# Future DUNE constraints on EFT

###### Abstract

In the near future, fundamental interactions at high-energy scales may be most efficiently studied via precision measurements at low energies. A universal language to assemble and interpret precision measurements is the so-called SMEFT, which is an effective field theory (EFT) where the Standard Model (SM) Lagrangian is extended by higher-dimensional operators. In this paper we investigate the possible impact of the DUNE neutrino experiment on constraining the SMEFT. The unprecedented neutrino flux offers an opportunity to greatly improve the current limits via precision measurements of the trident production and neutrino scattering off electrons and nuclei in the DUNE near detector. We quantify the DUNE sensitivity to dimension-6 operators in the SMEFT Lagrangian, and find that in some cases operators suppressed by an TeV scale can be probed. We also compare the DUNE reach to that of future experiments involving atomic parity violation and polarization asymmetry in electron scattering, which are sensitive to an overlapping set of SMEFT parameters.

91405 Orsay, FranceInstituto de Física, Universidade de São Paulo,

C.P. 66.318, 05315-970 São Paulo, BrazilInstituto de Física, Universidade de São Paulo,

C.P. 66.318, 05315-970 São Paulo, Brazil

## 1 Introduction

Current experimental measurements at high and low energies are well described by the Standard Model (SM) Lagrangian, which respects the local symmetry and contains only renormalizable interactions with canonical dimensions . At the same time, there are several theoretical arguments for the existence of physics beyond the SM (BSM) at some mass scale . It is reasonable to assume that there is a significant gap between the new physics and the weak scales: . If that is the case, the effects of new particles at the energy scale below can be described in a systematic way using the SM effective field theory (SMEFT). The framework consists in maintaining the SM particle content and symmetry structure, while abandoning the renormalizability requirement, so that interaction terms with are allowed Buchmuller:1985jz (). The higher dimensional operators constructed out of the SM fields describe, in a model independent way, all possible effects of heavy new physics.

In this sense, searching for new physics can be redefined as setting constraints on the Wilson coefficients of higher-dimensional operators in the SMEFT Lagrangian. There exist many previous studies along these lines, focusing mostly on the dimension-6 operators (which are expected to give leading contributions for ). A prominent place in this program is taken by precision measurements at very low energies, well below Han:2004az (); Han:2005pr (); Cirigliano:2009wk (); Carpentier:2010ue (); Filipuzzi:2012mg (); Berthier:2015gja (); Falkowski:2015krw (); Gonzalez-Alonso:2016etj (). These include neutrino scattering off nucleus or electron targets, atomic parity violation, parity-violating scattering of electrons, and a plethora of meson, nuclear, and tau decay processes. For certain dimension-6 operators the low-energy input offers a superior sensitivity compared to that achievable in colliders such as LEP or the LHC, see Ref. Falkowski:2017pss () for a recent summary. But even in those cases where the new physics reach of low-energy measurements is worse, they often play a vital role in lifting flat directions in the notoriously multi-dimensional parameter space of the SMEFT.

While efforts to get the best out of the existing data continue, it is also important to discuss what progress can be achieved in the future. This kind of studies facilitates planning new experiments and analyses, and allow one to understand the complementarity between the low-energy and collider programs. In this paper we focus on the future impact of the the Deep Underground Neutrino Experiment (DUNE) Acciarri:2015uup (). Given the intense neutrino beam, the massive far detector (FD), and the envisaged scale of the near detector (ND), DUNE will certainly offer a rich physics program. The main goal is to measure the parameters governing neutrino oscillations: the CP violating phase in the PMNS matrix and the neutrino mass ordering. Searches for proton decay and for neutrinos from core-collapse supernovas in our galaxy are also a part of the program.

From the effective theory perspective, the unprecedented neutrino flux in DUNE offers a unique opportunity to improve the limits on several dimension-6 operators in the SMEFT Lagrangian. This can be achieved via precision measurements of various abundant processes in the DUNE ND, such as the production of lepton pairs by a neutrino incident on a target nucleus (the so-called trident production), and neutrino scattering off electrons and nuclei. In this work we quantitatively study the potential of these processes to probe the dimension-6 operators. Moreover, since the neutrino couplings are related to the charged lepton ones due to the local symmetry of the SMEFT, we also include in our analysis other future experiments that do not involve neutrinos: parity-violating Møller scattering, atomic parity violation (APV), and polarization asymmetries in scattering of electrons on nuclei. Finally, we combine our projections with the constraints from existing measurements as described by the likelihood function obtained in Ref. Falkowski:2017pss (). The SMEFT sensitivity of current and future experiments is compared, with and without the input from DUNE and under different hypotheses about systematic errors.

