Background Electromagnetic Fields and NRQED Matching: Scalar Case
aa r X i v : . [ h e p - ph ] D ec Background Electromagnetic Fields and NRQED Matching: Scalar Case
Jong-Wan Lee ∗ and Brian C. Tiburzi
1, 2, 3, † Department of Physics, The City College of New York, New York, NY 10031, USA Graduate School and University Center, The City University of New York, New York, NY 10016, USA RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA (Dated: October 16, 2018)The low-energy structure of hadrons can be described systematically using effective field theory,and the parameters of the effective theory can be determined from lattice QCD computations. Re-cent work, however, points to inconsistencies between the background field method in lattice QCDand effective field theory matching conditions. We show that the background field problem neces-sitates inclusion of operators related by equations of motion. In the presence of time-dependentelectromagnetic fields, for example, such operators modify Green’s functions, thereby complicatingthe isolation of hadronic parameters which enter on-shell scattering amplitudes. The particularlysimple case of a scalar hadron coupled to uniform electromagnetic fields is investigated in detail.At the level of the relativistic effective theory, operators related by equations of motion are demon-strated to be innocuous. The same result does not hold in the non-relativistic effective theory,and inconsistencies in matching are resolved by carefully treating operators related by equations ofmotion. As uniform external fields potentially allow for surface terms, the problem is additionallyanalyzed on a torus where such terms are absent. Finite-size corrections are derived for chargedscalar correlation functions in uniform electric fields as a useful byproduct.
PACS numbers: 12.39.Hg, 13.40.Gp, 13.60.Fz, 14.20.Dh
I. INTRODUCTION
The electromagnetic structure of hadrons can be deter-mined directly from photon-hadron scattering cross sec-tions, but also affects low-energy quantities measured inhigh-precision experiments. A prominent example thathas sparked considerable recent interest is the protoncharge radius, which is a quantity appearing in the lowmomentum-transfer expansion of the proton electric formfactor. While the proton charge radius can be extractedfrom electron-proton scattering cross sections, this quan-tity additionally gives rise to the leading finite-size ef-fect in the spectrum of hydrogenic atoms. From high-precision measurements of the muonic hydrogen spec-trum [1, 2], the extracted proton radius is discrepantwith that from scattering data at the 7 σ level, see [3]for a comprehensive review. The systematic and unified treatment of low-energyhadron structure is afforded by effective field theory tech-niques. The description of proton-size effects in non-relativistic quantum electrodynamics (NRQED), for ex-ample, has been given in [8]. In the effective hadronictheory, low-energy interactions are systematically written ∗ [email protected] † [email protected] Another example, which is directly related to the present work,concerns the electromagnetic structure of the pion. Charged pionelectromagnetic polarizabilities determined from chiral pertur-bation theory [4] are discrepant with scattering experiments bya factor of two [5], which corresponds to ∼ . σ . These low-energy quantities also appear in high-precision physics, namelyas hadronic corrections to the anomalous magnetic moment ofthe muon [6, 7]. down with parameters that encompass hadronic struc-ture. Effective field theory matching allows one to relatethe universal low-energy parameters to physical observ-ables. In this way, one sees that the same parameterswhich enter the description of scattering cross sections,also enter high-precision low-energy quantities, such asproton-size corrections to the muonic hydrogen spec-trum. In principle, the parameters entering the effectivehadronic theory can be computed using lattice QCD, andsteps in this direction have been made. The present workconcerns the extension of effective field theory matchingto the case of background fields. To be clear, we findno problems with effective field theory matching of S -matrix elements, however, the extension of effective fieldtheory matching to theories in background fields involvesa subtlety.Background field calculations in lattice QCD repre-sent a fruitful method to determine hadronic properties,see [9–24]. In particular, electromagnetic polarizabilitiescan be accessed using the background field method on thelattice, while photon-hadron scattering computations arebeyond the current and foreseeable reach of lattice QCD.There is a theoretical need to understand the relationbetween parameters extracted from background field lat-tice calculations and those reported by the Particle DataGroup. In this respect, various groups calculating theneutron electric polarizability, for example, are not de-termining the same quantity. Before this issue can beaddressed, we must first understand how to match effec-tive theories in external fields.In the present work, we expose a subtlety in matchinghadronic effective field theories in electromagnetic fields.To highlight this subtlety, let us point to an inconsis-tency that results from incorrect matching conditions inexternal fields. Applying the NRQED method of [25] todetermine the initial energy shift of a charged scalar ina uniform electric field, we obtain the result (to be dis-cussed in Sec. IV C 2 below)∆ E = − (cid:20) πα E − Z M < r > (cid:21) ~E , (1)where M , α E , and < r > are the scalar’s mass, electricpolarizability, and charge radius, respectively. Appear-ance of the charge radius in the energy shift is rather sur-prising, because such virtual photon contributions shouldbe absent on strictly physical grounds. This result is tobe contrasted with the initial energy shift in the relativis-tic case [18] ∆ E = −
12 4 πα E ~E , (2)which turns out to be the correct result. Resolution ofthis inconsistency is one of the goals of this work. Wefind that resolution is possible by extending matching tothe Green’s functions, which requires retaining effectivefield theory operators related by equations of motion. Asa result, the matching of S -matrix elements determinedin [25] is completely unaffected, but can be modified inexternal fields. The modification accounts for the differ-ence between Eqs. (1) and (2).Throughout we consider the dynamics of a compositescalar coupled to electromagnetic fields, and our presen-tation is organized as follows. We begin in Sec. II with ademonstration that operators related by equations of mo-tion can modify Green’s functions. For time-dependentelectromagnetic fields, we provide an illustrative exam-ple that points to an obstruction in the extraction ofon-shell properties using background field correlators inlattice QCD. Despite this general obstruction, we show inSec. III that the particular case of a charged, relativistic,scalar particle coupled to uniform electromagnetic fieldshappens to be immune to such difficulties. To facilitatematching with the non-relativistic theory, we addition-ally compute one- and two-photon processes to relatethe parameters of the relativistic theory to observables,and obtain the correlation functions of charged relativis-tic scalars in uniform electric and magnetic fields. Nextin Sec. IV, we write down the non-relativistic theory ofa composite scalar using HQET power counting. We Here is where the exact definition of the initial energy becomesimportant. Technically we must determine the non-relativisticexpansion of the relativistic correlator to compare the initial en-ergy with that appearing in Eq. (1). Performing this expan-sion (also to be discussed in Sec. IV C 2 below), we obtain thenon-relativistic initial energy shift ∆ E = − h πα E + Z M i ~E .Notice that the additional term in the energy shift is already nec-essarily contained in the relativistic correlator employed in theanalysis of [18]. Here we employ the acronym HQET for any effective theorythat utilizes an expansion in inverse powers of a particle’s mass.This power counting is, of course, shared by its namesake, heavyquark effective theory. argue for the inclusion of an additional operator whichordinarily would be eliminated by use of the equationsof motion. The operator is shown to modify chargedscalar Green’s functions in uniform electric fields, andis required so that Green’s functions match between rel-ativistic and non-relativistic theories. To compute thenon-relativistic propagator, we employ both HQET andNRQED power counting, and obtain results consistentwith the non-relativistic reduction of the relativistic the-ory only when operators related by equations of mo-tion are retained in the non-relativistic theory. As a fi-nal check, we expand the relativistic theory in powersof the scalar’s mass for a brute-force determination ofthe matching coefficients at the level of the action. Cer-tain technical details are relegated to Appendices. Theproblem of a charged scalar hadron in an electric fieldis formulated on a Euclidean torus in Appendix A, andprecludes the possibility of surface terms. The NRQEDexpansion of the relativistic charged scalar propagator isdetermined in Appendix B. In the last section of the maintext, Sec. V, we summarize our findings.
