Large- N c analysis of two-nucleon neutrinoless double beta decay and charge-independence-breaking contact terms
Thomas R. Richardson, Matthias R. Schindler, Saori Pastore, Roxanne P. Springer
LLarge- N c analysis of two-nucleon neutrinoless double beta decay andcharge-independence-breaking contact terms Thomas R. Richardson ∗ and Matthias R. Schindler † Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208
Saori Pastore ‡ Department of Physics and McDonnell Center for the Space Sciences at Washington University in St. Louis, MO, 63130, USA
Roxanne P. Springer § Department of Physics, Duke University, Durham, NC 27708, USA (Dated: February 4, 2021)The interpretation of experiments that search for neutrinoless double beta decay relies on inputfrom nuclear theory. Cirigliano et al. recently showed that, for the light Majorana exchange formal-ism, effective field theory calculations require a nn → ppe − e − contact term at leading order. Theyestimated the size of this contribution by relating it to measured charge-independence-breaking(CIB) nucleon-nucleon interactions and making an assumption about the relative sizes of CIB oper-ators. We show that the assumptions underlying this approximation are justified in the limit of thenumber of colors N c being large. We also obtain a large- N c hierarchy among CIB nucleon-nucleoninteractions that is in agreement with phenomenological results. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ nu c l - t h ] F e b I. INTRODUCTION
Significant experimental efforts are underway to detect neutrinoless double beta (0 νββ ) decay [1–14], a process inwhich two neutrons are converted into two protons with the emission of two electrons but without the accompanyingemission of neutrinos. Neutrinoless double beta decay is a highly sensitive probe of lepton number violation (LNV)and, if detected, would be a clear demonstration that neutrinos are Majorana particles [15]. If this process is observed,it would also shed light on the neutrino mass hierarchy [16, 17] and on the matter-antimatter asymmetry in the universe[18].The inverse of the 0 νββ half-life can be expressed as (see Refs. [19–21] for reviews) (cid:104) T ν / (cid:105) − = G ν | M ν | m ββ , (1)where m ββ is the effective Majorana neutrino mass, G ν is a phase space factor, and M ν is the corresponding nuclearmatrix element (NME). This sensitivity to the NME requires a deep understanding of the nuclear physics involved.One important component in the calculation of the NMEs are multi-nucleon operators that encode the underlyingLNV mechanisms. While there are many models and methods that offer insight in this direction, effective fieldtheory (EFT) offers a systematic, model-independent way to study LNV and the corresponding one- and two-nucleonoperators that are required as input to many-body calculations. Each independent term in an effective Lagrangiancomes with a low-energy coefficient (LEC), into which all unresolved short-distance details are subsumed. TheseLECs need to be determined from a fit to data or a nonperturbative quantum chromodynamics (QCD) calculationlike those performed in lattice QCD (see, e.g., Refs. [22–26] for work related to double beta decays and Ref. [27] fora general review).An initial step towards the application of EFT to 0 νββ was taken in Ref. [28] in the context of chiral effectivefield theory. ChEFT refers to the generalization of chiral perturbation theory ( χ PT) [29–34] (see e.g., Refs. [35–40]for reviews)—the EFT of pions and single nucleons based on the approximate chiral symmetry of QCD—to two andmore nucleons. Recently, the EFT approach has received renewed attention focusing on the light-Majorana neutrinoexchange mechanism [41–44], where it has been shown that a contact term with the undetermined LEC g NNν is requiredat leading order (LO) [45, 46]. This term was absent in previous analyses. Additionally, Refs. [42, 45, 46] observedthat isospin symmetry dictates that g NNν is related to the LEC C of an operator parameterizing charge independencebreaking (CIB) in the two-nucleon system. The value of the LEC C is currently not determined by data. Only thelinear combination C + C , where C is the LEC of a second, independent CIB operator, has been extracted fromexperiment. To estimate the numerical impact of the contact term in nuclear matrix elements, Refs. [45, 46] assumedthat C ≈ C so that the value of g NNν can be approximated by g NNν ≈ ( C + C ). Exploring this assumption fromthe large- N c perspective is the main focus of this paper.While some lattice QCD calculations of double- β decay matrix elements in the two-nucleon system [22, 23] and 0 νββ calculations in the meson sector [23, 24] exist, a calculation of g NNν is currently not available. In the absence of latticeQCD calculations and sufficient data to determine g NNν , or equivalently the CIB LECs, the possibility of additionaltheoretical constraints is critical. Recently, Ref. [47] estimated the values of g NNν and C + C using a method analogousto the Cottingham formula [48, 49]. Their results support the assumption of Refs. [45, 46]. Here, a complementaryapproach based on the large- N c limit of QCD is explored. Constraints are obtained through the spin-flavor symmetrythat arises in the large- N c limit of QCD [50–53]. This method has been used to constrain nucleon-nucleon ( NN )interactions [54–58], including parity-violating couplings [59, 60], time-reversal-invariance-violating couplings [61, 62],as well as magnetic and axial couplings in the context of pionless EFT (EFT π/ ) [63]. Similar work has been done inboth the meson [64, 65] and single baryon sectors of chiral perturbation