Future DUNE constraints on EFT
Revised: March
DUNE constraints on EFT
Adam Falkowski 0 2
Giovanni Grilli di Cortona 0 1
Zahra Tabrizi 0 1
0 91405 Orsay , France
1 Instituto de F sica, Universidade de Sa~o Paulo
2 Laboratoire de Physique Theorique, CNRS, Univ. ParisSud, Universite ParisSaclay
In the near future, fundamental interactions at highenergy scales may be most e ciently studied via precision measurements at low energies. A universal language to assemble and interpret precision measurements is the socalled SMEFT, which is an e ective eld theory (EFT) where the Standard Model (SM) Lagrangian is extended by higherdimensional operators. In this paper we investigate the possible impact of the DUNE neutrino experiment on constraining the SMEFT. The unprecedented neutrino ux o ers an opportunity to greatly improve the current limits via precision measurements of the trident production and neutrino scattering o near detector. We quantify the DUNE sensitivity to dimension6 operators in the SMEFT Lagrangian, and nd that in some cases operators suppressed by an O(30) 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.
Beyond Standard Model; E ective Field Theories; Neutrino Physics

Future
1 Introduction Formalism 2 3
4
5
6
2.1
2.2
3.1
3.2
3.3
Neutrino interactions with charged leptons
Neutrino interactions with quarks
Neutrino scattering in DUNE
Trident production
Neutrino scattering o electrons
Neutrino scattering o nuclei
Related precision experiments
Future evolution of SMEFT constraints Conclusions 1 2
program is taken by precision measurements at very low energies, well below mZ [2{9].
These include neutrino scattering o
nucleus or electron targets, atomic parity violation,
parityviolating scattering of electrons, and a plethora of meson, nuclear, and tau decay
{ 1 {
processes. For certain dimension6 operators the lowenergy input o ers a superior
sensitivity compared to that achievable in colliders such as LEP or the LHC, see ref. [10] for a
recent summary. But even in those cases where the new physics reach of lowenergy
measurements is worse, they often play a vital role in lifting at directions in the notoriously
multidimensional parameter space of the SMEFT.
While e orts 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 lowenergy and collider programs. In this paper we focus on the future impact of the
Deep Underground Neutrino Experiment (DUNE) [11]. Given the intense neutrino beam,
the massive far detector (FD), and the envisaged scale of the near detector (ND), DUNE
will certainly o er 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 corecollapse
supernovas in our galaxy are also a part of the program.
From the e ective theory perspective, the unprecedented neutrino ux in DUNE o ers
a unique opportunity to improve the limits on several dimension6 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 socalled trident production), and neutrino scattering o electrons and
nuclei. In this work we quantitatively study the potential of these processes to probe the
dimension6 operators. Moreover, since the neutrino couplings are related to the charged
lepton ones due to the SU(2) local symmetry of the SMEFT, we also include in our analysis
other future experiments that do not involve neutrinos: parityviolating M ller 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. [10]. The SMEFT sensitivity
of current and future experiments is compared, with and without the input from DUNE
and under di erent 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 table 6 and
gure 4. section 6 presents our conclusions.
2
Formalism
If heavy BSM particles exist in nature, the adequate e ective theory at E & mZ is the
socalled SMEFT. It has the same particle spectrum and local symmetry as the SM, however
the Lagrangian admits higherdimensional (nonrenormalizable) interactions which encode
{ 2 {
BSM e ects. However, the SMEFT is not the optimal framework for dealing with
observables measured at E
mZ . At a few GeV scale relevant for DUNE, the propagating
particles are the light SM fermions together with the SU(3)C
U(1)em gauge bosons: the
photon and gluons. At these energies the W , Z, and Higgs bosons as well as the top quark
can be integrated out, and their e ects can be encoded in contact interactions between the
light particles. To mark the di erence from the SMEFT, we refer to the e ective theory
below mZ as the weak EFT (wEFT).1 It is an e ective theory with a limited validity range,
and at energies E
mZ it has to be matched to a more complete theory with the full SM
spectrum and the larger SU(3)C
U(1)Y local symmetry. If the SM were the
nal theory, that matching would uniquely predict the wEFT Wilson coe cients in terms
of the experimentally well known SM parameters. On the other hand, when the wEFT is
matched to the SMEFT, the wEFT Wilson coe cients deviate from the SM predictions
due to the e ects of the higherdimensional SMEFT operators. This way, measurements
of wEFT parameters in experiments with E
mZ provide nontrivial information about
new physics: they allow one to derive constraints on (and possibly to discover)
higherdimensional 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 modelindependent way as
constraints on the wEFT Wilson coe cients, but constraints on new physics are more
conveniently presented as a likelihood function for the SMEFT Wilson coe cients. Below we
summarize the wEFT interactions relevant for our analysis and the treelevel map between
the wEFT and SMEFT coe cients.
