A new leading contribution to neutrinoless double-beta decay

TL;DR

This paper identifies a crucial missing short-range operator in neutrinoless double-beta decay calculations within chiral effective field theory, which could significantly impact experimental interpretations.

Contribution

It introduces a leading-order short-range operator into the decay transition operator, addressing a gap in current theoretical models.

Findings

Necessity of including a short-range operator for accurate decay predictions

Strategies proposed for determining the operator's finite part

Potential impact on nuclear matrix elements and experimental results

Abstract

Within the framework of chiral effective field theory we discuss the leading contributions to the neutrinoless double-beta decay transition operator induced by light Majorana neutrinos. Based on renormalization arguments in both dimensional regularization with minimal subtraction and a coordinate-space cutoff scheme, we show the need to introduce a leading-order short-range operator, missing in all current calculations. We discuss strategies to determine the finite part of the short-range coupling by matching to lattice QCD or by relating it via chiral symmetry to isospin-breaking observables in the two-nucleon sector. Finally, we speculate on the impact of this new contribution on nuclear matrix elements of relevance to experiment.

Figures

Equations

$${\mathcal{L}}_{\Delta L=2}=-\frac{m_{\beta \beta }}{2}{\nu }_{eL}^{T}C{\nu }_{eL},$$

$$H_{\mathrm{eff}}=H_{\mathrm{strong}}+2G_F^2{V}_{ud}^2{m}_{\beta \beta }{\bar{e}}_{L}C{\bar{e}}_L^T{V}_{\nu },$$

$$V_{\nu ,0}(\mathbf{q})$$

$$V_{\nu ,CT}=-2g_{\nu }^{NN}{\tau }^{(1)+}{\tau }^{(2)+},$$

$$V_0(\mathbf{q})=\tilde{C}+V_{\pi }(\mathbf{q}),V_{\pi }(\mathbf{q})=-\frac{g_A^2}{4F_{\pi }^2}\frac{m_{\pi }^2}{{\mathbf{q}}^2+m_{\pi }^2},$$

$${\mathcal{A}}_{\Delta L=2}^{(\nu )}$$

$$K_E$$

$$\tilde{V}_{\nu}({\mathbf{q}})=\tau^{(1)+}\tau^{(2)+}\frac{1}{{\mathbf{q}}^{2}}\left(1-\frac{2}{3}g_{A}^{2}\mbox{\boldmath$\sigma$}^{(1)}\cdot\mbox{\boldmath$\sigma$}^{(2)}\right)~{}.$$

$${\mathcal{A}}_C^{(\mathrm{div})}$$

$$L_{\mathbf{p},{\mathbf{p}}^{\prime }}(\mu )$$

$${\mathcal{A}}_{\Delta L=2}^{(NN)}=K_{E^{\prime }}\frac{2g_{\nu }^{NN}}{{\tilde{C}}^2}{K}_E,$$

$$\mathcal{A}_{C}\to\delta\mathcal{A}_{C}+\left(\frac{m_{N}}{4\pi}\right)^{2}\Big{[}2\tilde{g}_{\nu}^{N\!N}(\mu)-\left(1+2g_{A}^{2}\right)L_{{\mathbf{p}},{\mathbf{p}}^{\prime}}(\mu)\Big{]}$$

$${\tilde{g}}_{\nu }^{NN}={\left(\frac{4\pi }{m_N\tilde{C}}\right)}^2{g}_{\nu }^{NN}.$$

$$\mu \frac{d{\tilde{g}}_{\nu }^{NN}}{d\mu }=\frac{1}{2}\left(1+2g_A^2\right),$$

$$\tilde{C}{\delta }^{(3)}(\mathbf{r})\to \frac{\tilde{C}(R_S)}{(\sqrt{\pi }{R}_S{)}^3}\exp \left(-\frac{r^2}{R_S^2}\right)\equiv \tilde{C}(R_S){\delta }_{R_S}^{(3)}(\mathbf{r}),$$

$${\mathcal{A}}_{\Delta L=2}^{(\nu )}=-\int d^3\mathbf{r}{\psi }_{{\mathbf{p}}^{\prime }}^{-}(\mathbf{r}{)}^{\ast }{V}_{\nu ,0}(\mathbf{r}){\psi }_{\mathbf{p}}^{+}(\mathbf{r}),$$

