7 pages, 4 figures, talk given at the 32nd International Symposium on Lattice Field Theory (Lattice 2014), June 23-28 2014, New York, USAWe report on our exploratory study for the evaluation of the parton distribution functions from lattice QCD, based on a new method proposed in Ref.~arXiv:1305.1539. Using the example of the nucleon, we compare two different methods to compute the matrix elements needed, and investigate the application of gauge link smearing. We also present first results from a large production ensemble and discuss the future challenges related to this method

Alexandrou, C

Cichy, K

Drach, V

Garcia-Ramos, E

Hadjiyiannakou, K

Jansen, K

Steffens, F

Wiese, C

We report on our exploratory study for the evaluation of the parton distribution functions from lattice QCD, based on a new method proposed in Ref. arXiv:1305.1539. Using the example of the nucleon, we compare two different methods to compute the matrix elements needed, and investigate the application of gauge link smearing. We also present first results from a large production ensemble and discuss the future challenges related to this method

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School of Engineering, Computing and Mathematics Faculty of Science and Engineering 2014-01-01 First results with twisted mass fermions towards the computation of parton distribution functions on the lattice C Alexandrou K Cichy V Drach School of Engineering, Computing and Mathematics E Garcia-Ramos K Hadjiyiannakou et al. See next page for additional authors Let us know how access to this document benefits you General rights All content in PEARL is protected by copyright law. Author manuscripts are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author. Take down policy If you believe that this document breaches copyright please contact the library providing details, and we will remove access to the work immediately and investigate your claim. Follow this and additional works at: https://pearl.plymouth.ac.uk/secam-research Recommended Citation Alexandrou, C., Cichy, K., Drach, V., Garcia-Ramos, E., Hadjiyiannakou, K., Jansen, K., Steffens, F., & Wiese, C. (2014) 'First results with twisted mass fermions towards the computation of parton distribution functions on the lattice', Proceedings of Science, . Retrieved from https://pearl.plymouth.ac.uk/secam-research/1207 This Article is brought to you for free and open access by the Faculty of Science and Engineering at PEARL. It has been accepted for inclusion in School of Engineering, Computing and Mathematics by an authorized administrator of PEARL. For more information, please contact openresearch@plymouth.ac.uk. Authors C Alexandrou, K Cichy, V Drach, E Garcia-Ramos, K Hadjiyiannakou, K Jansen, F Steffens, and C Wiese This article is available at PEARL: https://pearl.plymouth.ac.uk/secam-research/1207 arXiv:1411.0891v1  [hep-lat]  4 Nov 2014DESY-14-204, SFB/CPP-14-83, HU-EP-14/45, CP3-Origins-2014-036 DNRF90, DIAS-2014-36First results with twisted mass fermions towards thecomputation of parton distribution functions on thelatticeConstantia Alexandrou(a,b,c), Krzysztof Cichy(c,d), Vincent Drach(e), ElenaGarcia-Ramos(c, f ), Kyriakos Hadjiyiannakou(a), Karl Jansen(c), Fernanda Steffens(c),Christian Wiese∗ (c)(a) Department of Physics, University of Cyprus, P.O. Box 20537, 16 8 Nicosia, Cyprus(b) Computation-based Science and Technology Research Center, Cyprus Institute, 20 Kavafi Str.,Nicosia 2121, Cyprus(c) John von Neumann Institute for Computing (NIC), DESY, Platanen llee 6, D-15738 Zeuthen,Germany(d) Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland(e) CP3-Origins and the Danish Institute for Advanced Study DIAS, University of SouthernDenmark, Campusvej 55, DK-5230 Odense M, Denmark( f ) Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin,GermanyE-mail: christian.wiese@desy.deWe report on our exploratory study for the evaluation of the parton distribution functions fromlattice QCD, based on a new method proposed in Ref. arXiv:1305.1539. Using the example of thenucleon, we compare two different methods to compute the matrix elements needed, and investi-gate the application of gauge link smearing. We also presentfirst results from a large productionensemble and discuss the future challenges related to this method.The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/Hadron