2.1
Neutrino interactions with charged leptons
In order to characterize neutrino trident production and neutrinoelectron 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 e ective 4fermion operators:2
LwEFT
2
v2
( a
b) hgLabLcd(ec
ed) + gLabRcd(ecc
c i
ed) ;
(2.1)
where the sum over repeated generation indices a; b; c; d is implicit (only the rst two
generations are relevant for our purpose). Integrating out the W and Z bosons at tree
level in the SM yields the e ective Lagrangian
1Other names are frequently used in the literature to describe this e ective theory, e.g. the Fermi theory
or the LEFT [12].
of Weyl spinors f , f c: F = (f; f c)T . We follow the conventions and notation of ref. [13].
2For fermions we use the 2component spinor notation where a Dirac fermion F is represented as a pair
{ 3 {
Here gLW ` = gL, gZf = T 3
f
s2Yf , s2 = gY2 =(gL2 + gY2 ), and gL, gY are the gauge couplings
of SU(2)L
U(1)Y . Thus, at tree level, the SM predictions for the wEFT couplings in
eq. (2.1) are given by
gLabLc;dSM =
gLabRcd;SM = s2 ab cd:
1
2
+ s2
ab cd + ad bc;
For the numerical SM values we use gL22L1;1SM
= (gLVe;SM + gLAe;SM)=2 =
e
gL22R1;1SM = (gLV;SM
gLAe;SM)=2 = 0:2334 [
14
], which incorporates some loop correction
effects. For the remaining operators we use the analytic expression in terms of s2 evaluated
at the central value of the lowenergy Weinberg angle s2 = 0:23865 [
14
].
Going beyond the SM we have gLabXcd = gLabXcd;SM + gLabXcd, where
gLabXcd can be
calculated in terms of some highenergy parameters once the UV completion of the wEFT
is speci ed.3
Here we assume that the wEFT is matched at E
mZ to the SMEFT
Lagrangian truncated at the level of dimension6 operators. Moreover, for the purpose
of studying neutrino scattering at GeV energies, one can safely ignore the dimension5
SMEFT operators which give masses to the neutrinos. Thus we consider the Lagrangian
L = LSM +
Pi vc2i OD=6, where LSM is the SM Lagrangian, v = ( 2GF ) 1=2
i
' 246 GeV,
each OD=6 is a gaugeinvariant operator of dimension D=6, and ci are the corresponding
i
Wilson coe cients. We will work consistently up to linear order in ci, neglecting quadratic
p
and higher powers. In full generality, such a framework introduces 2499 new independent
free parameters [
16, 17
], 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 g to the Z and W interactions with leptons, and by Wilson coe cients
of 4lepton operators [18, 19]. The former are de ned via the Lagrangian
(2.3)
0:2730,
qgL2 + gY2 Z e
c
a
s2Qf + gZea e
R
c
a
X
f=e;
f a
T
f
3
s2Qf + gZfa fa;
L
(2.4)
where we display only the avordiagonal interactions. Not all the vertex corrections above
are independent, as in the dimension6 SMEFT there is the relation g
Z a
L
L
gZea = gW ea .
L
The vertex corrections can be expressed by a combination of dimension6 Wilson coe cients
in any operator basis (see e.g. [10] for the map to the socalled Warsaw basis), but it is
much more convenient to span the relevant parameter space with
g's. The remaining
parameters we make use here are Wilson coe cients of the 4lepton operators collected in
rise to neutrino interactions, but for completeness we also list the ones without `.
3In the neutrino literature BSM e ects in the wEFT Lagrangian are often referred to as nonstandard
interactions (NSI) and parametrized by ff0X . The translation between that language and our formalism
is simple: ieiX = gLiiX11, with ieiX de ned as in ref. [15].
