Composite Vector Particles in External Electromagnetic Fields

# Composite Vector Particles in External Electromagnetic Fields

Zohreh Davoudi111davoudi@mit.edu Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA    William Detmold222wdetmold@mit.edu Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
###### Abstract

Lattice quantum chromodynamics (QCD) studies of electromagnetic properties of hadrons and light nuclei, such as magnetic moments and polarizabilities, have proven successful with the use of background field methods. With an implementation of nonuniform background electromagnetic fields, properties such as charge radii and higher electromagnetic multipole moments (for states of higher spin) can be additionally obtained. This can be achieved by matching lattice QCD calculations to a corresponding low-energy effective theory that describes the static and quasi-static response of hadrons and nuclei to weak external fields. With particular interest in the case of vector mesons and spin-1 nuclei such as the deuteron, we present an effective field theory of spin-1 particles coupled to external electromagnetic fields. To constrain the charge radius and the electric quadrupole moment of the composite spin-1 field, the single-particle Green’s functions in a linearly varying electric field in space are obtained within the effective theory, providing explicit expressions that can be used to match directly onto lattice QCD correlation functions. The viability of an extraction of the charge radius and the electric quadrupole moment of the deuteron from the upcoming lattice QCD calculations of this nucleus is discussed.

preprint: MIT/CTP-4723

## I Introduction

Electromagnetic (EM) interactions serve as valuable probes by which to shed light on the internal structure of strongly interacting single and multi-hadron systems. They provide insight into the charge and current distributions inside the hadrons. These are conventionally characterized by EM form factors, and are accessible through experimental measurements of electron-hadron scattering as well as EM transitions. The static and quasi-static limits of form factors, known as EM moments and charge radii, are independently accessible through high-precision low-energy experiments, such as in the spectroscopy of electronic and muonic atoms. These two different experimental approaches can serve to test the accuracy of the obtained quantities, and an apparent discrepancy, such as the one reported on the charge radius of the proton Pohl et al. (2013); Carlson (2015), promotes investigations that can deepen our understanding of the underlying dynamics. In bound systems of nucleons, EM probes further serve as a tool to constrain the form of hadronic forces. As a primary example, the measurement of a nonvanishing electric quadrupole moment for the lightest nucleus, the deuteron, led to the establishment of the existence of tensor components in the nuclear forces Kellogg et al. (1940).

Since quantum chromodynamics (QCD) governs the interactions of quark and gluon constituents of hadrons, any theoretical determination of the EM properties of hadronic systems must tie to a QCD description. The spread of theoretical predictions based on QCD-inspired models, such as those reported on the EM moments of vector mesons Aliev and Savci (2004); Braguta and Onishchenko (2004); Choi and Ji (2004); Bhagwat and Maris (2008), highlights the importance of performing first-principles calculations that only incorporate the parameters of quantum electrodynamics (QED) and QCD as input. The only such calculations are those based on the method of lattice QCD (LQCD), and involve a numerical evaluation of the QCD path integral on a finite, discrete spacetime. By controlling/quantifying the associated systematics of these calculations, the QCD values of hadronic quantities can be obtained with systematically improvable uncertainties.

QED can be introduced in LQCD calculations, alongside with QCD, in the generation of gauge-field configurations. This, however, leads to large finite-volume (FV) effects arising from the long range of QED interactions Hayakawa and Uno (2008); Davoudi and Savage (2014); Borsanyi et al. (2014); Endres et al. (2015); Lucini et al. (2015). The numerical cost of a lattice calculation which treats photons as dynamical degrees of freedom has forbidden comprehensive first-principles studies of EM properties of hadrons and nuclei through this avenue.333Significant progress has been made in recent years on this front, resulting in increasingly more precise determinations of QED corrections to mass splittings among hadronic multiplets Blum et al. (2007); Basak et al. (2008); Blum et al. (2010); Portelli et al. (2010, 2011); Aoki et al. (2012); de Divitiis et al. (2013); Borsanyi et al. (2013); Drury et al. (2013); Borsanyi et al. (2014); Endres et al. (2015); Horsley et al. (2015), and recently more refined calculations of the hadronic light-by-light contribution to the muon anomalous magnetic moment, albeit at unphysical kinematics Blum et al. (2015). Alternatively, as is done in most studies of hadron structure, the matrix elements of the EM currents can be accessed through the evaluation of three-point correlation functions in a background of pure QCD gauge fields, with insertions of quark-level current operators between hadronic states.

