delta-baryon electromagnetic form factors in lattice QCD
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Alexandrou, C. et al. "delta-baryon electromagnetic form factors in
lattice QCD." Physical Review D 79.1 (2009): 014507.
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American Physical Society
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-baryon electromagnetic form factors in lattice QCD
C. Alexandrou,1T. Korzec,1G. Koutsou,1Th. Leontiou,1C. Lorce´,2J. W. Negele,3V. Pascalutsa,2 A. Tsapalis,4and M. Vanderhaeghen2
1Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus 2Institut fu¨r Kernphysik, Johannes Gutenberg-Universita¨t, D-55099 Mainz, Germany
3Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
4Institute of Accelerating Systems and Applications, University of Athens P.O. Box 17214, GR-10024, Athens, Greece
(Received 22 October 2008; published 21 January 2009)
We develop techniques to calculate the four electromagnetic form factors using lattice QCD, with particular emphasis on the subdominant electric quadrupole form factor that probes deformation of the. Results are presented for pion masses down to approximately 350 MeV for three cases: quenched QCD, two flavors of dynamical Wilson quarks, and three flavors of quarks described by a mixed action combining domain-wall valence quarks and dynamical staggered sea quarks. The magnetic moment of the is compared with chiral effective field theory calculations and the charge density distributions are discussed.
DOI:10.1103/PhysRevD.79.014507 PACS numbers: 11.15.Ha, 12.38.Aw, 12.38.Gc
I. INTRODUCTION
Lattice quantum chromodynamics (QCD) provides a well-defined framework to directly calculate hadron form factors from the fundamental theory of strong interactions. Form factors characterize the internal structure of hadrons, including their magnetic moment, their size, and their charge density distribution. Since the ð1232Þ decays strongly, experiments [1,2] to measure its form factors are harder and yield less precise results than for nucleons [3,4]. In this work, we compute form factors using lattice QCD more accurately than can be currently obtained from experiment.
A primary motivation for this work is to understand the role of deformation in baryon structure: whether any of the low-lying baryons have deformed intrinsic states and if so, why. Thus, a major achievement of this work is the devel-opment of lattice methods with sufficient precision to show, for the first time, that the electric quadrupole form factor is nonzero and hence the has a nonvanishing quadrupole moment and an associated deformed shape. Unlike the, the spin-1=2 nucleon cannot have a quadru-pole moment, so the experiment of choice to explore its deformation has been measurement of the nucleon to electric and Coulomb quadrupole transition form factors. Major experiments [5–7] have shown that these transition form factors are indeed nonzero, confirming the presence of deformation in either the nucleon, or both [8,9], and lattice QCD yields comparable nonzero results [10,11]. Our new calculation of the quadrupole form factor, coupled with the nucleon to transition form factors, should in turn shed light on the deformation of the nucleon. In order to evaluate the electromagnetic (EM) form factors to the required accuracy, we isolate the two domi-nant form factors and the subdomidomi-nant electric quadrupole form factor. This is particularly crucial for the latter since
without the construction of an optimized source for the sequential propagator it can not be extracted with the required precision. We note that this increases the compu-tational cost since additional inversions are needed. Our techniques are first tested in quenched QCD [12]. We then calculate form factors using two degenerate flavors of dynamical Wilson fermions, denoted by NF¼ 2, with
pion masses in the range of 700 MeV to 380 MeV [13,14]. Finally, we use a mixed action with chirally sym-metric domain-wall valence quarks and staggered sea quarks with two degenerate light flavors and one strange flavor [15], denoted by NF ¼ 2 þ 1, at a pion mass of 353 MeV. Using the results obtained with dynamical quarks, we extrapolate the magnetic moment to the physi-cal point. We extract the quark charge distributions in the , and discuss their quadrupole moment.
