We have initiated a program to compute the Compton amplitude from lattice QCD with the Feynman-Hellman method. This amplitude is related to the structure function via a Fredholm integral equation of the first kind. It is known that these types of equations are inherently ill--posed - they are, e.g., extremely sensitive to perturbations of the system. We discuss two methods which are candidates to handle these problems: the model free inversion based on singular value decomposition and one Bayesian type approach. We apply the Bayesian method to currently available lattice data for the Compton amplitude

collaboration, QCDSF-UKQCD-CSSM

Horsley, Roger

Nakamura, Yoshifumi

Perlt, Holger

Rakow, Paul E. L.

Schierholz, Gerrit

Somfleth, Kim

Young, Ross D.

Zanotti, James M.

English

arXiv

We have initiated a program to compute the Compton amplitude from lattice QCD
with the Feynman-Hellman method. This amplitude is related to the structure
function via a Fredholm integral equation of the first kind. It is known that
these types of equations are inherently ill--posed - they are, e.g., extremely
sensitive to perturbations of the system. We discuss two methods which are
candidates to handle these problems: the model free inversion based on singular
value decomposition and one Bayesian type approach. We apply the Bayesian
method to currently available lattice data for the Compton amplitude

Rakow, Paul EL

Young, Ross D

Zanotti, James M

University of Liverpool Repository

arXiv:2001.05366v1  [hep-lat]  15 Jan 2020DESY 20-005ADP-20-3/T1113LTH-1222Structure functions from the Compton amplitudeR. Horsleya, Y. Nakamurab, H. Perlt∗,c, P. E. L. Rakowd, G. Schierholze, K. Somfleth f ,R. D. Young f , J. M. Zanotti fQCDSF-UKQCD-CSSM collaborationa School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UKb RIKEN Advanced Institute for Computational Science, Kobe, Hyogo 650-0047, Japanc Institut für Theoretische Physik, Universität Leipzig, 04103 Leipzig, Germanyd Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, UKe Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germanyf CSSM, Department of Physics, The University of Adelaide, Adelaide SA 5005, AustraliaE-mail:holger.perlt@itp.uni-Leipzig.deWe have initiated a program to compute the Compton amplitude from lattice QCD with theFeynman-Hellman method. This amplitude is related to the structure function via a Fredholmintegral equation of the first kind. It is known that these types of equations are inherently ill–posed - they are, e.g., extremely sensitive to perturbations of the system. We discuss two methodswhich are candidates to handle these problems: the model free inversion based on singular valuedecomposition and one Bayesian type approach. We apply the Bayesian method to currentlyavailable lattice data for the Compton amplitude.37th International Symposium on Lattice Field Theory - Lattice201916-22 June 2019Wuhan, China∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/Structure functions H. Perlt1. IntroductionThe determination of the hadronic structure from first principles belongs to the key investi-gation topics in lattice QCD. Central to our understanding of hadron structure are the structurefunctions which describe the distribution of quarks and gluons inside hadrons. In the last yearssome promising approaches have been proposed, among them the calculation of the quasi particledistribution functions (for a review see [1]). Our group has initiated a program to compute thestructure functions from the forward Compton amplitude of the nucleon [2, 3]. A central motiva-tion for this is to overcome the issues of renormalization, operator mixing and the restriction tolight-cone operators.The starting point is the forward Compton amplitude of the nucleon [4],Tµν(p,q) = ρλλ ′∫d4xeiq·x〈p,λ ′|T Jµ(x)Jν (0)|p,λ 〉 , (1.1)which involves the time ordered product of electromagnetic currents sandwiched between nucleonstates of momentum p and polarization λ , where q is the momentum of the virtual photon and ρ isthe polarization density matrix. In view of our investigation below we consider only the unpolarizedstructure