Gluonic Lorentz violation and chiral perturbation theory
GGluonic Lorentz violation and chiral perturbation theory
J. P. Noordmans CENTRA, Departamento de F´ısica,Universidade do Algarve, 8005-139 Faro, Portugal (Dated: September 18, 2018)
Abstract
By applying chiral-perturbation-theory methods to the QCD sector of the Lorentz-violatingStandard-Model Extension, we investigate Lorentz violation in the strong interactions. In par-ticular, we consider the CPT-even pure-gluon operator of the minimal Standard-Model Extension.We construct the lowest-order chiral effective Lagrangian for three as well as two light quarkflavors. We develop the power-counting rules and construct the heavy-baryon chiral-perturbation-theory Lagrangian, which we use to calculate Lorentz-violating contributions to the nucleon selfenergy. Using the constructed effective operators, we derive the first stringent limits on many ofthe components of the relevant Lorentz-violating parameter. We also obtain the Lorentz-violatingnucleon-nucleon potential. We suggest that this potential may be used to obtain new limits fromatomic-clock or deuteron storage-ring experiments. a r X i v : . [ h e p - ph ] J a n . INTRODUCTION The detection of an experimental signal corresponding to the breakdown of Lorentzsymmetry [1] would be a major discovery and could potentially provide valuable informationabout a possible theory of quantum gravity. Although no such signal has been detected todate, there is still a large interest in the possibility that Lorentz symmetry might be violatedin nature. This is caused by the fact that some proposed models of quantum gravity involvemechanisms that allow for (spontaneous) Lorentz violation (LV) at Planck-scale energies[2]. Tiny remnants of such high-energy LV might be detectable at experimentally attainableenergies, in particular because there is little experimental background from conventionalLorentz-symmetric (LS) physics for many of the corresponding signals.Searches for LV are arguably [3] best performed in the context of a realistic effec-tive field theory (EFT) for general LV, called the Standard-Model Extension (SME) [4].Its particle-physics Lagrangian contains all possible operators that can be constructed us-ing the conventional standard-model fields, coupled to fixed background tensors. TheseLorentz-violating coefficients (LVCs) presumably originate from an underlying fundamentaltheory that (spontaneously) breaks Lorentz symmetry. Being the most general realistic EFTfor Lorentz-symmetry breaking, the SME is also the most general realistic EFT for CPTviolation [5].From a phenomenological point of view, the virtue of the SME lies in the fact that itprovides a means to explicitly calculate observable signals for Lorentz-symmetry breaking, aswell as a way to systematically identify unconstrained regions of the LV parameter space. Asa consequence, many stringent constraints on LVCs have been obtained experimentally [6].Particularly successful in this respect are low-energy precision tests of nuclear and hadronicsystems, providing severe limits on various effective nucleon and other hadronic parametersfor LV. However, since quantum chromodynamics (QCD) is nonperturbative at the relevantenergies, deriving direct bounds on the more fundamental quark and gluon parameters thatappear in the SME Lagrangian, is complicated. This amounts to a relatively small set ofdirect bounds on quark and gluon parameters [6].A promising approach, that is aimed at remedying this situation, is applying the well-established machinery of chiral perturbaton theory ( χ PT) [7] to the QCD sector of the SME[8, 9]. It is similar in spirit to studies of the breaking of parity [10] and of time-reversal2ymmetry [11]. In this work we extend this approach to the CPT-even pure-gluon sectorof the minimal Standard-Model Extension (mSME). The latter is the restriction of the fullSME to LV operators with mass dimension d ≤
4. In Sec. II, we will introduce the relevantmSME operator and discuss some pertinent properties of the corresponding LVC: k µνρσG .In Sec. III we construct the induced chiral effective Lagrangian in terms of the degreesof freedom that are relevant below the chiral-breaking scale Λ χ (cid:39) k µνρσG do not induceany kinetic nucleon terms. This leads us to conclude that nucleon bounds cannot directlyconstrain these ten parameters to the desired level of accuracy. On the other hand, suchbounds can be obtained by considering contributions of k µνρσG to a pure-photon operator.This will be considered in Sec. VI. In Sec. VII we show that additional and/or improvedbounds might be obtained by considering Cherenkov-like pion emission by protons and LVpion exchange between nucleons and its effect on for example the spin precession of thedeuteron. Finally, in Sec. VIII, we will summarize and present our conclusions. II. THE PURE GLUON CPT-EVEN MSME LAGRANGIAN
In the mSME there is one LV CPT-even coefficient that couples to a pure gluonoperator. This operator has mass-dimension four and is given by [4] L = − k µνρσG Tr ( G µν G ρσ ) , (1)where G µν = G aµν λ a is the gauge field strength of the SU (3) color gauge group (here, λ a /
2, with a = 1 , . . . ,
