# Pseudoscalar and vector meson form factors from lattice QCD

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Pseudoscalar and vector meson form factors from lattice QCD

J.N. Hedditch,1 W. Kamleh,1 B.G. Lasscock,1 D.B. Leinweber,1 A.G. Williams,1 and J.M. Zanotti2 1Department of Physics and Mathematical Physics and

Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, 5005, Australia

2School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK

We present a study of the pseudoscalar and vector meson form factors, calculated using the FatLink Irrelevant Clover (FLIC) action in the framework of Quenched Lattice QCD. Of particular interest is the determination of a negative quadrupole moment, indicating that the ρ meson is not spherically symmetric.

PACS numbers: 12.38.Gc,14.40.Aq Keywords: Mesons, form factors

arXiv:hep-lat/0703014v1 14 Mar 2007

I. INTRODUCTION

The important role that electromagnetic form factors play in our understanding of hadronic structure has been well documented for more than ﬁfty years. The reason for their popularity is that they encode information about the shape of hadrons, and provide valuable insights into their internal structure in terms of quark and gluon degrees of freedom.

Most of the attention, both experimentally and theoretically, has focused on the electromagnetic form factors of the nucleon (see Refs. [1, 2, 3, 4, 5] for recent reviews). The electromagnetic form factors of pseudoscalar mesons, especially the pion, being the lightest QCD bound state, have also been studied extensively [6, 7, 8, 9, 10, 11] in lattice QCD. More recently, there is a renewed interest in calculating the pion form factor on the lattice [12, 13, 14, 15, 16, 17, 18]. This is especially timely considering the new [19] and reanalysis of old [20] experimental data from JLab.

The vector meson form factors, on the other hand, have received less attention (see Refs. [21, 22, 23, 24, 25] for recent work). Of particular interest is the quadrupole moment of the ρ meson, where theoretical determinations can disagree by as much as a factor of two [22]. We aim to resolve this issue by performing the ﬁrst direct lattice calculation of the ρ-meson quadrupole form factor. Charge and magnetic form factors are also calculated and from these we extract the relevant static quantities, namely the mean square charge-radius and magnetic moment. We also analyse the dependence of light-quark contributions to these form factors on their environment and contrast these with a new calculation of the corresponding pseudoscalar-sector result.

Our aim is to reveal the electromagnetic structure of vector mesons and to study to what extent the qualitative quark model picture is consistent with quenched lattice QCD. Interestingly, it has been shown in a lattice calculation by Alexandrou et al. [26] that the distribution of charge in the vector meson is oblate, and therefore not consistent with the picture of a quark anti-quark in relative S-wave. By calculating the vector meson quadrupole

form factor we make a direct comparison with the ﬁndings of Ref. [26].

For each observable we calculate the quark sector contributions separately. Using this additional information we examine the environmental sensitivity of the lightquark contributions to the pseudoscalar and vector meson charge radii. We also evaluate the dominance of the light quark contributions to the K and K∗.

This paper builds on the preliminary work presented in Ref. [27]. In Section II A we introduce the theoretical formalism of meson form factors, including the techniques required to extract them from a lattice calculation. Section 3 contains details of our lattice simulation, while in Section 4 we present and discuss our results for both pseudoscalar and vector mesons. Finally, in Section 5 we summarise our ﬁndings and discuss future work.

II. THEORETICAL FORMALISM

A. Meson form factors

Meson form factors are extracted from matrix elements involving the vector (electromagnetic) current

M (p ′)|Jα|M (p) ,

(1)

where M (p) (M (p ′)) denotes a meson state with initial (ﬁnal) momentum p (p ′). The momentum transfer is qµ = (p′µ − pµ).

For a pion, the matrix element in Eq. (1) is described

by a single form factor

π(p ′)|Jα|π(p) =

1

[pα + pα′]Fπ(Q2) ,(2)

2 Eπ(p)Eπ(p ′)

where Q2 = −q2 is the invariant momentum transfer, and the energy of the pion with momentum p is Eπ(p) = m2π + p 2. The formula for the kaons are exactly analogous. The ρ-meson, on the other hand, is spin-1 and is described by three form factors [28],

ρ(p ′, s′)|J α|ρ(p, s) =

2

1

ǫ′τ⋆(p′, s′) J τασ(p′, p) ǫσ(p, s)(3)

2 Eρ(p)Eρ(p ′)

where ǫ and ǫ′ are the initial and ﬁnal polarisation vec-

tors, respectively, and

J τ ασ(p′, p) = − G1(Q2) gτ σ [pα + pα′] + G2(Q2)[gασqτ − gατ qσ] − G3(Q2) qσqτ pα + pα′ .

(4)

2m2ρ

The covariant vertex functions G1,2,3 can be rewritten in terms of the Sachs charge, magnetic and quadrupole

form factors [28, 29],

GQ(Q2) = G1(Q2) − G2(Q2) + (1 + η)G3(Q2) (5)

GM (Q2) = G2(Q2)

(6)

GC (Q2) = G1(Q2) + 2 η GQ(Q2) ,

(7)

3

where mρ is the mass of the vector-meson system calculated on the lattice and η = Q2/4m2ρ.

The charge qρ, magnetic moment µρ, and quadrupole moment Qρ are then extracted from GC , GM , and GQ, respectively at zero momentum transfer

e GC(0) = qρ

(8)

e GM (0) = 2mρµρ

(9)

e GQ(0) = m2ρQρ .

(10)

The formulae for the K∗ are exactly analogous.

B. Meson form factors on the lattice

The matrix elements in Eqs. (2) and (3) are obtained from ratios of three-point and two-point correlation func-

tions

Rα(p′, p) =

Gα(p ′, p, t2, t1) Gα(p, p ′, t2, t1) (11)

for pseudoscalar mesons, and

Rµαν (p′, p) =

Gαµν (p ′, p, t2, t1) Gανµ(p, p ′, t2, t1)

(12)

for vector mesons. Note repeated indices are not summed

over.

The two-point correlation function for the pseudoscalar mesons is

G(t2, p) = e−ip·x2 Ω|χ(x2)χ†(0)|Ω . (13)

x2

Similarly for the vector mesons,

Gµν (t2, p) = e−ip·x2 Ω|χµ(x2)χ†ν (0)|Ω . (14)

x2

The three-point correlation function for the pseudoscalar meson is

Gα(t2, t1, p ′, p) =

e−ip ′·(x2−x1)e−ip·x1 Ω| χ(x2)J α(x1)χ†(0) |Ω .

x1,x2

(15)

Similarly the three-point function for the vector meson is

Gαµν (t2, t1, p ′, p) =

e−ip ′·(x2−x1)e−ip·x1 Ω| χµ(x2)J α(x1)χ†ν (0) |Ω .

x1,x2

(16)

The Lorentz indices µ and ν are only present for the vector mesons, while α is the index of the electromagnetic current.

The ratios in Eqs.(11) and (12) are constructed in such a way as to remove the time-dependence and constants

of normalisation from the correlation functions at large time separations, t1 and t2 − t1.

These ratios diﬀer subtly from previous work [30], in that we are explicitly enforcing the parity of the terms through the choice of momenta (p′, p) and (p, p′) vs (p′, p)

and (−p, −p′). This requires two three-point propagators (with momentum-transfer q and −q) for each conﬁguration. However with the well established technique of averaging over U and U ∗ conﬁgurations [30, 31], there is no additional cost.

1. π-meson case

Since the pion has zero spin, the vertex is extraordinarily simple and takes the form given in Eq. (2). Here we show how this function is extracted from the ratio Eq. (11) by evaluating the correlation functions at large Euclidean times.

3 First we deﬁne the matrix elements as,

Ω|χµ(0)|π(p) =

1 λπ(p) ,

2Eπ (p)

π(p)|χ†µ(0)|Ω =

1 λ¯π(p) .

(17)

2Eπ (p)

Here λπ(p) and λ¯π(p) are the couplings of the interpolator to the pion with momentum p at the sink and source respectively. The momentum dependence allows for the use of smeared fermion sources and sinks. The bar allows for diﬀerent amounts of smearing at the source and sink.