functions. In the unphysical region (|p ·q|< q2/2) the relation of Tµν(p,q) to the structurefunctions F1(x,q2),F2(x,q2) is given by [4]Tµν(p,q) =(δµν −qµqνq2)4ω∫ 10dxωx1− (ωx)2 F1(x,q2)+(pµ − p ·qq2qµ)(pν − p ·qq2qν)8ω2p ·q∫ 10dx11− (ωx)2 F2(x,q2) , (1.2)with ω = 2p ·q/q2, discarding the subtraction term [2]. To simplify the numerical calculation, wemay choose µ = ν = 3 and p3 = q3 = q4 = 0. We then haveT33(p,q) = 4ω∫ 10dxωx1− (ωx)2 F1(x,q2)≡∫ 10dxK(x,ω)F1(x,q2) . (1.3)The matrix element T33(p,q) can be computed most efficiently by a simple extension of the Feynman–Hellmann method [2, 5].Performing a Taylor expansion of (1.3) leads to a simple relation between the moments t j =∫ 10 dxxj F1(x) of the structure function and the ω–dependent Compton amplitudeT33(ω) = 4(ω2 t1+ω4 t3+ · · ·+ω2M t2M−1+ . . .). (1.4)From these we then determine the moments of the parton distributions µ j from t j ∼ µ j/2 neglectinglogs and terms O(1/q2).2. Problems and solutions of the Fredholm integral equationFormula (1.3) is the basic relation for our investigation. It tells us how to extract the structurefunction F1(x,q2) given that we have available lattice data for the Compton amplitude T33(p,q).Unfortunately, it is a Fredholm integral equation of the first kind. Those equations are known to be1Structure functions H. Perltill–posed. E.g., they are extremely sensitive to very small perturbations of the data [6] – in our caseto the lattice results of T33(p,q). Additionally, the solutions are not guaranteed to be unique. Thereis no general solution method available. If one finds a successful numerical strategy at all it dependsalways on the specific kernel K. Therefore, a careful study of possible approaches is needed. Ananalogous problem arises in the reconstruction of Ioffe time pseudo particle distribution functions(pdf) and was investigated in great detail in [7].In order to test some possible numerical methods we generate mock data for the Comptonamplitude. As an example we choose a valence type up quark distributionx prefuv (x) = 5.107x0.8 (1− x)3 (2.1)chosen to satisfy the momentum sum rule∫ 10dxx prefuv (x) = 1/3 . (2.2)This function is then used to generate the T33 data via (1.3) and to compare with the results ofour tested inversion algorithms.The numerical inversion requires a discretization of (1.3)T33(ωi) =N∑j=1K(x j,ωi) p(x j)↔ T33,i =N∑j=1K ji p j , i = 1 . . .M , (2.3)where in general we have N 6= M.One basic method to solve (2.3) for the p j is the singular value decomposition (SVD) [8]. Ithas the advantage that one does not need to make any further input assumptions about the expectedform of the wanted p(x). On the other hand there is a certain freedom in omitting small singularvalues. Additionally, using our kernel K(x,ω) (1.3) we have cancellations of very large numberswhich increases with the number of included singular values. This demands very precise lattice datain order to get meaningful results. The result of the inversion x pSVDuv (x) is shown in the left panel of00.20.40.60.810 0.2 0.4 0.6 0.8 1xpval(x)xSVDinput00.20.40.60.810 0.2 0.4 0.6 0.8 1xpval(x)xBMCinputFigure 1: Left: The pdf as obtained from the SVD. The red curve is x prefuv (x) (2.1). The blueshadowed area shows the variation of the result x pSVDuv (x) due to a ±10% variation of T33. Right:The pdf from the BMC approach. The red curve is again the input (2.1). The shadowed area is the68% quantile.2Structure functions H. PerltFig. 1 for N = 50, M = 10 and 0< ω < 1. One recognizes a trend around the the input distributionwith some small oscillations . The integral using the mean value is∫ 10 dxx pSVDuv(x)≈ 0.33 .An alternative approach uses some prior model information concerning the distribution andtries to refine it according to the available data. It belongs to the class of Bayesian methods. Onevariant has been discussed in detail in [7]. We follow a slightly different procedure here (seealso [3]). Our model assumption is the general form of a valence quark type distributionpval(x,a,b,c) =axb (1− x)c Γ(b+ c+3)Γ(b+2)Γ(c+1). (2.4)We can determine the Compton amplitude from (1.3) analytically (for ω < 1)T val33 (ω) = 2−b−c−1√pi aω2Γ(b+ c+3)×3F˜2(1,b+22,b+32;12(b+ c+3),12(b+ c+4);ω2)(2.5)= c1(a,b,c)ω2+ c3(a,b,c)ω4+ c5(a,b,c)ω6+ . . . , (2.6)where 3F˜2 is a regularized hypergeometric function. The power expansion of Tval33 (ω) is given in(2.6). We proceed by first generating NMC Monte Carlo sets of model parameters {a,b,c}k=1,...,NMC .With these sets the quadratic deviations χ2kχ2k = ∑n, j(T33,n−T val33,(k)(ωn))C−1n j(T33, j −T val33,(k)(ω j))(2.7)are computed. T33,n are the data for ωn, whereas Tval33,(k)(ωn) is (2.5) calculated for one triple{a,b,c}k at ωn. C−1n j is the inverse covariance matrix of the data. The set χ2k is used to make aweighted random choice out of the total set {a,b,c}k by the likelihood exp(−χ2/2). This consti-tutes our sample parameter set from which we compute the means and the quantiles. Also in thiscase the model input is crucial: the final values are inside the MC sets and the χ2k should containreasonable small minimal values. We call this method a Bayesian Monte Carlo (BMC) approach.The resulting distribution is shown in the right panel of Fig. 1. The initial values of the parametersare selected uniformly distributed around some suitable values. An analogous procedure can beused to determine the moments via relation (1.4). In this case the moments t j play the role of theparameters and are obtained directly from this approach.Summarizing the SVD and the BMC approaches we favor the latter, because we recognize inthe SVD solution oscillations around the exact result although we use ideal mock data. For reallattice data which are far more scattered and which often have more significant uncertainties theSVD inversion gives very unstable results.3. First results from lattice dataNow we investigate these methods with our latest lattice data for the nucleon Compton am-plitude for the connected part of the combination u− d. We use 323 × 64(β = 5.5) lattices atthe SU(3)–flavour symmetric point (κl = κs) and Mpi ≈ 470 GeV. In this paper only data forq2 = 2.7, 3.5, 4.6 GeV2 are included. They are shown in the left panel of Fig. 2.3Structure functions H. Perlt-0.200.20.40.60.811.21.41.60 0.2 0.4 0.6 0.8 1Tu−d33ω2.7 GeV23.5 GeV24.6 GeV200.20.40.60.811.21.41.61.820 0.2 0.4 0.6 0.8 1Tu−d33ωq2 = 2.7GeV2dataFigure 2: Left: T u−d33 lattice data in the range 0< ω < 1 for the three q2 values used for the analysis.Right: Result for (3.1) together with the data for q2 = 2.7 GeV2.As a first step we determine the first moments using (1.4) as the defining relation. In this paperwe restrict ourselves to order ω12. In order to get information about the q2–dependence we applyour BMC procedure to each of the three data sets mentioned above. We compute the χ2k valuesfrom (2.7), now withT val33,(k)(ωn) = 46∑j=1t(k)2 j−1ω2 jn . (3.1)We select NMC sets by sampling {a,b,c}k uniformly from intervals suggested by phenomenologyand determine the moments t(a,b,c)(k)i according to their valence quark beavior. (A random se-lection with t(k)1 ≥ t(k)3 ≥ ·· · ≥ 0 as discussed in [3] leads to very similar results.) We generate100,000 MC data sets and from that we select a subset of 500 samples weighted by the likelihoodexp(−χ2/2). From this subset we compute the ti. The resulting Compton amplitude is given inthe right panel of Fig. 2 for q2 = 2.7 GeV2 where we observe a reasonable agreement with thedata. The moments themselves are presented in Fig. 3. They show their expected behavior withincreasing order. For the first moment we observe a slight dependence on q2.In the same spirit we try to obtain the complete particle distribution function. As data we use00.10.20.30.40.50.60.70.80 2 4 6 8 10 12〈xj〉 u−djq2 = 2.7GeV200.10.20.30.40.50.60.70.80.912 2.5 3 3.5 4 4.5 5〈x〉 u−dq2 [GeV2]Figure 3: Left: The first moments for q2 = 2.7 GeV2. The error bars are the quantiles encompassing68% of the data. Right: The first moment for q2 = 2.7,3.5 and 4.6 GeV2.4Structure functions H. Perlt00.20.40.60.811.21.41.60 0.2 0.4 0.6 0.8 1Tu−d33ωdata00.20.40.60.811.21.41.60 0.2 0.4 0.6 0.8 1xpu−d(x)xFigure 4: Left: The Compton amplitude (2.6) with parameters obtained with BMC