Composite Vector Particles in External Electromagnetic Fields
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.
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.
An alternative method, that has advantages over the aforementioned methods with regard to its simplicity, and potentially its computational costs, is the background field method. In this approach, a background EM field can be introduced in a LQCD calculation by imposing the gauge links onto the gauge links.444In order to reduce the computational cost, one may introduce the gauge links solely in the valence-quark sector of QCD. With this approximation, one can only reliably study those EM properties of the state that do not receive contributions from the sea quarks (receive no disconnected contributions). This is motivated by original experimental determinations of the static EM properties of hadrons and nuclei in external EM fields. By measuring the difference in the energy of the system with and without the background fields, and by matching to the knowledge of the Hamiltonian of the system deduced from the appropriate effective hadronic theory Caswell and Lepage (1986); Labelle (1992); Labelle et al. (1997); Kaplan et al. (1998a, b); Chen et al. (1999); Detmold and Savage (2004); Detmold et al. (2006, 2009); Hill et al. (2013); Lee and Tiburzi (2014a, b), the parameters of the low-energy Hamiltonian, i.e., those characterizing the coupling of the composite hadron to external fields, can be systematically constrained. This procedure has been successfully implemented to determine the magnetic moments of single hadrons and their electric and magnetic polarizabilities Bernard et al. (1982); Martinelli et al. (1982); Fiebig et al. (1989); Christensen et al. (2005); Lee et al. (2005, 2006); Detmold et al. (2006); Aubin et al. (2009); Detmold et al. (2009, 2010); Primer et al. (2014); Lujan et al. (2014). The utility of this method in accessing information about the structure of nuclei has been demonstrated recently through a determination of the magnetic moments and polarzabilities of nuclei with atomic number (at an unphysically heavy light-quark mass) Beane et al. (2014); Chang et al. (2015). It is desirable to gain further insight into the structure of these nuclei by studying their charge radii and quadrupole moments (for nuclei with spin ). These quantities require new developments that extend the implementation of uniform background fields to the case of nonuniform fields. We have recently presented such developments in Ref. Davoudi and Detmold (2015), providing the recipe for implementing general nonuniform background fields that satisfy the periodicity of a FV calculation.555See Refs. Lee and Alexandru (2011); Engelhardt (2011) for previous implementations of selected nonuniform, but nonperiodic, background EM fields in LQCD calculations of spin polarizabilities of the nucleon, and Ref. Bali and Endrodi (2015) for a periodic implementation of a plane-wave EM field in a LQCD calculation of the hadronic vacuum polarization function. In the present paper, motivated by interest in extracting the quadrupole moment of the deuteron, we provide the theoretical framework for performing a systematic matching between a suitable hadronic theory for spin-1 fields and the corresponding LQCD calculations in nonuniform background fields. Although LQCD studies of partial-wave mixing in the two-nucleon coupled channel can also reveal the noncentral feature of nuclear forces as demonstrated in Refs. Briceno et al. (2013); Briceño et al. (2013); Orginos et al. (2015), only a direct evaluation can incorporate the short-distance contributions to the quadrupole moment Kaplan et al. (1999); Chen et al. (1999).666Here we must distinguish the mass quadrupole moment of the deuteron from its electric quadrupole moment. It is the former that may be related to the S/D mixing in the deuteron channel. Although these two moments are comparable in the physical world, this might not be the case necessarily at unphysical values of quark masses. Other phenomenologically interesting quantities such as the (electric and magnetic) charge radii, which have been calculated so far through studies of the momentum dependence of the form factors,777See Ref. de Divitiis et al. (2012) for an alternative method to extract the form factors at zero momentum transfer by evaluating the derivatives of the correlation functions with respect to external momenta. This method circumvents the need for an extrapolation to zero momentum transfer, and has been extended in Ref. Tiburzi (2014a) to access the charge radii. can be also accessed via the nonuniform field technique. This formalism is equally applicable to the case of scalar and vector mesons so as long as they are nearly stable with regard to strong and EM interactions.888This assumption remains justified for several vector resonances such as the meson at heavy quark masses.
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.
To match to lattice correlation functions, it is of practical convenience to first deduce an effective NR Hamiltonian via the standard procedure of Foldy, Wouthuysen and Case Foldy and Wouthuysen (1950); Case (1954), as presented in Sec. III. We derive the quantum-mechanical wavefunctions of spin-1 particles in a linearly varying electric field (in space) and their corresponding Green’s functions in Sec. V, and show that, for a particular choice of the field, they are the Landau-level wavefunctions of a particle trapped in a harmonic potential. Despite their simple form, these NR Green’s functions can not be directly matched to LQCD correlation functions, unless a NR transformation is performed on the correlation functions, or alternatively, an inverse transformation is applied to NR Green’s functions, as discussed in Sec. V.1. This leads to at least two practical strategies to constrain the EM couplings of the low-energy theory, namely the quadrupole moment and the electric charge radius, as are presented in Secs. V.1 and V.2: one may try to match the transformed correlation function to the NR Green’s function directly, or alternatively, by projecting the NR Green’s functions onto given Landau eigenstates to identify the NR energy eigenvalues, and match them to the NR limit of energies extracted from the long-(Euclidean) time behavior of (spatially projected) LQCD correlation functions. Finally, the extracted quadrupole moment and charge radius must be extrapolated to their infinite-volume values by performing calculations in multiple volumes, or by determining their volume dependencies through an effective theory that is sensitive to the substructure of the hadron or nuclei, (e.g., chiral perturbation theory in the former case and pionless EFT in the latter). By inputting the knowledge of the charge radius and the quadrupole moment of the deuteron, we have investigated the range of validity of the results obtained in this paper under a single-particle effective theory of the deuteron given various electric field choices. The viability of an extraction of the deuteron’s quadruple moment and the charge radius within the framework of this paper from future LQCD calculations is then discussed, as presented in Sec. V.3. We conclude in Sec. VI by summarizing the results and commenting on future extensions. Additionally, the paper includes two appendices: appendix A is devoted to clarify the gauge dependency of the relativistic Green’s functions of Sec. IV, and Appendix B discusses the relation between the relativistic and NR Green’s functions through an example.
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
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
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
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,
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
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,
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,
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
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
These fully constrain the values of the four coefficients in the effective Lagrangian as following
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
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
It is also deduced from Eq. (3) that the field can be written in terms of the and fields,
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
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
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
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
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.
where the semi-relativistic Hamiltonian is
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
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
where the unitary transformation
removes the odd terms at in the transformed Hamiltonian, , leaving only the odd terms that are of or higher. The next transformation,
takes the odd operators in and builds a new Hamiltonian, , that is free of odd terms also at ,
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,
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
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
with the semi-relativistic Hamiltonian
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