SLAC-PUB-15759

Electron g-2 in Light-Front

Xingbo Zhaoa, Heli Honkanenb, Pieter Marisa, James P. Varya, Stanley J. Brodskyc

aDepartment of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA bThe Pennsylvania State University, 104 Davey Lab, University Park, PA 16802, USA cSLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

Abstract Basis Light-front Quantization has been proposed as a nonperturbative framework for solving quantum field theory. We apply this approach to and explicitly solve for the light-front of a physi- cal . Based on the resulting light-front wave function, we evaluate the electron anomalous magnetic moment. Nonperturbative is performed. Upon extrapolation to the infinite basis limit our numerical results agree with the Schwinger result obtained in perturbation theory to an accuracy of 0.06%.

Introduction. Nonperturbative approaches in quantum Light-front Hamiltonian in BLFQ basis. In light-front field theory are needed for many applications. One dynamics, physical processes are described in terms of important application is to study problems light-front coordinates, which consist of light-front time + 0 3 0 3 in strongly interacting systems, e.g., solving for the x x + x , the longitudinal direction x−=x x as ≡ 1 2 − structure in (QCD). well as the transverse co-ordinates x⊥={x , x }. We be- The Basis Light-front Quantization (BLFQ) approach gin with the light-front QED Hamiltonian which can be has recently been constructed [1, 2] as a nonperturba- derived from the ordinary Lagrangian through the stan- tive framework for quantum field theory in the Hamil- dard Legendre transform with the adoption of the light- tonian framework [3]. In previous work [2] this method cone gauge (A+=0), where the photon field has physi- has been applied to QED for the anomalous magnetic cal polarization and positive metric. The resulting QED moment of a physical electron in a cavity formed by an light-front Hamiltonian takes the following form, external transverse harmonic-oscillator potential, which 2 2 + ⊥ 2 1 + me (i∂ ) 1 j 2 j acts as a regulator of QED dynamics. In addition, the P− = d x⊥dx− Ψ¯ γ Ψ + A (i∂⊥) A 2 i∂+ 2 extension to strong time-dependent external field appli- Z " 2 2 + cations (tBLFQ) has been developed and successfully µ e + 1 + e µ γ ν +e j Aµ + j j + Ψ¯ γ Aµ γ AνΨ , applied [4, 5]. 2 (i∂+)2 2 i∂+ # In this work we employ BLFQ to compute the elec- (1) tron anomalous magnetic in .1 For an alternative where Ψ and Aµ are the fermion and gauge boson fields, nonperturbative calculation in light-front dynamics, see respectively. The first and second terms are their corre- Refs. [8, 9]. sponding kinetic terms, and the remaining three terms describe the interaction between the fermion and gauge boson fields. Specifically, these are the vertex in- Email addresses: [email protected] (Xingbo Zhao), [email protected] (Heli Honkanen), [email protected] (Pieter teraction, the instantaneous-photon interaction and the Maris), [email protected] (James P. Vary), instantaneous-fermion interaction in order of appear- [email protected] (Stanley J. Brodsky) ance in Eq. (1). The me and e are the bare electron mass 1 In addition, several improvements and corrections are carried out and the bare electromagnetic coupling constant, respec- over Ref. [2]: 1) we correct the used in [2] for extracting tively. In this work we only keep e and eγ ; i.e., two the anomalous magnetic moment by adopting the standard Pauli form | i | i factor operator [6]; 2) we correct a numerical error in [2], which led to Fock sectors, in our basis (see below). Consequently, an overestimate of the interaction terms in QED Light-front Hamilto- the instantaneous-photoninteraction does not contribute nian by a factor of 2; 3) we optimize the computational efficiency on since it involves Fock sectors with one more electron both the analytic and coding level and implement parallel computing so that the BLFQ calculation can take advantage of currently available (or positron). Moreover, the instantaneous fermion in- supercomputers. teraction either contributes to overall renormalization

Preprint submitted to Elsevier February 19, 2014

Published in arXiv:1402.4195.