II. OPERATORS RELATED BY EQUATIONSOF MOTION
Ordinarily operators related by equations of motionare redundant, and can be dropped from an effective fieldtheory. This has the desirable consequence of reducingthe number of low-energy parameters, which is essentialin economically parameterizing the model-independentphysics relevant at a given energy scale. In external fields,however, the issue becomes subtle, and our goal is toexpose the subtlety first in the context of a simplifiedexample.Consider the following toy-model Lagrange density fora charged composite scalar L = D µ Φ † D µ Φ − M Φ † Φ + C M Φ † Φ ∂ F + C M F (cid:0) D µ Φ † D µ Φ − M Φ † Φ (cid:1) , (3)where F µν is the electromagnetic field-strength tensor,with C and C as the dimensionless low-energy constantsof this model. Non-minimal photon couplings are al-lowed because we assume Φ is a composite particle withcharged constituents. The electromagnetic gauge covari-ant derivatives have the action D µ Φ = ∂ µ Φ + iZA µ Φ ,D µ Φ † = ∂ µ Φ † − iZA µ Φ † . (4)No power counting has been utilized in writing Eq. (3); inthis section, we merely select operators to illustrate ourpoint. It will prove useful to treat the electromagneticcoupling as small. To this end, we consider the param-eters C and C to be proportional to the square of theelectric charge, α = e π , and we will drop terms of O ( α )in what follows.For processes with only on-shell Φ states, it turns outthat observables, such as the amplitude for the Comp-ton scattering process γ + Φ → γ + Φ, depend only on aparticular linear combination of low-energy parameters, C + C . For virtual Φ states, the diagrammatic analy-sis is more involved; but, off-shell contributions from theequation-of-motion operator generally can be removed inthe renormalization of the theory. Because the diagram-matic approach is cumbersome, we handle the removalof redundant operators by employing field redefinitions,see [26] and references therein. For the toy model, weinvoke the field redefinitionΦ = (cid:18) − C M F (cid:19) Φ ′ , (5)which corresponds to dressing the scalar field with pho-tons. After field redefinition, the theory is described bythe Lagrange density L ′ = D µ Φ ′ D µ Φ ′ − M Φ ′† Φ ′ + C ′ M Φ ′† Φ ′ ∂ F + O ( α ) , (6)with C ′ = C + C . The operator related by equations ofmotion has now been removed. The coefficient C ′ canbe chosen so that Eq. (6) reproduces S -matrix elementsfor processes involving the composite scalar and photons.This procedure exposes that on-shell processes dependonly on C ′ . In Eq. (3), additional dependence on theparameter C that can arise from virtual Φ contributionsin loop diagrams must be cancelled by the counter-termsnecessary to renormalize the theory. In this way, thetheories described by Eqs. (3) and (6) are equivalent.Now consider the toy-model Lagrange density for thecase where F µν is a time-dependent external field. Theexplicit time-dependence introduced eliminates the pos-sibility of an on-shell condition. As a result, one cannotappeal to a renormalization prescription to fix the behav-ior of the two-point function at the single-particle pole,because there are no such poles. Consequently the pa-rameters C and C can be resolved at the level of theGreen’s function. Suppose we start with the reducedtheory described by Eq. (6). The propagator for Φ ′ wewrite as G ′ ( x, y ), with G ′ ( x, y ) = h | T (cid:8) Φ ′ ( x )Φ ′† ( y ) (cid:9) | i . (7)Starting with the theory in Eq. (3), on the other hand,the propagator for Φ we write as G ( x, y ). This propagator Notice that the charged particle Green’s function is gauge de-pendent. Implicitly included in the choice of external field isthe gauge, which is then fixed. We will derive results below forparticularly simple gauge choices; results in other gauges cansimilarly be derived. While appending an electromagnetic gaugelink between operators in the two-point function will lead one togauge invariant Green’s functions, these Green’s functions willthen depend on the path chosen to link the operators. Pathdependence arises because flux threads loops transverse to theelectromagnetic fields. can be deduced simply by utilizing the field redefinitionin Eq. (5) G ( x, y ) = h | T (cid:8) Φ( x )Φ † ( y ) (cid:9) | i = (cid:20) − C M [ F ( x ) + F ( y )] (cid:21) G ′ ( x, y ) , (8)where we have dropped contributions that are of order α . Notice this correlator necessarily has different timedependence. The difference between propagators has an importantconsequence for the background field method in latticeQCD computations. After computing correlation func-tions of the scalar particle on the lattice, we must matchthe behavior of the lattice-determined correlator withthe prediction from an effective hadronic theory. With-out an on-shell condition, this step is essential becausethe method hinges on the effective theory being able toreproduce the time dependence of the lattice correlatordata. We cannot simply assume that the external fieldpropagator will be given by Eq. (7). The correspondingeffective theory has been reduced by a field redefinition.The most general effective theory should start with allpossible operators, including operators that one wouldnormally remove via field redefinitions. Such a generaltheory will have different Green’s functions than its cor-responding reduced theory; and, in this way, two suchtheories are hence no longer equivalent.From the time dependence of the Green’s function,parameters associated with on-shell particles can be ex-tracted. In our toy-model example, one has access toboth parameters C ′ and C from the propagator, Eq. (8).In the corresponding effective theory, Eq. (3), only theparameter C ′ contributes to on-shell properties of Φ.Thus when considering time-dependent external field cor-relation functions, we must retain operators ordinarily One can also compute the Φ propagator directly by treatingthe operator with coefficient C in Eq. (3) as a perturbation. Inthis approach, one works in coordinate space, and utilizes theGreen’s function identity ( D y + M ) G ( x, y ) = iδ ( x − y ). Theresulting propagator is the same as Eq. (8), and eliminates pos-sible additional factors that could appear from carrying out thefield redefinition carefully at the level of the functional integral. It is useful to imagine the case of a time-independent magneticfield, for which one has an on-shell condition. In this case, theoperator with coefficient C does not modify the spectrum of thetheory as can be shown by taking the temporal Fourier transformof Eq. (8), G ( ~x, ~y | E ) = (cid:20) − C M [ F ( ~x ) + F ( ~y )] (cid:21) G ′ ( ~x, ~y | E ) . The residues at each energy pole, however, are different betweenthe reduced and unreduced theories. This difference reflects per-turbative corrections to the coordinate wavefunctions of energyeigenstates. Without the explicit coordinate dependence intro-duced by the magnetic field in this case, one could impose thestandard wavefunction renormalization condition which wouldlead to on-shell Green’s functions that match, both poles andresidues. removed by the equations of motion. Such operators af-fect the time dependence of Green’s functions, and theircoefficients are generally not related to physical proper-ties of the particle. In light of this observation, we con-sider effective theories of relativistic and non-relativisticcharged scalars in uniform electric and magnetic fields,and address possible contributions from operators relatedby the equations of motion.
III. RELATIVISTIC SCALAR QED
Moving on from the toy-model example, we considerthe fully relativistic action for a charged composite scalarΦ interacting with electromagnetic fields. This providesan effective hadronic theory that will ultimately be re-duced to a non-relativistic theory below, and has usefulapplications to pions and Helium–4. We detail the case ofthe scalar propagating in uniform electromagnetic fields;and, from our discussion above, we need to address pos-sible contributions arising from operators related by theequations of motion.
