aa r X i v : . [ h e p - t h ] M a y arXiv:1211.6614 Lorentz and CPT Violation inScalar-Mediated Potentials
Brett Altschul Department of Physics and AstronomyUniversity of South CarolinaColumbia, SC 29208
Abstract
In Lorentz- and CPT-violating effective field theories involving scalar and spinor fields,there exist forms of Lorentz violation that modify only the scalar-spinor Yukawa inter-action vertices. These affect low-energy fermion and antifermion scattering processesthrough modifications to the nonrelativistic Yukawa potentials. The modified potentialsinvolve novel combinations of momentum, spin, and Lorentz-violating background tensors. [email protected] Introduction
In recent years, a significant amount of attention has been paid to the possibility that thelaws of physics at the most fundamental level may not respect Lorentz and CPT symme-tries exactly. While there is thus far no compelling experimental reason to believe thatLorentz or CPT invariances are not exact, many candidate theories of quantum gravitysuggest the possibility of such symmetry violations, at least in certain regimes. The pos-sibilities for symmetry breaking include spontaneous breaking in string theory [1, 2] andelsewhere [3], mechanisms in loop quantum gravity [4, 5] and non-commutative geome-try [6, 7], Lorentz violation through spacetime-varying couplings [8, 9], and anomalousbreaking of Lorentz and CPT symmetries [10].Because any confirmed discovery of Lorentz violation would be a sure sign of newphysics—with a fundamentally different structure from anything previously observed—this subject remains quite interesting and an active area of both experimental and theoret-ical research. Most theoretical work is performed within the context of effective quantumfield theory. There is an effective field theory known as the standard model extension(SME) that contains all possible translation-invariant but Lorentz-violating operatorsthat may be constructed out of the fields of the standard model. (Generalizations to in-clude additional fields are straightforward.) Each Lorentz-violating operator that appearsin the SME Lagrangian is parameterized by a small background tensor [11, 12]. If theLorentz violation arises from spontaneous symmetry breaking, these background tensorsare essentially the vacuum expectation values of tensor-valued fields. Moreover, becausethe existence of CPT violation in a stable, unitary quantum field theory implies that theremust also be Lorentz violation [13], the SME is also the most general effective field theorydescribing CPT violation.The most frequently considered subset of the SME is the minimal SME, which con-tains only gauge-invariant, local, superficially renormalizable forms of Lorentz violation.The minimal SME has become the standard framework used for parameterizing the re-sults of experimental Lorentz tests. Recent searches for Lorentz violation have includedstudies of matter-antimatter asymmetries for trapped charged particles [14, 15, 16] andbound state systems [17, 18], measurements of muon properties [19, 20], analyses of thebehavior of spin-polarized matter [21], frequency standard comparisons [22, 23, 24, 25],Michelson-Morley experiments with cryogenic resonators [26, 27, 28, 29, 30], Doppler effectmeasurements [31, 32], measurements of neutral meson oscillations [33, 34, 35, 36, 37, 38],polarization measurements on the light from cosmological sources [39, 40, 41, 42], high-energy astrophysical tests [43, 44, 45, 46, 47], precision tests of gravity [48, 49], andothers. The results of these experiments set constraints on the various SME coefficients,and up-to-date information about most of these constraints may be found in [50].The least studied areas of the SME are those that involve scalar fields. There has been agood deal of theoretical investigation into the behavior of scalars in the (Lorentz-invariant)standard model—both the fundamental Higgs and composite pseudoscalar mesons in the1adronic sector. However, little attention has been paid to scalars by theorists studyingLorentz violation. For example, the one-loop renormalization of the SME Higgs sectorhas not yet been studied systematically. Although the one-loop renormalization of theAbelian [51], non-Abelian [52], and chiral [53] gauge theories with spinor matter that makeup parts of the SME were completed some time ago, only recently have the correspondingscalar field theories including Yukawa interactions [54] received similar treatments. Yetthe study of the renormalization of the