By inserting a complete set of energy and momentum states into Eq. (13), we can show that at large Euclidean time,

lim G(t2, p) = e−Eπ(p)t2 λπ(p)λ¯π(p) .

t2→∞

2Eπ (p)

(18)

Following the same treatment, one can show that the three-point function Eq. (15) at large Euclidean time is

lim Gα(t2, t1, p ′, p) =

t1 ,t2 −t1 →∞

e−Eπ(p ′)(t2−t1)e−Eπ(p)t1 λπ (p ′) π(p ′)|J α(0)|π(p) λ¯π(p) . 2 Eπ(p)Eπ(p ′)

(19)

Substituting these expressions into Eq. (11) and using Eq. (2), the ratio Rα(p′, p) simply reduces to

Rα(p′, p) =

1

[pα + pα′]Fπ(Q2) , (20)

2 Eπ(p)Eπ(p ′)

such that the large Euclidean time limits of the ratio Rα is a direct measure of Fπ(Q2) up to kinematical factors.

2. ρ-meson case

Following [28], we deﬁne the matrix element of the electromagnetic current for ρ-meson in terms of the covariant vertex functions G1,2,3 as in Eqs. (3) and (4).

The analogues of the matrix elements in Eq. (17) are

Ω| χµ(0) |ρ(p, s) = ρ(p, s)| χν†(0) |Ω =

1 λρ(p)ǫµ(p, s) 2Eρ(p)

1 λ¯ρ(p) ǫ⋆ν (p, s) . (21) 2Eρ(p)

The polarisation vectors obey the transversality condition

ǫµ(p, s) ǫ⋆ν(p, s) = − gµν − pµpν , (22)

s

m2ρ

because the vector meson current is a conserved current.

The evaluation of the two- and three-point functions proceeds as for our discussion of the pion. However the completeness relation includes a sum over spin-states. Using the transversality condition Eq. (22) the analogue of Eq. (18) becomes

lim G(t2, p) =

e−Eρ(p)t2 λρ(p)λ¯ρ(p)ǫµ(p, s) ǫ⋆(p, s)

t2→∞ s 2Eρ(p) ν

= − e−Eρ(p)t2 λρ(p)λ¯ρ(p) gµν − pµpν .

2Eρ(p)

m2ρ

(23)

Similarly using Eq. (3) we can evaluate the three-point function,

lim Gαµν (t2, t1, p ′, p) =

t1 ,t2 −t1 →∞

e−Eρ(p ′)(t2−t1)e−Eρ(p)t1 λρ(p ′)ǫµ(p′, s′) ǫ′⋆(p′, s′)J τ ασ(p′, p)ǫσ(p, s) λ¯ρ(p) ǫ⋆ (p, s) s,s′ 4Eρ(p)Eρ(p ′) τ ν

= e−Eρ(p ′)(t2−t1)e−Eρ(p)t1 λρ(p ′)λ¯ρ(p) gµτ − p′µp′τ J τ ασ gσν − pσpν .

4Eρ(p)Eρ(p ′)

m2ρ

m2ρ

4 (24)

Inserting the above expressions into the ratio in Eq. (12), together with our choice of momentum used in the simulations, namely p′ = (Eρ, px, 0, 0) (Eρ =

m2ρ + p2x) and p = (mρ, 0, 0, 0), it is possible to express Rµαν in terms of the Sachs form factors,

R0 =

p2x

GQ(Q2) + Eρ + mρ GC (Q2) ,

11

3mρ Eρmρ

2 Eρmρ

R0 = R0 = −

p2x

GQ(Q2) + Eρ + mρ GC (Q2) ,

22

33

6mρ Eρmρ

2 Eρmρ

R133 = R331 = 2

px GM (Q2) . Eρmρ

The individual form factors are isolated as follows:

GC (Q2) = 23 EρE+ρmmρρ R101 + R202 + R303 , (25)

GM (Q2) =

Eρmρ R3 + R3 ,

px

13

31

(26)

GQ(Q2) = mρ Eρmρ 2R0 − R0 − R0 . (27)

p2x

11

22

33

While we have used the subscript ρ to denote a vector

meson, the results are applicable to vector mesons in general, including the K∗ for example.

C. Extracting static quantities

The mean squared charge radius r2 is obtained from the charge form-factor through the following relation,

r2 = −6 ∂ G(Q2)

.

(28)

∂Q2

Q2 =0

To calculate the derivative the monopole form is used,

GC (Q2) =

1

.

Q2 Λ2

+1

(29)

Λ is referred to as the monopole mass. Inserting this form into Eq. (28) and rearranging provides

r2 = 6

1 −1 ,

(30)

Q2 GC (Q2)

valid for quantities with GC (Q2 = 0) = 1. As mentioned in Sec. II A, the charge (Eq. (8)),

magnetic moment (Eq. (9)), and quadrupole moment (Eq. (10)) can be extracted from the Sachs form factors

at zero momentum transfer. Since we perform our calculations at a single, ﬁnite value of Q2, we will need to

adjust our results to zero momentum transfer.

From studies of nucleon properties, it is observed that GM and GC have similar Q2-scaling at small Q2 [32]. In the following, we shall assume that this scaling also holds

for quark contributions to mesons. If GC (0) = 1, we have

GM (Q2)

GM (0) ≃ GC (Q2) .

(31)

Whilst a similar scaling could be used to relate our quadrupole form-factor to the quadrupole moment, we believe that the form-factor at our small ﬁnite Q2 ( ≃ 0.22GeV) will be of greater phenomenological interest.

We note that for a positively charged meson a negative value of GQ corresponds to an oblate deformation.

III. METHOD

The electromagnetic form factors are obtained using the three-point function techniques established by Leinweber, et al. in Refs. [30, 33, 34] and updated for smeared sources in Ref. [31]. Our quenched gauge ﬁelds are generated with the O(a2) mean-ﬁeld improved Luscher-Weisz plaquette plus rectangle gauge action [35] using the plaquette measure for the mean link. We use an ensemble of 379 quenched gauge ﬁeld conﬁgurations on 203 × 40 lattices with lattice spacing a = 0.128 fm. The gauge ﬁeld conﬁgurations are generated via the Cabibbo-Marinari pseudo-heat-bath algorithm [36] using a parallel algorithm with appropriate link partitioning [37].

We use the fat-link irrelevant clover (FLIC) Dirac operator [38] which provides a new form of nonperturbative O(a) improvement [39]. The improved chiral properties of FLIC fermions allow eﬃcient access to the light quark-mass regime [40], making them ideal for dynamical fermion simulations now underway [41]. For the vector current, we an O(a)-improved FLIC conserved vector current [31]. We use a smeared source at t2 = 8. Complete simulation details are described in Ref. [31].

Table I provides the κ-values used in our simulations, together with the calculated pseudoscalar and vector meson masses. While we refer to m2π in our ﬁgures and tables to infer the quark masses, we note that the critical value where the pion mass vanishes is κcr = 0.13135. Importantly the vector mesons remain bound at all quark masses considered in this calculation due to ﬁnite volume eﬀects. That is, the mass of the vector mesons is less than the energy of the lowest lying multi-hadron state with the appropriate quantum numbers.

The strange quark mass is chosen to be the third heaviest quark mass. This provides a pseudoscalar mass of 697

MeV which compares well with the experimental value of (2MK2 − Mπ2)1/2 = 693 MeV motivated by chiral perturbation theory. Two vector-meson interpolating ﬁelds are

considered, namely q¯γiq and q¯γiγ4q. Since results for the two interpolators agree, we simply present the results

for the q¯γiq interpolator, which displays a signiﬁcantly stronger signal.

The error analysis of the correlation function ratios

is performed via a second-order, single-elimination jackknife, with the χ2 per degree of freedom (χ2dof ) obtained via covariance matrix ﬁts. We perform a series of ﬁts

through the ratios after the current insertion at t = 14. By examining the χ2dof we are able to establish a valid window through which we may ﬁt in order to extract our observables. In all cases, we required a value of χ2dof no larger than 1.5. The values of the static quantities quoted

in this paper on a per quark-sector basis correspond to

values for single quarks of unit charge.

TABLE I: Meson masses for the respective values of the hopping parameter κ.

