Determining the leading-order contact term in neutrinoless double \boldsymbolβ decay
Vincenzo Cirigliano, Wouter Dekens, Jordy de Vries, Martin Hoferichter, Emanuele Mereghetti
LLA-UR-21-20994
Determining the leading-order contact term inneutrinoless double β decay Vincenzo Cirigliano, a Wouter Dekens, b Jordy de Vries, c,d,e,f
Martin Hoferichter, g Emanuele Mereghetti aa Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Department of Physics, University of California at San Diego, La Jolla, CA 92093,USA c Institute for Theoretical Physics Amsterdam and Delta Institute for TheoreticalPhysics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, TheNetherlands d Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands e Amherst Center for Fundamental Interactions, Department of Physics, University ofMassachusetts, Amherst, MA 01003 f RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York11973-5000, USA g Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Abstract
We present a method to determine the leading-order (LO) contact term contributingto the nn → ppe − e − amplitude through the exchange of light Majorana neutrinos. Ourapproach is based on the representation of the amplitude as the momentum integral ofa known kernel (proportional to the neutrino propagator) times the generalized forwardCompton scattering amplitude n ( p ) n ( p ) W + ( k ) → p ( p (cid:48) ) p ( p (cid:48) ) W − ( k ), in analogy to theCottingham formula for the electromagnetic contribution to hadron masses. We constructmodel-independent representations of the integrand in the low- and high-momentum regions,through chiral EFT and the operator product expansion, respectively. We then construct amodel for the full amplitude by interpolating between these two regions, using appropriatenucleon factors for the weak currents and information on nucleon–nucleon ( NN ) scattering inthe S channel away from threshold. By matching the amplitude obtained in this way to theLO chiral EFT amplitude we obtain the relevant LO contact term and discuss various sourcesof uncertainty. We validate the approach by computing the analog I = 2 NN contact termand by reproducing, within uncertainties, the charge-independence-breaking contribution tothe S NN scattering lengths. While our analysis is performed in the MS scheme, we expressour final result in terms of the scheme-independent renormalized amplitude A ν ( | p | , | p (cid:48) | ) ata set of kinematic points near threshold. We illustrate for two cutoff schemes how, using oursynthetic data for A ν , one can determine the contact-term contribution in any regularizationscheme, in particular the ones employed in nuclear-structure calculations for isotopes ofexperimental interest. a r X i v : . [ nu c l - t h ] F e b ontents A < . . . . . . . . . . . . . . . . . . . 134.2 High-momentum region: A > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 C C + C Z . . . . . . . . . . . . . . 266.3 nn → pp vector-like amplitude in chiral EFT and full theory . . . . . . . . . . . . 286.3.1 A V V in chiral EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3.2 A V V in the full theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.4 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.5 Charge-independence-breaking contribution to NN scattering . . . . . . . . . . . 35 nn → pp near threshold 368 Conclusions 40A Half-off-shell T matrix 42 A.1 The half-off-shell form factor in pionless EFT . . . . . . . . . . . . . . . . . . . . 42A.1.1 Half-off-shell T matrix in pionless EFT at NLO . . . . . . . . . . . . . . . 