Work supported in part by US Department of Energy under contract DE-AC02-76SF00515. factors, which do not affect the intrinsic structure of We truncate the basis states by eliminating states with the physical electron, or contains small-x divergences Nβ larger than a chosen cutoff Nmax. Increasing Nmax not which need to be cancelled by explicit fermion ex- only enhances the resolution but also provides a higher change contributions from higher Fock-sectors. Thus, ultraviolet cutoff and a lower infrared cutoff for the par- we defer the inclusion of the instantaneous interactions ticles’ transverse motion. and adopt the following Hamiltonian for this work, In the longitudinal direction we perform basis trun- cation by imposing (anti-)periodic boundary condition 2 2 + ⊥ 2 1 + me (i∂ ) 1 j 2 j for (fermions) bosons, such that the longitudinal mo- P− = d x⊥dx− Ψ¯ γ Ψ+ A (i∂⊥) A Z "2 i∂+ 2 mentum k for each Fock particle can µ only take (half-)integers. Being a good quantum num- +e j Aµ . (2) i ber for the QED Hamiltonian, the total longitudinal mo- In the second step we construct the Fock-sector ba- mentum summed over all Fock particles in a basis state, + + sis expansion. A physical electron, which is the focus P = i pi K, acts as an additional cutoff. Larger + ∝ of this work, receives contributions from the multiple P (K)P allows more possible partitions of longitudinal Fock-sectors, among Fock particles in a basis state and thus leads to a higher resolution in the longitudinal de- ephys = a e + b eγ + c eγγ + d eee¯ + ..., (3) grees of freedom. | i | i | i | i | i Therefore, in order to specify the truncated basis, and each Fock-sector itself consists of an infinite num- we need the following information: 1) Fock-sectors in- ber of basis states. For the purposeof numericalcalcula- cluded; 2) truncation parameters, Nmax (transverse) and tions we adopt both a Fock-sector truncation and limits K (longitudinal); 3) 2D-HO basis parameter b. The lon- on the basis states in each sector. In this work we make gitudinal period L is not needed due to the longitudinal the lowest nontrivial truncation by retaining the e and boost-invariance of light-front dynamics: the light-front | i eγ Fock-sectors only. This is sufficient for obtaining wave functions only depend on the longitudinalmomen- | i + + the (nonperturbative)electron wave function accurate to tum fraction xi = pi /P = ki/K. In this work we retain the leading order of the electromagnetic coupling con- only e and eγ Fock-sectors, and compare numeri- e2 | i | i stant α = 4π . cal results evaluated in bases of different Nmax and K. For each Fock-particle we employ a 2D-harmonic The 2D-HO parameter b is taken to match the physical oscillator (HO) wave function, Φnm(p⊥), to describe electron mass M, such that b=M=0.511 MeV. its transverse degrees of freedom and a plane-wave, Next we express our field operators in the BLFQ ba- ip+ x /2 e− − , to describe its longitudinal motion. For sis, specifically for the fermion and gauge boson field, each Fock-particle the (transverse) 2D-HO wave func- 2 tion carries the radial quantum number n and angu- 1 d p⊥ ip x Ψ(x) = bβ¯Φnm(p⊥)u(p, λ)e− · lar quantum number m, (as well as a parameter b, √2L Z (2π)2 Xβ¯  setting the scale of the HO wave functions, e.g., in ip x 2 2 + d†¯ Φnm∗ (p⊥)v(p, λ)e · , (5) exp[ (p⊥) /(2b )]). The (longitudinal)plane-wave car- β − + 2  ries one quantum number, k=p L/(2π), which is pro- 1 d p⊥ ip x + Aµ(x) = a¯ Φnm(p⊥)ǫµ(p, λ)e− · portional to the longitudinal momentum p , and L is the + (2π)2 β X¯ 2Lp Z length of the longitudinal “box” in which we compact- β  p ip x + a†Φ (p )ǫ (p, λ)e , (6) ify the longitudinal degrees of freedom of the system. β¯ nm∗ ⊥ µ∗ · With the additional quantum number λ for the helicity,  4 quantum numbers label each single particle state. where the u and v are the Dirac spinors for fermions In the transverse directions we implement the “N ” and anti-fermions, respectively. The ǫµ is the photon max + truncation in analogy with the 3D-HO truncation in nu- polarization vector. The p x = p x−/2 p⊥ x⊥ is the inner product between the· 3-momentum−p = p+, p clear many-bodytheory [10, 11]. Define a sum, Nβ, over { ⊥} and the coordinate x = x−, x⊥ . The b†, d† and a† the HO quantum numbers for all Fock particles, i, ina { } β¯ β¯ β¯ specific basis state (here we discuss only the quantum are creation operators for the fermion, anti-fermion and numbers of the transverse spatial motion), β , accord- gauge boson fields, respectively, with quantum numbers ing to, | i β¯ = k, n, m, λ . They satisfy the (anti-)commutation re- lations{ }