The paper is organized as follows. In Section 2 we explain our formalism. In particular, we review the EFT valid at a few GeV scale relevant for the DUNE experiment, and we describe its matching with the SMEFT. Section 3 is devoted to quantitative studies of various neutrino scattering processes in the DUNE ND. Other relevant experimental results which do not involve neutrinos are discussed in Section 4. In Section 5 we translate the projected constraints discussed in Sections 3 and 4 into the SMEFT language and we give our projections for future constraints. The main results are summarized in Tab. 6 and Fig. 4. Section 6 presents our conclusions.

## 2 Formalism

If heavy BSM particles exist in nature, the adequate effective theory at is the so-called SMEFT.
It has the same particle spectrum and local symmetry as the SM, however the Lagrangian admits higher-dimensional (non-renormalizable) interactions which encode BSM effects.
However, the SMEFT is not the optimal framework for dealing with observables measured at .
At a few GeV scale relevant for DUNE, the propagating particles are the light SM fermions together with the gauge bosons: the photon and gluons.
At these energies the , , and Higgs bosons as well as the top quark can be integrated out, and their effects can be encoded in contact interactions between the light particles.
To mark the difference from the SMEFT, we refer to the effective theory below as the weak EFT (wEFT).^{1}^{1}1Other names are frequently used in the literature to describe this effective theory, e.g. the Fermi theory or the LEFT Jenkins:2017jig ().
It is an effective theory with a limited validity range, and at energies it has to be matched to a more complete theory with the full SM spectrum and the larger local symmetry.
If the SM were the final theory, that matching would uniquely predict the wEFT Wilson coefficients in terms of the experimentally well known SM parameters.
On the other hand, when the wEFT is matched to the SMEFT, the wEFT Wilson coefficients deviate from the SM predictions due to the effects of the higher-dimensional SMEFT operators.
This way, measurements of wEFT parameters in experiments with provide non-trivial information about new physics: they allow one to derive constraints on (and possibly to discover) higher-dimensional SMEFT interactions.
The latter can be readily translated into constraints on masses and couplings of a large class of BSM theories.

The future DUNE results are best interpreted in a model-independent way as constraints on the wEFT Wilson coefficients, but constraints on new physics are more conveniently presented as a likelihood function for the SMEFT Wilson coefficients. Below we summarize the wEFT interactions relevant for our analysis and the tree-level map between the wEFT and SMEFT coefficients.

### 2.1 Neutrino interactions with charged leptons

In order to characterize neutrino trident production and neutrino–electron scattering in DUNE we will need the neutrino interactions with electrons and muons.
In the wEFT, neutrinos interact at tree level with charged leptons via the effective 4-fermion operators:^{2}^{2}2
For fermions we use the 2-component spinor notation where a Dirac fermion is represented as a pair of Weyl spinors , : .
We follow the conventions and notation of Ref. Dreiner:2008tw ().

(1) |

where the sum over repeated generation indices is implicit (only the first two generations are relevant for our purpose). Integrating out the and bosons at tree level in the SM yields the effective Lagrangian

Here , , , and , are the gauge couplings of . Thus, at tree level, the SM predictions for the wEFT couplings in Eq. (1) are given by

(3) |

For the numerical SM values we use , Olive:2016xmw (), which incorporates some loop correction effects. For the remaining operators we use the analytic expression in terms of evaluated at the central value of the low-energy Weinberg angle Olive:2016xmw ().

Going beyond the SM we have , where can be calculated in terms of some high-energy parameters once the UV completion of the wEFT is specified.^{3}^{3}3In the neutrino literature BSM effects in the wEFT Lagrangian are often referred to as non-standard interactions (NSI) and parametrized by .
The translation between that language and our formalism is simple: , with defined as in Ref. Farzan:2017xzy ().
Here we assume that the wEFT is matched at to the SMEFT Lagrangian truncated at the level of dimension-6 operators.
Moreover, for the purpose of studying neutrino scattering at GeV energies, one can safely ignore the dimension-5 SMEFT operators which give masses to the neutrinos.
Thus we consider the Lagrangian ,
where is the SM Lagrangian, GeV, each is a gauge-invariant operator of dimension =6, and are the corresponding Wilson coefficients.
We will work consistently up to linear order in , neglecting quadratic and higher powers.
In full generality, such a framework introduces 2499 new independent free parameters Grzadkowski:2010es (); Alonso:2013hga (), but working at tree level only a small subset of those is relevant for our analysis.
The relevant parameter space can be conveniently characterized
by a set of vertex corrections to the and interactions with leptons, and by Wilson coefficients of 4-lepton operators Gupta:2014rxa (); deFlorian:2016spz ().
The former are defined via the Lagrangian

