# An EFT toolbox for baryon and lepton number violating dinucleon to dilepton
decays

Xiao-Gang He<EMAIL_ADDRESS>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  Xiao-
Dong Ma<EMAIL_ADDRESS>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  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 to dilepton decays
($pp\to\ell^{+}\ell^{\prime+},pn\to\ell^{+}\bar{\nu}^{\prime},nn\to\bar{\nu}\bar{\nu}^{\prime}$)
with $\Delta B=\Delta L=-2$ in the framework of effective field theory. We
start by constructing a basis of dimension-12 (dim-12) operators mediating
such processes in the low energy effective field theory (LEFT) below the
electroweak scale. Then we consider their standard model effective field
theory (SMEFT) completions upwards and their chiral realizations in baryon
chiral perturbation theory (B$\chi$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
coefficients associated with the dim-12 operators in the SMEFT and the low
energy constants pertinent to B$\chi$PT. We find the current experimental
limits push the associated new physics scale larger than $1-3$ TeV, which is
still accessible to the future collider searches. Through weak isospin
symmetry, we find the current experimental limits on the partial lifetime of
transitions $pp\to\ell^{+}\ell^{\prime+},pn\to\ell^{+}\bar{\nu}^{\prime}$
imply stronger limits on $nn\to\bar{\nu}\bar{\nu}^{\prime}$ than their
existing lower bounds, which are improved by $2-3$ orders of magnitude.
Furthermore, assuming charged mode transitions are also dominantly generated
by the similar dim-12 SMEFT interactions, the experimental limits on $pp\to
e^{+}e^{+},e^{+}\mu^{+},\mu^{+}\mu^{+}$ lead to stronger limits on
$pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}$ with $\alpha,\beta=e,\mu$ than their
existing bounds. Conversely, the same assumptions help us to set a lower bound
on the lifetime of the experimentally unsearched mode $pp\to e^{+}\tau^{+}$
from that of $pn\to e^{+}\bar{\nu}_{\tau}$, i.e., $\Gamma^{-1}_{pp\to
e^{+}\tau^{+}}\gtrsim 2\times 10^{34}~{}\rm yr$.

###### Contents

  1. I Introduction
  2. II Operator basis for $pp\to\ell^{+}\ell^{\prime+},pn\to\ell^{+}\bar{\nu}^{\prime},nn\to\bar{\nu}\bar{\nu}^{\prime}$ transitions
    1. II.1 LEFT operators
    2. II.2 SMEFT completions
    3. II.3 Relations between SMEFT and LEFT operators
  3. III Chiral realizations
    1. III.1 Some basics of Chiral matching
    2. III.2 Decomposition of irreducible chiral symmetry
    3. III.3 Chiral matching for the operators
  4. IV Dinucleon and dilepton transition rate
  5. V Conclusion and outlook
  6. A Independent color tensors
  7. B LEFT operators
  8. C Reduction of the redundant operators in the LEFT
  9. D SMEFT operators
  10. E Reduction of the redundant operators in the SMEFT
  11. F Chiral basis construction
  12. G Chiral irreducible representations in terms of hadrons
  13. H LEFT and SMEFT contributions to $C^{pp}$, $C^{pn}$ and $C^{nn}$

##  I Introduction

The observed matter-antimatter asymmetry of the universe requires the
violation of baryon number, which as one of the three Sakharov conditions for
a successful baryogenesis mechanism Sakharov (1991); at the same time, the
extremely possible Majorana nature of neutrinos breaks the lepton number. Both
facts lead to the existence of a class of new physics scenarios beyond the
minimal standard model (SM) in which the baryon and/or lepton numbers are
violated explicitly or spontaneously to a certain degree, this is because the
baryon ($B$) and lepton ($L$) numbers are accidental global symmetries in the
SM but are violated through the quantum anomaly at an unobservable level ’t
Hooft (1976).111The difference $B-L$ is still an exact symmetry survived from
the anomaly cancellation in the SM. On the other hand, the violation of baryon
and/or lepton numbers would be expected in the grand unified theories (GUT)
Georgi and Glashow (1974); Babu and Barr (1993). Thus, the search of rare
baryon and/or lepton number violation 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 categorized into two classes in terms of the net baryon number being
changed by one unit ($\Delta B=1$) or two units ($\Delta B=2$). For the
$\Delta B=1$ case, the relevant processes are the single free or bound nucleon
decays, which could change the lepton numbers by $\Delta L=\pm 1$ or $\pm 3$
units, like the most well-known proton decay mode $p\to e^{+}\pi^{0}$, etc,
see Ref. Heeck and Takhistov (2020) and references therein for a thorough
discussion on these $\Delta B=1$ processes. Such processes have been searched
for experimentally for a long time but with null results Zyla  _et al._
(2020), which, however, tightly constrain the NP scenarios and push the NP
scale around the GUT scale unaccessible directly for the current and future
collider experiments.

On the other hand, for the $\Delta B=2$ case, the net lepton numbers can be
changed by either $\Delta L=0$ or $\Delta L=\pm 2$ units.222For the processes
with other larger even units of lepton numbers ($\Delta L=\pm 4,\pm
6,\cdots$), their effect would be expected to be severely suppressed due to
higher dimensionality of the relevant interactions Kobach (2016) and thus
omitted here. They are interesting because they can be searched for
experimentally with clean signatures and high precision, as well as because
there exist NP scenarios in which their contributions are dominant but that of
$\Delta B=1$ processes like proton decay are suppressed Nussinov and Shrock
(2002); Arnold  _et al._ (2013); Dev and Mohapatra (2015); Gardner and Yan
(2019); Girmohanta and Shrock (2020a). The neutron-antineutron oscillation
($n{\rm-}\bar{n}$) is a representative example for the $\Delta L=0$ case,
which has attracted a lot of attention in recent years both theoretically and
experimentally, see the review Phillips  _et al._ (2016) and references
therein for a summary on the state of the art of this process. For the $\Delta
L=\pm 2$ case, the interesting processes are the dinucleon to dilepton decays
in nuclei, including $pp\to\ell^{+}\ell^{\prime+}$,
$pn\to\ell^{+}\bar{\nu}^{\prime}$ and $nn\to\bar{\nu}\bar{\nu}^{\prime}$ with
$\ell(\ell^{\prime})=e,\mu,\tau$ and
$\bar{\nu}(\bar{\nu}^{\prime})=\bar{\nu}_{e},\bar{\nu}_{\mu},\bar{\nu}_{\tau}$,
respectively.333There are also $\Delta L=0$ dinucleon decays with the final
states being a pair of leptons conserving lepton flavor and number or two
mesons, like $pn\to\ell^{+}\nu$, $nn\to\nu\bar{\nu}$ and
$pp\to\pi^{+}\pi^{+},K^{+}K^{+}$ Girmohanta and Shrock (2019, 2020b). Such
processes get less attention than the above mentioned $\Delta B=1$ and
$n{\rm-}\bar{n}$ oscillation processes Arnold  _et al._ (2013); Bramante  _et
al._ (2015); Girmohanta and Shrock (2020a); Girmohanta (2020), but they may
open new avenues for the baryon number violation signals due to their distinct
theoretical origin and clear experimental signatures.444A timely brief status
report of the $\Delta B=2$ physics can be found in Bab (2020). The most
stringent lower limits on the partial lifetime of those dinucleon to dilepton
decays in oxygen 16O and carbon 12C nuclei are reported by the earlier Frejus
and KamLAND experiments Berger  _et al._ (1991); Araki  _et al._ (2006) and
the recent Super-K results Sussman  _et al._ (2018); Takhistov  _et al._
(2015), and are collected in Tab. 1 for our latter use.

Decay mode | Lifetime limit | Decay mode | Lifetime limit | Decay mode | Lifetime limit
---|---|---|---|---|---
$pp\to e^{+}e^{+}$ |  $4.2\times 10^{33}$ yr Sussman  _et al._ (2018) | $pn\to e^{+}\bar{\nu}^{\prime}$ |  $2.6\times 10^{32}$ yr Takhistov  _et al._ (2015) | $nn\to\bar{\nu}\bar{\nu}^{\prime}$ |  $1.4\times 10^{30}$ yr Araki  _et al._ (2006)
$pp\to e^{+}\mu^{+}$ |  $4.4\times 10^{33}$ yr Sussman  _et al._ (2018) | $pn\to\mu^{+}\bar{\nu}^{\prime}$ |  $2.2\times 10^{32}$ yr Takhistov  _et al._ (2015) |  |
$pp\to\mu^{+}\mu^{+}$ |  $4.4\times 10^{33}$ yr Sussman  _et al._ (2018) | $pn\to\tau^{+}\bar{\nu}^{\prime}$ |  $2.9\times 10^{31}$ yr Takhistov  _et al._ (2015) |  |
$pp\to e^{+}\tau^{+}$ | - |  |  |  |

Table 1:  The lower bound on the partial lifetime of the dinucleon to dilepton
transitions. Where the limits for the charged modes $(pp)$ and $(pn)$ were set
per oxygen nucleus 16O in water Cherenkov Super-K experiment at $90\%$
confidence level Takhistov  _et al._ (2015); Sussman  _et al._ (2018) while
the limit on the neutral mode $(nn)$ is obtained from the carbon nucleus 12C
in KamLAND experiment Araki  _et al._ (2006). The bounds on modes involving
anti-neutrinos are also valid for the cases with neutrinos since they are
treated as invisible states in the experimental search.

Confronted with the relatively less studies on those baryon and lepton number
violating $\Delta B=\Delta L=-2$ decays $pp\to\ell^{+}\ell^{\prime+}$,
$pn\to\ell^{+}\bar{\nu}^{\prime}$ and $nn\to\bar{\nu}\bar{\nu}^{\prime}$, it
is our goal in this paper to make a 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 the
decays, including the relevant degrees of freedom and symmetries in each EFT,
and the matching and renormalization group running procedures among different
EFTs. We start by constructing a basis mediating such processes using the
first two flavors of quarks $(u,d)$ and the charged and neutral leptons
$(\ell,\nu)$ that enjoys the QCD and QED gauge symmetries $SU(3)_{\rm C}\times
U(1)_{\rm EM}$ in the low energy effective field theory (LEFT) below the
electroweak scale $\Lambda_{\rm EW}$. At leading order in the LEFT, the
contribution arises from effective interactions of dimension-12 (dim-12)
$\Delta B=\Delta L=-2$ operators that involve six quark fields and two lepton
fields $(qqqqqqll)$, where $q$ denotes $u,d$ while $l$ stands for $\ell,\nu$.
By crossing symmetry, those operators also parametrize the general
interactions contributing to the baryon and lepton number violating conversion
processes $e^{-}p\to e^{+}\bar{p}\;,\;\bar{\nu}\bar{n}$ and
$e^{-}n\to\bar{\nu}\bar{p}$ in the electron-deuteron ($e$-d) scattering
Gardner and Yan (2019), the neutron exotic decay $n\to\bar{p}e^{+}\bar{\nu}$
He and Ma (2021) and the hydrogen-antihydrogen (${\rm H}{\rm-}\bar{\rm H}$)
oscillation Feinberg  _et al._ (1978).

Figure 1: The flow chart for an EFT calculation of the low energy observables.

To translate the experimental constraints at low energy to those on NP at a
high scale $\Lambda_{\rm NP}$, we have to climb up the ladder of energy scales
in Fig. 1. If there are no new particles with a mass at or below $\Lambda_{\rm
EW}$ , the standard model effective field theory (SMEFT) defined between some
NP scale $\Lambda_{\rm NP}$ and the electroweak scale $\Lambda_{\rm EW}$ is a
good starting point to parametrize the UV NP in a model-independent way. Based
on this logic, we then consider the leading order SMEFT completions of the
LEFT interactions in question and it happens that the relevant operators also
first appear at dim 12, and without counting lepton flavors we obtain 29
independent operators contributing to the decays. Along the way we perform a
tree-level matching between the SMEFT and LEFT interactions at $\Lambda_{\rm
EW}$, and it turns out that the interaction structures simplify significantly
due to the constraints of the SM gauge symmetry $SU(3)_{\rm C}\times
SU(2)_{\rm L}\times U(1)_{\rm Y}$ and many LEFT Wilson coefficients vanish at
this leading order.

To calculate the transition rate, at low energy which we take to be the chiral
symmetry breaking scale $\Lambda_{\chi}$, we perform the non-perturbative
matching of the LEFT interactions of quarks to those of nucleons and mesons
using the two-flavor baryon chiral perturbation theory (B$\chi$PT) formalism
Gasser  _et al._ (1988); Jenkins and Manohar (1991); Scherer (2003). In order
to realize this, we first organize the LEFT operators into irreducible
representations under the chiral group $SU(2)_{\rm L}^{f}\times SU(2)_{\rm
R}^{f}$ of chiral $u,d$ quarks and then construct the corresponding hadronic
operators using the spurion techniques with the non-perturbative QCD effect
being encoded in the so-called low energy constants (LECs). These LECs may be
extracted by chiral symmetry from other measured processes, or computed in
lattice theory (LQCD), or estimated based on the naive dimensional analysis
Manohar and Georgi (1984); Gavela  _et al._ (2016); Weinberg (1989).

Last we collect all pieces together to express the transition rate as the
function of the Wilson coefficients of effective interactions in the SMEFT and
the LECs. The merit in such an approach is that the uncertainties incurred in
the result may be estimated systematically. Taking the current experimental
limits into consideration, we find the NP scale is pushed around 1-3 TeV,
which is accessible to the future collider searches to complementarily study
the relevant NP. If we further assume these transitions are dominantly
generated by the similar dim-12 SMEFT operators, then the three different
types of transitions are correlated with each other via the weak isospin
symmetry, and the experimental limits on the partial lifetime of one type of
transitions may be translated into stronger limits on that of another type
than their existing experimental lower bounds. In addition, this correlation
helps us to set a lower bound on the lifetime of the experimentally unsearched
decay mode $pp\to e^{+}\tau^{+}$ from that of $pn\to e^{+}\bar{\nu}_{\tau}$,
and we find the partial lifetime is constrained to be $\Gamma^{-1}_{pp\to
e^{+}\tau^{+}}\gtrsim 2\times 10^{34}~{}\rm yr$.

This paper is organized as follows. In section II, we will first list at quark
level the operators for dinucleon and dilepton transitions. We then establish
the basis of dim-12 operators contributing to the dinucleon to dilepton
transitions with $\Delta B=\Delta L=-2$ in the LEFT in subsection II.1, and we
will do a similar job to establish the leading order dim-12 operators
contributing to the same transitions in the SMEFT in subsection II.2. In
subsection II.3, we give tree-level matching relations between LEFT and SMEFT
operators. Section III is devoted to the chiral realization of the six-part of
these dim-12 operators at $\Lambda_{\chi}$. After some brief explanation how
to perform matches to chiral theory from quark level operators in subsection
III.1, in subsections III.2 and III.3, we discuss the chiral irreducible
representation decompositions and matching operators. In section IV, we
calculate the decay rate from the series of EFTs and make predictions for the
sensitivity of such processes for future experimental searches. We summarize
our main results in section V. In Appendices A to H we list the operator basis
in each EFT obtained and chiral decompositions and matching leading to the
main results described in the text.

##  II Operator basis for
$pp\to\ell^{+}\ell^{\prime+},pn\to\ell^{+}\bar{\nu}^{\prime},nn\to\bar{\nu}\bar{\nu}^{\prime}$
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 obtained by Hermitian conjugation. To have
dinucleon to dilepton transitions of $\Delta B=\Delta L=-2$, the quark level
operators first appear at dim 12 and contain six quark fields made out of the
up and down quarks $(u,d)$ and two leptons ($\ell,\nu$). By the Fierz
transformation, these operators can be factorized as the pure quark sectors
multiplied 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:
$pp\to\ell^{+}\ell^{\prime+}$, $pn\to\ell^{+}\bar{\nu}^{\prime}$ and
$nn\to\bar{\nu}\bar{\nu}^{\prime}$, respectively. The corresponding lepton
currents for each type of transitions are defined as follows

$\displaystyle pp\to\ell\ell^{\prime}:~{}~{}$ $\displaystyle
j_{S,\pm}^{\ell\ell^{\prime}}=(\ell^{\rm T}CP_{\pm}\ell^{\prime})\;,$
$\displaystyle j_{V}^{\ell\ell^{\prime},\mu}=(\ell_{R}^{\rm
T}C\gamma^{\mu}\ell_{L}^{\prime})\;,$ $\displaystyle
j_{T,\pm}^{\ell\ell^{\prime},\mu\nu}=(\ell^{\rm
T}C\sigma^{\mu\nu}P_{\pm}\ell^{\prime})\;,$ (1) $\displaystyle
pn\to\ell\bar{\nu}^{\prime}:~{}~{}$ $\displaystyle
j_{S}^{\ell\nu^{\prime}}=(\ell^{\rm T}_{L}C\nu_{L}^{\prime})\;,$
$\displaystyle j_{V}^{\ell\nu^{\prime},\mu}=(\ell^{\rm
T}_{R}C\gamma^{\mu}\nu_{L}^{\prime})\;,$ $\displaystyle
j_{T}^{\ell\nu^{\prime},\mu\nu}=(\ell^{\rm
T}_{L}C\sigma^{\mu\nu}\nu_{L}^{\prime})\;,$ (2) $\displaystyle
nn\to\bar{\nu}\bar{\nu}^{\prime}:~{}~{}$ $\displaystyle
j_{S}^{\nu\nu^{\prime}}=(\nu^{\rm T}_{L}C\nu^{\prime}_{L})\;,$ $\displaystyle
j_{T}^{\nu\nu^{\prime},\mu\nu}=(\nu^{\rm
T}_{L}C\sigma^{\mu\nu}\nu^{\prime}_{L})\;,$ (3)

where $C$ is the charge conjugation matrix satisfying $C^{\rm
T}=C^{\dagger}=-C$ and $C^{2}=-1$, the charged lepton field is denoted by
$\ell,\ell^{\prime}\in\\{e,\mu,\tau\\}$ and the SM left-handed neutrinos by
$\nu_{L},\nu^{\prime}_{L}\in\\{\nu_{e},\nu_{\mu},\nu_{\tau}\\}$.555We do not
include the right-handed neutrinos since they are absent in the SM framework.
The chiral projection operators are abbreviated as $P_{\pm}\equiv
P_{R,L}=(1\pm\gamma_{5})/2$ and therefore
$\ell_{R,L}=P_{R,L}\ell=P_{\pm}\ell$. Here we see that for the diproton and
dineutron transitions $pp\to\ell^{+}\ell^{\prime+}$,
$nn\to\bar{\nu}\bar{\nu}^{\prime}$, the scalar lepton current is symmetric
under the exchange of the two leptons while the tensor lepton current is anti-
symmetric. Especially, for the transitions with identical charged leptons
$pp\to e^{+}e^{+},\mu^{+}\mu^{+}$ the tensor lepton current vanishes, and the
non-vanishing scalar and vector currents can be equivalently parametrized by
the four-component Dirac fields as

$\displaystyle j_{S}$ $\displaystyle=(\ell^{\rm T}C\ell)\;,$ $\displaystyle
j_{S,5}$ $\displaystyle=(\ell^{\rm T}C\gamma_{5}\ell)\;,$ $\displaystyle
j_{V,5}^{\mu}$ $\displaystyle=(\ell^{\rm T}C\gamma^{\mu}\gamma_{5}\ell)\;,$
(4)

and the vector current $(\ell^{\rm T}C\gamma^{\mu}\ell)=0$ from the same
reason as the tensor current.