42A.1.2 Behavior of f S and I 0) = (cid:104) h f ( p f ) | ˆΠ LLµν ( k, | h i ( p i ) (cid:105) , (2.8)one arrives at (cid:104) e e h f | S ∆ L =2eff | h i (cid:105) = (2 π ) δ (4) ( p e + p e + p f − p i ) (cid:16) G F V ud m ββ ¯ u L ( p ) u cL ( p ) (cid:17) × A ν , A ν = 2 (cid:90) d k (2 π ) T ( k, p ext ) k + i(cid:15) , T ( k, p ext ) ≡ g µν T µν ( k, p ext ) . (2.9)The hadronic amplitude A ν in Eq. (2.9) receives contributions from neutrino virtualities k ranging from the weak scale all the way down to the infrared (IR) scale of nuclear bound states.To estimate the LO contact term arising in chiral EFT, we will employ the representation (2.9)to obtain the amplitude in the “full theory,” and then match to the appropriate EFT expression.Since the contact term arises in the S channel, we will take as external states nn and pp inthe S state and T µν ( k, p ext ) will be thought of as the generalized forward Compton amplitude n ( p ) n ( p ) W + ( k ) → p ( p (cid:48) ) p ( p (cid:48) ) W − ( k ) . (2.10)Since the low-energy constants (LECs) do not depend on the IR details, we will perform thematching calculation at the simplest kinematic point, in which the two electrons are emittedwith zero three-momentum in the center-of-mass frame of the incoming neutron pair [40, 43].Explicitly we have p µ = ( E, p ) , p µ = ( E, − p ) , E = (cid:112) p + m n ,p (cid:48) µ = ( E (cid:48) , p (cid:48) ) , p (cid:48) µ = ( E (cid:48) , − p (cid:48) ) , E (cid:48) = (cid:113) p (cid:48) + m p , (2.11)where 2 E = 2 E (cid:48) + 2 m e . Free two-nucleon states with vanishing total three-momentum andindividual three-momenta given by ± q will be denoted by | q (cid:105) [23], so for example for the initialand final state we will have | i (cid:105) = | p (cid:105) and | f (cid:105) = | p (cid:48) (cid:105) , respectively. In what follows we will suppress the space-time label in the correlator: ˆΠ LLµν ( k, → ˆΠ LLµν ( k ). ardIntermediateSoftUSoft Contours of fixed Potential Figure 1: Schematic representation of the regions of neutrino virtuality contributing to theamplitude in Eq. (2.9). The boundaries between various regions are given by k F ∼ 100 MeV,Λ χ (cid:46) (cid:38) . The amplitude for the process nn → pp is given in Eq. (2.9) as the integral of the productof a massless propagator (remnant of the Majorana neutrino propagator) with the contractedhadronic tensor T ( k, p ext ) = g µν T µν . The neutrino four-momentum regions relevant for the in-tegration over d k are schematically depicted in Fig. 1. Denoting the Euclidean four-momentumsquared by k E ≡ ( k ) + k , one can introduce hard ( k E > Λ ), intermediate (Λ χ < k E < Λ ),and low-energy ( k E < Λ χ ) regions, separated by Λ χ (the breakdown scale of the low-energyhadronic EFT) and Λ (scale at which the OPE becomes reliable). The low-energy region fur-ther includes the soft ( | k | ∼ | k | ∼ k F ), potential ( | k | ∼ k F /m N , | k | ∼ k F ), and ultrasoft( | k | ∼ | k | (cid:28) k F ) regions, essential to reproduce the IR behavior of the amplitude.The basic idea behind our approach is that model-independent representations of the inte-grand in Eq. (2.9) can be constructed in the low-energy region (via pionless and chiral EFT) andin the hard region (via the OPE). Given this, a model for the full amplitude can be constructedby interpolating between these two regions. This approach uses model-dependent input for theintermediate momentum region, which we anchor to known constraints from QCD at low andhigh momenta.In practice, given the non-relativistic nature of the process of interest, we will not use k E asmatching variable. Instead, we decompose d k = dk d k , first