theory ( χ PT) [66–70].This paper is structured as follows. Section II contains a discussion of the results from Refs. [45, 46] relevant for thiswork. A large- N c analysis of one- and two-nucleon matrix elements is given in Sec. III, and the spurion construction inChEFT is discussed in Sec. IV. Complete but minimal sets of spurion operators for both the electromagnetic and weakinteractions are derived along with the large- N c scalings of the corresponding LECs in Sec. V. A large- N c hierarchyof CIB interactions in comparison to phenomenological descriptions is discussed in Sec. VI. The large- N c analyzedCIB Lagrangian is mapped onto to the CIB Lagrangian of Ref. [46] in Sec. VII and the consistency of the LNV andCIB LECs demonstrated. Finally, Sec. VIII summarizes the results. The appendices contain a detailed discussion ofan alternate large- N c scaling of the quark and nucleon charges, as well as a summary of relevant Fierz identities. II. BACKGROUND
In this section we introduce and discuss the relevant LNV and CIB Lagrangians. At leading order (LO) in the EFTpower counting, there is a contribution to the two-nucleon LNV transition operator from tree-level neutrino exchangebetween the nucleons. At the same order, there exist contributions from dressing the tree-level diagram by iterations ofthe LO NN interactions, which include contact terms and one-pion exchange diagrams [46]. A careful analysis [45, 46]of the resulting amplitude using renormalization arguments shows that an LNV amplitude that consists of only theabove contributions diverges logarithmically. Therefore, a leading-order contact operator must be included to obtainthe correct amplitude at this order. The contact term in the LO Lagrangian is [45, 46] L NN | ∆ L =2 | = (cid:16) √ G F V ud (cid:17) m ββ ¯ e L C ¯ e TL g NNν (cid:20)(cid:16) ¯ N u ˜ Q wL u † N (cid:17) −
16 Tr (cid:16) ˜ Q wL ˜ Q wL (cid:17) (cid:0) ¯ N τ a N (cid:1) (cid:21) + H.c. , (2)where N represents the doublet of nucleon fields, e L is the left-handed electron, the charge conjugation matrix is C = iγ γ , G F is the Fermi constant, V ud is an element of the Cabibbo-Kobayashi-Maskawa matrix, and˜ Q wL = τ + = 12 (cid:0) τ + iτ (cid:1) . (3)The matrix u is u = exp (cid:18) i F φ a τ a (cid:19) , (4)where the φ a ( a = 1 , ,
3) are the pion fields in Cartesian coordinates, the τ a are Pauli matrices in isospin space, and F is the pion decay constant in the chiral limit. The renormalization group (RG) requirement to include a contactterm at LO means that an additional unknown LEC, g NNν , must be determined in order to analyze and interpretcurrent and future measurements of 0 νββ decay.It has been shown in Refs. [45, 46] that chiral symmetry relates the LEC g NNν to an electromagnetic CIB isotensorLEC C . The CIB isotensor Lagrangian in ChEFT has received a significant amount of study [71–73]. In Ref. [46] itis written as L NNCIB = e (cid:26) C (cid:20)(cid:16) ¯ N u † ˜ Q R uN (cid:17) + (cid:16) ¯ N u ˜ Q L u † N (cid:17) −
16 Tr (cid:16) ˜ Q L + ˜ Q R (cid:17) (cid:0) ¯ N τ a N (cid:1) (cid:21) + C (cid:20) (cid:16) ¯ N u † ˜ Q R uN (cid:17) (cid:16) ¯ N u ˜ Q L u † N (cid:17) −
13 Tr (cid:16) U ˜ Q L U † ˜ Q R (cid:17) (cid:0) ¯ N τ a N (cid:1) (cid:21)(cid:27) , (5)where U = u and here ˜ Q L = ˜ Q R = 12 τ . (6)Additionally, many high-precision NN potentials, such as the Argonne v [74] and the CD-Bonn [75], as wellas several interactions derived from ChEFT [76–79] include short-range CIB and charge-symmetry-breaking (CSB)operators to reproduce the observed scattering data. In principle, determination of C from data also fixes the valueof the 0 νββ LEC g NNν . However, at present only the linear combination C + C is constrained by available data. Thecombination C − C is sensitive to two-nucleon-multi-pion interactions and is currently inaccessible. Reference [46]obtains an estimate of g NNν by assuming that the two LECs C and C are of the same size and sign, which implies g NNν ≈ ( C + C ). In the next sections we examine this assumption using large- N c scaling arguments. In particular,we show that the terms proportional to C − C are suppressed in the large- N c limit compared to those proportionalto C + C , thereby adding support to the assumptions that C and C are of the same size and sign and that g NNν canbe approximated by the sum of the CIB LECs divided by two.
III. LARGE- N c SCALING
In this section we outline the basic elements needed to perform large- N c analyses. The large- N c scaling of thesingle-nucleon matrix elements of an n -body operator O ( n ) IS with spin S and isospin I is provided by [54, 55] (cid:104) N | O ( n ) IS N nc | N (cid:105) (cid:46) N −| I − S | c , (7)where n denotes the number of quarks involved in the operator. In the large- N c limit, the Hamiltonian takes a Hartreeform [55, 80], H = N c (cid:88) n (cid:88) s,t v stn (cid:18) S i N c (cid:19) s (cid:18) I a N c (cid:19) t (cid:18) G ia N c (cid:19) n − s − t , (8)where the one-body operators are S i = q † σ i q, I a = q † τ a q, G ia = q † σ i τ a q. (9)The nucleon ground state is totally antisymmetric in the color degrees of freedom, and q is a colorless, bosonic quarkfield. The coefficients v stn are functions of momentum and at most scale as O ( N c ) [55]. In addition to single-nucleon matrix elements (also see Ref. [81] and references therein), these results were used in the study of two-nucleoninteractions via matrix elements of the form [54, 55] V ( p − , p + ) = (cid:104) N α ( p (cid:48) ) N β ( p (cid:48) ) | H | N γ ( p ) N δ ( p ) (cid:105) , (10)where the Greek subscripts indicate combined spin and isospin quantum numbers and p ± ≡ p (cid:48) ± p , (11)where p (cid:48) = p (cid:48) − p (cid:48) and p = p − p . The two-nucleon matrix elements factorize in the large- N c limit [54], (cid:104) N γ N δ | O O | N α N β (cid:105) N c →∞ −−−−−→ (cid:104) N γ | O | N α (cid:105) (cid:104) N δ | O | N β (cid:105) + crossed , (12)and the large- N c scaling of the two-nucleon matrix elements is determined by the large- N c dependence of the operators O and O ∈ { S i , I a , G ia , } , (cid:104) N (cid:48) | S i | N (cid:105) ∼ (cid:104) N | I a | N (cid:105) (cid:46) , (cid:104) N (cid:48) | G ia | N (cid:105) ∼ (cid:104) N | | N (cid:105) (cid:46) N c . (13)In addition, there can be a hidden large- N c suppression in the momentum dependence of the functions v stn [55].In t-channel diagrams, factors of p + only enter through relativistic corrections and are therefore suppressed by thenucleon mass, which scales as N c . Since the analysis in the t-channel is sufficient to establish the large- N c scaling,momenta are counted as [55] p − ∼ , p + ∼ N − c . (14)Finally, large- N c scaling is impacted by the number of pions involved in the process. In χ PT, pion fields are encodedin the exponential matrix u of Eq. (4). Expanding u in the number of pions, we see that each pion field is accompaniedby a factor of 1 /F . In the large- N c limit the decay constant F is O ( √ N c ) [80, 82]; each additional pion field in theexpansion of Eq. (4) yields a suppression by 1 / √ N c .In summary, the large- N c scaling of the LECs is determined by the spin-isospin structure of the matrix elements ofnucleon bilinear operators, the scaling of any relevant momentum factors, and additional suppressions from any pionfields. Finally, the overall factor of N c in the Hamiltonian of Eq. (8) reduces the scaling of the LECs by one power of N c .This approach has been used to analyze the large- N c behavior of NN interactions in the symmetry-even [54–58, 83, 84] and symmetry-odd sectors [59–62], three-nucleon forces [85], and the coupling of two nucleons to externalmagnetic and axial fields [63]. In this paper the large- N c scaling is used to establish relationships among LECsassociated with CIB NN operators.We briefly comment on the role of the ∆ in large- N c ChEFT. The nucleon and ∆ mass splitting is O (1 /N c );therefore, the nucleon and the ∆ resonance become degenerate in the large- N c limit. Including the ∆ resonance in χ PT leads to quantities that may depend on the ratio m ∆ − m N m π , (15)which depends on the order in which the large- N c and chiral limits are taken [34, 52, 86, 87]. The ∆ was shown to playan integral role in obtaining consistent large- N c scaling for pion-baryon scattering [88–91] and in the meson-exchangepicture of the NN interactions [56]. For the quantities of interest here, the ∆ can only appear in intermediate statesand, following earlier work that did not lead to any contradiction with available data [54, 55, 58], effects of the virtual∆ degrees of freedom are not considered explicitly in the following. The role of intermediate ∆ states in NN scatteringin the S channel and how they can be integrated out is discussed in Ref. [92]. IV. CHIRAL EFFECTIVE FIELD THEORY AND SPURION FIELDS
The CIB ChEFT Lagrangian is constructed using the spurion technique, the same technique used to construct themass term in the LO pion Lagrangian and to include the effects of virtual photons and leptons [93–101]. The CIBLagrangian of interest here contains terms with two insertions of the quark (or equivalently nucleon) charge matrix.The QCD Lagrangian for two flavors in terms of left- and right-handed quark fields with minimal coupling to anelectromagnetic potential is L = i ¯ q L /∂q L + i ¯ q R /∂q R − ¯ q L M † q R − ¯ q R M q L + ieA µ [¯ q L γ µ Q q q L + ¯ q R γ µ Q q q R ] , (16)where Q q = diag( , − ) is the quark charge matrix and the unit of charge e is factored out of Q q (see the last termsin Eq. (16)). The quark mass and the charge matrix terms break chiral symmetry explicitly. For the mass terms, thepattern of symmetry breaking can be mapped onto the effective Lagrangian by ( i ) assuming that the constant matrix M transforms under the chiral symmetry group as M (cid:55)→ RM L † , (17)where R and L are SU(2) matrices transforming the right- and left-handed components of the quark fields, respectively,and ( ii ) constructing all allowed terms that are chirally invariant with the assumed transformation behavior of thequark mass matrix. The same approach can be adopted for terms containing the charge matrix. First, Q is separatedinto two matrices, Q L and Q R , such that the electromagnetic part of the Lagrangian can be written as L EM = ieA µ [¯ q L γ µ Q L q L + ¯ q R γ µ Q R q R ] . (18)Next, the charge matrices are required to transform under the chiral symmetry group as Q R (cid:55)→ RQ R R † , (19) Q L (cid:55)→ LQ L L † . (20)At the nucleonic level, the Lagrangian can be written in terms of the nucleon doublet N = ( p, n ) T , which transformsunder chiral symmetry as N (cid:55)→ K ( L, R, u ) N, K ∈ SU (2) , (21)while the pion matrix u (cid:55)→ u (cid:48) = RuK † = KuL † . (22)In Ref. [46], the matrix containing the pion fields transforms as U (cid:55)→ LU R † , where u = U , while the transformationused here is U (cid:55)→ RU L † in accord with Ref. [30]. However, this difference does not impact the results. The constructionof all possible nucleon operators with two spurion insertions that are invariant under chiral transformations is simplifiedby using the combinations Q ± = 12 (cid:2) u † Q R u ± uQ L u † (cid:3) , (23)which transform under the chiral symmetry group as Q ± → KQ ± K † . (24)It is useful to separate these spurions into isoscalar and isovector components, Q ± = 12 Tr( Q ± ) + ˜ Q ± , (25)where ˜ Q ± = 12 Tr( Q ± τ a ) τ a , (26)so that the operators are written in terms of Tr( Q ± ) and ˜ Q ± . V. LARGE- N c SCALING OF NN INTERACTIONS WITH TWO SPURION FIELDS