A. Operators
To write down the effective theory for Φ, we accord-ingly enforce upon the action the usual C , P , and T invariance in addition to Lorentz and gauge invariance.The operators of this theory can be organized in powersof increasing mass dimension, however, we do not writedown all possible terms up to dimension eight. Instead,we keep only terms which turn out to be relevant in thenon-relativistic limit, specifically up to O ( M − ). Writingdown all of those such terms, we have L = D µ Φ † D µ Φ − M Φ † Φ − C M F Φ † Φ + C M [ ∂ µ F µν ] J ν + C M T µν D µ Φ † D ν Φ − C M [ ∂ ∂ µ F µν ] J ν . (9)We employ T µν for the electromagnetic stress-energy ten-sor, T µν = F ρ { µ F ν } ρ , where the curly braces denote sym-metrization and trace subtraction, O { µν } = 12 (cid:18) O µν + O νµ − g µν O αα (cid:19) . (10)The vector current is given by J µ = i (cid:0) Φ † [ D µ Φ] − [ D µ Φ † ]Φ (cid:1) , (11)where, in order to remove ambiguity throughout, we haveadopted the convention that bracketed derivatives onlyact inside the square brackets. To obtain the theory corresponding to a neutral scalar hadron,such as the neutral pion, one sets Z = 0, imposes Φ † = Φ, In writing down the effective theory in Eq. (9), weassumed the absence of surface terms to eliminate op-erators. At dimension six, for example, we utilized theidentity ∂ µ ( F µν J ν ) = [ ∂ µ F µν ] J ν + 2 iF µν D µ Φ † D ν Φ+ ZF Φ † Φ , (12)to exclude the operator iF µν D µ Φ † D ν Φ. For a uniformexternal field extending over all spacetime, it is not im-mediately obvious that surface terms vanish due to thelinearly rising four-vector potential A µ required to obtainsuch electromagnetic fields. For lattice applications, werestrict such electromagnetic fields to a Euclidean torusin Appendix A. In that case, there are no surface termsin both relativistic and non-relativistic theories, and in-finite spacetime results are recovered exponentially fastin the spacetime volume.The relativistic theory in Eq. (9) has been written,moreover, without any operators related by equations ofmotion. For spacetime varying external fields, the effectof such operators on Green’s functions is generally rathercomplicated. Ultimately we are concerned with deter-mining the behavior of Green’s functions in the effectivetheory to compare with lattice QCD data computed inuniform electric and magnetic fields. To this end, we de-rive specific results for the case of uniform electric andmagnetic fields. Unlike the toy-model example above,there are valuable simplifications in this case.For external electromagnetic fields, the additionalterms required in the effective theory appear in the La-grange density∆ L = C M Φ † ( D + M ) Φ + C M Φ † ( D + M ) Φ+ C M F Φ † ( D + M )Φ . (13)These operators have been simplified using integration byparts. In particular, an integration by parts shows thatthe last operator is equivalent to the equation-of-motionoperator appearing in the toy-model Lagrange density,Eq. (3), under the assumption that the electromagneticfields are uniform. The operators appearing in Eq. (13)can be removed with a field redefinition having the formΦ = " C M ( D + M )+ C M ( D + M ) + C − C C M F Φ ′ . (14) and includes the correct normalization factors. Accordingly theoperators with coefficients C and C vanish, and the theoryonly depends on C and C . Finally due to isospin breaking, thevalues of coefficients in the neutral hadron theory are generallyunrelated to those in the charged hadron theory. In terms of the redefined field, we accordingly have L + ∆ L = D µ Φ ′† D µ Φ ′ − M Φ ′† Φ ′ − C M F Φ ′† Φ ′ + C M T µν D µ Φ ′† D ν Φ ′ , (15)up to higher-order terms of mass-dimension ten. Noticethat the operators with coefficients C and C vanishin uniform external fields, and are not required in ourconsideration.The effect of the field redefinition Eq. (14) on Green’sfunctions happens to be innocuous. As in the toy-modelexample, we can compute the Φ propagator by first de-termining the Φ ′ propagator G ′ ( x, y ), and then appeal-ing to the field redefinition. Terms in the field redef-inition involving ( D + M ) n only produce contribu-tions to the Φ two-point function G ( x, y ) proportionalto (cid:0) D + M (cid:1) n − δ ( x − y ). Such singular contributionshave Φ and Φ † fields at the same spacetime point, andcan be removed by imposing a renormalization conditionon the vacuum energy. As a result, the Φ propagator hasthe form G ( x, y ) = (cid:18) − C − C C M F (cid:19) G ′ ( x, y ) , (16)for x µ = y µ . Thus the two-point functions in the reducedand unreduced theories only differ by an overall constant.For on-shell states, the overall constant can be fixed bythe wavefunction renormalization, i.e. the residue at thepole. As expected, the field redefinition does not changethe spectrum of the theory. In uniform electric fields,there is no on-shell condition for charged particles, how-ever, the overall constant crucially does not alter the timedependence of the correlation function. In this way, thereduced and unreduced theories yield the same predic-tion for the behavior of the correlation function, which isof practical importance for lattice QCD analyses. Based on our analysis, we can use the theory speci-fied by Eq. (9) to describe the dynamics of a chargedrelativistic scalar coupled non-minimally to electromag-netism. While additional terms, such as those in Eq. (13),are needed to determine the Green’s functions of thescalar propagating in external electromagnetic fields,these terms are not needed when we restrict our atten-tion to uniform fields. In a uniform magnetic field, thecorrelation functions are unchanged provided that wave-function renormalization has been accounted for in bothreduced and unreduced theories. In a uniform electricfield, which necessarily lacks the on-shell condition, theonly modification to the two-point function is an overallconstant. In lattice QCD computations, moreover, the overall normaliza-tion of the two-point correlation function is unknown. With thecontribution of the ground-state hadron isolated, the lattice cor-relator is proportional to the overlap factor between the chosenquark-level interpolating field and the ground-state hadron.
B. One- and Two-Photon Matching
To discuss matching between relativistic and non-relativistic theories below, it is efficacious to relate thelow-energy constants to observable quantities. The rela-tivistic scalar hadron theory given in Eq. (9) depends onfour unknown parameters, C – C . The operators withcoefficients C and C obviously only contribute to pro-cesses involving at least one virtual photon. To relatethese parameters to physical observables, we computeone- and two-photon processes. It is sufficient to treatprocesses with one virtual photon, and two real photonsin order to determine all four parameters.The scalar hadron’s interaction with a virtual pho-ton is described by the electromagnetic form factor, F ( q ), entering current matrix elements between thescalar hadron. These matrix elements have the form h Φ( p ′ ) | J µ e.m. | Φ( p ) i = ( p ′ + p ) µ F ( q ) , (17)on account of gauge invariance and Lorentz covariance.In the small momentum transfer limit, we may expandthe form factor to obtain F ( q ) = Z + 13! q < r > + 15! q < r > + · · · . (18)The form factor at vanishing momentum-transfer is con-strained by the Ward identity to be the total charge. Thefirst-order correction is conventionally parameterized bydefining a charge radius √ < r > , and the second-ordercorrection we define as being a higher moment of thecharge distribution, < r > . The physical interpretationof both of these quantities is complicated by relativisticeffects, however, one can identify them as moments ofthe transverse distribution of charge in the infinite mo-mentum frame [27]. Deriving the electromagnetic cur-rent from the relativistic action enables us to computethe scalar hadron’s form factor. In doing so, we find thesimple relations C M = 13! < r >, C M = 15! < r > . (19)The remaining two coefficients can be related to phys-ical observables by considering two-photon processes. Tothis end, we consider the real Compton scattering pro-cess, γ ( k ) + Φ( p ) → γ ( k ′ ) + Φ( p ′ ). Working in the lab-oratory frame, the forward and backward Compton am-plitudes are given in a low-energy expansion by [28] T ( θ = 0) = ~ε ′∗ · ~ε (cid:20) − Z M + 4 π ( α E + β M ) ω (cid:21) ,T ( θ = π ) = ~ε ′∗ · ~ε (cid:20) − Z M + 4 π ( α E − β M ) ωω ′ (cid:21) , (20)where α E and β M are the electric and magnetic polariz-abilities, respectively. Using the theory defined by Eq. (9)to compute the Compton amplitude at small photon en-ergy determines the values, C = πM ( α E − β M ) , C = 4 πM ( α E + β M ) . (21)Physically the Compton scattering cross section can bewritten as the coherent sum of contributions from pho-ton helicity preserving, ∆ λ = 0, and helicity flip, ∆ λ = 2,processes. The operator with coefficient C contributesexclusively to the former, while the operator with coeffi-cient C contributes exclusively to the latter. C. Uniform External Fields
To further aid in matching relativistic and non-relativistic theories below, we investigate the chargedparticle correlation functions in external magnetic andelectric fields. These external fields are chosen to be uni-form; and, because the correlation functions depend onthe gauge, particular gauges are employed. It is straight-forward to implement different gauge choices.