structure Christian Wiese1. IntroductionAn essential part in understanding QCD is to study the structu e of hadrons, e.g. the nucleon.Therefore, in the last decades a lot of effort was spent into ob aining the parton distribution function(PDF) of the nucleon. In particular,q(x) is the PDF that gives the probability of finding a quarkqcarrying a momentum fractionx of the parent hadron.On the experimental side, deep inelastic scattering is an important tool to access the structureof the nucleon. From the scattering cross section of these exp riments structure functions of thenucleon can be extracted. Perturbative QCD can then be used to rela e the experimental resultsobtained from different scales. However, perturbation theory does not give us any information aboutthe quark and gluon distributions themselves. For that one needs to assume input distributions,which are then fitted to the available data set. Different groups make different assumptions for theform of these input distribution and which data sets to include in the fit. This yields the PDFs albeitwith some model dependence.Therefore, a method to compute the quark distribution from first principles is highly desirable.On the lattice, however, it was up to now only possible to compute low moments of the PDFs (cf.e.g. [1]).Recently, a method to compute these distributions on a Euclidean lattice has been proposed in[2], and has first been applied in [3]. We present an exploratory study of this method using a setupof maximally twisted mass fermions with a focus on differentmethods to calculate the PDFs andsystematic errors appearing in the computations.2. Euclidean formulation of the light cone operatorThe quark distribution inside a nucleon is usually defined via matrix elements of the light coneoperator,q(x) =12π∫dξ−e−ixp+ξ−〈N|ψ(ξ−)ΓL (ξ−,0)ψ(0)|N〉, (2.1)whereξ− = ξ0−ξ 3√2andL (ξ−,0) is the Wilson line from 0 toξ−. Because deep inelastic scatteringis light cone dominated this expression has to be evaluated at ξ 2 ∼ 0. On a Euclidean lattice wewould thus have to selectξ 2 =~x2+ t2 ∼ 0, which is very hard to calculate due to a non-zero latticespacing.A new idea is to compute a quasi distribution ˜q which is purely spatial and uses nucleons withfinite momentum; hence, it can be computed on the lattice. Thequasi distribution is given byq̃(x,µ2, pz) =12π∫d∆z e−ixpz∆z〈N(pz)|ψ(∆z)γzL (∆z,0)ψ(0)|N(pz)〉µ2. (2.2)Here∆z is a distance in any spatial directionz and pz a momentum boost in this direction. Noticethat whenpz → ∞ we recover the usual definition of the quark distributions inthe infinite momen-tum frame, which are equivalent to the light cone distributions of Eq. (2.1). However, on the latticeone is limited to finite values ofpz. Thus, the proposal is to calculate ˜q at finite pz and relate it to2Hadron structure Christian WieseN(~pz, t0) N(~pz, ts)y= (~y,τ)∆ztFigure 1: Schematic picture of a possible Wick contraction of the quark fields in the three-point function.the usual distributions via perturbation theory [4],q̃(x,µ2, pz) =∫dy|y| Z(xy,ΛQCDpz,µpz)q(y,µ2)+O((mNpz)2). (2.3)3. Implementation and algorithmic testsThis work presents the first steps of the PDF calculation, which is the computation of the ma-trix elements appearing in Eq. (2.2), and some of the relatedlgorithmic and conceptual challenges.The first task is to compute the bare matrix element needed forthe quasi distribution,h(pz,∆z) = 〈N(pz)|ψ(∆z)γzL (∆z,0)ψ(0)|N(pz)〉 . (3.1)In order to compute these matrix elements a nucleon-nucleonthree-point function is required:C3pt(t,τ ,0) = 〈Γαβ Nα(~pz, t)O(τ)Nβ (~pz,0)〉, (3.2)whereΓαβ is a suitable parity projector. Here we will use the parity plus projectorΓ =1+γ42 .A nucleon field boosted with a three momentum can be defined viaa Fourier transformationof quark fields in position spaceNα(~pz, t) = ∑~xei~pz~xεabcuaα (x)(dbT(x)C γ5uc(x)). (3.3)In order to obtain a vanishing momentum transfer at the operator (Q2 = 0) we need to write itas follows:O(∆z,τ ,Q2 = 0) =∑~yψ(y+∆z)γzL (y+∆z,y)ψ(y), (3.4)with y= (~y,τ). After Wick contraction of the quark fields, the three-pointfunction can be expressedin terms of quark propagators. Fig. 1 shows a possible contraction.3Hadron structure Christian WieseFinally we can extract the matrix element via a ratio of a three- and a two-point functionC2pt(t,0;~pz) = Γαβ Nα(~pz, t)Nβ (~pz,0)C3pt(t,τ ,0;~pz)C2pt(t,0;~pz)0≪τ≪t=−ipzEh(pz,∆z), (3.5)whereE =√(pz)2+m2N is the total energy of the nucleon. As the flavor structure of the operatorwe will useu−d in order to have an isovector operator and avoid disconnected diagrams.An important conceptual problem one encounters when tryingto compute the matrix elementis the calculation of the propagator connecting the sink with the operator insertion (marked red inFig. 