In Sec. II, we present a general effective field theory (EFT) of composite vector particles coupled to perturbatively weak EM fields. Such effective theories have been worked out extensively in both classic and modern literature, with features and results that sometimes differ one another. Here we follow the most natural path, building up the Lagrangian of the theory from the most general set of nonminimal interactions (those arising from the composite nature of the fields) consistent with symmetries of the relativistically covariant theory, in an expansion in . denotes a typical scale of the hadronic theory which we take to be the physical mass of the composite particle. Since the organization of nonminimal couplings is only possible in the low-energy limit, this approach, despite its relativistically covariant formulation, can only be considered to be semi-relativistic. This means that the spin-1 field satisfies a relativistic dispersion relation in the absence of EM fields. However, once these external fields are introduced, one only accounts for those nonminimal interactions that will be relevant in the nonrelativistic (NR) Hamiltonian of the system at a given order in expansion (see Refs. Lee and Tiburzi (2014a, b) for a similar strategy in the case of spin-0 and spin- fields). We next match the low-energy parameters of the semi-relativistic Lagrangian to on-shell processes at low momentum transfers, and discuss subtleties when electromagnetism is only introduced through classical fields. The effective theory developed here relies on a -component representation of the vector fields which reveals a first order (with respect to time derivative) set of equations of motion (EOM). It resembles largely that presented in earlier literature by Sakata and Taketani Sakata and Taketani (1940), Young and Bludman Young and Bludman (1963), and Case Case (1954), but has also new features. In particular, it incorporates the most general nonminimal couplings at and therefore systematically includes operators that probe the electric and magnetic charge radii of the composite particle. The semi-relativistic Green’s functions are then constructed in Sec. IV for the case an electric field varying linearly in a spatial direction. These Green’s functions are related to the quantum-mechanical propagator of anharmonic oscillator and have no closed analytics forms, making it complicated to match them to LQCD calculations.

## Ii Composite Spin-1 Particles Coupled to External Electromagnetic Fields

Any relativistic description of massive vector particles, due to the requirement of Lorentz invariance, must introduce fields that have redundant degrees of freedom. The most obvious choice is to represent the spin-1 field by a Lorentz four-vector, , the so-called Proca field Proca (1936). The redundant degree of freedom of the Proca field, , can be eliminated using the EOM. These EOM are second order differential equations, and their reduced form, i.e., after the elimination of the redundant component, turns out to be non-Hermitian. Consequently, the solutions are in general nonorthogonal and difficult to construct in external EM fields Silenko (2004). To avoid these difficulties, an equivalent formalism can be adopted by casting the Proca equation into coupled first-order differential equations, known as the Duffin-Kemmer equations Duffin (1938); Kemmer (1939). This requires raising the number of degrees of freedom of the field and consequently introducing more redundancies. However, these redundant components can be eliminated in a straightforward manner, leading to EOM that can be readily solved (see the next section). There is a rich literature on relativistic spin-1 fields and their couplings to external EM fields via different first- and second-order formalisms, see for example Refs. Corben and Schwinger (1940); Vijayalakshmi et al. (1979); Santos and Van Dam (1986); Daicic and Frankel (1993); Khriplovich and Pomeransky (1998); Pomeransky and Sen’kov (1999); Silenko (2004, 2005, 2013). Here we follow closely the work of Young and Bludman Young and Bludman (1963) which is a generalization of first-order Sakata-Taketani equations for spin-1 fields Sakata and Taketani (1940). However, due to the spread of existing results, and occasionally inconsistencies among them, we independently work out the construction of an EFT for massive spin- fields towards our goal of deducing Green’s functions of spin-1 fields in a selected external field. In particular, the nonminimal couplings in our Lagrangian, as will be discussed shortly, are more general than those presented in all previous studies, and include all the possible terms needed to consistently match to not only the particle’s electric quadrupole moment but also its electric and magnetic charge radii at (we neglect terms that are proportional to the field-strength squared with coefficients that are matched to polarizabilities). Although fields and interactions have been described in a Lorentz-covariant relativistic framework, the nonminimal couplings to external fields can only be organized in an expansion in the mass of the particle, or in turn a generic hadronic scale above which the single-particle description breaks down.999Although the expansion parameter is taken to be the mass, the size of nonminimal interactions is indeed governed by the compositeness scale of the particle. In fact, as we will see shortly, when these compositeness scales, such as radii and moments, arise in matching the coefficients to on-shell processes, the factors of mass cancel. At low energies, one can truncate these nonminimal interactions at an order such that, after a full NR reduction, the effective theory incorporates information about as many low-energy parameters as one is interested in.