κ

0.12780 0.12830 0.12885 0.12940 0.12990 0.13205 0.13060 0.13080

amπ

0.5411(10) 0.5013(11) 0.4539(11) 0.4014(12) 0.3471(15) 0.3020(19) 0.2412(42) 0.1968(52)

amK

0.4993(11) 0.4782(11) 0.4539(11) 0.4285(11) 0.4044(12) 0.3862(13) 0.3671(19) 0.3574(16)

amρ

0.7312(30) 0.7067(36) 0.6797(46) 0.6537(49) 0.6309(56) 0.6160(64) 0.6039(71) 0.5982(80)

amK∗

0.7057(27) 0.6933(40) 0.6796(46) 0.6668(47) 0.6556(50) 0.6484(52) 0.6423(54) 0.6393(56)

5

obtained in the previous study reﬂects ﬁnite-volume effects attendant with the use of a small spatial volume.

By comparing the results for the up-quark contributions to the π and K (ρ and K∗) charge radii, it is possible to gain insights into the eﬀect that the presence of a heavier strange-quark has on the lighter up-quark in pseudoscalar (vector) mesons. Figures 2 and 3 show the quark sector contributions to the charge radii ( r2 ) of the pseudoscalar and vector mesons, respectively. The quark sector contributions to the charge radii for the pseudoscalar and vector meson are recorded in Tables II and III. From Fig. 2, we ﬁnd no evidence of environmental sensitivity in the light-quark contribution the pseudoscalar mesons. However in the vector sector, Fig. 3, we ﬁnd a consistently broader distribution of up-quark charge in the ρ compared to the up-quark in the K∗ at the smaller quark masses. The broadening of the charge distribution in the ρ is consistent with the hyperﬁne repulsion discussed above. The strange quark in the K∗ shows a particularly interesting environment sensitivity. While the strange quark mass is held ﬁxed, the distribution broadens as the light-quark regime is approached. This is consistent with the prediction of enhanced hyperﬁne repulsion as one of the quarks becomes light.

The strange neutral pseudoscalar and vector meson mean squared charge radii obtained from the weighted sum of the quark sector radii are displayed in Fig. 4. For the neutral strange mesons, we see a negative value for r2 , indicating that the negatively charged d-quark is lying further from the centre of mass on average than the s¯. We should expect just such a behaviour for two reasons, both stemming from the fact that the s¯ quark is considerably heavier than the d: the centre of mass must lie closer to the s¯, and the d-quark will also have a larger

IV. RESULTS

1. Charge radii

We begin the discussion of our results with the charge radii of the vector and pseudoscalar mesons. From the quark model we would expect a hyperﬁne interaction between the quark and anti-quark of the form mσqq·mσq¯q¯ . The interaction is repulsive where the spins are aligned, as in the vector mesons, and attractive where the spins are anti-aligned, as in the pseudoscalar mesons. In Fig. 1 we show the charge radii of the vector and pseudoscalar mesons. For comparison the charge radius of the proton is also shown. Indeed we ﬁnd that the charge radii of the vector mesons are consistently larger than the pseudoscalar mesons, and in fact similar to the charge radii of the proton, even at heavier quark masses. This is contrary to earlier lattice simulations with relatively small spatial extent [42], that have suggested that the π+, ρ+ and proton should have a very similar RMS charge radius at larger quark masses. It is possible that the agreement

FIG. 1: Strange and non-strange meson mean squared charge

radii for charged pseudoscalar and vector mesons. We also

include for comparison results for the proton taken from

Ref. [31]. The π and ρ-meson results are centred on the relevant value of m2π, other symbols are oﬀset horizontally for clarity.

6

TABLE II: Mean-square charge radius ( r2 ) for quarks of unit charge in units of fm2. m2π is given as a measure of the

input quark mass.

m2π (GeV2) 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uπ 0.216(5) 0.225(6) 0.240(8) 0.256(10) 0.274(14) 0.287(22) 0.304(44) 0.287(63)

uK 0.215(7) 0.224(7) 0.240(8) 0.257(9) 0.275(11) 0.289(12) 0.303(14) 0.306(15)

sK 0.242(7) 0.241(7) 0.240(8) 0.239(9) 0.239(10) 0.241(11) 0.243(13) 0.241(13)

TABLE III: Mean-square charge radius ( r2 ) for quarks of unit charge in units of fm2.

m2π (GeV2) 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ 0.268(9) 0.287(11) 0.315(16) 0.350(23) 0.397(36) 0.436(46) 0.492(72) 0.546(97)

uK∗ 0.271(11) 0.290(13) 0.315(16) 0.342(20) 0.372(26) 0.395(30) 0.417(35) 0.436(41)

sK∗ 0.309(12) 0.311(14) 0.315(16) 0.321(19) 0.331(23) 0.339(25) 0.353(28) 0.360(29)

FIG. 2: The quark sector contributions to the mean squared charge radius of the pseudoscalar mesons. The symbols are oﬀset horizontally for clarity.

Compton wavelength. Of course with exact isospin symmetry in our simulations, the non-strange charge neutral mesons have a zero electric charge radius.

To measure the environmental sensitivity of the lightquark sector more precisely, in Figs. 5 and 6 we show a ﬁt to the ratio of the light-quark contributions to the pseudoscalar and vector-mesons charge radii respectively. The diﬀerence is striking: for the pseudoscalar case we see no environment-dependence at all, whereas in the vector case we see that the presence of a strange quark acts to heavily suppress the light charge distribution. This is the eﬀect one predicts from a quark model, where the large mass of the s would act to suppress the hyperﬁne repulsion between the quark and anti-quark. It is also qualitatively consistent with eﬀective ﬁeld theory where the couplings of the light mesons are suppressed by the presence of the strange quark.

2. Magnetic moments

In Fig. 7 we present our results for the magnetic moments of the vector mesons. At the SU (3)ﬂavour limit, where we take the light quark ﬂavours to have the same mass as the strange quark, quark model arguments suggest the magnetic moment for a ρ+ should be -3 times the strange magnetic moment of the Λ (assuming no en-

FIG. 3: As for Figure 2 but for vector-mesons.

vironmental dependence). According to the particle data group [43], the magnetic moment of the Λ is −0.613 µN. Therefore we would naively expect a value of 1.84 µN for the magnetic moment of the ρ+, which is consistent with our ﬁndings.

In an earlier study, Anderson et al. [44] argued that the magnetic moment of the ρ-meson in natural magnetons (otherwise called the g-factor) should be approximately 2 at large quark masses. Converting our result to natural magnetons, we observe in Fig. 8 that our calculation of the ρ-meson g-factor (gρ) is fairly consistent with this. At light quark masses, however, we do see some evidence of chiral curvature, which would indicate that the linear chiral extrapolations of that paper should be considered with caution.

In Fig. 9 we present the quark sector contributions to the vector meson magnetic moments, the data is recorded in Table. IV. Here we observe a similar scenario to that observed earlier in the charge radius discussion, namely

7

FIG. 4: The mean squared charge radii for the neutral K0 and K0∗.

FIG. 6: As in Fig. 5 but for the vector-mesons.

FIG. 5: The ratio of the light quark contributions to the π and K mean squared charge radius.

that the u-quark contribution to the K∗ is consistently larger than the contribution from the heavier s-quark. We also ﬁnd that the contribution of the u-quark to the magnetic moment of a vector meson is suppressed when it is an environment of a heavier s-quark compared to when it is in the presence of another light quark. This is further supported when we consider the ratio of the contributions of a u-quark to the magnetic moments of the ρ and K∗ mesons, displayed in Fig. 10. This ratio is clearly greater than 1 below the SU (3)ﬂavour limit and is increasing for decreasing u-quark mass.

The magnetic moment of the vector meson, like the RMS charge radius, shows considerable environment dependence in the quark sector contributions. The larger contribution of a u-quark in a ρ relative to a K∗ is consistent with what we have already observed with the RMS charge radius, as follows: since r2 is larger for the uquark in a ρ meson than for the u-quark in a K∗, the eﬀective mass is reciprocally smaller for the u-quark in

FIG. 7: Charged vector meson magnetic moments.

a ρ. This smaller eﬀective mass gives rise in turn to a larger magnetic moment. Figure 10 shows this pattern. Figure 11 presents our results for the magnetic moment of the neutral K∗0 meson. As the d-quark becomes lighter than the s¯ we see the magnetic moment exhibiting a very linear negative slope. The magnitude of the magnetic moment is quite small, but clearly diﬀerentiable from zero everywhere except at the SU (3)ﬂavour limit where symmetry forces it to be exactly zero.