Nβ 2ni + mi + 1 . (4) ≡ | | bβ¯ , b†¯ = dβ¯ , d†¯ = [aβ¯, a†¯ ] = δβ¯β¯ . (7) Xi { β′ } { β′ } β′ ′ 2 Through Eqs. (2) and (7), we are able to write down the interactions. No further quantum fluctuations are in- light-front QED Hamiltonian in the BLFQ basis. Since troduced. Thus for basis states in the electron-photon we are interested in the mass eigenspectrum contributed sector the bare electron mass remains the same as the by the intrinsic rather than center-of-mass motion, we physical value. On the other hand, the basis states in the add an appropriate Lagrange multiplier term to the in- single electron sector couple to those in the electron- put light-front QED Hamiltonian. This has the effect photon sector and receive the self-energy correction. of shifting the states with excited center-of-mass mo- Therefore for these states, we need a mass counter- tion to high mass and the low-lying spectrum comprises term which we introduce via our iterative diagonaliza- states with lowest center-of-mass motion, following the tion scheme. techniques of nuclear many-body theory. The resulting low-lying states can be written as a simple product of Anomalous Magnetic Moment. Our calculated spec- internal and center-of-mass motion [10, 11] (see [12] trum includes both the physical electron state and for more details). electron-photon scattering states. The ground state is identified as the physical electron ( e ), and its Upon diagonalization of the resulting sparse Hamil- | phys i tonian matrix, one obtains its eigenvalue spectrum and eigenvalue has been renormalized to the mass of a phys- corresponding eigenvectors. In this work, the ground ical electron. The associated eigenvector (wave func- state of the Hamiltonian, with net fermion number be- tion) encodes all the information of intrinsic structure of the physical electron and can be employed to evalu- ing one (n f =1), is identified as the physical electron. ate observables. Its eigenvalue, Pe−, gives the electron mass according to 2 + 2 We focus on one specific observable: the electron M Pe−Pe P , where P is the total transverse mo- ≡ − ⊥ ⊥ mentum operator. anomalous magnetic moment, ae, which measures the deviation of the electron gyromagnetic ratio from g 2 the “normal” value, 2, namely, a s− . The elec- Renormalization. Before we are ready to obtain the e ≡ 2 electron wave function, one more technical detail needs tron spin gyromagnetic ratio gs is the ratio between the to be worked out: renormalization. In BLFQ, a nonper- electron’s magnetic moment, µ, and the product of the turbative approach, the renormalization procedures are electron’s spin, s=1/2, with the Bohr magneton, e/(2M), different from those adopted in perturbation theory [13]. e µ = g s . (8) In quantum field theory, the values for parameters in s 2M the Hamiltonian, the bare electron mass m and the bare e The finite electron anomalous magnetic moment a re- coupling constant e, are regulator (cutoff) dependent. e flects a nontrivial internal structure of the electron in Through renormalization, one establishes the exact con- QED: it originates from the relative motion between nection between these parameters and the theory’s regu- the constituent electron and the constituent photon (as lators. Since we omit Fock-sectors containing electron- well as higher Fock components in principle) inside a positron pairs in our bases, bare photons are not able physical electron. It was first calculated by Schwinger to fluctuate into electron-positron pairs and thus mod- in leading-order perturbation theory [7] with the result ify the physical charge of the electron. In this work we a = α . need only the electron mass renormalization. e 2π In QED, the a is defined by the Pauli form factor Guided by a sector-dependent renormalization ap- e F (q2) at the zero momentum transfer limit q2 0, proach [14, 15], our procedure for the electron mass 2 → renormalization is as follows: we numerically diagonal- 2 ae F2(q 0) , (9) ize the Hamiltonian matrix in an iterative scheme where ≡ → 2 µ µ we adjust the input bare electron mass in the Hamilto- where q = q qµ and q is the 4-momentum trans- nian in the single electron sector only, until the resulting ferred from a probe photon to the electron. We adopt mass for the ground (physical electron) state matches the Drell-Yan-West frame [6] where the incident elec- the physical electron mass of M=0.511MeV. The idea tron is directed along the 3-direction with 4-momentum µ + M2 ~ behind this procedure is the following: the mass coun- p = (p , + , 0 ) and the probe photon’s momentum p ⊥ terterm, the difference between the physical electron µ 2q p is in the transverse directions with q = (0, +· , ~q ), mass and the bare mass, compensates for the mass cor- p ⊥ where 2q p = q2 = q2 . In this frame the Pauli form rection due to the quantum fluctuations to higher Fock ⊥ factor can· be evaluated− as, sectors. The basis states in the electron-photon sec- tor, the highest Fock sector in our current truncation q1 iq2 2 + − F2(q ) = e↑ (~q ) J (0) e↓ (~0 ) , (10) scheme, generate the conventional one-loop self-energy − 2M h phys ⊥ | | phys ⊥ i 3 + + where ~q =(q1, q2) and J (0)=Ψ¯ (0)γ Ψ(0) is the elec- ⊥ µ ( ) Schwinger result tric charge density operator at x =0. The e↑ ↓ (~q ) phys 0.10 denotes the physical electron state with helicity| (anti-⊥ i + ) parallel to the longitudinal momentum (p ) direction 0.08 and (average) transverse center-of-mass momentum of ~q . The helicity-flip state and the states with nonzero 0.06 α = 1 ⊥ 137.036 (average) transverse momentum can be inferred from α = 1 from Eq. (10) π ↑ ~ ephys(0 ) by exploiting the transverse parity symme- 2 0.04 | ⊥ i e try [16, 12] and the boost invariance properties of light- / e front dynamics. a 0.02 p In BLFQ, we work with finite dimensional basis spaces. In order to obtain the electron anomalous mag- 0.00 netic moment ae in the limit of the infinite basis size, 4.0 + 1.78ln Nmax 6 −. + . we first calculate ae as a function of the truncation pa- 0 87 0 044 ln Nmax rameters, which also act as regulators, and then perform 5 extrapolations. The upper panel of Fig. 1 displays the 4 anomalous magnetic moment evaluated from Eq. (10) 2 Z at various selected basis sizes as discrete points and / 3 at both the physical electromagnetic coupling constant 1 1 1 2 α = 137.036 and an artificially enlarged α = π . We elect to relate the two basis space cutoffs and adopt Nmax = 1 K 1/2 for simplicity and convenience. The horizontal axis− in Fig. 1 is the N = K 1/2 of the basis. Results 0 max 10 100 1000 in bases at N =K 1/2 larger− than 50 are evaluated us- max Nmax = K 1/2 ing Hopper, a Cray− XE6 supercomputer, and Edison, a − Cray XC30 supercomputer, at the National Energy Re- Figure 1: (Color online) Upper panel: the (square root of) elec- search Scientific Computing Center (NERSC). Numer- tron anomalous magnetic moment ae (normalized to electron charge e) from Eq. (10) as a function of basis truncation parameter ical diagonalization of the Hamiltonian matrix is per- Nmax=K 1/2. All the data points are at odd Nmax/2. The dashed formed by ARPACK software [19]. The maximum ba- line indicates− the Schwinger result = 0.1125395. Lower panel: the sis dimensionality achieved so far is 28,027,289,920 at (inverse) electron wave-function renormalization factor Z2 as a func- tion of basis truncation parameter Nmax=K 1/2. The two-parameter Nmax = K 1/2=640. − − fits are based on data points with Nmax=K 1/2 > 150 only. The results in the upper panel of Fig. 1 suggest − that the anomalous magnetic moment directly evalu- ated from Eq. (10) tends to zero with increasing basis logarithmic divergence in 1/Z2 as a function of the trun- space cutoff. The origin of this behavior is that the cur- cation parameters (regulators), see the lower panel of rent Fock space truncation violates the condition Z1 = Fig. 1. Z2 [18], which would be a consequence of the Ward We next note that, due to our current Fock space trun- identity. Here Z1 is the renormalization factor for the cation, Z1 does not obtain the corresponding quantum vertex coupling the e and eγ sectors which remains fluctuation that would involve the eγγ sector. Hence, unity in the infinite basis| i limit| withi our Fock space trun- it seems reasonable to associate the| origini of the vanish- cation. Now, Z2 is the electron wave-function renor- ing (naive) anomalous magnetic moment from Eq. (10) malization which, in light-front dynamics, can be inter- with that of the vanishing renormalization constant Z2. preted as the probabilityof finding a constituent electron We therefore propose the following procedure to obtain out of a physical electron: the rescaled (“re”) Pauli form factor, 2 F (0) Z2 = e ephys , (11) re 2 |h | i| ae = F2 (0) = . (12) Xe Z2 | i where the summation runs over all the basis states in the After rescaling the Pauli form factor the (rescaled) e sector. In our Fock space truncation, Z receives a anomalous magnetic moment becomes independent of | i 2 contribution from the quantum fluctuation between the the coupling constant α (at fixed Nmax = K 1/2), as e and eγ sectors and consequently goes to zero in can be seen in Fig. 2, even though the naive− results for the| i infinite| basisi limit. Our numerical data suggest a the anomalous magnetic moment depend strongly on α 4 0.115 Schwinger result = 0.1125395... 0.1 0.11 α = 1