scalar sector is still not complete; gauge theorieswith scalar matter and theories with spontaneous gauge symmetry breaking have not beenadequately examined.This paper discusses how low-energy scalar-mediated interactions between fermionsand antifermions may be affected by Lorentz violation. At nonrelativistic energies, theseinteractions are described by modified Yukawa potentials. The changes to the Yukawapotential induced by Lorentz violation in the pure scalar propagation sector were pre-viously considered in [55]. However, as emphasized in [54], the Yukawa sector containsLorentz-violating operators that modify the scalar-spinor interactions. These operatorscan have a much more intricate structure than those considered in [55].The nonrelativistic limit is most relevant in low-energy hadronic physics. Understand-ing symmetry breaking in few-baryon systems is an important topic. Much of the researchin this area has focused on parity (P) violation, since this is the most strongly brokenspacetime symmetry in the standard model. Weak P violation in processes that are nor-mally dominated by the strong interaction has become a very interesting area of research,with experimental data coming from a number of different nuclear systems [56]. Thesame kinds of hadronic experiments, particularly those involving precision measurementsof neutron spin rotations, may be useful as tests of Lorentz and CPT invariances.P-violating observables in systems of interacting nucleons have traditionally been ana-lyzed in terms of interparticle potentials, as in the Desplanques-Donoghue-Holstein (DDH)model [57]. The DDH model uses a collection of one-boson exchange potentials, gener-ated by the exchange of both scalar and vector mesons. More recent analyses have usedeffective field theories, particularly a pionless [58] effective theory that should be reliableat energies less than m π /M N .The state of the art for P violation in field theory has moved beyond two-nucleonsystems. Three-nucleon observables, such as spin rotations in neutron-deuteron scatter-ing, have been analyzed using the pionless effective field theory [59, 60], as well as hybridmethods that use effective field theory to describe the symmetry violation along withmodified wave functions derived from the P-invariant portions of the interparticle po-tentials [61, 62]. These field theory methods have also been extended to deal with timereversal invariance violation [63], which is much weaker than P violation in the standardmodel.However, this paper will only consider two-body potentials. The rich additional struc-ture that arises when violations of isotropy and boost invariance are allowed may makethree-body physics more complicated (in contrast to the Lorentz-invariant pionless theory,2hich has no leading-order three-body interactions [64]). The potential theory providesa starting point for the analysis of observables that break Lorentz invariance. Our ap-proach to deriving the relevant potentials will be somewhat similar to the one used todetermine the low-energy forms of P violation in [65]—starting with the possible relativis-tic forms of symmetry violation, then reducing to a nonrelativistic theory and observingany redundancies.The outline of this paper is as follows. Section 2 describes the structure of the Lorentz-violating operators that appear in the scalar sector of a Lorentz-violating effective fieldtheory. Then section 3 examines how Lorentz-violating modifications to scalar-spinor cou-plings affect the Yukawa potentials between spin- particles. The results are summarizedin section 4, along with some additional remarks placing this work in context. The Lagrange density for the minimal SME contains local operators that can be con-structed out of the standard model’s scalar, spinor, and gauge fields. To maintain super-ficial renormalizability, only gauge invariant operators with mass dimensions of up to 4are included. In all cases that have been checked explicitly, these conditions are indeedsufficient to make the theories renormalizable at one-loop order. (As already noted, theYukawa theory discussed in this paper is among the cases for which such a check has beenmade.)For a single species of Dirac fermion, the minimal SME Lagrange density is L f = ¯ ψ ( i Γ µ ∂ µ − M ) ψ, (1)where Γ µ = γ µ + Γ µ = γ µ + c νµ γ ν + d νµ γ γ ν + e µ + if µ γ + 12 g λνµ σ λν (2)and M = m + im γ + M = m + im γ + a µ γ µ + b µ γ γ µ + 12 H µν σ µν . (3)These are the only operators satisfying the listed conditions that can exist