together withthe data. Right: The resulting valence type distribution function. The shaded area is the 68%quantile.the subset with q2 = 2.7 GeV2. Concerning the priors, we are guided by the success of the momentsdetermination above. We sample the first moment uniformly out of the interval [0 . . .1] and let theBMC method compute it from to the lattice data. This is supported by our model ansatz (2.4) since〈x〉=∫ 10dx x pval(x,a,b,c) = a . (3.2)For the parameters b and c we choose input intervals suggested by phenomenology. Other priorschemes will be investigated in a forthcoming paper.We find using the mean curve and its quantile borders∫ 10 dx x presu−d(x) = 0.58+25−26 , consistentwith the first moment given in Fig. 3. Additionally, inserting the resulting mean values of theparameters in (2.6) we find c1 ≈ 1.09 – also compatible with the moments. The results are shownin Fig. 4. One recognizes a strong similarity of the left panel in Fig. 4 with the right panel of Fig.2 which proves the consistency of both approaches. In order to demonstrate the effect of the BMCprocedure we show in Fig. 5 the change of the parameters from the uniformly input values (blue)to the final values (red). The histogram in the right panel demonstrates the transition from uniforminput to the peaked distribution triggered by the χ2k values. One recognizes that the procedure doesnot influence very much the values of the parameters b and c but significantly shrinks the range forparameter a towards the first moment.AcknowledgementsThe numerical configuration generation (using the BQCD lattice QCD program [9])) and dataanalysis (using the Chroma software library [10]) was carried out on the IBM BlueGene/Q and HPTesseract using DIRAC 2 resources (EPCC, Edinburgh, UK), the IBM BlueGene/Q (NIC, Jülich,Germany) and the Cray XC40 at HLRN (The North-German Supercomputer Alliance), the NCINational Facility in Canberra, Australia (supported by the Australian Commonwealth Government)and Phoenix (University of Adelaide). RH was supported by STFC through grant ST/P000630/1.HP was supported by DFG Grant No. PE 2792/2-1. PELR was supported in part by the STFCunder contract ST/G00062X/1. GS was supported by DFG Grant No. SCHI 179/8-1. RDY and5Structure functions H. Perlt-101234567a b cparameterfinalinput0.0 0.2 0.4 0.6 0.8 1.0 1.2012345aFigure 5: Left: The parameters (a,b,c) of the valence distribution (2.4). The blue lines are the inputvalue ranges, the red data points result from the BMC approach. Again the error bars are the 68%quantiles. Right: Histogram for parameter a. Blue: input parameter range. Yellow: parameterrange of the sample according to χ2 selection.JMZ were supported by the Australian Research Council Grants DP140103067 and DP190100297.We thank all funding agenciesReferences[1] K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019 (2019) 3036904 [arXiv:1811.07248[hep-lat]].[2] A. J. Chambers et al., Phys. Rev. Lett. 118 (2017) no.24, 242001 [arXiv:1703.01153 [hep-lat]].[3] R. D. Young, PoS(LATTICE2019) 278.[4] R. Devenish and A. Cooper-Sarkar, Deep Inelastic Scattering, Oxford University Press (2003,Oxford, UK); A.V. Manohar, in Lake Louise 1992, Symmetry and Spin in the Standard Model, p. 1-46[hep-ph/9204208]; K.-F. Liu, Phys. Rev. D 62 (2000) 074501 [hep-ph/9910306].[5] R. Horsley et al. Phys. Lett. B 714 (2012) 312 [arXiv:1205.6410 [hep-lat]]; A.J. Chambers et al.Phys. Rev. D 90 (2014) 014510 [arXiv:1405.3019 [hep-lat]]; A. J. Chambers et al. Phys. Rev. D 92(2015) no.11, 114517 [arXiv:1508.06856 [hep-lat]]; A. J. Chambers et al. Phys. Rev. D 96 (2017)no.11, 114509 [arXiv:1702.01513 [hep-lat]].[6] P. C. Hansen, Inverse Problems 8 (1992) 849.[7] J. Karpie, K. Orginos, A. Rothkopf and S. Zafeiropoulos, JHEP 1904 (2019) 057 [arXiv:1901.05408[hep-lat]].[8] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes, CambridgeUniversity Press (1989, Cambridge, UK).[9] T. R. Haar, Y. Nakamura and H. Stüben, EPJ Web Conf. 175 (2018) 14011 [arXiv:1711.03836[hep-lat]].[10] R. G. Edwards et al. [SciDAC and LHPC and UKQCD Collaborations], Nucl. Phys. Proc. Suppl. 140(2005) 832 [hep-lat/0409003].6