(4) | |||||

where we display only the flavor-diagonal interactions. Not all the vertex corrections above are independent, as in the dimension-6 SMEFT there is the relation . The vertex corrections can be expressed by a combination of dimension-6 Wilson coefficients in any operator basis (see e.g. Falkowski:2017pss () for the map to the so-called Warsaw basis), but it is much more convenient to span the relevant parameter space with ’s. The remaining parameters we make use here are Wilson coefficients of the 4-lepton operators collected in Table 1. Note that only the ones containing the left-handed lepton doublet give rise to neutrino interactions, but for completeness we also list the ones without .

One flavor () | Two flavors () |
---|---|

We are ready to write down the tree-level matching equations between the wEFT and SMEFT parameters. For the wEFT couplings relevant for our analysis we find

(5) |

(6) |

(7) |

(8) |

(9) |

A part of the results above may appear counterintuitive. For example depends on and , and likewise depends on . This happens because some dimension-6 SMEFT operators affect the observables , , and which traditionally serve as the input to determine the SM parameters , and from experiment. To take this into account one needs to absorb this effect into a redefinition of the SM parameters, which brings new terms into the matching equation. The most dramatic consequence is that and do not depend on new physics at all. That is because the corresponding 4-fermion SMEFT operator is responsible for the muon decay, from which the Fermi constant is experimentally determined.

### 2.2 Neutrino interactions with quarks

Another class of processes relevant in DUNE is the charged current (CC) and neutral current (NC) scattering of neutrinos on atomic nuclei. This can be characterized by 4-fermions wEFT interactions of neutrinos with up and down quarks:

(10) |

We do not display here the CC interactions of right-handed quark and chirality-violating NC interactions, as their effects for neutrino scattering in DUNE are suppressed. denotes the CKM matrix defined in such a way that most of new physics corrections affecting observables from which is experimentally determined are absorbed in its definition Gonzalez-Alonso:2016etj (). In the following we will use the numerical value , which is the central value of the fit in Gonzalez-Alonso:2016etj (), although more generally in a global analysis one should fit simultaneously with new physics parameters. In the SM the wEFT parameters in Eq. (10) take the values , , , , and . Neutrino scattering experiments typically measure ratios of NC to CC neutrino or anti-neutrino cross sections. For this reason, it is convenient to introduce the following notation:

(11) |

where the SM predictions are , , , Olive:2016xmw ().

With lepton doublets | Without lepton doublets |
---|---|

Turning to the SMEFT, much as in the 4-lepton case before, the space of relevant dimension-6 operators can be parameterized by vertex corrections and a set of 4-fermion operators. The former are defined by

(12) | |||||

The relevant 4-fermion operators are summarized in Table 2. Only the ones containing lepton doublets are relevant for neutrino interactions, but for future references we have also displayed the operators affecting only charged lepton couplings to quarks. Writing and matching at tree level to the SMEFT Lagrangian at we obtain the following relations between the parameters of the two theories:

(13) |

(14) |

(15) |

Above, we work in the approximation where the off-diagonal CKM parameters are treated as zero when they multiply dimension-6 SMEFT parameters; see Falkowski:2017pss () for more general expressions. Again, a somewhat counterintuitive expressions for follows from absorbing part new physics effects into the definition . The translation to the NSI language is and .

## 3 Neutrino scattering in DUNE

The DUNE design includes a near detector (ND) located at m from the proton beam target, as well as a far detector (FD) with kilotonnes of argon mass at km from the source. The ND is primarily designed to provide constraints on the systematic uncertainties in oscillation studies. On the other hand, thanks to the extremely high rate of neutrino interactions, it can be readily used for precision cross section measurements.

For the following analysis we assume + years of operation for neutrino+antineutrino beams and a near detector of 100 tonnes argon mass. Each beam of neutrinos (antineutrinos) consists of approximately beams and contamination of antineutrinos (neutrinos). In addition, each beam contains less than of and from pion decays. Our analysis is based on calculating the number of events at DUNE, for which we use the neutrino fluxes given in Ref. Alion:2016uaj (). The neutrino energies in DUNE range from GeV to GeV, however for our analysis we consider GeV bins between GeV and GeV as the contribution from the higher energies to the total flux is negligible. We calculate the expected number of events as:

(16) |

where is the neutrino flux, is the relevant cross section, is the neutrino energy, GeV and GeV. Here the number of targets is calculated for POT (proton on target) in (anti-)neutrino mode with a 120-GeV proton beam with 1.20 MW of power.