### ii.1 A semi-relativistic effective field theory

We start by writing down the most general Lorentz-invariant Lagrangian for a single massive spin-1 field, coupled to electromagnetism, that is invariant under charge conjugation, time reversal and parity. We choose to construct the Lagrangian out of a four-component field and a rank-two tensor (). However, as we shall see below, the EOM of the resulting theory constrain the number of independent degrees of freedom to those needed to describe the physical modes of a spin-1 field. The Lagrangian, in terms of and degrees of freedom, can be written as

 L=12W†μνWμν+M2V†αVα−12W†μν(DμVν−DνVμ)−12((DμVν)†−DνV†μ)Wμν+ ieC(0) Fμν V†μVν+ieC(2)1M2 ∂μFμν((DνVα)†Vα−V†αDνVα)+ ieC(2)2M2 ∂αFμν((DαVμ)†Vν−V†νDαVμ)+ieC(2)3M2 ∂νFμα((DμVα)†Vν−V†νDμVα)+O(1M4,F2), (1)

where denotes the covariant derivate, is the EM field strength tensor, denotes the photon gauge field, and refers to the electric charge of the particle. The superscripts on the coefficients denote the order of the corresponding terms in an expansion in . By we indicate any Lorentz-invariant term bilinear in and with appropriate numbers of covariant derivates and s such that the overall mass dimension is four when accompanied by . Similarly, corresponds to any Lorentz-invariant term with mass dimension four that contains two s. In particular, this latter include -type interactions that are of the same order in the inverse mass expansion as are the nonminimal terms we have considered, and whose coefficients are matched to electric and magnetic polarizabilities of the particle. By assuming a small external field strength, we can neglect these contributions. In order to access polarizabilities, Eq. (1) must be revisited to include such terms.

The coefficients of the leading contributions are fixed to reproduce the canonical normalization of the resulting kinetic term for massive spin-1 particles Proca (1936). We have taken advantage of the following property of the EM field strength tensor to eliminate redundant terms at . Additionally, the number of terms with a given Lorentz structure at each order can be considerably reduced by using the constraint of vanishing surface terms in the action. This constraint is not trivial in the presence of EM background fields which extend to infinite boundaries of spacetime (which is an unphysical but technically convenient situation). To rigorously define a field theory in the background of classical fields, one shall assume background fields are finite range, are adiabatically turned on in distant past and will be adiabatically turned off in far future. Mathematically, this means that one must accompany external fields by a factor of , where is positive and . This ensures that for any finite value of , the background field is independent and nonzero, while as , the field gradually vanishes. This procedure is particularly important when space-time dependent background fields are considered. This is because the sensibility of the expansion of nonminimal couplings in Eq. (1) when is guaranteed only if a mechanism similar to what described above is in place. In a calculation performed in a finite volume, such a procedure does not eliminate the contributions at the boundary. However, in this case one is free to choose the boundary conditions. For example, if periodic boundary conditions (PBCs) are imposed on the fields, the contributions of the surface terms to the action will in fact vanish just as in the infinite volume. As a result, the only relevant interactions in both scenarios have been already included in the Lagrangian in Eq. (1), with coefficients that could be meaningfully constrained by matching to on-shell processes in the infinite spacetime volume. To satisfy PBCs in a finite volume, certain quantization conditions must be imposed on the parameters of the background fields, which can be seen to also prevent potential large background field strengths at the boundaries of the volume, see Ref. Davoudi and Detmold (2015).