Structure functions from the Compton amplitude

We have initiated a program to compute the Compton amplitude from lattice QCD with the Feynman-Hellman method. This amplitude is related to the structure function via a Fredholm integral equation of the first kind. It is known that these types of equations are inherently ill– posed - they are, e.g., extremely sensitive to perturbations of the system. We discuss two methods which are candidates to handle these problems: the model free inversion based on singular value decomposition and one Bayesian type approach. We apply the Bayesian method to currently available lattice data for the Compton amplitude.R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, K. Somfleth, R. D. Young, J. M. Zanott

Perlt, H.

Horsley, R.

Nakamura, Y.

Rakow, P.

Schierholz, G.

Somfleth, K.

Young, R.

Zanotti, J.

Adelaide Research &amp; Scholarship

PoS(LATTICE2019)137Structure functions from the Compton amplitudeR. Horsleya, Y. Nakamurab, H. Perlt∗,c, P. E. L. Rakowd, G. Schierholze, K. Somfleth f ,R. D. Young f , J. M. Zanotti fQCDSF-UKQCD-CSSM collaborationa School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UKb RIKEN Advanced Institute for Computational Science, Kobe, Hyogo 650-0047, Japanc Institut für Theoretische Physik, Universität Leipzig, 04103 Leipzig, Germanyd Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, UKe Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germanyf CSSM, Department of Physics, The University of Adelaide, Adelaide SA 5005, AustraliaE-mail:holger.perlt@itp.uni-Leipzig.deWe have initiated a program to compute the Compton amplitude from lattice QCD with theFeynman-Hellman method. This amplitude is related to the structure function via a Fredholmintegral equation of the first kind. It is known that these types of equations are inherently ill–posed - they are, e.g., extremely sensitive to perturbations of the system. We discuss two methodswhich are candidates to handle these problems: the model free inversion based on singular valuedecomposition and one Bayesian type approach. We apply the Bayesian method to currentlyavailable lattice data for the Compton amplitude.37th International Symposium on Lattice Field Theory - Lattice201916-22 June 2019Wuhan, China∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/PoS(LATTICE2019)137Structure functions H. Perlt1. IntroductionThe determination of the hadronic structure from first principles belongs to the key investi-gation topics in lattice QCD. Central to our understanding of hadron structure are the structurefunctions which describe the distribution of quarks and gluons inside hadrons. In the last yearssome promising approaches have been proposed, among them the calculation of the quasi particledistribution functions (for a review see [1]). Our group has initiated a program to compute thestructure functions from the forward Compton amplitude of the nucleon [2, 3]. A central motiva-tion for this is to overcome the issues of renormalization, operator mixing and the restriction tolight-cone operators.The starting point is the forward Compton amplitude of the nucleon [4],Tµν(p,q) = ρλλ ′∫d4xeiq·x〈p,λ ′|T Jµ(x)Jν(0)|p,λ 〉 , (1.1)which involves the time ordered product of electromagnetic currents sandwiched between nucleonstates of momentum p and polarization λ , where q is the momentum of the virtual photon and ρ isthe polarization density matrix. In view of our investigation below we consider only the unpolarizedstructure functions. In the unphysical region (|p ·q|< q2/2) the relation of Tµν(p,q) to the structurefunctions F1(x,q2),F2(x,q2) is given by [4]Tµν(p,q) =(δµν −qµqνq2)4ω∫ 10dxωx1− (ωx)2 F1(x,q2)+(pµ −p ·qq2qµ)(pν −p ·qq2qν)8ω2p ·q∫ 10dx11− (ωx)2 F2(x,q2) , (1.2)with ω = 2p ·q/q2, discarding the subtraction term [2]. To simplify the numerical calculation, wemay choose µ = ν = 3 and p3 = q3 = q4 = 0. We then haveT33(p,q) = 4ω∫ 10dxωx1− (ωx)2 F1(x,q2)≡∫ 10dxK(x,ω)F1(x,q2) . (1.3)The matrix element T33(p,q) can be computed most efficiently by a simple extension of