In the following we describe the relevant observables at DUNE for trident production, elastic neutrino scattering on electrons and neutrino scattering off nuclei.

### 3.1 Trident production

We start with the trident production in DUNE ND. The neutrino trident events are the production of lepton pairs by a neutrino incident on a heavy nucleus: . The trident process has been previously observed by the CHARM-II Vilain:1994qy () and CCFR Mishra:1991bv () experiments. However, due to technological limitations in the detector design, these experiments could access the final state and the accuracy of the cross section measurement was only %.

The cross sections in the SM for different neutrino channels are taken from Magill:2016hgc (). We assume flat neutrino efficiencies, for and for and a detector resolution of for muons (electrons) Acciarri:2015uup (). The expected number of events for the trident channels we consider in our analysis of DUNE ND are given in Table 3.

beam | beam | ||
---|---|---|---|

We can now interpret the DUNE trident event in terms of constraints on the wEFT parameters. The general tree-level formula for the ratio of the trident cross section to its SM-predicted value reads

(17) |

The SM values of the wEFT couplings are given in Eq. (2.1). Here parameterize the deviations from the SM prediction, and we neglect the quadratic terms in ’s. The expressions for the relevant in terms of the SMEFT parameters are given in Eqs. (2.1)-(2.1). Note that, given , the processes and and their conjugates do not probe new physics at all (at least at tree level). In fact, they are just another measurement of , which of course cannot compete with the ultra-precise determination based on the muon decay. They may still be useful for calibration or normalization purposes, but by themselves they do not carry any information about BSM interactions.

The remaining trident processes do probe new physics, as indicated in the dependence of the appropriate wEFT couplings on the SMEFT coefficients. Here we focus on the trident process with a muon pair in the final state. Since the neutrino and antineutrino channels probe the same wEFT coefficient, we can define the following ratio:

(18) |

For , the expected contribution to the production is much smaller than the statistical error of the measurement and can be safely neglected. In the following we assume the measurement error for will be dominated by statistics, and that the central value is given by the SM prediction. If that is the case, the numbers in Table 3 translate to the following forecast:

(19) |

which in turn translate into the following measurement of the wEFT coefficients

(20) |

### 3.2 Neutrino scattering off electrons

We turn to neutrino scattering on electrons.
For the CC process the threshold energy is GeV and therefore its rate is negligible in DUNE.^{4}^{4}4Even if this process were abundant in DUNE it would not probe new physics for the reasons explained below Eq. (2.1).
We focus on the NC processes
and .
The total cross section can be expressed in terms of the wEFT parameters as

(21) |

where is the center-of-mass energy squared of the collision, is the incoming neutrino energy in the lab frame, and is the electron mass. Plugging the above cross sections into Eq. (16) and integrating over the incoming neutrino energy spectrum we obtain the total number of neutrino scattering events in the neutrino and antineutrino modes. The total number of the scattering events predicted by the SM is given in Table 4, where we also give the fractional contribution of and initiated processes, as well as the contribution due to the electron neutrino contamination. Comparing the results of Table 3 and Table 4 we note that the number of events for the neutrino-electron scattering is larger than the one for trident, even though the cross sections are of the same order for both. This is a consequence of the fact that the number of electron targets is times larger than the number of nucleus targets.

-mode | 0.898 | 0.059 | 0.040 | 0.003 | |
---|---|---|---|---|---|

-mode | 0.103 | 0.867 | 0.013 | 0.017 |

We define the following observables which can be measured in DUNE:

(22) |

where () is the fraction of () in the incoming beam in the neutrino () and antineutrino () modes. We assume the effect of the electron neutrino contamination of the beam can be estimated and subtracted away, and in the following analysis we approximate , , and . Writing , the deviation from the SM prediction can be expressed by the following combination of the wEFT parameters:

(23) |

We assume that the error of measuring will be statistically dominated and that the central values will coincide with the SM prediction. Given the number of events in each mode displayed in Table 4, we can forecast the following constraint on the wEFT parameter

(24) |

DUNE will by no means be the first probe of effective neutrino couplings to electrons. The scattering cross section of muon neutrinos and antineutrinos on electrons was measured e.g. in the CHARM Dorenbosch:1988is (), CHARM-II Vilain:1994qy (), and BNL-E734 Ahrens:1990fp () experiments with an accuracy. The existing constraints on the wEFT parameters are combined by PDG Olive:2016xmw ():

(25) |

where in the notation of Eq. (1)