An EFT toolbox for baryon and lepton number violating dinucleon to dilepton decays
AAn EFT toolbox for baryon and lepton number violating dinucleon todilepton decays
Xiao-Gang He
1, 2, 3, ∗ and Xiao-Dong Ma
2, 4, † Tsung-Dao Lee Institute, and School of Physics and Astronomy,Shanghai Jiao Tong University, 200240, China Department of Physics, National Taiwan University, Taipei 106, Taiwan Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan School of Nuclear Science and Technology,Lanzhou University, Lanzhou 730000, China
Abstract
In this paper we systematically consider the baryon ( B ) and lepton ( L ) number violating dinucleon todilepton decays ( pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) , nn → ¯ ν ¯ ν (cid:48) ) with ∆ B = ∆ L = − in the framework of effectivefield theory. We start by constructing a basis of dimension-12 (dim-12) operators mediating such processesin the low energy effective field theory (LEFT) below the electroweak scale. Then we consider their standardmodel effective field theory (SMEFT) completions upwards and their chiral realizations in baryon chiralperturbation theory (B χ PT) downwards. We work to the first nontrivial orders in each effective field theory,collect along the way the matching conditions, and express the decay rates in terms of the Wilson coefficientsassociated with the dim-12 operators in SMEFT and the low energy constants pertinent to B χ PT. We findthe current experimental limits push the associated new physics scale larger than − TeV, which is stillaccessible to the future collider searches. Through weak isospin symmetry, we find the current experimentallimits on the partial lifetimes of transitions pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) imply stronger limits on nn → ¯ ν ¯ ν (cid:48) than their existing lower bounds, which are improved by − orders of magnitude. Furthermore, assumingcharged mode transitions are also dominantly generated by the similar dim-12 SMEFT interactions, theexperimental limits on pp → e + e + , e + µ + , µ + µ + lead to stronger limits on pn → (cid:96) + α ¯ ν β with α, β = e, µ than their existing bounds. Conversely, the same assumptions help us to set a lower bound on the lifetime ofthe experimentally unsearched mode pp → e + τ + from that of pn → e + ¯ ν τ , i.e., Γ − pp → e + τ + (cid:38) × yrs . ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] F e b ONTENTS
I. Introduction 2II. Operator basis for pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) , nn → ¯ ν ¯ ν (cid:48) transitions 5A. LEFT operators 8B. SMEFT completions 10C. Relations between SMEFT and LEFT operators 12III. Chiral realizations 14A. Some basics of Chiral matching 14B. Decomposition of irreducible chiral symmetry 16C. Chiral matching for the operators 19IV. Dinucleon and dilepton transition rate 23V. Conclusion and outlook 29Acknowledgement 30Appendix A. Independent color tensors 31Appendix B. LEFT operators 32Appendix C. Reduction of the redundant operators in the LEFT 36Appendix D. SMEFT operators 39Appendix E. Reduction of the redundant operators in the SMEFT 40Appendix F. Chiral basis construction 42Appendix G. Chiral irreducible representations in terms of hadrons 47Appendix H. LEFT and SMEFT contributions to C pp , C pn and C nn . INTRODUCTION The observed matter-antimatter asymmetry of the universe requires the violation of baryonnumber, which as one of the three Sakharov conditions for a successful baryogengesis mecha-nism [1]; at the same time, the extremely possible Majorana nature of neutrinos breaks the leptonnumber. Both facts lead to the existence of a class of new physics scenarios beyond the minimalstandard model (SM) in which the baryon and/or lepton numbers are violated explicitly or sponta-neously to a certain degree, this is because the baryon ( B ) and lepton ( L ) numbers are accidentalglobal symmetries in the SM but are violated through the quantum anomaly in an unobservablelevel [2] . On the other hand, the violation of baryon and/or lepton numbers would be expectedin the grand unified theories (GUT) [3, 4]. Thus, the search of rare baryon and/or lepton numberviolation signals becomes more and more important than ever before for the pursuit of new physics(NP).Usually, the experimentally accessible baryon number violation signatures can be categorizedinto two classes in terms of the net baryon number being changed by one unit ( ∆ B = 1 ) ortwo units ( ∆ B = 2 ). For the ∆ B = 1 case, the relevant processes are the single free or boundnucleon decays, which could change the lepton numbers with ∆ L = ± or ∆ L = ± units, likethe most well-known proton decay mode p → e + π , etc, seeing Ref. [5] and references thereinfor a thorough discussion on these ∆ B = 1 processes. Such processes have been searched forexperimentally for a long time but with null results [6], which, however, tightly constrain the NPscenarios and push the NP scale around the GUT scale unaccessible directly for the current andfuture collider experiments.On the other hand, for the ∆ B = 2 case, the net lepton numbers can be changed by either ∆ L = 0 or ∆ L = ± units. They are interesting because they can be searched for experimentallywith clear signatures and high precision, as well as because there exist NP scenarios in which theircontributions are dominant but that of ∆ B = 1 processes like proton decay are suppressed [7–11].The neutron-antineutron oscillation ( n − ¯ n ) is a representative example for the ∆ L = 0 case, whichattracts a lot of attentions recent years both theoretically and experimentally, seeing the review [12]and references therein for a summary on the state of the art of this process. For the ∆ L = ± case,the interesting processes are the dinucleon to dilepton decays in nuclei, including pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) and nn → ¯ ν ¯ ν (cid:48) with (cid:96) ( (cid:96) ) = e, µ, τ and ¯ ν (¯ ν (cid:48) ) = ¯ ν e , ¯ ν µ , ¯ ν τ , respectively . Such processes The difference B − L is still an exact symmetry survived from the anomaly cancellation in the SM. There are also ∆ L = 0 dinucleon decays with the final states being a pair of leptons conserving flavor/number or ecay mode Lifetime limit Decay mode Lifetime limit Decay mode Lifetime limit pp → e + e + . × yrs [20] pn → e + ¯ ν (cid:48) . × yrs [21] nn → ¯ ν ¯ ν (cid:48) . × yrs [19] pp → e + µ + . × yrs [20] pn → µ + ¯ ν (cid:48) . × yrs [21] pp → µ + µ + . × yrs [20] pn → τ + ¯ ν (cid:48) . × yrs [21] pp → e + τ + - TABLE I. The lower bound on the partial lifetimes of the dinucleon to dilepton transitions. Where thelimits for the charged modes ( pp ) and ( pn ) were set per oxygen nucleus O in water Cherenkov Super-Kexperiment at confidence level [20, 21] while the limit on the neutral mode ( nn ) is obtained from thecarbon nucleus C in KamLAND experiment [19]. The bound on modes involving anti-neutrinos are alsovalid for the cases with neutrinos since they are treated as invisible states in the experimental search. get less attentions than the above mentioned ∆ B = 1 and n − ¯ n oscillation processes [8, 11, 15, 16],but they may open new avenues for the baryon number violation signals due to their distincttheoretical origin and clear experimental signatures. The most stringent lower limits on the partiallifetimes of those dinucleon to dilepton decays in oxygen O and carbon C nuclei are reportedby the earlier Frejus and KamLAND experiments [18, 19] and the recent Super-K results [20, 21],and are collected in Tab. I for our later use.Confronted with the relatively less studies on those baryon and lepton number violating ∆ B =∆ L = − decays pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) and nn → ¯ ν ¯ ν (cid:48) , it is our goal in this paper to makea comprehensive analysis via the model-independent framework of effective field theory (EFT).Our approach is pictorially explained in Fig. 1 which shows the series of EFTs relevant to thedecays, including the relevant degrees of freedom and symmetries in each EFT, and the matchingand renormalization group running procedures among different EFTs. We start by constructinga basis mediating such processes using the first two flavors of quarks ( u, d ) and the charged andneutral leptons ( (cid:96), ν ) that enjoys the QCD and QED gauge symmetries SU (3) C × U (1) EM in thelow energy effective field theory (LEFT) below the electroweak scale Λ EW . At leading order in theLEFT, the contribution arises from effective interactions of dimension-12 (dim-12) ∆ B = ∆ L = − operators that involve six quark fields and two lepton fields ( qqqqqqll ) , where q denotes u, d while l for (cid:96), ν . By crossing symmetry, those operators also parametrize the general interactionscontributing to the baryon and lepton number violating conversion modes e − p → e + ¯ p, ¯ ν ¯ n and two mesons, like pn → (cid:96) + ν , nn → ν ¯ ν and pp → π + π + , K + K + [13, 14]. A timely brief status report of the ∆ B = 2 physics can be found in [17]. IG. 1. The flow chart for an EFT calculation of the low energy observables. e − n → ¯ ν ¯ p in the electron-deuteron ( e -d) scattering [10], the neutron exotic decay mode n → ¯ pe + ¯ ν [22] and the hydrogen-antihydrogen ( H − ¯H ) oscillation [23].To translate the experimental constraints at low energy to those on NP at a high scale Λ NP , wehave to climb up the ladder of energy scales in Fig. 1. If there are no new particles with a mass ator below the electroweak scale Λ EW , the standard model effective field theory (SMEFT) definedbetween some NP scale Λ NP and the electroweak scale Λ EW is a good starting point to parametrizethe UV NP in a model-independent way. Based on this logic, we then consider the leading-orderSMEFT completions of the LEFT interactions in question and it happens that the relevant operatorsalso first appear at dim 12, and without counting lepton flavors we obtain 29 independent operatorscontributing to the decays. Along the way we perform a tree-level matching between the SMEFTand LEFT interactions at Λ EW , and it turns out that the interaction structures simplify significantlydue to the constraints of the SM gauge symmetry SU (3) C × SU (2) L × U (1) Y and many LEFTWilson coefficients vanish at this leading order.To calculate the transition rate, at low energy which we take to be the chiral symmetry break-ing scale Λ χ , we perform the non-perturbative matching of the LEFT interactions of quarks tothose of nucleons and mesons using the two-flavor baryon chiral perturbation theory (B χ PT) for-malism [24–26]. In order to realize this, we first organize the LEFT operators into irreduciblerepresentations under the chiral group SU (2) f L × SU (2) f R of chiral u, d quarks and then constructthe corresponding hadronic operators using the spurion techniques with the non-perturbative QCDeffect being encoded in the so-called low energy constants (LECs). These LECs may be extractedby chiral symmetry from other measured processes, or computed in lattice theory (LQCD), orestimated based on the naive dimensional analysis [27].4ast we collect all pieces together to express the transition rate as the function of the Wilsoncoefficients of effective interactions in SMEFT and the LECs. The merit in such an approach isthat the uncertainties incurred in the result may be estimated systematically. Taking the currentexperimental limits into consideration, we find the NP scale is pushed around 1-3 TeV, which isaccessible to the future collider searches to complementarily study the relevant NP. If we furtherassume these transitions are dominantly generated by the similar dim-12 SMEFT operators, thenwe find the three different types of transitions are correlated with each other via the weak isospinsymmetry, and the experimental limits on the partial lifetimes of one type transitions may betranslated into stronger limits on that of another type transitions than their existing experimentallower bounds. In addition, this correlation helps us to set a lower bound on the lifetime of theexperimentally unsearched decay mode pp → e + τ + from that of pn → e + ¯ ν τ , and we find thepartial lifetime is constrained to be Γ − pp → e + τ + (cid:38) × yrs .This paper is organized as follows. In section II, we will first list at quark level the operators fordinucleon and dilepton transitions. We then establish the basis of dim-12 operators contributingto the dinucleon to dilepton transitions with ∆ B = ∆ L = − in LEFT in subsection II A,and we will do a similar job to establish the leading order dim-12 operators contributing to thesame transitions in SMEFT in the subsection II B. In subsection II C, we give tree-level matchingrelations between LEFT and SMEFT operators. Section III is devoted to the chiral realizationof the six-part of these dim-12 operators at the chiral symmetry breaking scale. After some briefexplanation how to perform matches to chiral theory from quark level operators in subsection III A,in subsections III B and III C, we discuss the chiral irreducible representation decompositions andmatching operators. In Section IV, we calculate the decay rate from the series of EFTs and makepredictions for the sensitivity of such processes for future experimental searches. We summarizeour main results in section V. In Appendices A to H we list the operator basis in each EFT obtainedand chiral decompositions and matches leading to the main results described in the text. II. OPERATOR BASIS FOR pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) , nn → ¯ ν ¯ ν (cid:48) TRANSITIONS
Since all operators we are interested in violate baryon and lepton numbers and are thus non-Hermitian, we only list one half of them relevant to this work, and the other half is easily obtainedby Hermitian conjugation. To have dinucleon to dilepton transitions of ∆ B = ∆ L = − , thequark level operators first appear at dim 12 and contain six quark fields made out of the up and5own quarks ( u, d ) and two leptons ( (cid:96), ν ). By the Fierz transformation, these operators can befactorized as the pure quark sectors multiplying by the proper lepton currents. At the hadron level,according to the initial state nucleons or the final state leptons, by the electric charge conservation,we have three different types of transitions: pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) and nn → ¯ ν ¯ ν (cid:48) . Thecorresponding lepton currents for each type of transition are defined as follows pp → (cid:96)(cid:96) (cid:48) : j (cid:96)(cid:96) (cid:48) S, ± = ( (cid:96) T CP ± (cid:96) (cid:48) ) , j (cid:96)(cid:96) (cid:48) ,µV = ( (cid:96) T R Cγ µ (cid:96) (cid:48) L ) , j (cid:96)(cid:96) (cid:48) ,µνT, ± = ( (cid:96) T Cσ µν P ± (cid:96) (cid:48) ) , (1) pn → (cid:96) ¯ ν (cid:48) : j (cid:96)ν (cid:48) S = ( (cid:96) T L Cν (cid:48) L ) , j (cid:96)ν (cid:48) ,µV = ( (cid:96) T R Cγ µ ν (cid:48) L ) , j (cid:96)ν (cid:48) ,µνT = ( (cid:96) T L Cσ µν ν (cid:48) L ) , (2) nn → ¯ ν ¯ ν (cid:48) : j νν (cid:48) S = ( ν T L Cν (cid:48) L ) , j νν (cid:48) ,µνT = ( ν T L Cσ µν ν (cid:48) L ) , (3)where C is the charge conjugation matrix satisfying C T = C † = − C and C = − , thecharged lepton field is denoted by (cid:96), (cid:96) (cid:48) ∈ { e, µ, τ } and the SM left-handed neutrinos by ν L , ν (cid:48) L ∈{ ν e , ν µ , ν τ } . The chiral projection operators are abbreviated as P ± ≡ P R,L = (1 ± γ ) / and there-fore (cid:96) R,L = P R,L (cid:96) = P ± (cid:96) . Here we see that for the diproton and dineutron transitions pp → (cid:96) + (cid:96) (cid:48) + , nn → ¯ ν ¯ ν (cid:48) , the scalar lepton current is symmetric under the exchange of the two leptons whilethe tensor lepton current is anti-symmetric. Especially, for the transitions with identical chargedleptons pp → e + e + , µ + µ + the tensor lepton current vanishes, and the non-vanishing scalar andvector currents can be equivalently parametrized using the four-component Dirac fields as j S = ( (cid:96) T C(cid:96) ) , j S, = ( (cid:96) T Cγ (cid:96) ) , j µV, = ( (cid:96) T Cγ µ γ (cid:96) ) , (4)and the vector current ( (cid:96) T Cγ µ (cid:96) ) = 0 from the same reason as the tensor current.For the dinucleon to dilepton transitions, based on the above classification, we may first writea minimal hadron level effective Lagrangian consisting of nucleon currents N T Γ N (cid:48) with N, N (cid:48) ∈ ( p, n ) and the lepton currents in Eq. (1-3) into the following form L NN (cid:48) = (cid:88) a C ( pp ) a O ( pp ) a + (cid:88) b C ( pn ) b O ( pn ) b + (cid:88) c C ( nn ) c O ( nn ) c , (5)where a, b, c are just labels to distinguish different operators, and C ( NN (cid:48) ) a,b,c are the Wilson coeffi-cients associated with the relevant hadronic operators O ( NN (cid:48) ) a,b,c . Those operators O ( NN (cid:48) ) i are dim-6and composed of dinucleon and the above lepton currents, and can be written as pp → (cid:96) + (cid:96) (cid:48) + : O ( pp ) SL = ( p T Cp )( (cid:96) T L C(cid:96) (cid:48) L ) , O ( pp ) S L = ( p T Cγ p )( (cid:96) T L C(cid:96) (cid:48) L ) , We do not include the right-handed neutrinos since they are absent in the SM framework. ( pp ) SR = ( p T Cp )( (cid:96) T R C(cid:96) (cid:48) R ) , O ( pp ) S R = ( p T Cγ p )( (cid:96) T R C(cid:96) (cid:48) R ) , O ( pp ) V = ( p T Cγ µ γ p )( (cid:96) T R Cγ µ (cid:96) (cid:48) L ) , (6) pn → (cid:96) + ¯ ν (cid:48) : O ( pn ) SL = ( p T Cn )( (cid:96) T L Cν (cid:48) L ) , O ( pn ) S L = ( p T Cγ n )( (cid:96) T L Cν (cid:48) L ) , O ( pn ) VL = ( p T Cγ µ n )( (cid:96) T R Cγ µ ν (cid:48) L ) , O ( pn ) V L = ( p T Cγ µ γ n )( (cid:96) T R Cγ µ ν (cid:48) L ) , O ( pn ) T = ( p T Cσ µν n )( (cid:96) T L Cσ µν ν (cid:48) L ) , (7) nn → ¯ ν ¯ ν (cid:48) : O ( nn ) SL = ( n T Cn )( ν T L Cν (cid:48) L ) , O ( nn ) S L = ( n T Cγ n )( ν T L Cν (cid:48) L ) , (8)where there are no operators with a vector or tensor diproton/dineutron current since such currentsvanish, i.e., ( N T Cγ µ N ) = ( N T Cσ µν N ) = ( N T Cσ µν γ N ) = 0 for N = p, n .Since the dinucleon fields in the above operators must originate from the quarks, it is thusnecessary to start from the classification of relevant operators at the quark level. There are twoapproaches in constructing relevant quark level operators responsible to dinucleon to dileptontransitions. One is to use effective degrees of freedom below the electroweak scale to obtain allrelevant operators respecting SU (3) C × U (1) EM , this is the LEFT approach. And the other is to useeffective degrees of freedom of the SM to obtain all relevant operators respecting the SM gaugegroup SU (3) C × SU (2) L × U (1) Y , the SMEFT. We will obtain the operators in both approaches.In both the LEFT and SMEFT approaches the color SU (3) C symmetry must be respected, thesix quarks in all operators obtained must form color singlets. We denote a general six-quark fieldconfiguration as O ijklmn = q iI q jJ q kK q lL q mM q nN , (9)where the superscripts { i, j, k, l, m, n } are the color indices in fundamental representation of SU (3) C while the subscripts { I, J, K, L, M, N } encode the flavor and chiral information for eachquark field. To form a color invariant operator, the color indices must be contracted by a colortensor T ijklmn such that O ijklmn T ijklmn is invariant under SU (3) C . This color symmetry can beachieved by contracting the color indices ( i, j, k, l, m, n ) in the quark fields with the following fiveindependent color tensors T SSS { ij }{ kl }{ mn } = (cid:15) ikm (cid:15) jln + (cid:15) ikn (cid:15) jlm + (cid:15) ilm (cid:15) jkn + (cid:15) iln (cid:15) jkm ,T SAA { ij } [ kl ][ mn ] = (cid:15) imn (cid:15) jkl + (cid:15) ikl (cid:15) jmn , T SAA { kl } [ mn ][ ij ] = (cid:15) ijk (cid:15) mnl + (cid:15) ijl (cid:15) mnk ,T SAA { mn } [ ij ][ kl ] = (cid:15) ijm (cid:15) kln + (cid:15) ijn (cid:15) klm , T AAA [ ij ][ kl ][ mn ] = (cid:15) ijm (cid:15) kln − (cid:15) ijn (cid:15) klm . (10)We put the details for the color tensor constructions and their subtleties in Appendix A.7 . LEFT operators In the LEFT framework, the effective degrees of freedom are the SM light quarks ( u, d, s, c, b ) ,charged and neutral leptons ( e, µ, τ, ν e , ν µ , ν τ ) , and the effective interactions are governed by thehigher dimensional local operators Q di built out of those fields and satisfying the SM residualsymmetry SU (3) C × U (1) EM . The LEFT Lagrangian L LEFT is organized in terms of the canonicalmass dimension of the local operators L LEFT = L dim ≤ + (cid:88) dim 5 ,i ˆ C ,i Λ Q i + (cid:88) dim 6 ,i ˆ C ,i Λ Q i + · · · + (cid:88) dim 12 ,i ˆ C ,i Λ Q i + · · · , (11)where L dim ≤ is the renormalizable terms, and the Wilson coefficients ˆ C d,i together with the heavyscale Λ encode informations about the presumed fundamental physics. The systematic enumer-ation of operator bases up to dim 9 have been figured out in [28–31]. In our case, the relevantoperators appear first at dim 12 and can be parametrized as the product of six-quark sectors and alepton current shown in Eqs. (1-3).For the quark sectors, we can repeatedly apply the Fierz identities (FI) to reach as many scalarbilinear quark currents as possible. In this way, by the Lorentz symmetry, for the operators witha scalar lepton current their quark sectors can be factorized as three scalar quark currents, andfor the operators with a vector (tensor) lepton current their quark sectors can be factorized astwo scalar quark currents along with a single vector (tensor) quark current. Lastly, the color SU (3) C invariance can be done by contracting the free quark color indices using the independentcolor tensors constructed earlier. In doing so, one should be careful with operators containingseveral identical quark fields since the color relations in Eq. (A.4) and the FIs in Appendix C mayfurther restrict their independency. Combining the above points and excluding potential redundantoperators, and for an arbitrary flavor n f ( equals 3 for the real case) of lepton fields ( (cid:96), ν ) , the finalbases of the dim-12 operators mediating the transitions pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) , nn → ¯ ν ¯ ν (cid:48) aresummarized one by one as follows: (cid:4) Dim-12 operators contributing to pp → (cid:96) + (cid:96) (cid:48) + One can attach different lepton currents to operators already formed by six quarks to form theoperators. We will discuss for each types in the following. For a scalar current j (cid:96)(cid:96) (cid:48) S, ± , one justattaches it to some color singlet and Lorentz scalar six quark operators. An example of this classof operators, the Q ( pp ) S, ± LLL,a , is given in the following Q ( pp ) S, ± LLL,a = ( u i T L Cu jL )( u k T L Cd lL )( u m T L Cd nL ) j (cid:96)(cid:96) (cid:48) S, ± T SSS { ij }{ kl }{ mn } ,