$${\mathcal{A}}_{\Delta L=2}^{(NN)}=-\int d^3\mathbf{r}{\psi }_{{\mathbf{p}}^{\prime }}^{-}(\mathbf{r}{)}^{\ast }{V}_{\nu ,CT}(\mathbf{r}){\psi }_{\mathbf{p}}^{+}(\mathbf{r}),$$

$$O_1$$

$$O_2$$

$$\mu \frac{d}{d\mu }\frac{{\tilde{C}}_1+{\tilde{C}}_2}{2}$$

$${\mathcal{A}}_{\Delta L=2}^{(\nu )}=\int dr{\rho }_{\nu }(r),{\mathcal{A}}_{\Delta L=2}^{(NN)}=\int dr{\rho }_{NN}(r).$$

Full text

††preprint: LA-UR-18-21404††preprint: NIKHEF 2018-010

Introduction: Neutrinoless double-beta decay ($0\nu\beta\beta$) is the most sensitive laboratory probe of lepton number violation (LNV). In $0\nu\beta\beta$  $L$ is violated by two units when two neutrons in a nucleus turn into two protons, with the emission of two electrons and no neutrinos. The observation of $0\nu\beta\beta$ would demonstrate that neutrinos are Majorana fermions Schechter and Valle (1982), shed light on the mechanism of neutrino mass generation Minkowski (1977); Mohapatra and Senjanovic (1980); Gell-Mann et al. (1979), and give insight into leptogenesis scenarios for the generation of the matter-antimatter asymmetry in the universe Davidson et al. (2008).

$0\nu\beta\beta$ is actively being searched for in a number of even-even nuclei for which single-$\beta$ decay is energetically forbidden. Current experimental limits Gando et al. (2013); Agostini et al. (2013); Albert et al. (2014); Andringa et al. (2016); Gando et al. (2016); Elliott et al. (2016); Agostini et al. (2017); Albert et al. (2018); Alduino et al. (2017); Azzolini et al. (2018) on the half-lives are at the level of $T_{1/2}>5.3\times 10^{25}$ y for 76Ge Agostini et al. (2017) and $T_{1/2}>1.07\times 10^{26}$ y for 136Xe Gando et al. (2016), with next-generation ton-scale experiments aiming at improvements in sensitivity by two orders of magnitude.

$0\nu\beta\beta$ can be generated by a variety of dynamical LNV mechanisms, which in an effective field theory (EFT) approach to new physics are parametrized by $\Delta L=2$ operators of odd dimension greater than four Weinberg (1979); Babu and Leung (2001); Prezeau et al. (2003); de Gouvêa and Jenkins (2008); Lehman (2014); Graesser (2017); Cirigliano et al. (2017a). If the mass scale associated with LNV is much higher than the electroweak scale, the only low-energy manifestation of this new physics is a Majorana mass for light neutrinos, encoded in a single gauge-invariant dimension-five operator Weinberg (1979), which induces $0\nu\beta\beta$ through light Majorana-neutrino exchange Bilenky and Petcov (1987); Bilenky and Giunti (2015). To interpret positive or null $0\nu\beta\beta$ results in this minimal scenario it is crucial to have good control over the relevant hadronic and nuclear matrix elements. Current knowledge of these is not satisfactory Engel and Menéndez (2017), as various many-body approaches lead to estimates that differ by a factor of two to three and most calculations are not based on a modern EFT analysis. In Ref. Cirigliano et al. (2017b) a first step was presented towards the analysis of $0\nu\beta\beta$ induced by a light Majorana neutrino in the chiral EFT framework Bedaque and van Kolck (2002); Epelbaum et al. (2009); Machleidt and Entem (2011), which provides a systematic expansion of hadronic amplitudes in $p/\Lambda_{\chi}$, where $p\sim m_{\pi}\sim k_{F}\sim{\cal O}(100~{}{\rm MeV})$ and $\Lambda_{\chi}\sim 4\pi F_{\pi}\sim m_{N}\sim{{\cal O}(1~{}{\rm GeV})}$. The $0\nu\beta\beta$ transition operators were derived up to next-to-next-to-leading order (N2LO) in Weinberg’s power-counting scheme Weinberg (1990, 1991).