The Euler-Lagrange EOM arising from the Lagrangian in Eq. (1) are

 (I)   Wμν=DμVν−DνVμ+O(1M4,F2), (2)
 (II)   DμWμα+M2Vα+ieQ0C(0)FαμVμ=ieM2[2C(2)1∂μFμνDνVα+C(2)2∂2FανVν+ C(2)3(∂νFμαDμVν+∂αFμνDμVν+∂μ∂νFμαVν)]+O(1M4,F2), (3)

where in Eq. (2) (Eq. (3)) denotes any Lorentz-invariant term with mass dimension two (three) with at most one or field. Similarly, in Eq. (2) (Eq. (3)) denotes any Lorentz-invariant terms with mass dimension two (three) with at least two powers of the field strength tensor and at most one or field. Note that from the first equation, it is established that is an antisymmetric tensor up to corrections. We have anticipated this feature in writing down all possible terms at in the Lagrangian Eq. (1), as the nonantisymmetric piece of gives rise to contributions that are of higher orders. This also makes any term containing one and one field at redundant.

In writing the Lagrangian in Eq. (1), we have neglected terms of the type . These can be reduced to terms that have been already included in the Lagrangian at this order using the EOM. A number of inconsistencies might occur when the EOM operators are naively discarded in the presence of background fields. However, as is discussed in Refs. Lee and Tiburzi (2014a, b), the neglected terms in the Lagrangian only modify Green’s functions by overall spacetime-independent factors that can be safely neglected. The other sets of operators at that we have taken the liberty to exclude due to the constraint from the EOM are those containing at least one . These vanish up to corrections that scale as (see Eqs. (2) and (3) above), and therefore give rise to higher order terms, i.e., , in the Lagrangian.101010According to Refs. Lee and Tiburzi (2014a, b), the EOM operators in fact must be given special care only in the NR theory. The contribution from these operators to on-shell processes could be nontrivial in situations where QED is introduced through a background EM field. Given that we follow a direct NR reduction of the relativistic theory, all such subtleties will be automatically taken care of. In particular, it is notable that the semi-relativistic Lagrangian with a background electric field up to generates terms of the type in the NR Hamiltonian, see Sec. III. This is despite the fact that we have already neglected terms of in the semi-relativistic Lagrangian. These are the type of contributions that are shown to correspond to an EOM operator in the scalar NR Lagrangian, and will add to contributions that correspond to a polarizability shift in the energy of the NR particle. It is shown in Refs. Lee and Tiburzi (2014a, b) that by keeping track of these terms, inconsistencies that are observed in the second-order energy shifts of spin-0 and spin- particles in uniform external electric fields can be resolved. Although we do not explicitly work out the polarizability contributions in this paper, we expect the same mechanism to be in place with our framework for the case of spin-1 fields.

Before concluding the discussion of the semi-relativistic Lagrangian, it is worth pointing out that a number of pathologies have been noted in literature for relativistic theories of massive spin-1 (and higher) particles in background (EM or gravitational) fields. One issue that is most relevant to our discussion here is the emergence of superluminal modes from nonminimal couplings (such as quadrupole coupling) to EM fields, as noted by Velo and Zwanziger Velo and Zwanziger (1969). However, as is discussed in Ref. Porrati and Rahman (2008), the acasuality arising from nonminimal interactions are manifest as singularities (that can not be removed by any field redefinition) when one takes the limit. Therefore, the pathologies associated with these modes arise at a scale which is comparable or higher than the mass of the vector particle. Since the effective theory for nonminimal couplings already assumes a cutoff scale of , these pathologies are not relevant in our discussions. Thus, there in no contradiction to the existence of a well-defined low-energy effective theory that describes interactions of particles with any spin in external fields, as characterized by their EM moments, polarizabilities, and their higher static and quasi-static properties. With the assumption of weak external EM fields, other possibilities discussed in literature, such as the spontaneous EM superconductivity of vacuum due to the charged vector-particle condensation Ambjorn and Olesen (1989a, b); Chernodub (2011), will not be relevant in the framework of this paper.

In what follows, we carry out the matching to on-shell amplitudes at low-momentum transfer to constrain the values of the coefficients in the effective Lagrangian.

### ii.2 Matching the effective theory to on-shell amplitudes

Electromagnetic current and form-factor decomposition: The form-factor decomposition of the matrix elements of the EM current for spin-1 particles is well known, as is its connection to the EM multipole decomposition of NR charge and current densities, see for example Refs. Arnold et al. (1980); Lorce (2009). We briefly review the relevant discussions; this also serves as an introduction to our conventions.