The large- N c analysis discussed in Sec. III can be extended to include the spurion operators when an explicit formfor the spurion is chosen. The greatest possible large- N c scaling of a given CIB operator can be deduced from itsspin-flavor structure, which is used to guide the elimination of redundant operators when Eq. (26) is inserted in therelevant nucleon bilinears (see Appendix B). However, as discussed above, some operators may receive additional1 /N c suppressions when the leading term contains pion fields from the expansion of u . We will point out an explicitexample of this in the next subsection.One might attempt to obtain the large- N c scaling of g NNν , C , and C directly from Eqs. (2) and (5). However, theforms of the Lagrangians in Eqs. (2) and (5) are obtained by using Fierz identities to eliminate redundant operators.This procedure can obscure the correct large- N c scalings [59, 102]. Therefore, we present an alternative minimal basisin which the large- N c scaling of the LECs is manifest. The relationships between these LECs and the ones in Eqs. (2)and (5) are given in Sec. VII.Instead of working in the basis of Ref. [41], we will use the spurions defined in Eq. (23) and then translate betweenthe two bases after the LO-in- N c Lagrangian has been derived. When determining the large- N c scaling of generaloperator forms we will leave out electromagnetic or weak factors such as e or ( G F V ud ) . While these factors impactthe overall size of an operator, they will not be relevant for understanding the relative large- N c rankings amongoperators that have the same overall multiplicative factor. The most general set of operators for this analysis is givenby B = Tr( Q + ) (cid:0) N † Γ N (cid:1) ,B = Tr( Q + ) (cid:0) N † Γ N (cid:1) (cid:16) N † ˜ Q + Γ N (cid:17) ,B = (cid:16) N † ˜ Q + Γ N (cid:17) ,B = Tr( Q − ) (cid:0) N † Γ N (cid:1) ,B = Tr( Q − ) (cid:0) N † Γ N (cid:1) (cid:16) N † ˜ Q − Γ N (cid:17) ,B = (cid:16) N † ˜ Q − Γ N (cid:17) ,B = Tr( Q + ) Tr( Q − ) (cid:0) N † Γ N (cid:1) (cid:0) N † Γ N (cid:1) ,B = Tr( Q − ) (cid:16) N † ˜ Q + Γ N (cid:17) (cid:0) N † Γ N (cid:1) ,B = Tr( Q + ) (cid:16) N † ˜ Q − Γ N (cid:17) (cid:0) N † Γ N (cid:1) ,B = (cid:16) N † ˜ Q + Γ N (cid:17) (cid:16) N † ˜ Q − Γ N (cid:17) ,B = Tr (cid:16) ˜ Q + ˜ Q − (cid:17) (cid:0) N † Γ N (cid:1) = 12 Tr (cid:16) ˜ Q R + ˜ Q L (cid:17) (cid:0) N † Γ N (cid:1) ,B = Tr (cid:16) ˜ Q − ˜ Q − (cid:17) (cid:0) N † Γ N (cid:1) = Tr (cid:16) U ˜ Q L U † ˜ Q R (cid:17) (cid:0) N † Γ N (cid:1) ,B = Tr (cid:16) ˜ Q + ˜ Q − (cid:17) (cid:0) N † Γ N (cid:1) = Tr (cid:16) ˜ Q R − ˜ Q L (cid:17) (cid:0) N † Γ N (cid:1) , (27)where Γ can be , σ i , τ a , or σ i τ a . However, several of the operators that arise once all four of the possibilities for Γare inserted into the general forms of Eq. (27) will be redundant. The nucleon bilinears contained in operators from B , B , B , B , B , and B have the same structure as the operators from nucleon-nucleon scattering, so operatorswith Γ = and σ i τ a may start to contribute at LO in N c , while those with Γ = σ i and τ a are 1 /N c suppressed. TheFierz identity (cid:0) N † σ i τ a N (cid:1) = − (cid:0) N † N (cid:1) (28)can be used to eliminate σ i τ a in favor of , and since the corresponding LECs are of the same order there is no changein the scaling obtained. Similarly, the identity (cid:0) N † τ a N (cid:1) = − (cid:0) N † N (cid:1) − (cid:0) N † σ i N (cid:1) (29)shows that the bilinear with τ a is not independent of those containing and σ i in the operators of the form B , B , B , B , B , and B .For operators of the form B , B , B , B , B , B , and B , on the other hand, the insertion of Γ = τ a or σ i τ a creates terms containing products of Pauli matrices in isospin space in a single nucleon bilinear. The structure ofthese terms does not match directly onto the Hartree Hamiltonian of Eq. (8). But the terms can be rewritten using τ a τ b = δ ab + i(cid:15) abc τ c , (30)which generates structures that contain at most a single isospin Pauli matrix. Again, the large- N c scaling of theseterms can be determined from Eq. (13), and the forms with Γ = τ a and σ i τ a can be eliminated for this set ofoperators. There is one more redundancy. Operators with Γ = or σ i can be removed through the use of Eq. (26)and Fierz transformations. For B , B , and B , Γ = can be eliminated, while either choice is suitable for B , B , B , and B since both choices scale with N c in the same way. Again, Appendix B contains greater detail about thisprocedure. In the next two subsections, the explicit forms of the spurion fields for the electromagnetic and the weakcases, respectively, are considered. A. Electromagnetic Spurions