perform the k integral in theappropriate regions via Cauchy’s theorem, and then carry out the angular integrations in d k to reduce the amplitude to an integral over d | k | . To LO in the expansion in external momenta | p | , | p (cid:48) | ∼ Q (we denote by µ χ ∼ Q the soft scale in the EFT), we write the full amplitudeas an integral over the internal neutrino three-momentum k , which we split into a low- plusintermediate-momentum region and a high-momentum region A full ν = (cid:90) ∞ d | k | a full ( | k | ) = A < + A > , < = (cid:90) Λ0 d | k | a < ( | k | ) , A > = (cid:90) ∞ Λ d | k | a > ( | k | ) , (2.12)separated by the scale Λ that represents the onset of the asymptotic behavior for the current–current correlator, controlled by the OPE. This representation introduces model dependencethrough: (i) The choices made to extend the model-independent integrand a χ ( | k | ) dictated bychiral EFT in the region | k | < Λ χ to the function a < ( | k | ) valid up to | k | ∼ Λ. We will providea simple parameterization of a < ( | k | ) that reduces to a χ ( | k | ) for | k | < M π and incorporatesphenomenological input such as nucleon form factors of the weak current and resonance contri-butions to the strong-interaction potential. (ii) The choice of Λ that determines the boundaryof integration regions in the variable | k | . Once a representation for A full ν is obtained, along withan estimate of the associated uncertainties, we will estimate the LEC appearing in A χ EFT ν byenforcing the matching condition A χ EFT ν = A < + A > . (2.13)In the following sections, we will describe the construction of A <,> and the matching to A χ EFT ν ,starting with the spectral representation in Section 2.3. In this section we provide a spectral representation for the nn → pp amplitude, which will provevery useful in identifying and organizing the various intermediate-state contributions to A < andcarrying out the analysis in analogy to the Cottingham formula [55, 56].We begin by recalling some elements of the formal theory of scattering that we will use invarious parts of the discussion. We denote by ˆ H = ˆ H + ˆ V the total Hamiltonian, split into afree and interaction term ( ˆ V , not to be confused with the potential). Retarded and advancedGreen’s functions in the interacting theory are given byˆ G ± ( E ) = 1 E − ˆ H ± i(cid:15) = (cid:90) dt e iEt ˆ G ± ( t ) , ˆ G ± ( t ) = ∓ iθ ( ± t ) e − i ˆ Ht , (2.14)with analogous definitions for the free-theory ones, denoted by ˆ G (0) ± ( E ), with the replacementˆ H → ˆ H . The scattering operator ˆ T ( E ) is formally given byˆ T ( E ) = ˆ V (cid:16) I − ˆ G (0)+ ( E ) ˆ V (cid:17) − (2.15)and satisfies ˆ G + ( E ) ˆ V = ˆ G (0)+ ( E ) ˆ T ( E ). The scattering states (cid:104) f − | and | i + (cid:105) are related to thefree states (cid:104) f | and | i (cid:105) by | i + (cid:105) = (cid:16) I + ˆ G (0)+ ( E ) ˆ T ( E ) (cid:17) | i (cid:105) , (cid:104) f − | = (cid:104) f | (cid:16) ˆ T ( E (cid:48) ) ˆ G (0)+ ( E (cid:48) ) + I (cid:17) . (2.16)In terms of the scattering states, the amplitude for nn → pp can be written as A ν = (cid:90) d k (2 π ) (cid:104) f − | ˆ O LL ( k ) | i + (cid:105) , (2.17)7ith the weak transition operatorˆ O LL ( k ) ≡ (cid:90) dk π g µν ˆΠ LLµν ( k ) k + i(cid:15) . (2.18)From the definition of the correlator in Eq. (2.6) one obtains the following representation forˆΠ LLµν ( k ) in terms of Green’s functions: g µν ˆΠ LLµν ( k ) = i (2 π ) J Lµ (0) (cid:104) ˆ G + ( k ) δ (3) ( ˆ P − k + ) + ˆ G + ( k − ) δ (3) ( ˆ P − k − ) (cid:105) J Lµ (0) , (2.19)where ˆ P is the total three-momentum operator