1. Magnetic Field
A charged particle propagating in a uniform magneticfield can be projected onto states of definite energy. Forthis case, we choose to align the magnetic field with the z -direction, and accordingly choose the gauge potential A µ = − Bx δ µ . The Φ propagator has an infinite num-ber of poles corresponding to the various Landau levels.This feature is best exhibited by employing Schwinger’sproper-time trick [29]. The coordinate wavefunction ofthe n -th Landau level, ψ n ( x ), allows us to project outthis energy eigenstate due to orthogonality [30]. Thiscan be seen, for example, by computing the propagatorprojected at the sink G ( n ) B ( t,
0) = Z d~x ψ ∗ n ( x ) h | Φ( ~x, t )Φ † ( ~ , | i B , (22)which has the simple behavior G ( n ) B ( t,
0) = Z n e − iE n t , as-suming that t >
0. The energy E n is given by E n = h M + | ZB | (2 n + 1) − πβ M M B i / , (23)where we have replaced M ( C − C ) = 4 πβ M M toexpress the energy in terms of the magnetic polarizabil-ity. The expansion of the energy in powers of M enablesstraightforward comparison with the non-relativistic the-ory.
2. Electric Field
For a charged particle in a uniform electric field, how-ever, the situation is more involved. Specifying the four-vector potential A µ = − Et δ µ , and rescaling the Φ fieldleads to the Lagrange density L = Φ † ~p =0 (cid:20) − ∂ ∂t − ( ZEt ) − M + 4 πα E M E (cid:21) Φ ~p =0 , (24) where we have dropped terms of order E , and projectedthe field onto vanishing three momentum ~p = 0. We havealso rewritten the combination of low-energy parameters, M ( C + 4 C ) = 4 πα E M , in favor of the electric polar-izability. The propagator resulting from Eq. (24) con-tains singularities associated with the real-time produc-tion of any number of particle-antiparticle pairs. This isthe Schwinger mechanism.To avoid the Schwinger mechanism altogether, we workin Euclidean space. This choice is further motivated bylattice QCD computations which are necessarily carriedout in Euclidean space. With t = − iτ , and E = i E , wehave the Euclidean action density L E = Φ † ~p =0 (cid:20) − ∂ ∂τ + ( Z E τ ) + M + 4 πα E M E (cid:21) Φ ~p =0 . (25)From this action density, one can compute the two-pointfunction G E ( τ,
0) = Z d~x h | Φ( ~x, τ )Φ † ( ~ , | i E , (26)where τ > G E ( τ,
0) = 12 Z ∞ ds r Z E π sinh Z E s e − (cid:16) Z E τ Z E s + E E s (cid:17) , (27)where the quantity E E can roughly be termed the rel-ativistic initial energy, cf . the behavior of Eq. (25) at τ = 0, and is given by E E = (cid:2) M + 4 πα E M E (cid:3) / . (28)Notice that the electric polarizability produces a positiveshift of the initial energy in Euclidean space.Due to the lack of energy eigenstates, the long-timebehavior of the correlator in Eq. (27) does not exhibitthe exponential decay that is characteristic of correlationfunctions in Euclidean space. The logarithmic deriva-tive of the correlator grows in Euclidean time, whichroughly corresponds to the particle acquiring energy fromthe electric field. Unfortunately the proper-time inte-gration cannot be performed in closed form. The non-relativistic reduction of this propagator will be carriedout, and compared with the propagator computed fromthe non-relativistic effective theory. IV. NON-RELATIVISTIC SCALAR QED
For sufficiently low energies, one can formulate the ef-fective theory of a charged composite scalar using HQETpower counting. This theory is organized in inverse pow-ers of the particle’s mass M , which is treated as a largeenergy scale, see [32]. In considering the dynamics ofa charged scalar in external electromagnetic fields, wemust address the effects of operators related by the non-relativistic equations of motion. In a uniform electricfield, we find the non-relativistic effective theory requiresan additional such operator. A. Action and Relativistic Invariance
To write down the non-relativistic theory, we considerthe most general Lagrange density for a charged com-posite scalar φ interacting with electromagnetic fields.We impose Hermiticity, and invariance under P , T ,and gauge transformations. Including all terms up to O ( M − ), we find L = φ † " iD + c ~D M + c D [ ~ ∇ · ~E ]8 M + c ~D M + ic M { D i , [ ~ ∇ × ~B ] } M + c A ~B − ~E M − c A ~E M + c X [ iD , ~D · ~E + ~E · ~D ]8 M + c X [ ~D , ~D · ~E + ~E · ~D ]16 M + c X { ~D , [ ~ ∇ · ~E ] } M + c X [ ~ ∇ ~ ∇ · ~E ]16 M + ic X { D i , ( ~E × ~B ) i } M φ. (29)In the non-relativistic theory, the gauge covariant deriva-tive is specified by D = ∂ + iZA ,D i = ∇ i − iZA i . (30)The electric and magnetic fields ~E and ~B are givenby standard expressions, ~E = − ∂ ~A − ~ ∇ A and ~B = ~ ∇ × ~A , respectively. It will be useful in what followsto recall the commutators of two covariant derivatives,[ D i , D j ] = − iZǫ ijk B k and [ D i , D ] = − iZE i . Note thatthe product i ~B is time-reversal even and the factor i ne-cessitates the anti-commutator structure to satisfy Her-miticity. The other time-reversal even quantity involvingthe magnetic field can be eliminated by using Maxwell’sequation, ∂ ~B = − ~ ∇ × ~E . Anti-commutator terms withtwo derivatives are Hermitian without a factor of i , while c X and c X terms require a commutator for Hermiticity.Not all of the operator coefficients appearing in theeffective theory are independent parameters, becauseLorentz invariance implies relations between different or-ders in the 1 /M expansion [33, 34]. Such relations canbe deduced by performing an infinitesimal boost, and de-manding invariance order-by-order in 1 /M . This is thevariational method detailed in [25]. Parameterizing theboost with momentum ~q , we have the variations δ ~D = ~q D /M, δD = ~q · ~D/M, along with δ ~B = − ~q × ~E/M, δ ~E = ~q × ~B/M. We also require the transformation property of the scalarfield, which can be written to O ( M − ) in the form φ ( x ) → e − i~q · ~x " A i~q · ~D M + B i~q · ~E M + C (cid:8) i~q · ~D, ~D (cid:9) M + D ~q · [ ~ ∇ × ~B ]8 M + E ǫ ijk q i (cid:8) B j , D k (cid:9) M + F (cid:8) D , ~q · ~E (cid:9) M φ ( x ) , (31)where the parameters A – F remain to be determined.Boost invariance can be enforced order-by-order in1 /M provided the field transformation is specified by theparameters A = 1 , B = c D + 2 c X , C = 1 ,D = − c M , E = − c X , F = c X . (32)Furthermore, the coefficients of operators in the effectivetheory must satisfy the relations c = c = 1 , c M = 12 c D , c X − c X = 12 ( Z + c D ) ,c X = 0 , c X = 2 Zc D − c A . (33)Taking into account these relations, there are five uncon-strained parameters in the effective theory.With the exception of the operator having coefficient c X , the operators enumerated in Eq. (29) are identi-cal to the spin-independent operators found in [25]. Foron-shell processes involving φ , we furthermore have