For the dinucleon to dilepton transitions, from the above classification, we
may first write down a minimal hadron level effective Lagrangian consisting of
nucleon currents $N^{\rm T}\Gamma N^{\prime}$ ($N,N^{\prime}\in\\{p,n\\}$) as
well as the lepton currents in Eqs. (1-3),

$\displaystyle{\cal L}_{NN^{\prime}}=\sum_{a}C^{(pp)}_{a}{\cal
O}^{(pp)}_{a}+\sum_{b}C^{(pn)}_{b}{\cal
O}^{(pn)}_{b}+\sum_{c}C^{(nn)}_{c}{\cal O}^{(nn)}_{c}\;,$ (5)

where $a,b,c$ are just labels to distinguish different operators, and
$C^{(NN^{\prime})}_{a,b,c}$ are the Wilson coefficients associated with the
relevant hadronic operators ${\cal O}^{(NN^{\prime})}_{a,b,c}$. Those
operators ${\cal O}^{(NN^{\prime})}_{i}$ are dim-6 and composed of dinucleon
and the above lepton currents, and can be written as

$\displaystyle pp\to\ell^{+}\ell^{\prime+}:~{}$ $\displaystyle{{\cal
O}}^{(pp)S}_{L}=(p^{\rm T}Cp)(\ell^{\rm T}_{L}C\ell^{\prime}_{L})\;,$
$\displaystyle{{\cal O}}^{(pp)S}_{5L}=(p^{\rm T}C\gamma_{5}p)(\ell^{\rm
T}_{L}C\ell^{\prime}_{L})\;,$ $\displaystyle{{\cal O}}^{(pp)S}_{R}=(p^{\rm
T}Cp)(\ell^{\rm T}_{R}C\ell^{\prime}_{R})\;,$ $\displaystyle{{\cal
O}}^{(pp)S}_{5R}=(p^{\rm T}C\gamma_{5}p)(\ell^{\rm
T}_{R}C\ell^{\prime}_{R})\;,$ $\displaystyle{{\cal O}}^{(pp)V}=(p^{\rm
T}C\gamma_{\mu}\gamma_{5}p)(\ell^{\rm
T}_{R}C\gamma^{\mu}\ell^{\prime}_{L})\;,$ (6)
$\displaystyle{pn\to\ell^{+}\bar{\nu}^{\prime}}:~{}$ $\displaystyle{{\cal
O}}^{(pn)S}_{L}=(p^{\rm T}Cn)(\ell^{\rm T}_{L}C\nu^{\prime}_{L})\;,$
$\displaystyle{{\cal O}}^{(pn)S}_{5L}=(p^{\rm T}C\gamma_{5}n)(\ell^{\rm
T}_{L}C\nu_{L}^{\prime})\;,$ $\displaystyle{{\cal O}}^{(pn)V}_{L}=(p^{\rm
T}C\gamma_{\mu}n)(\ell^{\rm T}_{R}C\gamma^{\mu}\nu_{L}^{\prime})\;,$
$\displaystyle{{\cal O}}^{(pn)V}_{5L}=(p^{\rm
T}C\gamma_{\mu}\gamma_{5}n)(\ell^{\rm T}_{R}C\gamma^{\mu}\nu_{L}^{\prime})\;,$
$\displaystyle{{\cal O}}^{(pn)T}=(p^{\rm T}C\sigma_{\mu\nu}n)(\ell^{\rm
T}_{L}C\sigma_{\mu\nu}\nu_{L}^{\prime})\;,$ (7)
$\displaystyle{nn\to\bar{\nu}\bar{\nu}^{\prime}}:~{}$ $\displaystyle{{\cal
O}}^{(nn)S}_{L}=(n^{\rm T}Cn)(\nu^{\rm T}_{L}C\nu_{L}^{\prime})\;,$
$\displaystyle{{\cal O}}^{(nn)S}_{5L}=(n^{\rm T}C\gamma_{5}n)(\nu^{\rm
T}_{L}C\nu_{L}^{\prime})\;,$ (8)

where there are no operators with a vector or tensor diproton/dineutron
current since such currents vanish, i.e., $(N^{\rm T}C\gamma_{\mu}N)=(N^{\rm
T}C\sigma_{\mu\nu}N)=(N^{\rm T}C\sigma_{\mu\nu}\gamma_{5}N)=0$ for
$N\in\\{p,n\\}$.

Since the dinucleon fields in the above operators must originate from the
$u,d$ quarks, it is thus necessary to start from the classification of
relevant operators at the quark level. There are two approaches in
constructing relevant quark level operators responsible for dinucleon to
dilepton transitions. One is to use effective degrees of freedom below the
electroweak scale to obtain all relevant operators respecting $SU(3)_{\rm
C}\times U(1)_{\rm EM}$, this is the LEFT approach. And the other is to use
effective degrees of freedom of the SM to obtain all relevant operators
respecting the SM gauge symmetry $SU(3)_{\rm C}\times SU(2)_{\rm L}\times
U(1)_{\rm Y}$, the SMEFT. We will obtain the operators in both approaches.

In both the LEFT and SMEFT approaches the color $SU(3)_{\rm C}$ symmetry must
be respected, the six quarks in all operators obtained must form color
singlets. We denote a general six-quark field configuration as

$\displaystyle{\cal
O}^{ijklmn}=q^{i}_{I}q^{j}_{J}q^{k}_{K}q^{l}_{L}q^{m}_{M}q^{n}_{N}\;,$ (9)

where the superscripts $\\{i,j,k,l,m,n\\}$ are the color indices in
fundamental representation of $SU(3)_{\rm C}$ while the subscripts
$\\{I,J,K,L,M,N\\}$ encode the flavor and chiral information for each quark
field. To form a color invariant operator, the color indices must be
contracted by a color tensor $T_{ijklmn}$ such that ${\cal
O}^{ijklmn}T_{ijklmn}$ is invariant under $SU(3)_{\rm C}$. This color symmetry
can be achieved by contracting the color indices $\\{i,j,k,l,m,n\\}$ in the
quark fields with the following five independent color tensors

$\displaystyle T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}$
$\displaystyle=\epsilon_{ikm}\epsilon_{jln}+\epsilon_{ikn}\epsilon_{jlm}+\epsilon_{ilm}\epsilon_{jkn}+\epsilon_{iln}\epsilon_{jkm}\;,$
$\displaystyle T^{SAA}_{\\{ij\\}[kl][mn]}$
$\displaystyle=\epsilon_{imn}\epsilon_{jkl}+\epsilon_{ikl}\epsilon_{jmn}\;,\quad
T^{SAA}_{\\{kl\\}[mn][ij]}=\epsilon_{ijk}\epsilon_{mnl}+\epsilon_{ijl}\epsilon_{mnk}\;,$
$\displaystyle T^{SAA}_{\\{mn\\}[ij][kl]}$
$\displaystyle=\epsilon_{ijm}\epsilon_{kln}+\epsilon_{ijn}\epsilon_{klm}\;,\quad
T^{AAA}_{[ij][kl][mn]}=\epsilon_{ijm}\epsilon_{kln}-\epsilon_{ijn}\epsilon_{klm}\;.$
(10)

We put the details for the color tensor construction and their subtleties in
Appendix A.

###  II.1 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
$(e,\mu,\tau,\nu_{e},\nu_{\mu},\nu_{\tau})$, and the effective interactions
are governed by the higher dimensional local operators ${\cal Q}_{i}^{d}$
built out of those fields and satisfying the symmetry $SU(3)_{\rm C}\times
U(1)_{\rm EM}$. The LEFT Lagrangian ${\cal L}_{\rm LEFT}$ is organized in
terms of the canonical mass dimension of the local operators

$\displaystyle{\cal L}_{\rm LEFT}={\cal L}_{{\rm dim}\leq 4}+\sum_{{\dim
5},i}{\hat{C}_{5,i}\over\Lambda}{\cal Q}^{5}_{i}+\sum_{{\dim
6},i}{\hat{C}_{6,i}\over\Lambda^{2}}{\cal Q}^{6}_{i}+\cdots+\sum_{{\dim
12},i}{\hat{C}_{12,i}\over\Lambda^{8}}{\cal Q}^{12}_{i}+\cdots\;,$ (11)

where ${\cal L}_{{\rm dim}\leq 4}$ is the renormalizable terms, and the Wilson
coefficients $\hat{C}_{d,i}$ together with the heavy scale $\Lambda$ encode
informations about the presumed fundamental physics. The systematic
enumeration of operator bases up to dim 9 have been figured out in Jenkins
_et al._ (2018); Liao  _et al._ (2020a); Li  _et al._ (2020); Murphy (2020).
In our case, the relevant operators first appear at dim 12 and can be
parametrized as the product of six-quark sectors and a proper lepton current
given in Eqs. (1-3).

For the quark sectors, we can repeatedly apply the Fierz identities (FI) to
reach as many quark scalar bilinear currents as possible. In this way, by the
Lorentz symmetry, for the operators with a lepton scalar current their quark
sectors can be factorized as three quark scalar currents, and for the
operators with a lepton vector (tensor) current their quark sectors can be
factorized as two quark scalar currents along with a single quark vector
(tensor) current. As a non-trivial example, at the end of Appendix C, we will
employ the FIs to show how the operators with a lepton tensor current in quark
scalar-vector-vector bilinear structure are shifted into those with a scalar-
scalar-tensor structure outlined here. Lastly, the color $SU(3)_{\rm C}$
invariance can be done by contracting the free quark color indices using the
independent color tensors constructed earlier. In doing so, one should be
careful with operators containing several identical quark fields since the
color relations in Eq. (A.4) and the FIs in Appendix C may further restrict
their independency. Combining the above points and excluding potential
redundant operators, and for $n_{f}$ ( $n_{f}=3$ in the real case) flavors of
lepton fields $(\ell,\nu)$, the final bases of the dim-12 operators mediating
the transitions
$pp\to\ell^{+}\ell^{\prime+},pn\to\ell^{+}\bar{\nu}^{\prime},nn\to\bar{\nu}\bar{\nu}^{\prime}$
are summarized one by one as follows:

$\scriptscriptstyle\blacksquare$ Dim-12 operators contributing to
$pp\to\ell^{+}\ell^{\prime+}$
One can attach different lepton currents to operators already formed by six
quarks to form the operators. We will discuss for each type in the following.
For a scalar current $j_{S,\pm}^{\ell\ell^{\prime}}$, one just attaches it to
some color singlet and Lorentz scalar six quark operators. An example of this
class of operators, the ${\cal Q}_{1LLL,a}^{(pp)S,\pm}$, is given in the
following

$\displaystyle{\cal Q}_{1LLL,a}^{(pp)S,\pm}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}Cd_{L}^{n})j_{S,\pm}^{\ell\ell^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;.$

We find that there are 28 independent operators without referring to lepton
flavors which are listed in (B.1) in Appendix B.

For the operators with a vector lepton current
$j_{V}^{\ell\ell^{\prime},\mu}$, there are 19 independent operators, and for
the operators with a tensor lepton current $j_{T}^{\ell\ell^{\prime},\mu\nu}$,
we find there are 16 independent operators. Therefore there are total $63$
operators which we give them in Eqs. (B.1-B.3) in Appendix B. For $n_{f}$
flavors of charged leptons, there are $41n_{f}^{2}+6n_{f}$ operators.

As a cross-check, we also confirmed our above results (and following ones) by
the Hilbert series method Hanany  _et al._ (2011); Lehman and Martin (2015);
Henning  _et al._ (2016, 2017). In Appendix B, by a non-trivial example, we
also show how to reduce the redundant operators using the Fierz and Schouten
identities.

When restricting to the same flavor leptons with $\ell^{\prime}=\ell$, we find
there are $28+19=47$ independent operators from the scalar and vector lepton
currents, because the tensor lepton current vanishes for identical fields.
Besides the transitions $pp\to e^{+}e^{+},\mu^{+}\mu^{+}$ , such operators
also contribute to the ${\rm H}{\rm-}\bar{\rm H}$ oscillation process and have
been enumerated long ago by Caswell, Milutinovic and Senjanovic in Ref.
Caswell  _et al._ (1983). However, we find 13 out of total 60 operators in
their counting are redundant and all of them belong to the class with the
vector lepton current $j_{V}^{\ell\ell,\mu}$. In Appendix C, we show
explicitly the redundancy of the basis in Caswell  _et al._ (1983) and give
the correspondence of their basis (after excluding the redundant ones) with
the basis in Eqs. (B.1, B.2).

$\scriptscriptstyle\blacksquare$ Dim-12 operators contributing to
$pn\to\ell^{+}\nu^{\prime}$
For the operators with a scalar current $j_{S}^{\ell\nu^{\prime}}$, we find
there are 14 independent operators without counting lepton flavors. For the
operators with a vector current $j_{V}^{\ell\nu^{\prime},\mu}$, there are 24
independent operators. And for the operators with a tensor current
$j_{T}^{\ell\nu^{\prime},\mu\nu}$, there are 13 independent operators.

In total, there are 51 independent operators for the transition
$pn\to\ell^{+}\bar{\nu}^{\prime}$ which are listed in Eqs. (B.4-B.6) in
Appendix B. For $n_{f}$ flavors of charged leptons and neutrinos, there are
$51n_{f}^{2}$ operators. These operators are also responsible for the
conversions $e^{-}p\to\bar{\nu}\bar{n}$ and $e^{-}n\to\bar{\nu}\bar{p}$ in the
electron-deuteron scattering. In addition, we find they can contribute to the
unique neutron decay mode with the baryon number being changed by two units
$n\to\bar{p}e^{+}\bar{\nu}$ He and Ma (2021).

$\scriptscriptstyle\blacksquare$ Dim-12 operators contributing to
$nn\to\bar{\nu}\bar{\nu}^{\prime}$
For the operators with a scalar current $j_{S}^{\nu\nu^{\prime}}$, we find
there are 14 independent operators. And for the operators with a tensor
neutrino current $j_{T}^{\nu\nu^{\prime},\mu\nu}$, 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 $11n_{f}^{2}+3n_{f}$ operators. Excluding the scalar neutrino
currents, one can easily identity the remaining 7 quark operators in Eq. (B.7)
plus their parity partners are just the 14 operators contributing to the
neutron-antineutron oscillation Caswell  _et al._ (1983).

###  II.2 SMEFT completions

To connect with the NP scenarios at a higher scale than $\Lambda_{\rm EW}$,
the SMEFT can serve 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 is a good starting point for the
systematic EFT analysis of low energy processes. In this section, we consider
the leading order SMEFT completions for the $\Delta B=\Delta L=-2$ dinucleon
to dilepton transition operators discussed above. It happens that the relevant
SMEFT operators also first appear at dim 12 at leading order, and contain six
quark fields and two lepton fields. By the similar logic as the construction
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 generation of quark fields but without
restriction for the generation of the lepton fields. We denote the SM left-
handed lepton and quark doublet fields as $L(1,2,-1/2)$, $Q(3,2,1/6)$ and
right-handed up-type quark, down-type quark and charged lepton fields as
$u_{R}(3,1,2/3)$, $d_{R}(3,1,-1/3)$, $e_{R}(1,1,-1)$. We employ the front
Latin letters $(a,b,c,d,e,f)$ for the $SU(2)_{\rm L}$ indices and the middle
ones $(i,j,k,l,m,n)$ for the color $SU(3)_{\rm C}$ indices in fundamental
representations, respectively. Similar to the classification of operators in
the LEFT, we classify the relevant dim-12 SMEFT operators in terms of the
lepton currents, and the final results are summarized in Appendix D.

Here we briefly comment the procedures to reach the independent operators in
Appendix D. 1). Based on the $U(1)_{\rm Y}$ invariance, one can easily
identify the allowed field configurations with six quarks and two leptons. 2).
For each field configuration, we first use the Fierz transformation to fix the
Lorentz structure of the operator so that it takes the quark-lepton factorized
form ${\cal O}_{q}\times j_{L}$ in which the lepton current $j_{L}$ can be
either scalar, vector, or tensor type. For the scalar/vector/tensor lepton
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 as we did in section II.1. 3).
Followed by step 2), we consider the electroweak $SU(2)_{\rm L}$ invariance
which can be done by implementing the contractions using the rank-2 Levi-
Civita tensor $\epsilon_{ab}$. In doing so, the SI identity
$\epsilon_{ab}\epsilon_{cd}=\epsilon_{ac}\epsilon_{bd}+\epsilon_{ad}\epsilon_{cb}$
has to be considered carefully for the multiple $SU(2)_{\rm L}$ contractions
so as to reduce the redundant operators. 4). Last, the color $SU(3)_{\rm C}$
invariance can be done by contracting the free color indices using the
independent color tensors discussed in section A and Appendix A. If there are
multiple identical quark fields, the color relations in Eq. (A.4) and the FIs
in Appendix C must be taken into account to reduce the operators into the
minimal basis given in Appendix D. 5). We also count the number of independent
operators in each configuration using the Hilbert series method Henning  _et
al._ (2017) and confirmed our result.

The following is an example of using SM building blocks to build the SMEFT
dinucleon to dilepton operators which are different from those in the LEFT,

$\displaystyle{\cal O}_{Q^{6}L^{2}}^{S,(S)}$ $\displaystyle=(Q^{i{\rm
T}}_{a}CQ^{j}_{b})(Q^{k{\rm T}}_{c}CQ^{l}_{d})(Q^{m{\rm
T}}_{e}CQ^{n}_{f})(L^{\rm
T}_{g}CL_{h}^{\prime})\epsilon_{ab}\epsilon_{cd}\epsilon_{eg}\epsilon_{fh}T^{SAA}_{\\{mn\\}[kl][ij]}\;.$
(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

$\displaystyle{\cal O}_{Q^{6}L^{2}}^{S,(S)}$ $\displaystyle=4(u^{i{\rm
T}}_{L}Cu^{j}_{L})(u^{k{\rm T}}_{L}Cd^{l}_{L})(u^{m{\rm
T}}_{L}Cd^{n}_{L})T^{SAA}_{\\{ij\\}[kl][mn]}j_{S,-}^{\ell\ell^{\prime}}$
$\displaystyle-8(u^{i{\rm T}}_{L}Cd^{j}_{L})(u^{k{\rm
T}}_{L}Cd^{l}_{L})(u^{m{\rm
T}}_{L}Cd^{n}_{L})T^{SAA}_{\\{ij\\}[kl][mn]}j_{S}^{\ell\nu^{\prime}}$
$\displaystyle+4(d^{i{\rm T}}_{L}Cd^{j}_{L})(u^{k{\rm
T}}_{L}Cd^{l}_{L})(u^{m{\rm
T}}_{L}Cd^{n}_{L})T^{SAA}_{\\{ij\\}[kl][mn]}j_{S}^{\nu\nu^{\prime}}$
$\displaystyle=4{\cal O}_{1LLL,b}^{(pp)S,-}-8{\cal O}_{1LLL,b}^{(pn)S}+4{\cal
O}_{1LLL,b}^{(nn)S}\;.$ (13)

Therefore it is expected that the SMEFT approach will have less independent
operators than that can be constructed in the LEFT approach. Without counting
the lepton flavors, for operators with a scalar lepton current, we find there
are 12 independent operators. For operators with a vector lepton current, we
find there are 7 independent operators. And for operators with a tensor lepton
current, we find there are 10 independent operators. In total, there are 29
operators which are listed in Appendix D. For $n_{f}$ flavors of leptons there
are $18n_{f}^{2}+n_{f}$ operators. Except the sub-GeV scale dinucleon to
dilepton processes studied in this work, these SMEFT operators are crucial for
the model-independent study of the $\Delta B=\Delta L=2$ signals on colliders,
for example, the process $pp\to\ell^{+}\ell^{\prime+}+4~{}{\rm jets}$ at LHC
Bramante  _et al._ (2015) and $e^{-}p\to\ell^{+}+5~{}{\rm jets}$ in the future
electron-proton colliders like LHeC.

