The charge, magnetic and quadrupole form factors of vector mesons and the charge form factor of pseudo-scalar mesons are calculated in quenched lattice QCD. The charge radii and magnetic moments are derived. The quark sector contributions to the form factors are calculated separately and we highlight the environmental sensitivity of the light-quark contribution to charge radii.B. G. Lasscock, J. Hedditch, D. B. Leinweber, A. G. William

Hedditch, J.

Lasscock, B.

Leinweber, D.

Williams, A.

The charge, magnetic and quadrupole form factors of vector mesons and the charge form factor of pseudo-scalar mesons are calculated in quenched lattice QCD. The charge radii and magnetic moments are derived. The quark sector contributions to the form factors are calculated separately and we highlight the environmental sensitivity of the light-quark contribution to charge radii.QCDSF Collaboration: M. Gürtler, D. Brömmel, M. Göckeler, Ph. Hägler, R. Horsley, Y. Nakamura, D. Pleiter, P.E.L. Rakow, A. Schäfer, G. Schierholz, H. Stüben and J.M. Zanott

Gurtler, M.

Brommel, D.

Gockeler, M.

Hagler, P.

Horsley, R.

Nakamura, Y.

Pleiter, D.

Rakow, P.

Schafer, A.

Schierholz, G.

Stüben, H.

Zanotti, J.

