 points toa constant. Although there still is a schemedependence in the invariant step scaling func-tion through the remaining dependence on0, the value should be independent of thechoice of coupling used in the analysis. Thisis nicely illustrated in g. 3, where the choicesof g(L=4) and g(L) give consistent values forthe invariant step scaling function.We can therefore now give the rst pieceof information for the invariant matrix ele-ment itself, as needed in eq. (6), and quoteSUVINV(0) = 1:11(4) : (18)3 Renormalized matrix elementAs a next step we have to compute the renor-malized matrix element itself. Again, we willaFor the γ function we have taken an effective 3-loopparametrization as obtained by fitting our data to aneffective 3-loop form 2.always use the SF scheme and we remark thatthis is the rst time that a calculation of a2-quark matrix element is attempted in thisset-up. We rst tried to compute the matrixelement (within pion states) of the operatorof eq. (9). However, since this operator needsa non-vanishing momentum, we found, simi-lar as in 6, that the signal is very noisy. Wetherefore decided to switch to the operatorO00(x) =  (x)[γ0$D0 −133∑k=1γk$Dk]38 (x)(19)which has the advantage that it can be com-puted at zero momentum. Taking the bound-ary operators of eq. (11) we construct a cor-relation functionfM(x0) = a3∑xhO0O00(x)OT i ; (20)which again is to be normalized by f1,eq. (12), to take out the boundary wave-function contributions. Performing a transfermatrix decomposition, we nd that for largeenough values of x0 and staying far enoughfrom both boundariesf1 ’ 2e−mpiT ;fM(x0) ’ 2e−mpiT hjO00ji : (21)Assuming that there is a plateau region wherefM (x0)=f1 = const  hjO00ji, and inwhich the rst excited state gives essentiallyno contributions, we obtain the physical ma-trix element hxi after a suitable normaliza-tion (see 6):hxi  2mhjO00ji : (22)In order to extract the matrix element ofeq. (22) we have chosen lattices with T =3 fm and followed the correlation functionfM(x0) up to a distance of 1 fm in time di-rection. At this distance we are sure thatwe project on the pion states as an inspec-tion of the pseudoscalar and axial-vector cor-relation functions (from which we also ex-0010038: submitted to World Scientific on October 26, 2000 6For Publisher’s useFigure 4. Constrained continuum extrapolation ofthe renormalized matrix elementtracted the pion masses) showed. Indeed, for1 fm < x0 < 2 fm the correlation functionexhibits a plateau behaviour as can be seenin g. 2 of ref. 5.Once we have the bare matrix elementwe need to renormalize it. To this end wecomputed Z(1=0) with −10 = 1:436r0, r0 0:5 fm. We repeated such a calculation forvarious lattice sizes, choosing the values of such that Lmax = −10 is kept xed. In-terpolating the numerical simulation data weobtain in this way Z(1=0) in a range of lat-tice spacing 0:05  a  0:1, i.e. the rangeof a where the bare matrix element itself hasbeen computed.This allowed us to apply the renormal-ization factor at exactly the same values of where the matrix element has been com-puted. We show the matrix element renor-malized with Z12(1=0) in g. 4. Again allcalculations have been performed with twodierent discretizations, and we nd quiteconvincingly the same continuum limit.It might, however, not have escaped thereader’s attention that we have used thewrong renormalization factor, namely Z12,for renormalizing the operator O00(x) ofeq. (19). The continuum extrapolation of thesuch renormalized operator needs a correc-tion factor. In ref. 5 we have demonstratedthat this correction factor can be taken fromperturbation theory and amounts to a shift ofthe continuum renormalized matrix elementby a few per cent. Taking the correction fac-tor to be the same as found in 5, we can nallygive our main result:OrenINV = 0:222(24) (23)4 ConclusionIn this contribution we have demonstratedhow we can compute on the lattice, in a fullynon-perturbative fashion, a renormalizationgroup invariant matrix element OrenINV for thesecond moment of parton distributions of thetwist-2 non-singlet operator in the pion. Apreliminary value for OrenINV is given in eq. (23)and can be used now as integration constantto obtain the renormalized matrix elementat any scale in the preferred renormalizationscheme.We want to emphasize that the value ofthe renormalized matrix element is shiftedsubstantially from a value of about hxi(a =0:093) = 0:30 at  = 6 to hxi(a = 0) = 0:2in the continuum limit. Hence we experiencestrong lattice artefacts in the renormalizedmatrix element. Still, when the matrix ele-ment is run in the MS scheme to a scale of = 2:4 GeV, we nd hxi( = 2:4GeV)  0:3.Thus we nd that, by a conspiration of twoeects, our result agrees with the numberquoted in the pioneering work of 6. There-fore, the fact that within the quenched ap-proximation, used here exclusively, the num-ber from the lattice simulations is higher thanthe experimental value persists.AcknowledgementsThe work presented in this talk is part ofa most enjoyable and lively collaboration0010038: submitted to World Scientific on October 26, 2000 7For Publisher’s usewith M. Guagnelli and R. Petronzio. I alsowant to thank A. Bucarelli, F. Palombi, andA. Shindler for useful discussions and essen-tial contributions.References1. A. Bucarelli, F. Palombi, R. Petronzio,A. Shindler, Nucl. Phys. B552 (1999)3792. M. Guagnelli, K. Jansen, R. Petronzio,Nucl. Phys. B542 (1999) 3953. M. Guagnelli, K. Jansen, R. Petronzio,Phys. Lett. B457 (1999) 1534. M. Guagnelli, K. Jansen, R. Petronzio,Phys. Lett. B459 (1999) 5945. M. Guagnelli, K. Jansen, R. Petronzio,preprint, CERN-TH/2000-2596. C. Best et al., Phys. Rev. D56 (1997)27437. M. Lu¨scher, R. Narayanan, P. Weisz andU. Wol, Nucl. Phys. B384 (1992) 168S. Sint, Nucl. Phys. B421 (1994) 1350010038: submitted to World Scientific on October 26, 2000 8

Structure Functions on the lattice

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