II. LATTICE EVALUATION
The matrix element of the EM current jEM, hðpf; sfÞjj
EMjðpi; siÞi, can be parametrized in terms of
four multipole form factors that depend only on the mo-mentum transfer q2 Q2 ¼ ðpf piÞ2 [16] The
de-composition for the on shell matrix element is given by hðpf; sfÞjj EMjðpi; siÞi ¼ A uðpf; sfÞOuðpi; siÞ O ¼ g a1ðq2Þþa2ðq2Þ 2m ðp f þ p iÞ qq 4m2 c1ðq2Þþc2ðq2Þ 2m ðp f þ p i Þ ; (1)
where a1ðq2Þ, a2ðq2Þ, c1ðq2Þ, and c2ðq2Þ are known linear combinations of the electric charge form factor GE0ðq2Þ, the magnetic dipole form factor GM1ðq2Þ, the electric
quadrupole form factor GE2ðq2Þ, and the magnetic octu-pole form factor GM3ðq2Þ [17], and A is a known factor depending on the normalization of hadron states. These form factors can be extracted from correlation functions calculated in lattice QCD [17]. We calculate in Euclidean time the two- and three-point correlation functions in a frame where the final state is at rest:
Gðt; ~qÞ ¼X ~ xf X3 j¼1 ei ~xf ~q4 hJjðxfÞ Jjð0Þi G ð; t; ~qÞ ¼ X ~ xfx~ ei ~x ~q hJðxfÞjðxÞ Jð0Þi; (2)
where j is the electromagnetic current on the lattice, J
and J are the þ interpolating fields constructed from smeared quarks [12], 4 ¼14ð1 þ 4Þ, and k ¼ i45k. The form factors can then be extracted from
ratios of three- and two-point functions in which unknown normalization constants and the leading time dependence cancel R¼ G ð; t; ~qÞ Gðtf; ~0Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gðtf t; ~piÞGðt; ~0ÞGðtf; ~0Þ Gðtf t; ~0ÞGðt; ~piÞGðtf; ~piÞ v u u u t : (3) For sufficiently large tf t and t ti, this ratio exhibits a plateau Rð; t; ~qÞ ! ð; ~qÞ, from which the form factors are extracted, and we use the particular combinations
X3 k¼1 kkð4; ~qÞ ¼ K1GE0ðQ2Þ þ K2GE2ðQ2Þ; (4) X3 j;k;l¼1 jkljkð4; ~qÞ ¼ K3GM1ðQ2Þ; (5) X3 j;k;l¼1 jklj4kðj; ~qÞ ¼ K 4GE2ðQ2Þ: (6)
The connected part of each combination of three-point functions can be calculated efficiently using the method of sequential inversions [18]. These yield the isovector form factors. At present, it is not yet computationally feasible to calculate the contributions arising from discon-nected diagrams. A calculation of discondiscon-nected contribu-tions in the case of the electromagnetic form factors have shown that these are consistent with zero [19]. We note that these disconnected contributions are particularly hard to calculate not just because they require the all-to-all propa-gators but also because they are noise dominated [20]. We expect that, like in the case of nucleon electromagnetic form factors, the disconnected contributions to the electro-magnetic form factors are small. The known kinematical coefficients K1, K2, K3, K4are functions of the mass and energy as well as of and ~q. The combinations above are chosen such that all possible directions of and ~q con-tribute symmetrically to the form factors at a given Q2 [21]. The over-constrained system of Eqs. (4)–(6) is solved by a least-squares analysis, and GE2ðQ2Þ can also be iso-lated separately from Eq. (6).
The details of the simulations are summarized in TableI. In each case, the separation between the final and initial time is tf ti* 1 fm and Gaussian smearing is applied to
both source and sink to produce adequate plateaus by suppressing contamination from higher states having the quantum numbers of the ð1232Þ. For the mixed-action calculation, the domain-wall valence quark mass was chosen to reproduce the lightest pion mass obtained using NF¼ 2 þ 1 improved staggered quarks [21,23].
III. RESULTS
The results for GE0ðQ2Þ are shown in Fig.1as a function of Q2at the lightest pion mass for each of the three actions. For Wilson fermions, we use the conserved lattice current requiring no renormalization. The local current is used for the mixed action, and the renormalization constant ZV ¼
1:0992ð32Þ is determined by the condition that GE0ð0Þ
equals the charge of the in units of e. As can be seen,
TABLE I. Lattice parameters and results. Nconf denotes the number of lattice configurations,pffiffiffiffiffiffiffiffihr2igives the charge radius, þ is theþ magnetic moment in nuclear magnetons and Q3=2is theþ quadrupole moment.
Nconf m [GeV] m [GeV]
ffiffiffiffiffiffiffiffi hr2i p [fm] þ [N] Q3=2 Quenched Wilson,323 64, a ¼ 0:092 fm 200 0.563(4) 1.470(15) 0.6147(66) 1.720(42) 0.96(12) 200 0.490(4) 1.425(16) 0.6329(76) 1.763(51) 0.91(15) 200 0.411(4) 1.382(19) 0.6516(87) 1.811(69) 0.83(21)
NF¼ 2 Wilson, 243 40 (32 for lightest pion), a ¼ 0:077 fm
185 0.691(8) 1.687(15) 0.5279(61) 1.462(45) 0.80(21)
157 0.509(8) 1.559(19) 0.594(10) 1.642(81) 0.41(45)
200 0.384(8) 1.395(18) 0.611(17) 1.58(11) 0.46(35)
NF¼ 2 þ 1, Mixed action, 283 64, a ¼ 0:124 fm [22]
300 0.353(2) 1.533(27) 0.641(22) 1.91(16) 0.74(68)
C. ALEXANDROU et al. PHYSICAL REVIEW D 79, 014507 (2009)
all three calculations yield consistent results. The momen-tum dependence of the charge form factor is described well by a dipole form GE0ðQ2Þ ¼ 1=ð1 þQ22
E0Þ
2. To compare the
slopes at Q2 ¼ 0, we follow convention and show in Table I the so-called ‘‘rms radius’’ hr2i ¼ 6 d
dQ2GE0ðQ 2Þj
Q2¼0[17].