in a purelyfermionic theory. The Γ coefficients are dimensionless, while the M coefficients havedimension (mass) . However, some of the coefficients appearing in Γ and M are moreinteresting than others. Several, such as m , a , and f , may be eliminated from the theoryby a redefinition of the fermion field [66, 67].The species of interest in low-energy hadronic physics are composites formed fromthe fundamental quark and gluon fields. However, there are still Lorentz violation coef-ficients for the composite species; they are linear combinations of the coefficients for thefundamental fields. Many of the more precise bounds on SME coefficients are actuallyconstraints on the composite coefficients for protons and neutrons.3any of the terms present in (1–3) violate CPT as well as Lorentz symmetry. Of thecoefficients that cannot be eliminated by field redefinitions, those with odd numbers ofLorentz indices are also odd under CPT. However, the coefficients that have even numbersof indices are CPT invariant. It is thus possible to break Lorentz symmetry but leave CPTintact (although not vice versa). The full discrete symmetry properties of the minimalSME operators are discussed in [51, 67].To the fermionic theory may be appended one or more boson fields—of either scalaror gauged vector types. This also introduces new possible forms of Lorentz violation.However, there is a fundamental difference between the possibilities in scalar- and gauge-mediated interactions. In a gauge theory, whatever renormalizable Lorentz violation existsin the free fermion sector completely determines the Lorentz violation present at theboson-fermion vertex. The same quantity Γ µ appears in both the fermion propagatorand the vertex, because of gauge invariance. However, the situation is quite differentin a Yukawa theory. With no additional condition analogous to gauge invariance, thereis a completely independent set of Lorentz-violating operators that can appear in thefermion-scalar vertex.With the addition of a scalar field φ , the most general Lorentz-violating Lagrangedensity that does not lead to spontaneous symmetry breaking becomes L = ¯ ψ ( i Γ µ ∂ µ − M ) ψ + 12 ( ∂ µ φ )( ∂ µ φ ) + 12 K µν ( ∂ ν φ )( ∂ µ φ ) − µ φ − λ φ − ¯ ψGψφ. (4)The symmetric tensor K µν = K νµ represents the only kind of Lorentz violation that canbe introduced in the pure bosonic sector with a real scalar field. Much more intricate instructure is the operator G appearing in the Yukawa vertex term. G has essentially thesame structure as the M term in the pure scalar sector, G = g + ig ′ γ + G = g + ig ′ γ + I µ γ µ + J µ γ γ µ + 12 L µν σ µν . (5)The terms g and g ′ are the usual scalar and pseudoscalar Yukawa couplings, while theother terms are Lorentz violating. All the coefficients contained in G are dimensionless.The tensor term L µν is naturally antisymmetric. The discrete symmetries of the operatorsthat make up G are similar to the symmetries of the corresponding operators contained in M . If the φ field is a true scalar, the symmetries are exactly the same; for a pseudoscalarfield, the parity and time reversal behaviors are opposite between M and G , while thecharge conjugation properties are still the same. Ultimately, the I and J terms violateCPT as well as Lorentz symmetry, while L is CPT invariant.While the fact that this rich structure of operators could exist in the spinor-scalar cou-pling term was noted as part of the original formulation of the SME, very little attentionhas been paid to the G terms. There has been essentially no calculations of their phe-nomenalistic effects, and only recently [54] have the effects of these terms on the one-looprenormalization of the SME been considered.4 Modified Yukawa Potentials
Both the K and G terms will affect the Yukawa potentials for interacting fermions andantifermions, because both of them appear in the four-point correlation functions thatdescribe two-particle scattering. The purpose of this section will be to evaluate the inter-particle potentials that are associated with this scattering. Since the K term was alreadydiscussed in [55], the focus here will be on the effects of the I , J , and L terms that,together with the Lorentz-invariant g and g ′ terms, comprise G .In discussing the modified Yukawa potential, we shall only consider Lorentz-violatingeffects that are linear in the SME coefficients. Since Lorentz violation is known to be avery small phenomenon physically, higher-order effects should be minuscule. A similarleading-order analysis of