In this letter we demonstrate that Weinberg’s scheme for $0\nu\beta\beta$ assumed in Ref. Cirigliano et al. (2017b) breaks down and any consistent power counting requires a leading-order (LO) short-range $\Delta L=2$ operator, whose effect is missing in all current calculations. Our argument is based on renormalization. Using two different schemes (dimensional regularization with minimal subtraction and a coordinate-space cutoff) we show that once the strong nucleon-nucleon scattering amplitude is made finite and independent of the ultraviolet regulator, an additional $\Delta L=2$ contact operator with coupling $g_{\nu}^{N\!N}$ has to be introduced to make the $nn\to ppee$ amplitude finite and regulator-independent. The finite part of $g_{\nu}^{N\!N}$, which encodes hard-neutrino exchange, can be determined by (i) matching the chiral EFT $nn\to ppee$ amplitude to future lattice QCD calculations; (ii) relating it via chiral symmetry to electromagnetic low-energy constants (LECs) that control isospin-breaking in the two-nucleon sector. A combination of couplings involving $g_{\nu}^{N\!N}$ can be fit to nucleon-nucleon charge-independence-breaking (CIB) observables, confirming the LO scaling of this coupling. Based on this, we speculate on the impact of $g_{\nu}^{N\!N}$ on nuclear matrix elements of relevance to experiments.

The need for an LO short-range $\Delta L=2$ interaction: We consider a scenario in which LNV at low energy is dominated by the electron-neutrino Majorana mass

[TABLE]

where $C=i\gamma_{2}\gamma_{0}$ denotes the charge conjugation matrix.

The nuclear effective Hamiltonian can be written as

[TABLE]

in terms of the Fermi constant $G_{F}$ and the $V_{ud}$ element of the CKM matrix Cabibbo (1963); Kobayashi and Maskawa (1973). The neutrino potential $V_{\nu}$ can be obtained from two-nucleon irreducible diagrams mediating $nn\to ppee$ to a given order in $p/\Lambda_{\chi}$. Within Weinberg’s power counting the only LO contribution Cirigliano et al. (2017b) comes from the exchange of potential neutrinos, with $q^{0}\ll|{\mathbf{q}}|$,

[TABLE]

where $g_{A}\simeq 1.27$ is the nucleon axial coupling, $m_{\pi}$ the pion mass, and ${\mathbf{q}}$ the momentum transfer. N2LO terms arise from corrections to the single nucleon weak currents, irreducible one-loop diagrams, and contact interactions mediating $\pi\pi\to ee$, $n\to p\pi^{+}ee$, and $nn\to ppee$. In particular, the short-range potential includes a two-nucleon term Cirigliano et al. (2017b)

[TABLE]

where the LEC $g_{\nu}^{N\!N}$ is $\mathcal{O}((4\pi F_{\pi})^{-2})$ in Weinberg’s counting and $F_{\pi}=92.2$ MeV is the pion decay constant. However, it is known that Weinberg’s power counting leads to inconsistent results in nucleon-nucleon scattering Kaplan et al. (1996); Beane et al. (2002); Nogga et al. (2005); Long and Yang (2012) and nuclear processes mediated by external currents Pavón Valderrama and Phillips (2015), due to a conflict between naive dimensional analysis and nonperturbative renormalization. We therefore investigate the scaling of $g_{\nu}^{N\!N}$ by studying the amplitude $\mathcal{A}(nn\rightarrow ppee)\equiv\mathcal{A}_{\Delta L=2}$ with strong interactions, $H_{\rm strong}$, included nonperturbatively.

We work at LO in chiral EFT, and focus on the scattering of two neutrons to two protons in the ${}^{1}S_{0}$ wave, where $H_{\rm strong}$ has short-range and Yukawa components,

[TABLE]

with $\tilde{C}\sim\mathcal{O}(F_{\pi}^{-2},m_{\pi}^{2}F_{\pi}^{-4})$ Weinberg (1991); Kaplan et al. (1996); Beane et al. (2002). We have checked that transitions involving higher partial waves such as ${}^{3}P_{0,1}\rightarrow{}^{3}P_{0,1}$ are correctly renormalized and do not require enhanced $\Delta L=2$ counterterms.