8e find that there are 28 independent operators without referring to lepton flavors which we listin (B.1) in Appendix B.For the operators with a vector lepton current j (cid:96)(cid:96) (cid:48) ,µV , there are 19 independent operators, andfor the operators with a tensor lepton current j (cid:96)(cid:96) (cid:48) ,µνT , we find there are 16 independent operators.Therefore there are total operators which we give them in Eqs. (B.1-B.3) in Appendix B. For n f flavors of charged leptons, there are n f + 6 n f operators.As a cross-check, we also confirmed our above results (and following ones) by the Hilbertseries method [32]. In Appendix B, by a non-trivial example, we also show how to reduce theredundant operators using the Fierz and Schouten identities.When restricting to the same flavor leptons with (cid:96) (cid:48) = (cid:96) , we find there are
28 + 19 = 47 independent operators from the scalar and vector lepton currents, because the tensor lepton currentvanishes for identical fields. Besides the transitions pp → e + e + , µ + µ + , such operators can alsocontribute to the hydrogen-antihydrogen oscillation H − ¯H and have been enumerated long ago byCaswell, Milutinovic and Senjanovic in Ref. [33]. However, we find 13 out of total 60 operators intheir counting are redundant and all of them belong to the class with the vector lepton current j (cid:96)(cid:96),µV .In Appendix C, we show explicitly the redundancy of the basis in [33] and give the correspondenceof their basis (after excluding the redundant ones) with our basis in Eqs. (B.1, B.2). (cid:4) Dim-12 operators contributing to pn → (cid:96) + ν (cid:48) For the operators with a scalar current j (cid:96)ν (cid:48) S , we find there are 14 independent operators withoutcounting lepton flavors. For the operators with a vector current j (cid:96)ν (cid:48) ,µV , we find there are 24 inde-pendent operators. And for the operators with a tensor current j (cid:96)ν (cid:48) ,µνT , there are 13 independentoperators.In total, there are 51 independent operators for the mode pn → (cid:96) + ¯ ν (cid:48) which are listed inEqs. (B.4-B.6) in Appendix B. For n f flavors of charged leptons and neutrinos, there are n f operators. These operators are also responsible for the conversions e − p → ¯ ν ¯ n and e − n → ¯ ν ¯ p in the electron-deuteron scattering. In addition, we find they can contribute to the unique neutrondecay mode with the baryon number being changed by two units n → ¯ pe + ¯ ν [22]. (cid:4) Dim-12 operators contributing to nn → ¯ ν ¯ ν (cid:48) For the operators with a scalar current j νν (cid:48) S , we find there are 14 independent operators. And forthe operators with a tensor neutrino current j νν (cid:48) ,µνT , we find there are only 8 independent operators.In total, there are 22 independent operators which are listed in Eqs. (B.7, B.8) in Appendix B. For n f flavors of neutrinos there are n f + 3 n f operators. Excluding the scalar neutrino currents, one9an easily identity the 14 operators in Eq. (B.7) plus their parity partners are just the 14 operatorscontributing to the neutron-antineutron oscillation [33]. B. SMEFT completions
To connect with the NP scenarios at a higher scale than the electroweak scale Λ EW , the SMEFTcan be as a suitable bridge between the LEFT interactions and the unknown NP as shown in Fig. 1.It parametrizes the high scale UV NP in a model-independent way and therefore can be as a goodstarting point for the systematic EFT analysis of low energy processes. In this section, we considerthe leading order SMEFT completions for the ∆ B = ∆ L = − dinucleon to dilepton transitionoperators discussed above. It happens that the relevant SMEFT operators also first appear at dim12 at leading order, and contain six quark fields and two lepton fields. By the similar logic as theconstruction of the LEFT operators, we first factorize the operators as the convolution of the six-quark part and the proper lepton bilinear current through the Fierz rearrangement. Furthermore,since the nucleons made out of the up and down quarks, we only focus on the first generationof quark fields but without restriction for the generation of the lepton fields. We denote the SMleft-handed lepton and quark doublet fields as L (1 , , − / , Q (3 , , / and right-handed up-type quark, down-type quark and charged lepton singlet fields as u R (3 , , / , d R (3 , , − / , e R (1 , , − . We employ the front Latin letters ( a, b, c, d, e, f ) for the SU (2) L indices and themiddle ones ( i, j, k, l, m, n ) for the color SU (3) C indices in fundamental representations, respec-tively. Similar to the classification of operators in LEFT, we classify the relevant dim-12 SMEFToperators in terms of the lepton currents, and the final results are summarized in Appendix D.Here we briefly comment the procedures to obtain the independent operators in Appendix D.1). Based on the U (1) Y invariance, one can easily identify the allowed field configurations with sixquarks and two leptons. 2). For each field configuration, we first use the Fierz transformation tofix the Lorentz structure of the operator so that it takes the quark-lepton factorized form O q × j L inwhich the lepton current j L can be either scalar, vector, or tensor type. For the scalar/vector/tensorlepton current, the Fierz transformation can be used further to organize the corresponding six-quark part to be scalar-scalar-scalar/scalar-scalar-vector/scalar-scalar-tensor bilinear structures aswe did in Sec. II A. 3). Followed by step 2), we consider the electroweak SU (2) L invariance whichcan be done by implementing the contractions using the rank-2 Levi-Civita tensor (cid:15) ab . In doingso, the SI identity (cid:15) ab (cid:15) cd = (cid:15) ac (cid:15) bd + (cid:15) ad (cid:15) cb has to be considered carefully for the multiple SU (2) L SU (3) C invariance can bedone by contracting the free color indices using the independent color tensors discussed in Sec. Aand Appendix A. If there are multiple identical quark fields, the color relations in Eq. (A.4) andthe FIs in Appendix C must be taken into account to reduce the operators into the minimal basisgiven in Appendix D. 5). We also count the number of independent operators in each configurationusing the Hilbert series method [32] and confirmed our result.The following is an example of using SM building blocks to build the SMEFT dinucleon anddilepton operator which is different than those in the LEFT, O S, ( S ) Q L = ( Q i T a CQ jb )( Q k T c CQ ld )( Q m T e CQ nf )( L T g CL (cid:48) h ) (cid:15) ab (cid:15) cd (cid:15) eg (cid:15) fh T SAA { mn } [ kl ][ ij ] . (12)This time, u L and d L must appear at the same time so that the SM gauge symmetries are respected.Expanding Q into its u L and d L components, one obtains O S, ( S ) Q L = 4( u i T L Cu jL )( u k T L Cd lL )( u m T L Cd nL ) T SAA { ij } [ kl ][ mn ] j (cid:96)(cid:96) (cid:48) S, − − u i T L Cd jL )( u k T L Cd lL )( u m T L Cd nL ) T SAA { ij } [ kl ][ mn ] j (cid:96)ν (cid:48) S + 4( d i T L Cd jL )( u k T L Cd lL )( u m T L Cd nL ) T SAA { ij } [ kl ][ mn ] j νν (cid:48) S = 4 O ( pp ) S, − LLL,b − O ( pn ) S LLL,b + 4 O ( nn ) S LLL,b . (13)Therefore it is expected that the SMEFT approach will have less independent operators thanthat can be constructed in LEFT approach. Without counting the lepton flavors, for operators witha scalar lepton current, we find there are 12 independent operators. For operators with a vectorlepton current, we find there are 7 independent operators. And for operators with a tensor leptoncurrent, we find there are 10 independent operators. In total, we find there are 29 operators whichare listed in Appendix D. For n f flavors of leptons we find there are n f + n f operators. Exceptthe sub-GeV scale dinucleon to dilepton processes studied in this work, these SMEFT operators arecrucial for the model-independent study of the ∆ B = ∆ L = 2 signals on colliders, for example,the process pp → (cid:96) + (cid:96) (cid:48) + + 4 jets at LHC [15] and e − p → (cid:96) + + 5 jets in the future electron-protoncolliders like LHeC.In literature, Refs. [11, 16] also provide a bunch of the dim-12 operators contributing to dinu-cleon to dilepton transitions in SMEFT. We find the operators given in [11] are neither completenor independent as a basis. Specifically, the 28 operators listed in [11] can be covered by 21 Ref. [11] also considered operators with the SM singlet right-handed neutrinos, here we only focus on the SMEFTsubsets. O S, ( A ) u d L , , O S, ( A ) u d Q L , O T, ( A ) u d Q L , O T, ( A ) udQ L , O T, ( S ) Q L , O T, ( A ) u d e , O T, ( A ) u dQ e , in our basis are missed in their list.In Appendix E, we translate their operators as linear combinations of our above operators so thatone can easily recognize the redundancy and incompleteness of the operators in [11]. C. Relations between SMEFT and LEFT operators
As already mentioned earlier that some of the SMEFT operators contain several LEFT opera-tors, i.e. O S, ( S ) Q L = 4 O ( pp ) S, − LLL,b − O ( pn ) S LLL,b + 4 O ( nn ) S LLL,b . SMEFT approach will have less independentoperators than that can be constructed in LEFT approach at the same order. To have better idea onhow these operators are related to each other, in Tab. II, we perform a tree-level matching of thedim-12 SMEFT operators listed in Appendix D to the dim-12 LEFT operators listed in Appendix Bat the electroweak scale Λ EW . We have approximated the CKM factor V ud ≈ arising from themismatch of the flavor and mass eigenstates of the left-handed down quark d L . From Tab. II, it isobvious that the operators with the singlet charged lepton scalar/tensor current ( e T R C Γ e (cid:48) R ) can onlyexclusively contribute to the transition pp → (cid:96) + (cid:96) (cid:48) + , while the operators with an anti-symmetricscalar lepton current ‘S,(A)’ and the operators with a symmetric tensor current ‘T,(S)’ can onlycontribute to the transition pp → (cid:96) + ν (cid:48) . The remaining operators with a symmetric scalar leptoncurrent ‘S,(S)’ or with an anti-symmetric tensor current ‘T,(A)’ could contribute to both the threedifferent transition channels. Last but not least, one must be careful that the SMEFT Wilson co-efficients with a superscript ‘(A)’ vanish for identical lepton fields since the relevant operator isanti-symmetric for the two lepton fields.From Appendix D and Tab. II, we see there are many operators which can be constructed inLEFT, but not in SMEFT at leading dim-12 order, for instance, operators Q ( pp ) S, ± LLR , Q ( pn ) S LLR and Q ( nn ) S LLR for the three channels respectively. Those operators are not SU (2) L × U (1) Y invariant andcan only be generated by the higher dim-14 and/or dim-16 SMEFT SU (2) L × U (1) Y invariantoperators consisting of dim-12 fermion part ( qqqqqqll ) together with additional Higgs doublets.The physical effects from such operators are suppressed with additional factors like v / Λ and/or v / Λ relative to the dim-12 SMEFT operators and will be neglected in our numerical analysis.12 MEFT operators pp → (cid:96)(cid:96) (cid:48) pn → (cid:96) ¯ ν (cid:48) nn → ¯ ν ¯ ν (cid:48) O S, ( A ) u d L - C ( pn ) S RRR,a = − C S, ( A ) u d L - O S, ( A ) u d L - C ( pn ) , RRR,b = − C S, ( A ) u d L - O S, ( S ) u d Q L C ( pp ) S, − RRL,a = C S, ( S ) u d Q L C ( pn ) S RRL,a = − C S, ( S ) u d Q L C ( nn ) S RRL,a = C S, ( S ) u d Q L O S, ( A ) u d Q L - C ( pn ) S RRL,b = − C S, ( A ) u d Q L - O S, ( S ) u d Q L C ( pp ) S, − RRL,b = C S, ( S ) u d Q L C ( pn ) S RRL,c = − C S, ( S ) u d Q L C ( nn ) S RRL,b = C S, ( S ) u d Q L O S, ( A ) udQ L - C ( pn ) S LLR,c = − C S, ( A ) udQ L - O S, ( S ) udQ L C ( pp ) S, − LLR,b = 2 C S, ( S ) udQ L C ( pn ) S LLR,b = − C S, ( S ) udQ L C ( nn ) S LLR,b = 2 C S, ( S ) udQ L O S, ( S ) Q L C ( pp ) S, − LLL,b = 4 C S, ( S ) Q L C ( pn ) S LLL,b = − C S, ( S ) Q L C ( nn ) S LLL,b = 4 C S, ( S ) Q L O S, ( S ) u d e C ( pp ) S, +1 RRR,a = C S, ( S ) u d e - - O S, ( S ) u d e C ( pp ) S, +1 RRR,b = C S, ( S ) u d e - - O S, ( S ) u dQ e C ( pp ) S, +2 RRL,b = 2 C S, ( S ) u dQ e - - O S, ( S ) u Q e C ( pp ) S, +3 LLR,b = 4 C S, ( S ) u Q e - - O Vu d QeL C ( pp ) V RR,a = − C Vu d QeL C ( pn ) V RR,a = C Vu d QeL - O Vu d QeL C ( pp ) V RR,b = C Vu d QeL C ( pn ) V RR,b = − C Vu d QeL - O Vu d QeL C ( pp ) V RR,c = − C Vu d QeL C ( pn ) V RR,c = C Vu d QeL - O Vu dQ eL C ( pp ) V LR,c = 2 C Vu dQ eL C ( pn ) V RL,b = − C Vu dQ eL - O Vu dQ eL C ( pp ) V LR,d = 2 C Vu dQ eL C ( pn ) V RL,d = 2 C Vu dQ eL - O Vu dQ eL C ( pp ) V LR,e = − C Vu dQ eL C ( pn ) V RL,e = − C Vu dQ eL - O VuQ eL C ( pp ) V LL,c = 4 C VuQ eL C ( pn ) V LL,c = − C VuQ eL - O T, ( S ) u d Q L - C ( pn ) T RRL,a = − C T, ( S ) u d Q L - O T, ( A ) u d Q L C ( pp ) T, − RRL = C T, ( A ) u d Q L C ( pn ) T RRL,b = − C T, ( A ) u d Q L C ( nn ) T RRL = − C T, ( A ) u d Q L O T, ( S ) u d Q L - C ( pn ) T RRL,c = − C T, ( S ) u d Q L - O T, ( A ) udQ L C ( pp ) T, − LLR,b = 2 C T, ( A ) udQ L C ( pn ) T LLR,c = − C T, ( A ) udQ L C ( nn ) T LLR,b = − C T, ( A ) udQ L O T, ( S ) udQ L - C ( pn ) T LLR,b = − C T, ( S ) udQ L - O T, ( A ) udQ L C ( pp ) T, − LLR,c = − C T, ( A ) udQ L C ( pn ) T LLR,d = − C T, ( A ) udQ L C ( nn ) T LLR,c = − C T, ( A ) udQ L O T, ( S ) Q L - C ( pn ) T LLL,b = − C T, ( S ) Q L - O T, ( A ) u d e C ( pp ) T, +1 RRR = C T, ( A ) u d e - - O T, ( A ) u dQ e C ( pp ) T, +2 RRL,a = 2 C T, ( A ) u dQ e - - O T, ( A ) u dQ e C ( pp ) T, +2 RRL,c = 2 C T, ( A ) u dQ e - - TABLE II. The SMEFT dinucleon to dilepton operators and their matching onto the LEFT at the Λ EW .Where the notation for the Wilson coefficients is similar to the corresponding operators with the replacementof O by C , e.g., C T, ( A ) u dQ e for O T, ( A ) u dQ e , We do not show the explicit flavors of leptons in the abovematching but can be easily recognized through the corresponding operators. II. CHIRAL REALIZATIONSA. Some basics of Chiral matching
After establishing the operator basis for dinucleon to dilepton transitions in LEFT and SMEFT,the next step is to calculate the transition matrix elements and the decay rates. However, thehadronic matrix elements between the initial dinucleon state and the QCD vacuum state are not atrivial task due to their non-perturbative QCD nature. In order to obtain those matrix elements witha controllable uncertainty, fortunately, one can employ the successful effective chiral perturbationtheory of the low energy QCD and the spurion field techniques to shift the quark level interactionsinto the interactions among hadrons and leptons.In the QCD sector, the approximate chiral symmetry G χ = SU (2) f L × SU (2) fR of the two-flavorQCD Lagrangian under the limit of massless up and down quarks is spontaneously broken into itsisospin subgroup SU (2) V by the quark condensation (cid:104) ¯ qq (cid:105) at the scale Λ χ . The interaction of theresultant pseudo Nambu-Goldstone pion fields at low energy ( p < Λ χ ) is described by the χ PTwhich inherits the QCD chiral symmetry [34, 35], and the baryon extended χ PT termed as B χ PTis our main focus in this section. The (B) χ PT Lagrangian is organized in terms of the power ofthe soft momentum p relative to Λ χ . Introducing the proper external sources transforming underthe chiral group, the global chiral symmetry can be promoted to be a local one. Therefore, the(B) χ PT Ward identities can be easily formulated and the interaction of the hadrons with otherlight particles such as leptons and photon can be included. In the following we will use the two-flavor B χ PT formalism and the spurion field techniques to construct an equivalent effective chiralLagrangian for the dim-12 interactions in Sec. II A.Before performing the detailed non-perturbative matching for the LEFT interactions, we canexpect the six-quark part of those dim-12 operators will be transformed into proper nucleon currenttogether with pions and derivatives. For our purpose of capturing the leading order contributionsto the dinucleon to dilepton transitions, it is enough to consider the dim-6 terms composed of anucleon bilinear current and a lepton bilinear current without any pions and derivatives in Eqs.(6-8). The hadron level Wilson coefficients C ( NN (cid:48) ) a,b,c will be determined below by the chiral matchingto the quark level operators. Once these Wilson coefficients are obtained, it is straightforward to Usually the strange quark s can also be included in this framework to consider the larger group breaking SU (3) fL × SU (3) fR → SU (3) V [34, 35]. For our purpose, it is enough to only focus on the two-flavor case in which the chiralsymmetry breaking effect is relatively smaller. χ PT. For the light two-flavor quarks q = ( u, d ) T , the QCD-likeLagrangian with extended external sources can be parametrized as L = L M =0 QCD + q L l µ γ µ q L + q R r µ γ µ q R + [ q R ( s + ip ) q L + q R ( t µνl σ µν ) q L + h.c. ] , (14)where the flavor space × matrices { l µ = l † µ , r µ = r † µ , s = s † , p = p † , t µνr = t µν † l } are theexternal sources pertinent to the corresponding quark currents. Under the global chiral transfor-mation q L → ˆ Lq L and q R → ˆ Rq R with ( ˆ L, ˆ R ) ∈ G χ , the pure QCD part L M =0 QCD is invariant. Theintroduction of the external sources with proper transformation properties can promote the globalchiral symmetry to be a local one. In this way, the whole Lagrangian L can be made invariantunder the local chiral transformation q L → L ( x ) q L ≡ Lq L and q R → R ( x ) q R ≡ Rq R togetherwith the following transformations of the external sources χ → RχL † , l µ → Ll µ L † + iL∂ µ L † , r µ → Rr µ R † + iR∂ µ R † , t µνl → Rt µνl L † , (15)where χ ≡ B ( s + ip ) with B = −(cid:104) ¯ qq (cid:105) / (2 F π ) ≈ . . F π is the pion decay constant,and the quark condensate (cid:104) ¯ qq (cid:105) can be treated as an order parameter to measure the strength of thespontaneous chiral symmetry breaking.For the Lagrangian in Eq. 14, the equivalent chiral Lagrangian at low energy can be constructedby identifying the relevant degrees of freedom and to write down the most general chiral invariantLagrangian ordered in terms of the number of soft momenta. The relevant degrees of freedom arejust the light hadrons (pseudo-scalar pions and nucleons) and all possible non-QCD states (likeleptons and photon, etc) encoded in the external sources. Define the pseudo Nambu-Goldstonematrix U/u as U = u , u = exp (cid:18) i Π2 F (cid:19) , Π = π a τ a = π √ π + √ π − − π . (16)Then, under the chiral transformation ( L, R ) ∈ G χ , they transform as U → RU L † and u → Ruh † = huL † with the compensator matrix h as a function of U, L, R . Furthermore, we define thechiral vielbein as u µ = i (cid:0) u † ( ∂ µ − ir µ ) u − u ( ∂ µ − il µ ) u † (cid:1) , u † µ = u µ , (17)15hich transforms as u µ → hu µ h † under the chiral group. The power counting of these buildingblocks in terms of the soft momentum p as u = O ( p ) , u µ = O ( p ) , χ = O ( p ) , (18)Then the leading order mesonic chiral Lagrangian is at O ( p ) and take the form L = F u µ u µ + χ + ] , χ + = u † χu † + uχ † u , (19)where F is the pion decay constant at chiral limit.Next, we include the nucleons in this framework. Denote the nucleon doublet as Ψ = ( p, n ) T which transforms as Ψ → h Ψ under the chiral transformation. The covariant derivative of thenucleon doublet is D µ Ψ = ( ∂ µ + Γ µ )Ψ , Γ µ = 12 (cid:0) u † ( ∂ µ − ir µ ) u + u ( ∂ µ − il µ ) u † (cid:1) , (20)where Γ µ is the chiral connection which helps D µ Ψ to have the same transformation rule as Ψ . Thepower counting for the nucleon field is Ψ = O ( p ) , but D µ Ψ = O ( p ) , the latter is because thenucleon mass m N is compatible with the expansion scale Λ χ . However, ( i /D − m N )Ψ = O ( p ) ,then the leading order baryonic chiral Lagrangian takes L (1) πN = ¯Ψ (cid:16) i /D − m N + g A γ µ γ u µ (cid:17) Ψ , (21)where g A is the axial-nucleon coupling constant. For the following chiral matching of the dim-12operators, we will treat D µ Ψ as a higher term than Ψ through the naive dimensional analysis, andneglect their contribution to the leading order chiral realization of the relevant dim-12 operators. Apossible way out for saving the power counting rule of the nucleons is via the heavy baryon chiralperturbation formalism (HB χ PT) [25] but with the sacrifice of Lorentz covariance. The HB χ PTformalism is beyond our current scope and one can check Ref. [36] for the treatment of neutron-antineutron oscillation. A brief comment concerning the relation between the Lorentz covariantoperators and the HB χ PT reduction is given at the end of this section.