2 137.036 2

e 0.105 e / / e

0.09 e a

N = K− = a max 1/2 10 0.1 p Nmax = K−1/2 = 20 p Nmax = K−1/2 = 30 0.095 even Nmax/2 0.08 Nmax = K−1/2 = 40 odd Nmax/2 0.09 even Nmax/2 fit odd N /2 fit 0.07 max 0.001 0.01 0.1 1 10 100 0 0.05 0.1 0.15 0.2 −1/2 electron charge e (Nmax = K−1/2)

Figure 2: (Color online) The (square root of) electron anomalous mag- Figure 3: (Color online) The (square root of) electron anomalous mag- netic moment a (normalized to electron charge e) as a function elec- e netic moment ae (normalized to electron charge e) as a function of tron charge e at selected N = K 1/2. At each N = K 1/2, the 1 max max basis truncation parameter Nmax=K 1/2. We use α = . Linear variation is in the fourth significant− digit and not visible on the− figure. − 137.036 extrapolations use data points with Nmax=K 1/2>150. The arrow on the left y-axis indicates the Schwinger result−= 0.1125395.

(see Fig. 1). Furthermore, our results for the rescaled anomalous magnetic moment seem to increase mono- sults and (higher-order) perturbation theory. tonically with increasing N = K 1/2, and approach max − the Schwinger result from below, independent of α. Conclusion and Outlook. In this work we demonstrated In order to test the precision of BLFQ, we extrapolate the workflow of applying BLFQ to evaluate observables the (rescaled) anomalous magnetic moment to the infi- in the vacuum. Specifically, as a test problem, we ap- nite N = K 1/2 limit in Fig. 3. Here, the BLFQ plied this approach to QED and study the physical elec- max − results fall into two groups with even and odd Nmax/2, tron in bases truncated to e and eγ Fock-sectors. respectively. This odd-even effect originates from the We performed the electron| massi renormalization| i fol- oscillatory behavior of the (2D-HO) basis function in lowing a sector-dependent scheme [14]. We found that the transverse plane. In Fig. 3, we apply linear ex- the resulting (naive) electron anomalous magnetic mo- trapolation in 1/ √N = K 1/2 to data points with ment in this truncated basis approaches zero upon ex- max − Nmax=K>150 for the even (odd) Nmax/2 group individ- trapolation to the infinite basis limit, independent of the ually. The extrapolated ae at infinite Nmax = K 1/2 coupling constant. However, by rescaling the anoma- limit, is 0.112610 (0.112541), agreeing well with− the lous magnetic moment with the inverse of the electron Schwinger result with a relative deviation of +0.063% wave-function renormalization factor (Z2), we recover (+0.001%), for the even (odd) Nmax/2 group, respec- the Schwinger result to high precision (less than 0.1% tively. deviation), confirming the results found in another non- Our finding confirms the analytic result found in perturbative approach based on light-front wave equa- Refs. [14, 15], where a nonperturbative light-front wave tion formalism [14, 15]. equation approach is adopted and it was found that in On the computational aspects, we find the BLFQ ap- bases truncated to e and eγ two sectors, the nonper- proach may be parallelized following recent advances turbative results agree| i exactly| i with the Schwinger re- in computational low-energy nuclear physics. For fixed sult. This can be understood as follows: in the Fock- basis sizes, the (inverse) computational time (“speedup” sector truncation allowing