Considering Lorentz invariance, vector-current conservation and charge-conjugation invariance, the most general form of the matrix element of an EM current, , between on-shell vector particles can be written as

 ⟨p′,λ′|Jμ(q)|p,λ⟩=−e ϵ(λ′)α(p′)†[F1(Q2)Pμgαβ+F2(Q2)(gμβqα−gμαqβ)−F3(Q2)2M2qαqβPμ]ϵ(λ)β(p),

where denotes the initial state of a vector particle with momentum and polarization , and denotes its final state with momentum and polarization , and where the momentum transferred to the final state due to interaction with the EM current is . denotes the polarization vector of the particle with momentum . For massive on-shell particles runs from to . Additionally, and we have defined . Lorentz structures proportional to , and have been discarded by utilizing the following conditions on the polarization vectors: and . Although the right-hand sides of these conditions are modified in external electric, , and magnetic, , fields by terms of , this will not matter for calculating on-shell matrix element as long as the adiabatic procedure described above Eq. (2) is in place to eliminate surface terms in the Lagrangian. By introducing the external fields adiabatically, the asymptotic “in” and “out” states of the theory are free and the corresponding polarization vectors satisfy the noninteracting relations.

To relate the form factors in Eq. (LABEL:eq:current-decomp) at low to the low-energy EM properties of the spin-1 particle, one may interpret this current matrix element, when expressed in the Breit frame, as multipole decomposition of the classical electric and magnetic charge densities. These decompositions are defined through Sachs form factors,

 ρE(q)≡∫d3xeiq⋅xJ0cl(x)=e2S∑l=0, l even(−Q24M2)l2√4π2l+1l!(2l−1)!!GEl(Q2)Yl0(^0), (5)
 ρM(q)≡∫d3xeiq⋅x∇⋅(x×Jcl(x))=e2S∑l=0, l odd(−Q24M2)l2√4π2l+1(l+1)l!(2l−1)!!GMl(Q2)Yl0(^0), (6)

where and are the Sachs electric and magnetic form factors, respectively, and denotes the value of spin. If the particle was infinitely massive, such interpretation of the relativistic relation (LABEL:eq:current-decomp) would have been exact, and the current matrix element would be precisely the Fourier transform of some classical charge or current density distributed inside the hadron. However, away from this limit, there are small recoil effects at low energies that are hard to characterize in the hadronic theory. In the Breit frame, in which the energy of the transferred photon, , is zero, such effects are minimal as the initial and final states have the same energy. In fact, as is well known, by expressing Eq. (LABEL:eq:current-decomp) in this frame, and by taking the moving-frame polarizations vectors satisfying and , this matrix element resembles the classical forms in Eqs. (5) and (6). This enables one to directly relate the form factors and , to Sachs form factors and . For spin- particles this results in the relations

 GE0(Q2) ≡ GC(Q2)=F1(Q2)+23Q24M2GE2(Q2), (7) GE2(Q2) ≡ GQ(Q2)=F1(Q2)−F2(Q2)+(1+Q24M2)F3(Q2), (8) GM1(Q2) ≡ GM(Q2)=F2(Q2). (9)

The electric charge, electric quadrupole moment and magnetic dipole moment are defined as the zero momentum transfer limit of the Coulomb, , quadrupole, , and magnetic, , Sachs form factors, respectively,

 Q0 ≡ GC(0)=F1(0), (10) ¯¯¯¯Q2 ≡ GQ(0)=F1(0)−F2(0)+F3(0), (11) ¯¯¯μ1 ≡ GM(0)=F2(0). (12)

is the particle’s quadrupole moment in units of , and denotes its magnetic moment in units of . Additionally, the mean-squared electric and magnetic charge radii can be expressed, respectively, as the derivatives of the Coulomb and magnetic form factors with respect to at zero momentum transfer,

 ≡ −6edGC(Q2)dQ2∣∣∣Q2=0=−6edF1(Q2)dQ2∣∣∣Q2=0−e¯¯¯¯Q2M2, (13) ≡ −6edGM(Q2)dQ2∣∣∣Q2=0=−6edF2(Q2)dQ2∣∣∣Q2=0. (14)

The quadrupole charge radius can be defined similarly from the derivative of the quadrupole Sachs form factor, however the dependence on this radius only occurs at higher orders in than is considered below.