3. Quadrupole form-factors

The quadrupole form-factors of the ρ+ and K∗+ mesons are shown in Fig 12. We ﬁnd that the quadrupole form factor is less than zero, indicating that the spatial distribution of charge within the ρ and K∗ mesons is oblate. This is in accord with the ﬁndings of Alexandrou et al. [26] who observed a negative quadrupole moment for spin ±1 ρ-meson states in a density-density analy-

FIG. 8: The g-factor of the ρ meson.

8

TABLE IV: Magnetic moment for quarks of unit charge inside a vector meson in units of nuclear magnetons µN .

m2π (GeV2) 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ 1.71(2) 1.77(2) 1.84(3) 1.94(4) 2.04(6) 2.11(8) 2.20(11) 2.25(15)

uK∗ 1.73(2) 1.78(3) 1.84(3) 1.92(4) 1.99(5) 2.04(5) 2.10(6) 2.14(7)

sK∗ 1.82(3) 1.83(3) 1.84(3) 1.86(3) 1.88(4) 1.90(4) 1.92(5) 1.93(5)

FIG. 11: Neutral K∗-meson magnetic moment.

FIG. 9: Quark-sector contributions to the vector meson magnetic moments.

sis. We note that in a simple quark model, a negative quadrupole form factor requires that the quarks possess an admixture of s- and d-wave functions.

TABLE V: The quadrupole form-factor (in fm2) for quarks of unit charge inside a vector meson.

m2π (GeV)2 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ −0.0047(4) −0.0047(4) −0.0048(5) −0.0049(7) −0.0050(9) −0.0049(12) −0.0051(19) −0.0050(27)

uK∗ −0.0047(4) −0.0047(5) −0.0048(5) −0.0049(6) −0.0050(7) −0.0052(9) −0.0055(11) −0.0056(13)

sK∗ −0.0048(5) −0.0048(5) −0.0048(5) −0.0048(6) −0.0048(7) −0.0047(7) −0.0046(9) −0.0046(10)

FIG. 10: The ratio of the light-quark contributions to the magnetic moment of the ρ and K∗.

The quark sector contributions to the quadrupole form-factor are shown in Fig. 13. The corresponding data is contained in Table V. The ﬂavour independence of the results is remarkable.

We also ﬁnd that the ratio of the light-quark contri-

9

FIG. 12: Vector meson quadrupole form factors for ρ+ and K ∗+ .

FIG. 15: Quadrupole form-factor for neutral K∗ meson.

butions to the quadrupole form factor, shown in Fig. 14, is consistent with one within our statistics. In Fig. 15, we show the quadrupole form factor of the charge neutral K∗0 meson. We ﬁnd that the quadrupole moment of the K∗0 is non-trivial but just outside the one standard deviation level. The chiral trend towards positive values reﬂecting the negative charge of the larger d-quark contributions.

The lattice data for the quark sector contributions to the charge form factor is contained in Tables VI and VII for the pseudoscalar and vector mesons respectively. The magnetic and quadrupole form factors of the vector mesons is contained in Tables VIII and IX respectively.

FIG. 13: Quark-sector contributions to the quadrupole form factors.

FIG. 14: Environment-dependence for light-quark contribution to vector meson quadrupole form-factor.

V. CONCLUSIONS

We have established a formalism for determining the charge, magnetic and quadrupole Sachs form factors of vector mesons in lattice QCD. For the ﬁrst time the electric, magnetic, and quadrupole form factors of the light vector mesons have been calculated. The electric form factor of the pseudoscalar mesons have also been calculated.

With a large lattice volume and high statistics we have resolved a clear diﬀerence between the charge radii of the pseudoscalar and vector mesons. We argue that this is consistent with quark model predictions. Furthermore, we ﬁnd signiﬁcant environmental sensitivity of the lightquark contributions to the charge radii of the vector mesons.

We also presented a calculation of the magnetic moments of the vector mesons. We found that the magnetic moment of the ρ+ was consistent with the quark model predication of 1.84 µN at the SU (3)ﬂavour limit. We determine that there is also an environmental sensitivity in the magnitude of the light-quark contributions to the charged vector meson magnetic moments. We argue that

this is consistent with the environmental sensitivity in the light-quark contributions to the charge vector meson charge radii.

Finally, we have determined that the quadrupole form factor for a charged vector meson is negative in quenched Lattice QCD. This is consistent with previous calculations using density-density analysis. We ﬁnd that the ratio of quadrupole moment to mean square charge radius is 1:30, so the deformation is small but statistically signiﬁcant.

VI. APPENDIX

10

TABLE VII: As in Fig. VI but for the vector mesons.

m2π (GeV)2 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ 0.795(5) 0.784(7) 0.769(9) 0.750(12) 0.727(18) 0.708(22) 0.683(32) 0.660(40)

uK∗ 0.794(7) 0.783(8) 0.769(9) 0.754(11) 0.738(13) 0.727(15) 0.716(17) 0.707(19)

sK∗ 0.771(7) 0.771(8) 0.769(9) 0.766(10) 0.760(12) 0.756(13) 0.749(15) 0.745(15)

TABLE VI: The quark sector contributions to the charge form factor of the pseudoscalar mesons.

m2π (GeV)2 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uπ 0.833(4) 0.828(4) 0.822(5) 0.815(6) 0.810(8) 0.809(12) 0.812(22) 0.833(30)

uK 0.835(4) 0.830(5) 0.822(5) 0.813(6) 0.804(6) 0.798(7) 0.792(8) 0.791(8)

sK 0.818(4) 0.820(5) 0.822(5) 0.823(6) 0.825(6) 0.826(7) 0.826(8) 0.828(8)

Acknowledgments

We thank the Australian Partnership for Advanced Computing (APAC) and the South Australian Partnership for Advanced Computing (SAPAC) for generous grants of supercomputer time which have enabled this project. This work was supported by the Australian Research Council. J.Z. is supported by PPARC grant PP/D000238/1.

[1] J. Arrington, C. D. Roberts, and J. M. Zanotti (2006), nucl-th/0611050.

[2] K. de Jager (2006), nucl-ex/0612026. [3] H.-y. Gao, Int. J. Mod. Phys. E12, 1 (2003), nucl-

ex/0301002. [4] C. E. Hyde-Wright and K. de Jager, Ann. Rev. Nucl.

Part. Sci. 54, 217 (2004), nucl-ex/0507001. [5] C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen

(2006), hep-ph/0612014. [6] R. M. Woloshyn and A. M. Kobos, Phys. Rev. D33, 222

(1986). [7] W. Wilcox and R. M. Woloshyn, Phys. Rev. D32, 3282

(1985). [8] R. M. Woloshyn, Phys. Rev. D34, 605 (1986). [9] W. Wilcox and R. M. Woloshyn, Phys. Rev. Lett. 54,

2653 (1985). [10] T. Draper, R. M. Woloshyn, W. Wilcox, and K.-F. Liu,

Nucl. Phys. B318, 319 (1989). [11] T. Draper, R. M. Woloshyn, W. Wilcox, and K.-F. Liu,

Nucl. Phys. Proc. Suppl. 9, 175 (1989). [12] A. M. Abdel-Rehim and R. Lewis, Phys. Rev. D71,

014503 (2005), hep-lat/0410047. [13] F. D. R. Bonnet, R. G. Edwards, G. T. Fleming,

R. Lewis, and D. G. Richards (Lattice Hadron Physics), Phys. Rev. D72, 054506 (2005), hep-lat/0411028. [14] D. Br¨ommel et al. (QCDSF/UKQCD) (2006), heplat/0608021. [15] S. Capitani, C. Gattringer, and C. B. Lang (Bern-Graz-

TABLE VIII: The quark sector contributions to the magnetic form factor.

m2π (GeV)2 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ 1.360(16) 1.389(19) 1.418(25) 1.455(29) 1.484(37) 1.496(47) 1.500(60) 1.483(81)

uK∗ 1.373(19) 1.395(21) 1.418(25) 1.445(28) 1.469(31) 1.485(36) 1.501(41) 1.513(46)

sK∗ 1.406(21) 1.413(23) 1.418(25) 1.428(27) 1.435(28) 1.438(30) 1.440(31) 1.441(33)

Regensburg (BGR)), Phys. Rev. D73, 034505 (2006), hep-lat/0511040. [16] S. Hashimoto et al. (JLQCD), PoS LAT2005, 336 (2006), hep-lat/0510085. [17] Y. Nemoto (RBC), Nucl. Phys. Proc. Suppl. 129, 299 (2004), hep-lat/0309173. [18] J. van der Heide, J. H. Koch, and E. Laermann, Phys. Rev. D69, 094511 (2004), hep-lat/0312023. [19] T. Horn et al. (Fpi2), Phys. Rev. Lett. 97, 192001 (2006), nucl-ex/0607005. [20] V. Tadevosyan et al. (Fpi-1) (2006), nucl-ex/0607007.