B. Decomposition of irreducible chiral symmetry
In the matching to B χ PT for the effective operators in our case, the lepton current togetherwith the associated Wilson coefficient of a dim-12 operator in LEFT behaves as a fixed external16ource, thus we only have to cope with the six-quark sector of the operator. One of the key stepsfor the chiral matching is to identify irreducible chiral representations. We describe the proce-dures in the following. Suppose the quark sector has been decomposed into a sum of irreduciblerepresentations/tensors (irreps) of the chiral group P = θ uvwxyz ( q i T χ ,u C Γ q jχ ,v )( q k T χ ,w C Γ q lχ ,x )( q m T χ ,y C Γ q nχ ,z ) T color ijklmn , (22)where q χ i are the chiral quark doublets ( u, d ) T χ i with χ i being the proper chiral projectors P ± , Γ i are the Dirac gamma matrices, T color ijklmn a general color tensor discussed in Sec. II and Appendix A.The flavor indices { u, v, w, x, y, z } as dummy indices are summed over and take 1 or 2 for theup quark u or down quark d respectively. The set of pure numbers θ uvwxyz depends on the irrepunder consideration. θ is promoted as a spurion field that transforms properly together with chiraltransformations of quarks under G χ , so that P looks like a chiral invariant.The chiral counterparts of operator P are constructed out of the spurion field θ plus the hadronicdegrees of freedom { Ψ , D µ , u, u µ , χ, · · · } and share the same symmetry transformation propertiesas that of P , which include the chiral symmetry, the Lorentz and the global baryon/lepton trans-formation properties. Since P is chiral invariant and violates baryon number by two units, thematched operators must also be chiral invariant and contain exactly one spurion field θ and two nu-cleon fields Ψ s. Based on the chiral power counting property of the hadronic degrees of freedom,the obtained operators are ordered in terms of the number of soft momenta p and the dominantterms are those with least power of p . Last, for each independent operator we associate it with anunknown low energy constant (LEC) which accommodates the non-perturbative QCD dynamics.These LECs can be determined by fitting to the data, or calculated using the lattice QCD method,or estimated based on the naive dimensional analysis. In addition, for the LEFT operators belong-ing to the same chiral irrep, the chiral symmetry implies their chiral counterparts at a given chiralorder share the same LEC. Here we remark that the above procedures have been used previously tothe non-perturbative matching of the dim-9 operators mediating the nuclear and kaon neutrinolessdouble beta decay processes [37, 38] as well as the neutron-antineutron oscillations [36].With the above procedures, we can now match the operator basis in LEFT in Sec. II A onto theB χ PT at leading order of chiral expansion, i.e., at O ( p ) . We first transform the LEFT operatorbasis into a chiral basis in which each operator itself belongs to some irrep of the chiral group G χ . The chiral bases are shown in Tab. V, Tab. VI and Tab. VII in Appendix F for the operators The higher order terms can be constructed in the same style as did in [36]. pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) and nn → ¯ ν ¯ ν (cid:48) , respectively. In the tables,we list their relations with the LEFT operators in the first and second column and show their chiralirreps in the third column (where the subscripts ( a, b, c ) behind some irrep are used to distinguishdifferent irreps with the same chiral type) and the corresponding chiral spurion fields in lastcolumn. Except the gray sectors, which already include the parity conjugates, all the rest oneshave their parity conjugates with L ↔ R (and an additional exchange of + ↔ − for the tensoroperators in Tab. V). The parity conjugate of chiral operator P i is denoted by ˜ P i once it is needed.For those chiral operators expressed as a linear combination of two or more LEFT operators, theirequivalent definitions are given in Appendix F through fully symmetrizing all free quark flavorswith the same chirality. The relations between the Wilson coefficients of the chiral basis as thoseof the LEFT operators can be determined easily. The general expressions for the spurion fieldstake the form θ u L ··· u nL v R ··· v mR ( i ··· i n )( j ··· j m ) = θ v R ··· v mR u L ··· u nL ( j ··· j m )( i ··· i n ) = [ δ u ( i δ u i · · · δ u n i n ) ][ δ v ( j δ v j · · · δ v m j m ) ] , (23)where we take the symmetrization notation with the round brackets ( · · · ) . Symmetrization withrespect to a group of indices is defined by placing these indices between round brackets ( · · · ) , sowe have δ u ( i δ u i · · · δ u n i n ) = 1 n ! (cid:2) δ u i δ u i · · · δ u n i n + ( n ! −
1) permutations of ( i , · · · , i n ) (cid:3) , (24)where we take the normalization as in [39]. For example, for the operators with a scalar leptoncurrent belonging to the chiral ( L , R ) and ( L , R ) irreps in Tab. V, we have θ u L v L (11) = θ u L v R = δ u δ v ,θ u L v L w L x L y R z R (1112)(12) = 18 [ δ u δ v δ w δ x + δ u δ v δ w δ x + δ u δ v δ w δ x + δ u δ v δ w δ x ][ δ y δ z + δ y δ z ] ,θ u L v L w L x L y R z R (1122)(11) = 16 [ δ u δ v δ w δ x + δ u δ v δ w δ x + δ u δ v δ w δ x + 1 ↔ δ y δ z ,θ u L v L w L x L y R z R (1111)(22) = δ u δ v δ w δ x δ x δ z . (25)From Tabs. V-VII, we see that for the operators with a scalar lepton current, there are six types ofirreps under the chiral group: ( L , R ) , ( L , R ) , ( L , R ) plus their parity conjugates. The threeoperators P ( pp ) S ,a , P ( pn ) S ,a , P ( nn ) S ,a belong to the same irrep ( L , R ) and relate to each other by chiral For the operators belonging to the same irrep, they must have the similar chiral, Lorentz and color structures so thatthey can be related to each other through the action of the chiral transformation. ( L , R ) in which the nine operators are relatedto each other. However, for the irrep type ( L , R ) , there are three different irreps distinguishedby the subscripts a, b, c and each one contains three operators. For the operators with a vectorlepton current, there are also six types of irreps: ( L , R ) , ( L , R ) , ( L , R ) , ( L , R ) , ( L , R ) and ( L , R ) . One should be careful that there are four different irreps for the type ( L , R ) since the parity conjugates of the irrep ( L , R ) | a is different from itself and will be denoted as ( L , R ) | d . Last, for the operators with a tensor lepton current, there are still six types of irreps: ( L , R ) , ( L , R ) , ( L , R ) , ( L , R ) , ( L , R ) and ( L , R ) . Where the type ( L , R ) containsfour different irreps due to the parity conjugates of ( L , R ) | a,b are distinct from themselves forthe operators in Tab. V. Another interesting fact is that the LEFT operators belonging to the samechiral irrep will not mix with each other under the 1-loop QCD renormalization and have the sameanomalous dimensions since QCD preserves the chiral symmetry and quark flavors. In addition,the QCD renormalization for the operators related to each other by parity also is the same. Forthe operators with a scalar lepton current, the 1-loop QCD renormalization is identical to the dim-9 operators contributing to the n − ¯ n oscillation and can be found in [33, 40]. But there is noresult for the operators with a vector or tensor lepton current yet, and we will neglect the QCDrenormalization effect for the current work due to the involvement of considerable effort. C. Chiral matching for the operators
Based on the chiral irrep, we reorganize the chiral building blocks in terms of the power of softmomentum p and the explicit chiral left or right doublet indices such that they have only one or twofree indices, i.e., the building blocks are constructed to take the forms: X u χ and X u χ v χ with χ i = L, R . They transform as X u χ → ( g χ ) u χ ˆ u χ X ˆ u χ and X u χ v χ → ( g χ ) u χ ˆ u χ ( g χ ) u χ ˆ u χ X ˆ u χ ˆ v χ under chiral transformation g χ i ∈ SU (2) f L , R . Therefore, the first few building blocks with lowerchiral order are constructed as follows [36]: O ( p ) : ( U iτ ) x R y L , ( u Ψ) x R , ( u † Ψ) x L , O ( p ) : ( uu µ uiτ ) x R y L , ( uu µ u † iτ ) x R y R , ( u † u µ uiτ ) x L y L , O ( p ) : ( χiτ ) x R y L , ( χ † iτ ) x L y R , · · · , (26) The building blocks with three or more free indices are not independent and can be reduced into a product of X u χ sand X u χ v χ s. O ( p ) we just show a few examples, and the full list should include terms with two u µ s,field strength tensors for the vector external sources, etc. Due to the fact that xτ = τ x ∗ for x ∈ SU (2) , the other possible O ( p ) and O ( p ) building blocks are not independent: ( U † iτ ) x L y R = − ( U iτ ) y R x L and ( u † u µ u † iτ ) x L y R = − ( uu µ uiτ ) y R x L . Note that the two O ( p ) objects with thesame chirality are anti-symmetric under the exchange of the two indices. In the above, we neglectthose O ( p ) terms with derivatives acting on the nucleon field like ( uD µ Ψ) x R since they will notyield the leading operators in Eq. (5). Without consideration of the operators involving covariantderivatives acting on the nucleon fields, the leading order matching results for all the relevant chiralirreps are shown in Tab. VIII. Where the spurion fields are easily identified from Tabs. V-VII foreach specific operator, and g i are the unknown LECs parametrizing non-perturbative QCD effect.One should keep in mind that for each independent irrep there is a corresponding LEC.Here we again take the operator O S, ( S ) Q L as an example to show the relevant spurion fields andthe chiral matching result. From Eq. (13), the six-quark part of the matched three LEFT operators O ( pp ) S, − LLL,b , O ( nn ) S LLL,b , O ( pn ) S LLL,b can be rewritten as ( u i T L Cu jL )( u k T L Cd lL )( u m T L Cd nL ) T SAA { ij } [ kl ][ mn ] = ( δ u δ v ) T u L v L ≡ θ u L v L (11) T u L v L , ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cd nL ) T SAA { ij } [ kl ][ mn ] = 12 ( δ u δ v + δ u δ v ) T u L v L ≡ θ u L v L (12) T u L v L , ( d i T L Cd jL )( u k T L Cd lL )( u m T L Cd nL ) T SAA { ij } [ kl ][ mn ] = ( δ u δ v ) T u L v L ≡ θ u L v L (11) T u L v L , (27)where we defined the spurion fields as θ u L v L (11) = δ u δ v , θ u L v L (12) = 12 ( δ u δ v + δ u δ v ) , θ u L v L (11) = δ u δ v , (28)and T u L v L = T v L u L is a three-dimensional irrep tensor under the group SU (2) f L but a singlet under SU (2) f R , and takes the form T u L v L = 14 (cid:15) wx (cid:15) yz ( q i T Lu Cq jLv )( q k T Lw Cq lLx )( q m T Ly Cq nLz ) T SAA { mn } [ kl ][ ij ] G χ → L xu L yv T xy ∈ ( L , R ) . (29)According to the previous procedures, the leading order chiral realization of T u L v L is at O ( p ) and formed by two ( u † Ψ) u L s to have the same baryon number and chiral structure. In addi-tion, the Lorentz covariance further restricts T u L v L to be a scalar and have the general form ( u † ) u L a ( u † ) v L a Ψ b [ g × + ˆ g × γ ] Ψ b as shown in Tab. VIII. Then the complete matching for O S, ( S ) Q L together with its Wilson coefficient becomes C S, ( S ) Q L O S, ( S ) Q L → C S, ( S ) Q L (cid:16) θ u L v L (11) j (cid:96)(cid:96) (cid:48) S, − − θ u L v L (12) j (cid:96)ν (cid:48) S + θ u L v L (22) j νν (cid:48) S (cid:17) ( u † ) u L a ( u † ) v L b Ψ a [ g × + ˆ g × γ ] Ψ b . (30) O ( p ) terms in Tab. VIII and expand them to zeroth order to pion fields, we have O S × ,i → θ u L v L ( αβ ) [Ψ T u L C ( g × ,i + ˆ g × ,i γ )Ψ v L ] ,O S × → θ u L v L w L x L y R z R ( αβγρ )( στ ) (cid:15) y R w L (cid:15) z R x L [Ψ T u L C ( g × + ˆ g × γ )Ψ v L ] ,O V,µ × ,i → θ u L v R αβ [Ψ T u L Cγ µ ( g × ,i + ˆ g × ,i γ )Ψ v R ] ,O V,µ × ,i → g × ,i θ u L v L w L x R ( αβγ ) ρ (cid:15) x R w L [Ψ T u L Cγ µ γ Ψ v L ] ,O V,µ × → θ u L v L w L x R y R z R ( αβγ )( ρστ ) (cid:15) y R v L (cid:15) z R w L [Ψ T u L Cγ µ ( g × + ˆ g × γ )Ψ x R ] ,O T,µν × ,i = 12 (cid:15) ab [Ψ T a Cσ µν ( g × ,i + ˆ g × ,i γ )Ψ b ] ,O T,µν × ,i → θ u L v L w R x R ( αβ )( γρ ) (cid:15) x R v L [Ψ T u L Cσ µν ( g × ,i + ˆ g × ,i γ )Ψ w R ] , (31)and the similar expressions for the parity conjugates ˜ O S × ,i , ˜ O S × and ˜ O V,µ × ,i . Taking the specificexpressions of the spurion fields in Tabs. V-VII into consideration, we can obtain the matchingresults for the Wilson coefficients of the operators in Eqs. (6-8) as the functions of the LECsand the LEFT/SMEFT Wilson coefficients, the full matching results from the LEFT and SMEFToperators are listed in Appendix H.In the following, we show as an example the matching results from the SMEFT operators C S, ( S ) Q L and C T, ( S ) Q L . In terms of their LEFT counterparts Q ( pp,pn,nn ) S LLL and Q ( pn ) T LLL , we have C ( pp ) SR ( L ) = g × ,a (cid:18) C ( pp ) S, ± LLL,a + C ( pp ) S, ± LLL,b (cid:19) + · · · , C ( pp ) S R ( L ) = C ( pp ) SR ( L ) | g → ˆ g ,C ( pn ) SL = g × ,a (cid:18) C ( pn ) S LLL,a + C ( pn ) S LLL,b (cid:19) + · · · , C ( pn ) S L = C ( pn ) SL | g → ˆ g ,C ( pn ) T = g r × ,a (cid:18) C ( pn ) T LLL,a + C ( pn ) T LLL,b (cid:19) + · · · ,C ( nn ) SL = g × ,a (cid:18) C ( nn ) S LLL,a + C ( nn ) S LLL,b (cid:19) + · · · , C ( nn ) S L = C ( nn ) SL | g → ˆ g , (32)where · · · stand for contributions from other operators. For the tensor case C ( pn ) ,TL , we have usedthe identity σ µν γ = i(cid:15) µνρσ σ ρσ / to eliminate the operator ( p T Cσ µν γ n )( (cid:96) T R Cσ µν ν L ) in favor of ( p T Cσ µν n )( (cid:96) T R Cσ µν ν L ) which shifted the LECs to be g ri × i,x = g i × i,x − ˆ g i × i,x . After neglecting theQCD running effect and replacing the LEFT Wilson coefficients by the SMEFT ones C S, ( S ) Q L and C T, ( S ) Q L as the way shown in Tab. II, we find the above results are simplified to become C ( pp ) SL = 4 g × ,a C S, ( S ) Q L + · · · , C ( pp ) S L = C ( pp ) SL | g → ˆ g ,C ( pn ) SL = − g × ,a C S, ( S ) Q L + · · · , C ( pn ) S L = C ( pn ) SL | g → ˆ g , ( pn ) T = − g r × ,a C T, ( S ) Q L + · · · ,C ( nn ) SL = 4 g × ,a C S, ( S ) Q L + · · · , C ( nn ) S L = C ( nn ) SL | g → ˆ g . (33)Once the hadronic LEC g i is known one can obtain the dinucleon and dilepton transitions. Inthe following section we will discuss how this can be done and obtain constraints on LEFT andSMEFT operators.Before doing that, let us have some discussion about the LECs. By the parity invariance ofQCD, we expect the LECs of an operator and its parity conjugate are the same up to a sign deter-mined by the parity transformation property of the quark and corresponding hadron level operators,i.e., g i × j = ± g j × i . Particularly, for the scalar current case, we have g × ,i = − g × ,i , ˆ g × ,i = +ˆ g × ,i , g × = − g × , ˆ g × = +ˆ g × . (34)The numerical value of g × ,i and g × can be determined by the LQCD results for the n − ¯ n oscillation matrix elements [40]. This is because the quark sectors of the 14 operators with a scalarlepton current contributing to nn → ¯ ν ¯ ν (cid:48) transitions are exactly the 14 dim-9 operators mediating n − ¯ n oscillation. Neglecting the lepton current, the scalar chiral operators in the irreps ( L , R ) and ( L , R ) in Tab. VII have the following correspondence with the dim-9 chiral operators forthe n − ¯ n oscillation [40] Q = − P ( nn ) S ,b , Q = − P ( nn ) S ,b , Q = − P ( nn ) S ,b ,Q = P ( nn ) S , Q = − P ( nn ) S ,a , Q = − P ( nn ) S ,a . (35)After comparing the LQCD results on the n − ¯ n matrix elements from Q i and our chiral matchingresults for P ( nn ) Si in Eq. (31), we find g × ,a,c ∼ − × − GeV , g × ,b ∼ × − GeV , g × ∼ × − GeV , (36)where the above results are set at about 2 GeV and the uncertainty in [40] is neglected.Except the above LECs, the rest hadronic couplings, such as the ˆ g i and the ones related tothe vector and tensor current operators, have not been determined. For these LECs, we will usedimensional analysis as a guide to illustration. Since the transition from quarks to hadrons isthrough the non-perturbative QCD dynamics, the only relevant scale Λ QCD ∼ MeV will comeinto play. To make the dimensionality correct, one can take as a rough estimate the couplings tobe of order Λ ∼ . × − GeV . This is larger than the numbers in (36). But consideringthe large uncertainties involved, we can take it as a guide for estimate.22lternatively, these LECs can be estimated via the naive dimensional analysis by keeping trackof π factors and relating the hadronic matrix element with the chiral symmetry breaking scale Λ χ = 1190 MeV.One introduces “reduced” couplings for the hadron and quark level operators and to matchthem [27]. For a coupling constant g appearing in an interaction of dimensionality D in masswhich containing N field operators, the reduced coupling is (4 π ) − N Λ D − χ g . For our case, thehadronic operators involve two fields ( N = 2 ) with a coupling g N as given in Eqs.(6-8). The quarkoperators involve six quarks ( N = 6 ) with a coupling g q , therefore we would obtain g N / Λ χ = g q (Λ χ ) / (4 π ) by matching. Setting g q = 1 , one would have the hadronic coupling to be oforder (Λ χ ) / (4 π ) ∼ × − which is about 2 times the above dimensional estimate. For ournumerical estimate of the undetermined LECs in the next section, we assume their value to be thesimilar order as Λ .Last, we comment the heavy baryon χ PT formalism [25], which is a consistent framework forthe power counting of nucleon fields. In this framework, for our case, the anti-nucleon mode isintegrated out and the remaining heavy nucleon doublet is defined as N v ( x ) = e + imv · x P v + Ψ with P v ± ≡ (1 ± /v ) , where v is a reference velocity satisfying v = 1 and usually taken as v = (1 , ) .The chiral power counting for D µ N v ( x ) is O ( p ) as promised in this formalism. To leading orderof chiral matching, there should have no derivatives acting on the nucleon fields. Since N T v C N v = N T v P T v + CP v + N v = N T v CP v − P v + N v = 0 , (37)we find that the matched operators using heavy nucleon fields can be directly obtained from thematched operators using the relativistic nucleon fields in Tab. VIII by replacing the nucleon field Ψ by N v together with the omission of the operators with a scalar nucleon current. The use ofrelativistic formalism is its explicit Lorentz invariance and chiral symmetry, which are convenientfor the loop calculations. IV. DINUCLEON AND DILEPTON TRANSITION RATE
Combining the previous sections for the effective interactions from the SMEFT, to LEFT, thento B χ PT, in this section we will collect all pieces together and calculate the dinucleon to dileptondecay rate. Denote collectively
N N (cid:48) ∈ { pp, pn, nn } and l α l β ∈ { (cid:96) + (cid:96) (cid:48) + , (cid:96) + ¯ ν (cid:48) , ¯ ν ¯ ν (cid:48) } , then thetransition rate for the dinucleon N N (cid:48) to dilepton l α l β in nucleus can be estimated in the following23ay [41] Γ NN (cid:48) → l α l β = 1(2 π ) √ ρ N ρ N (cid:48) (cid:90) d k d k ρ N ( k ) ρ N (cid:48) ( k ) v rel . (1 − v · v ) σ ( N N (cid:48) → l α l β ) , (38) where ρ N ( k ) is the nucleon density distribution in momentum space and ρ N is the average nucleondensity defined as ρ N = (cid:82) d kρ N ( k ) / ( √ π ) . v ( v ) is the velocity of the nucleon N ( N (cid:48) ) . Thetotal cross-section for the free nucleon scattering process N ( k ) N (cid:48) ( k ) → l α ( p ) l β ( p ) is σ ( N N (cid:48) → l α l β ) = 1 S E E v rel . (cid:90) d Π (cid:12)(cid:12) M NN (cid:48) → l α l β (cid:12)(cid:12) , (39)where E ( E ) is the energy of the initial state nucleon N ( N (cid:48) ) , and S is a symmetry factor andequals 2 for identical final leptons l α = l β = { e + , µ + , ¯ ν e , ¯ ν µ , ¯ ν τ } , otherwise S = 1 . d Π is therelativistically invariant two-body phase space.The dinucleon collisions occur at low relative velocity v , they may be affected by some otherSM interaction resulting in modification of the cross sections. For example for pp → (cid:96) + α (cid:96) + β ,there is a repulsive force between the two protons due to electrodynamics which reduces the crosssection. The effect of exchange photons between protons is best captured by the Sommerfeldeffect [42, 43]. Because the repulsive nature of the electromagnetic force there is a reductionof the cross section, the original cross section σ is modified to ˜ σ = σSF with SF given by ( α em π/v ) / (exp[ α em π/v ] − . This reduction factor SF could be very severe if v is very small.For the case in question, the typical v of about . leads to SF ≈ . . Had v be 0.01, SF is furtherreduced to 0.26. Therefore the case we are considering the reduction is not severe. One expectssuch effects for np and nn cases will be smaller. We still use Eq. (39) as our order of magnitudeestimate.To a good approximation for the oxygen nuclei O, we treat the nucleons to be quasi-free andneglect the small effects due to the nucleon Fermi motion and nuclear binding energy, as well asthe above Sommerfeld suppression effect. The average nuclear matter density ρ N approximatelyequals .