{ 4 {
gLW e. This happens because some
dimension6 SMEFT operators a ect the observables GF , (0), and mZ which traditionally
serve as the input to determine the SM parameters gL, gY and v from experiment. To take
gL11L11 depends
{ 5 {
this into account one needs to absorb this e ect into a rede nition of the SM parameters,
which brings new terms into the matching equation. The most dramatic consequence is that
gL11L22 and gL22L11 do not depend on new physics at all. That is because the corresponding
4fermion SMEFT operator is responsible for the muon decay, from which the Fermi constant
GF is experimentally determined.
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
4fermions wEFT interactions of neutrinos with up and down quarks:
LwEFT
2Vv~2ud (1 + dLea )(ea
a)(u
d)
2
v2
( a
a)
X
q=u;d
gLaLq q
q + gLaRq(qc
The relevant LLQQ 4fermion 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 a ecting only charged lepton couplings to quarks. Writing
gLaXq = gLaXq;SM + gLaXq and matching at tree level to the SMEFT Lagrangian at E
mZ
{ 6 {
dea
notation:
We do not display here the CC interactions of righthanded quark and chiralityviolating
NC interactions, as their e ects for neutrino scattering in DUNE are suppressed. V~ud
denotes the CKM matrix de ned in such a way that most of new physics corrections
a ecting observables from which Vud is experimentally determined are absorbed in its
de nition [9]. In the following we will use the numerical value V~ud = 0:97451, which is the
central value of the t in [9], although more generally in a global analysis one should t V~ud
simultaneously with new physics parameters. In the SM the wEFT parameters in eq. (2.10)
take the values gLaLu;SM = 12
23s2 , gLaRu;SM =
23s2 , gLaLd;SM =
or antineutrino cross sections. For this reason, it is convenient to introduce the following
gLa=R
2
2
gLaLu=LR
+ gLaLd=LR
1 + dLea 2
2
;
a
tan L=R
gLaLu=LR ;
gLaLd=LR
where the SM predictions are (gL;SM)2 = 0:3034, (gR;SM)2 = 0:0302, tan L;SM =
0:80617,
tan R;SM =
Turning to the SMEFT, much as in the 4lepton case before, the space of relevant
dimension6 operators can be parameterized by vertex corrections and a set of 4fermion
operators. The former are de ned by
LSMEFT
we obtain the following relations between the parameters of the two theories:
as zero when they multiply dimension6 SMEFT parameters; see [10] for more general
expressions. Again, a somewhat counterintuitive expressions for dLe follows from
absorbing part new physics e ects into the de nition V~ud. In the neutrino literature the BSM
e ects in the wEFT Lagrangian are often referred to as nonstandard interactions (NSI)
and parametrized by ff0X . The translation to the NSI language is auadL = V~ud L
dea and
qaXa = gLaXq , with aa's de ned as in ref. [15].
3
Neutrino scattering in DUNE
The DUNE design includes a near detector (ND) located at 574 m from the proton beam
target, as well as a far detector (FD) with 34 kilotonnes of argon mass at 1297 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 3+3 years of operation for neutrino+antineutrino
beams and a near detector of 100 tonnes argon mass. Each beam of neutrinos
(antineutrinos) consists of approximately 90%
( ) beams and 10% contamination of antineutrinos
(neutrinos). In addition, each beam contains less than 1% of e and e from pion decays.
Our analysis is based on calculating the number of events at DUNE, for which we use the
neutrino uxes given in ref. [20]. The neutrino energies in DUNE range from 0:25 GeV to
{ 7 {
!
e ! e
beam
+
+
357
1:27
!
e ! e
beam
+
+
305
for each beam. We do not display the numbers for processes with one or two electrons in the nal
state, as this will be studied in a future publication [26], however we estimated that these have a
negligible e ect on the SMEFT ts. The
trident events have vanishing rates.
20 GeV, however for our analysis we consider 0:25 GeV bins between 0:25 GeV and 8:25 GeV
as the contribution from the higher energies to the total ux is negligible. We calculate
the expected number of events as:
N = time
# of targets
e ciency
Z Ef
Ei
dE
d (E )
dE
(E ) ;
where
is the neutrino ux,
is the relevant cross section, E is the neutrino energy,
Ei = 0:25 GeV and Ef = 8:25 GeV. Here the number of targets is calculated for 1:1
POT (proton on target) in (anti)neutrino mode with a 120GeV 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 o nuclei.