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Pseudoscalar and vector meson form factors from lattice QCD

J.N. Hedditch,1 W. Kamleh,1 B.G. Lasscock,1 D.B. Leinweber,1 A.G. Williams,1 and J.M. Zanotti2 1Department of Physics and Mathematical Physics and

Special Research Centre for the Subatomic Structure of Matter, University of Adelaide, 5005, Australia

2School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK

We present a study of the pseudoscalar and vector meson form factors, calculated using the FatLink Irrelevant Clover (FLIC) action in the framework of Quenched Lattice QCD. Of particular interest is the determination of a negative quadrupole moment, indicating that the ρ meson is not spherically symmetric.

PACS numbers: 12.38.Gc,14.40.Aq Keywords: Mesons, form factors

arXiv:hep-lat/0703014v1 14 Mar 2007

I. INTRODUCTION

The important role that electromagnetic form factors play in our understanding of hadronic structure has been well documented for more than ﬁfty years. The reason for their popularity is that they encode information about the shape of hadrons, and provide valuable insights into their internal structure in terms of quark and gluon degrees of freedom.

Most of the attention, both experimentally and theoretically, has focused on the electromagnetic form factors of the nucleon (see Refs. [1, 2, 3, 4, 5] for recent reviews). The electromagnetic form factors of pseudoscalar mesons, especially the pion, being the lightest QCD bound state, have also been studied extensively [6, 7, 8, 9, 10, 11] in lattice QCD. More recently, there is a renewed interest in calculating the pion form factor on the lattice [12, 13, 14, 15, 16, 17, 18]. This is especially timely considering the new [19] and reanalysis of old [20] experimental data from JLab.

The vector meson form factors, on the other hand, have received less attention (see Refs. [21, 22, 23, 24, 25] for recent work). Of particular interest is the quadrupole moment of the ρ meson, where theoretical determinations can disagree by as much as a factor of two [22]. We aim to resolve this issue by performing the ﬁrst direct lattice calculation of the ρ-meson quadrupole form factor. Charge and magnetic form factors are also calculated and from these we extract the relevant static quantities, namely the mean square charge-radius and magnetic moment. We also analyse the dependence of light-quark contributions to these form factors on their environment and contrast these with a new calculation of the corresponding pseudoscalar-sector result.

Our aim is to reveal the electromagnetic structure of vector mesons and to study to what extent the qualitative quark model picture is consistent with quenched lattice QCD. Interestingly, it has been shown in a lattice calculation by Alexandrou et al. [26] that the distribution of charge in the vector meson is oblate, and therefore not consistent with the picture of a quark anti-quark in relative S-wave. By calculating the vector meson quadrupole

form factor we make a direct comparison with the ﬁndings of Ref. [26].

For each observable we calculate the quark sector contributions separately. Using this additional information we examine the environmental sensitivity of the lightquark contributions to the pseudoscalar and vector meson charge radii. We also evaluate the dominance of the light quark contributions to the K and K∗.

This paper builds on the preliminary work presented in Ref. [27]. In Section II A we introduce the theoretical formalism of meson form factors, including the techniques required to extract them from a lattice calculation. Section 3 contains details of our lattice simulation, while in Section 4 we present and discuss our results for both pseudoscalar and vector mesons. Finally, in Section 5 we summarise our ﬁndings and discuss future work.

II. THEORETICAL FORMALISM

A. Meson form factors

Meson form factors are extracted from matrix elements involving the vector (electromagnetic) current

M (p ′)|Jα|M (p) ,

(1)

where M (p) (M (p ′)) denotes a meson state with initial (ﬁnal) momentum p (p ′). The momentum transfer is qµ = (p′µ − pµ).

For a pion, the matrix element in Eq. (1) is described

by a single form factor

π(p ′)|Jα|π(p) =

1

[pα + pα′]Fπ(Q2) ,(2)

2 Eπ(p)Eπ(p ′)

where Q2 = −q2 is the invariant momentum transfer, and the energy of the pion with momentum p is Eπ(p) = m2π + p 2. The formula for the kaons are exactly analogous. The ρ-meson, on the other hand, is spin-1 and is described by three form factors [28],

ρ(p ′, s′)|J α|ρ(p, s) =

2

1

ǫ′τ⋆(p′, s′) J τασ(p′, p) ǫσ(p, s)(3)

2 Eρ(p)Eρ(p ′)

where ǫ and ǫ′ are the initial and ﬁnal polarisation vec-

tors, respectively, and

J τ ασ(p′, p) = − G1(Q2) gτ σ [pα + pα′] + G2(Q2)[gασqτ − gατ qσ] − G3(Q2) qσqτ pα + pα′ .

(4)

2m2ρ

The covariant vertex functions G1,2,3 can be rewritten in terms of the Sachs charge, magnetic and quadrupole

form factors [28, 29],

GQ(Q2) = G1(Q2) − G2(Q2) + (1 + η)G3(Q2) (5)

GM (Q2) = G2(Q2)

(6)

GC (Q2) = G1(Q2) + 2 η GQ(Q2) ,

(7)

3

where mρ is the mass of the vector-meson system calculated on the lattice and η = Q2/4m2ρ.

The charge qρ, magnetic moment µρ, and quadrupole moment Qρ are then extracted from GC , GM , and GQ, respectively at zero momentum transfer

e GC(0) = qρ

(8)

e GM (0) = 2mρµρ

(9)

e GQ(0) = m2ρQρ .

(10)

The formulae for the K∗ are exactly analogous.

B. Meson form factors on the lattice

The matrix elements in Eqs. (2) and (3) are obtained from ratios of three-point and two-point correlation func-

tions

Rα(p′, p) =

Gα(p ′, p, t2, t1) Gα(p, p ′, t2, t1) (11)

for pseudoscalar mesons, and

Rµαν (p′, p) =

Gαµν (p ′, p, t2, t1) Gανµ(p, p ′, t2, t1)

(12)

for vector mesons. Note repeated indices are not summed

over.

The two-point correlation function for the pseudoscalar mesons is

G(t2, p) = e−ip·x2 Ω|χ(x2)χ†(0)|Ω . (13)

x2

Similarly for the vector mesons,

Gµν (t2, p) = e−ip·x2 Ω|χµ(x2)χ†ν (0)|Ω . (14)

x2

The three-point correlation function for the pseudoscalar meson is

Gα(t2, t1, p ′, p) =

e−ip ′·(x2−x1)e−ip·x1 Ω| χ(x2)J α(x1)χ†(0) |Ω .

x1,x2

(15)

Similarly the three-point function for the vector meson is

Gαµν (t2, t1, p ′, p) =

e−ip ′·(x2−x1)e−ip·x1 Ω| χµ(x2)J α(x1)χ†ν (0) |Ω .

x1,x2

(16)

The Lorentz indices µ and ν are only present for the vector mesons, while α is the index of the electromagnetic current.

The ratios in Eqs.(11) and (12) are constructed in such a way as to remove the time-dependence and constants

of normalisation from the correlation functions at large time separations, t1 and t2 − t1.