the Feynman–Hellmann method [2, 5].Performing a Taylor expansion of (1.3) leads to a simple relation between the moments t j =∫ 10 dxxj F1(x) of the structure function and the ω–dependent Compton amplitudeT33(ω) = 4(ω2 t1 +ω4 t3 + · · ·+ω2M t2M−1 + . . .). (1.4)From these we then determine the moments of the parton distributions µ j from t j ∼ µ j/2 neglectinglogs and terms O(1/q2).2. Problems and solutions of the Fredholm integral equationFormula (1.3) is the basic relation for our investigation. It tells us how to extract the structurefunction F1(x,q2) given that we have available lattice data for the Compton amplitude T33(p,q).Unfortunately, it is a Fredholm integral equation of the first kind. Those equations are known to be1PoS(LATTICE2019)137Structure functions H. Perltill–posed. E.g., they are extremely sensitive to very small perturbations of the data [6] – in our caseto the lattice results of T33(p,q). Additionally, the solutions are not guaranteed to be unique. Thereis no general solution method available. If one finds a successful numerical strategy at all it dependsalways on the specific kernel K. Therefore, a careful study of possible approaches is needed. Ananalogous problem arises in the reconstruction of Ioffe time pseudo particle distribution functions(pdf) and was investigated in great detail in [7].In order to test some possible numerical methods we generate mock data for the Comptonamplitude. As an example we choose a valence type up quark distributionx prefuv (x) = 5.107x0.8 (1− x)3 (2.1)chosen to satisfy the momentum sum rule∫ 10dxx prefuv (x) = 1/3 . (2.2)This function is then used to generate the T33 data via (1.3) and to compare with the results ofour tested inversion algorithms.The numerical inversion requires a discretization of (1.3)T33(ωi) =N∑j=1K(x j,ωi) p(x j)↔ T33,i =N∑j=1K ji p j , i = 1 . . .M , (2.3)where in general we have N 6= M.One basic method to solve (2.3) for the p j is the singular value decomposition (SVD) [8]. Ithas the advantage that one does not need to make any further input assumptions about the expectedform of the wanted p(x). On the other hand there is a certain freedom in omitting small singularvalues. Additionally, using our kernel K(x,ω) (1.3) we have cancellations of very large numberswhich increases with the number of included singular values. This demands very precise lattice datain order to get meaningful results. The result of the inversion x pSVDuv (x) is shown in the left panel of00.20.40.60.810 0.2 0.4 0.6 0.8 1xpval(x)xSVDinput00.20.40.60.810 0.2 0.4 0.6 0.8 1xpval(x)xBMCinputFigure 1: Left: The pdf as obtained from the SVD. The red curve is x prefuv (x) (2.1). The blueshadowed area shows the variation of the result x pSVDuv (x) due to a ±10% variation of T33. Right:The pdf from the BMC approach. The red curve is again the input (2.1). The shadowed area is the68% quantile.2PoS(LATTICE2019)137Structure functions H. PerltFig. 1 for N = 50, M = 10 and 0 < ω < 1. One recognizes a trend around the the input distributionwith some small oscillations . The integral using the mean value is∫ 10 dxx pSVDuv(x)≈ 0.33 .An alternative approach uses some prior model information concerning the distribution andtries to refine it according to the available data. It belongs to the class of Bayesian methods. Onevariant has been discussed in detail in [7]. We follow a slightly different procedure here (seealso [3]). Our model assumption is the general form of a valence quark type distributionpval(x,a,b,c) =axb (1− x)c Γ(b+ c+3)Γ(b+2)Γ(c+1). (2.4)We can determine the Compton amplitude from (1.3) analytically (for ω < 1)T val33 (ω) = 2−b−c−1√π aω2 Γ(b+ c+3)×3F̃2(1,b+22,b+32;12(b+ c+3),12(b+ c+4);ω2)(2.5)= c1(a,b,c)ω2 + c3(a,b,c)ω4 + c5(a,b,c)ω6 + . . . , (2.6)where 3F̃2 is a regularized hypergeometric function. The power expansion of Tval33 (ω) is given in(2.6). We proceed by first generating NMC Monte Carlo sets of model parameters {a,b,c}k=1,...,NMC .With these sets the quadratic deviations χ2kχ2k = ∑n, j(T33,n −T val33,(k)(ωn))C−1n j(T33, j −T