3.1
Trident production
resolution of
given in table 3.
(e)
E
We start with the trident production in DUNE ND. The trident events are the production
of lepton pairs by a neutrino incident on a heavy nucleus:
aN !
bec+ed N , and their
potential to probe new physics was emphasized in refs. [21, 22]. The trident process has
been previously observed by the CHARMII [23] and CCFR [24] experiments. However,
due to technological limitations in the detector design, these experiments could access the
+
nal state and the accuracy of the cross section measurement was only O(50)%.
The cross sections in the SM for di erent neutrino channels are taken from [25]. We
assume
at neutrino e ciencies, 85% for
( ) and 80% for e ( e) and a detector
=
0:2(0:15)=pE (GeV) for muons (electrons) [11]. The expected
number of events for the trident channels we consider in our analysis of DUNE ND are
We can now interpret the DUNE trident event in terms of constraints on the wEFT
parameters. The general treelevel formula for the ratio of the trident cross section to its
SMpredicted value reads
( b
SM( b
+
! a`c `d )
! a`c `d+)
=
( a
SM( a
+
! b`c `d )
! b`c `d+)
1+2 gLabLc;dSM gLabLcd +gLabRcd;SM gLabRcd :
(gLabLc;dSM)2 +(gLabRc;dSM)2
(3.2)
The SM values of the wEFT couplings gLabXcd;SM are given in eq. (2.3). Here gLabXcd
parameterize the deviations from the SM prediction, and we neglect the quadratic terms in
(3.1)
1021
For R , the expected e 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 R
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:
which in turn translate into the following measurement of the wEFT coe cients
R
= 1
0:039;
0:039 < 2 gL22L2;2SM gL22L22 + gL22R2;2SM gL22R22
(gL22L2;2SM)2 + (gL22R2;2SM)2
< 0:039:
3.2
Neutrino scattering o electrons
We turn to neutrino scattering on electrons. For the CC process
e
e the threshold
energy is m2 =2me
the NC processes
10:9 GeV and therefore its rate is negligible in DUNE.4 We focus on
!
e and
e
!
e . The total cross section can be expressed
in terms of the wEFT parameters as
g's. The expressions for the relevant gLabXcd in terms of the SMEFT parameters are given
in eqs. (2.5){(2.9). Note that, given
and e
and their conjugates do not probe new physics at all (at least at tree
level). In fact, they are just another measurement of GF , which of course cannot compete
with the ultraprecise 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 coe cients. Here we focus on the
trident process with a muon pair in the
nal state. Since the neutrino and antineutrino
channels probe the same wEFT coe cient, we can de ne the following ratio:
(
(
!
!
+) + (
+)SM + (
+
!
!
+)
+)SM
:
e =
e =
2 v4 (gL22L11)2 +
2 v4 (gL22R11)2 +
1
3
1
3
(gL22R11)2
(gL22L11)2
meE
meE
v4
v4
(gL22L11)2 +
(gL22R11)2 ;
(gL22R11)2 +
(gL22L11)2 ;
where s = 2meE is the centerofmass energy squared of the collision, E is the incoming
neutrino energy in the lab frame, and me is the electron mass. Plugging the above cross
sections into eq. (3.1) 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
4Even if this process were abundant in DUNE it would not probe new physics for the reasons explained
below eq. (2.9).
{ 9 {
R
e
s
s
!
1
3
1
3
(3.3)
(3.4)
(3.5)
(3.6)
Ntot
e
mode
mode
1:69
1:29
106
106
r
e
of DUNE ND. We also give the fractional contribution to the total rate of the four contributing
processes: e= e
! e= e
and e= e
! e= e .
the contribution due to the electron neutrino contamination. Comparing the results of
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 O(103) times
larger than the number of nucleus targets.