25 fm − for either proton or neutron. Then the transition rate reduces into Γ NN (cid:48) → l α l β = 1 S ρ N m N (cid:12)(cid:12) M NN (cid:48) → l α l β (cid:12)(cid:12) Π , (40)where we neglect the mass difference of the proton and nucleon and take both to be m N = ( m p + m n ) / . The two-body final state phase factor Π takes Π = 18 π [ λ (1 , δ α , δ β )] / , δ α = m α m N , λ ( x, y, z ) = x + y + z − xy + yz + zx ) . (41)24 C [ − GeV − ] WC [ − GeV − ] WC [ − GeV − ] C ( pp ) SL,R - C ( pp ) S L,R | ee,eµ,µµ . , . , . C ( pp ) V | ee,eµ,µµ , . , . C ( pn ) SL - C ( pn ) S L | eν,µν,τν . , . , C ( pn ) VL | eν,µν,τν . , . , - - C ( pn ) V L | eν,µν,τν , , C ( pn ) T | eν,µν,τν . , . , . C ( nn ) SL - C ( nn ) S L | ν α ν α ,ν α ν β (cid:54) = α . , . TABLE III. The upper limit on the Wilson coefficients (WC) of the dim-6 hadronic operators in Eqs. (6-8).Where we take the current experimental lower limit on the dinucleon to dilepton transitions in Tab. I to setthe limit.
Working on the center of mass frame of the two-nucleon system and neglecting the nucleons’velocity, from the effective interaction in Eq. (5), then the spin-averaged squared amplitudes are (cid:12)(cid:12)(cid:12) M pp → (cid:96) + α (cid:96) + β (cid:12)(cid:12)(cid:12) = 32 m N (cid:104) S (1 − δ α − δ β ) (cid:0)(cid:12)(cid:12) C ( pp ) S L (cid:12)(cid:12) + (cid:12)(cid:12) C ( pp ) S R (cid:12)(cid:12) (cid:1) + (cid:0) δ α + δ β − ( δ α − δ β ) (cid:1) (cid:12)(cid:12) C ( pp ) V (cid:12)(cid:12) − S (cid:112) δ α δ β Re (cid:2) C ( pp ) S L C ( pp ) S ∗ R (cid:3) +2 S (cid:0) (1 + δ α − δ β ) (cid:112) δ β Re (cid:2) C ( pp ) V C ( pp ) S ∗ R (cid:3) − ( L, δ α ↔ R, δ β ) (cid:1)(cid:105) + O ( v ) , (42) (cid:12)(cid:12)(cid:12) M pn → (cid:96) + α ¯ ν β (cid:12)(cid:12)(cid:12) = 8 m N (1 − δ α ) (cid:104)(cid:12)(cid:12) C ( pn ) S L (cid:12)(cid:12) + δ α (cid:12)(cid:12) C ( pn ) V L (cid:12)(cid:12) + (2 + δ α ) (cid:12)(cid:12) C ( pn ) VL (cid:12)(cid:12) + 4(1 + 2 δ α ) × (cid:12)(cid:12) C ( pn ) T (cid:12)(cid:12) − (cid:112) δ α Re (cid:2) C ( pn ) S L C ( pn ) V ∗ L − C ( pn ) VL C ( pn ) T ∗ (cid:3)(cid:105) + O ( v ) , (43) (cid:12)(cid:12) M nn → ¯ ν α ¯ ν β (cid:12)(cid:12) = 32 S m N (cid:12)(cid:12) C ( nn ) S L (cid:12)(cid:12) + O ( v ) , (44)where we see that the contribution to pp → (cid:96) + α (cid:96) + β ( nn → ¯ ν ¯ ν (cid:48) ) transition from the operators O ( pp ) SL,R ( O ( nn ) SL ) vanishes in that pp ( nn ) annihilation through such operators is p -wave ( ∝ v ),whereas the contribution from the vector operator O ( pp ) V is helicity-suppressed ( ∝ δ α,β ). For pn → ¯ ν ¯ ν (cid:48) , the vanishing from O ( pn ) SL has a similar reason.The partial lifetime characterizing the matter instability is the inverse of the rate ( τ / B i ) i ≡ Γ − i ,where B i is a branching ratio. Taking the experimental lower limits on the partial lifetimes inTab. I into consideration, and by the relation Γ − i = ( τ / B i ) i ≥ τ exp , we can obtain the constraintson the coefficients in Eqs. (42-44). Assuming one term dominates at a time, then the result isshown in Tab. III, where the upper limit on the Wilson coefficients is classified in terms of thefinal lepton states. We see the most stringent limit is for the operator O ( pp ) S L,R | ee,eµ,µµ in which C ( pp ) S L,R | ee,eµ,µµ ≤ . , . , . × − GeV − , this is because the strong experimental limits onthese decay modes. 25 MEFT WCs pp → e + e + , e + µ + , µ + µ + pn → e + ν, µ + ¯ ν, τ + ¯ ν nn → ¯ ν α ¯ ν α , ¯ ν α ¯ ν β (cid:54) = α Λ NP ≡ | C i | − [TeV] Λ NP ≡ | C i | − [TeV] Λ NP ≡ | C i | − [TeV] C S, ( A ) u d L - . , . , . × (cid:20) ˆ g × ,a Λ (cid:21) - C S, ( A ) u d L - . , . , . × (cid:20) ˆ g × ,a Λ (cid:21) - C S, ( S ) u d Q L , . , . , . × (cid:20) ˆ g × ,c Λ (cid:21) . , . , . × (cid:20) ˆ g × ,c Λ (cid:21) . , . × (cid:20) ˆ g × ,c Λ (cid:21) C S, ( A ) u d Q L - . , . , . × (cid:20) ˆ g × ,b Λ (cid:21) - C S, ( A ) udQ L - . , . , . × (cid:20) ˆ g × ,c Λ (cid:21) - C S, ( S ) udQ L . , . , . × (cid:20) ˆ g × ,b Λ (cid:21) . , . , . × (cid:20) ˆ g × ,b Λ (cid:21) . , . × (cid:20) ˆ g × ,b Λ (cid:21) C S, ( S ) Q L . , . , . × (cid:20) ˆ g × ,a Λ (cid:21) . , . , . × (cid:20) ˆ g × ,a Λ (cid:21) . , . × (cid:20) ˆ g × ,a Λ (cid:21) C S, ( S ) u d e . , . , . × (cid:20) ˆ g × ,a Λ (cid:21) - - C S, ( S ) u d e . , . , . × (cid:20) ˆ g × ,a Λ (cid:21) - - C S, ( S ) u dQ e . , . , . × (cid:20) ˆ g × ,b Λ (cid:21) - - C S, ( S ) u Q e . , . , . × (cid:20) ˆ g × ,c Λ (cid:21) - - C Vu d QeL , . , . , . × (cid:20) ˆ g × ,d Λ (cid:21) . , . , . × (cid:20) g × ,d Λ (cid:21) - C Vu d QeL . , . , . × (cid:20) ˆ g × ,d Λ (cid:21) . , . , . × (cid:20) g × ,d Λ (cid:21) - C Vu dQ eL . , . , . × (cid:20) ˆ g × ,c Λ (cid:21) . , . , . × (cid:20) g × ,c Λ (cid:21) - C Vu dQ eL . , . , . × (cid:20) ˆ g × ,b Λ (cid:21) . , . , . × (cid:20) g × ,b Λ (cid:21) - C Vu dQ eL . , . , . × (cid:20) ˆ g × ,c Λ (cid:21) . , . , . × (cid:20) g × ,c Λ (cid:21) - C VuQ eL . , . , . × (cid:20) ˆ g × ,a Λ (cid:21) . , . , . × (cid:20) g × ,a Λ (cid:21) - C T, ( S ) u d Q L , - . , . , . × (cid:20) g r × ,c Λ (cid:21) - C T, ( A ) u d Q L - . , . , . × (cid:20) g r × ,b Λ (cid:21) - C T, ( A ) udQ L - . , . , . × (cid:20) g r × ,a Λ (cid:21) - C T, ( S ) udQ L - . , . , . × (cid:20) g r × ,b Λ (cid:21) - C T, ( S ) Q L - . , . , . × (cid:20) g r × ,a Λ (cid:21) - TABLE IV. The constraint on the effective NP scale from the current experimental data in Tab. I. The flavorindex is suppressed and can be easily recognized in terms of the transition mode. One should keep in mindthat the SMEFT Wilson coefficients with a superscript ‘(A)’ vanish for identical lepton flavors. g i ∼ Λ ∼ . × − GeV as our benchmark value,therefore, the factor ( g i / Λ ) / is O (1) . Up to the O (1) hadronic LECs ratio, the associatedNP scale is found to be around . − . for all relevant operators. Here we see that, eventhe effective interactions are at dim 12, the matter instability puts a stringent limit on the NPscale. Similarly, we can set constraints on the LEFT operators. However, taking the assumptionof the NP scale much higher than Λ EW , the above constraints on the SMEFT interactions are moreilluminating in connection with NP scenarios, and thus we do not show the constraints on theLEFT interactions here for brevity.Furthermore, we consider the contribution to the transitions from the dim-12 operators O S, ( S ) Q L , O S, ( S ) u d Q L , and O S, ( S ) udQ L containing purely left-handed lepton fields. By weak isospin symmetry,they can contribute to both three transition modes, and in particular, they are the only possibleoperators contributing to the nn → ¯ ν ¯ ν (cid:48) modes at leading order. From the previous discussion, thetransition rates become ˜Γ pp → (cid:96) + α (cid:96) + β = Sπ (1 − δ α − δ β ) (cid:113) − δ α + δ β ) + ( δ α − δ β ) m N ρ N C αβ , ˜Γ pn → (cid:96) + α ¯ ν β = 1 π (1 − δ α ) m N ρ N C αβ , ˜Γ nn → ¯ ν α ¯ ν β = Sπ m N ρ N C αβ . (45)where we use a ‘tilde’ to represent such special contributions, and C αβ = 4ˆ g × ,a C S, ( S ) ,αβQ L + 2ˆ g × ,b C S, ( S ) ,αβudQ L + ˆ g × ,c (cid:0) C S, ( S ) ,αβu d Q L + C S, ( S ) ,αβu d Q L (cid:1) + ˆ g × C S, ( S ) ,αβu d Q L . Here we have added the lepton flavor indices for a more careful treatment . By the weak isospinsymmetry, we see the three transitions are related to each other. From Eq. (45), we obtain ˜Γ − pn → (cid:96) + α ¯ ν β = S (1 − δ α − δ β ) (cid:112) − δ α + δ β ) + ( δ α − δ β ) (1 − δ α ) ˜Γ − pp → (cid:96) + α (cid:96) + β , ˜Γ − nn → ¯ ν α ¯ ν β = (1 − δ α − δ β ) (cid:113) − δ α + δ β ) + ( δ α − δ β ) ˜Γ − pp → (cid:96) + α (cid:96) + β , For pn → (cid:96) + α ¯ ν β mode, the last term should take a minus sign. We neglect this sign difference here. Γ − nn → ¯ ν α ¯ ν β = S − (1 − δ α ) ˜Γ − pn → (cid:96) + α ¯ ν β . (46)Due to the stronger experimental limits on pp → (cid:96) + α (cid:96) + β and pn → (cid:96) + α ¯ ν β , through the above re-lations, we can set new stronger limits on the neutral modes nn → ¯ ν α ¯ ν β . Taking the experi-mental limits for the charged modes into consideration and requiring ˜Γ − pp → (cid:96) + α (cid:96) + β (cid:38) τ exp pp → (cid:96) + α (cid:96) + β and ˜Γ − pn → (cid:96) + α ¯ ν β (cid:38) τ exp pn → (cid:96) + α ¯ ν β , we obtain { ˜Γ − nn → ¯ ν e ¯ ν e , ˜Γ − nn → ¯ ν e ¯ ν µ , ˜Γ − nn → ¯ ν µ ¯ ν µ } (cid:38) × yrs , { ˜Γ − nn → ¯ ν e ¯ ν τ , ˜Γ − nn → ¯ ν µ ¯ ν τ } (cid:38) × yrs . (47)One can see the limits on nn → ¯ ν α ¯ ν β are improved by − orders of magnitude than the directexperimental search in Tab. I. On the other hand, if we assume the charged modes also exclusivelymediated by the same operators, then the experimental bounds on pp → (cid:96) + α (cid:96) + β imply the followingnew bounds on pn → (cid:96) + α ¯ ν β for α, β = e, µ flavors, { ˜Γ − pn → e + ¯ ν e , ˜Γ − pn → µ + ¯ ν µ } (cid:38) × yrs , { ˜Γ − pn → e + ¯ ν µ , ˜Γ − pn → µ + ¯ ν e } (cid:38) × yrs , (48)which are also stronger than the current experimental bounds by at least an order of magnitude.Conversely, the experimental limit on pn → (cid:96) + e ¯ ν τ can further translate into a bound on the transi-tion pp → e + τ + , which is also kinetically allowed but has not yet been searched for experimen-tally. Based on Eq. (46) and the limit on pn → (cid:96) + e ¯ ν τ in Tab. I, we obtain ˜Γ − pp → e + τ + (cid:38) × yrs , (49)we see this bound is even more stronger than any other ones due to the small phase space.In Fig. 2 we show the dependence of the partial lifetime on the NP scale. For simplicity, weonly consider the contribution from operator O S, ( S ) Q L and take Λ ≡ (cid:104) C S, ( S ) Q L (cid:105) − / . The relevanthadronic LEC ˆ g × ,a is set equal to . × − GeV . From the figure we see the partial lifetimeis very sensitive to the NP scale, because of the large power dependence( ∝ Λ ). For a futureexperimental sensitivity about yrs the NP scale is pushed towards or so.Finally, we make a brief comments on the concrete NP models and the collider signals. For agiven NP model, one can integrate out the heavy new physics states and match onto the dim-12SMEFT operators. In literature, there exist models contributing to dinucleon to dilepton transitionswith ∆ B = ∆ L = 2 but not giving rise to the ∆ B = 1 nucleon decays or ∆ B = 2 neutron-antineutron oscillation. In Refs. [8, 10], the authors considered a class of such models whichinvolve new scalar-fermion and scalar-quartic interactions, meanwhile, the left-right symmetric28 p → e + e + , μ + μ + ; nn →ν α ν α pp → e + μ + ; pn → e + ν , μ + ν ; n →ν α ν β≠α pp → e + τ + , pn →τ + ν Λ [ TeV ] P a r i t a lli f e t i m e Γ i - [ y r s ] Super - K bound: pp caseSuper - K bound: pn caseKamLAND bound: nn case Fu t u r e se n s i t i v i t y FIG. 2. The partial lifetime of the dinucleon to dilepton transitions as a function of the NP scale in theSMEFT. Where we assume the contribution from operator O S, ( S ) Q L and set Λ ≡ (cid:104) C S, ( S ) Q L (cid:105) − / . models with extra-dimensions considered in Refs. [11, 16] can also do the job. On the other hand,from the above limit of NP scale we have set, one may expect the search of the ∆ B = ∆ L = 2 NPsignals at the current/future high energy colliders to be interesting. Ref. [15] has made such a tryby studying the process pp → e + e + + 4 jets based on a dim-12 SMEFT operator (similar to theoperator O S, ( S ) Q L in our basis). However, the eight fermion operator considered in [15] cannot yet beprocessed by the FeynRules to MG5AMC framework, and the authors take a “stand-in” operatorfor estimation. Such a procedure could yield large uncertainty, and we would like to come back tothe collider signals in the future for a more precise analysis. V. CONCLUSION AND OUTLOOK
In this work we have made a thorough investigation on the baryon and lepton number violatingdinucleon to dilepton decays ( pp → (cid:96) + (cid:96) (cid:48) + , pn → (cid:96) + ¯ ν (cid:48) , nn → ¯ ν ¯ ν (cid:48) ) with ∆ B = ∆ L = − inthe framework of effective field theory. We first construct a basis of dim-12 operators mediatingsuch processes in the low energy effective field theory (LEFT) below the electroweak scale. Sucha basis not only contribute to the dinucleon decays studied in this work, it also serves as a startingpoint for model independent study of the hydrogen-antihydrogen oscillation and the low energybaryon number violating conversions e − p → e + ¯ p, ¯ ν ¯ n, e − n → e + ¯ n in electron-deuteron scattering.29hen we consider their leading-order standard model effective field theory (SMEFT) completionsupwards and obtain the SMEFT basis mediating such processes at dim 12. We find the SMEFTgauge symmetry has a strong constraint on the structure of the interactions. The dim-12 SMEFToperators are suitable for the high energy signature on colliders like LHC to search the excess ofevents with four jets plus two same-sign charged leptons.Next, we analyze the chiral structure of the LEFT operators and make a non-perturbative match-ing through the baryon chiral perturbation theory (B χ PT). In doing so, we construct a chiral basisin which each operator belongs to an irreducible representation of the two-flavor chiral group SU (2) fL × SU (2) fR , and then we construct the corresponding hadronic operators through the spu-rion techniques. Last, we express the dinucleon to dilepton decay rates in terms of the Wilsoncoefficients associated with the dim-12 operators in LEFT/SMEFT and the low energy constantspertinent to B χ PT. Our result is general in that it does not depend on dynamical details of physicsat a high scale that induce the effective interactions in SMEFT and in that it does not appeal toany hadronic models. We find the current experimental limits push the associated new physicsscale larger than a few TeV, a scale appealing to the future experimental searches. Due to theweak isospin symmetry, based on the experimental limits on pp → (cid:96) + α (cid:96) + β , pn → (cid:96) + α ¯ ν β , we improvethe lower limits on the partial lifetimes of the neutral transition modes nn → ¯ ν α ¯ ν β (except the ( α, β ) = ( τ, τ ) case) by − orders of magnitude than their current experimental sensitivity. Fur-thermore, assuming these transitions dominantly generated by the similar dimension-12 SMEFToperators, we find the limits on the partial lifetimes of modes pp → e + e + , e + µ + , µ + µ + are alsotransformed into stronger limits on pn → (cid:96) + α ¯ ν β with α, β = e, µ than their existing lower bounds.Our operator basis obtained in this work can be as a starting point for further investigation onthe related processes with ∆ B = ∆ L = 2 signals, in which the hydrogen-antihydrogen oscillationand the collider signals pp → (cid:96) + (cid:96) (cid:48) + + 4 jets and e − p → (cid:96) + + 5 jet are the most interesting ones.Both of these processes can be systematically studied in the current LEFT/SMEFT framework,and we will come back to these processes in the future publications. ACKNOWLEDGEMENT
The authors acknowledge G. Valencia for his valuable comments on the manuscript and usefuldiscussion concerning the collider aspect. The authors also thank J. Bramante for providing uswith their FeynRules code to clarify what they had done in [15]. XDM would like to thank F-S30u for his invitation as a visitor at Lanzhou Uni. where part of this work has been done. Thiswork was supported in part by the MOST (Grants No. 109-2112-M-002-017-MY3 and 109-2811-M-002-535), and in part by NSFC (Grants 11735010, 11975149, 12090064), by Key Laboratoryfor Particle Physics, Astrophysics and Cosmology, Ministry of Education, and Shanghai Key Lab-oratory for Particle Physics and Cosmology (Grant No. 15DZ2272100).