In literature, Refs. Girmohanta and Shrock (2020a); Girmohanta (2020) also
provide a bunch of dim-12 operators contributing to dinucleon to dilepton
transitions in the SMEFT. We find the operators given in Girmohanta and Shrock
(2020a) are neither complete nor independent as a basis. Specifically, the 28
operators666Ref. Girmohanta and Shrock (2020a) also considered operators with
the SM singlet right-handed neutrinos, here we only focus on the SMEFT
subsets. listed in Girmohanta and Shrock (2020a) can be covered by 21
operators in our counting and 8 of them are redundant. In addition, the 9
operators ${\cal O}_{u^{3}d^{3}L^{2}1,2}^{S,(A)}$, ${\cal
O}_{u^{2}d^{2}Q^{2}L^{2}2}^{S,(A)}$, ${\cal
O}_{u^{2}d^{2}Q^{2}L^{2}2}^{T,(A)}$, ${\cal O}_{udQ^{4}L^{2}3}^{T,(A)}$,
${\cal O}_{Q^{6}L^{2}}^{T,(S)}$, ${\cal O}_{u^{4}d^{2}e^{2}}^{T,(A)}$, ${\cal
O}_{u^{3}dQ^{2}e^{2}1,2}^{T,(A)}$ in our basis are missed in their list. In
Appendix E, we translate their operators as linear combinations of our above
operators so that one can easily recognize the redundancy and incompleteness
of the operators in Girmohanta and Shrock (2020a).

###  II.3 Relations between SMEFT and LEFT operators

As already mentioned earlier that some of the SMEFT operators contain several
LEFT operators, i.e. ${\cal O}_{Q^{6}L^{2}}^{S,(S)}=4{\cal
O}_{1LLL,b}^{(pp)S,-}-8{\cal O}_{1LLL,b}^{(pn)S}+4{\cal O}_{1LLL,b}^{(nn)S}$.
The SMEFT approach will have less independent operators than that can be
constructed in the LEFT approach at the same order. To have better idea on how
these operators are related to each other, in Tab. 2, we perform a tree-level
matching of the dim-12 SMEFT operators listed in Appendix D to the dim-12 LEFT
operators listed in Appendix B at the electroweak scale $\Lambda_{\rm EW}$. We
have approximated the CKM factor $V_{ud}\approx 1$ arising from the mismatch
of the flavor and mass eigenstates of the left-handed down quark $d_{L}$. From
Tab. 2, it is obvious that the operators with the singlet charged lepton
scalar/tensor current $(e_{R}^{\rm T}C\Gamma e_{R}^{\prime})$ can only
exclusively contribute to the transition $pp\to\ell^{+}\ell^{\prime+}$, while
the operators with an anti-symmetric scalar lepton current ‘S,(A)’ and the
operators with a symmetric tensor current ‘T,(S)’ can only contribute to the
transition $pn\to\ell^{+}\nu^{\prime}$. The remaining operators with a
symmetric scalar lepton current ‘S,(S)’ or with an anti-symmetric tensor
current ‘T,(A)’ could contribute to both the three different transition
channels. Last but not least, one must be careful that the SMEFT Wilson
coefficients with a superscript ‘(A)’ vanish for identical lepton fields since
the relevant operators are anti-symmetric for the two lepton fields.

From Appendix D and Tab. 2, we see there are many operators which can be
constructed in the LEFT, but not in the SMEFT at leading dim-12 order, for
instance, operators ${\cal Q}_{4LLR}^{(pp)S,\pm}$, ${\cal Q}_{4LLR}^{(pn)S}$
and ${\cal Q}_{4LLR}^{(nn)S}$ for the three channels respectively. Those
operators are not $SU(2)_{\rm L}\times U(1)_{\rm Y}$ invariant and can only be
generated by the higher dim-14 and/or dim-16 SMEFT $SU(2)_{\rm L}\times
U(1)_{\rm Y}$ invariant operators consisting of dim-12 fermion part
$(qqqqqqll)$ together with additional Higgs doublets. The physical effects
from such operators are suppressed with additional factors like
$v^{2}/\Lambda_{\rm NP}^{2}$ and/or $v^{4}/\Lambda_{\rm NP}^{4}$ relative to
the dim-12 SMEFT operators and will be neglected in our numerical analysis.

SMEFT operators | $pp\to\ell\ell^{\prime}$ | $pn\to\ell\bar{\nu}^{\prime}$ | $nn\to\bar{\nu}\bar{\nu}^{\prime}$
---|---|---|---
${\cal O}_{u^{3}d^{3}L^{2}1}^{S,(A)}$ | - | $C_{1RRR,a}^{(pn)S}=-2C_{u^{3}d^{3}L^{2}1}^{S,(A)}$ | -
${\cal O}_{u^{3}d^{3}L^{2}2}^{S,(A)}$ | - | $C_{1RRR,b}^{(pn),}=-2C_{u^{3}d^{3}L^{2}2}^{S,(A)}$ | -
${\cal O}_{u^{2}d^{2}Q^{2}L^{2}1}^{S,(S)}$ | $C_{3RRL,a}^{(pp)S,-}=C_{u^{2}d^{2}Q^{2}L^{2}1}^{S,(S)}$ | $C_{3RRL,a}^{(pn)S}=-2C_{u^{2}d^{2}Q^{2}L^{2}1}^{S,(S)}$ | $C_{3RRL,a}^{(nn)S}=C_{u^{2}d^{2}Q^{2}L^{2}1}^{S,(S)}$
${\cal O}_{u^{2}d^{2}Q^{2}L^{2}2}^{S,(A)}$ | - | $C_{3RRL,b}^{(pn)S}=-4C_{u^{2}d^{2}Q^{2}L^{2}2}^{S,(A)}$ | -
${\cal O}_{u^{2}d^{2}Q^{2}L^{2}3}^{S,(S)}$ | $C_{3RRL,b}^{(pp)S,-}=C_{u^{2}d^{2}Q^{2}L^{2}3}^{S,(S)}$ | $C_{3RRL,c}^{(pn)S}=-2C_{u^{2}d^{2}Q^{2}L^{2}3}^{S,(S)}$ | $C_{3RRL,b}^{(nn)S}=C_{u^{2}d^{2}Q^{2}L^{2}3}^{S,(S)}$
${\cal O}_{udQ^{4}L^{2}1}^{S,(A)}$ | - | $C_{3LLR,c}^{(pn)S}=-8C_{udQ^{4}L^{2}1}^{S,(A)}$ | -
${\cal O}_{udQ^{4}L^{2}2}^{S,(S)}$ | $C_{2LLR,b}^{(pp)S,-}=2C_{udQ^{4}L^{2}2}^{S,(S)}$ | $C_{3LLR,b}^{(pn)S}=-4C_{udQ^{4}L^{2}2}^{S,(S)}$ | $C_{2LLR,b}^{(nn)S}=2C_{udQ^{4}L^{2}2}^{S,(S)}$
${\cal O}_{Q^{6}L^{2}}^{S,(S)}$ | $C_{1LLL,b}^{(pp)S,-}=4C_{Q^{6}L^{2}}^{S,(S)}$ | $C_{1LLL,b}^{(pn)S}=-8C_{Q^{6}L^{2}}^{S,(S)}$ | $C_{1LLL,b}^{(nn)S}=4C_{Q^{6}L^{2}}^{S,(S)}$
${\cal O}_{u^{4}d^{2}e^{2}1}^{S,(S)}$ | $C_{1RRR,a}^{(pp)S,+}=C_{u^{4}d^{2}e^{2}1}^{S,(S)}$ | - | -
${\cal O}_{u^{4}d^{2}e^{2}2}^{S,(S)}$ | $C_{1RRR,b}^{(pp)S,+}=C_{u^{4}d^{2}e^{2}2}^{S,(S)}$ | - | -
${\cal O}_{u^{3}dQ^{2}e^{2}}^{S,(S)}$ | $C_{2RRL,b}^{(pp)S,+}=2C_{u^{3}dQ^{2}e^{2}}^{S,(S)}$ | - | -
${\cal O}_{u^{2}Q^{4}e^{2}}^{S,(S)}$ | $C_{3LLR,b}^{(pp)S,+}=4C_{u^{2}Q^{4}e^{2}}^{S,(S)}$ | - | -
${\cal O}_{u^{3}d^{2}QeL1}^{V}$ | $C_{1RR,a}^{(pp)V}=-C_{u^{3}d^{2}QeL1}^{V}$ | $C_{1RR,a}^{(pn)V}=C_{u^{3}d^{2}QeL1}^{V}$ | -
${\cal O}_{u^{3}d^{2}QeL2}^{V}$ | $C_{1RR,b}^{(pp)V}=C_{u^{3}d^{2}QeL2}^{V}$ | $C_{1RR,b}^{(pn)V}=-C_{u^{3}d^{2}QeL2}^{V}$ | -
${\cal O}_{u^{3}d^{2}QeL3}^{V}$ | $C_{1RR,c}^{(pp)V}=-C_{u^{3}d^{2}QeL3}^{V}$ | $C_{1RR,c}^{(pn)V}=C_{u^{3}d^{2}QeL3}^{V}$ | -
${\cal O}_{u^{2}dQ^{3}eL1}^{V}$ | $C_{4LR,c}^{(pp)V}=2C_{u^{2}dQ^{3}eL1}^{V}$ | $C_{4RL,b}^{(pn)V}=-2C_{u^{2}dQ^{3}eL1}^{V}$ | -
${\cal O}_{u^{2}dQ^{3}eL2}^{V}$ | $C_{4LR,d}^{(pp)V}=2C_{u^{2}dQ^{3}eL2}^{V}$ | $C_{4RL,d}^{(pn)V}=2C_{u^{2}dQ^{3}eL2}^{V}$ | -
${\cal O}_{u^{2}dQ^{3}eL3}^{V}$ | $C_{4LR,e}^{(pp)V}=-2C_{u^{2}dQ^{3}eL3}^{V}$ | $C_{4RL,e}^{(pn)V}=-2C_{u^{2}dQ^{3}eL3}^{V}$ | -
${\cal O}_{uQ^{5}eL}^{V}$ | $C_{1LL,c}^{(pp)V}=4C_{uQ^{5}eL}^{V}$ | $C_{2LL,c}^{(pn)V}=-4C_{uQ^{5}eL}^{V}$ | -
${\cal O}_{u^{2}d^{2}Q^{2}L^{2}1}^{T,(S)}$ | - | $C_{3RRL,a}^{(pn)T}=-4C_{u^{2}d^{2}Q^{2}L^{2}1}^{T,(S)}$ | -
${\cal O}_{u^{2}d^{2}Q^{2}L^{2}2}^{T,(A)}$ | $C_{3RRL}^{(pp)T,-}=C_{u^{2}d^{2}Q^{2}L^{2}2}^{T,(A)}$ | $C_{3RRL,b}^{(pn)T}=-2C_{u^{2}d^{2}Q^{2}L^{2}2}^{T,(A)}$ | $C_{3RRL}^{(nn)T}=-C_{u^{2}d^{2}Q^{2}L^{2}2}^{T,(A)}$
${\cal O}_{u^{2}d^{2}Q^{2}L^{2}3}^{T,(S)}$ | - | $C_{3RRL,c}^{(pn)T}=-4C_{u^{2}d^{2}Q^{2}L^{2}3}^{T,(S)}$ | -
${\cal O}_{udQ^{4}L^{2}1}^{T,(A)}$ | $C_{2LLR,b}^{(pp)T,-}=2C_{udQ^{4}L^{2}1}^{T,(A)}$ | $C_{3LLR,c}^{(pn)T}=-4C_{udQ^{4}L^{2}1}^{T,(A)}$ | $C_{2LLR,b}^{(nn)T}=-2C_{udQ^{4}L^{2}1}^{T,(A)}$
${\cal O}_{udQ^{4}L^{2}2}^{T,(S)}$ | - | $C_{3LLR,b}^{(pn)T}=-8C_{udQ^{4}L^{2}2}^{T,(S)}$ | -
${\cal O}_{udQ^{4}L^{2}3}^{T,(A)}$ | $C_{2LLR,c}^{(pp)T,-}=-2C_{udQ^{4}L^{2}3}^{T,(A)}$ | $C_{3LLR,d}^{(pn)T}=-4C_{udQ^{4}L^{2}3}^{T,(A)}$ | $C_{2LLR,c}^{(nn)T}=-2C_{udQ^{4}L^{2}3}^{T,(A)}$
${\cal O}_{Q^{6}L^{2}}^{T,(S)}$ | - | $C_{1LLL,b}^{(pn)T}=-16C_{Q^{6}L^{2}}^{T,(S)}$ | -
${\cal O}_{u^{4}d^{2}e^{2}}^{T,(A)}$ | $C_{1RRR}^{(pp)T,+}=C_{u^{4}d^{2}e^{2}}^{T,(A)}$ | - | -
${\cal O}_{u^{3}dQ^{2}e^{2}1}^{T,(A)}$ | $C_{2RRL,a}^{(pp)T,+}=2C_{u^{3}dQ^{2}e^{2}1}^{T,(A)}$ | - | -
${\cal O}_{u^{3}dQ^{2}e^{2}2}^{T,(A)}$ | $C_{2RRL,c}^{(pp)T,+}=2C_{u^{3}dQ^{2}e^{2}2}^{T,(A)}$ | - | -

Table 2: The SMEFT dinucleon to dilepton operators and their matching onto the
LEFT at the $\Lambda_{\rm EW}$. Where the notation for the Wilson coefficients
is similar to the corresponding operators with the replacement of ${\cal O}$
by $C$, e.g., $C_{u^{3}dQ^{2}e^{2}2}^{T,(A)}$ for ${\cal
O}_{u^{3}dQ^{2}e^{2}2}^{T,(A)}$, We do not show the explicit flavors of
leptons in the above matching but can be easily recognized through the
corresponding operators.

##  III Chiral realizations

###  III.1 Some basics of Chiral matching

After establishing the operator basis for dinucleon to dilepton transitions in
the LEFT and SMEFT, the next step is to calculate the transition matrix
elements and the decay rates. However, the hadronic matrix elements between
the initial dinucleon state and the QCD vacuum are not a trivial task due to
their non-perturbative QCD nature. In order to obtain those matrix elements
with a controllable uncertainty, fortunately, one can employ the successful
effective chiral perturbation theory of the low energy QCD and the spurion
field techniques to shift the quark level interactions into those interactions
among hadrons and leptons.

In the QCD sector, the approximate chiral symmetry $G_{\chi}=SU(2)_{\rm
L}^{f}\times SU(2)_{R}^{f}$ of the two-flavor QCD Lagrangian under the limit
of massless up and down quarks is spontaneously broken into its isospin
subgroup $SU(2)_{\rm V}$ by the quark condensation $\langle\bar{q}q\rangle$ at
the scale $\Lambda_{\chi}$ .777Usually the strange quark $s$ can also be
included in this framework to consider the larger group breaking pattern
$SU(3)_{L}^{f}\times SU(3)_{R}^{f}\to SU(3)_{V}$ Gasser and Leutwyler (1984,
1985). For our purpose, it is enough to only focus on the two-flavor case in
which the chiral symmetry breaking effect is relatively smaller. The
interaction of the resultant pseudo Nambu-Goldstone pion fields at low energy
($p<\Lambda_{\chi}$) is described by the $\chi$PT which inherits the QCD
chiral symmetry Gasser and Leutwyler (1984, 1985), and the baryon extended
$\chi$PT termed as B$\chi$PT is our main focus in this section. The
(B)$\chi$PT Lagrangian is organized in terms of the power of the soft momentum
$p$ relative to $\Lambda_{\chi}$. Introducing the proper external sources
transforming under the chiral group, the global chiral symmetry can be
promoted to be a local one. Therefore, the (B)$\chi$PT Ward identities can be
easily formulated and the interaction of hadrons with other light particles
such as leptons and photon can be included. In the following we will use the
two-flavor B$\chi$PT formalism and the spurion field techniques to construct
an equivalent effective chiral Lagrangian for the dim-12 interactions in
section II.1.

Before performing the detailed non-perturbative matching for the LEFT
interactions, we can expect the six-quark part of those dim-12 operators will
be transformed into proper nucleon current together with pions and
derivatives. For our purpose of capturing the leading order contributions to
the dinucleon to dilepton transitions, it is enough to consider the dim-6
terms composed of a nucleon bilinear current and a lepton bilinear current
without any pions and derivatives in Eqs. (6-8). The hadron level Wilson
coefficients $C^{(NN^{\prime})}_{a,b,c}$ will be determined below by the
chiral matching to the quark level operators. Once these Wilson coefficients
are obtained, it is straightforward to obtain the transition amplitudes and
henceforth the decay rates, which will be postponed in section IV for a
detailed analysis. Now we turn to the chiral matching and find the
relationship between the Wilson coefficients of the operators in Eqs. (6-8)
and those in the LEFT/SMEFT.