We de ne the following observables which can be measured in DUNE: (3.7) (3.8)
i
R e
xi
e + xi
xi SMe + xi SM ;
e
e
where xi (xi) is the fraction of
( ) in the incoming beam in the neutrino (i = ) and
antineutrino (i = ) modes. We assume the e ect of the electron neutrino contamination
of the beam can be estimated and subtracted away, and in the following analysis we
approximate x
= 0:9, x
= 0:1, and xi = 1
xi. Writing Ri e = 1 + Ri e, the deviation from
the SM prediction can be expressed by the following combination of the wEFT parameters:
Ri e = 2
(1 + 2xi) gL22L11g2211
LL;SM + (3
(1 + 2xi) (gL22L1;1SM)2 + (3
2xi) gL22R11g2211
2xi) (gL22R1;1SM)2
LR;SM :
We assume that the error of measuring Ri e 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
8:0
DUNE will by no means be the rst probe of e ective neutrino couplings to electrons.
The scattering cross section of muon neutrinos and antineutrinos on electrons was measured
e.g. in the CHARM [27], CHARMII [23], and BNLE734 [28] experiments with an O(1)%
accuracy. The existing constraints on the wEFT parameters are combined by PDG [
14
]:
gLVe =
0:040
0:015;
gLAe =
0:507
0:014;
(3.10)
where in the notation of eq. (2.1) gL22L11 = (gLVe +gLAe)=2 and gL22R11 = (gLVe
gLAe)=2. DUNE
is expected to dramatically improve on the existing constraints, as is evident in
gure 1,
even under the hypothesis that the systematic errors will greatly exceed the statistical ones.
HJEP04(218)
0.02
1
δgL2L211
eq. (2.1) from elastic neutrino scattering on electrons in DUNE ND, assuming the measurement
errors will be dominated by statistics. The dashed line shows the analogous constraints assuming 1%
systematic error on the Ri e measurements. The green region is allowed by the past
e scattering
experiments [
14
].
mode
mode
mode
mode
NtCotC
NtNotC
4:25
1:74
1:48
7:58
108
108
108
107
rCC
0.007
0.004
0.006
0.003
rCC
e
0.001
0.005
in 3 + 3 years of operation of the DUNE ND. The fractional contribution to the total rate of the
four contributing processes are also calculated.
3.3
Neutrino scattering o nuclei
We turn to discussing neutrino scattering o nuclei. To estimate the number of CC and
NC scattering events expected in the SM we use the cross sections quoted in ref. [20].
The results for the number of events and the fractional contributions of di erent neutrino
avors are given in table 5.
In order to reduce the impact of systematic uncertainties, experiments typically
measure the ratio of the NC and CC neutrino or antineutrino scattering cross sections on
nuclei,
NC= CC. For isoscalar target nuclei, when the incoming beam contains the
fraction x of neutrinos A and x
(1
x) of antineutrinos a of the same avor, the ratio of
NC to CC scattering events can be written as with
R aN
x
x
aN! aN + x
aN!ea N + x
aN!ea+N
aN! aN = (gLa )2 + r 1(gRa )2;
r =
x
x
aN!ea N + x
aN!ea N + x
aN!ea+N ;
aN!ea+N
(3.11)
(3.12)
HJEP04(218)
and the e ective couplings gLa=R de ned in eq. (2.11). This is a simple generalization
of the wellknown LlewellynSmith formula [29] usually presented in the limit x = 1 or
x = 0. For any value of x the dependence on the nuclear structure is contained only in the
factor r which can be separately measured in experiment. In the following analysis we use
eq. (3.11) for a
/
incoming beam, with x
= 0:9 in the mode, and x
= 0:1 in the
mode, in which case r
0:4. In reality, for the scattering process in DUNE
ND there will be important corrections to eq. (3.11). First of all, the 40Ar target nuclei
are not isoscalar, which implies corrections to the LlewellynSmith formula with a more
complicated dependence on the nuclear structure. In particular, R
N will also depend on
the ratio of neutrino e ective couplings to up and down quarks. Furthermore, the incoming
beam has an O(1)% admixture of electron neutrinos. We note that, in the general case
where the neutrino e ective couplings to quarks may depend on the lepton generation, the
dependence on the nuclear structure does not cancel in the ratio of NC to CC scattering
events when the incoming beam contains more than one neutrino
avor. The impact of
this DUNE measurement will crucially depend on how well these systematic e ects can be
controlled. Assuming they can be accurately constrained by dedicated measurements of
CC cross sections [11], we consider the best case scenario where the measurement errors
are dominated by statistics. Writing Ri
N = Ri
N;SM(1 +
Ri
N ), the deviation of the
ratio from the SM prediction can be constrained by DUNE as
9:6
Expanding eq. (3.11) linearly in the wEFT Wilson coe cients we get
R
i
N ' 2 gL;SM g
L + ri 1gR;SM g
(gL;SM)2 + ri 1(gR;SM)2R ;
(3.14)
where in terms of the parameters in eq. (2.10) we have gX;SM g
X = P
q=u;d gLXq;SM gLXq
(gX;SM)2 L .