The contributions to $\mathcal{A}_{\Delta L=2}$ from the exchange of a light neutrino ($\mathcal{A}_{\Delta L=2}^{(\nu)}$) are shown in Fig. 1. The blue ellipse denotes the iteration of the Yukawa potential $V_{\pi}({\mathbf{q}})$. The diagrams in the second and third rows include an infinite number of bubbles, dressed with iterations of $V_{\pi}$. Without loss of generality for our arguments, we use the kinematics $n({\mathbf{p}})\ n(-{\mathbf{p}})\to p({\mathbf{p}}^{\prime})\ p(-{\mathbf{p}}^{\prime})\ e({\mathbf{p}_{e1}}=0)\ e({\mathbf{p}_{e2}}=0)$, with $|{\mathbf{p}}|=1$ MeV and correspondingly $|{\mathbf{p}}^{\prime}|=38$ MeV.

$\mathcal{A}_{\Delta L=2}^{(\nu)}$ can be expressed in terms of the Yukawa “in” and “out” wavefunctions $\chi_{{\mathbf{p}}}^{\pm}({\mathbf{r}})$ and the propagators $G^{\pm}_{E}({\mathbf{r}},{\mathbf{r}}^{\prime})=\langle{\mathbf{r}}^{\prime}|(E-T-V_{\pi}\pm i0^{+})^{-1}|{\mathbf{r}}\rangle$ Kaplan et al. (1996); Long and Yang (2012). Observing that the bubble diagrams in Fig. 1 are related to $G^{+}_{E}({\mathbf{0}},{\mathbf{0}})$, while the triangles dressed by Yukawas are related to $\chi^{+}_{{\mathbf{p}}}({\mathbf{0}})$ and $\chi^{-}_{{\mathbf{p}}^{\prime}}({\mathbf{0}})^{*}=\chi^{+}_{{\mathbf{p}}^{\prime}}({\mathbf{0}})$ Kaplan et al. (1996), the LO amplitude reads

[TABLE]

where $\mathcal{A}_{A}$, $\mathcal{A}_{B}$, and $\mathcal{A}_{C}$ denote the first diagram in the first, second, and third rows of Fig. 1, respectively (without the wavefunctions at ${\mathbf{0}}$, in the case of $\mathcal{A}_{B}$ and $\mathcal{A}_{C}$). $\mathcal{\bar{A}}_{B}$ is similar to $\mathcal{A}_{B}$ and not shown in Fig. 1.

To study the renormalization of the $\Delta L=2$ amplitude, we now discuss the divergence structure of $\mathcal{A}_{\Delta L=2}^{(\nu)}$. $\chi^{+}_{{\mathbf{p}}}({\mathbf{0}})$ is finite and the divergence in $G^{+}_{E}({\mathbf{0}},{\mathbf{0}})$ is absorbed by $\tilde{C}^{-1}$, so that $K_{E}$ is finite and scheme-independent Kaplan et al. (1996). We note that:

(i) All diagrams in $\mathcal{A}_{A}$ are finite. The tree level is finite and each $V_{\pi}$ iteration improves the convergence by bringing in a factor of $d^{3}{\mathbf{k}}/({\mathbf{k}}^{2})^{2}$, where one ${\mathbf{k}}^{2}$ comes from the pion propagator and the other from the two-nucleon propagator.

(ii) All the diagrams in $\mathcal{A}_{B}$ and $\bar{\mathcal{A}}_{B}$ are finite. The first loop goes as $d^{3}{\mathbf{k}}/({\mathbf{k}}^{2})^{2}$, while $V_{\pi}$ insertions further improve the convergence.

(iii) The first two-loop diagram in $\mathcal{A}_{C}$ has a logarithmic divergence, which stems from an insertion of the most singular component of the neutrino potential, namely

[TABLE]

The two-loop diagram with insertion of $V_{\nu,0}-\tilde{V}_{\nu}$ and higher-loop diagrams are convergent.

We focus on $\mathcal{A}_{C}$ and write $\mathcal{A}_{C}=\mathcal{A}_{C}^{(\rm div)}+\delta\mathcal{A}_{C}$. In dimensional regularization,

[TABLE]

where $\Delta\equiv\left(1/(4-d)-\gamma+\log 4\pi\right)/2$. The divergence for $d\to 4$ can be removed by introducing $g_{\nu}^{N\!N}$ at LO. The counterterm amplitude, shown in the fourth line of Fig. 1, reads

[TABLE]

and we can renormalize $\mathcal{A}_{\Delta L=2}$ by replacing $\mathcal{A}_{C}\to\mathcal{A}_{C}+2g_{\nu}^{N\!N}/\tilde{C}^{2}$ in Eq. (6). In the $\overline{\rm MS}$ scheme,

[TABLE]

after defining the dimensionless coupling

[TABLE]