We start from the basics of the $\chi$PT. For the light two-flavor quarks
$q=(u,~{}d)^{\rm T}$, the QCD-like Lagrangian with extended external sources
is parametrized as

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{QCD}}^{M=0}+\overline{q_{L}}l_{\mu}\gamma^{\mu}q_{L}+\overline{q_{R}}r_{\mu}\gamma^{\mu}q_{R}-\left[\overline{q_{R}}(s-ip)q_{L}-\overline{q_{R}}(t_{l}^{\mu\nu}\sigma_{\mu\nu})q_{L}+\text{h.c.}\right]\;,$
(14)

where the flavor space $2\times 2$ matrices
$\\{l_{\mu}=l_{\mu}^{\dagger},~{}r_{\mu}=r_{\mu}^{\dagger},~{}s=s^{\dagger},~{}p=p^{\dagger},~{}t_{r}^{\mu\nu}=t_{l}^{\mu\nu\dagger}\\}$
are the external sources related to the corresponding quark currents. Under
the global chiral transformation $q_{L}\to\hat{L}q_{L}$ and
$q_{R}\rightarrow\hat{R}q_{R}$ with $(\hat{L},\hat{R})\in G_{\chi}$, the pure
QCD part $\mathcal{L}_{\text{QCD}}^{M=0}$ is invariant. The introduction of
the external sources with proper transformation properties can promote the
global chiral symmetry to be a local one. In this way, the whole Lagrangian
$\mathcal{L}$ can be made invariant under the local chiral transformation
$q_{L}\to L(x)q_{L}\equiv Lq_{L}$ and $q_{R}\rightarrow R(x)q_{R}\equiv
Rq_{R}$ together with the following transformations of the external sources

$\displaystyle\chi\to R\chi L^{\dagger}\;,$ $\displaystyle l_{\mu}\to
Ll_{\mu}L^{\dagger}+iL\partial_{\mu}L^{\dagger}\;,$ $\displaystyle r_{\mu}\to
Rr_{\mu}R^{\dagger}+iR\partial_{\mu}R^{\dagger}\;,$ $\displaystyle
t_{l}^{\mu\nu}\to Rt_{l}^{\mu\nu}L^{\dagger}\;,$ (15)

where $\chi\equiv 2B(s+ip)$ with
$B=-\langle\bar{q}q\rangle/(2F_{\pi}^{2})\approx 2.8~{}\rm GeV$. $F_{\pi}$ is
the pion decay constant, and the quark condensate $\langle\bar{q}q\rangle$ can
be treated as an order parameter to measure the strength of the spontaneous
chiral symmetry breaking.

For the Lagrangian in Eq. 14, the equivalent chiral Lagrangian at low energy
can be constructed by identifying the relevant degrees of freedom and to write
down the most general chiral invariant Lagrangian ordered in terms of the
number of soft momenta. The relevant degrees of freedom are just the light
hadrons (pseudo-scalar pions and nucleons) and all possible non-QCD states
(like leptons and photon) encoded in the external sources. Define the pseudo
Nambu-Goldstone matrix $U$ as

$\displaystyle U=u^{2}\;,$ $\displaystyle u={\rm exp}\left({i\Pi\over
2F_{0}}\right)\;,$
$\displaystyle\Pi=\pi^{a}\tau^{a}=\begin{pmatrix}\pi^{0}&\sqrt{2}\pi^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}\end{pmatrix}\;.$ (16)

Then, under the chiral transformation $(L,R)\in G_{\chi}$, they transform as
$U\to RUL^{\dagger}$ and $u\to Ruh^{\dagger}=huL^{\dagger}$ with the
compensator matrix $h\in SU(2)_{V}$ as a function of $U,L,R$. Furthermore, we
define the chiral vielbein as

$\displaystyle
u_{\mu}=i\left(u^{\dagger}(\partial_{\mu}-ir_{\mu})u-u(\partial_{\mu}-il_{\mu})u^{\dagger}\right)\;,~{}~{}u_{\mu}^{\dagger}=u_{\mu}\;,$
(17)

which transforms as $u_{\mu}\to hu_{\mu}h^{\dagger}$ under the chiral group.
The power counting of these building blocks in terms of the soft momentum $p$
is

$\displaystyle u={\cal O}(p^{0})\;,$ $\displaystyle u_{\mu}={\cal
O}(p^{1})\;,$ $\displaystyle\chi={\cal O}(p^{2})\;.$ (18)

Then the leading order mesonic chiral Lagrangian is at ${\cal O}(p^{2})$ and
takes the form

$\displaystyle\mathcal{L}_{2}=\frac{F^{2}_{0}}{4}{\rm
Tr}[u_{\mu}u^{\mu}+\chi_{+}]\;,$ $\displaystyle\chi_{+}=u^{\dagger}\chi
u^{\dagger}+u\chi^{\dagger}u\;,$ (19)

where $F_{0}$ is the pion decay constant in the chiral limit. Here we see the
tensor external source does not enter into the leading chiral Lagrangian but
rather first appears at ${\cal O}(p^{4})$ Cata and Mateu (2007).

Next, we include the nucleons in this framework. Denote the nucleon doublet as
$\Psi=(p,n)^{\rm T}$ which transforms as $\Psi\to h\Psi$ under the chiral
transformation. The covariant derivative of the nucleon doublet is

$\displaystyle D_{\mu}\Psi=(\partial_{\mu}+\Gamma_{\mu})\Psi\;,$
$\displaystyle\Gamma_{\mu}=\frac{1}{2}\left(u^{\dagger}(\partial_{\mu}-ir_{\mu})u+u(\partial_{\mu}-il_{\mu})u^{\dagger}\right)\;,$
(20)

where $\Gamma_{\mu}$ is the chiral connection which helps $D_{\mu}\Psi$ to
have the same transformation rule as $\Psi$. The power counting for the
nucleon field is $\Psi={\cal O}(p^{0})$, and also $D_{\mu}\Psi={\cal
O}(p^{0})$, this latter result is because the nucleon mass $m_{N}$ is
comparable with the expansion scale $\Lambda_{\chi}$. However,
$(i\not{D}-m_{N})\Psi={\cal O}(p^{1})$, then the leading order baryonic chiral
Lagrangian takes

$\displaystyle{\cal L}_{\pi
N}^{(1)}=\bar{\Psi}\left(i\not{D}-m_{N}+{g_{A}\over
2}\gamma^{\mu}\gamma_{5}u_{\mu}\right)\Psi\;,$ (21)

where $g_{A}$ is the axial-nucleon coupling constant. For the chiral matching
of the dim-12 operators below, we will treat $D_{\mu}\Psi$ as a higher term
than $\Psi$ through the naive dimensional analysis, and neglect their
contribution to the leading order chiral realization of the relevant dim-12
operators. A possible way out for saving the power counting rule of the
nucleons is via the heavy baryon chiral perturbation formalism (HB$\chi$PT)
Jenkins and Manohar (1991) but with the sacrifice of Lorentz covariance. The
HB$\chi$PT formalism is beyond our current scope and one can check Ref.
Bijnens and Kofoed (2017) for the treatment of neutron-antineutron
oscillation. A brief comment concerning the relation between the Lorentz
covariant operators and their HB$\chi$PT reduction is given at the end of this
section.

###  III.2 Decomposition of irreducible chiral symmetry

In the matching onto B$\chi$PT for the effective operators in our case, the
lepton current together with the associated Wilson coefficient of a dim-12
operator in the LEFT behaves as a fixed external source, thus we only have to
cope with the six-quark sector of the operator. One of the key steps for the
chiral matching is to identify irreducible chiral representations. We describe
the procedures in the following. Suppose the quark sector has been decomposed
into a sum of irreducible representations/tensors (irreps) of the chiral group

$\displaystyle P=\theta^{uvwxyz}(q^{i{\rm
T}}_{\chi_{1},u}C\Gamma_{1}q^{j}_{\chi_{2},v})(q^{k{\rm
T}}_{\chi_{3},w}C\Gamma_{2}q^{l}_{\chi_{4},x})(q^{m{\rm
T}}_{\chi_{5},y}C\Gamma_{3}q^{n}_{\chi_{6},z})T_{ijklmn}^{\rm color}\;,$ (22)

where $q_{\chi_{i}}$ are the chiral quark doublets $(u,d)^{\rm T}_{\chi_{i}}$
with $\chi_{i}$ being the proper chiral projectors $P_{\pm}$, $\Gamma_{i}$ are
the Dirac gamma matrices, $T_{ijklmn}^{\rm color}$ a general color tensor
discussed in section 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 the up quark $u$ or down
quark $d$ respectively. The set of pure numbers $\theta^{uvwxyz}$ depends on
the irrep under consideration. $\theta$ is promoted as a spurion field that
transforms properly together with chiral transformations of the quarks under
$G_{\chi}$, so that $P$ looks like a chiral invariant.

The chiral counterparts of operator $P$ are constructed out of the spurion
field $\theta$ plus the hadronic degrees of freedom
$\\{\Psi,D_{\mu},u,u_{\mu},\chi,\cdots\\}$ and share the same symmetry
transformation properties as that of $P$, which include the chiral symmetry,
the Lorentz and the global baryon/lepton transformation properties. Since $P$
is chiral invariant and violates baryon number by two units, the matched
operators must also be chiral invariant and contain exactly one spurion field
$\theta$ and two nucleon fields $\Psi$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 dominant terms are
those with least power of $p$. Last, for each independent operator we
associate it with an unknown LEC which accommodates the non-perturbative QCD
dynamics. These LECs can be determined by fitting to the data, or calculated
using the LQCD method, or estimated based on the naive dimensional analysis.
In addition, for the LEFT operators belonging to the same chiral irrep, the
chiral symmetry implies their chiral counterparts at a given chiral order
share the same LEC. Here we remark that the above procedures have been used
previously to the non-perturbative matching of the dim-9 operators mediating
the nuclear and kaon neutrinoless double beta decay processes Liao  _et al._
(2020b); Graesser (2017) as well as the neutron-antineutron oscillations
Bijnens and Kofoed (2017).

With the above procedures, we can now match the operator basis in the LEFT in
section II.1 onto B$\chi$PT at leading order of the chiral expansion, i.e., at
${\cal O}(p^{0})$.888The higher order terms can be constructed in the same
style as was done in Bijnens and Kofoed (2017). We first transform the LEFT
operator basis into a chiral basis in which each operator itself belongs to
some irrep of the chiral group $G_{\chi}$. The chiral bases are shown in Tab.
5, Tab. 6 and Tab. 7 in Appendix F for the operators contributing to the
transition $pp\to\ell^{+}\ell^{\prime+}$, $pn\to\ell^{+}\bar{\nu}^{\prime}$
and $nn\to\bar{\nu}\bar{\nu}^{\prime}$, respectively. In the tables, we list
their relations with the LEFT operators in the first and second column and
show their chiral irreps in the third column (where the subscripts $(a,b,c)$
behind some irrep are used to distinguish different irreps with the same
chiral type)999For the operators belonging to the same irrep, they must have
the similar chiral, Lorentz and color structures so that they can be related
to each other through the action of the chiral transformation. and the
corresponding chiral spurion fields in last column. Except the gray sectors,
which already include the parity conjugates, all the rest ones have their
parity conjugates with $L\leftrightarrow R$ (and an additional exchange of
$+\leftrightarrow-$ for the tensor operators in Tab. 5). The parity conjugate
of chiral operator $P_{i}$ is denoted by $\tilde{P}_{i}$ once it is needed.
For those chiral operators expressed as a linear combination of two or more
LEFT operators, their equivalent definitions are given in Appendix F through
fully symmetrizing all free quark flavors with the same chirality. The
relations between the Wilson coefficients of the chiral basis as those of the
LEFT operators can be determined easily. The general expression for the
spurion fields takes the form

$\displaystyle\theta^{u_{1L}\cdots u_{nL}v_{1R}\cdots v_{mR}}_{(i_{1}\cdots
i_{n})(j_{1}\cdots j_{m})}=\theta^{v_{1R}\cdots v_{mR}u_{1L}\cdots
u_{nL}}_{(j_{1}\cdots j_{m})(i_{1}\cdots
i_{n})}=[\delta_{(i_{1}}^{u_{1}}\delta_{i_{2}}^{u_{2}}\cdots\delta_{i_{n})}^{u_{n}}][\delta_{(j_{1}}^{v_{1}}\delta_{j_{2}}^{v_{2}}\cdots\delta_{j_{m})}^{v_{m}}]\;,$
(23)

where we take the symmetrization notation with the round brackets $(\cdots)$.
Symmetrization with respect to a group of indices is defined by placing these
indices between round brackets $(\cdots)$, so we have

$\displaystyle\delta_{(i_{1}}^{u_{1}}\delta_{i_{2}}^{u_{2}}\cdots\delta_{i_{n})}^{u_{n}}={1\over
n!}\left[\delta_{i_{1}}^{u_{1}}\delta_{i_{2}}^{u_{2}}\cdots\delta_{i_{n}}^{u_{n}}+(n!-1){\rm~{}permutations~{}of~{}}(i_{1},\cdots,i_{n})\right]\;,$
(24)

where we take the normalization as in Rinaldi  _et al._ (2019a). For example,
for the operators with a scalar lepton current belonging to the chiral $({\bf
3}_{L},{\bf 1}_{R})$ and $({\bf 5}_{L},{\bf 3}_{R})$ irreps in Tab. 5, we have

$\displaystyle\theta^{u_{L}v_{L}}_{(11)}$
$\displaystyle=\theta^{u_{L}v_{R}}_{11}=\delta^{u}_{1}\delta^{v}_{1}\;,$
$\displaystyle\theta^{u_{L}v_{L}w_{L}x_{L}y_{R}z_{R}}_{(1112)(12)}$
$\displaystyle={1\over
8}[\delta^{u}_{1}\delta^{v}_{1}\delta^{w}_{1}\delta^{x}_{2}+\delta^{u}_{1}\delta^{v}_{1}\delta^{w}_{2}\delta^{x}_{1}+\delta^{u}_{1}\delta^{v}_{2}\delta^{w}_{1}\delta^{x}_{1}+\delta^{u}_{2}\delta^{v}_{1}\delta^{w}_{1}\delta^{x}_{1}][\delta^{y}_{1}\delta^{z}_{2}+\delta^{y}_{2}\delta^{z}_{1}]\;,$
$\displaystyle\theta^{u_{L}v_{L}w_{L}x_{L}y_{R}z_{R}}_{(1122)(11)}$
$\displaystyle={1\over
6}[\delta^{u}_{1}\delta^{v}_{1}\delta^{w}_{2}\delta^{x}_{2}+\delta^{u}_{1}\delta^{v}_{2}\delta^{w}_{1}\delta^{x}_{2}+\delta^{u}_{1}\delta^{v}_{2}\delta^{w}_{2}\delta^{x}_{1}+1\leftrightarrow
2]\delta^{y}_{1}\delta^{z}_{1}\;,$
$\displaystyle\theta^{u_{L}v_{L}w_{L}x_{L}y_{R}z_{R}}_{(1111)(22)}$
$\displaystyle=\delta^{u}_{1}\delta^{v}_{1}\delta^{w}_{1}\delta^{x}_{1}\delta^{x}_{2}\delta^{z}_{2}\;.$
(25)

From Tabs. 5-7, we see that for the operators with a scalar lepton current,
there are six types of irreps under the chiral group: $({\bf 3}_{L},{\bf
1}_{R})$, $({\bf 5}_{L},{\bf 3}_{R})$, $({\bf 7}_{L},{\bf 1}_{R})$ plus their
parity conjugates. The three operators
$P_{1,a}^{(pp)S},P_{1,a}^{(pn)S},P_{1,a}^{(nn)S}$ belong to the same irrep
$({\bf 7}_{L},{\bf 1}_{R})$ and relate to each other by the chiral symmetry,
and similar situation appears in the irrep $({\bf 5}_{L},{\bf 3}_{R})$ in
which the nine operators are related to each other. However, for the irrep
type $({\bf 3}_{L},{\bf 1}_{R})$, there are three different irreps
distinguished by the subscripts $a,b,c$ and each one contains three operators.
For the operators with a vector lepton current, there are also six types of
irreps: $({\bf 2}_{L},{\bf 2}_{R})$, $({\bf 4}_{L},{\bf 2}_{R})$, $({\bf
4}_{L},{\bf 4}_{R})$, $({\bf 6}_{L},{\bf 2}_{R})$, $({\bf 2}_{L},{\bf 4}_{R})$
and $({\bf 2}_{L},{\bf 6}_{R})$. One should be careful that there are four
different irreps for the type $({\bf 2}_{L},{\bf 2}_{R})$ since the parity
conjugate of the irrep $({\bf 2}_{L},{\bf 2}_{R})|_{a}$ is different from
itself and will be denoted as $({\bf 2}_{L},{\bf 2}_{R})|_{d}$ . Last, for the
operators with a tensor lepton current, there are still six types of irreps:
$({\bf 1}_{L},{\bf 1}_{R})$, $({\bf 3}_{L},{\bf 1}_{R})$, $({\bf 3}_{L},{\bf
3}_{R})$, $({\bf 5}_{L},{\bf 1}_{R})$, $({\bf 1}_{L},{\bf 3}_{R})$ and $({\bf
1}_{L},{\bf 5}_{R})$. Where the type $({\bf 3}_{L},{\bf 3}_{R})$ contains four
different irreps due to the parity conjugates of $({\bf 3}_{L},{\bf
3}_{R})|_{a,b}$ are distinct from themselves for the operators in Tab. 5.
Another interesting fact is that the LEFT operators belonging to the same
chiral irrep will not mix with each other under the QCD renormalization and
have the same anomalous dimensions since the QCD preserves the chiral symmetry
and quark flavors. In addition, the QCD renormalization for the operators
related to each other by parity is also the same. For the operators with a
scalar lepton current, the 1-loop QCD renormalization is identical to the
dim-9 operators contributing to the $n{\rm-}\bar{n}$ oscillation and can be
found in Caswell  _et al._ (1983); Rinaldi  _et al._ (2019b). But there is no
result for the operators with a vector or tensor lepton current yet, and we
will neglect the QCD renormalization effect for the current work due to the
involvement of considerable effort. However, from the 1-loop anomalous
dimension matrix result for the dim-9 $n{\rm-}\bar{n}$ oscillation operators
given in Rinaldi  _et al._ (2019b), we can estimate the 1-loop QCD correction
for those operators with a lepton scalar current. From the electroweak scale
$\Lambda_{\rm EW}$ to the scale $\Lambda_{\chi}$, we find the running effect
could have substantial impact on some scalar lepton current operators but the
influence on the derived NP scale is at most ${\cal O}(30~{}\%)$ due to the
high power dependence ($C_{i}\propto\Lambda^{-8}_{{\rm NP}}$). We will
systematically explore in the future work their renormalization effect.