A. INDEPENDENT COLOR TENSORS
In this appendix, we give details the independent color tensors to contract with six quarks in acolor SU (3) C invariant way. Denoting a general six-quark field configuration as O ijklmn = q iI q jJ q kK q lL q mM q nN , (A.1)where the superscripts { i, j, k, l, m, n } are the color indices in fundamental representation of SU (3) C while the subscripts { I, J, K, L, M, N } encode the flavor and chiral information for eachquark field. To form a color invariant operator, the color indices must be contracted by a colortensor T ijklmn such that O ijklmn T ijklmn is invariant under SU (3) C . Since all quark fields belongto the fundamental representation of the color group, the color tensor T ijklmn must be a linear com-bination of two rank-3 totally anti-symmetric tensor (cid:15) xyz s. By the Schouten identities (SI) [44] (cid:15) imn (cid:15) jkl = − (cid:15) ijm (cid:15) kln + (cid:15) ikm (cid:15) jln − (cid:15) ilm (cid:15) jkn ,(cid:15) ijm (cid:15) kln = (cid:15) ikm (cid:15) jln + (cid:15) ikn (cid:15) jlm − (cid:15) ilm (cid:15) jkn − (cid:15) iln (cid:15) jkm − (cid:15) ijn (cid:15) klm , (A.2)we can split the m -index and n -index into two epsilon tensors via the first SI, in turn, there are sixindependent combinations remained and the second SI further reduces them into five independentones. By symmetrizing or anti-symmetrizing pairs of indices ( ij ) , ( kl ) , ( mn ) , we can choose thefollowing five independent color tensors: T SSS { ij }{ kl }{ mn } = (cid:15) ikm (cid:15) jln + (cid:15) ikn (cid:15) jlm + (cid:15) ilm (cid:15) jkn + (cid:15) iln (cid:15) jkm ,T SAA { ij } [ kl ][ mn ] = (cid:15) imn (cid:15) jkl + (cid:15) ikl (cid:15) jmn = (cid:15) ikm (cid:15) jln − (cid:15) ikn (cid:15) jlm − (cid:15) ilm (cid:15) jkn + (cid:15) iln (cid:15) jkm ,T SAA { kl } [ mn ][ ij ] = (cid:15) ijk (cid:15) mnl + (cid:15) ijl (cid:15) mnk = (cid:15) ikm (cid:15) jln − (cid:15) ikn (cid:15) jlm + (cid:15) ilm (cid:15) jkn − (cid:15) iln (cid:15) jkm ,T SAA { mn } [ ij ][ kl ] = (cid:15) ijm (cid:15) kln + (cid:15) ijn (cid:15) klm = (cid:15) ikm (cid:15) jln + (cid:15) ikn (cid:15) jlm − (cid:15) ilm (cid:15) jkn − (cid:15) iln (cid:15) jkm ,T AAA [ ij ][ kl ][ mn ] = 13 ( (cid:15) ijm (cid:15) kln − (cid:15) ijn (cid:15) klm − (cid:15) ijk (cid:15) mnl + (cid:15) ijl (cid:15) mnk + (cid:15) ikl (cid:15) jmn − (cid:15) imn (cid:15) jkl )= (cid:15) ijm (cid:15) kln − (cid:15) ijn (cid:15) klm . (A.3)31here the subscripts in curly bracket { ij } and squared bracket [ kl ] indicate separately the sym-metrization and anti-symmetrization under the exchange of two color indices i ↔ j within. In theabove, the T SSS { ij }{ kl }{ mn } and T AAA [ ij ][ kl ][ mn ] are separately totally symmetric and anti-symmetric underthe exchange of any pairs of the arguments, and T SAA { ij } [ mn ][ kl ] is symmetric for the latter two pairs ofindices. In addition, we have the constraints for exchanging two indices among two different pairsof indices T SSS { ik }{ jl }{ mn } = − T SSS { ij }{ kl }{ mn } + 3 T SAA { mn } [ ij ][ kl ] , T SAA { ij } [ km ][ ln ] = T SSS { ij }{ kl }{ mn } + T SAA { ij } [ kl ][ mn ] , T SAA { ik } [ jl ][ mn ] = − T SAA { ij } [ kl ][ mn ] + T SAA { kl } [ mn ][ ij ] + 2 T AAA [ ij ][ kl ][ mn ] , T SAA { in } [ kl ][ mj ] = − T SAA { ij } [ kl ][ mn ] − T SAA { mn } [ ij ][ kl ] − T AAA [ ij ][ kl ][ mn ] , T AAA [ ik ][ jl ][ mn ] = T SAA { ij } [ kl ][ mn ] + T SAA { kl } [ mn ][ ij ] . (A.4)These relations are useful to reduce redundant operators, and will be used repeatedly in latersections to reach the minimal basis for the dim-12 ∆ B = ∆ L = − operators both in LEFT andin SMEFT . B. LEFT OPERATORS
The full list for dim-12 operators inducing dinucleon to dilepton transitions. (cid:4)
Dim-12 operators contributing to pp → (cid:96) + (cid:96) (cid:48) + For the operators with a scalar current j (cid:96)(cid:96) (cid:48) S, ± , we find there are 28 independent operators which canbe parametrized as follows Q ( pp ) S, ± LLL,a = ( u i T L Cu jL )( u k T L Cd lL )( u m T L Cd nL ) j (cid:96)(cid:96) (cid:48) S, ± T SSS { ij }{ kl }{ mn } , Q ( pp ) S, ± LLL,b = ( u i T L Cu jL )( u k T L Cd lL )( u m T L Cd nL ) j (cid:96)(cid:96) (cid:48) S, ± T SAA { ij } [ kl ][ mn ] , Q ( pp ) S, ± LLR,a = ( u i T L Cu jL )( u k T L Cd lL )( u m T R Cd nR ) j (cid:96)(cid:96) (cid:48) S, ± T SSS { ij }{ kl }{ mn } , Q ( pp ) S, ± LLR,b = ( u i T L Cu jL )( u k T L Cd lL )( u m T R Cd nR ) j (cid:96)(cid:96) (cid:48) S, ± T SAA { ij } [ kl ][ mn ] , Q ( pp ) S, ± LLR,a = ( u i T L Cd jL )( u k T L Cd lL )( u m T R Cu nR ) j (cid:96)(cid:96) (cid:48) S, ± T SSS { ij }{ kl }{ mn } , Q ( pp ) S, ± LLR,b = ( u i T L Cd jL )( u k T L Cd lL )( u m T R Cu nR ) j (cid:96)(cid:96) (cid:48) S, ± T SAA { mn } [ ij ][ kl ] , Q ( pp ) S, ± LLR = ( u i T L Cu jL )( u k T L Cu lL )( d m T R Cd nR ) j (cid:96)(cid:96) (cid:48) S, ± T SSS { ij }{ kl }{ mn } , (B.1)together with their parity partners with L ↔ R .32or the operators with a vector lepton current j (cid:96)(cid:96) (cid:48) ,µV , there are 19 independent operators whichare chosen to take Q ( pp ) V LL,a = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cγ µ u nR ) j (cid:96)(cid:96) (cid:48) ,µV T SSS { ij }{ kl }{ mn } , Q ( pp ) V LL,b = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cγ µ u nR ) j (cid:96)(cid:96) (cid:48) ,µV T SAA { ij } [ kl ][ mn ] , Q ( pp ) V LL,c = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cγ µ u nR ) j (cid:96)(cid:96) (cid:48) ,µV T SAA { mn } [ kl ][ ij ] , Q ( pp ) V LL,a = ( u i T L Cu jL )( u k T L Cd lL )( u m T L Cγ µ d nR ) j (cid:96)(cid:96) (cid:48) ,µV T SSS { ij }{ kl }{ mn } , Q ( pp ) V LL,b = ( u i T L Cu jL )( u k T L Cd lL )( u m T L Cγ µ d nR ) j (cid:96)(cid:96) (cid:48) ,µV T SAA { ij } [ kl ][ mn ] , Q ( pp ) V LR,a = ( u i T L Cu jL )( u k T R Cd lR )( u m T L Cγ µ d nR ) j (cid:96)(cid:96) (cid:48) ,µV T SSS { ij }{ kl }{ mn } , Q ( pp ) V LR,b = ( u i T L Cu jL )( u k T R Cd lR )( u m T L Cγ µ d nR ) j (cid:96)(cid:96) (cid:48) ,µV T SAA { ij } [ kl ][ mn ] , Q ( pp ) V LR,a = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ u nR ) j (cid:96)(cid:96) (cid:48) ,µV T SSS { ij }{ kl }{ mn } , Q ( pp ) V LR,b = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ u nR ) j (cid:96)(cid:96) (cid:48) ,µV T SAA { ij } [ kl ][ mn ] , Q ( pp ) V LR,c = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ u nR ) j (cid:96)(cid:96) (cid:48) ,µV T SAA { kl } [ mn ][ ij ] , Q ( pp ) V LR,d = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ u nR ) j (cid:96)(cid:96) (cid:48) ,µV T SAA { mn } [ ij ][ kl ] , Q ( pp ) V LR,e = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ u nR ) j (cid:96)(cid:96) (cid:48) ,µV T AAA [ ij ][ kl ][ mn ] , (B.2)together with the parity partners for operators Q V − with L ↔ R .For the operators with a tensor lepton current j (cid:96)(cid:96) (cid:48) ,µνT , we find there are 16 independent operatorswhich are chosen to take Q ( pp ) T, − LLL = ( u i T L Cσ µν u jL )( u k T L Cd lL )( u m T L Cd nL ) j (cid:96)(cid:96) (cid:48) ,µνT, − T SAA { mn } [ ij ][ kl ] , Q ( pp ) T, − LLR,a = ( u i T L Cσ µν u jL )( u k T L Cd lL )( u m T R Cd nR ) j (cid:96)(cid:96) (cid:48) ,µνT, − T SAA { kl } [ mn ][ ij ] , Q ( pp ) T, − LLR,b = ( u i T L Cσ µν u jL )( u k T L Cd lL )( u m T R Cd nR ) j (cid:96)(cid:96) (cid:48) ,µνT, − T SAA { mn } [ ij ][ kl ] , Q ( pp ) T, − LLR,c = ( u i T L Cσ µν u jL )( u k T L Cd lL )( u m T R Cd nR ) j (cid:96)(cid:96) (cid:48) ,µνT, − T AAA [ ij ][ kl ][ mn ] , Q ( pp ) T, +2 LLR,a = ( u i T L Cu jL )( u k T L Cd lL )( u m T R Cσ µν d nR ) j (cid:96)(cid:96) (cid:48) ,µνT, + T SSS { ij }{ kl }{ mn } , Q ( pp ) T, +2 LLR,b = ( u i T L Cu jL )( u k T L Cd lL )( u m T R Cσ µν d nR ) j (cid:96)(cid:96) (cid:48) ,µνT, + T SAA { ij } [ kl ][ mn ] , Q ( pp ) T, − LLR = ( u i T L Cd jL )( u k T L Cσ µν d lL )( u m T R Cu nR ) j (cid:96)(cid:96) (cid:48) ,µνT, − T SAA { mn } [ ij ][ kl ] , Q ( pp ) T, +3 LLR = ( u i T L Cd jL )( u k T L Cd lL )( u m T R Cσ µν u nR ) j (cid:96)(cid:96) (cid:48) ,µνT, + T SAA { ij } [ kl ][ mn ] , (B.3)together with the parity partners for operators Q T − with L ↔ R and − ↔ + .As a non-trivial example for the reduction of redundant operators, we consider the above tensorcurrent operator Q ( pp ) T, +3 LRR in Eq. (B.3) with the replacement of the color tensor T SAA { ij } [ kl ][ mn ] by33 SSS { ij }{ kl }{ mn } , then the new operator is reduced as follows u i T L Cu jL )( u k T R Cd lR )( u m T R Cσ µν d nR ) j (cid:96)(cid:96) (cid:48) ,µνT, + T SSS { ij }{ kl }{ mn } FI = − u i T L Cu jL ) (cid:2) ( u m T R Cd lR )( u k T R Cσ µν d nR ) + ( u k T R Cu mR )( d l T R Cσ µν d nR ) (cid:3) j (cid:96)(cid:96) (cid:48) ,µνT, + T SSS { ij }{ kl }{ mn } SI = ( u i T L Cu jL )( u k T R Cd lR )( u m T R Cσ µν d nR ) j (cid:96)(cid:96) (cid:48) ,µνT, + (cid:0) T SSS { ij }{ kl }{ mn } + 3 T SAA { ij } [ kl ][ mn ] (cid:1) + ( u i T L Cu jL )( u k T R Cu lR )( d m T R Cσ µν d nR ) j (cid:96)(cid:96) (cid:48) ,µνT, + (cid:0) T SSS { ij }{ kl }{ mn } − T SAA { ij } [ kl ][ mn ] (cid:1) (= 0) , where the second step uses the FI in Appendix C and the third step exploits the SI in Eq. (A.4),and the terms in last line vanish due to mismatched color symmetry. We see this new operator isequivalent to Q ( pp ) T, +3 LRR and therefore redundant. All other operators with different color tensors orLorentz structures beyond the above lists can be reduced in a similar manner. (cid:4)
Dim-12 operators contributing to pn → (cid:96) + ν (cid:48) For the operators with a scalar current j (cid:96)ν (cid:48) S , there are 14 independent operators which can beparametrized as follows Q ( pn ) S LLL,a = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cd nL ) j (cid:96)ν (cid:48) S T SSS { ij }{ kl }{ mn } , Q ( pn ) S LLL,b = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cd nL ) j (cid:96)ν (cid:48) S T SAA { ij } [ kl ][ mn ] , Q ( pn ) S LLR = ( u i T L Cu jL )( u k T L Cd lL )( d m T R Cd nR ) j (cid:96)ν (cid:48) S T SSS { ij }{ kl }{ mn } , Q ( pn ) S LLR,a = ( u i T L Cd jL )( u k T L Cd lL )( u m T R Cd nR ) j (cid:96)ν (cid:48) S T SSS { ij }{ kl }{ mn } , Q ( pn ) S LLR,b = ( u i T L Cd jL )( u k T L Cd lL )( u m T R Cd nR ) j (cid:96)ν (cid:48) S T SAA { ij } [ kl ][ mn ] , Q ( pn ) S LLR,c = ( u i T L Cd jL )( u k T L Cd lL )( u m T R Cd nR ) j (cid:96)ν (cid:48) S T SAA { mn } [ ij ][ kl ] , Q ( pn ) S LLR = ( d i T L Cd jL )( u k T L Cd lL )( u m T R Cu nR ) j (cid:96)ν (cid:48) S T SSS { ij }{ kl }{ mn } , (B.4)together with their parity partners with L ↔ R .For the operators with a vector current j (cid:96)ν (cid:48) ,µV , we find there are 24 independent operators whichare parametrized as follows Q ( pn ) V LL,a = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T SSS { ij }{ kl }{ mn } , Q ( pn ) V LL,b = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T SAA { ij } [ kl ][ mn ] , Q ( pn ) V LL,c = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T SAA { mn } [ ij ][ kl ] , Q ( pn ) V LL,a = ( u i T L Cd jL )( u k T L Cd lL )( d m T L Cγ µ u nR ) j (cid:96)ν (cid:48) ,µV T SSS { ij }{ kl }{ mn } , Q ( pn ) V LL,b = ( u i T L Cd jL )( u k T L Cd lL )( d m T L Cγ µ u nR ) j (cid:96)ν (cid:48) ,µV T SAA { ij } [ kl ][ mn ] , Q ( pn ) V LL,c = ( u i T L Cd jL )( u k T L Cd lL )( d m T L Cγ µ u nR ) j (cid:96)ν (cid:48) ,µV T SAA { mn } [ ij ][ kl ] , ( pn ) V LR = ( u i T L Cu jL )( d k T R Cd lR )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T SSS { ij }{ kl }{ mn } , Q ( pn ) V LR,a = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T SSS { ij }{ kl }{ mn } , Q ( pn ) V LR,b = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T SAA { ij } [ kl ][ mn ] , Q ( pn ) V LR,c = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T SAA { kl } [ mn ][ ij ] , Q ( pn ) V LR,d = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T SAA { mn } [ ij ][ kl ] , Q ( pn ) V LR,e = ( u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ d nR ) j (cid:96)ν (cid:48) ,µV T AAA [ ij ][ kl ][ mn ] , (B.5)together with their parity partners with L ↔ R .For the operators with a tensor current j (cid:96)ν (cid:48) ,µνT , there are 13 independent operators which can beparametrized as follows Q ( pn ) TLLL,a = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cσ µν d nL ) j (cid:96)ν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] , Q ( pn ) TLLL,b = ( u i T L Cd jL )( u k T L Cd lL )( u m T L Cσ µν d nL ) j (cid:96)ν (cid:48) ,µνT T SAA { mn } [ ij ][ kl ] , Q ( pn ) T LLR = ( u i T L Cσ µν u jL )( u k T L Cd lL )( d m T R Cd nR ) j (cid:96)ν (cid:48) ,µνT T SAA { mn } [ ij ][ kl ] , Q ( pn ) T LLR,a = ( u i T L Cd jL )( u k T L Cσ µν d lL )( u m T R Cd nR ) j (cid:96)ν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] , Q ( pn ) T LLR,b = ( u i T L Cd jL )( u k T L Cσ µν d lL )( u m T R Cd nR ) j (cid:96)ν (cid:48) ,µνT T SAA { kl } [ mn ][ ij ] , Q ( pn ) T LLR,c = ( u i T L Cd jL )( u k T L Cσ µν d lL )( u m T R Cd nR ) j (cid:96)ν (cid:48) ,µνT T SAA { mn } [ ij ][ kl ] , Q ( pn ) T LLR,d = ( u i T L Cd jL )( u k T L Cσ µν d lL )( u