electromagnetic potentials was conducted in [68].Lorentz violation in the pure fermion sector (and in the fermion-gauge interactionsector) is relatively well constrained, at least for the first generation fermions that makeup the stable constituents of everyday matter. For this reason, we shall neglect the Γ and M terms (even though some small nonzero Γ and M terms could be generatedfrom G by radiative corrections [54]).However, if the coefficients of the operators involved were not too small, the forms ofLorentz violation described by Γ and M would affect fermionic scattering in a significantway. These pure fermion sector terms would affect both the amputated matrix element forthe one-boson exchange process that dominates low-energy scattering and the dispersionrelations for the external particles, which would in turn affect the kinematics of a reaction.In fact, the changes to scattering and decay rates due to changes in particle velocities andavailable phase space may be as large as or larger than the changes arising from Lorentzviolation in the invariant matrix element itself [69, 70, 71]. When Γ and M are neglected, it is possible to use standard external fermion and an-tifermion spinor states for the calculation of a matrix element. Since the behaviors of the G operators depend in nontrivial ways on the spins of the external particles, it is simplestto perform the matrix element calculation using explicit spinor eigenstates. Using theDirac-Pauli basis for the Dirac matrices and a relativistic normalization convention, theDirac spinor u s ( p ) (corresponding to momentum p and spin s ) is u s ( p ) = r E ( E + m )2 m (cid:20) ξ s~σ · ~pE + m ξ s (cid:21) , (6)where, ξ s is a two-component spinor. In the nonrelativistic limit, this becomes u s ( p ) = √ m (cid:20) ξ s~σ · ~p m ξ s (cid:21) . (7)5sing the explicit spinors, it is possible to calculate the fermion bilinears that appear ina scattering amplitude. In particular, if the external particles are nonrelativistic, so thatterms with more than a single power of p/m may be neglected,¯ u s ′ ( p ′ ) Gu s ( p ) = 2 m (cid:20) ξ † s ′ , − ξ † s ′ ~σ · ~p ′ m (cid:21) ( g + iγ g ′ + G ) (cid:20) ξ s~σ · ~p m ξ s (cid:21) (8)= 2 m ( g + I ) ξ † s ′ ξ s + ig ′ ( p j − p ′ j ) ξ † s ′ σ j ξ s + 2 m ( J j + ǫ jkl L kl ) ξ † s ′ σ j ξ s − I j h ( p j + p ′ j ) ξ † s ′ ξ s − iǫ jkl ( p k − p ′ k ) ξ † s ′ σ l ξ s i − J ( p j + p ′ j ) ξ † s ′ σ j ξ s − L j h i ( p j − p ′ j ) ξ † s ′ ξ s − ǫ jkl ( p k + p ′ k ) ξ † s ′ σ l ξ s i . (9)If the two fermions involved are of different species, direct scattering is the only possiblechannel. We shall henceforth assume that the particles involved in a scattering event areindeed distinguishable. However, in the scattering of identical particles, the usual methodof replacing the scattering amplitude f ( θ, φ ) with f ( θ, φ ) − f ( π − θ, π + φ ) will give thecorrect result.For antiparticle scattering, we shall similarly assume that the annihilation scatteringchannel is not available, and only a single diagram contributes to the potential. We mayalso take advantage of the fact that v s ( p ) = γ u − s ( p ), where the subscript − s on the spinor u ( p ) indicates a spinor with spin projections that are opposite those of u s ( p ). Then wehave that¯ v s ( p ) Gv s ′ ( p ′ ) = − ¯ u − s ( p ) γ Gγ u − s ′ ( p ′ ) (10)= − m ( g − I ) ξ †− s ξ − s ′ + ig ′ ( p j − p ′ j ) ξ †− s σ j ξ − s ′ + 2 m ( J j − ǫ jkl L kl ) ξ †− s σ j ξ − s ′ − I j h ( p j + p ′ j ) ξ †− s ξ − s ′ + iǫ jkl ( p k − p ′ k ) ξ †− s σ l ξ − s ′ i − J ( p j + p ′ j ) ξ †− s σ j ξ − s ′ − L j h i ( p j − p ′ j ) ξ †− s ξ − s ′ + ǫ jkl ( p k + p ′ k ) ξ †− s σ l ξ − s ′ i . (11)The remaining products of Pauli spinors can be simplified further. The inner product ξ †− s ξ − s ′ simply equals ξ † s ′ ξ s = δ ss ′ . However, because of the spin reversal present in ξ − s ,the matrix element ξ †− s σ j ξ − s ′ is equal to − ξ † s ′ σ j ξ s .Note that present in (9) [and (11)] are almost all the possible vector structures thatcan be constructed at first order in the momenta. There are three independent three-vector operators that may be constructed: the total momenta along the incoming andoutgoing lines from a vertex ( ~p + ~p ′ ), the momentum transfer ( ~q = ~p − ~p ′ ), and the spinoperator ( ~σ ). Each of these three may form a dot product with an isotropy-breakingbackground vector; either of the momentum observables may form a dot product with thespin; or there may be a triple product with a background vector, the