###  III.3 Chiral matching for the operators

Based on the chiral irrep, we reorganize the chiral building blocks in terms
of the power of soft momentum $p$ and the explicit chiral left or right
doublet indices such that they have only one or two free indices, i.e., the
building blocks are constructed to take the forms: $X_{u_{\chi_{1}}}$ and
$X_{u_{\chi_{1}}v_{\chi_{2}}}$ with $\chi_{i}=L,R$.101010The building blocks
with three or more free indices are not independent and can be reduced into a
product of $X_{u_{\chi_{1}}}$s and $X_{u_{\chi_{1}}v_{\chi_{2}}}$s. They
transform as
$X_{u_{\chi_{1}}}\to(g_{\chi_{1}})_{u_{\chi_{1}}\hat{u}_{\chi_{1}}}X_{\hat{u}_{\chi_{1}}}$
and
$X_{u_{\chi_{1}}v_{\chi_{2}}}\to(g_{\chi_{1}})_{u_{\chi_{1}}\hat{u}_{\chi_{1}}}(g_{\chi_{2}})_{u_{\chi_{2}}\hat{u}_{\chi_{2}}}X_{\hat{u}_{\chi_{1}}\hat{v}_{\chi_{2}}}$
under chiral transformation $g_{\chi_{i}}\in SU(2)_{\rm L,R}^{f}$. Therefore,
the first few building blocks with lower chiral order are constructed as
follows Bijnens and Kofoed (2017):

$\displaystyle{\cal O}(p^{0}):~{}$ $\displaystyle(Ui\tau^{2})_{x_{R}y_{L}}\;,$
$\displaystyle(u\Psi)_{x_{R}}\;,$ $\displaystyle(u^{\dagger}\Psi)_{x_{L}}\;,$
$\displaystyle{\cal O}(p^{1}):~{}$
$\displaystyle(uu_{\mu}ui\tau^{2})_{x_{R}y_{L}}\;,$
$\displaystyle(uu_{\mu}u^{\dagger}i\tau^{2})_{x_{R}y_{R}}\;,$
$\displaystyle(u^{\dagger}u_{\mu}ui\tau^{2})_{x_{L}y_{L}}\;,$
$\displaystyle{\cal O}(p^{2}):~{}$ $\displaystyle(\chi
i\tau^{2})_{x_{R}y_{L}}\;,$
$\displaystyle(\chi^{\dagger}i\tau^{2})_{x_{L}y_{R}}\;,$
$\displaystyle\cdots\;,$ (26)

where for ${\cal O}(p^{2})$ we just show a few examples, and the full list
should include terms with two $u_{\mu}$s, field strength tensors for the
vector external sources, etc. Due to the fact that $x\tau^{2}=\tau^{2}x^{*}$
for $x\in SU(2)$, the other possible ${\cal O}(p^{0})$ and ${\cal O}(p^{1})$
building blocks are not independent:
$(U^{\dagger}i\tau^{2})_{x_{L}y_{R}}=-(Ui\tau^{2})_{y_{R}x_{L}}$ and
$(u^{\dagger}u_{\mu}u^{\dagger}i\tau^{2})_{x_{L}y_{R}}=-(uu_{\mu}ui\tau^{2})_{y_{R}x_{L}}$.
Note that the two ${\cal O}(p^{1})$ objects with the same chirality are anti-
symmetric under the exchange of the two indices. In the above, we neglect
those ${\cal O}(p^{0})$ terms with derivatives acting on the nucleon field
like $(uD_{\mu}\Psi)_{x_{R}}$ since, at leading ${\cal O}(p^{0})$ order of the
matched operators, they are actually redundant and can be transformed into
those non-derivative operators plus higher order terms (${\cal O}(p^{\geq
1})$) via the equation of motion (EoM) of nucleon fields and integration by
parts (IBP) relations. At the same time they can not yield the operators in
Eq. (5) used for the analysis in this paper. For instance, we consider the
lepton vector current operators, one possible leading ${\cal O}(p^{0})$ order
operator with derivative acting on the nucleon field takes $(\Psi^{{\rm
T}}_{a}iD^{\mu}\Psi_{b})P(u)j_{V,\mu}$ with $P(u)$ a polynomial of pion field
$u$. It can be reduced as follows

$\displaystyle 2\Psi^{{\rm T}}_{a}CiD^{\mu}\Psi_{b}P(u)j_{V,\mu}$
$\displaystyle=\Psi^{{\rm
T}}_{a}C(i\not{D}\gamma^{\mu}+\gamma^{\mu}i\not{D})\Psi_{b}P(u)j_{V,\mu}$
$\displaystyle\overset{\rm EoM}{=}\Psi^{{\rm
T}}_{a}C(i\not{D}\gamma^{\mu}+m_{N}\gamma^{\mu})\Psi_{b}P(u)j_{V,\mu}+{\cal
O}(p^{1})$ $\displaystyle\overset{\rm IBP}{=}-\Psi^{{\rm
T}}_{a}C(\overleftarrow{\not{D}}\gamma^{\mu}-m_{N}\gamma^{\mu})\Psi_{b}P(u)j_{V,\mu}-\Psi^{{\rm
T}}_{a}C\gamma^{\nu}\gamma^{\mu}\Psi_{b}\partial_{\nu}[P(u)j_{V,\mu}]+{\cal
O}(p^{1})\;,$

where in the last line, the first term can be further reduced by the EoM to be
a derivative-free operator, the second term is again ${\cal O}(p^{1})$ due to
the derivative acting on the pion and external source. The other ${\cal
O}(p^{0})$ terms with derivatives acting on the nucleon field can be reduced
in a similar fashion.

Without consideration of the operators involving covariant derivatives acting
on the nucleon fields, the leading order matching results for all the relevant
chiral irreps are shown in Tab. 8. Where the spurion fields are easily
identified from Tabs. 5-7 for each 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 ${\cal O}_{Q^{6}L^{2}}^{S,(S)}$ as an example
to show the relevant spurion fields and the chiral matching result. From Eq.
(13), the six-quark part of the matched three LEFT operators ${\cal
O}_{1LLL,b}^{(pp)S,-},{\cal O}_{1LLL,b}^{(nn)S},{\cal O}_{1LLL,b}^{(pn)S}$ can
be rewritten as

$\displaystyle(u^{i{\rm T}}_{L}Cu^{j}_{L})(u^{k{\rm
T}}_{L}Cd^{l}_{L})(u^{m{\rm
T}}_{L}Cd^{n}_{L})T^{SAA}_{\\{ij\\}[kl][mn]}=(\delta^{u}_{1}\delta^{v}_{1})T_{u_{L}v_{L}}\equiv\theta^{u_{L}v_{L}}_{(11)}T_{u_{L}v_{L}}\;,$
$\displaystyle(u^{i{\rm T}}_{L}Cd^{j}_{L})(u^{k{\rm
T}}_{L}Cd^{l}_{L})(u^{m{\rm
T}}_{L}Cd^{n}_{L})T^{SAA}_{\\{ij\\}[kl][mn]}={1\over
2}(\delta^{u}_{1}\delta^{v}_{2}+\delta^{u}_{2}\delta^{v}_{1})T_{u_{L}v_{L}}\equiv\theta^{u_{L}v_{L}}_{(12)}T_{u_{L}v_{L}}\;,$
$\displaystyle(d^{i{\rm T}}_{L}Cd^{j}_{L})(u^{k{\rm
T}}_{L}Cd^{l}_{L})(u^{m{\rm
T}}_{L}Cd^{n}_{L})T^{SAA}_{\\{ij\\}[kl][mn]}=(\delta^{u}_{2}\delta^{v}_{2})T_{u_{L}v_{L}}\equiv\theta^{u_{L}v_{L}}_{(11)}T_{u_{L}v_{L}}\;,$
(27)

where we have defined the spurion fields as

$\displaystyle\theta^{u_{L}v_{L}}_{(11)}=\delta^{u}_{1}\delta^{v}_{1}\;,$
$\displaystyle\theta^{u_{L}v_{L}}_{(12)}={1\over
2}(\delta^{u}_{1}\delta^{v}_{2}+\delta^{u}_{2}\delta^{v}_{1})\;,$
$\displaystyle\theta^{u_{L}v_{L}}_{(11)}=\delta^{u}_{2}\delta^{v}_{2}\;,$ (28)

and $T_{u_{L}v_{L}}=T_{v_{L}u_{L}}$ is a three-dimensional irrep tensor under
the group $SU(2)_{\rm L}^{f}$ but a singlet under $SU(2)_{\rm R}^{f}$, and
takes the form

$\displaystyle T_{u_{L}v_{L}}={1\over 4}\epsilon_{wx}\epsilon_{yz}(q^{i{\rm
T}}_{Lu}Cq^{j}_{Lv})(q^{k{\rm T}}_{Lw}Cq^{l}_{Lx})(q^{m{\rm
T}}_{Ly}Cq^{n}_{Lz})T^{SAA}_{\\{ij\\}[kl][mn]}\overset{G_{\chi}}{\to}L_{u}^{x}L_{v}^{y}T_{xy}\in({\bf
3}_{L},{\bf 1}_{R})\;.$ (29)

According to the previous procedures, the leading order chiral realization of
$T_{u_{L}v_{L}}$ is at ${\cal O}(p^{0})$ and formed by two
$(u^{\dagger}\Psi)_{u_{L}}$s to have the same baryon number and chiral
structure. In addition, the Lorentz covariance further restricts
$T_{u_{L}v_{L}}$ to be a scalar and have the general form
$(u^{\dagger})_{u_{L}a}(u^{\dagger})_{v_{L}a}\Psi_{b}\left[g_{3\times
1}+\hat{g}_{3\times 1}\gamma_{5}\right]\Psi_{b}$ as shown in Tab. 8. Then the
complete matching for ${\cal O}_{Q^{6}L^{2}}^{S,(S)}$ together with its Wilson
coefficient becomes

$\displaystyle C_{Q^{6}L^{2}}^{S,(S)}{\cal O}_{Q^{6}L^{2}}^{S,(S)}\to
4C_{Q^{6}L^{2}}^{S,(S)}\left(\theta^{u_{L}v_{L}}_{(11)}j_{S,-}^{\ell\ell^{\prime}}-2\theta^{u_{L}v_{L}}_{(12)}j_{S}^{\ell\nu^{\prime}}+\theta^{u_{L}v_{L}}_{(22)}j_{S}^{\nu\nu^{\prime}}\right)(u^{\dagger})_{u_{L}a}(u^{\dagger})_{v_{L}b}\Psi_{a}\left[g_{3\times
1}+\hat{g}_{3\times 1}\gamma_{5}\right]\Psi_{b}\;.$ (30)

Keeping the ${\cal O}(p^{0})$ terms in Tab. 8 and expanding them to zeroth
order in the pion fields, we have

$\displaystyle O^{S}_{3\times 1,i}$
$\displaystyle\to\theta^{u_{L}v_{L}}_{(\alpha\beta)}[\Psi^{\rm
T}_{u_{L}}C(g_{3\times 1,i}+\hat{g}_{3\times 1,i}\gamma_{5})\Psi_{v_{L}}]\;,$
$\displaystyle O^{S}_{5\times 3}$
$\displaystyle\to\theta^{u_{L}v_{L}w_{L}x_{L}y_{R}z_{R}}_{(\alpha\beta\gamma\rho)(\sigma\tau)}\epsilon_{y_{R}w_{L}}\epsilon_{z_{R}x_{L}}[\Psi^{\rm
T}_{u_{L}}C(g_{5\times 3}+\hat{g}_{5\times 3}\gamma_{5})\Psi_{v_{L}}]\;,$
$\displaystyle O^{V,\mu}_{2\times 2,i}$
$\displaystyle\to\theta^{u_{L}v_{R}}_{\alpha\beta}[\Psi^{\rm
T}_{u_{L}}C\gamma^{\mu}(g_{2\times 2,i}+\hat{g}_{2\times
2,i}\gamma_{5})\Psi_{v_{R}}]\;,$ $\displaystyle O^{V,\mu}_{4\times 2,i}$
$\displaystyle\to g_{4\times
2,i}\theta^{u_{L}v_{L}w_{L}x_{R}}_{(\alpha\beta\gamma)\rho}\epsilon_{x_{R}w_{L}}[\Psi^{\rm
T}_{u_{L}}C\gamma^{\mu}\gamma_{5}\Psi_{v_{L}}]\;,$ $\displaystyle
O^{V,\mu}_{4\times 4}$
$\displaystyle\to\theta^{u_{L}v_{L}w_{L}x_{R}y_{R}z_{R}}_{(\alpha\beta\gamma)(\rho\sigma\tau)}\epsilon_{y_{R}v_{L}}\epsilon_{z_{R}w_{L}}[\Psi^{\rm
T}_{u_{L}}C\gamma^{\mu}(g_{4\times 4}+\hat{g}_{4\times
4}\gamma_{5})\Psi_{x_{R}}]\;,$ $\displaystyle O^{T,\mu\nu}_{1\times 1,i}$
$\displaystyle={1\over 2}\epsilon^{ab}[\Psi^{\rm
T}_{a}C\sigma^{\mu\nu}(g_{1\times 1,i}+\hat{g}_{1\times
1,i}\gamma_{5})\Psi_{b}]\;,$ $\displaystyle O^{T,\mu\nu}_{3\times 3,i}$
$\displaystyle\to\theta^{u_{L}v_{L}w_{R}x_{R}}_{(\alpha\beta)(\gamma\rho)}\epsilon_{x_{R}v_{L}}[\Psi^{\rm
T}_{u_{L}}C\sigma^{\mu\nu}(g_{3\times 3,i}+\hat{g}_{3\times
3,i}\gamma_{5})\Psi_{w_{R}}]\;,$ (31)

and the similar expressions for the parity conjugates $\tilde{O}^{S}_{1\times
3,i}$, $\tilde{O}^{S}_{3\times 5}$ and $\tilde{O}^{V,\mu}_{4\times 2,i}$.
Taking the specific expressions of the spurion fields in Tabs. 5-7 into
consideration, we can obtain the matching results for the Wilson coefficients
of the operators in Eqs. (6-8) as the function of the LECs and the LEFT/SMEFT
Wilson coefficients, the full matching results from the LEFT and SMEFT
operators are listed in Appendix H.

In the following, we show as an example the matching results from the SMEFT
operators $C_{Q^{6}L^{2}}^{S,(S)}$ and $C_{Q^{6}L^{2}}^{T,(S)}$. In terms of
their LEFT counterparts ${\cal Q}_{1LLL}^{(pp,pn,nn)S}$ and ${\cal
Q}_{1LLL}^{(pn)T}$ , we have

$\displaystyle C_{R(L)}^{(pp)S}=g_{3\times 1,a}\left({3\over
5}C_{1LLL,a}^{(pp)S,\pm}+C_{1LLL,b}^{(pp)S,\pm}\right)+\cdots\;,$
$\displaystyle C_{5R(L)}^{(pp)S}=C_{R(L)}^{(pp)S}|_{g\to\hat{g}}\;,$
$\displaystyle C_{L}^{(pn)S}=g_{3\times 1,a}\left({9\over
5}C_{1LLL,a}^{(pn)S}+C_{1LLL,b}^{(pn)S}\right)+\cdots\;,$ $\displaystyle
C_{5L}^{(pn)S}=C_{L}^{(pn)S}|_{g\to\hat{g}}\;,$ $\displaystyle
C^{(pn)T}=g_{1\times 1,a}^{r}\left({1\over
3}C_{1LLL,a}^{(pn)T}+C_{1LLL,b}^{(pn)T}\right)+\cdots\;,$ $\displaystyle
C_{L}^{(nn)S}=g_{3\times 1,a}\left({3\over
5}C_{1LLL,a}^{(nn)S}+C_{1LLL,b}^{(nn)S}\right)+\cdots\;,$ $\displaystyle
C_{5L}^{(nn)S}=C_{L}^{(nn)S}|_{g\to\hat{g}}\;,$ (32)

where $\cdots$ stand for contributions from other operators. For the tensor
case $C_{L}^{(pn),T}$, we have used the identity
$\sigma^{\mu\nu}\gamma_{5}=i\epsilon^{\mu\nu\rho\sigma}\sigma_{\rho\sigma}/2$
to eliminate the operator $(p^{\rm
T}C\sigma_{\mu\nu}\gamma_{5}n)(\ell_{R}^{\rm T}C\sigma^{\mu\nu}\nu_{L})$ in
favor of $(p^{\rm T}C\sigma_{\mu\nu}n)(\ell_{R}^{\rm
T}C\sigma^{\mu\nu}\nu_{L})$ with the shifted LECs $g_{i\times
i,x}^{r}=g_{i\times i,x}-\hat{g}_{i\times i,x}$. After neglecting the QCD
running effect and replacing the LEFT Wilson coefficients by the SMEFT ones
$C_{Q^{6}L^{2}}^{S,(S)}$ and $C_{Q^{6}L^{2}}^{T,(S)}$ as the way shown in Tab.
2, we find the above results are simplified to become

$\displaystyle C_{L}^{(pp)S}=4g_{3\times 1,a}C_{Q^{6}L^{2}}^{S,(S)}+\cdots\;,$
$\displaystyle C_{5L}^{(pp)S}=C_{L}^{(pp)S}|_{g\to\hat{g}}\;,$ $\displaystyle
C_{L}^{(pn)S}=-8g_{3\times 1,a}C_{Q^{6}L^{2}}^{S,(S)}+\cdots\;,$
$\displaystyle C_{5L}^{(pn)S}=C_{L}^{(pn)S}|_{g\to\hat{g}}\;,$ $\displaystyle
C^{(pn)T}=-16g_{1\times 1,a}^{r}C_{Q^{6}L^{2}}^{T,(S)}+\cdots\;,$
$\displaystyle C_{L}^{(nn)S}=4g_{3\times 1,a}C_{Q^{6}L^{2}}^{S,(S)}+\cdots\;,$
$\displaystyle C_{5L}^{(nn)S}=C_{L}^{(nn)S}|_{g\to\hat{g}}\;.$ (33)

Once the hadronic LEC $g_{i}$ is known one can obtain the dinucleon and
dilepton transitions. In the following section we will discuss how this can be
done and obtain constraints on the LEFT and SMEFT operators.

Before doing that, let us have some discussion about the LECs. By the parity
invariance of QCD, we expect the LECs of an operator and its parity conjugate
are the same up to a sign determined by the parity transformation property of
the quark and the corresponding hadron level operators, i.e., $g_{i\times
j}=\pm g_{j\times i}$. Particularly, for the scalar current case, we have

$\displaystyle g_{3\times 1,i}=-g_{1\times 3,i}\;,$
$\displaystyle{\hat{g}}_{3\times 1,i}=+\hat{g}_{1\times 3,i}\;,$
$\displaystyle g_{3\times 5}=-g_{5\times 3}\;,$
$\displaystyle{\hat{g}}_{3\times 5}=+\hat{g}_{5\times 3}\;.$ (34)

The numerical value of $g_{3\times 1,i}$ and $g_{3\times 5}$ can be determined
by the LQCD results for the $n{\rm-}\bar{n}$ oscillation matrix elements
Rinaldi  _et al._ (2019b). This is because the quark sectors of the 14
operators with a scalar lepton current contributing to
$nn\to\bar{\nu}\bar{\nu}^{\prime}$ transitions are exactly the 14 dim-9
operators mediating $n{\rm-}\bar{n}$ oscillation. Neglecting the lepton
current, the scalar chiral operators in the irreps $({\bf 1}_{L},{\bf 3}_{R})$
and $({\bf 5}_{L},{\bf 3}_{R})$ in Tab. 7 have the following correspondence
with the dim-9 chiral operators for the $n{\rm-}\bar{n}$ oscillation Rinaldi
_et al._ (2019b)

$\displaystyle Q_{1}=-4\tilde{P}_{1,b}^{(nn)S}\;,$ $\displaystyle
Q_{2}=-4\tilde{P}_{2,b}^{(nn)S}\;,$ $\displaystyle Q_{3}=-4P_{3,b}^{(nn)S}\;,$
$\displaystyle Q_{5}=P_{4}^{(nn)S}\;,$ $\displaystyle
Q_{6}=-4P_{2,a}^{(nn)S}\;,$ $\displaystyle Q_{7}=-4P_{3,a}^{(nn)S}\;.$ (35)

After comparing the LQCD results on the $n{\rm-}\bar{n}$ matrix elements from
$Q_{i}$ and our chiral matching results for $P_{i}^{(nn)S}$ in Eq. (31), we
find

$\displaystyle g_{1\times 3,a,c}\sim-6\times 10^{-6}~{}\rm GeV^{6}\;,$
$\displaystyle g_{1\times 3,b}\sim 1\times 10^{-5}~{}\rm GeV^{6}\;,$
$\displaystyle g_{5\times 3}\sim 5\times 10^{-6}~{}\rm GeV^{6}\;,$ (36)

at the scale $\Lambda\sim 2$ GeV or so and the uncertainty in Rinaldi  _et
al._ (2019b) is neglected.