m T R Cd nR ) j (cid:96)ν (cid:48) ,µνT T AAA [ ij ][ kl ][ mn ] , Q ( pn ) T LLR = ( d i T L Cσ µν d jL )( u k T L Cd lL )( u m T R Cu nR ) j (cid:96)ν (cid:48) ,µνT T SAA { mn } [ ij ][ kl ] , Q ( pn ) T RRL = ( u i T R Cu jR )( u k T R Cd lR )( d m T L Cσ µν d nL ) j (cid:96)ν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] , Q ( pn ) T RRL,a = ( u i T R Cd jR )( u k T R Cd lR )( u m T L Cσ µν d nL ) j (cid:96)ν (cid:48) ,µνT T SSS { ij }{ kl }{ mn } , Q ( pn ) T RRL,b = ( u i T R Cd jR )( u k T R Cd lR )( u m T L Cσ µν d nL ) j (cid:96)ν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] , Q ( pn ) T RRL,c = ( u i T R Cd jR )( u k T R Cd lR )( u m T L Cσ µν d nL ) j (cid:96)ν (cid:48) ,µνT T SAA { mn } [ ij ][ kl ] , Q ( pn ) T RRL = ( d i T R Cd jR )( u k T R Cd lR )( u m T L Cσ µν u nL ) j (cid:96)ν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] . (B.6) (cid:4) Dim-12 operators contributing to nn → ¯ ν ¯ ν (cid:48) For the operators with a scalar current j νν (cid:48) S , there are 14 independent operators which are parametrizedas follows Q ( nn ) S LLL,a = ( d i T L Cd jL )( d k T L Cu lL )( d m T L Cu nL ) j νν (cid:48) S T SSS { ij }{ kl }{ mn } , Q ( nn ) S LLL,b = ( d i T L Cd jL )( d k T L Cu lL )( d m T L Cu nL ) j νν (cid:48) S T SAA { ij } [ kl ][ mn ] , ( nn ) S LLR,a = ( d i T L Cd jL )( d k T L Cu lL )( d m T R Cu nR ) j νν (cid:48) S T SSS { ij }{ kl }{ mn } , Q ( nn ) S LLR,b = ( d i T L Cd jL )( d k T L Cu lL )( d m T R Cu nR ) j νν (cid:48) S T SAA { ij } [ kl ][ mn ] , Q ( nn ) S LLR,a = ( d i T L Cu jL )( d k T L Cu lL )( d m T R Cd nR ) j νν (cid:48) S T SSS { ij }{ kl }{ mn } , Q ( nn ) S LLR,b = ( d i T L Cu jL )( d k T L Cu lL )( d m T R Cd nR ) j νν (cid:48) S T SAA { mn } [ ij ][ kl ] , Q ( nn ) S LLR = ( d i T L Cd jL )( d k T L Cd lL )( u m T R Cu nR ) j νν (cid:48) S T SSS { ij }{ kl }{ mn } , (B.7)together with their parity partners with L ↔ R .And for the operators with a tensor neutrino current j νν (cid:48) ,µνT , there are only 8 independent oper-ators which are parametrized as follows Q ( nn ) T LLL = ( d i T L Cσ µν d jL )( d k T L Cu lL )( d m T L Cu nL ) j νν (cid:48) ,µνT T SAA { mn } [ kl ][ ij ] , Q ( nn ) T LLR,a = ( d i T L Cσ µν d jL )( d k T L Cu lL )( d m T R Cu nR ) j νν (cid:48) ,µνT T SAA { kl } [ mn ][ ij ] , Q ( nn ) T LLR,b = ( d i T L Cσ µν d jL )( d k T L Cu lL )( d m T R Cu nR ) j νν (cid:48) ,µνT T SAA { mn } [ ij ][ kl ] , Q ( nn ) T LLR,c = ( d i T L Cσ µν d jL )( d k T L Cu lL )( d m T R Cu nR ) j νν (cid:48) ,µνT T AAA [ ij ][ kl ][ mn ] , Q ( nn ) T LLR = ( d i T L Cu jL )( d k T L Cσ µν u lL )( d m T R Cd nR ) j νν (cid:48) ,µνT T SAA { mn } [ ij ][ kl ] , Q ( nn ) T RRL,a = ( d i T R Cd jR )( d k T R Cu lR )( d m T L Cσ µν u nL ) j νν (cid:48) ,µνT T SSS { ij }{ kl }{ mn } , Q ( nn ) T RRL,b = ( d i T R Cd jR )( d k T R Cu lR )( d m T L Cσ µν u nL ) j νν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] , Q ( nn ) T RRL = ( d i T R Cu jR )( d k T R Cu lR )( d m T L Cσ µν d nL ) j νν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] . (B.8) C. REDUCTION OF THE REDUNDANT OPERATORS IN THE LEFT
The dim-12 operators contributing to H − ¯H oscillation and pp → e + e + transitions in the LEFTwere given first in Ref. [33]. For the operators with the scalar lepton current, their results areconsistent with ours, and the 28 operators in their paper can be easily identified with the resultsshown in Eq. (B.1). For the operators with a vector current, they count 32 operators and 13 ofthem are redundant and will be reduced in the following. In doing so, we first notice that the colortensors in Ref. [33] have the following one-to-one correspondence with our notation ( T S ) ijklmn = T SSS { ij }{ kl }{ mn } , ( T A ) ijklmn = T SAA { mn } [ kl ][ ij ] , ( ˜ T A ) ijklmn = T AAA [ ij ][ kl ][ mn ] . (C.1)The relations in Eq. (A.4) imply the following corresponding relations T S ) ikjlmn = 3( T A ) ijklmn − ( T S ) ijklmn , ( T A ) mjklin = −
2( ˜ T A ) ijklmn − ( T A ) ijklmn − ( T A ) mnklij . (C.2)Since the two lepton fields in the vector current have different chirality, Ref. [33] parametrizedall such operators using four scalar fermion bilinears in which the two lepton fields are combinedseparately with two quark fields to make scalar currents. By the Fierz identity ( q m T L C(cid:96) L )( q n T R C(cid:96) R ) = −
12 ( q m T L Cγ µ q nR )( (cid:96) T R Cγ µ (cid:96) L ) , (C.3)we can rewrite the operators with a pair of ( (cid:96) L , (cid:96) R ) in Ref. [33] to have a factorized vector leptoncurrent as follows ( O H ¯ HH ) χ χ LR = −
12 ( u i T χ Cu jχ )( u k T χ Cu lχ )( d m T L Cγ µ d nR )( (cid:96) T R Cγ µ (cid:96) L )( T S ) ijklmn , ( O H ¯ HH ) χ χ LR = −
12 ( u i T χ Cu jχ )( d k T χ Cd lχ )( u m T L Cγ µ u nR )( (cid:96) T R Cγ µ (cid:96) L )( T S ) ijklmn , ( O H ¯ HH ) χ χ LR = −
12 ( u i T χ Cu jχ )( u k T χ Cd lχ )( u m T L Cγ µ d nR )( (cid:96) T R Cγ µ (cid:96) L )( T S ) mnklij , ( O H ¯ HH ) χ χ LR = −
12 ( u i T χ Cu jχ )( u k T χ Cd lχ )( u m T L Cγ µ d nR )( (cid:96) T R Cγ µ (cid:96) L )( T A ) mnklij , ( O H ¯ HH ) χ χ RL = −
12 ( u i T χ Cu jχ )( u k T χ Cd lχ )( d m T L Cγ µ u nR )( (cid:96) T R Cγ µ (cid:96) L )( T S ) mnklij , ( O H ¯ HH ) χ χ RL = + 12 ( u i T χ Cu jχ )( u k T χ Cd lχ )( d m T L Cγ µ u nR )( (cid:96) T R Cγ µ (cid:96) L )( T A ) mnklij , ( O H ¯ HH ) χ χ LR = −
12 ( u i T χ Cd jχ )( u k T χ Cd lχ )( u m T L Cγ µ u nR )( (cid:96) T R Cγ µ (cid:96) L )( T S ) ijklmn , ( O H ¯ HH ) χ χ LR = −
12 ( u i T χ Cd jχ )( u k T χ Cd lχ )( u m T L Cγ µ u nR )( (cid:96) T R Cγ µ (cid:96) L )( T A ) ijklmn , ( O H ¯ HH ) χ χ LR = −
12 ( u i T χ Cd jχ )( u k T χ Cd lχ )( u m T L Cγ µ u nR )( (cid:96) T R Cγ µ (cid:96) L )( ˜ T A ) ijklmn , ( O H ¯ HH ) χ χ LR = −
12 ( u i T χ Cd jχ )( u k T χ Cd lχ )( u m T L Cγ µ u nR )( (cid:96) T R Cγ µ (cid:96) L )( T A ) mnklij , (C.4)where the convention for the operators is taken from Ref. [33] with a minor change e ↔ (cid:96) .Under the exchange of the chirality L ↔ R , we see easily that ( O H ¯ HHi ) χ χ RL = ( O H ¯ HHi ) χ χ LR for i = 1 , , , and ( O H ¯ HHj ) χ χ RL = − ( O H ¯ HHj ) χ χ LR for j = 7 , . Considering ( χ χ ) ∈{ LL, LR, RL, RR } , there are totally 40 operators in the above. The 32 operators counted in [33]can be obtained after taking into account the following eight obvious relations ( O H ¯ HHi ) LRLR = ( O H ¯ HHi ) RLLR , i = 1 , , , ( O H ¯ HH ) χ χ LR = − ( O H ¯ HH ) χ χ LR , χ , = L, R , ( O H ¯ HH ) χχLR = ( O H ¯ HH ) χχLR − O H ¯ HH ) χχLR , χ = L, R , (C.5)37here the last relation is obtained by exploiting the first relation in Eq. (C.2) and the Fierz identity ( ψ T1 χ Cψ χ )( ψ T3 χ Cψ χ ) = − ( ψ T1 χ Cψ χ )( ψ T2 χ Cψ χ ) − ( ψ T1 χ Cψ χ )( ψ T3 χ Cψ χ ) , χ = L, R .
Now we show that the remaining 32 operators in Eq. (C.4) after modulo the relations inEq. (C.5) can be further reduced to the 19 operators shown in Eqs. (B.2). Using the followingFierz identities ( ψ T1 L Cψ L )( ψ T3 L Cγ µ ψ R ) = − ( ψ T1 L Cψ L )( ψ T2 L Cγ µ ψ R ) − ( ψ T2 L Cψ L )( ψ T1 L Cγ µ ψ R ) , ( ψ T1 R Cψ R )( ψ T3 L Cγ µ ψ R ) = − ( ψ T1 R Cψ R )( ψ T3 L Cγ µ ψ R ) − ( ψ T2 R Cψ R )( ψ T3 L Cγ µ ψ R ) , (C.6)and the relations in Eq. (C.2), we finally obtain the following relations among the remaining 32operators ( O H ¯ HH ) χLLR = ( O H ¯ HH ) χLLR − O H ¯ HH ) χLLR , χ = L, R , ( O H ¯ HH ) χRLR = ( O H ¯ HH ) χRRL − O H ¯ HH ) χRRL , χ = L, R , ( O H ¯ HH ) χLLR = ( O H ¯ HH ) χLRL − O H ¯ HH ) χLRL , χ = L, R , ( O H ¯ HH ) χRLR = ( O H ¯ HH ) χRLR − O H ¯ HH ) χRLR , χ = L, R , ( O H ¯ HH ) RχLR = ( O H ¯ HH ) RχLR − O H ¯ HH ) χRLR , χ = L, R , ( O H ¯ HH ) LχRL = ( O H ¯ HH ) LχLR + 3( O H ¯ HH ) χLLR , χ = L, R , ( O H ¯ HH ) RχLR = +( O H ¯ HH ) RχLR + 2( O H ¯ HH ) RχLR + ( O H ¯ HH ) RχLR , χ = L, R , ( O H ¯ HH ) LχRL = − ( O H ¯ HH ) LχLR + 2( O H ¯ HH ) LχLR + ( O H ¯ HH ) LχLR , χ = L, R , (C.7)On top of the relations in Eq. (C.5), the above relations give 13 new constraints which thereforereduce the 32 operators into 19 independent operators as we claimed. Choosing the following 19independent operators ( O H ¯ HH , ) LχLR , ( O H ¯ HH , ) RχRL , ( O H ¯ HH , , ) χχLR , χ = L, R ; ( O H ¯ HH , , , ) LRLR , ( O H ¯ HH ) RLLR . One can easily find they have a one-to-one correspondence with the ones given in Eq. (B.2): ( O H ¯ HH , ) LLLR + L ↔ R ⇔ Q ( pp ) V LL,a,b + L ↔ R , ( O H ¯ HH , ) LRLR + L ↔ R ⇔ Q ( pp ) V LR,a,b + L ↔ R , ( O H ¯ HH , , ) LLLR + ( O H ¯ HH , , ) RRLR ⇔ Q ( pp ) V LL,a,b,c + L ↔ R , ( O H ¯ HH , , , ) LRLR + ( O H ¯ HH ) RLLR ⇔ Q ( pp ) V LR,a,b,c,d,e . (C.8)38 . SMEFT OPERATORS For operators with a scalar lepton current, there are 12 independent operators chosen as follows O S, ( A ) u d L = ( u i T R Cd jR )( u k T R Cd lR )( u m T R Cd nR )( L T a CL (cid:48) b ) (cid:15) ab T SSS { ij }{ kl }{ mn } , O S, ( A ) u d L = ( u i T R Cd jR )( u k T R Cd lR )( u m T R Cd nR )( L T a CL (cid:48) b ) (cid:15) ab T SAA { ij } [ kl ][ mn ] , O S, ( S ) u d Q L = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a CQ nb )( L T c CL (cid:48) d ) (cid:15) ac (cid:15) bd T SSS { ij }{ kl }{ mn } , O S, ( A ) u d Q L = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a CQ nb )( L T c CL (cid:48) d ) (cid:15) ab (cid:15) cd T SAA { ij } [ kl ][ mn ] , O S, ( S ) u d Q L = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a CQ nb )( L T c CL (cid:48) d ) (cid:15) ac (cid:15) bd T SAA { mn } [ kl ][ ij ] , O S, ( A ) udQ L = ( u i T R Cd jR )( Q k T a CQ lb )( Q m T c CQ nd )( L T e CL (cid:48) f ) (cid:15) ab (cid:15) cd (cid:15) ef T SAA { ij } [ kl ][ mn ] , O S, ( S ) udQ L = ( u i T R Cd jR )( Q k T a CQ lb )( Q m T c CQ nd )( L T e CL (cid:48) f ) (cid:15) ab (cid:15) ce (cid:15) df T SAA { mn } [ kl ][ ij ] , O S, ( S ) Q L = ( Q i T a CQ jb )( Q k T c CQ ld )( Q m T e CQ nf )( L T g CL (cid:48) h ) (cid:15) ab (cid:15) cd (cid:15) eg (cid:15) fh T SAA { mn } [ kl ][ ij ] , O S, ( S ) u d e = ( u i T R Cu jR )( u k T R Cd lR )( u m T R Cd nR )( e T R Ce (cid:48) R ) T SSS { ij }{ kl }{ mn } , O S, ( S ) u d e = ( u i T R Cu jR )( u k T R Cd lR )( u m T R Cd nR )( e T R Ce (cid:48) R ) T SAA { ij } [ kl ][ mn ] , O S, ( S ) u dQ e = ( u i T R Cu jR )( u k T R Cd lR )( Q m T a CQ nb )( e T R Ce (cid:48) R ) (cid:15) ab T SAA { ij } [ kl ][ mn ] , O S, ( S ) u Q e = ( u i T R Cu jR )( Q k T a CQ lb )( Q m T c CQ nd )( e T R Ce (cid:48) R ) (cid:15) ab (cid:15) cd T SAA { ij } [ kl ][ mn ] , (D.1)where the first superscript ‘S’ ( and the following ‘V’ and ‘T’ ) is used to represent the relevantoperator with a scalar (vector and tensor for the following ones) lepton current, and the bracketsuperscripts ‘(S)/(A)’ indicate the flavor symmetric/anti-symmetric property of the lepton currentunder the exchange of the two lepton fields.For operators with a vector lepton current, we find there are independent 7 operators chosen asfollows O Vu d QeL = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a Cγ µ u nR )( e T R Cγ µ L (cid:48) b ) (cid:15) ab T SSS { ij }{ kl }{ mn } , O Vu d QeL = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a Cγ µ u nR )( e T R Cγ µ L (cid:48) b ) (cid:15) ab T SAA { ij } [ kl ][ mn ] , O Vu d QeL = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a Cγ µ u nR )( e T R Cγ µ L (cid:48) b ) (cid:15) ab T SAA { mn } [ kl ][ ij ] , O Vu dQ eL = ( u i T R Cd jR )( Q k T a CQ lb )( Q m T c Cγ µ u nR )( e T R Cγ µ L (cid:48) d ) (cid:15) ab (cid:15) cd T SAA { ij } [ kl ][ mn ] , O Vu dQ eL = ( u i T R Cd jR )( Q k T a CQ lb )( Q m T c Cγ µ u nR )( e T R Cγ µ L (cid:48) d ) (cid:15) ab (cid:15) cd T SAA { mn } [ kl ][ ij ] , O Vu dQ eL = ( u i T R Cd jR )( Q k T a CQ lb )( Q m T c Cγ µ u nR )( e T R Cγ µ L (cid:48) d ) (cid:15) ab (cid:15) cd T AAA [ ij ][ kl ][ mn ] , O VuQ eL = ( Q i T a CQ jb )( Q k T c CQ ld )( Q m T e Cγ µ u nR )( e T R Cγ µ L (cid:48) f ) (cid:15) ab (cid:15) cd (cid:15) ef T SAA { mn } [ kl ][ ij ] , (D.2)39here the lepton current mixes the lepton doublet field L and singlet field e R and therefore no anyflavor symmetry property.For operators with a tensor lepton current, there are 10 independent operators which are chosenas follows O T, ( S ) u d Q L = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a Cσ µν Q nb )( L T c Cσ µν L (cid:48) d ) (cid:15) ab (cid:15) cd T SSS { ij }{ kl }{ mn } , O T, ( A ) u d Q L = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a Cσ µν Q nb )( L T c Cσ µν L (cid:48) d ) (cid:15) ac (cid:15) bd T SAA { ij } [ kl ][ mn ] , O T, ( S ) u d Q L = ( u i T R Cd jR )( u k T R Cd lR )( Q m T a Cσ µν Q nb )( L T c Cσ µν L (cid:48) d ) (cid:15) ab (cid:15) cd T SAA { mn } [ kl ][ ij ] , O T, ( A ) udQ L = ( u i T R Cd jR )( Q k T a CQ lb )( Q m T c Cσ µν Q nd )( L T e Cσ µν L (cid:48) f ) (cid:15) ab (cid:15) ce (cid:15) df T SAA { ij } [ kl ][ mn ] , O T, ( S ) udQ L = ( u i T R Cd jR )( Q k T a CQ lb )( Q m T c Cσ µν Q nd )( L T e Cσ µν L (cid:48) f ) (cid:15) ab (cid:15) cd (cid:15) ef T SAA { mn } [ kl ][ ij ] , O T, ( A ) udQ L = ( u i T R Cd jR )( Q k T a CQ lb )( Q m T c Cσ µν Q nd )( L T e Cσ µν L (cid:48) f ) (cid:15) ab (cid:15) ce (cid:15) df T AAA [ ij ][ kl ][ mn ] , O T, ( S ) Q L = ( Q i T a CQ jb )( Q k T c CQ ld )( Q m T e Cσ µν Q nf )( L T g Cσ µν L (cid:48) h ) (cid:15) ab (cid:15) cd (cid:15) ef (cid:15) gh T SAA { mn } [ kl ][ ij ] , O T, ( A ) u d e = ( u i T R Cσ µν u jR )( u k T R Cd lR )( u m T R Cd nR )( e T R Cσ µν e (cid:48) R ) T SAA { mn } [ ij ][ kl ] , O T, ( A ) u dQ e = ( u i T R Cσ µν u jR )( u k T R Cd lR )( Q m T a CQ nb )( e T R Cσ µν e (cid:48) R ) (cid:15) ab T SAA { kl } [ mn ][ ij ] , O T, ( A ) u dQ e = ( u i T R Cσ µν u jR )( u k T R Cd lR )( Q m T a CQ nb )( e T R Cσ µν e (cid:48) R ) (cid:15) ab T AAA [ ij ][ kl ][ mn ] . (D.3) E. REDUCTION OF THE REDUNDANT OPERATORS IN THE SMEFT