1). Due to momentum projection there is a spatial sum on both ends of the propagator. For thispurpose an all-to-all propagator is needed. However, the computation using point sources wouldtakeV = L3×T sets of inversion, which is too much computational effort.Here we test two different approaches to compute the all-to-a l propagator: the sequentialmethod, where we use standard methods to compute the two quark propagators that do not have theoperator insertion and contract the result to a sequential source. This can again be inverted and thencontracted with the operator to obtain the final result. However, in order to use this method the sinkhas to be fixed, which requires a new inversion for each momentum. Thus, six sets of inversionshave to be computed for each flavor, accounting for positive and negative momenta in all threespatial directions. For higher momenta this would have to berep ated.The second method is the stochastic method, where we use source that containZ4 noise onone single time-slice (cf. e.g. [5]). In addition we use diluted sources, which means that the noise isonly on one single spin and color slice. After inverting these sources, the obtained sink can be usedtogether with the source to estimate the all-to-all propagator. As a drawback we add stochasticnoise to the result, however, for this method one set of inversions for each flavor is sufficient toaccount for all momenta.For first initial tests we use a 163 × 32 twisted mass ensemble withβ = 3.9 [6] generatedby the tmLQCD software package [7]. We choose a twisted mass prameter ofµ = 0.004, whichcorresponds to a pion mass ofmPS≈ 340 MeV. For the matrix element the source-sink separation is6a and we only compute momentumpz = 12πL . We show only the real part of the matrix elements.The imaginary part has a non-vanishing contribution, however, it is relatively small and can atthe moment be neglected. Fig. 2 shows the comparison betweenth sequential and the stochasticmethod. For the stochastic method we show a different numberof noise vectors. Note that six noisevectors would compare to the computational cost of the sequential method.Both methods yield results that are compatible within errors. One cannot directly comparethe errors of both methods. However, taking into account thediff rent costs of inversions, bothmethods are approximately equal.In [3] the authors applied HYP smearing [8] to the gauge links. This is known to bring the nec-essary renormalization factors closer to the corresponding tree level value. To study the influenceof HYP smearing we apply one and two steps of smearing and showthe results in Fig. 3.The results indicate that gauge link smearing increases thevalu of the matrix elements, espe-cially for larger∆z, however, it does not decrease the noise of the result, as it was observed e.g. forthe gluon moment [9]. Thus, for the following computations we ill not use gauge link smearingand try to compute the renormalization factor for each∆zseparately.4Hadron structure Christian Wiese00.511.52−8 −6 −4 −2 0 2 4 6 8matrixelementhu−d∆zsequential method3 stoch. vectors2 stoch. vectors1 stoch. vectorFigure 2: Comparison of the sequential method with the stochastic method (with several numbers of noisevectors). The computational cost of the sequential method equals six noise vectors of the stochastic method.We show the real part of the matrix element.00.511.52−8 −6 −4 −2 0 2 4 6 8matrixelementhu−d∆zno HYP, real part1 step HYP, real part2 step HYP, real partFigure 3: Different number of gauge link smearing steps for the real part of the matrix element.5Hadron structure Christian Wiese00.511.52−10 −5 0 5 10matrixelementhu−d∆zpz = 1 2πLpz = 2 2πLFigure 4: Real part of the matrix element for the first two momenta with 1000 measurements.4. Results from Nf = 2+1+1 ensembleAfter these initial tests we continued with a larger ETMC production ensemble [10]. We de-cided to use the stochastic method because, although