One-photon amplitude from the effective theory: The next step is to evaluate the one-photon amplitude from the effective Lagrangian in Eq. (1). Explicitly, the following quantity

 Γαβμ≡−⟨Vα(p′)|L[V†,V,W†,W,A]|Vβ(p)Aμ(q)⟩, (15)

must be evaluated from the Lagrangian in Eq. (1) to match to Eq. (LABEL:eq:current-decomp). In obtaining this on-shell amplitude, the condition of the orthogonality of the momentum vectors to their corresponding polarization vectors can be used once again. Moreover, we use the EOM (see Eq. (2)) to convert fields to fields. A straightforward but slightly lengthy calculation gives

 Γαβμ = −e{[Q0+C(2)1q2M2]gαβPμ−C(2)3qαqβM2Pμ+ (16) [Q0+C(0)+(C(2)2−12C(2)3)q2M2](gμβqα−gμαqβ)}.

By comparing Eqs. (16) and (LABEL:eq:current-decomp), and with the aid of Eqs. (10)-(14), the following relations can be deduced,

 (17) (18) F3(Q2)=2C(2)3+O(Q2M2)=(−Q0+¯¯¯¯Q2+¯¯¯μ1)+O(Q2M2). (19)

These fully constrain the values of the four coefficients in the effective Lagrangian as following

 C(0)=¯¯¯μ1−Q0, (20) (21) (22) C(2)3=12(−Q0+¯¯¯¯Q2+¯¯¯μ1). (23)

With nonminimal interactions being constrained by the on-shell amplitudes, Eq. (1) can now be utilized to study properties of spin-1 particles in external fields. This is pursued in the next section through analyzing the EOM of the vector particle in time-independent but otherwise general and fields and their reduced forms in the NR limit.

## Iii Equations of Motion in External Fields and their Nonrelativistic Reductions

To be able to find the physical solutions of the EOM, one must first eliminate the redundant degrees of freedom of the spin-1 field in Eqs. (2) and (3). This can be established by eliminating and , with , in favor of the remaining 6 components of the fields, namely

 Vi  and  ϕi≡1MWi0. (24)

Our choice here is justified by noting that these latter are the only dynamical components of the fields (according to Eqs. (2) and (3), the time derivatives of and are absent from the EOM). From Eq. (2) it is manifest that the fields are related to the derivative of the fields

 Wij=DiVj−DjVi. (25)

It is also deduced from Eq. (3) that the field can be written in terms of the and fields,

 V0=−D⋅ϕM−ieC(0)M2E⋅V+O(1M3), (26)

where and . and refer to the scalar and vector EM potentials, respectively. The bold-faced quantities now represent ordinary three-vectors; as a result from here on we do not distinguish the upper and lower indices and let them all represent cartesian spatial indices. The terms that originate from the LHS of Eq. (3) contribute to at or higher. As can be seen from the EOM for the dynamical fields (see below), such terms give rise to contributions that are of or higher and will be neglected in our analysis. By taking into account these relations, and further by assuming time-independent external fields, the coupled EOM for the and fields can be written as

 idϕdt=MV+eQ0φϕ+1MD×D×V−ieC(0)MB×V+eC(0)M2E(D⋅ϕ) −2eC(2)1M2[(¯¯¯¯¯∇⋅E)ϕ+iM(¯¯¯¯¯∇×B⋅D)V]−ieC(2)2M3(¯¯¯¯¯∇2B)×V+eC(2)3M2[(ϕ⋅¯¯¯¯¯∇)E+ ¯¯¯¯¯∇(E⋅ϕ)+iM¯¯¯¯¯∇(B×D)⋅V+iM¯¯¯¯¯∇k(B×D)Vk−iM(V⋅¯¯¯¯¯∇)(¯¯¯¯¯∇×B)]+O(1M4,F2), (27)
 idVdt=Mϕ+eQ0φV−1MD(D⋅ϕ)−eC(0)M2D(E⋅V)+O(1M4,F2), (28)

where we have transformed the field to . The line over the derivatives indicates that the operator acts solely on the electric or magnetic field and not on the spin-1 fields following them.

These equations can be cast into an elegant matrix form. This can be achieved by introducing the following matrices

 S1=⎛⎜⎝00000−i0i0⎞⎟⎠,   S2=⎛⎜⎝00i000−i00⎞⎟⎠,   S3=⎛⎜⎝0−i0i00000⎞⎟⎠, (29)

with the properties: and , where is the three-dimensional Levi-Civita tensor. These matrices are closely related to the notion of spin in a NR theory as will become clear shortly.111111These are the analogues of Pauli matrices for spin- particles. In the following, the EOM are further analyzed by separating the case of electric and magnetic fields. This is solely to keep the presentation tractable, and the results for the case of nonvanishing electric and magnetic fields can be straightforwardly obtained following the same procedure.