spin, and one of themomenta. The only possible structures that are missing are contractions of the momentaand spin with symmetric, traceless three-tensors; however, these structures cannot exist6ecause there is no such symmetric, traceless tensor that can be constructed at first orderin G .The terms involving background three-vectors manifestly break isotropy. Moreover,Lorentz boost invariance normally prevents the appearance of the ~p + ~p ′ terms. Theaverage velocity ~v av = ( ~v + ~v ′ ) (which is, of course, the passing velocity ~v av ≈ ~v ≈ ~v ′ ina glancing collision) is measured in the specific laboratory frame in which the calculationhas been performed. The dot product of this velocity with the spin is not invariant undernonrelativistic Galilean transformations, and this signifies the failure of Lorentz boostsymmetry. In contrast, the difference ~v − ~v ′ , which appears in the Lorentz-invarianttheory in conjunction with the pseudoscalar g ′ term, transforms covariantly under Galielantransformations For the scattering of two nonidentical fermions, with the exchange of a single scalar bosonbetween them, the matrix element is i M = ¯ u as ′ a ( p ′ a ) G a u as a ( p a ) − iq − µ + iǫ ¯ u bs ′ b ( p ′ b ) G b u bs b ( p b ) . (12)The indices a and b denote the identities of the species involved. However, for simplicity,we shall assume that there is only Lorentz violation for one species of particle (the onelabeled a ). Including Lorentz violation for both is a straightforward generalization. Weshall consider both scalar and pseudoscalar couplings for the second particle, however.For nonrelativistic scattering, the interaction may be described using a potential, suchthat V ( ~r ) = Z d q (2 π ) e i~q · ~r i M ( ~q ) , (13)in the limit where q = 0. The integrand involves the usual scalar Yukawa amplitudeproportional to 1 / ( ~q + µ ), as well as terms with additional factors of q j . The spatialextent of the interactions is therefore determined by the Yukawa potential function andits derivative, f ( ~r ) = − e − µr πr (14) g j ( ~r ) = ∂ j f ( ~r ) = e − µr πr (cid:18) µ + 1 r (cid:19) x j . (15)Although higher powers of ~q (and thus additional spatial derivatives) will be largelyneglected in this paper, we notice that the next order term is h jk ( ~r ) = ∂ j ∂ k f ( r ) = e − µr πr (cid:20)(cid:18) µ + 1 r (cid:19) δ jk − (cid:18) µ r + 3 µr + 3 r (cid:19) x j x k (cid:21) + 13 δ ( ~r ) δ jk . (16)7patial potentials with this shape already appear in the Lorentz-invariant theory, comingfrom terms with pseudoscalar g ′ couplings at both vertices. It is also possible to have afirst-order Lorentz-violating potential with this shape, if one vertex involves a G termand the other vertex a g ′ .The function f contributes to that part of the potential that comes from vertex termsin which q does not appear. These generate a potential V f ( ~r ) = [˜ g − I j ( v aav ) j − J ( v aav ) j σ aj + ˜ J j σ aj + ǫ jkl ˜ L j ( v aav ) k σ al ] g b f ( ~r ) . (17)The terms in square in brackets in (17) comes from the vertex with the a species, whilethe g b term comes from the b vertex. The species labels are omitted on the Lorentzviolation coefficients, since there is assumed to be no Lorentz violation in the vertex withthe b species. The scalar ˜ g and three-vectors ˜ J j and ˜ L j marked with tildes denote thecombinations ˜ g = g + I (18)˜ J j = J j + ǫ jkl L kl (19)˜ L j = L j = − L j , (20)which are the combinations that are observable in experiments with nonrelativistic fer-mions conducted in a single Lorentz frame. They are analogous to the tilde-markedcoefficients defined in other sectors of the SME, although the ones defined in (18–20)are defined to be dimensionless, unlike the tilde coefficients in other sectors, which mosttypically have units of mass. These combinations of indistinguishable terms exist becausethe leading order fermionic matrix elements of even Dirac operators (which involve onlythe large components of the Dirac spinors) are unchanged when multiplied by γ . So if E is an even operator, the nonrelativistic matrix elements of E and γ E are identical. Notehowever, that different combinations exist for antifermions, because γ is equivalent to − G is still quite a bit richer than the corresponding structure for matrixelements of M . Because M appears in the bilinear propagation Lagrangian for fermionspecies, its physical matrix elements always involved particles with identical incomingand outgoing momenta. So a matrix element such as ¯ u s ′ ( p ) M u s ( p ) lacks any terms thatdepend on ~p − ~p ′ .The part of the interaction potential with the more complicated g j ( ~r ) spatial depen-dence arises from those terms in the scattering amplitude with just a single factor of q j .This factor can appear at either vertex. At the a vertex, either a Lorentz-violating termor the g ′ a coupling may be responsible; or at the b vertex, there may be a coupling g ′ b .Taken together, these terms generate a potential V g ( ~r ) = (cid:26) m a h g ′ a σ aj − ˜ L j − ǫ jkl I k σ al i g b − m b h ˜ g − I k ( v aav ) k J ( v aav ) k σ ak + ˜ J k σ ak + ǫ kln ˜ L k ( v aav ) l σ an i g ′ b σ bj o g j ( ~r ) . (21)For completeness, we may mention the potential term that arises when a factor of q appears at both vertices. This potential has the spatial shape h jk ( ~r ), so that V h ( ~r ) = 14 m a m b h g ′ a σ aj − ˜ L j − ǫ jln I l σ an i (cid:0) g ′ b σ bk (cid:1) h jk ( ~r ) . (22)However, this is not a complete description of the potential at this order. Other termswith multiple factors of p/m have also been neglected (for example, in the normalizationof the Dirac spinors). The one-boson scalar exchange also generates potentials between antifermions and otherparticles. We shall now look explicitly at the case where the species- a particle is anantifermion, while the species- b particle remains a fermion. The two species are stilldifferent, so there is no annihilation scattering. In this case, the potential is derived froma matrix element similar to (12): i M = − ¯ v as a ( p a ) G a v as ′ a ( p ′ a ) − iq − µ + iǫ ¯ u bs ′ b ( p ′ b ) G b u bs b ( p b ) , (23)with the usual overall minus sign for antiparticle scattering, coming from the fields’ anti-commutation. This cancels the overall minus sign from (10).There are additional minus signs associated with the I and L terms. These can beread off directly from the explicit expression (11). However, they can be derived moststraightforwardly simply by noting that exactly those operators associated with I and L are odd under fermion-antifermion charge conjugation.Therefore, the potentials, up to linear order in p/m , are V ∗ f ( ~r ) = [˜ g ∗ + I j ( v aav ) j − J ( v aav ) j σ aj + ˜ J ∗ j σ aj − ǫ jkl ˜ L j ( v aav ) k σ al ] g b f ( ~r ) (24) V ∗ g ( ~r ) = (cid:26) m a h g ′ a σ aj + ˜ L j + ǫ jkl I k σ al i g b − m b h ˜ g ∗ + I k ( v aav ) k − J ( v aav ) k σ ak + ˜ J ∗ k σ ak − ǫ kln ˜ L k ( v aav ) l σ an i g ′ b σ bj o g j ( ~r ) . (25)Because of the sign changes associated with charge conjugation, two additional linearcombinations of coefficients are relevant for nonrelativistic experiments with antiparticles:˜ g ∗ = g − I (26)˜ J ∗ j = J j − ǫ jkl L kl , (27)where the star superscript notation continues to follow [50].9 Conclusions
The potential V f + V g provides a description of the O ( p/m ) nonrelativistic interactionsbetween two fermion species—one with Lorentz violation present in the scalar-spinor ver-tex and one without. For a Lorentz-invariant fermion and Lorentz-violating antifermion,the equivalent potential is V ∗ f + V ∗ g , which is related by charge conjugation in the Lorentz-violating a sector. The Lorentz-violating structure of these terms is evident in severalways. Dependences on specific projections of the spin and momenta break spatial isotropy,and structures such as ~v av · ~σ break boost invariance.This study has already demonstrated several important properties of the Lorentz-violating operators that form G . In most sectors of the SME, there are operators thatare not physically observable. However, all the terms that compose G appear in thenonrelativistic potentials, making observable contributions to the energy. While it maynot be surprising that terms from G , which only affect interactions, not free particlepropagation, cannot be eliminated from observables in the same fashion as m , a , or f ,neither is it obvious that such is the case.The potentials derived in this paper provide a fairly general formalism for study-ing violations of fundamental symmetries in low-energy, potential-dominated interactionprocesses. As noted in section 3.1, most of the observables than can be constructed at O ( p/m ) are included in the potentials, and those that are not included cannot descendfrom a renormalizable relativistic quantum field theory. The study of symmetry violationin low-energy processes is an active area of hadronic research, and it may be possible toplace constraints on completely new SME parameters through studies of