val33,(k)(ω j))(2.7)are computed. T33,n are the data for ωn, whereas Tval33,(k)(ωn) is (2.5) calculated for one triple{a,b,c}k at ωn. C−1n j is the inverse covariance matrix of the data. The set χ2k is used to make aweighted random choice out of the total set {a,b,c}k by the likelihood exp(−χ2/2). This consti-tutes our sample parameter set from which we compute the means and the quantiles. Also in thiscase the model input is crucial: the final values are inside the MC sets and the χ2k should containreasonable small minimal values. We call this method a Bayesian Monte Carlo (BMC) approach.The resulting distribution is shown in the right panel of Fig. 1. The initial values of the parametersare selected uniformly distributed around some suitable values. An analogous procedure can beused to determine the moments via relation (1.4). In this case the moments t j play the role of theparameters and are obtained directly from this approach.Summarizing the SVD and the BMC approaches we favor the latter, because we recognize inthe SVD solution oscillations around the exact result although we use ideal mock data. For reallattice data which are far more scattered and which often have more significant uncertainties theSVD inversion gives very unstable results.3. First results from lattice dataNow we investigate these methods with our latest lattice data for the nucleon Compton am-plitude for the connected part of the combination u− d. We use 323 × 64(β = 5.5) lattices atthe SU(3)–flavour symmetric point (κl = κs) and Mπ ≈ 470 GeV. In this paper only data forq2 = 2.7, 3.5, 4.6 GeV2 are included. They are shown in the left panel of Fig. 2.3PoS(LATTICE2019)137Structure functions H. Perlt-0.200.20.40.60.811.21.41.60 0.2 0.4 0.6 0.8 1Tu−d33ω2.7 GeV23.5 GeV24.6 GeV200.20.40.60.811.21.41.61.820 0.2 0.4 0.6 0.8 1Tu−d33ωq2 = 2.7GeV2dataFigure 2: Left: T u−d33 lattice data in the range 0<ω < 1 for the three q2 values used for the analysis.Right: Result for (3.1) together with the data for q2 = 2.7 GeV2.As a first step we determine the first moments using (1.4) as the defining relation. In this paperwe restrict ourselves to order ω12. In order to get information about the q2–dependence we applyour BMC procedure to each of the three data sets mentioned above. We compute the χ2k valuesfrom (2.7), now withT val33,(k)(ωn) = 46∑j=1t(k)2 j−1 ω2 jn . (3.1)We select NMC sets by sampling {a,b,c}k uniformly from intervals suggested by phenomenologyand determine the moments t(a,b,c)(k)i according to their valence quark beavior. (A random se-lection with t(k)1 ≥ t(k)3 ≥ ·· · ≥ 0 as discussed in [3] leads to very similar results.) We generate100,000 MC data sets and from that we select a subset of 500 samples weighted by the likelihoodexp(−χ2/2). From this subset we compute the ti. The resulting Compton amplitude is given inthe right panel of Fig. 2 for q2 = 2.7 GeV2 where we observe a reasonable agreement with thedata. The moments themselves are presented in Fig. 3. They show their expected behavior withincreasing order. For the first moment we observe a slight dependence on q2.In the same spirit we try to obtain the complete particle distribution function. As data we use00.10.20.30.40.50.60.70.80 2 4 6 8 10 12〈xj〉 u−djq2 = 2.7GeV200.10.20.30.40.50.60.70.80.912 2.5 3 3.5 4 4.5 5〈x〉 u−dq2 [GeV2]Figure 3: Left: The first moments for q2 = 2.7 GeV2. The error bars are the quantiles encompassing68% of the data. Right: The first moment for q2 = 2.7,3.5 and 4.6 GeV2.4PoS(LATTICE2019)137Structure functions H. Perlt00.20.40.60.811.21.41.60 0.2 0.4 0.6 0.8 1Tu−d33ωdata00.20.40.60.811.21.41.60 0.2 0.4 0.6 0.8 1xpu−d(x)xFigure 4: Left: The Compton amplitude (2.6) with parameters obtained with BMC together withthe data. Right: The resulting valence type distribution function. The shaded area is the 68%quantile.the subset with q2 = 2.7 GeV2. Concerning the priors, we are guided by the success of the momentsdetermination above. We sample the first moment uniformly out of the interval [0 . . .1] and let theBMC method compute it from to the