### iii.1 An external electric field

For the case of an electric field with no time variation, the EOM for the and fields can be rewritten as

 idϕdt=MV+eQ0φϕ−1M(S⋅D)2V+eC(0)M2[E⋅D−SiSjEjDi]ϕ−2eC(2)1M2(¯¯¯¯¯∇⋅E)ϕ+ (30)
 idVdt=Mϕ+eQ0φV−1M[D2−(S⋅D)2]ϕ−eC(0)M2[D⋅E−SjSiDiEj]V+O(1M4,F2), (31)

with the aid of spin matrices in Eq. (29). These two equations can be represented by a single EOM for a 6-component vector, conveniently defined as

 ψ≡1√2(ϕ+Vϕ−V). (32)

This equation resembles a Schrödinger equation for the field , 121212For this wavefunction, the expectation values of operators are defined by (33) This imposes the condition of pseudo-Hermiticity on the Hamiltonian, , which is clearly the case for the Hamiltonians in Eqs. (35) and (48). See Ref. Case (1954) for more details.

 iddtψ=ˆH(E)SRψ, (34)

where the semi-relativistic Hamiltonian is

 ˆH(E)SR=Mσ3+eQ0ˆφ+(σ3+iσ2)ˆπ22M−iσ2M(S⋅ˆπ)2+e2M2(1+σ1)× [iC(0)[ˆE⋅ˆπ−SiSjˆEjˆπi]−2C(2)1(¯¯¯¯¯∇⋅ˆE)+2C(2)3[¯¯¯¯¯∇⋅ˆE−12(SiSj+SjSi)¯¯¯¯¯∇iˆEj]] −ieC(0)2M2(1−σ1)[ˆπ⋅ˆE−SiSjˆπjˆEi]+O(1M4,F2). (35)

is the conjugate momentum operator corresponding to the spatial covariant derivative, , and the coordinate is consequently promoted to a quantum-mechanical operator, (as is any space-dependent function such as the electric field). The s are the Pauli matrices and act either on an implicit unity matrix or the spin-1 matrices through a direct multiplication.

The Hamiltonian in Eq. (35) is comprised of

 ˆHSR=ˆE(−1)+ˆE(0)+⋯+ˆO(1)+ˆO(2)+…, (36)

where and denote operators that are proportional to (even) and (odd), respectively. The superscript on these operators denote the order at which they contribute in a expansion. The odd operators couple the upper and lower components of the wavefunction in the EOMs. These equations can be decoupled order by order in the expansion using the familiar Foldy-Wouthuysen-Case (FWC) transformation Foldy and Wouthuysen (1950); Case (1954)). Explicitly, one has

 ˆH′=U(1)−1ˆHSRU(1), (37)

where the unitary transformation

 U(1)≡eiˆS(1)≡e−σ32MˆO(1), (38)

removes the odd terms at in the transformed Hamiltonian, , leaving only the odd terms that are of or higher. The next transformation,

 U(2)≡eiˆS(2)≡e−σ32MˆO(2), (39)

takes the odd operators in and builds a new Hamiltonian, , that is free of odd terms also at ,

 ˆH′′=U(2)−1ˆH′U(2). (40)

By iteratively performing this transformation, all the odd operators can be eliminated up to the order one desires. Through this procedure, the NR reduction of the semi-relativistic theory can be systematically obtained.

Following the above procedure, we find that the NR Hamiltonian for the case of a nonzero field up to is131313A useful formula is the Baker-Campbell-Hausdorff relation,

 ˆH(E)NR = Mσ3+eQ0φ+σ3ˆπ22M−σ3ˆπ48M3−σ3(S.ˆπ)42M3+σ3{ˆπ2,(S⋅ˆπ)2}4M3 (41) −eC(0)4M2[S⋅(ˆE×ˆπ)−S⋅(ˆπ×ˆE)]−e(C(0)+6C(2)1−2C(2)3)6M2¯¯¯¯¯∇⋅ˆE −e(−C(0)+2C(2)3)4M2[SiSj+SjSi−23S2δij]¯¯¯¯¯∇iˆEj+O(1M4,F2).