Except the above LECs, the rest of the hadronic couplings, such as the
$\hat{g}_{i}$ and the ones related to the vector and tensor current operators,
have not been determined. For these LECs, we will use dimensional analysis as
a guide to illustration. Since the transition from quarks to hadrons is
through the non-perturbative QCD dynamics, the only relevant scale
$\Lambda_{\rm QCD}\sim 200$ MeV will come into play. To make the
dimensionality correct, one can take as a rough estimate the couplings to be
of order $\Lambda_{\rm QCD}^{6}\sim 6.4\times 10^{-5}~{}\rm GeV^{6}$. This is
larger than the numbers in (36). But considering the large uncertainties
involved, we can take it as a guide for estimate.

Alternatively, these LECs can be estimated via the naive dimensional analysis
by keeping track of $4\pi$ factors Manohar and Georgi (1984); Gavela  _et al._
(2016) and relating the hadronic matrix element to the chiral symmetry
breaking scale $\Lambda_{\chi}=1190$ MeV. One introduces “reduced” couplings
for the hadron and quark level operators and to match them Weinberg (1989).
For a coupling constant $g$ appearing in an interaction of dimensionality $D$
in mass which containing $N$ field operators, the reduced coupling is
$(4\pi)^{2-N}\Lambda_{\chi}^{D-4}g$. For our case, the hadronic operators
involve two fields ($N=2$) with a coupling $g_{N}$ as given in Eqs. (6-8). The
quark operators involve six quarks ($N=6$) with a coupling $g_{q}$, therefore
we would obtain $g_{N}/\Lambda_{\chi}=g_{q}(\Lambda_{\chi})^{5}/(4\pi)^{4}$ by
matching. Setting $g_{q}=1$, one would have the hadronic coupling to be of
order $(\Lambda_{\chi})^{6}/(4\pi)^{4}\sim 11\times 10^{-5}$ which is about 2
times the above dimensional estimate. For our numerical estimate of the
undetermined LECs in the next section, we assume their value to be the similar
order as $\Lambda_{\rm QCD}^{6}$ .

Last, we comment the heavy baryon $\chi$PT formalism Jenkins and Manohar
(1991), which is a consistent framework for the power counting of nucleon
fields. In this framework, for our case, the anti-nucleon mode is integrated
out and the remaining heavy nucleon doublet is defined as ${\cal
N}_{v}(x)=e^{+imv\cdot x}P_{v+}\Psi$ with $P_{v\pm}\equiv{1\over
2}(1\pm\not{v})$, where $v$ is a reference velocity satisfying $v^{2}=1$ and
usually taken as $v=(1,\bf 0)$. The chiral power counting for $D_{\mu}{\cal
N}_{v}(x)$ is ${\cal O}(p^{1})$ as promised in this formalism. To leading
order of chiral matching, there should have no derivatives acting on the
nucleon fields. Since

$\displaystyle{\cal N}^{\rm T}_{v}C{\cal N}_{v}={\cal N}^{\rm
T}_{v}P_{v+}^{\rm T}CP_{v+}{\cal N}_{v}={\cal N}^{\rm T}_{v}CP_{v-}P_{v+}{\cal
N}_{v}=0\;,$ (37)

we find that the matched operators using heavy nucleon fields can be directly
obtained from the matched operators using the relativistic nucleon fields in
Tab. 8 by replacing the nucleon field $\Psi$ by ${\cal N}_{v}$ together with
the omission of the operators with a scalar nucleon current. The use of
relativistic formalism is its explicit Lorentz invariance and chiral symmetry,
which are convenient for the loop calculations.

##  IV Dinucleon and dilepton transition rate

Combining the previous sections for the effective interactions from the SMEFT,
to LEFT, then to B$\chi$PT, in this section we will collect all pieces
together and calculate the dinucleon to dilepton decay rate. Denote
collectively $NN^{\prime}\in\\{pp,pn,nn\\}$ and
$l_{\alpha}l_{\beta}\in\\{\ell^{+}\ell^{\prime+},\ell^{+}\bar{\nu}^{\prime},\bar{\nu}\bar{\nu}^{\prime}\\}$,
then the decay rate for dinucleon $NN^{\prime}$ to dilepton
$l_{\alpha}l_{\beta}$ transition in nucleus can be estimated in the following
way Goity and Sher (1995)

$\displaystyle\Gamma_{NN^{\prime}\to
l_{\alpha}l_{\beta}}={1\over(2\pi)^{3}\sqrt{\rho_{N}\rho_{N^{\prime}}}}\int
d^{3}k_{1}d^{3}k_{2}\rho_{N}(k_{1})\rho_{N^{\prime}}(k_{2})v_{\rm
rel.}(1-\mathbf{v_{1}\cdot v_{2}})\sigma(NN^{\prime}\to
l_{\alpha}l_{\beta})\;,$ (38)

where $\rho_{N}(k)$ is the nucleon density distribution in momentum space and
$\rho_{N}$ is the average nucleon density defined as $\rho_{N}=\int
d^{3}k\rho_{N}(k)/(\sqrt{2\pi})^{3}$. $\mathbf{v_{1}}(\mathbf{v_{2}})$ is the
velocity of the nucleon $N(N^{\prime})$. The total cross-section for the free
nucleon scattering process $N(k_{1})N^{\prime}(k_{2})\to
l_{\alpha}(p_{1})l_{\beta}(p_{2})$ is

$\displaystyle\sigma(NN^{\prime}\to l_{\alpha}l_{\beta})={1\over S}{1\over
4E_{1}E_{2}v_{\rm rel.}}\int d\Pi_{2}\overline{\left|{\cal M}_{NN^{\prime}\to
l_{\alpha}l_{\beta}}\right|}^{2},$ (39)

where $E_{1}(E_{2})$ is the energy of the initial state nucleon
$N(N^{\prime})$, and $S$ is a symmetry factor and equals 2 for identical final
leptons
$l_{\alpha}=l_{\beta}=\\{e^{+},\mu^{+},\bar{\nu}_{e},\bar{\nu}_{\mu},\bar{\nu}_{\tau}\\}$,
otherwise $S=1$. $d\Pi_{2}$ is the relativistically invariant two-body phase
space.

The dinucleon collisions occur at low relative velocity $v$, they may be
affected by some other SM interaction resulting in modification of the cross
sections. For example for $pp\to\ell^{+}_{\alpha}\ell^{+}_{\beta}$, there is a
repulsive force between the two protons due to electrodynamics which reduces
the cross section. The effect of exchange photons between protons is best
captured by the Sommerfeld effect Hisano  _et al._ (2005); He (2009). Because
the repulsive nature of the electromagnetic force there is a reduction of the
cross section, the original cross section $\sigma$ is modified to
$\tilde{\sigma}=\sigma SF$ with $SF$ given by $(\alpha_{\rm em}\pi/v)/({\rm
exp}[\alpha_{\rm em}\pi/v]-1)$. This reduction factor $SF$ could be very
severe if $v$ is very small. For the case in question, the typical $v$ of
about $0.1$ leads to $SF\approx 0.9$. Had $v$ be 0.01, $SF$ is further reduced
to 0.26. Therefore the case we are considering the reduction is not severe.
One expects such effects for $np$ and $nn$ cases will be smaller. We still use
Eq. (39) as our order of magnitude estimate.

To a good approximation for the oxygen nuclei 16O, we treat the nucleons to be
quasi-free and neglect the small effects due to the nucleon Fermi motion and
nuclear binding energy, as well as the above Sommerfeld suppression effect.
The average nuclear matter density $\rho_{N}$ approximately equals
$0.25~{}{\rm fm}^{-3}$ for either proton or neutron. Then the transition rate
reduces into

$\displaystyle\Gamma_{NN^{\prime}\to l_{\alpha}l_{\beta}}={1\over
S}{\rho_{N}\over 4m_{N}^{2}}\overline{\left|{\cal M}_{NN^{\prime}\to
l_{\alpha}l_{\beta}}\right|}^{2}\Pi_{2}\;,$ (40)

where we have neglected the mass difference between proton and neutron and
taken both to be $m_{N}=(m_{p}+m_{n})/2$. The two-body final state phase
factor $\Pi_{2}$ takes

$\displaystyle\Pi_{2}={1\over
8\pi}[\lambda(1,\delta_{\alpha},\delta_{\beta})]^{1/2}\;,~{}\delta_{\alpha}={m_{\alpha}^{2}\over
4m_{N}^{2}}\;,~{}\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+yz+zx)\;.$ (41)

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

$\displaystyle\overline{\left|{\cal
M}_{pp\to\ell^{+}_{\alpha}\ell^{+}_{\beta}}\right|}^{2}$
$\displaystyle=32m_{N}^{4}\left[S^{2}\left(1-\delta_{\alpha}-\delta_{\beta}\right)\big{(}\big{|}C_{5L}^{(pp)S}\big{|}^{2}+\big{|}C_{5R}^{(pp)S}\big{|}^{2}\big{)}\right.$
$\displaystyle\left.+\left(\delta_{\alpha}+\delta_{\beta}-(\delta_{\alpha}-\delta_{\beta})^{2}\right)\big{|}C^{(pp)V}\big{|}^{2}-4S^{2}\sqrt{\delta_{\alpha}\delta_{\beta}}{\rm
Re}\big{[}C_{5L}^{(pp)S}C_{5R}^{(pp)S*}\big{]}\right.$
$\displaystyle\left.+2S\big{(}\left(1+\delta_{\alpha}-\delta_{\beta}\right)\sqrt{\delta_{\beta}}{\rm
Re}\big{[}C^{(pp)V}C_{5R}^{(pp)S*}\big{]}-(L,\delta_{\alpha}\leftrightarrow
R,\delta_{\beta})\big{)}\right]+{\cal O}(v^{2})\;,$ (42)
$\displaystyle\overline{\left|{\cal
M}_{pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}}\right|}^{2}$
$\displaystyle=8m_{N}^{4}(1-\delta_{\alpha})\left[\big{|}C_{5L}^{(pn)S}\big{|}^{2}+\delta_{\alpha}\big{|}C_{5L}^{(pn)V}\big{|}^{2}+(2+\delta_{\alpha})\big{|}C_{L}^{(pn)V}\big{|}^{2}+4(1+2\delta_{\alpha})\right.$
$\displaystyle\left.\times\big{|}C^{(pn)T}\big{|}^{2}-2\sqrt{\delta_{\alpha}}{\rm
Re}\big{[}C_{5L}^{(pn)S}C_{5L}^{(pn)V*}-6C_{L}^{(pn)V}C^{(pn)T*}\big{]}\right]+{\cal
O}(v^{2})\;,$ (43) $\displaystyle\overline{\left|{\cal
M}_{nn\to\bar{\nu}_{\alpha}\bar{\nu}_{\beta}}\right|}^{2}$
$\displaystyle=32S^{2}m_{N}^{4}\big{|}C_{5L}^{(nn)S}\big{|}^{2}+{\cal
O}(v^{2})\;,$ (44)

where we see that the contribution to $pp\to\ell^{+}_{\alpha}\ell^{+}_{\beta}\
(nn\to\bar{\nu}\bar{\nu}^{\prime})$ transition from the operators ${\cal
O}^{(pp)S}_{L,R}\ ({\cal O}^{(nn)S}_{L})$ vanishes in that $pp\ (nn)$
annihilation through such operators is $p$-wave ($\propto v^{2}$), whereas the
contribution from the vector operator ${\cal O}^{(pp)V}$ is helicity-
suppressed ($\propto\delta_{\alpha,\beta}$). For
$pn\to\bar{\nu}\bar{\nu}^{\prime}$, the vanishing from ${\cal O}^{(pn)S}_{L}$
has a similar reason.

The partial lifetime characterizing the matter instability is the inverse of
the rate $(\tau/{\cal B}_{i})_{i}\equiv\Gamma^{-1}_{i}$, where ${\cal B}_{i}$
is a branching ratio. Taking the experimental lower limits on the partial
lifetime in Tab. 1 into consideration, and by the relation
$\Gamma^{-1}_{i}=(\tau/{\cal B}_{i})_{i}\geq\tau_{\rm exp}$, we can obtain the
constraints on the coefficients in Eqs. (42-44). Assuming one term dominates
at a time, then the result is shown in Tab. 3, where the upper limit on the
Wilson coefficients is classified in terms of the final state leptons. We see
the most stringent limit is for the operator ${\cal
O}_{5L,R}^{(pp)S}|_{ee,e\mu,\mu\mu}$ in which
$C_{5L,R}^{(pp)S}|_{ee,e\mu,\mu\mu}\leq 6.8,9.4,6.6\times 10^{-32}~{}\rm
GeV^{-2}$, this is because the strong experimental limits on these decay
modes.

WC | [${10^{-30}\rm~{}GeV^{-2}}$] | WC | [${10^{-30}\rm~{}GeV^{-2}}$] | WC | [${10^{-30}\rm~{}GeV^{-2}}$]
---|---|---|---|---|---
$C_{L,R}^{(pp)S}$ | - | $C_{5L,R}^{(pp)S}|_{ee,e\mu,\mu\mu}$ | $0.068,~{}0.094,~{}0.067$ | $C^{(pp)V}|_{ee,e\mu,\mu\mu}$ | $350,~{}1.7,~{}1.7$
$C_{L}^{(pn)S}$ | - | $C_{5L}^{(pn)S}|_{e\nu,\mu\nu,\tau\nu}$ | $0.77,~{}0.84,~{}22$ | $C_{L}^{(pn)V}|_{e\nu,\mu\nu,\tau\nu}$ | $0.55,~{}0.59,~{}13$
- | - | $C_{5L}^{(pn)V}|_{e\nu,\mu\nu,\tau\nu}$ | $2800,~{}15,~{}23$ | $C^{(pn)T}|_{e\nu,\mu\nu,\tau\nu}$ | $0.39,~{}0.42,~{}6.6$
$C_{L}^{(nn)S}$ | - | $C_{5L}^{(nn)S}|_{\nu_{\alpha}\nu_{\alpha},\nu_{\alpha}\nu_{\beta\neq\alpha}}$ | $3.7,~{}5.3$ |  |

Table 3: 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. 1 to set the bound.

SMEFT WCs | $pp\to e^{+}e^{+},~{}e^{+}\mu^{+},~{}\mu^{+}\mu^{+}$ | $pn\to e^{+}\nu,~{}\mu^{+}\bar{\nu},~{}\tau^{+}\bar{\nu}$ | $nn\to\bar{\nu}_{\alpha}\bar{\nu}_{\alpha},~{}\bar{\nu}_{\alpha}\bar{\nu}_{\beta\neq\alpha}$
---|---|---|---
$\Lambda_{\rm NP}\equiv|C_{i}|^{-{1\over 8}}~{}[\rm TeV]$ | $\Lambda_{\rm NP}\equiv|C_{i}|^{-{1\over 8}}~{}[\rm TeV]$ | $\Lambda_{\rm NP}\equiv|C_{i}|^{-{1\over 8}}~{}[\rm TeV]$
$C_{u^{3}d^{3}L^{2}1}^{S,(A)}$ | - | $2.04,~{}2.02,~{}1.34\times\left[{\hat{g}_{1\times 3,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{u^{3}d^{3}L^{2}2}^{S,(A)}$ | - | $1.89,~{}1.87,~{}1.25\times\left[{\hat{g}_{1\times 3,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{u^{2}d^{2}Q^{2}L^{2}1,3}^{S,(S)}$ |  $2.35,~{}2.26,~{}2.36\times\left[{\hat{g}_{3\times 1,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ |  $1.89,~{}1.87,~{}1.25\times\left[{\hat{g}_{3\times 1,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ |  $1.43,~{}1.37\times\left[{\hat{g}_{3\times 1,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$
$C_{u^{2}d^{2}Q^{2}L^{2}2}^{S,(A)}$ | - | $2.07,~{}2.04,~{}1.36\times\left[{\hat{g}_{1\times 3,b}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{udQ^{4}L^{2}1}^{S,(A)}$ | - | $2.25,~{}2.23,~{}1.48\times\left[{\hat{g}_{1\times 3,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{udQ^{4}L^{2}2}^{S,(S)}$ | $2.57,~{}2.46,~{}2.57\times\left[{\hat{g}_{3\times 1,b}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $2.07,~{}2.04,~{}1.36\times\left[{\hat{g}_{3\times 1,b}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $1.56,~{}1.49\times\left[{\hat{g}_{3\times 1,b}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$
$C_{Q^{6}L^{2}}^{S,(S)}$ | $2.80,~{}2.69,~{}2.81\times\left[{\hat{g}_{3\times 1,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $2.25,~{}2.23,~{}1.48\times\left[{\hat{g}_{3\times 1,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $1.70,~{}1.63\times\left[{\hat{g}_{3\times 1,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$
$C_{u^{4}d^{2}e^{2}1}^{S,(S)}$ | $2.21,~{}2.12,~{}2.21\times\left[{\hat{g}_{1\times 3,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | - | -
$C_{u^{4}d^{2}e^{2}2}^{S,(S)}$ | $2.35,~{}2.26,~{}2.36\times\left[{\hat{g}_{1\times 3,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | - | -
$C_{u^{3}dQ^{2}e^{2}}^{S,(S)}$ | $2.57,~{}2.46,~{}2.57\times\left[{\hat{g}_{1\times 3,b}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | - | -
$C_{u^{2}Q^{4}e^{2}}^{S,(S)}$ | $2.80,~{}2.69,~{}2.81\times\left[{\hat{g}_{1\times 3,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | - | -
$C_{u^{3}d^{2}QeL1,3}^{V}$ | $0.808,~{}1.58,~{}1.58\times\left[{\hat{g}_{2\times 2,d}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $1.81,~{}1.79,~{}1.22\times\left[{g_{2\times 2,d}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{u^{3}d^{2}QeL2}^{V}$ | $0.704,~{}1.37,~{}1.37\times\left[{\hat{g}_{2\times 2,d}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $1.58,~{}1.56,~{}1.06\times\left[{g_{2\times 2,d}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{u^{2}dQ^{3}eL1}^{V}$ | $0.837,~{}1.63,~{}1.63\times\left[{\hat{g}_{2\times 2,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $1.88,~{}1.86,~{}1.27\times\left[{g_{2\times 2,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{u^{2}dQ^{3}eL2}^{V}$ | $0.881,~{}1.72,~{}1.72\times\left[{\hat{g}_{2\times 2,b}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $1.98,~{}1.96,~{}1.33\times\left[{g_{2\times 2,b}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{u^{2}dQ^{3}eL3}^{V}$ | $0.881,~{}1.72,~{}1.72\times\left[{\hat{g}_{2\times 2,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $1.98,~{}1.96,~{}1.33\times\left[{g_{2\times 2,c}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{uQ^{5}eL}^{V}$ | $0.961,~{}1.88,~{}1.88\times\left[{\hat{g}_{2\times 2,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | $2.16,~{}2.13,~{}1.45\times\left[{g_{2\times 2,a}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{u^{2}d^{2}Q^{2}L^{2}1,3}^{T,(S)}$ | - | $2.25,~{}2.23,~{}1.35\times\left[{g_{1\times 1,c}^{r}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{u^{2}d^{2}Q^{2}L^{2}2}^{T,(A)}$ | - | $1.89,~{}1.87,~{}1.33\times\left[{g_{3\times 3,b}^{r}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{udQ^{4}L^{2}1}^{T,(A)}$ | - | $2.07,~{}2.04,~{}1.45\times\left[{g_{3\times 3,a}^{r}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{udQ^{4}L^{2}2}^{T,(S)}$ | - | $2.46,~{}2.43,~{}1.72\times\left[{g_{1\times 1,b}^{r}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -
$C_{Q^{6}L^{2}}^{T,(S)}$ | - | $2.68,~{}2.65,~{}1.88\times\left[{g_{1\times 1,a}^{r}\over\Lambda_{\rm QCD}^{6}}\right]^{1\over 8}$ | -

Table 4: The constraint on the effective NP scale from the current
experimental data in Tab. 1. The flavor index is suppressed and can be easily
recognized in terms of the transition mode. One should keep in mind that the
SMEFT Wilson coefficients with a superscript ‘(A)’ vanish for identical lepton
flavors.