Ref. [11] provides a bunch of the dim-12 operators contributing to dinucleon to dilepton tran-sitions in SMEFT, but the operators are neither complete nor independent as a basis. By usingthe color tensor relations in Eq. (A.4) and the SI for the SU (2) L group and FIs in Appendix C, inthe following, we translate their operators as linear combinations of our operators so that one caneasily recognize the redundancy and incompleteness in [11]: O ( pp )1 = + O S, ( S ) u d e − O S, ( S ) u d e , O ( pp )2 = + O S, ( S ) u d e , O ( pp )3 = + O S, ( S ) u d e ; (E.1) O ( pp )4 = + O S, ( S ) u dQ e ; (E.2) O ( pp )5 = + O S, ( S ) u Q e , O ( pp )6 = − O S, ( S ) u Q e ; (E.3) O ( pp,np )7 = − (cid:0) C Vu d QeL − C Vu d QeL (cid:1) , ( pp,np )8 = − (cid:0) C Vu d QeL − C Vu d QeL + 3 C Vu d QeL (cid:1) , O ( pp,np )9 = − (cid:0) C Vu d QeL − C Vu d QeL (cid:1) , O ( pp,np )10 = − (cid:0) C Vu d QeL + 4 C Vu d QeL + C Vu d QeL (cid:1) , O ( pp,np )11 = − (cid:0) C Vu d QeL − C Vu d QeL − C Vu d QeL (cid:1) ; (E.4) O ( pp,np )12 = − (cid:0) C Vu dQ eL + C Vu dQ eL + C Vu dQ eL (cid:1) , O ( pp,np )13 = − C Vu dQ eL , O ( pp,np )14 = − C Vu dQ eL , O ( pp,np )16 = + 32 (cid:0) C Vu dQ eL + C Vu dQ eL + 2 C Vu dQ eL (cid:1) , O ( pp,np )17 = + 32 C Vu dQ eL ; (E.5) O ( pp,np )15 = − O SuQ eL , O ( pp,np )24 = + 32 O SuQ eL ; (E.6) O ( pp,np,nn )18 = − (cid:16) O S, ( S ) u d Q L − O S, ( S ) u d Q L − O T, ( S ) u d Q L + 3 O T, ( S ) u d Q L (cid:17) , O ( pp,np,nn )19 = − (cid:16) O S, ( S ) u d Q L − O T, ( S ) u d Q L (cid:17) , O ( pp,np,nn )20 = − (cid:16) O S, ( S ) u d Q L − O T, ( S ) u d Q L (cid:17) , O ( pp,np,nn )22 = + 116 (cid:16) O S, ( S ) u d Q L − O S, ( S ) u d Q L + 3 O T, ( S ) u d Q L − O T, ( S ) u d Q L (cid:17) , O ( pp,np,nn )23 = + 116 (cid:16) O S, ( S ) u d Q L + 3 O T, ( S ) u d Q L (cid:17) ; (E.7) O ( pp,np,nn )21 = − (cid:16) O S, ( S ) udQ L − O T, ( S ) udQ L (cid:17) , O ( pp,np,nn )25 = + 38 (cid:16) O S, ( A ) udQ L − O T, ( A ) udQ L (cid:17) , O ( pp,np,nn )26 = + 116 (cid:16) O S, ( S ) udQ L + 3 O T, ( S ) udQ L (cid:17) , O ( pp,np,nn )27 = +4 O S, ( A ) udQ L + O T, ( A ) udQ L ; (E.8) O ( pp,np,nn )28 = − (cid:16) O S, ( S ) Q L − O T, ( S ) Q L (cid:17) . (E.9)41 . CHIRAL BASIS CONSTRUCTION In Tabs. V-VII, we rewrite the operator basis in LEFT into a chiral basis which shows theexplicit chiral transformation properties. This chiral basis is obtained from LEFT basis by sym-metrizing quark flavors with the same chirality modulo the anti-symmetric chiral singlet bilinear D ijχ = (cid:15) uv ( q i T χ,u Γ q jχ,v ) = ( u i T χ Γ d jχ ) − i ↔ j . In doing so, most of the operators with color ten-sors T SAA { ij } [ kl ][ mn ] and T AAA [ ij ][ kl ][ mn ] in the LEFT already belong to the chiral irreps under SU (2) f L × SU (2) fR as shown in Tabs. V-VII. For the remaining operators, especially these with color ten-sor T SSS { ij }{ kl }{ mn } , the chiral irrep ones are defined in terms of the above flavor symmetrizationprocedures as follows: (cid:4) Dim-12 operators contributing to pp → (cid:96) + (cid:96) (cid:48) + P ( pp ) S, ± ,a = 15 ( u i T L Cu jL ) (cid:2) u k T L Cd lL )( u m T L Cd nL ) + ( u k T L Cu lL )( d m T L Cd nL ) (cid:3) j (cid:96)(cid:96) (cid:48) S, ± T SSS { ij }{ kl }{ mn } ,P ( pp ) S, ± ,a = 13 (cid:104) u i T L Cd jL )( u k T L Cd lL ) + ( u i T L Cu jL )( d k T L Cd lL ) (cid:105) ( u m T R Cu nR ) j (cid:96)(cid:96) (cid:48) S, ± T SSS { ij }{ kl }{ mn } ,P ( pp ) V ,a = 15 (cid:104) u i T L Cd jL )( u k T L Cd lL )( u m T L Cγ µ u nR ) + ( u i T L Cu jL )( d k T L Cd lL )( u m T L Cγ µ u nR )+2( u i T L Cu jL )( u k T L Cd lL )( d m T L Cγ µ u nR ) (cid:105) j (cid:96)(cid:96) (cid:48) ,µV T SSS { ij }{ kl }{ mn } ,P ( pp ) V ,b = 13 ( u k T L Cd lL ) (cid:104) u i T L Cd jL )( u m T L Cγ µ u nR ) + ( u i T L Cu jL )( d m T L Cγ µ u nR ) (cid:105) j (cid:96)(cid:96) (cid:48) ,µV T SAA { ij } [ kl ][ mn ] ,P ( pp ) V ,a = 15 ( u i T L Cu jL ) (cid:2) u k T L Cd lL )( u m T L Cγ µ d nR ) + ( u k T L Cu lL )( d m T L Cγ µ d nR ) (cid:3) j (cid:96)(cid:96) (cid:48) ,µV T SSS { ij }{ kl }{ mn } ,P ( pp ) V ,a = 13 ( u i T L Cu jL ) (cid:2) u k T R Cd lR )( u m T L Cγ µ d nR ) + ( d k T R Cd lR )( u m T L Cγ µ u nR ) (cid:3) j (cid:96)(cid:96) (cid:48) ,µV T SSS { ij }{ kl }{ mn } ,P ( pp ) V ,a = 19 (cid:104) u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ u nR ) + 2( u i T L Cu jL )( u k T R Cd lR )( d m T L Cγ µ u nR )+2( u i T L Cd jL )( u k T R Cu lR )( u m T L Cγ µ d nR ) + ( u i T L Cu jL )( u k T R Cu lR )( d m T L Cγ µ d nR ) (cid:105) j (cid:96)(cid:96) (cid:48) ,µV T SSS { ij }{ kl }{ mn } ,P ( pp ) V ,b = 13 ( u k T R Cd lR ) (cid:104) u i T L Cd jL )( u m T L Cγ µ u nR ) + ( u i T L Cu jL )( d m T L Cγ µ u nR ) (cid:105) j (cid:96)(cid:96) (cid:48) ,µV T SAA { ij } [ kl ][ mn ] ,P ( pp ) V ,c = 13 ( u i T L Cd jL ) (cid:2) u k T R Cd lR )( u m T L Cγ µ u nR ) + ( u k T R Cu lR )( u m T L Cγ µ d nR ) (cid:3) j (cid:96)(cid:96) (cid:48) ,µV T SAA { kl } [ mn ][ ij ] ,P ( pp ) T, − ,a = 12 (cid:104) ( u i T L Cσ µν u jL )( u k T L Cd lL ) + ( u i T L Cσ µν d jL )( u k T L Cu lL ) (cid:105) ( u m T R Cd nR ) j (cid:96)(cid:96) (cid:48) ,µνT, − T SAA { kl } [ mn ][ ij ] . (cid:4) Dim-12 operators contributing to pn → (cid:96) + ¯ ν (cid:48) P ( pn ) S ,a = 15 ( u i T L Cd jL ) (cid:2) u k T L Cd lL )( u m T L Cd nL ) + 3( u k T L Cu lL )( d m T L Cd nL ) (cid:3) j (cid:96)ν (cid:48) S T SSS { ij }{ kl }{ mn } ,P ( pn ) S ,a = 13 (cid:104) u i T L Cd jL )( u k T L Cd lL ) + ( u i T L Cu jL )( d k T L Cd lL ) (cid:105) ( u m T R Cd mR ) j (cid:96)ν (cid:48) S T SSS { ij }{ kl }{ mn } ,P ( pn ) V ,a = 15 (cid:104) u i T L Cd jL )( u k T L Cd lL )( u m T L Cγ µ d nR ) + ( u i T L Cu jL )( d k T L Cd lL )( u m T L Cγ µ d nR )+2( u i T L Cu jL )( u k T L Cd lL )( d m T L Cγ µ d nR ) (cid:105) j (cid:96)ν (cid:48) ,µV T SSS { ij }{ kl }{ mn } , ( pn ) V ,b = 13 ( u k T L Cd lL ) (cid:104) u i T L Cd jL )( u m T L Cγ µ d nR ) + ( u i T L Cu jL )( d m T L Cγ µ d nR ) (cid:105) j (cid:96)ν (cid:48) ,µV T SAA { ij } [ kl ][ mn ] ,P ( pn ) V ,a = 15 (cid:104) u i T L Cd jL )( u k T L Cd lL )( d m T L Cγ µ u nR ) + ( u i T L Cu jL )( d k T L Cd lL )( d m T L Cγ µ u nR )+2( u i T L Cd jL )( d k T L Cd lL )( u m T L Cγ µ u nR ) (cid:105) j (cid:96)ν (cid:48) ,µV T SSS { ij }{ kl }{ mn } ,P ( pn ) V ,b = 13 ( u k T L Cd lL ) (cid:104) u i T L Cd jL )( d m T L Cγ µ u nR ) + ( d i T L Cd jL )( u m T L Cγ µ u nR ) (cid:105) j (cid:96)ν (cid:48) ,µV T SAA { ij } [ kl ][ mn ] ,P ( pn ) V ,a = 19 (cid:104) u i T L Cd jL )( u k T R Cd lR )( u m T L Cγ µ d nR ) + 2( u i T L Cu jL )( u k T R Cd lR )( d m T L Cγ µ d nR )+2( u i T L Cd jL )( d k T R Cd lR )( u m T L Cγ µ u nR ) + ( u i T L Cu jL )( d k T R Cd lR )( d m T L Cγ µ u nR ) (cid:105) j (cid:96)ν (cid:48) ,µV T SSS { ij }{ kl }{ mn } ,P ( pn ) V ,b = 13 ( u k T R Cd lR ) (cid:104) u i T L Cd jL )( u m T L Cγ µ d nR ) + ( u i T L Cu jL )( d m T L Cγ µ d nR ) (cid:105) j (cid:96)ν (cid:48) ,µV T SAA { ij } [ kl ][ mn ] ,P ( pn ) V ,c = 13 ( u i T L Cd jL ) (cid:2) u k T R Cd lR )( u m T L Cγ µ d nR ) + ( d k T R Cd lR )( u m T L Cγ µ u nR ) (cid:3) j (cid:96)ν (cid:48) ,µV T SAA { kl } [ mn ][ ij ] ,P ( pn ) T ,a = 16 ( u k T L Cd lL ) (cid:104) u i T L Cd jL )( u m T L Cσ µν d nL ) + ( u i T L Cu jL )( d m T L Cσ µν d nL )+( d i T L Cd jL )( u m T L Cσ µν u nL ) (cid:105) j (cid:96)ν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] ,P ( pn ) T ,a = 16 (cid:104) u i T L Cd jL )( u k T L Cσ µν d lL ) + ( u i T L Cu jL )( d k T L Cσ µν d lL ) + ( d i T L Cd jL )( u k T L Cσ µν u lL ) (cid:105) × ( u m T R Cd nR ) j (cid:96)ν (cid:48) ,µνT T SAA { ij } [ kl ][ mn ] , ˆ P ( pn ) T ,a = 13 (cid:104) u i T R Cd jR )( u k T R Cd lR ) + ( u i T R Cu jR )( d k T R Cd lR ) (cid:105) ( u m T L Cσ µν d nL ) j (cid:96)ν (cid:48) ,µνT T SSS { ij }{ kl }{ mn } . (cid:4) Dim-12 operators contributing to nn → ¯ ν ¯ ν (cid:48) P ( nn ) S ,a = 15 ( d i T L Cd jL ) (cid:2) d k T L Cu lL )( d m T L Cu nL ) + ( d k T L Cd lL )( u m T L Cu nL ) (cid:3) j νν (cid:48) S T SSS { ij }{ kl }{ mn } ,P ( nn ) S ,a = 13 (cid:104) d i T L Cu jL )( d k T L Cu lL ) + ( d i T L Cd jL )( u k T L Cu lL ) (cid:105) ( d m T R Cd nR ) j νν (cid:48) S T SSS { ij }{ kl }{ mn } ,P ( nn ) T ,a = − (cid:104) ( d i T L Cσ µν d jL )( d k T L Cu lL ) + ( d i T L Cσ µν u jL )( d k T L Cd lL ) (cid:105) ( d m T R Cu nR ) j νν (cid:48) ,µνT T SAA { kl } [ mn ][ ij ] . They are converted into the linear combinations of the LEFT operators shown in Tabs. V-VII viathe Schouten identities in Eq. (A.4) and the Fierz identities in Eq. (C.6) and their similar cousins.43 hiral basis LEFT basis Chiral irrep. Chiral spurion P ( pp ) S, ± ,a (cid:16) Q ( pp ) S, ± LLL,a − Q ( pp ) S, ± LLL,b (cid:17) ( L , R ) θ u L v L w L x L y L z L (111122) P ( pp ) S, ± ,b Q ( pp ) S, ± LLL,b ( L , R ) | a θ u L v L (11) P ( pp ) S, ± ,a Q ( pp ) S, ± LLR,a ( L , R ) θ u L v L w L x L y R z R (1112)(12) P ( pp ) S, ± ,b Q ( pp ) S, ± LLR,b ( L , R ) | b θ u L v L (11) P ( pp ) S, ± ,a Q ( pp ) S, ± LLR,a − Q ( pp ) S, ± LLR,b ( L , R ) θ u L v L w L x L y R z R (1122)(11) P ( pp ) S, ± ,b Q ( pp ) S, ± LLR,b ( L , R ) | c θ u R v R (11) P ( pp ) S, ± Q ( pp ) S, ± LLR ( L , R ) θ u L v L w L x L y R z R (1111)(22) P ( pp ) V ,a (cid:16) Q ( pp ) V LL,a − Q ( pp ) V LL,b − Q ( pp ) V LL,c (cid:17) ( L , R ) θ u L v L w L x L y L z R (11122)1 P ( pp ) V ,b (cid:16) Q ( pp ) V LL,b − Q ( pp ) V LL,c (cid:17) ( L , R ) | a θ u L v L w L x R (112)1 P ( pp ) V ,c Q ( pp ) V LL,c ( L , R ) | a θ u L v R P ( pp ) V ,a (cid:16) Q ( pp ) V LL,a − Q ( pp ) V LL,b (cid:17) ( L , R ) θ u L v L w L x L y L z R (11112)2 P ( pp ) V ,b Q ( pp ) V LL,b ( L , R ) | a θ u L v L w L x R (111)2 P ( pp ) V ,a Q ( pp ) V LR,a − Q ( pp ) V LR,b ( L , R ) θ u L v L w L x R y R z R (111)(122) ˜ P ( pp ) V ,a − (cid:16) Q ( pp ) V RL,a − Q ( pp ) V RL,b (cid:17) ( L , R ) θ u L v L w L x R y R z R (122)(111) P ( pp ) V ,b Q ( pp ) V LR,b ( L , R ) | b θ u L v L w L x R (111)2 ˜ P ( pp ) V ,b Q ( pp ) V RL,b ( L , R ) | b θ u L v R w R x R P ( pp ) V ,a (cid:16) Q ( pp ) V LR,a + 3 Q ( pp ) V LR,b − Q ( pp ) V LR,c − Q ( pp ) V LR,d + 2 Q ( pp ) V LR,e (cid:17) ( L , R ) θ u L v L w L x R y R z R (112)(112) P ( pp ) V ,b (cid:16) Q ( pp ) V LR,b − Q ( pp ) V LR,d + 2 Q ( pp ) V LR,e (cid:17) ( L , R ) | b θ u L v L w L x R (112)1 P ( pp ) V ,c (cid:16) Q ( pp ) V LR,c + Q ( pp ) V LR,d − Q ( pp ) V LR,e (cid:17) ( L , R ) | b θ u L v R w R x R P ( pp ) V ,d Q ( pp ) V LR,d ( L , R ) | b θ u L v R P ( pp ) V ,e Q ( pp ) V LR,e ( L , R ) | c θ u L v R P ( pp ) T, − Q ( pp ) T, − LLL ( L , R ) | a θ u L v L w L x L (1112) P ( pp ) T, − ,a Q ( pp ) T, − LLR,a + Q ( pp ) T, − LLR,c ( L , R ) | b θ u L v L w L x L (1112) P ( pp ) T, − ,b Q ( pp ) T, − LLR,b ( L , R ) | a θ u L v L w R x R (11)(12) P ( pp ) T, − ,c Q ( pp ) T, − LLR,c ( L , R ) | d θ u L v L (11) P ( pp ) T, − Q ( pp ) T, − LLR ( L , R ) | a θ u L v L w R x R (12)(11) ˆ P ( pp ) T, − ,a Q ( pp ) T, − RRL,a ( L , R ) | c θ u R v R w R x R (1112) ˆ P ( pp ) T, − ,b Q ( pp ) T, − RRL,b ( L , R ) | b θ u L v L w R x R (12)(11) ˆ P ( pp ) T, − Q ( pp ) T, − RRL ( L , R ) | b θ u L v L w R x R (11)(12) TABLE V. The chiral basis and their chiral irreps under G χ for the operators contributing to pp → (cid:96) + (cid:96) (cid:48) + . hiral basis LEFT basis Chiral irrep. Chiral spurion P ( pn ) S ,a (cid:16) Q ( pn ) S LLL,a − Q ( pn ) S LLL,b (cid:17) ( L , R ) θ u L v L w L x L y L z L (111222) P ( pn ) S ,b Q ( pn ) S LLL,b ( L , R ) | a θ u L v L (12) P ( pn ) S Q ( pn ) S LLR ( L , R ) θ u L v L w L x L y R z R (1112)(22) P ( pn ) S ,a Q ( pn ) S LLR,a − Q ( pn ) S LLR,c ( L , R ) θ u L v L w L x L y R z R (1122)(12) P ( pn ) S ,b Q ( pn ) S LLR,b ( L , R ) | b θ u L v L (12) P ( pn ) S ,c Q ( pn ) S LLR,c ( L , R ) | c θ u R v R (12) P ( pn ) S Q ( pn ) S LLR ( L , R ) θ u L v L w L x L y R z R (1222)(11) P ( pn ) V ,a (cid:16) Q ( pn ) V LL,a − Q ( pn ) V LL,b − Q ( pn ) V LL,c (cid:17) ( L , R ) θ u L v L w L x L y L z R (11122)2 P ( pn ) V ,b (cid:16) Q ( pn ) V LL,b − Q ( pn ) V LL,c (cid:17) ( L , R ) | a θ u L v L w L x R (112)2 P ( pn ) V ,c Q ( pn ) V LL,c ( L , R ) | a θ u L v R P ( pn ) V ,a (cid:16) Q ( pn ) V LL,a + 6 Q ( pn ) V LL,b − Q ( pn ) V LL,c (cid:17) ( L , R ) θ u L v L w L x L y L z R (11222)1 P ( pn ) V ,b (cid:16) Q ( pn ) V LL,b + Q ( pn ) V LL,c (cid:17) ( L , R ) | a θ u L v L w L x R (122)1 P ( pn ) V ,c