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Interpreting the Neutron's Electric Form Factor: Rest Frame Charge

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Interpreting the Neutron's Electric Form Factor: Rest Frame Charge JLAB-THY-98-48 Interpreting the Neutron’s Electric Form Factor: Rest Frame Charge Distribution or Foldy Term? Nathan Isgur Jefferson Lab 12000 Jefferson Avenue, Newport News, Virginia, 23606 Abstract The neutron’s electric form factor contains vital information on nucleon structure, but its interpretation within many models has been obscured by relativistic effects. I demonstrate that, to leading order in the relativistic ex- arXiv:hep-ph/9812243v1 3 Dec 1998 pansion of a constituent quark model, the Foldy term cancels exactly against a contribution to the Dirac form factor F1 to leave intact the naive inter- n pretation of GE as arising from the neutron’s rest frame charge distribution. 1 I. INTRODUCTION In 1962, Sachs showed [1] that the combinations of elastic nucleon form factors (N = p or n) 2 N N Q N GE = F1 − 2 F2 (1) 4mN N N N GM = F1 + F2 (2) have simple interpretations as the spatial Fourier transforms of the nucleons’ charge and Q~ magnetization distributions in the Breit frame (where momentum ~p = − 2 is scattered to ′ Q~ N N momentum ~p =+ 2 ). Here F1 and F2 are the Dirac and Pauli form factors, respectively, defined by µν ′ ′ µ ′ ′ N µ σ qν N hN(~p ,s )|jem(0)|N(~p, s)i =u ¯(~p ,s )[F1 γ + i F2 ]u(~p, s) (3) 2mN ′ N N 2 2 where qν = pν − pν and F1 and F2 are functions of Q = −q . These form factors obviously contain vital information on the internal composition of the nucleons. Although it has proven elusive experimentally, the electric form factor of the n n neutron GE is particularly fascinating in this respect. In pion-nucleon theory, GE would arise from a π− cloud with convection currents producing the anomalous magnetic moments p n F2 = 1.79 ≡ µp − 1 and F2 = −1.91 ≡ µn. In contrast, in a valence quark model, the 1 nucleon magnetic moments arise from the underlying charged spin- 2 constituents with the famous SU(6) relation µ 3 p = − (4) µn 2 and with a scale set by mN µp = ≃ 3 (5) md 1 where md ≃ mu ≃ 3 mN is a valence quark effective mass. Within this model it has been n 2 argued that in the SU(6) limit GE(Q ) would be identically zero, but that the spin-spin 2 forces which produce the SU(6)-breaking ∆ − N splitting create a charge segregation inside n the neutron and lead to a nonzero GE [2–4]. The effect arises because the spin-spin forces push d quarks to the periphery of the neutron and pull the u quark to the center. Thus both the π− cloud picture and the hyperfine-perturbed quark model predict a negative neutron charge radius, as observed. Nonrelativistically, the squared charge radius is simply the charge-weighted mean square position of the constituents. More generally 1 Gp (Q2) ≡ 1 − r2 Q2 + ··· (6) E 6 Ep and 1 Gn (Q2) ≡ − r2 Q2 + ··· (7) E 6 En define the proton and neutron charge radii, with N 2 GM (Q ) 1 2 2 =1 − rMN Q + ··· (8) µN 6 p p defining the corresponding magnetic radii. Experimentally, the three form factors GE, GM , n and GM are reasonably well known, although new measurements should soon determine them with much greater precision [5]. From these measurements, which cover Q2 in the multi-GeV2 range, we have learned that all three form factors have roughly similar shapes: p 2 n 2 p 2 GM (Q ) GM (Q ) 2 GE(Q ) ≃ ≃ ≃ GD(Q ) (9) µp µn where the dipole form factor 2 1 GD(Q ) ≡ 2 2 2 (10) [1 + Q /Mdipole] 2 2 with Mdipole = 0.71 GeV . I note in passing that this observation makes little sense in a picture where the nucleons have a point-like core surrounded by a pion cloud (since in such n p p a picture GM is purely pionic while GE and GM are mixtures of a “bare” proton and a pion cloud). In contrast, Eq. (9) is very natural in a valence quark model where, to leading order 3 2 in a relativistic expansion, GD(Q ) would simply be the Fourier transform of the square of the ground state spatial wavefunction. Recent advances in experimental technique should lead to a clear measurement of the n 2 2 2 neutron’s electric form factor GE(Q ) for Q in the GeV range in the next few years 2 [6]. Its charge radius rEn is known from low energy neutron-electron elastic scattering to be −0.113 ± 0.005 fm2 [7], but the electron-neutron scattering measurements needed n 2 to determine GE in the GeV range (in order to roughly map