For the electromagnetic case, it is useful to write the Lagrangian in terms of the nucleon charge matrix, Q = 12 (cid:0) + τ (cid:1) . (31)The difference between using the nucleon charge matrix and using the quark charge matrix amounts to a shift by anunobservable constant [100]. Here, the nucleon charge matrix is independent of N c , which implies that the up anddown quark charges are N c -dependent. The alternative choice that the quark charges are constant and the nucleoncharge depends on N c is discussed in Appendix A.Using Eq. (31) as the spurion field in Eq. (27) yields operators with a clear spin-flavor structure. Setting Q R = Q L = Q gives ˜ Q ± = 14 (cid:2) u † τ u ± uτ u † (cid:3) . (32)The corresponding traces of operators are [101] Tr( Q + ) = 1 , (33)Tr( Q − ) = 0 , (34)Tr (cid:16) ˜ Q + ˜ Q − (cid:17) = Tr (cid:16) ˜ Q (cid:17) = 1 / , (35)Tr (cid:16) ˜ Q − ˜ Q − (cid:17) = Tr (cid:16) U ˜ QU † ˜ Q (cid:17) , (36)Tr (cid:16) ˜ Q + ˜ Q − (cid:17) = 0 . (37)Since Tr (cid:16) ˜ Q + ˜ Q − (cid:17) (cid:0) N † O N (cid:1) ∼ Tr( Q + ) (cid:0) N † O N (cid:1) , (38)the operators from B can be absorbed into those from B . Therefore, the only independent operators are thosefrom B , B , B , B , B , B , and B . The operators from B vanish at least through O ( φ ) when u is expandedand can be neglected at this order. After eliminating redundancies (see Appendix B) the remaining operators are O , = (cid:0) N † N (cid:1) , (39) O , = (cid:0) N † σ i N (cid:1) , (40) O = (cid:0) N † N (cid:1) (cid:16) N † ˜ Q + N (cid:17) , (41) O = (cid:16) N † σ i ˜ Q + N (cid:17) (cid:16) N † σ i ˜ Q + N (cid:17) , (42) O = (cid:16) N † σ i ˜ Q − N (cid:17) (cid:16) N † σ i ˜ Q − N (cid:17) , (43) O = (cid:16) N † ˜ Q − N (cid:17) (cid:0) N † N (cid:1) , (44) O , = Tr (cid:16) U ˜ QU † ˜ Q (cid:17) (cid:0) N † N (cid:1) , (45) O , = Tr (cid:16) U ˜ QU † ˜ Q (cid:17) (cid:0) N † σ i N (cid:1) , (46)where the first subscript i in O i,j indicates the B i from which each operator originates, and the second index j , wherenecessary, refers to a specific operator within the B i , j = 1 , , , , σ i , τ a , σ i τ a , respectively.Finally, as discussed in Sec. III, each additional pion field introduces a factor of 1 /F ∼ / √ N c . Expanding eachoperator to second order in the pion fields to determine the maximum N c scaling of the corresponding LECs yields O , = (cid:0) N † N (cid:1) + · · · , (47) O , = (cid:0) N † σ i N (cid:1) + · · · , (48) O = 12 (cid:18) − F φ a φ a (cid:19) (cid:0) N † N (cid:1) (cid:0) N † τ N (cid:1) + 14 F φ φ a (cid:0) N † N (cid:1) (cid:0) N † τ a N (cid:1) + · · · , (49) O = 14 (cid:18) − F φ a φ a (cid:19) (cid:0) N † σ i τ N (cid:1) + 14 F φ φ a (cid:0) N † σ i τ N (cid:1) (cid:0) N † σ i τ a N (cid:1) + · · · , (50) O = 14 F (cid:15) ab (cid:15) cd φ a φ c (cid:0) N † σ i τ b N (cid:1) (cid:0) N † σ i τ d N (cid:1) + · · · , (51) O = − F (cid:15) ab φ a (cid:0) N † τ b N (cid:1) (cid:0) N † N (cid:1) + · · · , (52) O , = (cid:20) − F ( φ a φ a − φ φ ) (cid:21) (cid:0) N † N (cid:1) + · · · , (53) O , = (cid:20) − F ( φ a φ a − φ φ ) (cid:21) (cid:0) N † σ i N (cid:1) + · · · (54)where the ellipses indicate additional pion fields. The scaling of the LECs ¯ C i,j multiplying O i,j in the Lagrange densityis given by ¯ C , ∼ N c , (55)¯ C , ∼ N − c , (56)¯ C ∼ , (57)¯ C ∼ N c , (58)¯ C ∼ , (59)¯ C ∼ N − / c , (60)¯ C , ∼ N c , (61)¯ C , ∼ N − c . (62)The operators O , and O , differ only at the multi-pion level. Therefore, differences between the two will be 1/ N c suppressed. The same holds for the operators O , and O , . The operator O provides a concrete example ofan earlier point: the generic spin-flavor structure of the operator, before expanding u in the number of pion fields,indicates that it could be O ( N c ), but the first nonzero term has two pion fields and is thus suppressed by an additionalfactor of 1 /N c .The Lagrangian at LO and next-to-leading order (NLO) in the large- N c expansion is L LO-in- N c = e (cid:26)(cid:104) ¯ C , + ¯ C , Tr (cid:16) U ˜ QU † ˜ Q (cid:17)(cid:105) (cid:0) N † N (cid:1) + ¯ C (cid:16) N † σ i ˜ Q + N (cid:17) (cid:27) , (63) L NLO-in- N c = e (cid:26) ¯ C (cid:0) N † N (cid:1) (cid:16) N † ˜ Q + N (cid:17) + ¯ C (cid:16) N † σ i ˜ Q − N (cid:17) (cid:27) . (64)The ¯ C i and ¯ C i,j are LECs that have to be determined from comparison to data or from a calculation in terms of theunderlying QCD degrees of freedom. Expanding the matrices u and U in the number of pion fields also creates termsat higher order in the large- N c counting than indicated by the subscript on the left side; see the discussion in Sec. III.In Sec. VII, we will map the form of the Lagrangian in Eqs. (63) and (64) to the one used in Eq. (5) to determine thelarge- N c scaling of the LECs in Eq. (5). B. Weak Spurions
For weak interactions, Q L is given by Eq. (3) while Q R = 0, which gives Q ± = ˜ Q ± = ± uQ L u † = ± uτ + u † . (65)As a result, all traces in Eqs. (27) vanish and therefore operators from B , B , B , B , B , B , B , B , B , and B do not contribute. Since ˜ Q + = − ˜ Q − , the only nonvanishing term is (cid:0) N † uτ + u † Γ N (cid:1) , (66)and the structures B , B , and B become identical. As pointed out in Ref. [41], the two operators correspondingto Γ = and σ i in this term are related through a Fierz identity and are not independent at O ( φ ). The authors ofRef. [41] choose to retain Γ = ; that is, the operator (cid:0) N † τ + N (cid:1) . According to Eq. (13), this operator does not appearat LO in the large- N c expansion. However, eliminating the operator (cid:0) N † σ i τ + N (cid:1) through the Fierz transformation (cid:0) N † σ i τ + N (cid:1) = − (cid:0) N † τ + N (cid:1) (67)introduces a hidden LO-in- N c contribution in the term proportional to (cid:0) N † τ + N (cid:1) . As a result, after removing theoverall factor of N c from the Hartree Hamiltonian as discussed in Sec. III, g NNν is of LO in the large- N c expansion, g NNν ∼ O ( N c ). This result by itself does not justify the assumptions underlying the approximation g NNν ≈ ( C + C )proposed in Refs. [45, 46]. However, an inconsistency in the large- N c scaling of g NNν versus ( C + C ) would cast doubton the approximation. As will be shown in Sec. VII, ( C + C ) ∼ O ( N c ), consistent with the LO scaling of g NNν foundhere.