theoperator equivalence φ † [ iD , ~D · ~E + ~E · ~D ]8 M φ eom = − φ † [ ~D , ~D · ~E + ~E · ~D ]16 M φ, (34)which arises from applying the HQET equations of mo-tion. Consequently, there are only four independent pa-rameters required to describe on-shell process. As we willsee, however, the remaining parameter c X is necessaryto describe the Green’s functions in a uniform electricfield.Notice we did not write down all possible operatorsrelated by the equations of motion. For example, opera-tors of the form φ † ( iD ) n φ for n > G ( t ′ , t ) bysingluar terms involving derivatives of delta functions,( D ) n − δ ( t ′ − t ). For this reason, these operators havebeen excluded. Beyond such operators, there are furtherterms, for example the operator ~E φ † iD φ , which canmodify the time dependence of Green’s functions in anon-constant electric field. We have omitted this opera-tor, however, because it only modifies Green’s functionsby an overall constant in uniform electric fields. In writ-ing Eq. (29), we are claiming that the operator with co-efficient c X is the only operator related by the equationsof motion that is required to address the case of uniformelectromagnetic fields at O ( M − ). The appearance of ad-ditional equation-of-motion operators for uniform fieldsat O ( M − ) has not been considered. B. One- and Two-Photon Matching
To determine the phenomenological values of the non-relativistic effective field theory coefficients, we performone- and two-photon matching similar to that carried outabove in the relativistic case. This is the scalar analogueof non-relativistic effective field theory matching carriedout in [25, 35]. The resulting matching conditions willadditionally establish the relations between relativisticand non-relativistic low-energy constants. As above, werestrict our attention to processes involving either onevirtual photon, or two real photons.For virtual photon scattering with the φ , we use therelativistic decomposition of the form factor given inEq. (18). Kinematically, the momentum transfer squaredhas the non-relativistic expansion q = − ~q + 14 M (cid:0) ~q + 2 ~q · ~p (cid:1) + · · · . (35)The matrix element of the charge density operator in turnhas the non-relativistic expansion h ~p ′ | J | ~p i = Z − ~q < r > + ~q < r > + ( ~q + 2 ~q · ~p ) M (cid:20) Z M < r > (cid:21) , (36)in an arbitrary frame. In the above expression, wehave accounted for the differing normalization betweenrelativistic and non-relativistic states, see Eq. (58) be-low. Computation of the same matrix element using theHQET action in Eq. (29), produces the relations c D = 43 M < r >, c X = 215 M < r >,c X − c X = 12 (cid:18) Z + 43 M < r > (cid:19) , c X = 0 . (37)The latter two relations are required by the impositionof Lorentz invariance, see Eq. (33). Matching the spatialcurrent matrix element in an arbitrary frame confirmsthe relation c M = c D .Evaluation of the real Compton scattering amplitudeis simplified in the non-relativistic limit. In the labo-ratory frame, the final-state photon frequency satisfiesthe condition ω ′ = ω + O ( ω/M ). Computing the Comp-ton amplitude up to O ( M − ) accuracy, we determine thematching conditions16 πM α E = Zc D − c A − c A , πM β M = c A , (38) which relate low-energy constants to the electric andmagnetic polarizabilities.From one- and two- photon processes, we have thus de-termined the four on-shell parameters of the effective the-ory in terms of physical observables. The parameter c X cannot be determined in this way, because physical pro-cesses only depend on the linear combination c X − c X , cf . Eq. (34). Comparing the matching conditions betweenrelativistic and non-relativistic theories enables us to re-late the low-energy constants of the two effective theories.From single-photon matching, we find the relations c D = 8 C , c X = 16 C , (39)which shows that these low-energy constants are deter-mined entirely from virtual photon couplings in the rela-tivistic theory. The two-photon matching conditions pro-duce the relations c A = 2( C − C ) , c A = 8(2 ZC − C ) . (40)Notice the parameter c A has a piece ∝ C that arisesfrom a relativistic operator contributing exclusively tovirtual photon processes. This produces exact cancela-tion of the c D term contributing to α E in Eq. (38), whichis required because α E can be determined from Comptonscattering with two real photons.The remaining on-shell parameters of the non-relativistic theory are constrained by Lorentz invariance.For completeness, the remaining relations between non-relativistic and relativistic low-energy constants are c M = 4 C , c X − c X = 12 ( Z + 8 C ) , c X = 8 C . (41)As far as on-shell processes are concerned, we can employthe effective theory in Eq. (29) omitting the operator withcoefficient c X . This is not the case when we consider theGreen’s functions in a uniform electric field. We now turnour attention to background electromagnetic fields. C. Uniform External Fields
We consider the non-relativistic effective theory inbackground electromagnetic fields. First we show thatthere are no complications for the case of a uniform mag-netic field. For a uniform electric field, we expose the dif-ficulty of dropping the operator with coefficient c X . Wethen determine this coefficient by matching Green’s func-tions calculated with HQET power counting and the cor-responding HQET expansion of the relativistic Green’sfunction. The matching condition for c X is verified byrepeating the Green’s function matching with NRQEDpower counting.
1. Magnetic Field
To match Green’s functions, let us first consider thecase of a uniform magnetic field specified, as above, bythe vector potential ~A = − x B ˆ x . In such an externalfield, the HQET action reduces to L = φ † ~p ⊥ =0 (cid:20) i∂ − H + H M + c A B M (cid:21) φ ~p ⊥ =0 , (42)up to terms of order M − . Notice we have projected ontothe sector of zero transverse momentum ~p ⊥ = ( p , p ) =0, for simplicity; and, H is the harmonic oscillator Hamil-tonian given by H = M (cid:2) − ∂ + ( ZB ) x (cid:3) . Expandingin the oscillator basis, we see the energy eigenvalues havethe form E NR n = | ZB | M ( n + 12 ) − ( ZB ) M ( n + 12 ) −
12 4 πβ M B , (43)having traded the low-energy constant c A for the mag-netic polarizability β M through the matching condition,Eq. (38). Comparing with the relativistic spectrum fromEq. (23), we see they agree E n − M = E NR n + O ( M − ) , (44)to the order we are working in the HQET expansion.Because the single-particle wave-functions of the Landaulevels also agree, the two-point correlation function calcu-lated in HQET matches with the non-relativistic expan-sion of the relativistic correlation function. As expected,no operators related by equations of motion are requiredfor this case.