Adelaide Research &amp; Scholarship

 PUBLISHED VERSION    QCDSF Collaboration; Gürtler, M.; Brömmel, D.; Göckeler, Meinulf; Hägler, Philipp; Horsley, Roger; Nakamura, Yoshifumi; Pleiter, Dirk; Rakow, Paul E. L.; Schäfer, Andreas; Schierholz, Gerrit; Stüben, Hinnerk; Zanotti, James Michael  Vector meson electromagnetic form factors, P o S - Proceedings of Science (Lattice 2008), 2008; 051:1-051:7. Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.                   http://hdl.handle.net/2440/77056  PERMISSIONS  http://pos.sissa.it/POSauthors.html    • All contributions are published in PoS under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. Every author submitting his/her contribution to PoS retains the copyright, and upon submission (s)he applies this license to his/her work.    24rthApril 2013 PoS(LATTICE 2008)051Vector meson electromagnetic form factorsQCDSF Collaboration: M. Gürtler∗,a D. Brömmel,b M. Göckeler,c Ph. Hägler,aR. Horsley,d Y. Nakamura,e D. Pleiter,e P.E.L. Rakow, f A. Schäfer,c G. Schierholz,c,eH. Stübeng and J.M. ZanottidaInstitut für Theoretische Physik T39, Physik-Department der TU München, 85747 Garching,GermanybSchool of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UKcInstitut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, GermanydSchool of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UKeJohn von Neumann-Institut für Computing NIC / DESY, 15738 Zeuthen, Germanyf Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, UKgKonrad-Zuse-Zentrum für Informationstechnik Berlin, 14195 Berlin, GermanyEmail: martin.guertler@ph.tum.deWe present preliminary results for the electromagnetic form factors of vector mesons in lat-tice QCD. The analysis is based on two- and three-point functions, calculated with two-flavourclover-improved configurations for a lattice spacing of a ∼ 0.08 fm. The pions on these lat-tices are too heavy (mpi ≥ 400 MeV) to allow for ρ meson decay. We obtain a charge ra-dius〈r2〉∼ 0.49(5) fm2, a g-factor gρ ∼ 1.6(1), and a small negative quadrupole momentµQ =−0.017(2) fm2.The XXVI International Symposium on Lattice Field TheoryJuly 14-19 2008Williamsburg, Virginia, USA∗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 2008)051Vector meson electromagnetic form factors M. Gürtler1. IntroductionMost of our knowledge of the internal structure of hadrons is encoded in (generalised) formfactors. They allow to study quantities like charge radii or multipole moments.A number of lattice results is available for the nucleon, eg., [1] and the pion [2] electromagneticform factors, and a first attempt in the quenched approximation to compute the form factors ofvector mesons can be found in [3]. All these analyses are based on suitable hadron three-pointfunctions. Another interesting ansatz based on density-density correlators can be found in [4]. Arecent overview of the subject can be found in[5].In this contribution, we present results for vector meson form factors in unquenched latticeQCD employing the standard three-point functions, obtained for non-perturbatively clover im-proved Wilson fermions and Wilson gauge action.2. Vector meson form factorsWe consider the interaction of a vector meson with an electromagnetic current Jα . In thissection, all quantities are expressed in Minkowski space, gµν = diag(−1,1,1,1). The relevantmatrix element 〈p′,s′ |Jα | p,s〉 can be parametrisedN〈p′,s′ |Jα | p,s〉= −G1(Q2)(ε ′∗ · ε)Pα−G2(Q2)[(ε ′∗ ·q)εα − (ε ·q)ε ′∗α]+G3(Q2)(ε ·q)(ε ′∗ ·q)Pα/(2mρ)2. (2.1)Here s(s′) is the spin and p(p′) the momentum of the incoming (outgoing) vector meson, P = p+ p′,and N = 2√Eρ(~p′)Eρ(~p) is a normalisation factor. The meson’s polarisation λ ∈ {0,+,−} isencoded in the polarisation vectors ε(λ ,~p), which obey pµεµ(~p,λ ) = 0 and the completenessrelation∑λ=0,+,−εi(~p,λ )∗ε j(~p,λ ) =−gi j + pi p j/m2.The form factors Gi depend on the momentum transfer Q2 = −q2 = −(p′− p)2. They can berewritten in terms of the Sachs charge, magnetic and quadrupole form factors:GC(Q2) = G1 (Q2)+2/3 η GQ (Q2)GM(Q2) = G2 (Q2)GQ(Q2) = G1 (Q2)−G2 (Q2)+(1+η)G3 (Q2) , (2.2)where η = Q2/(2mρ)2. The momentum transfer dependence of the electric form factor GC containsinformation on the charge radius〈r2〉=−6 ∂GC∂ (Q2)∣∣∣∣Q2=0, (2.3)while GM gives the magnetic moment of the ρ mesonµρ =e2mρGM(Q2 = 0).2PoS(LATTICE 2008)051Vector meson electromagnetic form factors M. GürtlerThe forward limit of GM ≡G2 itself is also called the ρ meson’s g-factor gρ . In the simulation,it can be accessed only from an extrapolation in q2 → 0 because the G2 contribution in Eq. (2.1)vanishes in the forward case. Information about the shape of the vector meson is encoded in thequadrupole moment µQ = e/m2ρGQ(0); for µQ < 0 the meson is oblate, and for µQ > 0 its shape isprolate. Also µQ can only be determined from a q2 → 0 extrapolation.3. Lattice calculationOn the lattice, the calculation of form factors requires an analysis of suitable three-point func-tions. A meson with momentum ~p is created at time t = 0, interacts with the electromagneticcurrent at intermediate time τ , and gets annihilated at time t = tsink. Its creation is described by aninterpolating operator χ†µ = ¯dγµu, which has non-vanishing overlap with a charged ρ meson:〈Ω∣∣χµ(0)∣∣ρ(~p,s)〉=√2Eρ(~p)−1λρ(~p)εµ(p,s).Here, λρ describes the overlap of χµ with the ρ meson, and εµ is the polarisation vector. Thecorresponding three-point function is given byGαµν(t,τ,~p′,~p) = ∑~x,~ξe−i~p′(~x−~ξ )e−i~p~ξ〈Ω∣∣χµ(x)Jα(ξ )χ†ν(0)∣∣Ω〉 ,which can be written for large time separations aslimτ→∞t−τ→∞Gαµν(t,τ,~p′,~p) =e−Eρ (~p′)(t−τ)e−Eρ (~p)τ4Eρ(~p′)Eρ(~p)λρ(~p′)¯λρ(~p)×(gµτ −p′µ p′τm2ρ)Jτασ(gσν −pσ pνm2ρ), (3.1)where we have neglected the exponentially suppressed contributions from excited states, and whereJτασ is defined via Jα = ε ′∗τ Jτασ εσ . Two-point functions are defined byGµν(t,~p) = ∑~xe−i~p~x〈Ω∣∣χµ(x)χ†ν(0)∣∣Ω〉 .For a sufficiently large separation of source and sink they readlimt→∞Gµν(t,~p) =−e−Eρ (~p) t2Eρ(~p)λρ(~p)¯λρ(~p)(gµν −pµ pνm2ρ). (3.2)Consequently, the overlap factors λρ and the exponentials from Eqs. (3.1) and (3.2) cancel in theratioRαµν(τ,~p′,~p) =Gαµν(t,τ,~p′,~p)Gµµ(t,~p′)√Gνν(t− τ,~p)Gµµ(τ,~p′)Gµµ(t,~p′)Gνν(τ,~p)Gµµ(t− τ,~p′)Gνν(t,~p), (3.3)which is independent of τ for large time separations. Thus, we obtain a system of linear equationsRαµν(~p,~p′) = ∑iciGi (3.4)3PoS(LATTICE 2008)051Vector meson electromagnetic form factors M. GürtlerVolume β κ mpi [MeV] mρ [ MeV] a[fm] tsink163 32 5.29 0.13500 929(2) 1228(4) 0.089 14163 32 5.29 0.13550 784(3) 1090(8) 0.084 14243 48 5.29 0.13590 591(2) 965(6) 0.080 18243 48 5.29 0.13620 406(6) 898(10) 0.077 18Table 1: Compilation of parameters.with easily computable coefficients ci of the Gi for each value of Q2. This overdetermined systemis solved by χ2 minimisation to finally obtain the form factors Gi.For our computation, we use QCDSF configurations, generated with two flavours of cloverimproved Wilson fermions and the Wilson gauge action. The improvement coefficient cSW wasdetermined non-perturbatively[6], giving cSW = 1.9192 at β = 5.29. Parameters, lattice constantsetc. are summarised in Tab. 1. The scale was set using the nucleon mass. Jacobi smearing is applied(κ = 0.21,N = 60) in order to enhance the ground state overlap of our source and sink operators.We use the sequential source technique for the calculation of the three-point functions. Thesink time tsink is fixed to the values in Tab. 1, where the source for the secondary inversion is con-structed from the quark propagator. Without further inversions, three point functions with insertionsof arbitrary operators on all time slices can be computed.Because each value of the sink momentum ~p′ = 2pi/Ls~n′ (Ls being the spatial lattice extent) re-quires separate secondary inversions and is thus expensive, we use only three values for~n′, namely(0,0,0), (−1,0,0), and (0,−1,0). For the source momentum, we take 2pi/Ls~n, with ~n assuming(0,0,0), (1,0,0), (1,1,0), (1,1,1), (2,0,0), (2,1,1), (2,2,1) and all possible permutations of thevector components.There is no need to consider the disconnected part of the three-point functions, because theircontributions from a gauge configuration U and its complex conjugate U∗ (which have the sameweight) cancel in the ensemble average.The physical ρ meson is a resonance and decays predominantly into two pions. In contrast, thevector mesons on the lattices we consider here are stable, because their mass is too small to createtwo pions in a relative P wave. This means that the quantities we compute here are not directlyaccessible to any experiment, and should rather be considered as being exemplary for a genericspin-1 particle.4. ResultsFig. 1 shows the Sachs form factors (2.2) for the smallest pion mass we have analysed so far.On this lattice, the smallest momentum transfer we have is Q2min ∼ 0.44 GeV2. This is relativelylarge, and will be improved in the future by the use of partially twisted boundary conditions [7].The electric form factor is fitted with the phenomenologically motivated monopole ansatzGC(Q2) = GC(0)1+Q2/m2C.In the forward case, Q2 = 0, this is equal to the electric charge. Because we use the non-conservedlocal vector current γµ , then measured electric form factor in the forward case GunrenC (0) gives us4PoS(LATTICE 2008)051Vector meson electromagnetic form factors M. Gürtler-1-0.500.511.520 0.5 1 1.5 2 2.5Q2[GeV2]β = 5.29 κ = 0.13620 Vol 24348 mpi = 406MeVGM(t), dipole fitGC(t), monopole fitGQ(t), linear fitFigure 1: Example plot of form factors, together with fits described in Sec. 4.the renormalisation factor ZV = 1/GunrenC (0), that has to be applied to all three-point functions.However, we do not fix GC(0) in the monopole fit. The charge radius is computed according toEq. (2.3).For the magnetic form factor, a monopole ansatz would be inadequate, because this would leadto divergencies in the magnetisation density one obtains after Fourier transforming to coordinatespace. Thus, we use a dipole ansatzGM(Q2) = GM(0)(1+Q2/m2M)2 . (4.1)The quadrupole form factor shows relatively large error bars, and is for all but the smallestnon-vanishing momentum transfer compatible with zero. To get a first idea about the quadrupolemoment, we use an ansatz linear in Q2 to extrapolate the lattice data to Q2 = 0. An analysiswith partially twisted boundary conditions will be helpful here, too, in order to decide about themagnitude of the quadrupole moment.The results for charge radii, magnetic and quadrupole moment for all analysed lattices aresummarised in