These ratios diﬀer subtly from previous work [30], in that we are explicitly enforcing the parity of the terms through the choice of momenta (p′, p) and (p, p′) vs (p′, p)

and (−p, −p′). This requires two three-point propagators (with momentum-transfer q and −q) for each conﬁguration. However with the well established technique of averaging over U and U ∗ conﬁgurations [30, 31], there is no additional cost.

1. π-meson case

Since the pion has zero spin, the vertex is extraordinarily simple and takes the form given in Eq. (2). Here we show how this function is extracted from the ratio Eq. (11) by evaluating the correlation functions at large Euclidean times.

3 First we deﬁne the matrix elements as,

Ω|χµ(0)|π(p) =

1 λπ(p) ,

2Eπ (p)

π(p)|χ†µ(0)|Ω =

1 λ¯π(p) .

(17)

2Eπ (p)

Here λπ(p) and λ¯π(p) are the couplings of the interpolator to the pion with momentum p at the sink and source respectively. The momentum dependence allows for the use of smeared fermion sources and sinks. The bar allows for diﬀerent amounts of smearing at the source and sink.

By inserting a complete set of energy and momentum states into Eq. (13), we can show that at large Euclidean time,

lim G(t2, p) = e−Eπ(p)t2 λπ(p)λ¯π(p) .

t2→∞

2Eπ (p)

(18)

Following the same treatment, one can show that the three-point function Eq. (15) at large Euclidean time is

lim Gα(t2, t1, p ′, p) =

t1 ,t2 −t1 →∞

e−Eπ(p ′)(t2−t1)e−Eπ(p)t1 λπ (p ′) π(p ′)|J α(0)|π(p) λ¯π(p) . 2 Eπ(p)Eπ(p ′)

(19)

Substituting these expressions into Eq. (11) and using Eq. (2), the ratio Rα(p′, p) simply reduces to

Rα(p′, p) =

1

[pα + pα′]Fπ(Q2) , (20)

2 Eπ(p)Eπ(p ′)

such that the large Euclidean time limits of the ratio Rα is a direct measure of Fπ(Q2) up to kinematical factors.

2. ρ-meson case

Following [28], we deﬁne the matrix element of the electromagnetic current for ρ-meson in terms of the covariant vertex functions G1,2,3 as in Eqs. (3) and (4).

The analogues of the matrix elements in Eq. (17) are

Ω| χµ(0) |ρ(p, s) = ρ(p, s)| χν†(0) |Ω =

1 λρ(p)ǫµ(p, s) 2Eρ(p)

1 λ¯ρ(p) ǫ⋆ν (p, s) . (21) 2Eρ(p)

The polarisation vectors obey the transversality condition

ǫµ(p, s) ǫ⋆ν(p, s) = − gµν − pµpν , (22)

s

m2ρ

because the vector meson current is a conserved current.

The evaluation of the two- and three-point functions proceeds as for our discussion of the pion. However the completeness relation includes a sum over spin-states. Using the transversality condition Eq. (22) the analogue of Eq. (18) becomes

lim G(t2, p) =

e−Eρ(p)t2 λρ(p)λ¯ρ(p)ǫµ(p, s) ǫ⋆(p, s)

t2→∞ s 2Eρ(p) ν

= − e−Eρ(p)t2 λρ(p)λ¯ρ(p) gµν − pµpν .

2Eρ(p)

m2ρ

(23)

Similarly using Eq. (3) we can evaluate the three-point function,

lim Gαµν (t2, t1, p ′, p) =

t1 ,t2 −t1 →∞

e−Eρ(p ′)(t2−t1)e−Eρ(p)t1 λρ(p ′)ǫµ(p′, s′) ǫ′⋆(p′, s′)J τ ασ(p′, p)ǫσ(p, s) λ¯ρ(p) ǫ⋆ (p, s) s,s′ 4Eρ(p)Eρ(p ′) τ ν

= e−Eρ(p ′)(t2−t1)e−Eρ(p)t1 λρ(p ′)λ¯ρ(p) gµτ − p′µp′τ J τ ασ gσν − pσpν .

4Eρ(p)Eρ(p ′)

m2ρ

m2ρ

4 (24)

Inserting the above expressions into the ratio in Eq. (12), together with our choice of momentum used in the simulations, namely p′ = (Eρ, px, 0, 0) (Eρ =

m2ρ + p2x) and p = (mρ, 0, 0, 0), it is possible to express Rµαν in terms of the Sachs form factors,

R0 =

p2x

GQ(Q2) + Eρ + mρ GC (Q2) ,

11

3mρ Eρmρ

2 Eρmρ

R0 = R0 = −

p2x

GQ(Q2) + Eρ + mρ GC (Q2) ,

22

33

6mρ Eρmρ

2 Eρmρ

R133 = R331 = 2

px GM (Q2) . Eρmρ

The individual form factors are isolated as follows:

GC (Q2) = 23 EρE+ρmmρρ R101 + R202 + R303 , (25)

GM (Q2) =

Eρmρ R3 + R3 ,

px

13

31

(26)

GQ(Q2) = mρ Eρmρ 2R0 − R0 − R0 . (27)

p2x

11

22

33

While we have used the subscript ρ to denote a vector

meson, the results are applicable to vector mesons in general, including the K∗ for example.

C. Extracting static quantities

The mean squared charge radius r2 is obtained from the charge form-factor through the following relation,

r2 = −6 ∂ G(Q2)

.

(28)

∂Q2

Q2 =0

To calculate the derivative the monopole form is used,

GC (Q2) =

1

.

Q2 Λ2

+1

(29)

Λ is referred to as the monopole mass. Inserting this form into Eq. (28) and rearranging provides

r2 = 6

1 −1 ,

(30)

Q2 GC (Q2)

valid for quantities with GC (Q2 = 0) = 1. As mentioned in Sec. II A, the charge (Eq. (8)),

magnetic moment (Eq. (9)), and quadrupole moment (Eq. (10)) can be extracted from the Sachs form factors

at zero momentum transfer. Since we perform our calculations at a single, ﬁnite value of Q2, we will need to

adjust our results to zero momentum transfer.

From studies of nucleon properties, it is observed that GM and GC have similar Q2-scaling at small Q2 [32]. In the following, we shall assume that this scaling also holds

for quark contributions to mesons. If GC (0) = 1, we have

GM (Q2)

GM (0) ≃ GC (Q2) .

(31)

Whilst a similar scaling could be used to relate our quadrupole form-factor to the quadrupole moment, we believe that the form-factor at our small ﬁnite Q2 ( ≃ 0.22GeV) will be of greater phenomenological interest.

We note that for a positively charged meson a negative value of GQ corresponds to an oblate deformation.

III. METHOD

The electromagnetic form factors are obtained using the three-point function techniques established by Leinweber, et al. in Refs. [30, 33, 34] and updated for smeared sources in Ref. [31]. Our quenched gauge ﬁelds are generated with the O(a2) mean-ﬁeld improved Luscher-Weisz plaquette plus rectangle gauge action [35] using the plaquette measure for the mean link. We use an ensemble of 379 quenched gauge ﬁeld conﬁgurations on 203 × 40 lattices with lattice spacing a = 0.128 fm. The gauge ﬁeld conﬁgurations are generated via the Cabibbo-Marinari pseudo-heat-bath algorithm [36] using a parallel algorithm with appropriate link partitioning [37].

We use the fat-link irrelevant clover (FLIC) Dirac operator [38] which provides a new form of nonperturbative O(a) improvement [39]. The improved chiral properties of FLIC fermions allow eﬃcient access to the light quark-mass regime [40], making them ideal for dynamical fermion simulations now underway [41]. For the vector current, we an O(a)-improved FLIC conserved vector current [31]. We use a smeared source at t2 = 8. Complete simulation details are described in Ref. [31].

Table I provides the κ-values used in our simulations, together with the calculated pseudoscalar and vector meson masses. While we refer to m2π in our ﬁgures and tables to infer the quark masses, we note that the critical value where the pion mass vanishes is κcr = 0.13135. Importantly the vector mesons remain bound at all quark masses considered in this calculation due to ﬁnite volume eﬀects. That is, the mass of the vector mesons is less than the energy of the lowest lying multi-hadron state with the appropriate quantum numbers.