Note that, as expected, this Hamiltonian is invariant under parity and time-reversal, and is no longer proportional to and . Additionally, by utilizing the matching conditions in Eq. (20), (21) and (23), one finds

 C(0) = ¯¯¯μ1−Q0, (42) C(0)+6C(2)1−2C(2)3 = (43) −C(0)+2C(2)3 = ¯¯¯¯Q2. (44)

Since the most general effective Lagrangian was used, with low-energy coefficients that are directly matched to the low-energy EM properties of the spin-1 particle, the expected NR interactions are automatically produced with the desired coefficients: the value of gives the correct coefficient of the spin-orbit interaction in Eq. (41). Moreover, the coefficients of the Darwin term, , and the quadrupole interaction, , are correctly produced to be proportional to the particle’s mean-squared electric charge radius and the quadrupole moment, respectively.

The coefficient of the Darwin (contact) term we have obtained here differs that obtained by Young and Bludman Young and Bludman (1963) which is found to be (this reference assumes ). This is only a definitional issue as if one defines the electric charge radius in Eq. (13) to be the derivative of the form factor with respect to at (instead of the derivative of the Sachs form factor, , that has been adopted here), both results agree.141414We note however that from a physical point of view, these are the Sachs form factors that are directly related to the NR charge and current distributions inside the hadrons, see Eqs. (5) and (6), and so the current definitions appear more natural (for a discussion of different definitions and associated confusions see Ref. Friar et al. (1997)). With our definition of the charge radius, the coefficient of the Darwin term for spin-0 and spin-1 particles Lee and Tiburzi (2014a) turns out to be the same, both having the value of , which is a convenient feature. After accounting for this difference, the Hamiltonian in Eq. (41) is in complete agreement with those presented in Refs. Sakata and Taketani (1940); Case (1954); Young and Bludman (1963), and extends the results in the literature by including all the operators at . The NR Hamiltonian in Eq. (41) applies straightforwardly to scalar particles in an external electric field by setting .

### iii.2 An external magnetic field

Eqs. (27) and (28) for the case of an external magnetic field that is constant in time can be rewritten as

 iddtϕ=MV+eQ0φϕ−1M(S⋅D)2V−eC(0)M(S⋅B)V−2eC(2)1M3(S⋅¯¯¯¯¯∇)(B⋅D)V −eC(2)2M3¯¯¯¯¯∇2(S⋅B)V+2eC(2)3M3[¯¯¯¯¯∇k(S⋅B)Dk−12(SiSj+SjSi)¯¯¯¯¯∇j(S⋅B)Di]V −eC(2)3M3[¯¯¯¯¯∇k(S⋅¯¯¯¯¯∇)Bk−SiSj¯¯¯¯¯∇i(S⋅¯¯¯¯¯∇)Bj]V+O(1M4,F2), (45)
 iddtV=Mϕ+eQ0φV−1M[D2−(S⋅D)2]ϕ−eQ0M(S⋅B)ϕ+O(1M4,F2), (46)

with the help of spin-1 matrices in Eq. (29). In terms of the 6-component field introduced in Eq. (32), the EOM reads

 iddtψ=ˆH(B)SRψ, (47)

with the semi-relativistic Hamiltonian

 ˆH(B)SR=Mσ3+eQ0ˆφ+(σ3+iσ2)ˆπ22M−iσ2M(S⋅ˆπ)2−(σ3−iσ2)eC(0)2M(S⋅ˆB) −(σ3+iσ2)eQ02M(S⋅ˆB)−(σ3−iσ2)e2M3[2iC(2)1(S⋅¯¯¯¯¯∇)(ˆB⋅ˆπ)+C(2)2¯¯¯¯¯∇2(S⋅ˆB) −2iC(2)3[¯¯¯¯¯∇k(S⋅ˆB)ˆπk−12(SiSj+SjSi)¯¯¯¯¯∇i(S⋅ˆB)ˆπj]+ C(2)3[¯¯¯¯¯∇k(S⋅¯¯¯¯¯∇)ˆBk−SiSj¯¯¯¯¯∇i(S⋅¯¯¯¯¯∇)ˆBj]]+O(1M4,F2). (48)

The decoupling of the EOM for the upper and lower three components of can be performed via the FWC procedure as detailed above. The result is

 ˆH(B)<