table 2. In Fig. 2 we plot these results as functions of m2pi and attempt a linearextrapolation to the chiral limit. The (squared) charge radius 〈r2〉 behaves as one would expectfrom the pion cloud picture, showing the growth of the pion cloud with decreasing quark mass.The value of the charge radius is similar to the value known from the nucleon, and larger thanthe pion charge radius. This would also be the expectation from quark models, that attribute thelarger size of a vector meson to the repulsive hyperfine interaction of the aligned spins, whereasthe anti-aligned spins in the pion experience hyperfine attraction. Our linear extrapolation gives acharge radius of〈r2〉= 0.49(5) fm2.5PoS(LATTICE 2008)051Vector meson electromagnetic form factors M. Gürtlerκ volume〈r2〉[ fm2] µM µQ[ fm2]0.13500 16332 0.33(5) 2.1(3) 0.002(5)0.13550 16332 0.42(8) 1.7(3) −0.004(2)0.13590 24348 0.38(3) 1.8(2) −0.007(3)0.13620 24348 0.49(4) 1.7(3) −0.015(4)Table 2: Charge radii, magnetic and quadrupole moments on the analysed lattices.0.10.20.30.40.511.522.53−0.03−0.010.010 0.2 0.4 0.6 0.8 1m2pi [GeV2]〈r2〉[fm2]gρµQ[fm2]Figure 2: Chiral extrapolation for charge radius, g-factor and quadrupole moment.For the g-factor we obtain gρ = 1.6(1). This is compatible with expectations based on sum rulecalculations, gρ = 2.0(3)[8]. The quark model calculation in [9] gives gρ = 1.92, and [10] obtaingρ = 2.01, based on Dyson-Schwinger equations. It is considerably smaller than the quenchedlattice result gρ ∼ 2.3 of [3]. Their result, however, was not based on an extrapolation in Q2, buton an estimate assuming GC ∝ GM, using only one non-zero value of the momentum transfer.A phenomenologically interesting quantity is the quadrupole moment. For large quark masses,and for large momentum transfers, the quadrupole form factor is hardly distinguishable from zero.For small quark masses, however, we see a trend of GQ becoming increasingly negative, beingclearly different from zero for our lightest pion and small values of the momentum transfer. Thelinear extrapolation gives a value of µQ = −0.017(2) fm2. The negative sign is interpreted as theρ meson having oblate shape, an observation also found in [3], and [4]. The statistical uncertainty,and the systematic uncertainties from the extrapolations to the forward limit are still rather large.Our results should therefore be considered as preliminary. Calculations at smaller quark massesand smaller values of the momentum transfer using, e.g., partially twisted boundary conditions, are6PoS(LATTICE 2008)051Vector meson electromagnetic form factors M. Gürtlerclearly desirable to reach firm conclusions.5. SummaryWe have computed the electromagnetic vector meson form factors in a dynamical two-flavourlattice simulation. We determined the charge radius of the ρ meson from a monopole fit of the q2dependence of the electric form factor, obtaining〈r2〉= 0.49(5) fm2 from a linear extrapolationto vanishing quark mass. From a dipole extrapolation of the magnetic form factor, we obtained ag-factor of gρ = 1.6(1) as forward limit of the magnetic form factor. The result for the quadrupoleform factor implies an oblate shape of the ρ meson, manifesting itself in a quadrupole moment ofµQ = −0.017(2) fm2. We are currently improving our study by repeating the analysis for moreensembles and by employing partially twisted boundary conditions.6. AcknowledgementThe numerical calculations have been performed on the Hitachi SR8000 at LRZ (Munich), theAPE1000 and apeNEXT at NIC/DESY (Zeuthen) and the BlueGeneL’s at NIC/Jülich and EPCC.This work was supported by the DFG (Forschergruppe Gitter-Hadronen- Phänomenologie, Tran-sregional Collaborative Research Centre SFB/TRR 55 and Emmy-Noether program) and by the EUI3HP under contract number RII3-CT-2004-506078. P. H. thanks the Excellence Cluster Universeat Technische Universität München for support.References[1] M. Göckeler et al. [QCDSF/UKQCD Collaboration], PoS LAT2007 (2007) 161 [arXiv:0710.2159[hep-lat]].[2] D. Brömmel et al. [QCDSF/UKQCD Collaboration], Eur. Phys. J. C 51 (2007) 335[arXiv:hep-lat/0608021].[3] J. N. Hedditch, W. Kamleh, B. G. Lasscock, D. B. Leinweber, A. G. Williams and J. M. Zanotti, Phys.Rev. D 75 (2007) 094504 [arXiv:hep-lat/0703014].[4] C. Alexandrou, P. de Forcrand and A. Tsapalis, Phys. Rev. D 66 (2002) 094503[arXiv:hep-lat/0206026], C. Alexandrou and G. Koutsou, arXiv:0809.2056 [hep-lat].[5] J. Zanotti, PoS LATTICE 2008 (2008) 007[6] K. Jansen and R. Sommer [ALPHA collaboration], Nucl. Phys. B 530 (1998) 185 [Erratum-ibid. B643 (2002) 517] [arXiv:hep-lat/9803017].[7] C. T. Sachrajda and G. Villadoro, Phys. Lett. B 609 (2005) 73 [arXiv:hep-lat/0411033].[8] A. Samsonov, JHEP 0312 (2003) 061 [arXiv:hep-ph/0308065].[9] H. M. Choi and C. R. Ji, Phys. Rev. D 70 (2004) 053015 [arXiv:hep-ph/0402114].[10] M. S. Bhagwat and P. Maris, Phys. Rev. C 77 (2008) 025203 [arXiv:nucl-th/0612069].7

Vector meson electromagnetic form factors

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Vector meson electomagnetic form factors

Vector meson electomagnetic form factors

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Vector meson electomagnetic form factors

Abstract

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Adelaide Research & Scholarship