We present preliminary results from the first calculation of the pion electromagnetic form factor at physical light quark masses. This form factor parameterises the deviations from the behaviour of a point-like particle when a photon hits the pion. These deviations result from the internal structure of the pion and can thus be calculated in QCD. We use three sets (different lattice spacings) of n_f=2+1+1 lattice configurations generated by the MILC collaboration. The Highly Improved Staggered Quark formalism (HISQ) is used for all of the sea and valence quarks. Using lattice configurations with u/d quark masses very close to the physical value is an advantage, as we avoid the chiral extrapolation. We study the shape of the vector (f_+) form factor in the q^2 range from 0 to -0.12 GeV^2 and extract the mean square radius, &#60;r^2_v&#62;. The shape of the vector form factor and the resulting radius is compared with experiment

Bursa, Francis

Davies, Christine

Donald, Gordon

Dowdall, Rachel

Koponen, Jonna

English

arXiv

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Pion electromagnetic form factor from full lattice QCD

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     Koponen, Jonna, Bursa, Francis, Davies, Christine, Donald, Gordon, and Dowdall, Rachel (2013) Pion electromagnetic form factor from full lattice QCD. In: 31st International Symposium on Lattice Field Theory LATTICE 2013 , 29 July – 3 Aug 2013, Mainz, Germany.  Copyright © 2013 The Authors    http://eprints.gla.ac.uk/90856/     Deposited on: 26 June 2014                         Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk PoS(LATTICE 2013)282Pion electromagnetic form factor from full latticeQCDHPQCD CollaborationJonna Koponen∗SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UKE-mail: jonna.koponen@glasgow.ac.ukFrancis BursaSUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UKChristine DaviesSUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UKGordon DonaldSUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UKSchool of Mathematics, Trinity College, Dublin 2, IrelandRachel DowdallDAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UKWe present preliminary results from the first calculation of the pion electromagnetic form factor atphysical light quark masses. This form factor parameterises the deviations from the behaviour of apoint-like particle when a photon hits the pion. These deviations result from the internal structureof the pion and can thus be calculated in QCD. We use three sets (different lattice spacings) ofn f = 2+1+1 lattice configurations generated by the MILC collaboration. The Highly ImprovedStaggered Quark formalism (HISQ) is used for all of the sea and valence quarks. Using latticeconfigurations with u/d quark masses very close to the physical value is an advantage, as we avoidthe chiral extrapolation. We study the shape of the vector ( f+) form factor in the q2 range from 0to −0.12 GeV2 and extract the mean square radius, 〈r2v〉. The shape of the vector form factor andthe resulting radius is compared with experiment.31st International Symposium on Lattice Field Theory LATTICE 2013July 29 – August 3, 2013Mainz, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LATTICE 2013)282Pion electromagnetic form factor from full lattice QCD Jonna Koponen 0 0.1 0.2 0.3 0.4 0.5 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5< r2 > /  fm2- ln (Mπ2 /GeV2)ETMCHPQCD, this workNA7, exptFigure 1: Dependence of the charge mean square radius on the pion mass. Note that we plot the radius as afunction of ln(m2pi), not m2pi . The experimental result is from [3] and the ETMC lattice results are from [1].1. MotivationThe electromagnetic form factor of the charged pi meson parameterises the deviations fromthe behaviour of a point-like particle when struck by a photon. These deviations arise from theinternal structure of the pion: constituent quarks and their strong interaction. The form factor canbe calculated in Lattice QCD, but it is desirable to work at the physical pion mass to avoid chiralextrapolation. Plots of lattice determinations of the pion charge radius as a function of pion mass– like Fig. 1, or Fig. 11 in [1] or Fig. 4 in the very recent review paper [2] – show very clearlythat the extrapolation to physical pion mass plays a key role, if the pion masses one works at aremuch heavier than the physical mass. On the experimental side, the vector form factor has beenmeasured by NA7 collaboration [3] in a pi – e scattering experiment, which allows a comparisonbetween theory and experiment.2. Lattice configurationsWe use the n f = 2+1+1 HISQ (Highly Improved Staggered Quark) physical pion mass latticeconfigurations