2. Electric Field: HQET
Turning our attention to the case of a uniform elec-tric field, we first write the non-relativistic action in Eu-clidean space. Specifying a uniform electric field throughthe vector potential ~A = −E τ ˆ x , as above, we have theHQET action density L E = φ † ~p =0 (cid:20) ∂∂τ + ( Z E τ ) M − ( Z E τ ) M + c NR E M (cid:21) φ ~p =0 , (45)having projected onto the sector of vanishing three-momentum for ease. Above, we employ the abbreviation c NR = − c A − c A − Zc X , (46)for the linear combination of low-energy constants mul-tiplying the electric-field-squared operator. Correctionsto this action are of order M − . The coefficient c NR issurprising for two reasons. First it depends on the linearcombination of low-energy parameters − c A − c A =16 πM α E − Zc D , where the left-hand side of the equa-tion makes use of the matching condition in Eq. (38).The E shift of the action depends on the electric polar-izability α E as it must, however, there is also a contribu-tion from the charge radius, c D , which is physically im-possible because the external field cannot probe virtual photon couplings. The second surprise is the appear-ance of the contribution proportional to c X which arisesfrom the operator related by the equations of motion. Toobtain the correct physics, the first surprising feature ofEq. (46) actually requires the second surprising featurefor cancellation of the offending term.Let us further scrutinize the appearance of the coupling c X in Eq. (46). Notice the equation of motion equiva-lence shown in Eq. (34) becomes invalid in a uniformelectric field due to the striking difference in evaluatingthe two terms h iD , ~D · ~E + ~E · ~D i = − Z ~E , h ~D , ~D · ~E + ~E · ~D i = 0 . (47)Because the latter operator vanishes in a uniform electricfield, the corresponding coupling c X disappears from theaction. Consequently the Green’s function does not de-pend on the linear combination c X − c X which enterson-shell processes. Instead it depends on the parameter c X , which we have yet to determine.The fact that the equation-of-motion-related operatoris relevant to the uniform electric field case is furtherevidenced by considering the field redefinition that canbe employed for its removal. To remove the operatorwith coefficient c X from the HQET action, Eq. (29), weinvoke the field redefinition φ = − c X ~D · ~E + ~E · ~D M ! φ ′ . (48)Rewritten in terms of the φ ′ field, the equation-of-motionoperator has been removed, and the related operator nowhas coefficient c X − c X . In a uniform electric field, therelated operator vanishes by Eq. (47). The Green’s func-tions for the fields φ and φ ′ , however, are different dueto the field redefinition employed in Eq. (48). In particu-lar, the Euclidean time correlation functions in a uniformelectric field G E ( τ,
0) = Z d~x h | φ ( ~x, τ ) φ † (0 , | i E ,G ′E ( τ,
0) = Z d~x h | φ ′ ( ~x, τ ) φ ′† (0 , | i E , (49)are related by G E ( τ,
0) = (cid:18) c X Z E τ M (cid:19) G ′E ( τ, . (50)Hence the correlation functions have visibly differenttime dependence. This difference between correlation Without the equation-of-motion operator, we would set its co-efficient to zero, c X = 0, and accordingly the shift of the initialenergy arising from c NR in Eq. (45) appears in Minkowski spaceexactly as shown in Eq. (1). c X term present in the action, Eq. (45),in perturbation theory.Having argued that the c X term belongs in the HQETaction for a uniform electric field, we must determinethis parameter. A way to determine c X is to start withthe fully relativistic scalar propagator in an electric field,and perform the HQET expansion. Matching the behav-ior of the propagator order-by-order in M − will yieldthe value of this parameter. Thought of in this way,the external electric field problem requires an additionalmatching relation due to the lack of an on-shell condition.The coefficient c X , which cannot be resolved from on-shell processes, can be determined at the level of Green’sfunctions.Computing the Euclidean two-point correlation func-tion for φ directly from Eq. (45), we arrive at G E ( τ,
0) = θ ( τ ) " − ( Z E ) τ M + ( Z E ) τ M + ( Z E ) τ M − ( Z E ) τ M − c NR E τ M . (51)On the other hand, carrying out the 1 /M expansion ofthe relativistic correlation function G E ( τ,
0) in Eq. (27),and appropriately accounting for the difference in nor-malization factors [see Eq. (58) below], we find the differ-ence between relativistic and non-relativistic correlatorsis given by∆ G E ( τ,
0) = θ ( τ ) E τ M ( c NR − c R ) , (52)where the coefficient c R arises from the relativistic corre-lation function, and is given by c R = 32 πM α E + 4 Z . (53)This coefficient produces a perturbative correction to thenon-relativistic initial energy having the form ∆ E = − (cid:16) πα E + Z M (cid:17) ~E , in Minkowski space. This resultis to be contrasted with that in Eq. (1), which was ob-tained by incorrectly dropping the equation-of-motionoperator.Requiring that the correlation functions match de-mands that c NR = c R , and allows us to determine c X = − c D − Z. (54)Having determined this final parameter, the time-dependence of the HQET propagator in a uniform electricfield is fully specified. In practice, the HQET expansionis insufficient to describe lattice QCD data. While theexternal electric field may be weak, large corrections willarise in the long-time limit of the correlator. To thisend, it is efficacious to include the Euclidean time τ inthe power counting, and thus we turn to NRQED.
3. Electric Field: NRQED
The parameter c X can also be determined from car-rying out the matching of correlation functions usingNRQED power counting. This power counting, more-over, leads to a useful expansion of the relativistic corre-lation function that potentially could simplify the anal-ysis of lattice QCD data. HQET and NRQCD effectivetheories share the same Lagrange density, however, theordering of operators is different. Instead of countingpowers of 1 /M , the NRQCD counting is organized inpowers of the small velocity v [36].For a charged particle in a uniform electromagneticfield, we employ NRQED power counting in which D and ~D both count as O ( v ). Consequently explicit fac-tors of the time t count as O ( v − ), while the electric andmagnetic fields, ~E and ~B , count as O ( v ). Keeping allterms of the HQET Lagrange density in uniform electro-magnetic fields up to order v , we have L = φ † " iD + ~D M + ~D M + ~D M + c NR ~E M + c A ~B M φ. (55)Notice at this order, a further term from the HQET La-grange density is required. This term is the next-orderrelativistic correction to the kinetic energy, and is theonly term at O ( M − ). In NRQED, this term contributesat O ( v ) which is the same order required to determinethe electric and magnetic polarizabilities.To carry out the matching, we work in Euclidean space.For a uniform electric field, the Euclidean NRQED actiondensity is given by L E = φ † ~p =0 (cid:20) ∂∂τ + ( Z E τ ) M − ( Z E τ ) M + ( Z E τ ) M + c NR E M (cid:21) φ ~p =0 , (56)in the sector of vanishing three-momentum. Because theaction involves only a first-order differential operator, wecan easily determine the Green’s function G E ( τ,
0) = θ ( τ ) exp (cid:20) − ( Z E ) τ M + ( Z E ) τ M − ( Z E ) τ M − c NR E τ M (cid:21) . (57)Notice that the first term in the exponential is O ( v ),while the second term is O ( v ), and the last two terms areboth O ( v ). These latter terms need not be exponenti-ated, but can be expanded out to O ( v ). The O ( v ) termwas derived in the original proposal for treating chargedhadrons in external fields [37]. The present result pro-vides a useful extension including relativistic correctionsin a systematic way.1The NRQED expansion of the relativistic propagator G E ( τ,