and we have introduced the four-vectors k µ ± = ˜ p µ ± k µ , ˜ p µ = 12 ( p i + p f ) µ = ( ˜ E, ˜ p ) . (2.20)The labels in p µi,f refer to the initial and final states between which ˆΠ LLµν ( k ) is evaluated. Sincewe are considering two-nucleon external states with vanishing total three-momentum (and totalmomentum is conserved at each vertex) we have ˜ p = 0 and hence k ± = ± k . In Eq. (2.19)the dependence on k is very simple, as k appears only through the energy denominators ofˆ G + ( k ± ). Performing the integration over k in Eq. (2.18) with Cauchy’s theorem, one arrivesat ˆ O LL ( k ) = 1 | k | J Lµ (0) ˆ G + ( ˜ E − | k | ) (2 π ) (cid:104) δ (3) ( ˆ P − k + ) + δ (3) ( ˆ P − k − ) (cid:105) J Lµ (0) . (2.21)Further inserting a complete set of states between the current operators in Eq. (2.21) leads tothe spectral representation for the amplitude A ν = − (cid:88) n (cid:90) d k (2 π ) | k | (cid:34) (cid:104) f − | J Lµ | n ( k + ) (cid:105)(cid:104) n ( k + ) | J Lµ | i + (cid:105)| k | + ( E n ( k + ) − ˜ E ) − i(cid:15) + (cid:104) f − | J Lµ | n ( k − ) (cid:105)(cid:104) n ( k − ) | J Lµ | i + (cid:105)| k | + ( E n ( k − ) − ˜ E ) − i(cid:15) (cid:35) . (2.22)The representations (2.17) and (2.22) are quite general. The asymptotic behavior of theintegrand in Eq. (2.22) at large | k | is dictated by the OPE for ˆΠ LLµν ( k ) or, equivalently, ˆ O LL ( k ).An explicit calculation to be described below shows the behavior d k / | k | , so the amplitude inthe full theory is finite. Moreover, Eq. (2.22) shows that once | k | > k F , so that k /m N is abovethe typical nuclear binding energies, one expects ( E n ( k ± ) − ˜ E ) > | k | . The matrix elements in the numerator are alsoexpected to have a smooth behavior in | k | , dictated by single- and multi-hadron form factors, asshown by explicit EFT calculations. Based on these considerations, we conclude that a smoothinterpolation between the calculable regimes of | k | (cid:46) Λ χ and | k | (cid:38) Λ is adequate. For each term in Eq. (2.19), one can close the contour in the upper or lower k plane so that the integral isgiven by the residue at the k pole from the neutrino propagator in Eq. (2.18). The summation is over intermediate states | n ( k ± ) (cid:105) of total three-momentum k ± , enforced by the δ -functionsin Eq. (2.21). Therefore, for an N -particle intermediate state (cid:80) n involves phase space integrals over the N − k ± ) and carries non-zero mass dimension. For example, fortwo-nucleon intermediate states, using non-relativistic normalizations for the states (cid:104) p n | p (cid:48) n (cid:105) = (2 π ) δ (3) ( p n − p (cid:48) n )one has (cid:80) n → (cid:82) d p n / (2 π ) , where p n is the relative momentum of the two-nucleon pair. In general thesummation (cid:80) n | n ( k ± ) (cid:105)(cid:104) n ( k ± ) | carries mass dimension − 8n order to make the integrand in Eqs. (2.17) and (2.22) more explicit, we use the expressionfor the scattering states (2.16) in Eq. (2.17) and arrive at A ν = (cid:90) d k (2 π ) (cid:104) f | (cid:16) ˆ T ( E (cid:48) ) ˆ G (0)+ ( E (cid:48) ) + I (cid:17) ˆ O LL ( k ) (cid:16) I + ˆ G (0)+ ( E ) ˆ T ( E ) (cid:17) | i (cid:105) (2.23)= (cid:90) d k (2 π ) (cid:40) (cid:104) f | ˆ O LL ( k ) | i (cid:105) + (cid:88) m (cid:104) f | ˆ T ( E (cid:48) ) | m (cid:105) (cid:104) G (0)+ ( E (cid:48) ) (cid:105) m (cid:104) m | ˆ O LL ( k ) | i (cid:105) + (cid:88) m (cid:104) f | ˆ O LL ( k ) | m (cid:105) (cid:104) G (0)+ ( E ) (cid:105) m (cid:104) m | ˆ T ( E ) | i (cid:105) + (cid:88) m,m (cid:48) (cid:104) f | ˆ T ( E (cid:48) ) | m (cid:48) (cid:105) (cid:104) G (0)+ ( E (cid:48) ) (cid:105) m (cid:48) (cid:104) m (cid:48) | ˆ O LL ( k ) | m (cid:105) (cid:104) G (0)+ ( E ) (cid:105) m (cid:104) m | ˆ T ( E ) | i (cid:105) (cid:41) , where (cid:104) G (0)+ ( E ) (cid:105) m = 1 E − E (0) m + i(cid:15) , (2.24) E (0) m denotes the energy associated with the free Hamiltonian ˆ H , and (cid:104) a | ˆ O LL ( k ) | b (cid:105) = − (cid:88) n | k | (cid:34) (cid:104) a | J Lµ | n ( k ) (cid:105)(cid:104) n ( k ) | J Lµ | b (cid:105)| k | + ( E n ( k ) − ˜ E ab ) − i(cid:15) + (cid:104) a | J Lµ | n ( − k ) (cid:105)(cid:104) n ( − k ) | J Lµ | b (cid:105)| k | + ( E n ( − k ) − ˜ E ab ) − i(cid:15) (cid:35) , (2.25)with ˜ E ab = ( E a + E b ) / 2. In general the sum over hadronic intermediate states in Eqs. (2.23)and (2.25) involves | m (cid:105) , | m (cid:48) (cid:105) , | n (cid:105) , ∈ { NN, NN π, ... } , i.e., both elastic | NN (cid:105) contributions andinelastic contributions | m (cid:105) , | m (cid:48) (cid:105) , | n (cid:105) (cid:54) = | NN (cid:105) . Equations (2.23)–(2.25) make it explicit whichdynamical input is needed for the evaluation of the nn → pp amplitude:1. One needs the matrix elements of the current–current operator ˆ O LL ( k ) among two-nucleonand possibly other intermediate states, namely (cid:104) m (cid:48) | ˆ O LL ( k ) | m (cid:105) , that can be further decom-posed according to Eq. (2.25).2. One needs the T -matrix elements (cid:104) m | ˆ T ( E ) | i (cid:105) and (cid:104) f | ˆ T ( E (cid:48) ) | m (cid:48) (cid:105) involving arbitrary inter-mediate states (cid:104) m | , | m (cid:48) (cid:105) and on-shell two-nucleon states | i (cid:105) = | p (cid:105) (with E = p /m N ) and (cid:104) f | = (cid:104) p (cid:48) | (with E (cid:48) = p (cid:48) /m N ). When considering the elastic contributions, these reduceto the so-called half-off-shell (HOS) T -matrix elements (cid:104) p m | ˆ T ( E ) | p (cid:105) and (cid:104) p (cid:48) | ˆ T ( E (cid:48) ) | p m (cid:48) (cid:105) ,involving loop momenta p m and p m (cid:48) . While it is well known that the HOS T -matrix ele-ments by themselves are not physical quantities (see for example the discussion in Ref. [74]),they enter Eq. (2.23) in such a way that the full physical amplitude A ν is free of off-shellambiguities (see Appendix A for an explicit check of this point).To LO in chiral EFT the amplitude A ν is saturated by elastic contributions, with all inputsin Eqs. (2.23)–(2.25) given to leading chiral order. The LO chiral input provides a good rep-resentation of the low-momentum part of the integrand but misrepresents the high-momentumcomponent. In this language, the ultraviolet (UV) divergence and the need for a LO contactterm arises from the 1 / | k | behavior of the integrand, as discussed in Section 3.9n the other hand, in our estimate of the full amplitude to be described in Section 4, wewill start from Eqs. (2.23)–(2.25) and use representations of (cid:104) m (cid:48) | ˆ O LL ( k ) | m (cid:105) , (cid:104) p m | ˆ T ( E ) | p (cid:105) , and (cid:104) p (cid:48) | ˆ T ( E (cid:48) ) | p m (cid:48) (cid:105) that go beyond leading chiral order to construct a UV convergent integrand.Motivated by the leading chiral EFT analysis and the analogy with the Cottingham approachto the pion and nucleon electromagnetic mass splitting, we expect the elastic two-nucleon inter-mediate state to provide the dominant contribution. While we will mostly focus on the elasticchannel, we will also estimate the effect of the leading NN π inelastic channel as we expect thisto be one of the dominant sources of uncertainty in our final result.