Next, we consider the above limits on the implications of SMEFT Wilson
coefficients and the relevant NP scales. Based on the matching results in Eqs.
(H.11-H.20) in Appendix H, and assuming one term active in the matching result
at a time, then the limits in Tab. 3 translate into limits on the SMEFT Wilson
coefficients as shown in Tab. 4. In obtaining the results, we have taken our
previous estimation of the LECs $g_{i}\sim\Lambda_{\rm QCD}^{6}\sim 6.4\times
10^{-5}~{}\rm GeV^{6}$ as our benchmark value, therefore, the factor
$(g_{i}/\Lambda_{\rm QCD}^{6})^{1/8}$ is ${\cal O}(1)$. Up to the ${\cal
O}(1)$ hadronic LECs ratio, the associated NP scale is found to be around
$0.8-2.7~{}\rm TeV$ for all relevant operators. Here we see that, even the
effective interactions are at dim 12, the matter instability puts a stringent
limit on the NP scale. Similarly, we can set constraints on the LEFT
operators. However, taking the assumption of the NP scale much higher than
$\Lambda_{\rm EW}$, the above constraints on the SMEFT interactions are more
illuminating in connection with NP scenarios, and thus we do not show the
constraints on the LEFT interactions here for brevity.

Furthermore, we consider the contribution to the transitions from the dim-12
operators ${\cal O}_{Q^{6}L^{2}}^{S,(S)}$, ${\cal
O}_{u^{2}d^{2}Q^{2}L^{2}1,3}^{S,(S)}$ and ${\cal O}_{udQ^{4}L^{2}2}^{S,(S)}$
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 possible operators contributing to the $nn\to\bar{\nu}\bar{\nu}^{\prime}$
modes at leading order. From the previous discussion, the transition rates
become

$\displaystyle\tilde{\Gamma}_{pp\to\ell_{\alpha}^{+}\ell_{\beta}^{+}}$
$\displaystyle={S\over\pi}\left(1-\delta_{\alpha}-\delta_{\beta}\right)\sqrt{1-2(\delta_{\alpha}+\delta_{\beta})+(\delta_{\alpha}-\delta_{\beta})^{2}}m_{N}^{2}\rho_{N}C_{\alpha\beta}^{2}\;,$
$\displaystyle\tilde{\Gamma}_{pn\to\ell_{\alpha}^{+}\bar{\nu}_{\beta}}$
$\displaystyle={1\over\pi}(1-\delta_{\alpha})^{2}m_{N}^{2}\rho_{N}C_{\alpha\beta}^{2}\;,$
$\displaystyle\tilde{\Gamma}_{nn\to\bar{\nu}_{\alpha}\bar{\nu}_{\beta}}$
$\displaystyle={S\over\pi}m_{N}^{2}\rho_{N}C_{\alpha\beta}^{2}\;,$ (45)

where we use a ‘tilde’ to represent such special contributions, and

$\displaystyle C_{\alpha\beta}=4\hat{g}_{3\times
1,a}C_{Q^{6}L^{2}}^{S,(S),\alpha\beta}+2\hat{g}_{3\times
1,b}C_{udQ^{4}L^{2}2}^{S,(S),\alpha\beta}+\hat{g}_{3\times
1,c}\big{(}C_{u^{2}d^{2}Q^{2}L^{2}1}^{S,(S),\alpha\beta}+C_{u^{2}d^{2}Q^{2}L^{2}3}^{S,(S),\alpha\beta}\big{)}+{\hat{g}_{3\times
5}\over 6}C_{u^{2}d^{2}Q^{2}L^{2}1}^{S,(S),\alpha\beta}\;.$

Here we have added the lepton flavor indices for a more careful
treatment111111For $pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}$ mode, the last
term should take a minus sign. We neglect this sign difference here.. By the
weak isospin symmetry, we see the three transitions are related to each other.
From Eq. (45), we obtain

$\displaystyle\tilde{\Gamma}_{pn\to\ell_{\alpha}^{+}\bar{\nu}_{\beta}}^{-1}$
$\displaystyle={S\left(1-\delta_{\alpha}-\delta_{\beta}\right)\sqrt{1-2(\delta_{\alpha}+\delta_{\beta})+(\delta_{\alpha}-\delta_{\beta})^{2}}\over(1-\delta_{\alpha})^{2}}\tilde{\Gamma}_{pp\to\ell_{\alpha}^{+}\ell_{\beta}^{+}}^{-1}\;,$
$\displaystyle\tilde{\Gamma}_{nn\to\bar{\nu}_{\alpha}\bar{\nu}_{\beta}}^{-1}$
$\displaystyle=\left(1-\delta_{\alpha}-\delta_{\beta}\right)\sqrt{1-2(\delta_{\alpha}+\delta_{\beta})+(\delta_{\alpha}-\delta_{\beta})^{2}}\tilde{\Gamma}_{pp\to\ell_{\alpha}^{+}\ell_{\beta}^{+}}^{-1}\;,$
$\displaystyle\tilde{\Gamma}_{nn\to\bar{\nu}_{\alpha}\bar{\nu}_{\beta}}^{-1}$
$\displaystyle=S^{-1}(1-\delta_{\alpha})^{2}\tilde{\Gamma}_{pn\to\ell_{\alpha}^{+}\bar{\nu}_{\beta}}^{-1}\;.$
(46)

Due to the stronger experimental limits on
$pp\to\ell^{+}_{\alpha}\ell^{+}_{\beta}$ and
$pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}$, through the above relations, we can
set new stronger limits on the neutral modes
$nn\to\bar{\nu}_{\alpha}\bar{\nu}_{\beta}$. Taking the experimental limits for
the charged modes into consideration and requiring
$\tilde{\Gamma}_{pp\to\ell_{\alpha}^{+}\ell_{\beta}^{+}}^{-1}\gtrsim\tau_{pp\to\ell_{\alpha}^{+}\ell_{\beta}^{+}}^{\rm
exp}$ and
$\tilde{\Gamma}_{pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}}^{-1}\gtrsim\tau_{pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}}^{\rm
exp}$, we obtain

$\displaystyle\\{\tilde{\Gamma}^{-1}_{nn\to\bar{\nu}_{e}\bar{\nu}_{e}},\tilde{\Gamma}^{-1}_{nn\to\bar{\nu}_{e}\bar{\nu}_{\mu}},\tilde{\Gamma}^{-1}_{nn\to\bar{\nu}_{\mu}\bar{\nu}_{\mu}}\\}\gtrsim
4\times 10^{33}{~{}\rm yr}\;,$
$\displaystyle\\{\tilde{\Gamma}^{-1}_{nn\to\bar{\nu}_{e}\bar{\nu}_{\tau}},\tilde{\Gamma}^{-1}_{nn\to\bar{\nu}_{\mu}\bar{\nu}_{\tau}}\\}\gtrsim
2\times 10^{32}{~{}\rm yr}\;.$ (47)

One can see the limits on $nn\to\bar{\nu}_{\alpha}\bar{\nu}_{\beta}$ are
improved by $2-3$ orders of magnitude than the direct experimental search in
Tab. 1. On the other hand, if we assume the charged modes also exclusively
mediated by the same operators, then the experimental bounds on
$pp\to\ell^{+}_{\alpha}\ell^{+}_{\beta}$ imply the following new bounds on
$pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}$ for $\alpha,\beta=e,\mu$ flavors,

$\displaystyle\\{\tilde{\Gamma}^{-1}_{pn\to
e^{+}\bar{\nu}_{e}},\tilde{\Gamma}^{-1}_{pn\to\mu^{+}\bar{\nu}_{\mu}}\\}\gtrsim
8\times 10^{33}{~{}\rm yr}\;,$ $\displaystyle\\{\tilde{\Gamma}^{-1}_{pn\to
e^{+}\bar{\nu}_{\mu}},\tilde{\Gamma}^{-1}_{pn\to\mu^{+}\bar{\nu}_{e}}\\}\gtrsim
4\times 10^{33}{~{}\rm yr}\;,$ (48)

which are also stronger than the current experimental bounds by at least an
order of magnitude. Conversely, the experimental limit on
$pn\to\ell^{+}_{e}\bar{\nu}_{\tau}$ can further translate into a bound on the
transition $pp\to e^{+}\tau^{+}$, which is also kinetically allowed but has
not yet been searched for experimentally. Based on Eq. (46) and the limit on
$pn\to\ell^{+}_{e}\bar{\nu}_{\tau}$ in Tab. 1, we obtain

$\displaystyle\tilde{\Gamma}^{-1}_{pp\to e^{+}\tau^{+}}\gtrsim 2\times
10^{34}~{}\rm yr\;,$ (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, we only consider the contribution from operator ${\cal
O}_{Q^{6}L^{2}}^{S,(S)}$ and take
$\Lambda\equiv\left[C_{Q^{6}L^{2}}^{S,(S)}\right]^{-1/8}$. The relevant
hadronic LEC $\hat{g}_{1\times 3,a}$ is set equal to $6.4\times 10^{-5}~{}\rm
GeV$. From the figure we see the partial lifetime is very sensitive to the NP
scale, because of the large power dependence($\propto\Lambda^{16}$). For a
future experimental sensitivity about $10^{40}$ yr the NP scale is pushed
towards $5~{}{\rm TeV}$ or so.

Figure 2: The partial lifetime of the dinucleon to dilepton transitions as a
function of the NP scale in the SMEFT. Where we assume the contribution from
operator ${\cal O}_{Q^{6}L^{2}}^{S,(S)}$ and set
$\Lambda\equiv\left[C_{Q^{6}L^{2}}^{S,(S)}\right]^{-1/8}$.

Finally, we make a brief comments on the concrete NP models and the collider
signals. For a given NP model, one can integrate out the heavy new physics
states and match onto the dim-12 SMEFT operators. In literature, there exist
models contributing to dinucleon to dilepton transitions with $\Delta B=\Delta
L=2$ but not giving rise to the $\Delta B=1$ nucleon decays or $\Delta B=2$
neutron-antineutron oscillation. In Refs. Arnold  _et al._ (2013); Gardner and
Yan (2019), the authors considered a class of such models which involve new
scalar-fermion and scalar-quartic interactions, meanwhile, the left-right
symmetric models with extra-dimensions considered in Refs. Girmohanta and
Shrock (2020a); Girmohanta (2020) 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
$\Delta B=\Delta L=2$ NP signals at the current/future high energy colliders
to be interesting. Ref. Bramante  _et al._ (2015) has made such a try by
studying the process $pp\to e^{+}e^{+}+4~{}{\rm jets}$ based on a dim-12 SMEFT
operator (similar to the operator ${\cal O}_{Q^{6}L^{2}}^{S,(S)}$ in our
basis). However, the eight fermion operator considered in Bramante  _et al._
(2015) cannot yet be processed by the FeynRules to MG5AMC framework, and the
authors take a “stand-in” operator for estimation. Such a procedure could
yield large uncertainty, and we would like to come back to the 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 violating dinucleon to dilepton decays
($pp\to\ell^{+}\ell^{\prime+},pn\to\ell^{+}\bar{\nu}^{\prime},nn\to\bar{\nu}\bar{\nu}^{\prime}$)
with $\Delta B=\Delta L=-2$ in the framework of effective field theory. We
first construct a basis of dim-12 operators mediating such processes in the
low energy effective field theory (LEFT) below the electroweak scale. Such a
basis not only contribute to the dinucleon decays studied in this work, it
also serves as a starting point for model independent study of the hydrogen-
antihydrogen oscillation and the low energy baryon number violating
conversions $e^{-}p\to
e^{+}\bar{p},\bar{\nu}\bar{n},e^{-}n\to\bar{\nu}\bar{p}$ in electron-deuteron
scattering. Then we consider their leading-order standard model effective
field theory (SMEFT) completions upwards and obtain the SMEFT basis mediating
such processes at dim 12. We find the SMEFT gauge symmetry has a strong
constraint on the structure of the interactions. The dim-12 SMEFT operators
are suitable for the high energy signature on colliders like LHC to search the
excess of events with four jets plus two same-sign charged leptons.

Next, we analyze the chiral structure of the LEFT operators and make a non-
perturbative matching through the baryon chiral perturbation theory
(B$\chi$PT). In doing so, we construct a chiral basis in which each operator
belongs to an irreducible representation of the two-flavor chiral group
$SU(2)_{L}^{f}\times SU(2)_{R}^{f}$, and then we construct the corresponding
hadronic operators through the spurion techniques. Last, we express the
dinucleon to dilepton decay rates in terms of the Wilson coefficients
associated with the dim-12 operators in the LEFT/SMEFT and the low energy
constants pertinent to B$\chi$PT. Our result is general in that it does not
depend on dynamical details of physics at a high scale that induce the
effective interactions in the SMEFT and in that it does not appeal to any
hadronic models. We find the current experimental limits push the associated
new physics scale larger than a few TeV, a scale appealing to the future
experimental searches. Due to the weak isospin symmetry, based on the
experimental limits on
$pp\to\ell^{+}_{\alpha}\ell^{+}_{\beta},pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}$,
we improve the lower limits on the partial lifetimes of the neutral transition
modes $nn\to\bar{\nu}_{\alpha}\bar{\nu}_{\beta}$ (except the
$(\alpha,\beta)=(\tau,\tau)$ case) by $2-3$ orders of magnitude than their
current experimental sensitivity. Furthermore, assuming these transitions
dominantly generated by the similar dim-12 SMEFT operators, we find the limits
on the partial lifetime of $pp\to e^{+}e^{+},e^{+}\mu^{+},\mu^{+}\mu^{+}$ are
also transformed into stronger limits on
$pn\to\ell^{+}_{\alpha}\bar{\nu}_{\beta}$ ($\alpha,\beta=e,\mu$) than their
existing lower bounds.

Our operator basis obtained in this work is a starting point for further
investigation on the related processes with $\Delta B=\Delta L=2$ signals, in
which the hydrogen-antihydrogen oscillation and the collider signals
$pp\to\ell^{+}\ell^{\prime+}+4~{}\rm jets$ and $e^{-}p\to\ell^{+}+5~{}\rm jet$
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 useful discussion concerning the collider aspect. The authors
also thank J. Bramante for providing us with their FeynRules code to clarify
what they had done in Bramante  _et al._ (2015). XDM would like to thank F-S
Yu for his invitation as a visitor at Lanzhou Uni. where part of this work has
been done. This work 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 Laboratory for Particle Physics,
Astrophysics and Cosmology, Ministry of Education, and Shanghai Key Lab-
oratory for Particle Physics and Cosmology (Grant No. 15DZ2272100).