provided by MILC Collaboration [4]. Three ensembles (different lattice spacings)are used in this study – see details in Table 1. We use the HISQ action for valence quarks as well,and using the same light quark mass as in the sea. We have Lmpi ≈ 4 for the coarse (a= 0.12 fm)and fine (a = 0.088 fm) lattices, so finite volume effects are expected to be very small. We havegood statistics with 1000 configurations (for the very coarse and coarse ensembles) and four timesources per configuration.3. Form FactorsOn the lattice, form factors are extracted from 3-point correlators – see Fig. 2. We have twopseudoscalar mesons, the initial and final state pions, time T apart, and we use a 1-link spatial2PoS(LATTICE 2013)282Pion electromagnetic form factor from full lattice QCD Jonna KoponenSet a/fm aml ams amc mpi /MeV (L/a)3×Lt/a Nconf T/a1 0.15 0.00235 0.0647 0.831 133 323×48 1000 9, 12, 152 0.12 0.00184 0.0507 0.628 133 483×64 1000 12, 15, 183 0.088 0.00120 0.0363 0.432 128 643×96 223 16, 21, 26Table 1: Details of the MILC 2+1+1 flavor lattice configurations used in this study. Set 1 is very coarse, set2 is coarse and set 3 is fine ensemble. The second column is the lattice spacing ([5], using w0 to determinethe scale) and columns 3-6 list the sea quark masses and the pion mass. Columns 7 and 8 give the size ofthe lattice and the number of configurations used. Four time sources are used per configuration to get goodstatistics. T in the 9th column is the separation between the initial and final state mesons. We use multiplevalues of T to improve extraction of the ground state matrix elements.t’pipiTtJpi("p1) pi("p2)Figure 2: 2pt and 3pt correlation functions. J denotes the current, p1 is the momentum of the initial pionand p2 is the momentum of the pion in the final state. Three different values of T , the separation betweenthe initial and final state mesons, were used in this study.vector current (in the staggered formulation we need a 1-link operator to make a taste singlet, asboth pions are Goldstone mesons). A phase at the boundary (twisted boundary condition, [6]) isused to give the quarks momentum: a twistΦ(x+ eˆ jL) = ei2piθ jΦ(x) (3.1)is equivalent to the quark having a momentump j =2piθ jL. (3.2)θ can be tuned to get the desired q2, the four-momentum transfer defined asq2 = (E(~p2)−E(~p1))2− (~p2−~p1) · (~p2−~p1). (3.3)We calculate the form factor f+(q2) in the space-like (negative) region of q2 near zero (this is therange where experimental data is available). In addition to the 3-point correlators we also need theamplitudes (and energies) from the meson 2-point correlators.4. Fitting the correlatorsWe fit the 2-point and 3-point correlators simultaneously. We use multi-exponential fits toreduce systematic errors from the excited states, varying the number of exponentials. We take afit that gives a good χ2 and check that increasing the number of exponentials does not change theresult. The preliminary results presented in this paper are from a 5 exponential fit. Bayesian priors3PoS(LATTICE 2013)282Pion electromagnetic form factor from full lattice QCD Jonna Koponenare used to constrain the fit parameters. We fit all q2 values simultaneously to take into account thecorrelations. The fit functions for the 2-point correlators are of the formC2pt(p, t) =∑ib2i (~p)fn(Ei(~p), t)+∑ib′i2(~p)fo(E ′i(~p), t) (4.1)withfn(E, t) = e−Et + e−E(Lt−t),fo(E, t) = (−1)t/a[e−Et + e−E(Lt−t)]. (4.2)Here “fn” are the “normal” states and “fo” are the opposite parity states that appear due to staggeredquark formulation. The 3-point correlators are fitted withC3pt(~p1,~p2, t,T ) =∑i, jbi(~p1)fn(Ei(~p1), t)Jnni, j (~p1,~p2)b j(~p2)fn(E j(~p2),T − t) (4.3)−∑i, jbi(~p1)fn(Ei(~p1), t)Jnoi, j (~p1,~p2)b′j(~p2)fo(E′j(~p2),T − t)+(n↔ o). (4.4)Note that the amplitudes bi and the energies Ei, E ′i are the same as in the 2-point correlators.5. Vector form factorThe matrix element relevant for the pion form factor is〈pi(~p1)|J|pi(~p2)〉= Z√4E0(~p1)E0(~p2)J0,0(~p1,~p2), (5.1)where J0,0 is the ground state amplitude of the 3-point correlator. The form factor f+ is related tothe matrix element as〈pi(~p1)|Vi|pi(~p2)〉= f+(q2)(~p1 +~p2)i. (5.2)A renormalisation constant Z is needed for the vector current: we normalise the current by demand-ing that f+(0) = 1. Our results for the form factor as a function of q2 for different lattice ensemblesare shown in Fig. 3 along with the experimental data points.6. Continuum extrapolationWe do a simultaneous extrapolation to zero lattice spacing and physical pion mass by fittingthe results from the very coarse, coarse and fine lattice to the pole formf (q2) =11−q2〈r2〉/6 (6.1)allowing for a2 and mpi dependence:〈r2〉= A(1+Ba2 +Ca4)+ cJ ln(m2pi/µ2), (6.2)where