lattice data. This is supported by our model ansatz (2.4) since〈x〉=∫ 10dx x pval(x,a,b,c) = a . (3.2)For the parameters b and c we choose input intervals suggested by phenomenology. Other priorschemes will be investigated in a forthcoming paper.We find using the mean curve and its quantile borders∫ 10 dx x presu−d(x) = 0.58+25−26 , consistentwith the first moment given in Fig. 3. Additionally, inserting the resulting mean values of theparameters in (2.6) we find c1 ≈ 1.09 – also compatible with the moments. The results are shownin Fig. 4. One recognizes a strong similarity of the left panel in Fig. 4 with the right panel of Fig.2 which proves the consistency of both approaches. In order to demonstrate the effect of the BMCprocedure we show in Fig. 5 the change of the parameters from the uniformly input values (blue)to the final values (red). The histogram in the right panel demonstrates the transition from uniforminput to the peaked distribution triggered by the χ2k values. One recognizes that the procedure doesnot influence very much the values of the parameters b and c but significantly shrinks the range forparameter a towards the first moment.AcknowledgementsThe numerical configuration generation (using the BQCD lattice QCD program [9])) and dataanalysis (using the Chroma software library [10]) was carried out on the IBM BlueGene/Q and HPTesseract using DIRAC 2 resources (EPCC, Edinburgh, UK), the IBM BlueGene/Q (NIC, Jülich,Germany) and the Cray XC40 at HLRN (The North-German Supercomputer Alliance), the NCINational Facility in Canberra, Australia (supported by the Australian Commonwealth Government)and Phoenix (University of Adelaide). RH was supported by STFC through grant ST/P000630/1.HP was supported by DFG Grant No. PE 2792/2-1. PELR was supported in part by the STFCunder contract ST/G00062X/1. GS was supported by DFG Grant No. SCHI 179/8-1. RDY and5PoS(LATTICE2019)137Structure functions H. Perlt-101234567a b cparameterfinalinput0.0 0.2 0.4 0.6 0.8 1.0 1.2012345aFigure 5: Left: The parameters (a,b,c) of the valence distribution (2.4). The blue lines are the inputvalue ranges, the red data points result from the BMC approach. Again the error bars are the 68%quantiles. Right: Histogram for parameter a. Blue: input parameter range. Yellow: parameterrange of the sample according to χ2 selection.JMZ were supported by the Australian Research Council Grants DP140103067 and DP190100297.We thank all funding agenciesReferences[1] K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019 (2019) 3036904 [arXiv:1811.07248[hep-lat]].[2] A. J. Chambers et al., Phys. Rev. Lett. 118 (2017) no.24, 242001 [arXiv:1703.01153 [hep-lat]].[3] R. D. Young, PoS(LATTICE2019) 278.[4] R. Devenish and A. Cooper-Sarkar, Deep Inelastic Scattering, Oxford University Press (2003,Oxford, UK); A.V. Manohar, in Lake Louise 1992, Symmetry and Spin in the Standard Model, p. 1-46[hep-ph/9204208]; K.-F. Liu, Phys. Rev. D 62 (2000) 074501 [hep-ph/9910306].[5] R. Horsley et al. Phys. Lett. B 714 (2012) 312 [arXiv:1205.6410 [hep-lat]]; A.J. Chambers et al.Phys. Rev. D 90 (2014) 014510 [arXiv:1405.3019 [hep-lat]]; A. J. Chambers et al. Phys. Rev. D 92(2015) no.11, 114517 [arXiv:1508.06856 [hep-lat]]; A. J. Chambers et al. Phys. Rev. D 96 (2017)no.11, 114509 [arXiv:1702.01513 [hep-lat]].[6] P. C. Hansen, Inverse Problems 8 (1992) 849.[7] J. Karpie, K. Orginos, A. Rothkopf and S. Zafeiropoulos, JHEP 1904 (2019) 057 [arXiv:1901.05408[hep-lat]].[8] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes, CambridgeUniversity Press (1989, Cambridge, UK).[9] T. R. Haar, Y. Nakamura and H. Stüben, EPJ Web Conf. 175 (2018) 14011 [arXiv:1711.03836[hep-lat]].[10] R. G. Edwards et al. [SciDAC and LHPC and UKQCD Collaborations], Nucl. Phys. Proc. Suppl. 140(2005) 832 [hep-lat/0409003].6

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Structure functions from the Compton amplitude

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Structure functions from the Compton amplitude

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