VI. LARGE- N c HIERARCHY OF CHARGE-INDEPENDENCE-BREAKING INTERACTIONS
Before focusing on the isotensor terms and their relation to g NNν , we will analyze the large- N c scaling of generalCIB NN interactions using the results of Sec. V A. In the absence of external pions, the operators in Eqs. (47) - (54)that contain pions only contribute through pion-loop diagrams that are of higher order in the chiral power countingthan is considered in this analysis. Adopting the conventions in Ref. [103] (also see Ref. [104]), the NN interactions,including the effects of virtual photons, are divided into four classes characterized by the following isospin structures:(I) isospin invariant and charge symmetric: , (cid:126)τ · (cid:126)τ ,(II) CIB but not charge-symmetry-breaking (CSB), which have the isotensor form: τ τ − (cid:126)τ · (cid:126)τ ,(III) CSB (and thus CIB) terms that are symmetric in spin and isospin indices: τ + τ ,(IV) CSB with isospin mixing (these vanish on nn and pp systems, but not np , and only occur in L (cid:54) = 0 partialwaves): τ − τ , ( (cid:126)τ × (cid:126)τ ) .The subscripts in the expressions above denote nucleon bilinears one and two. Refs. [73, 104, 105] use dimensionalanalysis to argue that the size of these interactions is such that Class (I) > Class (II) > Class (III) > Class (IV).Neglecting the operators O and O because they contain at least one pion field, the independent contact operatorsgenerated by the spurion formalism fall into the categories(I) O , , O , , (68)(II) O , (69)(III) O . (70)As discussed in Sec. V A, the pionless parts of the operators O , and O , are identical to O , and O , , respectively.Class (I) and (II) interactions appear at the same order in the large- N c expansion, while Class (III) terms aresuppressed by 1 /N c . It may be unexpected that the large- N c analysis suggests that the isospin-invariant Class(I) interactions appear at the same order as CIB terms. Recall, though, that the operators considered here areaccompanied by factors of e in the Lagrangian. The Class (I) terms derived here are therefore O ( e )-suppressedcorrections to the dominant isospin-invariant interactions. Taking into account the additional e suppression of the0isospin-violating terms, our results are not in contradiction with the expectations of Refs. [73, 104, 105] that someClass (I) terms are larger than Class (II) terms. Contact operators leading to Class (IV) CIB contain two derivativesand are of higher order in the EFT expansion. Taking into account the scaling of the momenta in Eq. (14), theseterms are at most O ( N c ). Additionally, at the level of the NN Lagrangian, the two operators that lead to the Class(IV) potential given in [104] are related by Fierz transformations and are not independent at the two-derivative orderin the EFT expansion. But previous work [106–108] has shown that formally Fierz-equivalent operators can lead toambiguities when used in deriving potentials with local regulators.
VII. LARGE- N c JUSTIFICATION FOR g NNν ≈ ( C + C ) [46] To connect Eq. (63) to Eq. (5), it is helpful to rearrange the LO-in- N c Lagrangian (Eq. (63)) as L LO-in- N c = e (cid:26) (cid:2) C , + ¯ C , − ¯ C (cid:3) Tr (cid:16) ˜ Q (cid:17) (cid:0) N † N (cid:1) + ¯ C (cid:20)(cid:16) N † σ i ˜ Q + N (cid:17) −
16 Tr (cid:16) ˜ Q (cid:17) (cid:0) N † σ i τ a N (cid:1) (cid:21)(cid:27) , (71)where the second term proportional to ¯ C is now a symmetric traceless isotensor. The included trace term appearsat the same order in the large- N c expansion. This rearrangement also produces a NLO-in- N c contribution such thatEq. (64) becomes L NLO-in- N c = e (cid:26) (cid:2) C , − ¯ C , − ¯ C (cid:3) Tr (cid:16) ˜ Q − (cid:17) (cid:0) N † N (cid:1) + ¯ C (cid:0) N † N (cid:1) (cid:16) N † ˜ Q + N (cid:17) + ¯ C (cid:20)(cid:16) N † σ i ˜ Q − N (cid:17) −
16 Tr (cid:16) ˜ Q − (cid:17) (cid:0) N † σ i τ a N (cid:1) (cid:21)(cid:27) . (72)We now consider the isotensor CIB term proportional to ¯ C in more detail, and relate it to the terms used in Ref. [46],see Eq. (5). Fierz transformations are used to rewrite the leading terms (see Eq. (B8)). This uncovers LO-in- N c scaling in terms that naively appear to be of higher order. The resulting Lagrangian is L ∆ I =2LO-in- N c = − e ¯ C (cid:20)(cid:16) N † ˜ Q + N (cid:17) −
16 Tr (cid:16) ˜ Q (cid:17) (cid:0) N † τ a N (cid:1) (cid:21) . (73)Using the definition of the spurion fields in Eqs. (23), the Lagrangian of Eq. (5) can be written as L NNCIB = e (cid:26) ( C + C ) (cid:20)(cid:16) N † ˜ Q + N (cid:17) −
16 Tr (cid:16) ˜ Q (cid:17) (cid:0) N † τ a N (cid:1) (cid:21) + ( C − C ) (cid:20)(cid:16) N † ˜ Q − N (cid:17) −
16 Tr (cid:16) ˜ Q − (cid:17) (cid:0) N † τ a N (cid:1) (cid:21)(cid:27) . (74)Comparison with Eq. (73) shows that 12 ( C + C ) = − C , (75)which demonstrates that C + C ∼ N c . A similar transformation for the isotensor term in Eq. (72) shows that12 ( C − C ) = − C , (76)demonstrating that C − C is 1/ N c suppressed relative to C + C . Inverting these equations gives C = − C − C = − C [1 + O (1 /N c )] , (77) C = − C + 3 ¯ C = − C [1 + O (1 /N c )] . (78)These results support the assumption of Ref.[46] that the LECs in the CIB Lagrangian are of the same size and sign,and that therefore the neutrinoless LEC can be approximated as g NNν ≈ ( C + C ).1 VIII. CONCLUSION
The renormalization group analysis of Refs. [45, 46] showed that, for light-Majorana exchange, an LNV contactterm is required at leading order in ChEFT. The presence of this term impacts the calculation of nuclear matrixelements relevant for 0 νββ decay. Neither sufficient data nor lattice QCD results are currently available to determinethe size of the corresponding LEC, g NNν . To estimate the contribution of the LNV contact term to nuclear matrixelements, Refs. [45, 46] assumed that the two CIB LECs C and C are of the same size and sign, which allowed themto approximate g NNν ≈ ( C + C ) / N c analyses of the LNV and CIB NN operators appearing at the first nonvanishing orderin ChEFT power counting. Our results show that the assumptions underlying the approximations of g NNν used inRefs. [45, 46] are consistent with ordering based upon the large- N c limit, lending additional support to the numericalestimates for matrix elements found there. They are also in line with the recent results of Ref. [47].Our analysis also shows a hierarchy of the different classes of CIB NN interactions as defined in Refs. [103]. Theordering obtained does not contradict phenomenological expectations [73, 104, 105]. However, as is generally the case,the large- N c results should not be treated as precise predictions. The ordering of LECs is based on expansions in1/ N c and the assumption that other numerical factors are of natural size. For example, symmetries not captured bythe large- N c expansion may lead to unnaturally small parameters. We hope that this work will help guide many-bodystudies of LNV in heavier elements, as well as the interpretation of neutrinoless double beta decay experiments. ACKNOWLEDGMENTS
We thank Emanuele Mereghetti for useful discussions. We thank the Institute for Nuclear Theory at the University ofWashington for its kind hospitality and stimulating research environment during the INT-18-2a program “FundamentalPhysics with Electroweak Probes of Light Nuclei.” This research was supported in part by the INT’s U.S. Departmentof Energy grant No. DE-FG02-00ER41132. This material is based upon work supported by the U.S. Department ofEnergy, Office of Science, Office of Nuclear Physics, under Award Numbers DE-SC0019647 (TRR and MRS), DE-SC0021027 (SP), and DE-FG02-05ER41368 (RPS).