8, are the corresponding generators) and k µνρσG is a real tensor thatparametrizes the LV. The operator is even under charge conjugation and after either a parityor a time-reversal transformation, it gains a factor ( − µ ( − ν ( − ρ ( − σ , with ( − µ = 1if µ = 0 and ( − µ = − k µνρσG in Eq. (1) is real and has the symmetries of the Riemann curvaturetensor, i.e. k µνρσG = − k µνσρG = k ρσµνG , k µ [ νρσ ] G = 0 , (2)where the square brackets indicate total anti-symmetrization of the enclosed indices. Thesecond relation holds for anti-symmetrization of any group of three indices. It follows fromthe fact that, using the first two relations in Eq. (2), we can write k µ [ νρσ ] G = (cid:15) αβγδ k αβγδG (cid:15) µνρσ .Therefore, a corresponding nonzero part of k µνρσG does not violate Lorentz symmetry andcan be absorbed in the conventional ¯ θ term of QCD. Additionally, we can take k µνρσG to havea vanishing double trace, i.e. ( k G ) µνµν = 0, since such a trace part also does not violateLorentz invariance and can be absorbed in the conventional LS gauge term.These considerations show that k µνρσG has 19 independent real, physical, and LV com-ponents. These can be grouped into two groups of 9 and 10 parameters, respectively, bydecomposing k µνρσG as k µνρσG = E µνρσ + W µνρσ , (3)where E µνρσ = 12 ( η µρ k νσ + η νσ k µρ − η νρ k µσ − η µσ k νρ ) , (4a) k µν = η αβ k µανβG , (4b)and η µν is the Minkowski metric tensor. This decomposition is similar to the Ricci decom-position of the Riemann curvature tensor, with E µνρσ the semi-traceless part build in termsof the Ricci curvature k µν and W µνρσ the fully traceless Weyl tensor (the would-be Ricciscalar vanishes because k µνρσG is doubly traceless). A convenient way of writing E µνρσ and W µνρσ is E µνρσ = 12 (cid:16) k µνρσG + ˘ k µνρσG (cid:17) , W µνρσ = 12 (cid:16) k µνρσG − ˘ k µνρσG (cid:17) , (5)with ˘ k µνρσG = (cid:15) µναβ (cid:15) ρσγδ ( k G ) αβγδ and (cid:15) µνρσ the Levi-Civita tensor with (cid:15) = +1. Someadditional intuition can be gained by comparing k µνρσG to its U (1) photon analogue, k µνρσF ,which has been studied in much greater detail than k µνρσG . Using Eq. (5) it is easy to see4hat the 10 independent components of W µνρσ are the gluon analogues of the birefringentcomponents of k µνρσF , which cause the vacuum to have an effective refractive index [13]. Incontrast, the 9 independent components of E µνρσ should be compared to the non-birefringentpart of k µνρσF [13]. Both E µνρσ and W µνρσ obey Eq. (2).Presently, the only reported bound on k µνρσG is obtained through quantum mixing of k µν with the LVC c µν of the electron [14]. The resulting bound is given by | ˜ k tr | = 23 | k | < × − . (6)This leaves 18 of the 19 independent CPT-even mSME pure-gluon parameters unbounded.However, one expects at least some of them to contribute to effective LV parameters fornucleons and hadrons, for which stringent limits have been obtained [6]. Therefore, we willconsider the EFT of QCD, chiral perturbation theory, which is formulated in terms of these(effective) degrees of freedom. III. THE EFFECTIVE CHIRAL LAGRANGIAN
We construct the low-energy effective chiral Lagrangian corresponding to Eq. (1) inthe formalism of Gasser and Leutwyler [15]. For a pedagogical introduction see Ref. [16]. Asany approach to χ PT, it is based on the observation that the QCD Lagrangian, containingonly gluons and the lightest three quarks, is approximately invariant under global SU (3) L × SU (3) R × U (1) V transformations of the quark fields (disregarding the axial U (1) A symmetry,which is broken by quantum anomalies). From the absence of parity doubling in the hadronspectrum one deduces that the axial SU (3) A part of the SU (3) L × SU (3) R symmetry mustbe spontaneously broken, leaving SU (3) V × U (1) V as the remaining symmetry group. Thepseudo-Goldstone bosons associated with the symmetry breaking are identified with the light J P = 0 − mesons, which have a small mass compared to the J P = 1 − vector mesons andthe J P =
12 + baryons. The nonzero masses of the pseudoscalar mesons originate from thefact that in the QCD Lagrangian the SU (3) A symmetry is explicitly broken by the (small)quark masses.The effective Lagrangian is constructed in terms of the low-energy degrees of freedom,i.e. the light-meson fields and the baryons, such that it contains all possible operators thatobey the symmetries of the QCD Lagrangian [7]. To correctly implement these symmetries,5he light-meson fields are collected in the unitary matrix U = exp( iφ a ( x ) λ a /F ) , (7)where φ a are the pseudo-Goldstone fields, λ a are the Gell-Mann matrices, and F (cid:39) Λ χ / (4 π )is the pion-decay constant in the limit of vanishing quark masses, i.e. the chiral limit. Thematrix U transforms as U → RU L † (8)under chiral transformations. Here, the global matrices R and L are independent SU (3)matrices. We disregard interactions with external fields in this work, since they appear onlyin higher-order effects (however, see Sec. VI). Consistent introduction of such interactionswould require Eq. (8) to become a local transformation.To apply the QCD symmetries to the effective baryon Lagrangian, one defines theunitary square root of the matrix U in Eq. (7) by u , i.e. u = U . The chiral transformationof u leads to the definition of the unitary matrix K = K ( L, R, U ) by u → u (cid:48) = √ RU L † ≡ RuK − , or K = ( RU L † ) − Ru = u (cid:48)† Ru = u (cid:48) Lu † . Subsequently, the