0) in Eq. (27) requires Laplace’s method. Thetechnical details of the expansion are presented in Ap-pendix B. Up to O ( v ), we obtain the same form of thecorrelation function as in Eq. (57) with the exceptionthat the coefficient c NR is replaced by c R . Matching theNRQED correlators then requires c NR = c R , which con-sequently leads to the value of c X obtained in Eq. (54)above. We have established that NRQED matching ofthe Green’s functions yields the same result. D. Non-Relativistic Expansion of Relativistic QED
As a final check of our results, we determine the pa-rameter c X using a brute-force expansion of the rela-tivistic Lagrange density with a careful treatment of theequations of motion. While such an expansion is ratherantithetical to the effective field theory mindset, it can becarried out straightforwardly for a scalar particle, and thenon-relativistic matching can thus be performed directlyat the level of the action. We consider the expansion to O ( M − ).To reduce the relativistic theory in Eq. (9) to the non-relativistic theory in Eq. (29), we need the relation be-tween the relativistic scalar field Φ and non-relativisticscalar field φ , which isΦ( x ) = e − iMt [4( M − ~D )] / φ ( x ) . (58)This relation has already been used throughout to con-vert between the relativistic and non-relativistic normal-ization of states.After inserting the relation between relativistic andnon-relativistic fields into the relativistic action, Eq. (9),we perform the 1 /M -expansion keeping all terms up to O ( M − ). Many of the terms in Eq. (29) automaticallyappear in the expansion, however, there are also addi-tional terms. Explicitly, we have L = φ † " iD + ~D M − D M + (cid:8) iD , ~D (cid:9) M + c D [ ~ ∇ · ~E ]8 M + ~D M − (cid:8) D , ~D (cid:9) M + c A ~B − ~E M − c A ~E M + ic D M (cid:8) D i , [ ~ ∇ × ~B ] i (cid:9) + c D M (cid:8) iD , [ ~ ∇ · ~E ] (cid:9) − c D M (cid:2) iD , ~D · ~E + ~E · ~D (cid:3) φ, (59)where we have rewritten the relativistic parameters interms of the non-relativistic ones. To arrive at the aboveform, we utilized the identity (cid:8) D i , [ ∂ E i ] (cid:9) = (cid:2) D , ~D · ~E + ~E · ~D (cid:3) − iZ ~E , (60)to remove an operator involving the time derivative ofthe electric field. To arrive at the HQET Lagrange density from Eq. (59),we need to invoke field redefinitions. As we have seen,care must be applied in field redefinitions to preservethe time dependence of Green’s functions. Defining the O ( M − ) equation of motion operator ✷ by ✷ = iD + ~D M + c D [ ~ ∇ · ~E ]8 M , (61)we see that a large class of such field redefinitions takethe general form φ = X j =1 f j ✷ j φ ′ , (62)where f j are arbitrary coefficients. Because the Green’sfunction G ′ ( x, y ) of the redefined field satisfies the equa-tion ✷ y G ′ ( x, y ) = iδ ( x − y ), the Green’s function of theoriginal field is related by G ( x,
0) = − i X j =1 f j ✷ j − δ ( x ) G ′ ( x, , (63)and will not be altered aside from singular behavior atthe point x µ = 0.To produce Eq. (61) as the equation of motion oper-ator for the redefined field φ ′ , we require that the fieldredefinition have the explicit form φ = (cid:20) − ✷ M + 3 ✷ M − ✷ M (cid:21) φ ′ . (64)In terms of the redefined field φ ′ , the Lagrange densityto O ( M − ) becomes L = φ ′† " iD + ~D M + c D [ ~ ∇ · ~E ]8 M + ~D M + ic D M (cid:8) D i , [ ~ ∇ × ~B ] i (cid:9) + c A ~B − ~E M − c A ~E M − D ~D D M + − c D − Z M (cid:2) iD , ~D · ~E + ~E · ~D (cid:3) φ ′ , (65)which is nearly identical to the HQET Lagrange densityin Eq. (29). The last term appearing above is preciselythe equation of motion operator we retained in HQET,with a coefficient c X , moreover, which agrees with thatobtained by matching Green’s functions in uniform elec-tric fields, see Eq. (54). The second-to-last term is a newoperator. Unlike the last term, however, this new opera-tor can be removed by a field redefinition without alteringthe time dependence of Green’s functions. The requisitefield redefinition does not fall into the class consideredabove in Eq. (62). In fact, we must take φ ′ = − ~D M ✷ ! φ ′′ , (66)2so that the Lagrange density rewritten in terms of thefield φ ′′ is missing the D ~D D operator. This final fieldredefinition allows us to see the simple relation betweenGreen’s functions G ′ ( x,
0) = " i ~D M δ ( x ) G ′′ ( x, , (67)which is valid up to O ( M − ). Aside from a singularcontribution at x µ = 0, there is no modification to theGreen’s function. Alternatively we can treat the newoperator appearing Eq. (65) in perturbation theory tocompute the Green’s function, and arrive at the sameconclusion. Nonetheless, brute-force expansion of the rel-ativistic scalar action confirms the matching conditionsdetermined above. V. SUMMARY
Our primary goal is to understand matching in effec-tive field theories with the inclusion of classical externalfields. Understanding effective field theories in externalfields is particularly relevant for lattice QCD computa-tions of certain hadronic observables. On the surface,there are inconsistencies between effective field theorymatching of S -matrix elements, and the external fieldcorrelation functions which should depend on the samephysical parameters.We uncover a potential stumbling block for effectivefield theories in classical external fields in Sec. II. In anexternal field with arbitrary spacetime dependence, wefind that the effective field theory must include opera-tors related by equations of motion. Lacking an on-shellcondition, the Green’s functions are the only theoreti-cal constructs available to extract physical parameters ofthe effective theory. The Green’s functions, however, arealtered by the field redefinitions necessary to remove op-erators related by equations of motion. Starting with themost general effective field theory including such opera-tors, one cannot pass to the reduced theory. As a result,the Green’s functions generally depend on unphysical pa-rameters which must be isolated to extract physical cou-plings of the effective field theory.Despite this general obstruction, we consider the par-ticularly simple case of the uniform external field problemfor a charged scalar hadron, such as the pion. This is un-dertaken in Sec. III. Due to the simplicity of the actionin uniform electric fields, we demonstrate that operatorsrelated by equations of motion do not alter the space-time dependence of Green’s functions. Because of thisfortuitous situation, the relativistic effective theory canbe written down without including operators related bythe equations of motion. The same is not true in thenon-relativistic effective theory of the charged scalar.Applying equations of motion to reduce the non-relativistic theory leads to inconsistencies, see Eq. (47).The culprit of these inconsistencies is the field redefi-nition required to remove operators that are ordinarily redundant. Even in uniform electric fields, the requiredfield redefinition in the non-relativistic theory alters theGreen’s function in an essential way, see Eq. (50). Re-taining operators related by equations of motion necessi-tates additional matching conditions. The coefficientsof equation-of-motion operators cannot be determinedfrom matching S -matrix elements. One must appealto matching at the level of the Green’s function, whichcontains information beyond that entering on-shell pro-cesses. Using the Green’s function computed in the rela-tivistic effective field theory provides a way to determinecoefficients of operators related by equations of motionin the non-relativistic theory. Expanding the relativis-tic Green’s function in an external electric field, we ob-tain the coefficient of an equation-of-motion operator inthe non-relativistic theory. This Green’s function match-ing is exhibited by employing both HQET and