Appendix A: Alternative Electric Charge Scaling
The choice of keeping the nucleon charge independent of N c , while the quark charges scale with N c , has theadvantage that anomaly cancellations persist in a large- N c extended standard model [109, 110]. The up and downquark charges in units of e are then given by q u = N c + 12 N c , q d = 1 − N c N c , (A1)where N c is odd but arbitrary. This choice leads to a proton with electric charge of one in units of e when it is takento consist of ( N c + 1) up quarks and ( N c −
1) down quarks. Similarly, the neutron has electric charge 0 when thenumbers of quark flavors are switched.In the meson sector of χ PT, it is customary to use the quark charge matrix when constructing the spurion coun-terterms. However, when nucleons are included it is conventional to use the nucleon charge matrix. The terms in thepion Lagrangian are then replaced accordingly, but this only amounts to the addition of an unobservable constantterm. When going to large- N c , it is reasonable to ask if this is still the case when the charge matrices with differentlarge- N c scalings are interchanged. To answer this question, the quark and nucleon charge matrices are generalized[101], Q = α + βτ . (A2)The leading order operator in the pion Lagrangian is e C Tr (cid:0) QU QU † (cid:1) = e C Tr (cid:0) α + β τ U τ U † (cid:1) . (A3)The first term is indeed an unobservable constant shift, and the second term leads to the electromagnetic pion masssplitting when U is expanded to O ( φ ), i.e. δm π = 2 e F C . (A4)2For quark charges that scale as Eq. (A1), the quark and nucleon charge matrices become Q q-scaling q = 12 N c (cid:2) + N c τ (cid:3) , (A5) Q q-scaling N = 12 (cid:2) + τ (cid:3) , (A6)where the superscript indicates that the quark q = u, d charges scale with N c .Alternatively, it was argued that for baryons containing O ( N c ) strange quarks, quantization conditions requirethe quark charges be fixed to their physical values and independent of N c [111]. However, for this choice, anomaliesin an SU( N c )-extended standard model do not cancel [109, 110] and the nucleon charge becomes N c -dependent andunbounded as N c → ∞ . Nevertheless, as shown in the following, such a choice does not change our conclusions. Thequark and nucleon charge matrices are then Q q-fixed q = 16 (cid:2) + 3 τ (cid:3) , (A7) Q q-fixed N = 16 (cid:2) N c + 3 τ (cid:3) , (A8)where the superscript indicates that the quark charges q are fixed as N c changes. Regardless of whether the quark ornucleon charge matrices are chosen to scale with N c , the coefficient β = . Therefore, both choices lead to the samepion mass splitting.Based on the argument that a single flavor trace operator in the meson sector of χ PT corresponds to a singleclosed loop in large- N c QCD, it might be expected that the LEC C scales at most as C ∼ N c . Using the typicaldiagrammatic arguments in Fig. 1, adding a photon in the loop does not modify the color structure, so it still consistsof a single sum over all colors but it does pick up a factor of e . Therefore, the pion mass splitting in Eq. (A4) is atmost O ( N c ) when e is taken to be fixed and after accounting for the suppression due to F . However, it was shown(see, e.g., Ref. [109, 110]) that for electroweak effects to be finite, the electromagnetic coupling can be rescaled likethe strong coupling, i.e. e ∼ N − / c . In this case, the mass splitting will be O (1 /N c ). FIG. 1. Leading order diagram in large- N c QCD which is O ( e N c ). When the nucleon charge is chosen to have the N c dependence given by Eq. (A8), the large- N c behavior of theoperators in Eqs. (27) needs to be reexamined for possible changes. The operators that contain Tr( Q ± ) are nowmultiplied by an overall factor of N c for each insertion of the trace. This leads to¯ C , ∼ N c , (A9)¯ C , ∼ N c , (A10)¯ C ∼ N c , (A11)¯ C ∼ N / c , (A12)while the large- N c scaling of the other LECs remains unchanged. The operators relevant for the classification of theCIB terms are still O , , O , , O , and O . To obtain the traceless form of the Class (II) interactions, the term inthe Lagrangian containing O can be rewritten as( N † σ i τ N ) = ( N † σ i τ N ) −