out the neutron’s electric structure with a resolution of 10% of the proton’s size) have been plagued by the lack of a free neutron target [6]. Fortunately, the method of Arnold, Carlson, and Gross [8], which n n uses spin observables sensitive to GE −GM interference, opens up new methods for measuring n GE, and recent advances in accelerator, target, and detector technology are beginning to exploit these new methods. In anticipation of these measurements, there has been renewed discussion about their 2 interpretation. I focus here on the belief that the measured rEn is explained by the “Foldy term” [10]. I.e., using Eqs. (1) and (7), 2 2 3µn 2 2 rEn = r1n + 2 ≡ r1n + rFoldy,n , (11) 2mN 2 n 1 2 2 where r1n is the “charge radius” associated with F1 ≃ − 6 r1nQ + ···. The second term in Eq. (11), called the Foldy term, appears to arise as a relativistic correction associated with the neutron’s magnetic moment and so to have nothing to do with the neutron’s rest frame charge distribution. It has the value −0.126 fm2, nearly coinciding with the measured value. On this basis it has been argued that any “true” charge distribution effect must be very 2 small. In this paper I will show that while the Foldy term closely resembles rEn numerically, it does not “explain it”. Indeed, I will demonstrate that, in the relativistic approximation to the constituent quark model in which the Foldy term first appears, it is cancelled exactly 2 by a contribution to the Dirac form factor F1 leaving rEn correctly interpreted as arising entirely from the rest frame internal charge distribution of the neutron. 4 II. THE INTERPRETATION OF THE NEUTRON CHARGE RADIUS IN A CONSTITUENT QUARK MODEL The relationship (1) between the Sachs form factor GE and the Dirac and Pauli form factors F1 and F2 is relativistic in origin. Unfortunately, relativistic constituent models of the nucleon are notoriously difficult: rest frame models are difficult to boost and infinite- momentum-frame (or light-cone) quark models have trouble constructing states of definite P n J . This could be the reason that the interpretation of GE has not been clarified in the context of such models. While an accurate constituent quark model of nucleon structure must certainly be fully relativistic, the issue at hand can be resolved by using a relativistic expansion around the 2 2 2 nonrelativistic limit. This is possible because the Foldy term rFoldy,n arises at order Q /m n 2 and so its character may be exposed by an expansion of GE to order 1/m . I will also exploit symmetries of the problem available in certain limits which will make the discussion independent of the details of models. I begin with a simple “toy model” in which a “toy neutron”n ˜SD¯ is composed of a ¯ 1 scalar antiquark S of mass mS and charge −eD and a spin- 2 Dirac particle D of mass mD and charge eD bound by flavor and momentum independent forces into a rest frame nonrelativistic S-wave. The calculation begins by noting that, from their definitions, n˜S¯D 2 Qzˆ Qzˆ G (Q )= hn˜ ¯ (+ , +)|ρ |n˜ ¯ (− , +)i (12) E SD 2 em SD 2 and n˜S¯D 2 mn˜SD¯ Qzˆ 1+i2 Qzˆ G (Q )= hn˜ ¯ (+ , +)|j |n˜ ¯ (− , −)i . (13) M Q SD 2 em SD 2 It is immediately clear that the calculation of these form factors requires boosting the rest Qzˆ 2 frame S-wave bound state to momenta ± 2 . Doing so can introduce a host of 1/m effects in the boosted counterpart of the S-wave state and it can also produce new P -wave-like components by Wigner-rotation of the D-quark spinors [9]. I will show that the latter 5 effect is subleading, and will deal with the former effect by exploiting an effective charge- conjugation symmetry of the system for mD = mS ≡ m. Since µn˜SD¯ involves the limit of Eq. (13) as Q → 0, to the required order in 1/m it simply takes on its nonrelativistic value eDmn˜S¯D µn˜S¯D = (14) mD where of course mn˜SD¯ = mS + mD in this limit. The Foldy term is thus well-defined: 2 eD rF oldy,n˜S¯D = . (15) 2mDmn˜S¯D n˜SD¯ 2 2 We next compute GE (Q ) directly from Eq. (12). To leading order in 1/m , ρem remains a one-body current, and the impulse approximation is valid. Within this approximation, we make use of the relation 2 ′ Q ˜ ˜ ′ hD(~p + Qz,sˆ )|ρem|D(~p, s)i = eD(1 − 2 )hD(~p + Qzˆ)|ρem|D(~p)iδss + ρspin−flip (16) 8mD where eDQ ′ ′ ρspin−flip ≡ 2 (p−δs +δs− − p+δs −δs+) (17) 4mD and where D˜ is a fictitious scalar quark with the mass and charge of D.
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