##  Appendix A Independent color tensors

In this appendix, we give details the independent color tensors to contract
with six quarks in a color $SU(3)_{\rm C}$ invariant way. Denoting a general
six-quark field configuration as

$\displaystyle{\cal
O}^{ijklmn}=q^{i}_{I}q^{j}_{J}q^{k}_{K}q^{l}_{L}q^{m}_{M}q^{n}_{N}\;,$ (A.1)

where the superscripts $\\{i,j,k,l,m,n\\}$ are the color indices in
fundamental representation of $SU(3)_{\rm C}$ while the subscripts
$\\{I,J,K,L,M,N\\}$ encode the flavor and chiral information for each quark
field. To form a color invariant operator, the color indices must be
contracted by a color tensor $T_{ijklmn}$ such that ${\cal
O}^{ijklmn}T_{ijklmn}$ is invariant under $SU(3)_{\rm C}$ . Since all quark
fields belong to the fundamental representation of the color group, the color
tensor $T_{ijklmn}$ must be a linear combination of two rank-3 totally anti-
symmetric tensor $\epsilon_{xyz}$s. By the Schouten identities (SI) Liao and
Ma (2020)

$\displaystyle\epsilon_{imn}\epsilon_{jkl}$
$\displaystyle=-\epsilon_{ijm}\epsilon_{kln}+\epsilon_{ikm}\epsilon_{jln}-\epsilon_{ilm}\epsilon_{jkn}\;,$
$\displaystyle\epsilon_{ijm}\epsilon_{kln}$
$\displaystyle=\epsilon_{ikm}\epsilon_{jln}+\epsilon_{ikn}\epsilon_{jlm}-\epsilon_{ilm}\epsilon_{jkn}-\epsilon_{iln}\epsilon_{jkm}-\epsilon_{ijn}\epsilon_{klm}\;,$
(A.2)

we can split the $m$-index and $n$-index into two epsilon tensors via the
first SI, in turn, there are six independent combinations remained and the
second SI further reduces them into five independent ones. By symmetrizing or
anti-symmetrizing pairs of indices $(ij),(kl),(mn)$, we can choose the
following five independent color tensors:

$\displaystyle T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}$
$\displaystyle=\epsilon_{ikm}\epsilon_{jln}+\epsilon_{ikn}\epsilon_{jlm}+\epsilon_{ilm}\epsilon_{jkn}+\epsilon_{iln}\epsilon_{jkm}\;,$
$\displaystyle T^{SAA}_{\\{ij\\}[kl][mn]}$
$\displaystyle=\epsilon_{imn}\epsilon_{jkl}+\epsilon_{ikl}\epsilon_{jmn}=\epsilon_{ikm}\epsilon_{jln}-\epsilon_{ikn}\epsilon_{jlm}-\epsilon_{ilm}\epsilon_{jkn}+\epsilon_{iln}\epsilon_{jkm}\;,$
$\displaystyle T^{SAA}_{\\{kl\\}[mn][ij]}$
$\displaystyle=\epsilon_{ijk}\epsilon_{mnl}+\epsilon_{ijl}\epsilon_{mnk}=\epsilon_{ikm}\epsilon_{jln}-\epsilon_{ikn}\epsilon_{jlm}+\epsilon_{ilm}\epsilon_{jkn}-\epsilon_{iln}\epsilon_{jkm}\;,$
$\displaystyle T^{SAA}_{\\{mn\\}[ij][kl]}$
$\displaystyle=\epsilon_{ijm}\epsilon_{kln}+\epsilon_{ijn}\epsilon_{klm}=\epsilon_{ikm}\epsilon_{jln}+\epsilon_{ikn}\epsilon_{jlm}-\epsilon_{ilm}\epsilon_{jkn}-\epsilon_{iln}\epsilon_{jkm}\;,$
$\displaystyle T^{AAA}_{[ij][kl][mn]}$ $\displaystyle={1\over
3}\left(\epsilon_{ijm}\epsilon_{kln}-\epsilon_{ijn}\epsilon_{klm}-\epsilon_{ijk}\epsilon_{mnl}+\epsilon_{ijl}\epsilon_{mnk}+\epsilon_{ikl}\epsilon_{jmn}-\epsilon_{imn}\epsilon_{jkl}\right)$
$\displaystyle=\epsilon_{ijm}\epsilon_{kln}-\epsilon_{ijn}\epsilon_{klm}\;.$
(A.3)

where the subscripts in curly bracket $\\{ij\\}$ and squared bracket $[kl]$
indicate separately the symmetrization and anti-symmetrization under the
exchange of two color indices $i\leftrightarrow j$ within. In the above, the
$T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}$ and $T^{AAA}_{[ij][kl][mn]}$ are
separately totally symmetric and anti-symmetric under the exchange of any
pairs of the arguments, and $T^{SAA}_{\\{ij\\}[mn][kl]}$ is symmetric for the
latter two pairs of indices. In addition, we have the constraints for
exchanging two indices among two different pairs of indices

$\displaystyle 2T^{SSS}_{\\{ik\\}\\{jl\\}\\{mn\\}}$
$\displaystyle=-T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}+3T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle 2T^{SAA}_{\\{ij\\}[km][ln]}$
$\displaystyle=T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}+T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle 2T^{SAA}_{\\{ik\\}[jl][mn]}$
$\displaystyle=-T^{SAA}_{\\{ij\\}[kl][mn]}+T^{SAA}_{\\{kl\\}[mn][ij]}+2T^{AAA}_{[ij][kl][mn]}\;,$
$\displaystyle 2T^{SAA}_{\\{in\\}[kl][mj]}$
$\displaystyle=-T^{SAA}_{\\{ij\\}[kl][mn]}-T^{SAA}_{\\{mn\\}[ij][kl]}-2T^{AAA}_{[ij][kl][mn]}\;,$
$\displaystyle 2T^{AAA}_{[ik][jl][mn]}$
$\displaystyle=T^{SAA}_{\\{ij\\}[kl][mn]}+T^{SAA}_{\\{kl\\}[mn][ij]}\;.$ (A.4)

These relations are useful to reduce redundant operators, and will be used
repeatedly in latter sections to reach the minimal basis for the dim-12
$\Delta B=\Delta L=-2$ operators both in the LEFT and in the SMEFT .

##  Appendix B LEFT operators

The full list for dim-12 operators inducing dinucleon to dilepton transitions.

$\scriptscriptstyle\blacksquare$ Dim-12 operators contributing to
$pp\to\ell^{+}\ell^{\prime+}$
For the operators with a scalar current $j_{S,\pm}^{\ell\ell^{\prime}}$, we
find there are 28 independent operators which can be parametrized as follows

$\displaystyle{\cal Q}_{1LLL,a}^{(pp)S,\pm}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}Cd_{L}^{n})j_{S,\pm}^{\ell\ell^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{1LLL,b}^{(pp)S,\pm}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}Cd_{L}^{n})j_{S,\pm}^{\ell\ell^{\prime}}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{2LLR,a}^{(pp)S,\pm}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S,\pm}^{\ell\ell^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{2LLR,b}^{(pp)S,\pm}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S,\pm}^{\ell\ell^{\prime}}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{3LLR,a}^{(pp)S,\pm}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cu_{R}^{n})j_{S,\pm}^{\ell\ell^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{3LLR,b}^{(pp)S,\pm}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cu_{R}^{n})j_{S,\pm}^{\ell\ell^{\prime}}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{4LLR}^{(pp)S,\pm}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cu_{L}^{l})(d_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S,\pm}^{\ell\ell^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
(B.1)

together with their parity partners with $L\leftrightarrow R$.

For the operators with a vector lepton current
$j_{V}^{\ell\ell^{\prime},\mu}$, there are 19 independent operators which are
chosen to take

$\displaystyle{\cal Q}^{(pp)V}_{1LL,a}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{1LL,b}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{1LL,c}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SAA}_{\\{mn\\}[kl][ij]}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{2LL,a}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{2LL,b}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{3LR,a}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{3LR,b}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{4LR,a}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{4LR,b}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{4LR,c}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SAA}_{\\{kl\\}[mn][ij]}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{4LR,d}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}^{(pp)V}_{4LR,e}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\ell^{\prime},\mu}T^{AAA}_{[ij][kl][mn]}\;,$
(B.2)

together with the parity partners for operators ${\cal Q}^{V}_{1-3}$ with
$L\leftrightarrow R$.

For the operators with a tensor lepton current
$j_{T}^{\ell\ell^{\prime},\mu\nu}$, we find there are 16 independent operators
which are chosen to take

$\displaystyle{\cal Q}_{1LLL}^{(pp)T,-}$ $\displaystyle=(u_{L}^{i{\rm
T}}C\sigma_{\mu\nu}u_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}Cd_{L}^{n})j_{T,-}^{\ell\ell^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{2LLR,a}^{(pp)T,-}$ $\displaystyle=(u_{L}^{i{\rm
T}}C\sigma_{\mu\nu}u_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{T,-}^{\ell\ell^{\prime},\mu\nu}T^{SAA}_{\\{kl\\}[mn][ij]}\;,$
$\displaystyle{\cal Q}_{2LLR,b}^{(pp)T,-}$ $\displaystyle=(u_{L}^{i{\rm
T}}C\sigma_{\mu\nu}u_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{T,-}^{\ell\ell^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{2LLR,c}^{(pp)T,-}$ $\displaystyle=(u_{L}^{i{\rm
T}}C\sigma_{\mu\nu}u_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{T,-}^{\ell\ell^{\prime},\mu\nu}T^{AAA}_{[ij][kl][mn]}\;,$
$\displaystyle{\cal Q}_{2LLR,a}^{(pp)T,+}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}C\sigma_{\mu\nu}d_{R}^{n})j_{T,+}^{\ell\ell^{\prime},\mu\nu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{2LLR,b}^{(pp)T,+}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}C\sigma_{\mu\nu}d_{R}^{n})j_{T,+}^{\ell\ell^{\prime},\mu\nu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{3LLR}^{(pp)T,-}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}C\sigma_{\mu\nu}d_{L}^{l})(u_{R}^{m{\rm
T}}Cu_{R}^{n})j_{T,-}^{\ell\ell^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{3LLR}^{(pp)T,+}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}C\sigma_{\mu\nu}u_{R}^{n})j_{T,+}^{\ell\ell^{\prime},\mu\nu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
(B.3)

together with the parity partners for operators ${\cal Q}^{T}_{1-3}$ with
$L\leftrightarrow R$ and $-\leftrightarrow+$.

As a non-trivial example for the reduction of redundant operators, we consider
the above tensor current operator ${\cal Q}_{3LRR}^{(pp)T,+}$ in Eq. (B.3)
with the replacement of the color tensor $T^{SAA}_{\\{ij\\}[kl][mn]}$ by
$T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}$, then the new operator is reduced as
follows

$\displaystyle 2(u_{L}^{i{\rm T}}Cu_{L}^{j})(u_{R}^{k{\rm
T}}Cd_{R}^{l})(u_{R}^{m{\rm
T}}C\sigma_{\mu\nu}d_{R}^{n})j_{T,+}^{\ell\ell^{\prime},\mu\nu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}$
$\displaystyle\overset{\rm FI}{=}-2(u_{L}^{i{\rm
T}}Cu_{L}^{j})\left[(u_{R}^{m{\rm T}}Cd_{R}^{l})(u_{R}^{k{\rm
T}}C\sigma_{\mu\nu}d_{R}^{n})+(u_{R}^{k{\rm T}}Cu_{R}^{m})(d_{R}^{l{\rm
T}}C\sigma_{\mu\nu}d_{R}^{n})\right]j_{T,+}^{\ell\ell^{\prime},\mu\nu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}$
$\displaystyle\overset{\rm SI}{=}(u_{L}^{i{\rm T}}Cu_{L}^{j})(u_{R}^{k{\rm
T}}Cd_{R}^{l})(u_{R}^{m{\rm
T}}C\sigma_{\mu\nu}d_{R}^{n})j_{T,+}^{\ell\ell^{\prime},\mu\nu}\left(T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}+3T^{SAA}_{\\{ij\\}[kl][mn]}\right)$
$\displaystyle~{}~{}~{}~{}+(u_{L}^{i{\rm T}}Cu_{L}^{j})(u_{R}^{k{\rm
T}}Cu_{R}^{l})(d_{R}^{m{\rm
T}}C\sigma_{\mu\nu}d_{R}^{n})j_{T,+}^{\ell\ell^{\prime},\mu\nu}\left(T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}-3T^{SAA}_{\\{ij\\}[kl][mn]}\right)(=0)\;,$

where in the second step we have used the FIs in Appendix C and the third step
the SI in Eq. (A.4), and the terms in last line vanish due to mismatched color
symmetry. We see this new operator is equivalent to ${\cal
Q}_{3LRR}^{(pp)T,+}$ and therefore redundant. All other operators with
different color tensors or Lorentz structures beyond the above lists can be
reduced in a similar manner.

$\scriptscriptstyle\blacksquare$ Dim-12 operators contributing to
$pn\to\ell^{+}\nu^{\prime}$
For the operators with a scalar current $j_{S}^{\ell\nu^{\prime}}$, there are
14 independent operators which can be parametrized as follows

$\displaystyle{\cal Q}_{1LLL,a}^{(pn)S}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}Cd_{L}^{n})j_{S}^{\ell\nu^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{1LLL,b}^{(pn)S}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}Cd_{L}^{n})j_{S}^{\ell\nu^{\prime}}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{2LLR}^{(pn)S}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(d_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S}^{\ell\nu^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{3LLR,a}^{(pn)S}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S}^{\ell\nu^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{3LLR,b}^{(pn)S}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S}^{\ell\nu^{\prime}}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{3LLR,c}^{(pn)S}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S}^{\ell\nu^{\prime}}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{4LLR}^{(pn)S}$ $\displaystyle=(d_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cu_{R}^{n})j_{S}^{\ell\nu^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
(B.4)

together with their parity partners with $L\leftrightarrow R$.

For the operators with a vector current $j_{V}^{\ell\nu^{\prime},\mu}$, we
find there are 24 independent operators which are parametrized as follows

$\displaystyle{\cal Q}_{1LL,a}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{1LL,b}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{1LL,c}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{2LL,a}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(d_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{2LL,b}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(d_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{2LL,c}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(d_{L}^{m{\rm
T}}C\gamma_{\mu}u_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{3LR}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cu_{L}^{j})(d_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{4LR,a}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{4LR,b}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{4LR,c}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SAA}_{\\{kl\\}[mn][ij]}\;,$
$\displaystyle{\cal Q}_{4LR,d}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{4LR,e}^{(pn)V}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\gamma_{\mu}d_{R}^{n})j_{V}^{\ell\nu^{\prime},\mu}T^{AAA}_{[ij][kl][mn]}\;,$
(B.5)

together with their parity partners with $L\leftrightarrow R$.

For the operators with a tensor current $j_{T}^{\ell\nu^{\prime},\mu\nu}$,
there are 13 independent operators which can be parametrized as follows

$\displaystyle{\cal Q}_{LLL,a}^{(pn)T}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\sigma_{\mu\nu}d_{L}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{LLL,b}^{(pn)T}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{L}^{m{\rm
T}}C\sigma_{\mu\nu}d_{L}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{2LLR}^{(pn)T}$ $\displaystyle=(u_{L}^{i{\rm
T}}C\sigma_{\mu\nu}u_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(d_{R}^{m{\rm
T}}Cd_{R}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{3LLR,a}^{(pn)T}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}C\sigma_{\mu\nu}d_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{3LLR,b}^{(pn)T}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}C\sigma_{\mu\nu}d_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{kl\\}[mn][ij]}\;,$
$\displaystyle{\cal Q}_{3LLR,c}^{(pn)T}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}C\sigma_{\mu\nu}d_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{3LLR,d}^{(pn)T}$ $\displaystyle=(u_{L}^{i{\rm
T}}Cd_{L}^{j})(u_{L}^{k{\rm T}}C\sigma_{\mu\nu}d_{L}^{l})(u_{R}^{m{\rm
T}}Cd_{R}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{AAA}_{[ij][kl][mn]}\;,$
$\displaystyle{\cal Q}_{4LLR}^{(pn)T}$ $\displaystyle=(d_{L}^{i{\rm
T}}C\sigma_{\mu\nu}d_{L}^{j})(u_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cu_{R}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{2RRL}^{(pn)T}$ $\displaystyle=(u_{R}^{i{\rm
T}}Cu_{R}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(d_{L}^{m{\rm
T}}C\sigma_{\mu\nu}d_{L}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{3RRL,a}^{(pn)T}$ $\displaystyle=(u_{R}^{i{\rm
T}}Cd_{R}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\sigma_{\mu\nu}d_{L}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{3RRL,b}^{(pn)T}$ $\displaystyle=(u_{R}^{i{\rm
T}}Cd_{R}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\sigma_{\mu\nu}d_{L}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{3RRL,c}^{(pn)T}$ $\displaystyle=(u_{R}^{i{\rm
T}}Cd_{R}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\sigma_{\mu\nu}d_{L}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{4RRL}^{(pn)T}$ $\displaystyle=(d_{R}^{i{\rm
T}}Cd_{R}^{j})(u_{R}^{k{\rm T}}Cd_{R}^{l})(u_{L}^{m{\rm
T}}C\sigma_{\mu\nu}u_{L}^{n})j_{T}^{\ell\nu^{\prime},\mu\nu}T^{SAA}_{\\{ij\\}[kl][mn]}\;.$
(B.6)

$\scriptscriptstyle\blacksquare$ Dim-12 operators contributing to
$nn\to\bar{\nu}\bar{\nu}^{\prime}$
For the operators with a scalar current $j_{S}^{\nu\nu^{\prime}}$, there are
14 independent operators which are parametrized as follows

$\displaystyle{\cal Q}_{1LLL,a}^{(nn)S}$ $\displaystyle=(d_{L}^{i{\rm
T}}Cd_{L}^{j})(d_{L}^{k{\rm T}}Cu_{L}^{l})(d_{L}^{m{\rm
T}}Cu_{L}^{n})j_{S}^{\nu\nu^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{1LLL,b}^{(nn)S}$ $\displaystyle=(d_{L}^{i{\rm
T}}Cd_{L}^{j})(d_{L}^{k{\rm T}}Cu_{L}^{l})(d_{L}^{m{\rm
T}}Cu_{L}^{n})j_{S}^{\nu\nu^{\prime}}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{2LLR,a}^{(nn)S}$ $\displaystyle=(d_{L}^{i{\rm
T}}Cd_{L}^{j})(d_{L}^{k{\rm T}}Cu_{L}^{l})(d_{R}^{m{\rm
T}}Cu_{R}^{n})j_{S}^{\nu\nu^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{2LLR,b}^{(nn)S}$ $\displaystyle=(d_{L}^{i{\rm
T}}Cd_{L}^{j})(d_{L}^{k{\rm T}}Cu_{L}^{l})(d_{R}^{m{\rm
T}}Cu_{R}^{n})j_{S}^{\nu\nu^{\prime}}T^{SAA}_{\\{ij\\}[kl][mn]}\;,$
$\displaystyle{\cal Q}_{3LLR,a}^{(nn)S}$ $\displaystyle=(d_{L}^{i{\rm
T}}Cu_{L}^{j})(d_{L}^{k{\rm T}}Cu_{L}^{l})(d_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S}^{\nu\nu^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
$\displaystyle{\cal Q}_{3LLR,b}^{(nn)S}$ $\displaystyle=(d_{L}^{i{\rm
T}}Cu_{L}^{j})(d_{L}^{k{\rm T}}Cu_{L}^{l})(d_{R}^{m{\rm
T}}Cd_{R}^{n})j_{S}^{\nu\nu^{\prime}}T^{SAA}_{\\{mn\\}[ij][kl]}\;,$
$\displaystyle{\cal Q}_{4LLR}^{(nn)S}$ $\displaystyle=(d_{L}^{i{\rm
T}}Cd_{L}^{j})(d_{L}^{k{\rm T}}Cd_{L}^{l})(u_{R}^{m{\rm
T}}Cu_{R}^{n})j_{S}^{\nu\nu^{\prime}}T^{SSS}_{\\{ij\\}\\{kl\\}\\{mn\\}}\;,$
(B.7)

together with their parity partners with $L\leftrightarrow R$.

And for the operators with a tensor neutrino current
$j_{T}^{\nu\nu^{\prime},\mu\nu}$, there are only 8 independent operators which
are parametrized as follows

$\displaystyle{\cal Q}_{1LLL}^{(nn)T}$ $\displaystyle=(d_{L}^{i{\rm
T}}C\sigma_{\mu\nu}d_{L}^{j})(d_{L}^{k{\rm T}}Cu_{L}^{l})(d_{L}^{m{\rm
T}}Cu_{L}^{n})j_{T}^{\nu\nu^{\prime},\mu\nu}T^{SAA}_{\\{mn\\}[kl][ij]}\;,$
$\displaystyle{\cal Q}_{2LLR,a}^{(nn)T}$ $\displaystyle=(d_{L}^{i{\rm
T}}C\sigma_{\mu\nu}d_{L}^{j})(d_{L}^{k{\rm T}}Cu_{L}^{l})(d_{R}^{m{\rm