12 + baryon octet,described by eight Dirac spinors B a , with a = 1 , . . . ,
8, is represented by the traceless 3 × B = B a λ a √ , (9)that transforms under global SU (3) L × SU (3) R as B → KBK † . (10)The chiral covariant derivative of B is defined using the chiral connection Γ µ = (cid:2) u † ∂ µ u + u∂ µ u † (cid:3) and is given by D µ B = ∂ µ B + [Γ µ , B ] . (11)The final building block of the Lagrangian we need is the chiral vielbein, given by u µ = i (cid:2) u † ∂ µ u − u∂ µ u † (cid:3) , (12)that transforms as u µ → Ku µ K † under chiral transformations. Both Γ µ and u µ becomedependent on external fields, when one includes them.Based on the chiral-transformation properties of the different building blocks of theLagrangian one builds all operators that have the symmetry properties of the QCD La-grangian. These operators can be ordered by powers of the expansion parameter q/ Λ χ ,6here q ∼ m π (cid:28) Λ χ is the typical momentum of the process. We will discuss the powercounting in more detail in the next section.The lowest order light-meson and baryon Lagrangians are then given by [15, 17, 18] L φ = F (cid:2) ∂ µ U ( ∂ µ U ) † (cid:3) + F B (cid:2) M U † + U M † (cid:3) , (13a) L φB = Tr (cid:2) ¯ B ( i /D − m ) B (cid:3) + D (cid:2) ¯ Bγ µ γ { u µ , B } (cid:3) + F (cid:2) ¯ Bγ µ γ [ u µ , B ] (cid:3) , (13b)respectively. Here, M = diag( m u , m d , m s ) is the quark-mass matrix. The mass term in QCDLagrangian breaks chiral symmetry, However, it would be invariant under chiral transfor-mations if M would transform as M → R M L † . This property is mimicked by Eq. (13a) andit exemplifies how symmetry-breaking terms are incorporated into the formalism [17].Furthermore, B , D , and F are low-energy constants (LECs) whose size cannot bedetermined using symmetry arguments. However, an order-of-magnitude estimate can begiven, using naive dimensional analysis (NDA) [19]. In this case NDA gives B = O (Λ χ ),which leads for example to m π = O (( m u + m d )Λ χ ), which agrees fairly well with the actualpion mass. For this reason an insertion of the quark-mass matrix in the Lagrangian countsas O ( q ) for the power counting. The coefficients D and F are experimentally determinedto be D = 0 .
80 and F = 0 .
50 (at tree level) [20]. This also agrees well with NDA estimates,which give
D, F = O (1).In the same way the conventional QCD Lagrangian gives rise to a LS low-energyeffective Lagrangian, the LV operators in the QCD sector of the SME can be related toeffective operators in a LV chiral Lagrangian [8, 9]. Being a pure gluon operator, Eq. (1) istrivially invariant under chiral transformations of the quark fields. The lowest-order relevanteffective operators that capture this property, as well as the C, P, and T characteristics ofEq. (1), are given by L k G φ = F r k µν Tr (cid:2) ( ∂ µ U ) † ∂ ν U (cid:3) , (14a) L k G φB = ir k µν Tr (cid:2) ¯ Bγ µ D ν B (cid:3) + ir m ˜ W µνρσ Tr (cid:2) ¯ Bσ µν [ u ρ , D σ B ] (cid:3) + ir m ˜ W µνρσ Tr (cid:2) ¯ Bσ µν { u ρ , D σ B } (cid:3) , (14b)where ˜ W µνρσ = (cid:15) µναβ W ρσαβ . Additional operators with the correct symmetry properties canbe constructed at the present chiral order. However, they can all be shown to be redundantup to higher-order terms, using the leading-order equations of motion [8, 21], or by using7ymmetries of the Lorentz indices of the LV coefficients. We omitted any pure pion termsinvolving W µνρσ , since they are at least two orders higher in the chiral expansion than theterm in Eq. (14a). In the final two terms of Eq. (14b) we included a factor 1 /m such thatNDA designates all LECs r , . . . , r to be of order O (1).For applications to experimental observations, the explicit lowest order operators interms of the physical pion and nucleon fields are the most likely to be relevant. They canbe found directly from Eqs. (13) and (14) and are given by L π = 12 ( ∂ µ π ) · ( ∂ µ π ) − m π π , (15a) L πN = ¯ N (cid:20) i /D − m N − g A F π γ µ γ ( τ · ∂ µ π ) (cid:21) N , (15b) L k G π = r k µν ( ∂ µ π ) · ( ∂ ν π ) , (15c) L k G πN = ir k µν ¯ N γ µ ∂ ν N + i r + r m N ˜ W µνρσ ¯ N σ µν ( τ · ∂ ρ π ) ∂ σ N , (15d)where N = ( p, n ) T is the nucleon doublet, τ · π = (cid:18) π √ π + √ π − − π (cid:19) , and F π = 92 . W µνρσ does not contributeto any kinetic pion or nucleon terms (nor to any other kinetic terms in Eqs. (13a) and (13b)).This is reminiscent of a different LVC that was considered in Ref. [8]. As in that paper,also here it has great consequences for the way limits can be set on W µνρσ . As we will see,for k µν we can set limits using the very precise bounds that follow from clock-comparisonexperiments [6]. However, to leading order, these do not pertain to W µνρσ , because it doesnot contribute to properties of free protons and neutrons.8 V. POWER COUNTING AND THE HEAVY-BARYON APPROACH