meson-mediatedinteractions between baryons.Naturally, further generalizations of the results discussed here are also possible. Theremay be Lorentz violation at both vertices, and accounting for this possibility is entirelystraightforward, as are accounting for the additional diagrams that appear when theexternal particles are associated with the same species. However, it is possible that the V f and V g potentials may not actually include the predominant effects, even when themomentum transfer in a collision is very low. The g ′ a g ′ b term in V h is O ( p /m ), but it isLorentz invariant; so it would be no surprise if that term were substantially larger thanLorentz-violating terms that are nominally lower order in p/m . The g ′ a g ′ b term is in factthe dominant term in standard model interactions involving pseudoscalar mesons whenP violation is small.Lorentz violation for the external fermion states has also been neglected, althoughif such Lorentz violation exists, it will modify the interparticle potentials further. Thepurely fermionic Γ and M terms in the SME Lagrangian were neglected because suchterms, which would affect freely propagating fermions, are rather well constrained for first-generation species. However, Lorentz violation in the scalar sector is a separate matter,and the effects of Lorentz violation in the scalar sector on the Yukawa potential have10lready been studied [55]. The effect of a (CPT-even) tensor K µν is to modify f ( r ) to f K ( ~r ) = − e − µr πr (cid:20) K jj − K jk (cid:18) µr + 1 r (cid:19) x j x k (cid:21) . (28)When the vertex interactions involve the Lorentz-invariant g ′ term, K leads to a furthermodified version of the derivative g Kj ( ~r ) = ∂ j f K ( ~r ) = g j ( ~r ) (cid:20) K kk − K kl (cid:18) µr + 1 r (cid:19) x k x l (cid:21) + 12 f ( ~r ) K kl (cid:20)(cid:18) µr + 2 r (cid:19) x j x k x l − (cid:18) µr + 1 r (cid:19) ( δ jk x l + δ jl x k ) (cid:21) . (29)The modified g Kj ( ~r ) is not needed in the ~q -dependent terms that are themselves Lorentzviolating; any resulting changes to the potentials calculated in section 3 would be higherorder in the small Lorentz violation coefficients.Finally, this work provides another stepping stone in the general analysis of Lorentzviolation in scalar theories. With the advent of the Large Hadron Collider, it appearsthat it is finally possible to see direct evidence of the Higgs boson [72, 73]. This natu-rally opens up the possibility of studying the Lorentz symmetry behavior of fundamentalscalar fields. The era of direct experimental studies of the Higgs particle is just begin-ning, and the theoretical foundation for understanding such studies needs to be prepared.While spontaneous gauge symmetry breaking is one of the most important features ofthe standard model, its complexity has limited studies of SME scalar fields to particularsub-topics. In addition to the renormalization of the Yukawa and pure scalar sectors,the tree-level quantization of the theory in the spontaneously broken phase [74, 75] hasalready been studied. Certain specific quantum corrections, originating in the Faddeev-Popov ghost sector [76], or involving higher powers of the SME coefficients [77], have alsobeen examined. This paper presents some new results that increase our understandingof the scalar sector of the SME. While particles interacting via direct Higgs exchange aregenerally not at low nonrelativistic energies, this work nonetheless gives a new windowinto the dynamics of Higgs interactions. References [1] V. A. Kosteleck´y, S. Samuel, Phys. Rev. D , 683 (1989).[2] V. A. Kosteleck´y, R. Potting, Nucl. Phys. B , 545 (1991).[3] B. Altschul, V. A. Kosteleck´y, Phys. Lett. B , 106 (2005).[4] R. Gambini, J. Pullin, Phys. Rev. D , 124021 (1999).115] J. Alfaro, H. A. Morales-T´ecotl, L. F. Urrutia, Phys. Rev. D , 103509 (2002).[6] I. Mocioiu, M. Pospelov, R. Roiban, Phys. Lett. B , 390 (2000).[7] S. M. Carroll, J. A. Harvey, V. A. Kosteleck´y, C. D. Lane, T. Okamoto, Phys. Rev.Lett. , 141601 (2001).[8] V. A. Kosteleck´y, R. Lehnert, M. J. Perry, Phys. Rev. D , 123511 (2003).[9] A. Ferrero, B. Altschul, Phys. Rev. D , 125010 (2009).[10] F. R. Klinkhamer, C. Rupp, Phys. Rev. D , 045020 (2004).[11] D. Colladay, V. A. Kosteleck´y, Phys. Rev. D , 6760 (1997).[12] D. Colladay, V. A. Kosteleck´y, Phys. Rev. D , 116002 (1998).[13] O. W. Greenberg, Phys. Rev. Lett. , 231602 (2002).[14] R. Bluhm, V. A. Kosteleck´y, N. Russell, Phys. Rev. Lett. , 1432 (1997).[15] G. Gabrielse, A. Khabbaz, D. S. Hall, C. Heimann, H. Kalinowsky, W. Jhe, Phys.Rev. Lett. , 3198 (1999).