The strange quark mass is chosen to be the third heaviest quark mass. This provides a pseudoscalar mass of 697

MeV which compares well with the experimental value of (2MK2 − Mπ2)1/2 = 693 MeV motivated by chiral perturbation theory. Two vector-meson interpolating ﬁelds are

considered, namely q¯γiq and q¯γiγ4q. Since results for the two interpolators agree, we simply present the results

for the q¯γiq interpolator, which displays a signiﬁcantly stronger signal.

The error analysis of the correlation function ratios

is performed via a second-order, single-elimination jackknife, with the χ2 per degree of freedom (χ2dof ) obtained via covariance matrix ﬁts. We perform a series of ﬁts

through the ratios after the current insertion at t = 14. By examining the χ2dof we are able to establish a valid window through which we may ﬁt in order to extract our observables. In all cases, we required a value of χ2dof no larger than 1.5. The values of the static quantities quoted

in this paper on a per quark-sector basis correspond to

values for single quarks of unit charge.

TABLE I: Meson masses for the respective values of the hopping parameter κ.

κ

0.12780 0.12830 0.12885 0.12940 0.12990 0.13205 0.13060 0.13080

amπ

0.5411(10) 0.5013(11) 0.4539(11) 0.4014(12) 0.3471(15) 0.3020(19) 0.2412(42) 0.1968(52)

amK

0.4993(11) 0.4782(11) 0.4539(11) 0.4285(11) 0.4044(12) 0.3862(13) 0.3671(19) 0.3574(16)

amρ

0.7312(30) 0.7067(36) 0.6797(46) 0.6537(49) 0.6309(56) 0.6160(64) 0.6039(71) 0.5982(80)

amK∗

0.7057(27) 0.6933(40) 0.6796(46) 0.6668(47) 0.6556(50) 0.6484(52) 0.6423(54) 0.6393(56)

5

obtained in the previous study reﬂects ﬁnite-volume effects attendant with the use of a small spatial volume.

By comparing the results for the up-quark contributions to the π and K (ρ and K∗) charge radii, it is possible to gain insights into the eﬀect that the presence of a heavier strange-quark has on the lighter up-quark in pseudoscalar (vector) mesons. Figures 2 and 3 show the quark sector contributions to the charge radii ( r2 ) of the pseudoscalar and vector mesons, respectively. The quark sector contributions to the charge radii for the pseudoscalar and vector meson are recorded in Tables II and III. From Fig. 2, we ﬁnd no evidence of environmental sensitivity in the light-quark contribution the pseudoscalar mesons. However in the vector sector, Fig. 3, we ﬁnd a consistently broader distribution of up-quark charge in the ρ compared to the up-quark in the K∗ at the smaller quark masses. The broadening of the charge distribution in the ρ is consistent with the hyperﬁne repulsion discussed above. The strange quark in the K∗ shows a particularly interesting environment sensitivity. While the strange quark mass is held ﬁxed, the distribution broadens as the light-quark regime is approached. This is consistent with the prediction of enhanced hyperﬁne repulsion as one of the quarks becomes light.

The strange neutral pseudoscalar and vector meson mean squared charge radii obtained from the weighted sum of the quark sector radii are displayed in Fig. 4. For the neutral strange mesons, we see a negative value for r2 , indicating that the negatively charged d-quark is lying further from the centre of mass on average than the s¯. We should expect just such a behaviour for two reasons, both stemming from the fact that the s¯ quark is considerably heavier than the d: the centre of mass must lie closer to the s¯, and the d-quark will also have a larger

IV. RESULTS

1. Charge radii

We begin the discussion of our results with the charge radii of the vector and pseudoscalar mesons. From the quark model we would expect a hyperﬁne interaction between the quark and anti-quark of the form mσqq·mσq¯q¯ . The interaction is repulsive where the spins are aligned, as in the vector mesons, and attractive where the spins are anti-aligned, as in the pseudoscalar mesons. In Fig. 1 we show the charge radii of the vector and pseudoscalar mesons. For comparison the charge radius of the proton is also shown. Indeed we ﬁnd that the charge radii of the vector mesons are consistently larger than the pseudoscalar mesons, and in fact similar to the charge radii of the proton, even at heavier quark masses. This is contrary to earlier lattice simulations with relatively small spatial extent [42], that have suggested that the π+, ρ+ and proton should have a very similar RMS charge radius at larger quark masses. It is possible that the agreement

FIG. 1: Strange and non-strange meson mean squared charge

radii for charged pseudoscalar and vector mesons. We also

include for comparison results for the proton taken from

Ref. [31]. The π and ρ-meson results are centred on the relevant value of m2π, other symbols are oﬀset horizontally for clarity.

6

TABLE II: Mean-square charge radius ( r2 ) for quarks of unit charge in units of fm2. m2π is given as a measure of the

input quark mass.

m2π (GeV2) 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uπ 0.216(5) 0.225(6) 0.240(8) 0.256(10) 0.274(14) 0.287(22) 0.304(44) 0.287(63)

uK 0.215(7) 0.224(7) 0.240(8) 0.257(9) 0.275(11) 0.289(12) 0.303(14) 0.306(15)

sK 0.242(7) 0.241(7) 0.240(8) 0.239(9) 0.239(10) 0.241(11) 0.243(13) 0.241(13)

TABLE III: Mean-square charge radius ( r2 ) for quarks of unit charge in units of fm2.

m2π (GeV2) 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ 0.268(9) 0.287(11) 0.315(16) 0.350(23) 0.397(36) 0.436(46) 0.492(72) 0.546(97)

uK∗ 0.271(11) 0.290(13) 0.315(16) 0.342(20) 0.372(26) 0.395(30) 0.417(35) 0.436(41)

sK∗ 0.309(12) 0.311(14) 0.315(16) 0.321(19) 0.331(23) 0.339(25) 0.353(28) 0.360(29)

FIG. 2: The quark sector contributions to the mean squared charge radius of the pseudoscalar mesons. The symbols are oﬀset horizontally for clarity.

Compton wavelength. Of course with exact isospin symmetry in our simulations, the non-strange charge neutral mesons have a zero electric charge radius.

To measure the environmental sensitivity of the lightquark sector more precisely, in Figs. 5 and 6 we show a ﬁt to the ratio of the light-quark contributions to the pseudoscalar and vector-mesons charge radii respectively. The diﬀerence is striking: for the pseudoscalar case we see no environment-dependence at all, whereas in the vector case we see that the presence of a strange quark acts to heavily suppress the light charge distribution. This is the eﬀect one predicts from a quark model, where the large mass of the s would act to suppress the hyperﬁne repulsion between the quark and anti-quark. It is also qualitatively consistent with eﬀective ﬁeld theory where the couplings of the light mesons are suppressed by the presence of the strange quark.

2. Magnetic moments

In Fig. 7 we present our results for the magnetic moments of the vector mesons. At the SU (3)ﬂavour limit, where we take the light quark ﬂavours to have the same mass as the strange quark, quark model arguments suggest the magnetic moment for a ρ+ should be -3 times the strange magnetic moment of the Λ (assuming no en-

FIG. 3: As for Figure 2 but for vector-mesons.

vironmental dependence). According to the particle data group [43], the magnetic moment of the Λ is −0.613 µN. Therefore we would naively expect a value of 1.84 µN for the magnetic moment of the ρ+, which is consistent with our ﬁndings.

In an earlier study, Anderson et al. [44] argued that the magnetic moment of the ρ-meson in natural magnetons (otherwise called the g-factor) should be approximately 2 at large quark masses. Converting our result to natural magnetons, we observe in Fig. 8 that our calculation of the ρ-meson g-factor (gρ) is fairly consistent with this. At light quark masses, however, we do see some evidence of chiral curvature, which would indicate that the linear chiral extrapolations of that paper should be considered with caution.

In Fig. 9 we present the quark sector contributions to the vector meson magnetic moments, the data is recorded in Table. IV. Here we observe a similar scenario to that observed earlier in the charge radius discussion, namely

7

FIG. 4: The mean squared charge radii for the neutral K0 and K0∗.