13 ( N † σ i τ a N ) + 13 ( N † σ i τ a N ) . (A13)The first two terms on the right-hand side combine to form the Class (II) interaction. The last term is absorbed as an O ( N c ) contribution into the Class (I) interaction. With the alternative large- N c scaling of the charges, the classes ofcharge dependence are then altered such that (II) and (III) are the same order in N c while they are both suppressedby N − c relative to (I). This also indicates that the correspondence between the LNV operator and the CIB contactterm remains intact regardless of the choice taken for the scaling of the nucleon charge with N c .3 Appendix B: Fierz identities and the elimination of redundant operators
The operators that contain only traces of the spurions have the form ( N † Γ N ) , where Γ can be , σ i , τ a , or σ i τ a .The Fierz identities in Eqs. (28) and (29) from Sec. V, (cid:0) N † σ i τ a N (cid:1) = − (cid:0) N † N (cid:1) , (28) (cid:0) N † τ a N (cid:1) = − (cid:0) N † N (cid:1) − (cid:0) N † σ i N (cid:1) , (29)reduce the number of independent operators from four to two; (cid:0) N † N (cid:1) , (B1) (cid:0) N † σ i N (cid:1) , (B2)where the first operator is LO-in- N c , and the second is 1 /N c suppressed.For the operators involving the traceless part of the spurion field in the bilinears, the spurions in Eq. (26) areexpanded and the products of Pauli matrices reduced using Eq. (30). All of the suppressions arising from the presenceof pion fields are contained in coefficients defined by c a, ± = Tr (cid:16) ˜ Q ± τ a (cid:17) . Therefore, Γ = 1 and σ i , respectively, leadto (cid:16) N † ˜ Q ± N (cid:17) (cid:16) N † ˜ Q ± N (cid:17) = c a, ± c b, ± (cid:0) N † τ a N (cid:1) (cid:0) N † τ b N (cid:1) , (B3) (cid:16) N † ˜ Q ± σ i N (cid:17) (cid:16) N † ˜ Q ± σ i N (cid:17) = c a, ± c b, ± (cid:0) N † τ a σ i N (cid:1) (cid:0) N † τ b σ i N (cid:1) . (B4)For Γ = τ c , (cid:16) N † ˜ Q ± τ c N (cid:17) (cid:16) N † ˜ Q ± τ c N (cid:17) = c a, ± c a, ± (cid:0) N † N (cid:1) − c a, ± c a, ± (cid:0) N † τ b N (cid:1) + c a, ± c b, ± (cid:0) N † τ a N (cid:1) (cid:0) N † τ b N (cid:1) , (B5)and the operator (cid:0) N † τ b N (cid:1) can be removed using the Fierz identity of Eq. (29) to obtain (cid:16) N † ˜ Q ± τ c N (cid:17) (cid:16) N † ˜ Q ± τ c N (cid:17) = 3 c a, ± c a, ± (cid:0) N † N (cid:1) + c a, ± c a, ± (cid:0) N † σ i N (cid:1) + c a, ± c b, ± (cid:0) N † τ a N (cid:1) (cid:0) N † τ b N (cid:1) . (B6)The first two terms in Eq. (B6) have the same bilinear structure as the operators O , and O , , respectively. Theircontributions can be absorbed into a redefinition of the LECs of these operators. The third term in Eq. (B6) isEq. (B3). For Γ = σ i τ c , Eq. (B6) appears again except that the last term is Eq. (B4) instead of Eq. (B3). This showsthat Γ = τ c and σ i τ c do not yield additional independent operators and can be neglected.Additional relationships exist among some of the operators corresponding to Γ = and Γ = σ i . For B , B , and B , Γ = can be eliminated by applying Fierz transformations to Eq. (B3) along with the decomposition in Eq. (26).Using Tr (cid:16) ˜ Q ± (cid:17) = 2 c a, ± c a, ± , (B7)the Fierz transformation for Eq. (B3) leads to − (cid:20)(cid:16) N † ˜ Q ± N (cid:17) (cid:16) N † ˜ Q ± N (cid:17) −
16 Tr (cid:16) ˜ Q ± ˜ Q ± (cid:17) (cid:0) N † τ a N (cid:1) (cid:21) = (cid:16) N † ˜ Q ± N (cid:17) (cid:16) N † σ i ˜ Q ± N (cid:17) −
16 Tr (cid:16) ˜ Q ± ˜ Q ± (cid:17) (cid:0) N † σ i τ a N (cid:1) , (B8)which can be arranged, with the help of additional Fierz transformations, to be (cid:16) N † ˜ Q ± N (cid:17) (cid:16) N † ˜ Q ± N (cid:17) = − (cid:16) N † ˜ Q ± N (cid:17) (cid:16) N † σ i ˜ Q ± N (cid:17) −
12 Tr (cid:16) ˜ Q ± ˜ Q ± (cid:17) (cid:0) N † N (cid:1) −
16 Tr (cid:16) ˜ Q ± ˜ Q ± (cid:17) (cid:0) N † σ i N (cid:1) . (B9)4Therefore, the choice of Γ = can be eliminated from B , B , and B in favor of combinations of Γ = σ i andoperators from B , B , and B .Following the same procedure for operators from B , B , B , and B results inTr( Q ± ) (cid:16) N † ˜ Q ± N (cid:17) (cid:0) N † N (cid:1) = c a, ± Tr( Q ± ) (cid:0) N † N (cid:1) , (B10)Tr( Q ± ) (cid:16) N † ˜ Q ± σ i N (cid:17) (cid:0) N † σ i N (cid:1) = c a, ± Tr( Q ± ) (cid:0) N † N (cid:1) , (B11)for Γ = and σ i , respectively. When Γ = τ b , Fierz transformations yieldTr( Q ± ) (cid:16) N † ˜ Q ± τ b N (cid:17) (cid:0) N † τ b N (cid:1) = − c a, ± Tr( Q ± ) (cid:104) (cid:0) N † N (cid:1) + (cid:0) N † σ i N (cid:1) (cid:105) . (B12)Similarly, Γ = σ i τ b leads toTr( Q ± ) (cid:16) N † ˜ Q ± σ i τ b N (cid:17) (cid:0) N † σ i τ b N (cid:1) = − c a, ± Tr( Q ± ) (cid:0) N † N (cid:1) . 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