A,B,C are fit parameters and cJ = 1/(8pi2F2) is a fixed constant from NLO chiral perturba-tion theory (here F is the pion decay constant in the chiral limit). The form factor extrapolated to4PoS(LATTICE 2013)282Pion electromagnetic form factor from full lattice QCD Jonna Koponen 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05-0.25 -0.2 -0.15 -0.1 -0.05  0f +(q2)q2 / GeV2NA7, exptvery coarsecoarsefineFigure 3: Shape of the pion electromagnetic form factor. Experimental results by NA7 Collaboration arefrom [3], lattice results by HPQCD from this work. The lines with error bands are from fits to experimentaldata (grey colour, fit from [3]) and to our lattice results (red colour, this work).real world is then the a= 0, physical mpi part of the fit function, also shown in Fig. 3 with the errorbands. The chiral log gives only a very small correction, as the pion masses are very close to thephysical value. The slope at q2 = 0 gives the mean square value of the charge radius:〈r2v〉= 6d f+(q2)dq2|q2=0. (6.3)Our preliminary result is 〈r2v〉= 0.40(3) fm2. Comparison to other lattice calculations and experi-ment in Fig. 4 shows good agreement.The form factor f+(q2) can be viewed as a Fourier transform of the electric charge distribution.Hence the charge density can be calculated from the vector form factor once its functional form isknown. In the non-relativistic limit the charge density isρ(R) =32piR〈r2v〉exp(− R√〈r2v〉/6)(6.4)Using our result 〈r2v〉= 0.40 fm2 gives the charge density plotted in Fig. 5.7. SummaryWe have presented here preliminary results from a full Lattice QCD calculation of the pionvector electromagnetic form factor at physical pion mass. The use of twisted boundary conditions5PoS(LATTICE 2013)282Pion electromagnetic form factor from full lattice QCD Jonna Koponen0.30.350.40.450.50.55<r2> /  f m2expt, NA7HPQCDUKQCD/RBCETMCMainzJLQCD/TWQCDQCDSF/UKQCDnf=2+1+1nf=2+1nf=2nf=2nf=2nf=2mpimin=128 MeVmpimin=330 MeVmpimin=260 MeVmpimin=280 MeVmpimin=290 MeVmpimin=400 MeVFigure 4: Vector mean square radius. The experimental result is from [3], and the HPQCD result is fromthis work. Other lattice results are from [7, 1, 8, 9, 10] (from left to right). n f is the number of flavors andmminpi is the smallest pion mass used in that calculation. The results shown here are each group’s final resultafter continuum and chiral extrapolation.Figure 5: Non-relativistic charge density calculated using the pole form and 〈r2v〉 from our fit. We omit zdirection here for clarity and plot the charge density against x and y.6PoS(LATTICE 2013)282Pion electromagnetic form factor from full lattice QCD Jonna Koponenallows us to calculate the form factor in the q2 range from 0 to −0.12 GeV2 where experimentaldata is available. We also determine the charge radius: our preliminary result is 〈r2v〉= 0.40(3) fm2.Comparison with experiment shows very good agreement.AcknowledgementsWe are grateful to MILC for the use of their gauge configurations and the MILC code. We usedthe Darwin Supercomputer in Cambridge as part of the DiRAC facility jointly funded by STFC,BIS and the Universities of Cambridge and Glasgow.References[1] ETM Collaboration, R. Frezzotti, V. Lubicz, and S. Simula, PRD 79 (2009) 074506[2] B.B. Brandt, arXiv 1310.6389 [hep-lat][3] NA7 Collaboration, S.R. Amendolia et al., Nucl. Phys. B277 (1986) 168[4] MILC Collaboration, A. Bazavov, C. Bernard, C. DeTar, W. Freeman, S. Gottlieb, U. M. Heller, J. E.Hetrick, J. Laiho, L. Levkova, M. Oktay, J. Osborn, R.L. Sugar, D. Toussaint, R.S. Van de Water, PRD82 (2010) 074501, arXiv:1004.0342; and A. Bazavov, C. Bernard, C. DeTar, W. Freeman, S. Gottlieb,U. M. Heller, J. E. Hetrick, J. Komijani, J. Laiho, L. Levkova, J. Osborn, R. L. Sugar, D. Toussaint, R.S. Van de Water, R. Zhou, PRD 87 (2013) 054505, arXiv:1212.4768[5] HPQCD Collaboration, R. J. Dowdall, C. T. H. Davies, G. P. Lepage, C. McNeile, PRD 88 (2013)074504, arXiv:1303.1670[6] D. Guadagnoli, F. Mescia, and S. Simula, PRD 73 (2006) 114504[7] UKQCD and RBC Collaborations, P.A. Boyle, J.M. Flynn, A. Jüttner, C. Kelly, H. Pedroso de Lima,C.M. Maynard, C.T. Sachrajda, and J.M. Zanotti, JHEP 07 (2008) 112[8] B.B. Brandt, A. Jüttner, H. Wittig, arXiv 1306.2916 [hep-lat],V. Gülpers, G. von Hippel and H. Wittig, PoS (Lattice 2012) 181 and arXiv 1309.2104 [hep-lat][9] JLQCD and TWQCD Collaborations, T. Kaneko, S. Aoki, G. Cossu, X. Feng, H. Fukaya, S.Hashimoto, J. Noaki, and T. Onogi,PoS (Lattice 2008) 158[10] QCDSF and UKQCD Collaborations, D. Brömmel et al., Eur. Phys. J. C51 (2007) 3357

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Pion electromagnetic form factor from full lattice QCD

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