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Electron G-2 in Light-Front Quantization SLAC-PUB-15759 Electron g-2 in Light-Front Quantization 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 Quantum Electrodynamics and explicitly solve for the light-front wave function of a physi- cal electron. Based on the resulting light-front wave function, we evaluate the electron anomalous magnetic moment. Nonperturbative mass renormalization 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 bound state 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 − hadron structure in Quantum Chromodynamics(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 vacuum.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 energy 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 operator 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 quantum number 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, momentum 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.
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