The momentum dependence of GM1ðQ2Þ is displayed in Fig.2. In order to extract the magnetic moment an extrapo-lation to zero momentum transfer is necessary. Both an exponential form, GM1eQ2=2M1, and a dipole describe the
Q2 dependence well, and we adopt the exponential form because of its faster decay at large Q2, in accord with perturbative arguments. The larger spatial volume for the quenched and mixed-action cases yields smaller and more densely spaced values of the lattice momenta and corre-spondingly more precise determination of the form factor than for the smaller volume used with dynamical Wilson fermions. In Fig. 2, we show the best exponential fit and error band for the mixed action and quenched results. As can be seen, results in the quenched theory and for NF ¼ 2 Wilson fermions are within the error band. The magnetic moment in natural units is given by ¼ GM1ð0Þe=ð2mÞ, where m is the mass measured on the lattice and GM1ð0Þ is from the exponential fits. In Table I, we give the values of the þ magnetic moment in nuclear magnetons e=ð2mNÞ, with mN the physical nucleon mass. The magnetic moments of the þ and þþ are accessible to experiments [1,2], which presently
suffer from large uncertainties.
The magnetic moment as function of m2 is shown in Fig.3, together with a comparison to a chiral effective field theory (ChEFT) result [24]. The ChEFT result has one free parameter (a low-energy constant) that has been fitted to lattice data, shown by the central line. We also estimate the uncertainty of the ChEFT expansion (expansion in pion mass) by the error band in Fig. 3. The uncertainty of the ChEFT calculation vanishes in the chiral limit because in this limit one simply has the value of the low-energy constant, which lies within the broad experimental error band þ¼ 2:7þ1:01:3ðstatÞ 1:5ðsystÞ 3:0ðtheoryÞN
[1]. In this work, we do not consider the uncertainty in the fit value of the low-energy constant due to the lattice errors, as the calculations are still performed for pion masses where the is stable (on the right side of the kink). A calculation for pion mass values where the
0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 Q2 in GeV2 G E0
quenched Wilson, mπ = 410 MeV
hybrid, mπ = 353 MeV
dynamical Wilson, mπ = 384 MeV
FIG. 1 (color online). The electric charge form factor versus Q2. The green (red) line and error band show a dipole fit to the mixed-action (quenched) results.
FIG. 3 (color online). The magnetic dipole moment in nuclear magnetons. The value at the physical pion mass (filled square) is shown with statistical and systematic errors [1]. The solid and dashed curves show the results of ChEFT with the theoretical error estimate [24]. 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Q2 in GeV2 G M1
quenched Wilson, mπ = 410 MeV
hybrid, mπ = 353 MeV
dynamical Wilson, mπ = 384 MeV
FIG. 2 (color online). The magnetic dipole form factor. The green (red) line and error band show an exponential fit to the mixed-action (quenched) results.
becomes unstable will be a challenge for future calcula-tions. The moments using an approach similar to ours are calculated only in the quenched approximation [17,25,26]. Our magnetic moment results agree with recent back-ground field calculations using dynamical improved Wilson fermions [27], which supersede previous quenched background field results [28]. The spatial length Lsof our
lattices satisfies Lsm >4 in all cases except at the lightest
pion mass with NF ¼ 2 Wilson fermions, for which
Lsm ¼ 3:6. For that point, the magnetic moment falls
slightly below the error band, consistent with the fact that Ref. [27] shows that finite volume effects decrease the magnetic moment.