A consistent power-counting scheme is necessary to turn the obtained effective theoryinto a practical tool. We have to know which (loop) diagrams contribute if a certain level ofprecision is required. To quantify this, one first defines the chiral index ∆, which representsthe importance of an operator in the Lagrangian [7, 15]. In terms of this chiral index a chiraldimension ν is defined, which specifies the significance of a renormalized Feynman diagram.A diagram of chiral dimension ν will contribute at order O ( q ν ), where q is a small quantityof the order of the pion mass.In the Lorentz invariant case, the chiral index for operators with at most two baryonfields is given by ∆ = d + f / − , (16)where f ≤ d is determined by the number of(covariant) derivatives plus twice the number of light-quark masses (since m q is proportionalto m π ). For a LV operator, we use the same definition of the chiral index. This does notdirectly account for the presence of the small LVC. However, since the coefficients for LVmust be heavily suppressed, the LS contributions will essentially always dominate over theLV ones, at least for energies that are relevant in the present context. Therefore, one neverneeds to compare chiral indices of LV interactions to those of LS interactions.A generic diagram will now contribute at the following chiral order [23]: ν = 2 N L + I B − N B + 2 + (cid:88) i ∆ i , (17)where N L , I B , and N B are the number of independent loops, internal baryon lines, and thetotal number of baryon vertices, respectively, while i runs over the different interactions thatcontribute to the diagram. For diagrams with exactly one baryon in the initial and finalstate, it holds that N B = I B + 1 (there are no closed fermion loops in the low-energy EFT)and ν becomes ν = 2 N L + 1 + (cid:88) i ∆ i , (18)Again, with a LV insertion, the diagram will be suppressed with respect to any diagramwithout such an insertion and we can use the same definition of ν for LV diagrams.It is well-known that the diagrams in relativistic meson-baryon theory only obey thepower-counting in Eq. (18) if the theory is properly renormalized [24]. Otherwise loop9 a) (b) (c) FIG. 1. Three loop diagrams that contribute to the nucleon self energy. The dots representconventional χ PT vertices, while the squares represent (different) LV insertions into the pion andnucleon propagators. The solid (dashed) lines are nucleon (pion) propagators. calculations will receive contributions of order m , which is not a small quantity, in fact m / Λ χ = O (1). These contributions will upset the power counting in Eq. (18). This holdsfor LS, as well as LV loop diagrams. For example, the nucleon self energy will receivea contribution from the diagram in Fig. 1(b), which involves a LV insertion in the pionpropagator, originating from Eq. (15c). The power counting predicts that this diagram willstart to contribute at order O ( q ν ) = O ( q ). However, if we calculate the value of the diagramusing dimensional regularization, we find that it gives a termΣ LV ( p = m ) = (cid:18) ˜ g A m π πm N ˜ F π − ˜ g A m π π m N ˜ F π (cid:19) r k µν p µ p ν + · · · , (19)where p µ is the nucleon momentum, m ph denotes the physical nucleon mass (to order q ),the tildes indicate renormalized quantities and the dots represent other LV contributions(not necessarily of higher order). We employed the modified minimal substraction schemeof chiral perturbation theory [15], commonly denoted by (cid:102) MS, and for simplicity took therenormalization parameter µ = m N . The first term in parentheses indeed is of order O ( q ),as one would expect from Eq. (18). However, the second term is of order O ( q ) and thusdoes not obey the assumed power counting. This already happens in the LS case, where asimilar contribution to the nucleon mass appears when the (cid:102) MS scheme is employed in therelativistic theory to calculate the nucleon self energy [24]. The power counting can be madeconsistent by absorbing additional finite terms by counter terms. Infrared regularization [25]and the extended on-mass-shell scheme [26] are examples of a systematic application of suchan approach. However, here we will employ a different approach called heavy-baryon chiralperturbation theory (HB χ PT) [27].The fact that the power-counting is upset in the relativistic theory in the (cid:102)
MS scheme,can be traced to the fact that the baryon mass is not small, i.e. time derivatives of the(static) heavy-baryon fields are of order m / Λ χ = O (1) (while all other terms in the baryon10ovariant derivative are of order O ( q )). To remedy this, in the heavy-baryon formalism thenucleon momentum is usually separated into a large and a small piece like p µ = m v µ + k µ ,where v µ represents a fixed baryon velocity, which obeys v = 1 and k µ is a small residualmomentum. For the LS case, k µ also parametrizes how far the nucleon is off-shell, since p = m if k = 0. However, in the LV case, the dispersion relation of the baryons, whichfollows from Eqs. (13b) and (14b), is ˜ p = m , with ˜ p µ = p µ + r k µν p ν . It is therefore moreconvenient to define the momentum separation by˜ p µ = m v µ + ˜ k µ , (20)where v = 1 still holds, ˜ k µ = k µ + r k µν k ν , and k µ remains to be a small residual momentum.It follows that v · ˜ k = − ˜ k m . We then define a new heavy-baryon field by B v = 12 (1 + /v ) e im ˆ v µ x µ B , (21)with ˆ v µ defined such that ( η µν + r k µν )ˆ v ν = v µ , i.e. to LO in LV ˆ v µ = ( η µν − r k µν ) v ν .Derivatives of these fields will give the small residual momentum such that all derivativescan be counted as O ( q ). The