This coupling obeys the renormalization-group equation (RGE)

[TABLE]

confirming that $\tilde{g}_{\nu}^{N\!N}\sim\mathcal{O}(1)$. Since $\tilde{C}(\mu=m_{\pi})\approx-0.9/F_{\pi}^{2}$, we find that $g_{\nu}^{N\!N}\sim\mathcal{O}\left(F_{\pi}^{-2}\right)$ instead of $\mathcal{O}\left((4\pi F_{\pi})^{-2}\right)$. A similar enhancement also occurs in four-nucleon couplings induced by higher-dimensional LNV operators. Treating $V_{\pi}$ as a subleading correction Kaplan et al. (1998); Beane et al. (2002) is equivalent to working to LO in pionless EFT, and does not affect our conclusions about the importance of $g_{\nu}^{N\!N}$ Cirigliano et al. (2017b). Details on how to obtain $\delta\mathcal{A}_{C}$ will be provided in future work Cirigliano et al. (2018).

$\mathcal{A}_{\Delta L=2}$** in a cutoff scheme:** The need for an LO counterterm can be demonstrated also in a coordinate-space scheme that makes no direct reference to Feynman diagrams. In this approach we regulate the short-range part of $V_{0}$ with a smeared $\delta$-function,

[TABLE]

and obtain $\psi^{-}_{{\mathbf{p}}^{\prime}}({\mathbf{r}})$ and $\psi^{+}_{{\mathbf{p}}}({\mathbf{r}})$ by solving the Schrödinger equation. We determine $\tilde{C}(R_{S})$ by requiring that the ${}^{1}S_{0}$ scattering length be reproduced ($\tilde{C}\approx-0.4/F_{\pi}^{2}$ at $R_{S}=0.8$ fm). We find that $1/\tilde{C}(R_{S})$ has linear ($1/R_{S}$) and logarithmic divergences Beane et al. (2002) and that the ${}^{1}S_{0}$ phase shifts at nonzero momentum are indeed $R_{S}$-independent.

We then compute

[TABLE]

where $V_{\nu,0}({\mathbf{r}})$ is obtained by Fourier-transforming the ${}^{1}S_{0}$ projection of Eq. (3). In Fig. 2 we plot $\mathcal{A}_{\Delta L=2}^{(\nu)}$ as a function of $R_{S}$. The plot displays a logarithmic dependence on $R_{S}$ (analogous to the $\log\mu$ dependence in Eq. (10)) as well as milder power-like behavior. Therefore, to obtain a physical, regulator-independent amplitude one needs to include an LO counterterm, given in $r$-space by $V_{\nu,CT}({\mathbf{r}})=-2\,g_{\nu}^{N\!N}(R_{S})\delta_{R_{S}}^{(3)}({\mathbf{r}})$. The corresponding amplitude,

[TABLE]

is also regulator-dependent. As expected from Eq. (9), we find its leading divergent behavior to be well reproduced by $1/\tilde{C}(R_{S})^{2}$. We can then make $\mathcal{A}_{\Delta L=2}=\mathcal{A}_{\Delta L=2}^{(\nu)}+\mathcal{A}_{\Delta L=2}^{(N\!N)}$ finite for $R_{S}\to 0$ and $R_{S}$-independent by choosing $\tilde{g}_{\nu}^{N\!N}(R_{S})=-(a/2)(1+2g_{A}^{2})\log R_{S}+b+cR_{S}+\cdots$, with the coefficient of the logarithm quite close to the $\overline{\textrm{MS}}$ expectation $a=1$.

Relating $g_{\nu}^{N\!N}$ to electromagnetic isospin violation: The finite part of $g_{\nu}^{N\!N}$ can be obtained by matching the chiral EFT amplitude to a lattice QCD calculation performed at the same kinematic point, as it is done in the strong-interacting sector Barnea et al. (2015). First lattice results related to double-beta decay are starting to appear Nicholson et al. (2016); Shanahan et al. (2017).