FIG. 6: As in Fig. 5 but for the vector-mesons.

FIG. 5: The ratio of the light quark contributions to the π and K mean squared charge radius.

that the u-quark contribution to the K∗ is consistently larger than the contribution from the heavier s-quark. We also ﬁnd that the contribution of the u-quark to the magnetic moment of a vector meson is suppressed when it is an environment of a heavier s-quark compared to when it is in the presence of another light quark. This is further supported when we consider the ratio of the contributions of a u-quark to the magnetic moments of the ρ and K∗ mesons, displayed in Fig. 10. This ratio is clearly greater than 1 below the SU (3)ﬂavour limit and is increasing for decreasing u-quark mass.

The magnetic moment of the vector meson, like the RMS charge radius, shows considerable environment dependence in the quark sector contributions. The larger contribution of a u-quark in a ρ relative to a K∗ is consistent with what we have already observed with the RMS charge radius, as follows: since r2 is larger for the uquark in a ρ meson than for the u-quark in a K∗, the eﬀective mass is reciprocally smaller for the u-quark in

FIG. 7: Charged vector meson magnetic moments.

a ρ. This smaller eﬀective mass gives rise in turn to a larger magnetic moment. Figure 10 shows this pattern. Figure 11 presents our results for the magnetic moment of the neutral K∗0 meson. As the d-quark becomes lighter than the s¯ we see the magnetic moment exhibiting a very linear negative slope. The magnitude of the magnetic moment is quite small, but clearly diﬀerentiable from zero everywhere except at the SU (3)ﬂavour limit where symmetry forces it to be exactly zero.

3. Quadrupole form-factors

The quadrupole form-factors of the ρ+ and K∗+ mesons are shown in Fig 12. We ﬁnd that the quadrupole form factor is less than zero, indicating that the spatial distribution of charge within the ρ and K∗ mesons is oblate. This is in accord with the ﬁndings of Alexandrou et al. [26] who observed a negative quadrupole moment for spin ±1 ρ-meson states in a density-density analy-

FIG. 8: The g-factor of the ρ meson.

8

TABLE IV: Magnetic moment for quarks of unit charge inside a vector meson in units of nuclear magnetons µN .

m2π (GeV2) 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ 1.71(2) 1.77(2) 1.84(3) 1.94(4) 2.04(6) 2.11(8) 2.20(11) 2.25(15)

uK∗ 1.73(2) 1.78(3) 1.84(3) 1.92(4) 1.99(5) 2.04(5) 2.10(6) 2.14(7)

sK∗ 1.82(3) 1.83(3) 1.84(3) 1.86(3) 1.88(4) 1.90(4) 1.92(5) 1.93(5)

FIG. 11: Neutral K∗-meson magnetic moment.

FIG. 9: Quark-sector contributions to the vector meson magnetic moments.

sis. We note that in a simple quark model, a negative quadrupole form factor requires that the quarks possess an admixture of s- and d-wave functions.

TABLE V: The quadrupole form-factor (in fm2) for quarks of unit charge inside a vector meson.

m2π (GeV)2 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ −0.0047(4) −0.0047(4) −0.0048(5) −0.0049(7) −0.0050(9) −0.0049(12) −0.0051(19) −0.0050(27)

uK∗ −0.0047(4) −0.0047(5) −0.0048(5) −0.0049(6) −0.0050(7) −0.0052(9) −0.0055(11) −0.0056(13)

sK∗ −0.0048(5) −0.0048(5) −0.0048(5) −0.0048(6) −0.0048(7) −0.0047(7) −0.0046(9) −0.0046(10)

FIG. 10: The ratio of the light-quark contributions to the magnetic moment of the ρ and K∗.

The quark sector contributions to the quadrupole form-factor are shown in Fig. 13. The corresponding data is contained in Table V. The ﬂavour independence of the results is remarkable.

We also ﬁnd that the ratio of the light-quark contri-

9

FIG. 12: Vector meson quadrupole form factors for ρ+ and K ∗+ .

FIG. 15: Quadrupole form-factor for neutral K∗ meson.

butions to the quadrupole form factor, shown in Fig. 14, is consistent with one within our statistics. In Fig. 15, we show the quadrupole form factor of the charge neutral K∗0 meson. We ﬁnd that the quadrupole moment of the K∗0 is non-trivial but just outside the one standard deviation level. The chiral trend towards positive values reﬂecting the negative charge of the larger d-quark contributions.

The lattice data for the quark sector contributions to the charge form factor is contained in Tables VI and VII for the pseudoscalar and vector mesons respectively. The magnetic and quadrupole form factors of the vector mesons is contained in Tables VIII and IX respectively.

FIG. 13: Quark-sector contributions to the quadrupole form factors.

FIG. 14: Environment-dependence for light-quark contribution to vector meson quadrupole form-factor.

V. CONCLUSIONS

We have established a formalism for determining the charge, magnetic and quadrupole Sachs form factors of vector mesons in lattice QCD. For the ﬁrst time the electric, magnetic, and quadrupole form factors of the light vector mesons have been calculated. The electric form factor of the pseudoscalar mesons have also been calculated.

With a large lattice volume and high statistics we have resolved a clear diﬀerence between the charge radii of the pseudoscalar and vector mesons. We argue that this is consistent with quark model predictions. Furthermore, we ﬁnd signiﬁcant environmental sensitivity of the lightquark contributions to the charge radii of the vector mesons.

We also presented a calculation of the magnetic moments of the vector mesons. We found that the magnetic moment of the ρ+ was consistent with the quark model predication of 1.84 µN at the SU (3)ﬂavour limit. We determine that there is also an environmental sensitivity in the magnitude of the light-quark contributions to the charged vector meson magnetic moments. We argue that

this is consistent with the environmental sensitivity in the light-quark contributions to the charge vector meson charge radii.

Finally, we have determined that the quadrupole form factor for a charged vector meson is negative in quenched Lattice QCD. This is consistent with previous calculations using density-density analysis. We ﬁnd that the ratio of quadrupole moment to mean square charge radius is 1:30, so the deformation is small but statistically signiﬁcant.

VI. APPENDIX

10

TABLE VII: As in Fig. VI but for the vector mesons.

m2π (GeV)2 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ 0.795(5) 0.784(7) 0.769(9) 0.750(12) 0.727(18) 0.708(22) 0.683(32) 0.660(40)

uK∗ 0.794(7) 0.783(8) 0.769(9) 0.754(11) 0.738(13) 0.727(15) 0.716(17) 0.707(19)

sK∗ 0.771(7) 0.771(8) 0.769(9) 0.766(10) 0.760(12) 0.756(13) 0.749(15) 0.745(15)

TABLE VI: The quark sector contributions to the charge form factor of the pseudoscalar mesons.

m2π (GeV)2 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uπ 0.833(4) 0.828(4) 0.822(5) 0.815(6) 0.810(8) 0.809(12) 0.812(22) 0.833(30)

uK 0.835(4) 0.830(5) 0.822(5) 0.813(6) 0.804(6) 0.798(7) 0.792(8) 0.791(8)

sK 0.818(4) 0.820(5) 0.822(5) 0.823(6) 0.825(6) 0.826(7) 0.826(8) 0.828(8)

Acknowledgments

We thank the Australian Partnership for Advanced Computing (APAC) and the South Australian Partnership for Advanced Computing (SAPAC) for generous grants of supercomputer time which have enabled this project. This work was supported by the Australian Research Council. J.Z. is supported by PPARC grant PP/D000238/1.

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m2π (GeV)2 0.6956(26) 0.5970(26) 0.4895(24) 0.3828(23) 0.2862(25) 0.2166(27) 0.1382(48) 0.0920(49)

uρ 1.360(16) 1.389(19) 1.418(25) 1.455(29) 1.484(37) 1.496(47) 1.500(60) 1.483(81)

uK∗ 1.373(19) 1.395(21) 1.418(25) 1.445(28) 1.469(31) 1.485(36) 1.501(41) 1.513(46)

sK∗ 1.406(21) 1.413(23) 1.418(25) 1.428(27) 1.435(28) 1.438(30) 1.440(31) 1.441(33)

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