baryon propagator no longer contains the large baryon mass.This causes loop diagrams to obey the power counting in Eq. (18) without absorbing anyfinite terms by counter terms. Additionally, in HB χ PT, the Dirac matrices can be eliminatedin favor of the simpler velocity v µ and the covariant spin vector S µ = i γ σ µν v ν with S =(0 , (cid:126) Σ / (cid:126) Σ = γ γ (cid:126)γ , for v = (1 , (cid:126) L HB = Tr (cid:2) ¯ B ( iv · ∂ ) B (cid:3) + D Tr (cid:2) ¯ BS µ { u µ , B } (cid:3) + F Tr (cid:2) ¯ BS µ [ u µ , B ] (cid:3) + · · · , (22a) L k G HB = ir k µν Tr (cid:2) ¯ Bv µ ∂ ν B (cid:3) + 2 r W µνρσ v µ v σ Tr (cid:2) ¯ BS ν [ u ρ , B ] (cid:3) +2 r W µνρσ v µ v σ Tr (cid:2) ¯ BS ν { u ρ , B } (cid:3) + · · · , (22b)where we dropped the subscript v on B v . We only kept the leading term for each LVC andthe dots represent the higher-order terms and terms with more pions. The propagator ofthe baryon field becomes i/ ( v · ˜ k ), which indeed no longer contains a contribution from m .In terms of nucleon and pion fields, the HB χ PT Lagrangian is L HB = ¯ N ( iv · ∂ ) N − g A F π ¯ N S µ ( τ · ∂ µ π ) N + · · · , (23a) L k G HB = ir k µν ¯ N v µ ∂ ν N + 2( r + r ) W µνρσ v µ v σ ¯ N S ν ( τ · ∂ ρ π ) N + · · · , χ PT Lagrangian, we getΣ = 3 g A m π πF π (cid:18) − (cid:18) r − r (cid:19) k µν v µ v ν (cid:19) , (24)which only contains terms of order O ( q ). This shows that in HB χ PT these diagrams thusobey the power-counting rule in Eq. (18) (at least to the present order), as expected.
V. LIMITS FROM NUCLEON OBSERVABLES
The best limits on the semi-traceless part of k µνρσG come from the fact that it contributesto the effective c µν parameter for the proton and the neutron. In other words, the firstoperator in Eq. (15d) has the form ic µν ¯ ψγ µ ∂ ν ψ and this operator has been studied intensivelyfor the cases that ψ represents the proton or the neutron. The bounds on the componentsof the neutron and proton c µν translate almost directly to bounds on the correspondingcomponents of k µν .One has to keep in mind, however, that the operator in Eq. (15d) contains a LEC r whose size can only be estimated by NDA. Moreover, the effective c µν parameters forthe proton and neutron will receive additional contributions from other LV coefficients withthe same symmetry properties, in particular from several quark parameters, discussed inRef. [9]. It is impossible to completely disentangle the contributions from different coeffi-cients, using just the proton and neutron bounds. The best one can do is obtain a boundon the isospin even (odd) part of c µν by considering the sum (difference) of the neutron andproton coefficients ( k µν contributes to the isospin even part).On the other hand, it seems hard to imagine that the contributions from differentLVCs conspire to cancel to the level of the stringent proton and neutron bounds. Especiallybecause they all come with LECs that are not related by symmetry arguments. To beconservative, we have therefore set an order of magnitude bound on the components of k µν that is two orders of magnitude weaker than the best bound on the corresponding protonor neutron parameter. The results are summarized in Table I. In fact, only nine of the tencomponents in the table are independent, since we did not incorporate the tracelessness of k µν . 12 ensor component Limit Ref. k T T − [28, 29] k T J − [30] k JK − [31] k XX , k Y Y − [31] k ZZ − [30]TABLE I. Order-of-magnitude bounds on the LV components of k µν = η αβ k αµβνG in the Sun-centered inertial reference frame [6], with J, K ∈ {
X, Y, Z } . In the right-most column we referencethe papers where the corresponding bounds on c µν for the proton or the neutron were obtained. VI. LIMITS FROM PHOTON OBSERVABLES
In this section, we look at the fully traceless part of k µνρσG , defined in Eq. (3). As canbe seen from Eqs. (14a) and (14b) there are no kinetic terms in the lowest-order Lagragianwhen one does not include external fields. However, upon inclusion of electromagnetic fields W µνρσ induces the operator L EM = r F W µνρσ F µν F ρσ , (25)with F µν the photon field strength. The lowest-order Feynman diagram that induces thisoperator involves a quark-loop in the photon propagator, where a gluon is exchanged betweenthe internal quark lines. Therefore, the NDA estimate for the LEC r F is given by r F = O ( α/ (4 π )), with α the fine-structure constant. The operator in Eq. (25) has exactly thesame form as a photon operator [4] that involves the parameter k µνρσF . The fully tracelesspart of this coefficient, defined as in Eq. (3), causes birefringence of light in vacuum [13]. Byinvestigating the light from distant gamma-ray bursts (GRBs), very stringent limits havebeen set on the birefringent part of k µνρσF , which we denote here by W µνρσF . We see nowthat these bounds are actually bounds on W µνρσF + r F W µνρσ . Because of the symmetryproperties in Eq. (2) these are the only CPT-even mSME coefficients that, to leading orderin LV, contribute to birefringent effects in photons [32].Using bounds on birefringent photon coefficients [33] one can thus obtain bounds oncertain combinations of components of W µνρσ . We conservatively estimate these bounds tobe five orders of magnitude weaker than the limits on k µνρσF , i.e. three orders to account for13 ensor components Limit W T Y XZ , W T XY Z , W