d
0.004
0.002
ΝΜ gR 0.000
∆
0.002
0.004
0.004
0.002
will be dominated by statistics. The dashed line shows the analogous constraints assuming 0.1%
L and g
R in
systematic error on the Ri
N measurements. The green region is allowed by the past N scattering
In the past, electron neutrino scattering on nuclei was measured in the CHARM
experiment [30], albeit with a poor accuracy, which is currently being improved thanks to the
COHERENT experiment [31]. Much more information is available regarding the scattering
cross sections of muon neutrinos on nuclei [32{34], which probe the interaction terms in
eq. (2.10). The combined analysis in the PDG yields the constraints on the parameter
combinations de ned in eq. (2.11): (gL )
= 4:56+00::4227 [
14
]. One can see that the couplings gL=R are
probed with a relative accuracy of order 1%, which should be improved in DUNE. In this
case the DUNE errors are unlikely to be statistics dominated, due to the nuclear structure
dependent corrections to the LlewellynSmith formula discussed above. However, even a
O(0:1)% accuracy would lead to a dramatic improvement of the existing constraints, as is
shown in gure 2.
Finally, it should be noted that in a year run in the neutrino mode, the intense
neutrino source will provide approximately 108 total CC+NC neutrino interactions in a 100t
near detector; and almost 0.4 times in the antineutrino mode. Hence, thanks to the
extremely high rate of neutrino interactions, one expects the near detector to be systematics
dominated within the rst year of the data taking.
In section 5 we will use the results in eqs. (3.5), (3.9) and (3.13) in section 5 to estimate
the impact of the DUNE ND measurements on the global SMEFT t.5 We note in passing
that e ective 4fermion interactions of neutrinos with quarks can also be probed via matter
e ects in neutrino oscillations [15, 35{37], see also [38{47] for projected DUNE constraints.
Estimating the sensitivity of oscillation processes to the wEFT parameters is beyond the
scope of this paper.
4
Related precision experiments
In the SMEFT framework, the neutrino couplings are correlated with the charged lepton
ones due to the underlying SU(2)L symmetry. In this context, it is important to have in
mind a more global picture, and include in the analysis other relevant experimental results
which do not necessarily involve neutrinos.
An important source of information about lepton couplings are the measurements of
the e+e
! ea+ea di erential cross sections in LEP1 [48] and LEP2 [49] experiments.
The LEP1 data place permille level constraints on the vertex corrections gZf , f = q; `,
while the LEP2 data probe gLW f , and the 4fermion LLLL and LLQQ operators at a
percent level. We will take into account the LEP input by directly using the likelihood
function given in ref. [50].
At lower energies, some relevant information is provided by parityviolating M ller
scattering, which probes the electron's axial selfcoupling in the wEFT:
LwEFT
1
2v2 gAeeV [ (e
e)(e
e) + (ec
ec)(ec
ec)] :
(4.1)
gAeeV = 0:0190
racy on sin2
gAeeV = 0:0225
couplings
The SLAC E158 measurement [51] can be translated into the current constraint
0:0027 [
14
]. In the near future, the MOLLER collaboration in JLAB
will signi cantly reduce the error on the parameter gAeeV .
The projected accu
W (mZ ) quoted in ref. [
52
] can be recasted into the future constraint
0:0006, which will reduce the current error by almost a factor of 5.
In the SMEFT context, there are many additional probes of the neutrino couplings
to quarks, as they are correlated with charged lepton couplings. For example, the wEFT
LwEFT
1
2v2 gAeqV (e
e
e
c
ec)(q
q + qc
qc)
(4.2)
can be probed at low energies by measurements of APV [53] and polarization asymmetries
in scattering of electrons on nuclei [54, 55]. The PDG quotes the combined constraints
5The 0:1 (1)% systematic errors mentioned in gures 1 and 2 and table 6 are the systematic errors in
the measurement of R de ned in Eqs (3.3), (3.7) and (3.11). For the DUNE measurements we de ne
2 = X
R
2
&
R
1
2 + 2
1
sys
;
with
R's and their errors de ned in Eqs (3.5), (3.83.9) and (3.133.14). The main sources of systematic
uncertainties in DUNE will be on the beam
ux normalization and detector performance. A careful study
on the e ect of di erent systematic uncertainties of DUNE in the measurement of the SM parameters is
necessary and will be done in a future publication [26].