We now discuss a complementary estimate based on the fact that the short-range operators and associated LECs arising in $0\nu\beta\beta$ and electromagnetic processes are closely related Cirigliano et al. (2017b). In the electromagnetic case, the short-range hadronic operators arise from amplitudes in the underlying theory involving two insertions of the electromagnetic current with exchange of hard virtual photons Moussallam (1997); Ananthanarayan and Moussallam (2004). In the $\Delta L=2$ case, up to a proportionality factor, the same operators are generated by the insertion of two weak currents with exchange of hard neutrinos. This comes about because the neutrino propagator and weak vertices combine to give a massless gauge-boson propagator in Feynman gauge, multiplied by $8G_{F}^{2}V_{ud}^{2}m_{\beta\beta}\bar{e}_{L}e_{L}^{c}$ Cirigliano et al. (2017b). The LECs needed for $0\nu\beta\beta$ are therefore related to the LECs associated with the isospin $I=2$ component of the product of two electromagnetic currents, which belongs to the $5_{L}\times 1_{R}$ irreducible representation of chiral $SU(2)_{L}\times SU(2)_{R}$.

Only two independent four-nucleon operators that transform as $I=2$ objects exist:

[TABLE]

where $\mathcal{Q}_{L}=u^{\dagger}Q_{L}u$, $\mathcal{Q}_{R}=uQ_{R}u^{\dagger}$, u=\exp(i\mbox{\boldmath\tau}\cdot\mbox{\boldmath\pi}/(2F_{\pi})), and $Q_{L,R}$ are “spurions” transforming under the chiral group as $Q_{L}\rightarrow LQ_{L}L^{\dagger}$, $Q_{R}\rightarrow RQ_{R}R^{\dagger}$. In the electromagnetic case $Q_{L}=Q_{R}=\tau_{3}/2$, while in $0\nu\beta\beta$ $Q_{L}=\tau^{+}$, $Q_{R}=0$. In our conventions $O_{1}$ enters the $\Delta L=2$ Lagrangian with coefficient $2G_{F}^{2}V_{ud}^{2}m_{\beta\beta}g_{\nu}^{N\!N}$. Defining the electromagnetic LECs multiplying $O_{1,2}$ as $e^{2}C_{1,2}/4$, chiral symmetry dictates $g^{N\!N}_{\nu}=C_{1}$.

In the electromagnetic case, $O_{1}$ and $O_{2}$ only differ at the multipion level, and an isospin-breaking two-nucleon observable, such as the $I=2$ combination of scattering lengths $a_{\rm CIB}=(a_{nn}+a_{pp})/2-a_{np}$, only constrains the sum $C_{1}+C_{2}$. Extracting this combination from data provides a rough estimate of $g_{\nu}^{N\!N}$ under the assumption $C_{1}\sim C_{2}$. As in the $\Delta L=2$ case, we introduce the dimensionless couplings $\tilde{C}_{i}\equiv[4\pi/(m_{N}\tilde{C})]^{2}C_{i}$ and compute the scattering lengths $a_{pp,nn,np}$ including the leading sources of isospin breaking – the Coulomb potential and pion mass splitting – and $\tilde{C}_{1}+\tilde{C}_{2}$. Similarly to the $\Delta L=2$ case, we find that $\tilde{C}_{1}+\tilde{C}_{2}$ needs to be promoted to LO and obeys the RGE

[TABLE]

while, of course, $\tilde{C}_{1}$ has the same RGE as $\tilde{g}_{\nu}^{N\!N}$. By fitting to $a_{\rm CIB}$ using $a_{np}=-23.7$ fm, $a_{nn}=-18.9$ fm, and $a_{pp}=-7.8$ fm, we find $(\tilde{C}_{1}+\tilde{C}_{2})/2\approx 2.5$ at $\mu=m_{\pi}$ in the $\overline{\rm MS}$ scheme. Using instead the $R_{S}$ scheme, we find $(\tilde{C}_{1}+\tilde{C}_{2})/2\approx 2.0$ at $R_{S}=0.5$ fm. 111Our result is consistent with analyses based on chiral Walzl et al. (2001); Piarulli et al. (2015, 2016); Reinert et al. (2017) and phenomenological potentials such as AV18 Wiringa et al. (1995), which also find that, except at very low energies, long- and short-range components of the CIB interaction induce effects of similar size. This estimate, based on data and chiral symmetry, again confirms that $g_{\nu}^{N\!N}\sim\mathcal{O}(F_{\pi}^{-2})$.