T XXY , W
T XXZ , W
T Y XY − W T Y T Y , W
T ZT Z , W
T XT Y , W
T XT Z , W
T Y T Z − TABLE II. Order-of-magnitude bounds on the LV components of W µνρσ in the Sun-centered inertialreference frame [6], obtained by comparing to results in Ref. [33]. r F = O ( α/ (4 π )) and two orders for the uncertainty in r F and partial cancellations betweencoefficients. The resulting bounds are collected in Table II.These limits are not independent, in fact, they essentially come from just three mea-surements. However, any further cancellation between the different components seemsunlikely. Using Eq. (5) one can easily translate Table II to the corresponding limits on k µνρσG . One finds that of the components of W µνρσ in Table II, the top row correspondsto k aG = k G , k G , k G , k G , k G , respectively, while the bottom row corresponds to k G , . . . , k G ,respectively, with k aG = (cid:0) k G , k G , k G − k G , k G − k G , k G + k G , k G − k G , k G + k G ,k G + k G , k G − k G , k G − k G (cid:1) , (26)defined analogously to k a for the photon [13]. VII. POTENTIAL IMPROVEMENTS
The limits in Tables I and II are already quite strict. In fact, they can seem more thansufficient, when one compares them to a reasonable guess for the size of the dimensionlessLVCs: M ew /M pl (cid:39) − with M ew and M pl the electroweak scale and the Planck scale,respectively. However, one does not know what mechanism, if any, would induce theseoperators and what the associated mass scales are. There are even models where the LVparameters scale with some power of the temperature of the universe [34]. Also, all boundson k µν are based on one operator and therefore depend on one (renormalized) LEC. Similarly,all bounds on W µνρσ depend on r F (plus loop corrections). It is desirable to get boundsfrom different effective operators that involve the same LVCs, but different (renormalized)LECs. We consider one option for each coefficient in the following.14 . k µν and Cherenkov-like pion emission In addition to the nucleon operator in Eq. (15d), the semi-traceless part of k µνρσG alsoinduces the kinetic pion operator in Eq. (15c). Such a pion operator has been studiedbefore on several occasions [35]. One of the potential observational consequences of thisoperator is that it induces an effective refractive index for the vacuum, in the sense thatthe maximal attainable velocity of the pion will be smaller or larger than the speed oflight. If the coefficients have the correct sign, then Cherenkov-like processes can occur, e.g.protons with an energy above some threshold E th will start emitting (neutral) pions untiltheir energy falls below E th . An easy way to see this is by realizing that such a processrequires the pion to have a spacelike momentum (for simplicity we assume that the protonhas a conventional kinetic term). The LV dispersion relation for the pion is then given by p + r k µν p µ p ν − m π = 0 and therefore the threshold condition becomes | (cid:126)p | > m π r k µν ˆ p µ ˆ p ν , (27)with ˆ p µ = p µ / | (cid:126)p | . Clearly, this is only a physical threshold if r k µν ˆ p µ ˆ p ν > k µν of r k µν ˆ p µ ˆ p µ (cid:46) − . (28)This has the potential of improving at least some of the limits in Tab. I.Notice that actually obtaining limits from Cherenkov-like processes requires some the-oretical work. Strictly speaking, one has to calculate the actual emission rate and demon-strate that it does not vanish. But more importantly, one has to verify that the theorywith spacelike momenta can be made consistent, since spacelike momenta correspond tonegative energies in some observer frames (notably the restframe of the decaying particle).For a different LVC it has been shown that such a theory can nevertheless be quantized anddoes not contain runaway stability issues [12]. For the present parameter this remains to be15hown, however. B. W µνρσ and the nucleon-nucleon potential The fully traceless part of k µνρσG does not appear in any kinetic term for baryons orlight mesons. The lowest-order term allowed by all the symmetries is the one in Eq. (15d),which is a pion-nucleon interaction term. A similar situation was identified in Ref. [8] for adifferent LVC. As in Ref. [8], we can calculate the nucleon-nucleon potential that follows fromconsidering one-pion exchange between the two nucleons, with one of the vertices originatingform the LV πN operator in Eq. (15d). The resulting potential is given by V LV = − g A F ( r + r ) W i j τ · τ ( σ i σ m + σ m σ i ) q j q m q + m π , (29)where σ , ( τ , ) are spin (isospin) operators corresponding the interacting nucleons and q = p − p (cid:48) is the momentum transfer that flows from nucleon 1 to nucleon 2, while p and p (cid:48) are the relative momenta of the incoming and outgoing nucleon pair in the center-of-massframe. As usual, the Latin indices run only over spatial directions.We thus see that W µνρσ induces an isospin even two-body operator in the nucleon-nucleon potential. We leave the detailed study of this operator for future work. Obviously,one should expect physical consequences of such a term in nuclear systems with two or morenucleons. For example clock-comparison experiments will most likely be able to providelimits on several components of W µνρσ . However, a multipole decomposition of W i j showsthat it only has parts with an angular momentum quantum number of l = 2. And there-fore, at least to leading order, W i j will contribute only to clock-comparison experimentsinvolving nuclei with a nuclear spin of I ≥