only for the quantum fluctua- factor) almost scales linearly with the number of pro- tion into eγ sector, the resulting nonperturbative light- cessors. It is thus conceivable that this method will front wavefunction| i encodes the identical information greatly benefit from anticipated advances in supercom- on the structure of the physical electron, compared to puter technology. that from leading-order perturbation theory. The higher- Since the electron light-front wave function encodes order contributions only contribute to the electron wave- all the information on the electron structure, it can be function renormalization factor, Z2. Once we include employed to evaluate other observables which “mea- higher Fock sectors in our BLFQ calculations we expect sure” the electron structure in QED, such as the elec- to see deviations between the nonperturbative BLFQ re- tromagnetic form factors, the generalized parton distri- 5 bution functions (GPDs) [21], etc. Also, we have initi- [15] V. A. Karmanov, J. F. Mathiot and A. V. Smirnov, Phys. Rev.D ated applications of this method to other systems, such 86, 085006 (2012). as positronium, for which we add a positron into the [16] S. J. Brodsky, S. Gardner and D. S. Hwang, Phys. Rev. D 73, 036007 (2006). current single electron system. Indeed, initial positron- [17] S. J. Brodsky, D. S. Hwang, B. -Q. Ma and I. Schmidt, Nucl. ium calculations are already underway [22, 23, 24]. In Phys. B 593, 311 (2001) [hep-th/0003082]. addition to these “stationary” problems, electron light- [18] S. J. Brodsky, V. A. Franke, J. R. Hiller, G. McCartor, S. A. Pas- ton and E. V. Prokhvatilov, Nucl. Phys. B 703 (2004) 333 [hep- front solutions also find applications in the recently de- ph/0406325]. veloped time-dependent Basis Light-front Quantization [19] R. B. Lehoucq and D. C. Sorensen and C. Yang, ARPACK Users (tBLFQ) approach [4], where the single electron states Guide: Solution of Large Scale Eigenvalue Problems by Implic- as well as the electron-photon scattering states are em- itly Restarted Arnoldi Methods, (1997). [20] D. Mustaki, S. Pinsky, J. Shigemitsu and K. Wilson, Phys. Rev. ployed to solve the photon-emission problem in a strong D 43, 3411 (1991). and time-dependent laser field. Ultimately, our goal is [21] D. Chakrabarti, X. Zhao, H. Honkanen, R. Manohar, P. Maris to apply this method to QCD and solve for the hadron and J. P. Vary, 2014, in preparation. spectrum and structure. [22] P. Maris, P. Wiecki, Y. Li, X. Zhao and J. P. Vary, Acta Phys. Polon. Supp. 6, 321 (2013). We acknowledge valuable discussions with K. [23] P. W. Wiecki, Y. Li, X. Zhao, P. Maris and J. P. Vary, Tuchin, P. Hoyer, P. Wiecki and Y. Li. This work was arXiv:1311.2908 [nucl-th]. supported in part by the Department of Energy under [24] Y. Li, P. W. Wiecki, X. Zhao, P. Maris and J. P. Vary, Grant Nos. DE-FG02-87ER40371 and DESC0008485 arXiv:1311.2980 [nucl-th]. (SciDAC-3/NUCLEI) and by the National Science Foundation under Grant No. PHY-0904782. A portion of the computational resources were provided by the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. H. H is supported by US DOE Contract Number DE-FG02-93ER40771 and S. J. B. is supported by US DOE Contract Number DE-AC02- 76SF00515. This paper is SLAC-PUB-15759.

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