[16] H. Dehmelt, R. Mittleman, R. S. Van Dyck, Jr., P. Schwinberg, Phys. Rev. Lett. ,4694 (1999).[17] R. Bluhm, V. A. Kosteleck´y, N. Russell , Phys. Rev. Lett. , 2254 (1999).[18] D. F. Phillips, M. A. Humphrey, E. M. Mattison, R. E. Stoner, R. F. C. Vessot, R.L. Walsworth , Phys. Rev. D , 111101(R) (2001).[19] R. Bluhm, V. A. Kosteleck´y, C. D. Lane, Phys. Rev. Lett. , 1098 (2000).[20] V. W. Hughes, et al. , Phys. Rev. Lett. , 111804 (2001).[21] B. R. Heckel, E. G. Adelberger, C. E. Cramer, T. S. Cook, S. Schlamminger, U.Schmidt, Phys. Rev. D , 092006 (2008).[22] C. J. Berglund, L. R. Hunter, D. Krause, Jr., E. O. Prigge, M. S. Ronfeldt, S. K.Lamoreaux, Phys. Rev. Lett. , 1879 (1995).[23] V. A. Kosteleck´y, C. D. Lane, Phys. Rev. D , 116010 (1999).[24] D. Bear, R. E. Stoner, R. L. Walsworth, V. A. Kosteleck´y, C. D. Lane, Phys. Rev.Lett. , 5038 (2000).[25] P. Wolf, F. Chapelet, S. Bize, A. Clairon, Phys. Rev. Lett. , 060801 (2006).1226] H. M¨uller, et al. , Phys. Rev. Lett. , 050401 (2007).[27] S. Herrmann, A. Senger, K. M¨ohle, E. V. Kovalchuk, A. Peters, in CPT and LorentzSymmetry IV , edited by V. A. Kosteleck´y (World Scientific, Singapore, 2008), p. 9.[28] S. Herrmann, et al. , Phys. Rev. D 80, 105011 (2009).[29] Ch. Eisele, A. Yu. Nevsky, S. Schiller, Phys. Rev. Lett. 103, 090401 (2009).[30] H. M¨uller, Phys. Rev. D , 045004 (2005).[31] G. Saathoff, S. Karpuk, U. Eisenbarth, G. Huber, S. Krohn, R. Mu˜noz Horta, S.Reinhardt, D. Schwalm, A. Wolf, G. Gwinner, Phys. Rev. Lett. , 190403 (2003).[32] C. D. Lane, Phys. Rev. D , 016005 (2005).[33] V. A. Kosteleck´y, Phys. Rev. Lett. , 1818 (1998).[34] V. A. Kosteleck´y, Phys. Rev. D , 016002 (1999).[35] Y. B. Hsiung, Nucl. Phys. Proc. Suppl. , 312 (2000).[36] K. Abe et al. , Phys. Rev. Lett. , 3228 (2001).[37] J. M. Link et al. , Phys. Lett. B , 7 (2003).[38] B. Aubert et al. , Phys. Rev. Lett. , 251802 (2006).[39] S. M. Carroll, G. B. Field, Phys. Rev. Lett. , 2394 (1997).[40] V. A. Kosteleck´y, M. Mewes, Phys. Rev. Lett. , 251304 (2001).[41] V. A. Kosteleck´y, M. Mewes, Phys. Rev. Lett. , 140401 (2006).[42] V. A. Kosteleck´y, M. Mewes, Phys. Rev. Lett. , 011601 (2007).[43] F. W. Stecker, S. L. Glashow, Astropart. Phys. , 97 (2001).[44] T. Jacobson, S. Liberati, D. Mattingly, Nature , 1019 (2003).[45] B. Altschul, Phys. Rev. Lett. , 201101 (2006).[46] B. Altschul, Phys. Rev. D , 083003 (2006).[47] F. R. Klinkhamer, M. Risse, Phys. Rev. D , 016002 (2008); addendum Phys. Rev.D , 117901 (2008).[48] J. B. R. Battat, J. F. Chandler, C. W. Stubbs, Phys. Rev. Lett. , 241103 (2007).1349] H. M¨uller, S. W. Chiow, S. Herrmann, S. Chu, K.-Y. Chung, Phys. Rev. Lett. ,031101 (2008).[50] V. A. Kosteleck´y, N. Russell, arXiv:0801.0287v5.[51] V. A. Kosteleck´y, C. D. Lane, A. G. M. Pickering, Phys. Rev. D , 056006 (2002).[52] D. Colladay, P. McDonald, Phys. Rev. D, , 105002 (2007).[53] D. Colladay, P. McDonald, Phys. Rev. D, , 125019 (2009).[54] A. Ferrero, B. Altschul, Phys. Rev. D , 065030 (2011).[55] B. Altschul, Phys. Lett. B , 679 (2006).[56] B. R. Holstein, Nucl. Phys. A , 160c (2010).[57] B. Desplanques, J. F. Donoghue, B. R. Holstein, Ann. Phys. (NY) , 499 (1980).[58] S.-L. Zhu, C. M. Maekawa, B. R. Holstein, M. J. Ramsey-Musolf, U. van Kolck, Nucl.Phys. A , 435 (2005).[59] H. W. Greisshammer, M. R. Schindler, R. P. Springer, Eur. Phys. J. A , 7 (2012).[60] J. Vanasse, Phys. Rev. C , 014001 (2012).[61] R. Schiavilla, M. Viviani, L. Girlanda, A. Kievsky, L. E. Marcucci, Phys. Rev. C ,014002 (2008); , 029902 (E) (2011).[62] Y.-H. Song, R. Lazauskas, V. Gudkov, Phys. Rev. C , 015501 (2011).[63] Y.-H. Song, R. Lazauskas, V. Gudkov, Phys. Rev. C , 065503 (2011).[64] H. W. Greisshammer, M. R. Schindler, Eur. Phys. J. A , 73 (2010)[65] L. Girlanda, Phys. Rev. C , 067001 (2008).[66] D. Colladay, P. McDonald, J. Math. Phys. , 3554 (2002).[67] B. Altschul, J. Phys. A , 13757 (2006).[68] Q. G. Bailey, V. A. Kosteleck´y, Phys. Rev. D , 076006 (2004).[69] D. Colladay, V. A. Kosteleck´y, Phys. Lett. B
209 (2001).[70] B. Altschul, Phys. Rev. D , 056005 (2004).[71] B. Altschul, Phys. Rev. D , 091902(R) (2011).1472] G. Aad, et al. (ATLAS Collaboration), Phys. Lett. B , 1 (2012).[73] S. Chatrchyan, et al. (CMS Collaboration), Phys. Lett. B , 30 (2012).[74] D. L. Anderson, M. Sher, I. Turan, Phys. Rev. D , 016001 (2004).[75] B. Altschul, Phys. Rev. D , 045008 (2012).[76] B. Altschul, Phys. Rev. D , 045004 (2006).[77] M. Gomes, J. R. Nascimento, A. Yu. Petrov, A. J. da Silva, Phys. Rev. D81