1. The best corresponding sensitivity for LVwas obtained in a co-magnetometer experiment involving
Cs, which has I = 7 / − GeV on a dimensionful LV parameter wasset. Naively we thus expect a bound in the order of 10 − on ( r + r ) W i j from suchexperiments.A related possibility is to study Eq. (29) in the context of the spin precession ofthe deuteron or other light nuclei in a storage ring [38]. Especially the deuteron has theadvantage that one does not have to assume a nuclear model to calculate the physicalobservables. Studying the sidereal variation of the spin-precession frequency of the deuteron16ould provide limits on W µνρσ that are complementary to those quoted in Table II and topotential bounds from clock-comparison experiments. Most likely, such experiments willnot improve on the results in Table II, however, they have the added benefit that they willbe laboratory bounds, which generally involve less assumptions than astrophysical bounds,such as those in Table II. VIII. SUMMARY AND CONCLUSION
In this paper, we constructed the chiral effective Lagrangian that is induced by thepure gluon LV operator of mass dimension four, which is part of the mSME Lagrangian. Wewrote down the dominant operators in the context of a three-flavor SU (3) as well as a two-flavor SU (2) formalism. Relations between the LECs in these two formalisms are yet to beobtained, but are not very relevant for obtaining order-of-magnitude limits on the LVCs. Wedeveloped the power-counting rules and showed that in a relativistic meson-baryon theoryin the (cid:102) MS scheme, loop contributions to the LVCs upset the power counting, analogous toLS contributions in the conventional theory. To deal with this situation, we wrote down thedominant operators in heavy-baryon chiral perturbation theory and used them to calculatethe LV contribution to the nucleon self energy to order O ( q ).The symmetries of the mSME gluon operator constrain the form of the chiral effectiveoperators in such a way that kinetic hadron terms can be written down for nine of thenineteen independent components of k µνρσG . We showed that these can be bounded by clock-comparison experiments. The resulting limits are collected in Tab. I. We also suggestedthat additional and improved bounds can be obtained by considering Cherenkov-like pionemission by high-energy protons.From our constructed chiral effective Lagrangian we concluded that bounds on theremaining 10 components of k µνρσG , collected in W µνρσ , cannot be obtained from considera-tions of free nucleon properties. Therefore, presently available analyses of clock-comparisonexperiments do not pertain to these components of k µνρσG . On the other hand, W µνρσ doesinduce a photon operator that causes the birefringence of light in vacuum. Therefore wewere able to translate existing photon bounds to bounds on W µνρσ . These are collected inTable II. Additional and complementary bounds on the fully traceless part of k µνρσG can mostlikely be obtained by considering the effect of the nucleon-nucleon potential in Eq. (29) on17lock-comparison experiments and storage ring experiments involving the deuteron. ACKNOWLEDGMENTS
We thank J. de Vries and R. Potting for helpful suggestions. This work is sup-ported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia of Portugal (FCT) through projectsUID/FIS/00099/2013 and SFRH/BPD/101403/2014 and program POPH/FSE. [1] A. Einstein, Annalen Phys. , 891 (1905) [Annalen Phys. , 194 (2005)]; E. P. Wigner,Annals Math. , 149 (1939) [Nucl. Phys. Proc. Suppl. , 9 (1989)]; S. Weinberg, The QuantumTheory of Fields , Vol. 1 (Cambridge University Press, Cambridge, 1995).[2] V. A. Kosteleck´y and S. Samuel, Phys. Rev. D , 683 (1989); V. A. Kosteleck´y and R. Potting,Nucl. Phys. B , 545 (1991); J. R. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Gen.Rel. Grav. , 1257 (1999); R. Gambini and J. Pullin, Phys. Rev. D , 124021 (1999);C. P. Burgess, J. M. Cline, E. Filotas, J. Matias and G. D. Moore, JHEP , 043 (2002).[3] V. A. Kosteleck´y, arXiv:0802.0581 [gr-qc].[4] D. Colladay and V. A. Kosteleck´y, Phys. Rev. D , 6760 (1997); Phys. Rev. D , 116002(1998).[5] O. W. Greenberg, Phys. Rev. Lett. , 231602 (2002).[6] V. A. Kosteleck´y and N. Russell, Rev. Mod. Phys. , 11 (2011) [2016 edition:arXiv:0801.0287v9 [hep-ph]].[7] S. Weinberg, Physica A , 327 (1979); H. Leutwyler, Annals Phys. , 165 (1994).[8] J. P. Noordmans, J. de Vries and R. G. E. Timmermans, Phys. Rev. C , 025502 (2016).[9] R. Kamand, B. Altschul and M. R. Schindler, arXiv:1608.06503 [hep-ph].[10] D. B. Kaplan and M. J. Savage, Nucl. Phys. A , 653 (1993); ibid . , 833(E) (1994); ibid . , 679(E) (1994).[11] J. de Vries, E. Mereghetti, R. G. E. Timmermans and U. van Kolck, Annals Phys. , 50(2013); J. Bsaisou, U. G. Meiner, A. Nogga and A. Wirzba, Annals Phys. , 317 (2015).[12] D. Colladay, P. McDonald, J. P. Noordmans and R. Potting, arXiv:1610.00169 [hep-th].[13] V. A. Kosteleck´y and M. Mewes, Phys. Rev. Lett. , 251304 (2001).