The electric quadrupole form factor is particularly inter-esting because it can be related to the shape of a hadron, and lattice calculations for each of the three actions are shown in Fig.4with exponential fits for the quenched and mixed-action cases. Just as the electric form factor for a spin-1=2 nucleon can be expressed precisely as the trans-verse Fourier transform of the transtrans-verse quark charge density in the infinite momentum frame [29], a proper field-theoretic interpretation of the shape of the ð1232Þ can be obtained by considering the quark transverse charge densities in this frame [30–32]. With respect to the direc-tion of the average baryon momentum P, the transverse charge density in a spin-3=2 state with transverse polariza-tion s?is defined as Ts?ð ~bÞ Z d 2q~ ? ð2 Þ2ei ~q? ~b 1 2Pþ Pþ;q~? 2 ; s? Jþð0Þ Pþ; ~ q? 2 ; s? ; (7) where the photon transverse momentum ~q?satisfies ~q2? ¼
Q2, Jþ J0þ J3, and ~b specifies the quark position in the xy plane relative to the center of mass. Choosing the transverse spin vector along the x axis, the quadrupole moment of this two-dimensional charge distribution is defined as [21]:
Qs? eZ d2bðb~ 2x b2yÞTs?ð ~bÞ: (8) In terms of the EM form factors [21],
Q3=2¼ 1
2f2½GM1ð0Þ 3e þ ½GE2ð0Þ þ 3eg e m2: (9) The term proportional to ½GM1ð0Þ 3e is an electric quadrupole moment induced in the moving frame due to the magnetic dipole moment. For a spin-3=2 particle with-out internal structure, GM1ð0Þ ¼ 3e, GE2ð0Þ ¼ 3e [21,33], and the quadrupole moment of the transverse charge density vanishes. Hence, Qs?, and thus the
defor-mation of the two-dimensional transverse charge density, is only sensitive to the anomalous parts of the spin-3=2 magnetic dipole and electric quadrupole moments, and vanishes for a particle without internal structure. The analogous property holds for a spin-1 particle [32], indi-cating the generality of this description in terms of trans-verse densities.
Figure5shows the transverse density Ts?for aþwith transverse spin s?¼ þ3=2 calculated from the fit to the
quenched Wilson lattice results for the form factors (which has the smallest statistical errors of the three
cal-0 0.5 1 1.5 −4 −3 −2 −1 0 Q2 in GeV2 G E2
quenched Wilson, mπ = 410 MeV
hybrid, mπ = 353 MeV
dynamical Wilson, mπ = 384 MeV
FIG. 4 (color online). The electric quadrupole form factor. The notation is the same as that in Fig.1. The value of GE2, in units of e=ð2m2
Þ, at Q2¼ 0 are 0:810 0:291 for the quenched
calculation, 0:87 0:67 for the dynamical Wilson case, and 2:06þ1:27
2:35for the hybrid calculation.
FIG. 5 (color online). Quark transverse charge density in aþ polarized along the x axis, with s?¼ þ3=2. The light (dark)
regions correspond with the largest (smallest) values of the density.
C. ALEXANDROU et al. PHYSICAL REVIEW D 79, 014507 (2009)
culations). It is seen that theþ quark charge density is elongated along the axis of the spin (prolate). This prolate deformation is robust in the sense that the values for Q3=2 obtained from Eq. (9) and given in TableIare all consis-tently positive. The connection between the results on deformation in the current formulation and previous ones will be discussed in a forthcoming publication [21]. In the case of the magnetic octupole form factor [21], which is related to the magnetic octupole moment O ¼ GM3ð0Þe=2m3, our statistics are insufficient to distinguish the result from zero.
IV. CONCLUSIONS
In summary, a formalism for the accurate evaluation of the electromagnetic form factors as functions of q2 has been developed and used in quenched QCD and full QCD with two and2 þ 1 flavors. The charge radius and mag-netic dipole moment were determined as a function of m2 and the dipole moment was compared with the result of chiral effective theory. The electric quadrupole form factor was evaluated for the first time with sufficient accuracy to distinguish it from zero. The lattice calculations show that
the quark density in a þ of transverse spin projection þ3=2 is elongated along the spin axis.
ACKNOWLEDGMENTS
A. T. acknowledges support by the University of Cyprus. This work is supported in part by the Cyprus Research Promotion Foundation (RPF) under Contract No. ENEK/ENIX/0505-39, the EU Integrated Infrastructure Initiative Hadron Physics (I3HP) under Contract No. RII3-CT-2004-506078, and the U.S. Department of Energy (D.O.E.) Office of Nuclear Physics under Contract Nos. DE-FG02-94ER40818 and DE-FG02-04ER41302. This research used computational resources provided by RFP under Contract No. EPYAN/ 0506/08, the IBM machine at NIC, Ju¨lich, Germany, the National Energy Research Scientific Computing Center supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098 and the MIT Blue Gene computer supported by the DOE under Grant No. DE-FG02-05ER25681. Dynamical staggered quark configurations and forward domain-wall quark propagators were provided by the MILC and LHPC collaborations, respectively.
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![FIG. 3 (color online). The magnetic dipole moment in nuclear magnetons. The value at the physical pion mass (filled square) is shown with statistical and systematic errors [1]](https://thumb-eu.123doks.com/thumbv2/123doknet/14674365.557596/4.918.81.450.699.1014/online-magnetic-dipole-nuclear-magnetons-physical-statistical-systematic.webp)