NRQEDpower counting, for which we obtain identical results inSecs. IV C 2 and IV C 3, respectively. An ultimate checkof our results is achieved by performing a brute-force ex-pansion of the relativistic Lagrange density. This expan-sion explicitly allows us to track the field redefinitionsnecessary to arrive at the non-relativistic theory. Thebrute-force expansion in Sec. IV D confirms the necessityof including an equation-of-motion operator, as well asverifies the value determined for its coefficient.In our investigation, we additionally determine newresults that should be useful to lattice QCD computa-tions in external fields. To remove potential surface termsin the uniform field problem, we formulate the effectivefield theory on a Euclidean torus in Appendix A, wheresuch terms are absent. In the process, we determine aclosed-form expression for the finite-size artifacts affect-ing Green’s functions due to electroperiodic boundaryconditions, see Eq. (A6). This result should prove usefulin addressing finite-size effects in lattice QCD. Comput-ing the Green’s function of a charged scalar in uniformelectric fields, we employ NRQED power counting, andarrive at a functional form that we suspect will be highlyuseful in fitting lattice QCD data. The proper-time in-tegration required in determining the relativistic corre-lator results in cumbersome numerical fits; however, theNRQED expansion gives a systematically improvable re-sult for the correlator that does not require numericalintegration, see Eq. (57). We intend to learn whetherthe NRQED approach is beneficial. Finally our studyfocuses exclusively on the case of a charged scalar forsimplicity. The phenomenologically relevant case of spin-half hadrons must be treated in a similar fashion. Havingexposed the technical challenges, we leave this case to fu-ture work. Appendix A: Euclidean Torus
In this Appendix, we investigate the charged particlepropagator on a torus in order to remove potential sur-face terms that could arise for uniform external fields.3We note that the case of a charged particle propagatorin a uniform magnetic field has already been consideredon a torus in [30]. Thus we restrict our attention to thecase of a uniform electric field in Euclidean space, whichrequires only a simple generalization of the magnetic pe-riodic group, see [38] for clear exposition of the magneticcase.To compute the charged particle propagator, we workin the sector of vanishing transverse momentum, ~p ⊥ =( p , p ) = (0 , x -direction is taken to have length L , while the temporal direction has length β = 1 /T .On a Euclidean torus, the four-vector potential A F Vµ = −E x δ µ is not periodic, however, it is periodic up to agauge transformation A F V ( x + β ˆ x ) = A F V ( x ) + ∂ Λ( x ) , (A1)where Λ( x ) = −E βx . The gauge transformed scalarfield then obeys what we call electroperiodic boundaryconditions Φ F V ( x + L ˆ x ) = Φ F V ( x ) , Φ F V ( x + β ˆ x ) = e − iZ E βx Φ F V ( x ) . (A2)Consistency of these boundary conditions requires quan-tization of the field strength, namely Z E βL = 2 πn φ ,where n φ is the flux quantum of the torus [39–41]. Theboundary conditions satisfied by the fields ensure thatthere are no surface terms for any gauge invariant oper-ators appearing in the action.The finite volume propagator which satisfies the appro-priate electroperiodic boundary conditions can be con-structed from images of the infinite volume propagator.Explicitly we have G F V E ( x ′ , x ) = 1 L X ν,n e − πin ( x ′ − x ) /L e iZ E βνx ′ ×G E (cid:18) x ′ + νβ − n n φ β, x − n n φ β (cid:19) , (A3)where the sum over the temporal winding number ν , andsum over the periodic momentum index n both extendfrom −∞ to ∞ . Above we use the notation G E ( τ ′ , τ ) = 12 Z ∞ ds e − sE E h τ ′ , s | τ, i E , (A4)with h τ ′ , s | τ, i = r Z E π sinh Z E× e − Z E Z E s [ ( τ ′ + τ ) cosh Z E s − τ ′ τ ] . (A5)Notice that G E ( τ,
0) determined from Eq. (A4) agreeswith Eq. (27). To simplify the finite volume propagatorin Eq. (A3), we set ( x , x ) = (0 ,
0) and perform thecompact integral over x ′ , with x ′ = τ . This procedureyields G F V E ( τ ) = X ν G E ( τ, − νβ ) , (A6) where we define G F V E ( τ ) = R L dx ′ G F V E ( x ′ , ν = 0, corresponds to the infinite temporal extent limitemployed in the main text above. Contributions withnon-zero winding number correspond to finite-size correc-tions, and these vanish exponentially with β . In practice,these corrections are useful to know in order to addressfinite-size effects on correlation functions calculated withlattice QCD.In the non-relativistic theory, the field φ F V satisfies avariant of electroperiodic boundary conditions by virtueof Eq. (58), namely φ F V ( x + L ˆ x ) = φ F V ( x ) ,φ F V ( x + β ˆ x ) = e βM e − iZ E βx φ F V ( x ) . (A7)As a consequence, all gauge invariant operators in theNRQED action are periodic, and one need not worryabout surface terms. The non-relativistic correlationfunction can be constructed from electroperiodic im-ages as was done for the relativistic correlator. The fi-nal result for the non-relativistic x ′ -integrated correlator G F V E ( τ ) = R L dx ′ G F V E ( x ′ ,
0) is given by G F V E ( τ ) = X ν e − νβM G E ( τ, − νβ ) , (A8)which is quite similar to Eq. (A6) above. Appendix B: NRQED Expansion of the RelativisticPropagator
One of the alternate ways to determine the parameter c X , which enters the non-relativistic effective theory, isto start with the fully relativistic scalar propagator inan electric field and perform the NRQED expansion. Toperform the NRQED expansion of Eq. (27), we note thevelocity scaling of the parameters µ = E E τ ∼ O ( v − ) ,ζ = Z E τ ∼ O ( v − ) . (B1)We must be careful about sub-leading corrections, how-ever, because the initial energy E E = M + πα E E + · · · contains a subleading term which contributes at O ( v )to the parameter µ . Using the relation between therelativistic and non-relativistic scalar fields in Eq. (58),the relativistic G E and non-relativistic G E Euclidean two-point correlation functions are in turn related by G E ( τ ) = 2 M e Mτ (cid:20) ζ µ (cid:21) / G E ( τ ) . (B2) One must be careful to note the conjugate field does not sat-isfy the complex conjugate of the boundary conditions. Instead,we have φ F V † ( x + L ˆ x ) = φ F V † ( x ) and φ F V † ( x + β ˆ x ) = e − βM e iZ E βx φ F V † ( x ). s = Z E s . After this rescaling,we have the relativistic propagator in the form G E ( τ ) = 12 √ πZ E Z ∞ d s exp (cid:20) − f ( s ) (cid:21) , (B3)where f ( s ) = µ ζ s + ζ coth s + ln sinh s . (B4)Because terms in the exponent become large and negativeas v →
0, the bulk of the integrand arises from s wherethe exponent is a maximum. This maximum occurs atthe value coth s = 1 + p ζ + µ )2 ζ > . (B5)Using Laplace’s method, we expand f ( s ) about s , andthe relativistic correlator can be written in the form G E ( τ ) = e − f ( s ) p πZ E f ′′ ( s ) Z ∞− s √ f ′′ ( s ) dS e − S × exp − ∞ X j =3 f ( j ) ( s ) j ! S p f ′′ ( s ) ! j . (B6)The lower bound of integration has the form − s p f ′′ ( s ) = − p µ + · · · ∼ O ( v − ), and cantherefore be extended to −∞ up to corrections that are exponentially small. To compute the propagator to O ( v ) accuracy, we require j max = 6. Performing theGaussian integration, and then expanding to this order,we find G E ( τ ) = e − (cid:16) µ + ζ µ (cid:17) M " − ζ µ (cid:18) − ζ µ (cid:19) − ζ µ (cid:18) − ζ µ + 17 ζ µ − ζ µ (cid:19) + O ( v ) . (B7)Finally appending the conversion factors in Eq. (B2)and expanding to the same order, we obtain the non-relativistic correlator G E ( τ ) = e − ζ µ + ζ µ − ζ µ − ζ µ − ( E E − M ) τ , (B8)which is exactly the same as Eq. (57) with the replace-ment c NR → c R . ACKNOWLEDGMENTS
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