14] C. D. Carone, M. Sher and M. Vanderhaeghen, Phys. Rev. D , 077901 (2006).[15] J. Gasser and H. Leutwyler, Annals Phys. , 142 (1984); Nucl. Phys. B , 465 (1985).[16] S. Scherer and M. R. Schindler, A Primer for Chiral Perturbation Theory , (Heidelberg, 2012).[17] H. Georgi,
Weak interactions and Modern Particle Theory , (Menlo Park, 1984).[18] A. Krause, Helv. Phys. Acta , 3 (1990).[19] A. Manohar and H. Georgi, Nucl. Phys. B , 189 (1984); H. Georgi and L. Randall, Nucl.Phys. B , 241 (1986).[20] B. Borasoy, Phys. Rev. D , 054021 (1999).[21] N. Fettes, U. G. Meissner, M. Mojzis and S. Steininger, Annals Phys. , 273 (2000) Erratum:[Annals Phys. , 249 (2001)].[22] M. Mai, P. C. Bruns, B. Kubis and U. G. Meissner, Phys. Rev. D , 094006 (2009).[23] S. Weinberg, The quantum theory of fields , Vol. 2: (Cambridge University Press, Cambridge,1996).[24] J. Gasser, M. E. Sainio and A. Svarc, Nucl. Phys. B , 779 (1988).[25] T. Becher and H. Leutwyler, Eur. Phys. J. C , 643 (1999).[26] J. Gegelia and G. Japaridze, Phys. Rev. D , 114038 (1999); T. Fuchs, J. Gegelia andS. Scherer, Eur. Phys. J. A , 35 (2004).[27] E. E. Jenkins and A. V. Manohar, Phys. Lett. B , 558 (1991); V. Bernard, N. Kaiser,J. Kambor and U. G. Meissner, Nucl. Phys. B , 315 (1992).[28] S. R. Coleman and S. L. Glashow, Phys. Lett. B , 249 (1997).[29] A. Kosteleck´y and M. Mewes, Phys. Rev. D , 096006 (2013).[30] P. Wolf, F. Chapelet, S. Bize and A. Clairon, Phys. Rev. Lett. , 060801 (2006).[31] M. Smiciklas, J. M. Brown, L. W. Cheuk and M. V. Romalis, Phys. Rev. Lett. , 171604(2011).[32] One could argue that there is also a contribution coming from a W -boson loop. This wouldinvolve a contribution from k µνρσW and would be suppressed by a factor q /M W (cid:39) − ,where q (cid:39) − GeV is the typical energy of photons coming from GRBs. Additionally, thereare contributions from a mass-dimension three CPT-odd coefficient. However, for this LVCthe birefringence is energy independent and therefore the studies of GRBs are not sensitiveto it. This CPT-odd coefficient has been bounded by looking at the isotropy of the CMBpolarization and determined to be of order k µAF < − GeV [6].
33] V. A. Kosteleck´y and M. Mewes, Phys. Rev. D , 015020 (2009); V. A. Kosteleck´y andM. Mewes, Phys. Rev. Lett. , 201601 (2013).[34] M. de Cesare, N. E. Mavromatos and S. Sarkar, Eur. Phys. J. C , no. 10, 514 (2015).[35] J. P. Noordmans and K. K. Vos, Phys. Rev. D , 101702 (2014); B. Altschul, Phys. Rev. D , 105018 (2008); Phys. Rev. D , 105007 (2016).[36] A. Aab et al. [Pierre Auger Collaboration], Astrophys. J. , no. 1, 15 (2015); T. Abu-Zayyad et al. [Telescope Array Collaboration], Astrophys. J. , 88 (2013).[37] S. K. Peck, D. K. Kim, D. Stein, D. Orbaker, A. Foss, M. T. Hummon and L. R. Hunter,Phys. Rev. A , 012109 (2012).[38] D. Eversmann et al. [JEDI Collaboration], Phys. Rev. Lett. , 094801 (2015)., 094801 (2015).