both methods yield equal results, the stochas-tic method is more flexible concerning the study of larger momenta. The matrix elements are com-puted on a 323×64 lattice withNf = 2+1+1 flavors of maximally twisted mass fermions. Thisensemble hasβ = 1.95, which corresponds to a lattice spacing ofa ≈ 0.078 fm and the twistedmass parameterµ = 0.0055, which is a pion mass ofmPS≈ 373 MeV. All the results presented arecomputed with a source-sink separation of 10a. With our current statistics ofNcon f = 1000 we areable to extract the matrix element for the first two momenta. We display the result in Fig. 4. Notethat the value for∆z= 0, which can be identified with the local vector current atQ2 = 0, has to berenormalized withZV , which is for this ensembleZV = 0.627(4) [11]. After renormalization theconditionFu−d1 (Q2 = 0) = 1 (see e.g. [12]) is fulfilled within errors.5. Conclusion and outlookWe have investigated a new method for the computation of quasi parton distributions and haveshown that using a stochastic estimator for the all-to-all propagator is a well-suited method for thecomputation of the necessary matrix elements. Smearing is shown not to improve the statisticaluncertainty and was thus dropped.Given the first results obtained using 1000 measurements, weare currently increasing statis-tics, which can reach up to 30 000 measurements.6Hadron structure Christian WieseAt present, we are also working on the next steps of the project. This includes the implemen-tation of the matching factorsZ of Eq. (2.3). The factors were already computed to one loop in[4],however the implementation and numerical solution of the equation are not easily done. In additionwe are working on implementing the finite nucleon mass correction [13].In parallel, we are continuing to study systematic effects that might occur in the simulation,e.g. we study the influence of excited states on the matrix elements by varying the source-sink sep-aration. We also plan to include more momenta, in particularwhen the mentioned larger statisticsis available.Another challenge is the renormalization of the quantitieshat are involved. One possibilityis to compute separate renormalization factors for the matrix elements with different∆z. We arein the process of testing different ideas for this renormalization process, details of which will bepresented elsewhere.Finally, it would be attractive to use the new ETMC ensemble [14] to compute the quarkdistribution directly at the physical value of the pion mass.References[1] C. Alexandrou, EPJ Web Conf.73 (2014) 01013 [arXiv:1404.5213 [hep-lat]].[2] X. Ji, Phys. Rev. Lett.110 (2013) 262002 [arXiv:1305.1539 [hep-ph]].[3] H. W. Lin, J. W. Chen, S. D. Cohen and X. Ji, [arXiv:1402.1462 [hep-ph]].[4] X. Xiong, X. Ji, J. H. Zhang and Y. Zhao, Phys. Rev. D90 (2014) 014051 [arXiv:1310.7471 [hep-ph]].[5] C. Alexandrouet al. [ETM Collaboration], Eur. Phys. J. C74 (2014) 2692 [arXiv:1302.2608[hep-lat]].[6] K. Cichy, V. Drach, E. Garcia-Ramos, G. Herdoiza and K. Jansen, Nucl. Phys. B869 (2013) 131[arXiv:1211.1605 [hep-lat]].[7] K. Jansen and C. Urbach, Comput. Phys. Commun.180 (2009) 2717 [arXiv:0905.3331 [hep-lat]].[8] A. Hasenfratz and F. Knechtli, Phys. Rev. D64 (2001) 034504 [hep-lat/0103029].[9] C. Alexandrou, V. Drach, K. Hadjiyiannakou, K. Jansen, B. Kostrzewa and C. Wiese,PoS(LATTICE 2013)289 [arXiv:1311.3174 [hep-lat]].[10] R. Baron, P. Boucaud, J. Carbonell, A. Deuzeman, V. Drach, F. Farchioni, V. Gimenez andG. Herdoizaet al., JHEP1006 (2010) 111 [arXiv:1004.5284 [hep-lat]].[11] C. Alexandrou, M. Constantinou, T. Korzec, H. Panagopoulos and F. Stylianou, Phys. Rev. D86(2012) 014505 [arXiv:1201.5025 [hep-lat]].[12] P. Hagler, Phys. Rept.490 (2010) 49 [arXiv:0912.5483 [hep-lat]].[13] F. M. Steffens, M. D. Brown, W. Melnitchouk and S. Sanches, Phys. Rev. C86 (2012) 065208[arXiv:1210.4398 [hep-ph]].[14] A. Abdel-Rehim, P. Boucaud, N. Carrasco, A. Deuzeman, P. Dimopoulos, R. Frezzotti, G. Herdoizaand K. Jansenet al., PoS(LATTICE 2013)264 [arXiv:1311.4522 [hep-lat]].7

First results with twisted mass fermions towards the computation of parton distribution functions on the lattice

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First results with twisted mass fermions towards the computation of parton distribution functions on the lattice

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