Numerical impact: To roughly estimate the impact of the contact term, we assume for concreteness $C_{1}=C_{2}$ and hence $g_{\nu}^{N\!N}=(C_{1}+C_{2})/2$ at some $\bar{R}_{S}$ or $\bar{\mu}^{-1}$ in the range $0.002-0.8$ fm. The total two-nucleon amplitude $\mathcal{A}_{\Delta L=2}=\mathcal{A}_{\Delta L=2}^{(\nu)}+\mathcal{A}_{\Delta L=2}^{(N\!N)}$ then becomes independent of the regulator, as illustrated in Fig. 2, where the widths of the horizontal bands reflect the ambiguity in the choice of the point $\bar{R}_{S}$ or $\bar{\mu}$ where $C_{1}=C_{2}$ is assumed. (They do not account for the uncontrolled error of the assumption itself.) The relative size of the two components depends on $R_{S}$, with $\mathcal{A}_{\Delta L=2}^{(N\!N)}/\mathcal{A}_{\Delta L=2}^{(\nu)}\sim 30\%$ at $R_{S}\sim 0.1$ fm, decreasing to $\sim 10\%$ at $R_{S}\sim 0.6$ fm. More insight can be obtained by plotting the matrix-element densities $\rho_{\nu}$ and $\rho_{N\!N}$ defined as

[TABLE]

Figure 3 (top panel) shows that $\rho_{N\!N}(r)$ is concentrated at smaller distances than $\rho_{\nu}(r)$, and its contribution to the amplitude is thus partially diluted.

We have performed a similar analysis for $A=6,12$ nuclei, using Variational Monte Carlo nuclear wavefunctions Pastore et al. (2018) based on the AV18 two-nucleon Wiringa et al. (1995) and IL7 three-nucleon Pieper (2008) interactions. The mismatch between the short-range behaviors of existing strong-interaction potentials and our $0\nu\beta\beta$ interaction introduces additional model dependence, which we mitigate by: (i) Considering an alternative extraction of $(C_{1}+C_{2})/2$ from the phase-shift analysis of Refs. Piarulli et al. (2015, 2016) 222 $C_{1}+C_{2}$ is related to the CIB coefficient $C_{0}^{IT}$ of Refs. Piarulli et al. (2015, 2016) by $(C_{1}+C_{2})/2=-6C_{0}^{IT}/e^{2}$., which employs the same regulator (13) with $R_{S}\simeq 0.6-0.8$ fm, approximately the range of AV18’s short-range part. (ii) Simply replacing our $V_{\nu,CT}({\mathbf{r}})$ with AV18’s short-range CIB potential.

For $\Delta I=0$ transitions such as the 6He $\rightarrow$ 6Be shown in Fig. 3 (middle panel), we find $\mathcal{A}_{\Delta L=2}^{(N\!N)}/\mathcal{A}_{\Delta L=2}^{(\nu)}\sim 10\%$, similarly to the $nn\rightarrow ppee$ case. In realistic $0\nu\beta\beta$ transitions, however, the total nuclear isospin changes by two units, $\Delta I=2$. This implies the presence of a node in $\rho_{\nu}(r)$ due to the orthogonality of the initial and final spatial wavefunctions. The resulting partial cancellation between the regions with r\hskip 1.99997pt\raisebox{2.15277pt}{<}\raisebox{-3.00003pt}{\sim}\hskip 1.99997pt2 fm and r\hskip 1.99997pt\raisebox{2.15277pt}{>}\raisebox{-3.00003pt}{\sim}\hskip 1.99997pt2 fm Pastore et al. (2018) leads to a relative enhancement of the short-range contribution, as illustrated in Fig. 3 (bottom panel) for 12Be $\rightarrow$ 12C. Numerically we find $\mathcal{A}_{\Delta L=2}^{(N\!N)}/\mathcal{A}_{\Delta L=2}^{(\nu)}\sim 25\%$ (our fit), $\sim 55\%$ (fit from Refs. Piarulli et al. (2015, 2016)), and $\sim 60\%$ (AV18 representation of the short-range CIB potential). Because the node in the density is a robust feature of $\Delta I=2$ transition Šimkovic et al. (2008); Menéndez et al. (2009), we expect the effects in 12Be $\rightarrow$ 12C and experimentally relevant transitions to be of comparable size.

Conclusion: The above arguments suggest that the new short-range $\Delta L=2$ potential identified in this work can significantly impact $0\nu\beta\beta$ phenomenology and its implications for Majorana neutrino masses. We hope this will stimulate work towards a more controlled determination of $g_{\nu}^{N\!N}$ from lattice QCD and an assessment of the impact of the short-range potential in nuclei of experimental interest.