Q ( pn ) V LL,c ( L , R ) | a θ u L v R P ( pn ) V Q ( pn ) V LR ( L , R ) θ u L v L w L x R y R z R (111)(222) ˜ P ( pn ) V −Q ( pn ) V RL ( L , R ) θ u L v L w L x R y R z R (222)(111) P ( pn ) V ,a (cid:16) Q ( pn ) V LR,a − Q ( pn ) V LR,b − Q ( pn ) V LR,c + Q ( pn ) V LR,d − Q ( pn ) V LR,e (cid:17) ( L , R ) θ u L v L w L x R y R z R (112)(122) ˜ P ( pn ) V ,a − (cid:16) Q ( pn ) V RL,a − Q ( pn ) V RL,b − Q ( pn ) V RL,c + Q ( pn ) V RL,d − Q ( pn ) V RL,e (cid:17) ( L , R ) θ u L v L w L x R y R z R (122)(112) P ( pn ) V ,b (cid:16) Q ( pn ) V LR,b − Q ( pn ) V LR,d + 2 Q ( pn ) V LR,e (cid:17) ( L , R ) | b θ u L v L w L x R (112)2 ˜ P ( pn ) V ,b (cid:16) Q ( pn ) V RL,b − Q ( pn ) V RL,d + 2 Q ( pn ) V RL,e (cid:17) ( L , R ) | b θ u L v R w R x R P ( pn ) V ,c (cid:16) Q ( pn ) V LR,c − Q ( pn ) V LR,d + 2 Q ( pn ) V LR,e (cid:17) ( L , R ) | b θ u L v R w R x R ˜ P ( pn ) V ,c (cid:16) Q ( pn ) V RL,c − Q ( pn ) V RL,d + 2 Q ( pn ) V RL,e (cid:17) ( L , R ) | b θ u L v L w L x R (122)1 P ( pn ) V ,d Q ( pn ) V LR,d ( L , R ) | b θ u L v R ˜ P ( pn ) V ,d −Q ( pn ) V RL,d ( L , R ) | b θ u L v R P ( pn ) V ,e Q ( pn ) V LR,e ( L , R ) | c θ u L v R ˜ P ( pn ) V ,e −Q ( pn ) V RL,e ( L , R ) | c θ u L v R P ( pn ) T ,a (cid:16) Q ( pn ) T LLL,a − Q ( pn ) T LLL,b (cid:17) ( L , R ) | a θ u L v L w L x L (1122) P ( pn ) T ,b Q ( pn ) T LLL,b ( L , R ) | a P ( pn ) T Q ( pn ) T LLR ( L , R ) | a θ u L v L w R x R (11)(22) P ( pn ) T ,a (cid:16) Q ( pn ) T LLR,a − Q ( pn ) T LLR,b (cid:17) ( L , R ) | b θ u L v L w L x L (1122) P ( pn ) T ,b Q ( pn ) T LLR,b ( L , R ) | b P ( pn ) T ,c Q ( pn ) T LLR,c ( L , R ) | a θ u L v L w R x R (12)(12) P ( pn ) T ,d −Q ( pn ) T LLR,d ( L , R ) | d θ u L v L (12) P ( pn ) T Q ( pn ) T LLR ( L , R ) | a θ u L v L w R x R (22)(11) ˆ P ( pn ) T Q ( pn ) T RRL ( L , R ) | b θ u L v L w R x R (22)(11) ˆ P ( pn ) T ,a Q ( pn ) T RRL,a − Q ( pn ) T RRL,c ( L , R ) | c θ u R v R w R x R (1122) ˆ P ( pn ) T ,b Q ( pn ) T RRL,b ( L , R ) | b θ u L v L w R x R (12)(12) ˆ P ( pn ) T ,c Q ( pn ) T RRL,c ( L , R ) | c ˆ P ( pn ) T Q ( pn ) T RRL ( L , R ) | b θ u L v L w R x R (11)(22) TABLE VI. The chiral basis and their chiral irreps under G χ for the operators contributing to pn → (cid:96) + ¯ ν (cid:48) . hiral basis LEFT basis Chiral irrep. Chiral spurion P ( nn ) S ,a (cid:16) Q ( nn ) S LLL,a − Q ( nn ) S LLL,b (cid:17) ( L , R ) θ u L v L w L x L y L z L (112222) P ( nn ) S ,b Q ( nn ) S LLL,b ( L , R ) | a θ u L v L (22) P ( nn ) S ,a Q ( nn ) S LLR,a ( L , R ) θ u L v L w L x L y R z R (1222)(12) P ( nn ) S ,b Q ( nn ) S LLR,b ( L , R ) | b θ u L v L (22) P ( nn ) S ,a Q ( nn ) S LLR,a − Q ( nn ) S LLR,b ( L , R ) θ u L v L w L x L y R z R (1122)(22) P ( nn ) S ,b Q ( nn ) S LLR,b ( L , R ) | c θ u R v R (22) P ( nn ) S Q ( nn ) S LLR ( L , R ) θ u L v L w L x L y R z R (2222)(11) P ( nn ) T −Q ( nn ) T LLL ( L , R ) | a θ u L v L w L x L (1222) P ( nn ) T ,a − (cid:16) Q ( nn ) T LLR,a + Q ( nn ) T LLR,c (cid:17) ( L , R ) | b θ u L v L w L x L (1222) P ( nn ) T ,b −Q ( nn ) T LLR,b ( L , R ) | a θ u L v L w R x R (22)(12) P ( nn ) T ,c Q ( nn ) T LLR,c ( L , R ) | d θ u L v L (22) P ( nn ) T −Q ( nn ) T LLR ( L , R ) | a θ u L v L w R x R (12)(22) ˆ P ( nn ) T ,a −Q ( nn ) T RRL,a ( L , R ) | c θ u R v R w R x R (1222) ˆ P ( nn ) T ,b −Q ( nn ) T RRL,b ( L , R ) | b θ u L v L w R x R (12)(22) ˆ P ( nn ) T −Q ( nn ) T RRL ( L , R ) | b θ u L v L w R x R (22)(12) TABLE VII. The chiral basis and their chiral irreps under G χ for the operators contributing to nn → ¯ ν ¯ ν (cid:48) . . CHIRAL IRREDUCIBLE REPRESENTATIONS IN TERMS OF HADRONS Ope. type Chi. irrep Chi. order Matching operator S ca l a r c u rr e n t: O S q u a r k × j S ( L , R ) | i p O S × ,i = θ u L v L ( αβ ) ( u † ) u L a ( u † ) v L b [Ψ T a C ( g × ,i + ˆ g × ,i γ )Ψ b ]( L , R ) p O S × = θ u L v L w L x L y R z R ( αβγρ )( στ ) ( Uiτ ) y R w L ( Uiτ ) z R x L ( u † ) u L a ( u † ) v L b [Ψ T a C ( g × + ˆ g × γ )Ψ b ]( L , R ) p ( × ) O S × = θ u L v L w L x L y L z L ( αβγρστ ) ( u † u µ uiτ ) w L x L ( u † u µ uiτ ) y L z L ( u † ) u L a ( u † ) v L b [Ψ T a C ( g × + ˆ g × γ )Ψ b ]( L , R ) | i p ˜ O S × ,i = θ u R v R ( αβ ) u u R a u v R b [Ψ T a C ( g × ,i + ˆ g × ,i γ )Ψ b ]( L , R ) p ˜ O S × = θ u R v R w R x R y L z L ( αβγρ )( στ ) ( Uiτ ) w R y L ( Uiτ ) x R z L u u R a u v R b [Ψ T a C ( g × + ˆ g × γ )Ψ b ]( L , R ) p ( × ) ˜ O S × = θ u R v R w R x R y R z R ( αβγρστ ) ( uu µ u † iτ ) w R x R ( uu µ u † iτ ) y R z R u u R a u v R b [Ψ T a C ( g × + ˆ g × γ )Ψ b ] V ec t o r c u rr e n t: O V , µ q u a r k × j V , µ ( L , R ) | i p O V,µ × ,i = θ u L v R αβ ( u † ) u L a u v R b [Ψ T a Cγ µ ( g × ,i + ˆ g × ,i γ )Ψ b ]( L , R ) | i p O V,µ × ,i = g × ,i θ u L v L w L x R ( αβγ ) ρ ( Uiτ ) x R w L ( u † ) u L a ( u † ) v L b [Ψ T a Cγ µ γ Ψ b ]( L , R ) p O V,µ × = θ u L v L w L x R y R z R ( αβγ )( ρστ ) ( Uiτ ) y R v L ( Uiτ ) z R w L ( u † ) u L a u x R b [Ψ T a Cγ µ ( g × + ˆ g × γ )Ψ b ]( L , R ) p ( × ) O V,µ × = θ u L v L w L x L y L z R ( αβγρσ ) τ ( Uiτ ) z R w L ( u † u µ uiτ ) x L y L ( u † ) u L a ( u † ) v L b [Ψ T a C ( g × + ˆ g × γ )Ψ b ]( L , R ) | i p ˜ O V,µ × ,i = − g × ,i θ u R v R w R x L ( αβγ ) ρ ( Uiτ ) w R x L u u R a u v R b (Ψ T a Cγ µ γ Ψ b )( L , R ) p ( × ) ˜ O V,µ × = − θ u R v R w R x R y R z L ( αβγρσ ) τ ( Uiτ ) w R z L ( uu µ u † iτ ) x R y R u u R a u v R b [Ψ T a C ( g × + ˆ g × γ )Ψ b ] T e n s o r c u rr e n t: O T , µ ν q u a r k × j µ ν T ( L , R ) | i p O T,µν × ,i = (cid:15) ab [Ψ T a Cσ µν ( g × ,i + ˆ g × ,i γ )Ψ b ]( L , R ) p ( × ) O T,µν × = θ u L v L ( αβ ) ( u † u µ ) u L a ( u † ) v L b [Ψ T a Cγ ν ( g × ,T + ˆ g × ,T γ )Ψ b ] − µ ↔ ν ( L , R ) | i p O T,µν × ,i = θ u L v L w R x R ( αβ )( γρ ) ( Uiτ ) x R v L ( u † ) u L a u w R b [Ψ T a Cσ µν ( g × ,i + ˆ g × ,i γ ))Ψ b ]( L , R ) | i p ( × ) O T,µν × ,i = g × ,i θ u L v L w L x L ( αβγρ ) ( u † u µ uiτ ) w L x L ( u † ) u L a ( u † ) v L b (Ψ T a Cγ ν γ Ψ b ) − µ ↔ ν ( L , R ) p ( × ) ˜ O T,µν × = θ u R v R ( αβ ) ( uu µ ) u R a u v R b [Ψ T a Cγ ν ( g × ,T + ˆ g × ,T γ )Ψ b ] − µ ↔ ν ( L , R ) | i p ( × ) ˜ O T,µν × ,i = g × ,i θ u R v R w R x R ( αβγρ ) ( uu µ u † iτ ) w R x R u u R a u v R b (Ψ T a Cγ ν γ Ψ b ) − µ ↔ ν TABLE VIII. The leading order chiral matching of the chiral irrep six-quark part in our assumed frameworkwithout including terms containing nucleon derivatives. Where in the third column we show the leadingchiral order of each matched operator, and the cross ( × ) behind p , indicate the matched operators cannotcontribute to the leading order dim-6 interactions in Eq. (5) . . LEFT AND SMEFT CONTRIBUTIONS TO C pp , C pn AND C nn Taking the specific expressions of the spurion fields in Tabs. (V-VII) into the leading ordermatching results in Eq. 31, and combining with the relevant LEFT Wilson coefficients and leptoncurrents, we obtain the final matching results for the operators in Eqs. (6-8) as the functions of theLECs and the LEFT Wilson coefficients as follows C ( pp ) SR ( L ) = (cid:20) g × ,a (cid:18) C ( pp ) S, ± LLL,a + C ( pp ) S, ± LLL,b (cid:19) + g × ,b C ( pp ) S, ± LLR,b + g × ,c (cid:16) C ( pp ) S, ± RRL,a + C ( pp ) S, ± RRL,b (cid:17) + g × (cid:18) − C ( pp ) S, ± LLR,a + 16 C ( pp ) S, ± LLR,a + C ( pp ) S, ± LLR (cid:19)(cid:21) + (cid:18) L ↔ Rg i × j ↔ g j × i (cid:19) , (H.1) C ( pp ) S R ( L ) = C ( pp ) SR ( L ) | g → ˆ g , (H.2) C ( pp ) V = (cid:20) ˆ g × ,a (cid:18) C ( pp ) V LL,a + 13 C ( pp ) V LL,b + C ( pp ) V LL,c (cid:19) + (cid:18) L ↔ R ˆ g × ,a ↔ ˆ g × ,d (cid:19)(cid:21) + ˆ g × ,b (cid:18) C ( pp ) V LR,d − (cid:16) C ( pp ) V LR,a − C ( pp ) V LR,b + C ( pp ) V LR,c (cid:17)(cid:19) + ˆ g × ,c (cid:18) C ( pp ) V LR,e + 23 (cid:16) C ( pp ) V LR,a − C ( pp ) V LR,b + C ( pp ) V LR,c (cid:17)(cid:19) + ˆ g × (cid:18) (cid:16) C ( pp ) V LR,a + C ( pp ) V RL,a (cid:17) − C ( pp ) V LR,a (cid:19) + (cid:20) g × ,a (cid:18) (cid:18) C ( pp ) V LL,a + C ( pp ) V LL,b (cid:19) − (cid:18) C ( pp ) V LL,a + C ( pp ) V LL,b (cid:19)(cid:19) + (cid:18) L ↔ Rg i × j ↔ g j × i (cid:19)(cid:21) − g × ,b (cid:18)(cid:16) C ( pp ) V LR,a + C ( pp ) V LR,b (cid:17) + 13 (cid:16) C ( pp ) V LR,a − C ( pp ) V LR,b (cid:17)(cid:19) − g × ,b (cid:18)(cid:16) C ( pp ) V RL,a + C ( pp ) V RL,b (cid:17) − (cid:16) C ( pp ) V LR,a + C ( pp ) V LR,c (cid:17)(cid:19) , (H.3) C ( pn ) SL = (cid:20) g × ,a (cid:18) C ( pn ) S LLL,a + C ( pn ) S LLL,b (cid:19) + g × ,b C ( pn ) S LLR,b + g × ,c (cid:16) C ( pn ) S RRL,a + C ( pn ) S RRL,c (cid:17) + g × (cid:18) C ( pn ) S LLR − C ( pn ) S LLR,a + 12 C ( pn ) S LLR (cid:19)(cid:21) + (cid:18) L ↔ Rg i × j ↔ g j × i (cid:19) , (H.4) C ( pn ) S L = C ( pn ) SL | g → ˆ g , (H.5) C ( pn ) VL = (cid:20) g × ,a (cid:18) C ( pn ) V LL,a + 13 C ( pn ) V LL,b + C ( pn ) V LL,c − C ( pn ) V LL,a + 13 C ( pn ) V LL,b − C ( pn ) V LL,c (cid:19) + (cid:18) L ↔ Rg × ,a ↔ g × ,d (cid:19)(cid:21) + g × ,b (cid:20)(cid:18) C ( pn ) V LR,d + 13 (cid:16) C ( pn ) V LR,a + C ( pn ) V LR,b + C ( pn ) V LR,c (cid:17)(cid:19) + ( L ↔ R ) (cid:21) + g × ,c (cid:20)(cid:18) C ( pn ) V LR,e − (cid:16) C ( pn ) V LR,a + C ( pn ) V LR,b + C ( pn ) V LR,c (cid:17)(cid:19) + ( L ↔ R ) (cid:21) + g × (cid:18) C ( pn ) V LR + C ( pn ) V RL − (cid:16) C ( pn ) V LR,a + C ( pn ) V RL,a (cid:17)(cid:19) , (H.6) C ( pn ) V L = (cid:20) ˆ g × ,a (cid:18) C ( pn ) V LL,a + 13 C ( pn ) V LL,b + C ( pn ) V LL,c + C ( pn ) V LL,a − C ( pn ) V LL,b + C ( pn ) V LL,c (cid:19) + (cid:18) L ↔ R ˆ g × ,a ↔ ˆ g × ,d (cid:19)(cid:21) + ˆ g × ,b (cid:20)(cid:18) C ( pn ) V LR,d + 13 (cid:16) C ( pn ) V LR,a + C ( pn ) V LR,b + C ( pn ) V LR,c (cid:17)(cid:19) − ( L ↔ R ) (cid:21) ˆ g × ,c (cid:20)(cid:18) C ( pn ) V LR,e − (cid:16) C ( pn ) V LR,a + C ( pn ) V LR,b + C ( pn ) V LR,c (cid:17)(cid:19) − ( L ↔ R ) (cid:21) + ˆ g × (cid:18) C ( pn ) V LR − C ( pn ) V RL − (cid:16) C ( pn ) V LR,a − C ( pn ) V RL,a (cid:17)(cid:19) − (cid:20) g × ,a (cid:18) C ( pn ) V LL,a + C ( pn ) V LL,b + 65 C ( pn ) V LL,a − C ( pn ) V LL,b (cid:19) + (cid:18) L ↔ Rg i × j ↔ g j × i (cid:19)(cid:21) − (cid:20) g × ,b (cid:16) C ( pn ) V LR,a + C ( pn ) V LR,b − C ( pn ) V RL,a − C ( pn ) V RL,c (cid:17) + (cid:18) L ↔ Rg i × j ↔ g j × i (cid:19)(cid:21) , (H.7) C ( pn ) T = g r × ,a (cid:18) C ( pn ) T LLL,a + C ( pn ) T LLL,b (cid:19) + g r × ,b (cid:18) C ( pn ) T LLR,a + C ( pn ) T LLR,b (cid:19) + g r × ,c (cid:16) C ( pn ) T RRL,a + C ( pn ) T RRL,c (cid:17) − g r × ,a (cid:18) C ( pn ) T LLR − C ( pn ) T LLR,c + C ( pn ) T LLR (cid:19) − g r × ,b (cid:18) C ( pn ) T RRL − C ( pn ) T RRL,b + C ( pn ) T RRL (cid:19) , (H.8) C ( nn ) SL = (cid:20) g × ,a (cid:18) C ( nn ) S LLL,a + C ( nn ) S LLL,b (cid:19) + g × ,b C ( nn ) S LLR,b + g × ,c (cid:16) C ( nn ) S RRL,a + C ( nn ) S RRL,b (cid:17) + g × (cid:18) − C ( nn ) S LLR,a + 16 C ( nn ) S LLR,a + C ( nn ) S LLR (cid:19)(cid:21) + (cid:18) L ↔ Rg i × j ↔ g j × i (cid:19) , (H.9) C ( nn ) S L = C ( nn ) SL | g → ˆ g . (H.10) After neglecting the QCD running effect and taking the matching result of the LEFT and SMEFTinteractions in Tab. II into consideration, we find the above results simplify considerably and take C ( pp ) SL = 4 g × ,a C S, ( S ) Q L + 2 g × ,b C S, ( S ) udQ L + g × ,c (cid:16) C S, ( S ) u d Q L + C S, ( S ) u d Q L (cid:17) + g × C S, ( S ) u d Q L , (H.11) C ( pp ) SR = g × ,a (cid:18) C S, ( S ) u d e + C S, ( S ) u d e (cid:19) + 2 g × ,b C S, ( S ) u dQ e + 4 g × ,c C S, ( S ) u Q e ,C ( pp ) S L ( R ) = C ( pp ) SL ( R ) | g → ˆ g , (H.12) C ( pp ) V = 4ˆ g × ,a C VuQ eL − ˆ g × ,d (cid:18) C Vu d QeL − C Vu d QeL + C Vu d QeL (cid:19) − g × ,b (cid:18) C Vu dQ eL − C Vu dQ eL (cid:19) + 2ˆ g × ,c (cid:18) C Vu dQ eL − C Vu dQ eL (cid:19) − g × ,a (cid:18) C Vu d QeL − C Vu d QeL (cid:19) + 23 g × ,b C Vu dQ eL , (H.13) C ( pn ) SL = − (cid:16) g × ,a C S, ( S ) Q L + 2 g × ,b C S, ( S ) udQ L + g × ,c (cid:16) C S, ( S ) u d Q L + C S, ( S ) u d Q L (cid:17)(cid:17) − (cid:18) g × ,a (cid:18) C S, ( A ) u d L + C S, ( A ) u d L (cid:19) + 2 g × ,b C S, ( A ) u d Q L + 4 g × ,c C S, ( A ) udQ L (cid:19) + 23 g × C S, ( S ) u d Q L , (H.14) C ( pn ) S L = C ( pn ) SL | g → ˆ g , (H.15) C ( pn ) VL = 4 g × ,a C VuQ eL + g × ,d (cid:18) C Vu d QeL − C Vu d QeL + C Vu d QeL (cid:19) − g × ,b (cid:18) C Vu dQ eL − C Vu dQ eL (cid:19) + 2 g × ,c (cid:18) C Vu dQ eL − C Vu dQ eL (cid:19) , (H.16) C ( pn ) V L = − g × ,a C VuQ eL + ˆ g × ,d (cid:18) C Vu d QeL − C Vu d QeL + C Vu d QeL (cid:19) g × ,b (cid:18) C Vu dQ eL − C Vu dQ eL (cid:19) − g × ,c (cid:18) C Vu dQ eL − C Vu dQ eL (cid:19) , − g × ,a (cid:18) C Vu d QeL − C Vu d QeL (cid:19) + 43 g × ,b C Vu dQ eL , (H.17) C ( pn ) T = − g r × ,a C T, ( S ) Q L − g r × ,b C T, ( S ) udQ L − g r × ,c (cid:16) C T, ( S ) u d Q L + C T, ( S ) u d Q L (cid:17) − g r × ,a C T, ( A ) udQ L − g r × ,b C T, ( A ) u d Q L , (H.18) C ( nn ) SL = 4 g × ,a C S, ( S ) Q L + 2 g × ,b C S, ( S ) udQ L + g × ,c (cid:16) C S, ( S ) u d Q L + C S, ( S ) u d Q L (cid:17) + g × C S, ( S ) u d Q L , (H.19) C ( nn ) S L = C ( nn ) SL | g → ˆ g . 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