0.005
ed gAV 0.000
δ
0.005
APV
Ra225
P2C
P2H
Future Weak Charge Constraints
SoLID
0.005
parameters gAedV and gAedV . The dashed lines show the projected 95 % CL constraints separately from
P2 with the hydrogen (blue) and carbon (black) target, SoLID (brown), and APV in 225Ra (red).
0:005 and 2gAeuV
gAedV =
0:708
0:016, where the SM predicts
ggAAeeuuVV;SM+ 2=gAedV = 0:489
0:1887 and gAedV;SM = 0:3419 [
14
]. These should be signi cantly improved in
the near future, thanks to precise determination of APV in 225Ra ions [56], and improved
measurement of polarization asymmetries in the Qweak [54], MESA P2 targets [57], and
SoLID [58] experiments. To estimate the future bounds, for APV and Qweak we translate
the projected sensitivity to s2 into a constraint on the relevant combination of gAeqV , for P2
we assume the relative uncertainty on the weak charge measurement of 1:7% (0:3%) for the
hydrogen (carbon) target, while for SoLID we use the likelihood extracted from
gure 3 of
ref. [59]. De ning gAeqV
level: gAeuV = (0:29
gAeqV;SM, we estimate the future constraints will be permille
0:65) 10 3, gAedV = ( 0:54: 0:88) 10 3, as illustrated in gure 3.
At higher energies, where the wEFT is no longer valid and the full SMEFT formalism
needs to be used, leptonquark interactions can be probed in colliders.
Existing LEP
measurements [48, 49] of electronquark interactions are compiled in [10] with the conclusion
that, in combination with the lowenergy measurements, chiralityconserving
2electron2quark operators are constrained in a modelindependent way at a percent level. The
sensitivity improves greatly when lepton pair DrellYan production at the LHC is taken
into account, and in fact 2lepton2quark operators can be accessed for all three lepton
generations [60{62]. For eeqq and eedd operators the current sensitivity is at a permille
level, and is expected to improve to an O(10 4) level after the highluminosity run is
completed [
62
]. That level of precision may be di cult to beat in DUNE. However, a full
modelindependent analysis of the LHC constraints has not been attempted yet, due to a
large number of di erent SMEFT operators contributing to the DrellYan process. In this
analysis we focus on lowenergy precision measurements, and we do not include the LHC
constraints. It is reasonable to expect that the LHC will leave at or poorly constrained
directions in this parameter space, along which Wilson coe cient much larger than
can be present without con icting the data. The task of lifting these at directions would
then be left for DUNE and other lowenergy precision experiments.
5
Future evolution of SMEFT constraints
In this section we translate the projected lowenergy constraints discussed in sections 3 and 4
into the SMEFT language. The goal is to illustrate the place of DUNE in the landscape
of lowenergy precision observables, and to quantify its sensitivity to new physics at
highenergy scales. With this goal in mind, we consider deformations of the SM where only a
single SMEFT parameter is nonzero at a time. Then we compare the constraints on that
parameter using the current and future lowenergy experiments. For the current ones we
use the likelihood function compiled in ref. [10]. This is subsequently combined with the
future DUNE constraints based on the projections in section 3, and with the future APV and
polarization asymmetry constraints discussed in section 4. To illustrate DUNE's impact
on the precision program we separately show the future projections with and without the
DUNE input. Furthermore, the DUNE projections are shown for 3 scenarios: (very)
optimistic with the error dominated by statistics, more realistic with 10 3 systematic errors,
and pessimistic with 10 2 systematic errors.
Our results are summarized in table 6 where we display the constraints on those Higgs
basis parameters for which the projected error is reduced by at least a factor of 5 compared
to the current one. We can see that DUNE will potentially have a dramatic impact on
constraining the SMEFT parameter space. Without the DUNE input, the precision that
can be achieved is typically an order of magnitude worse (for the operators we consider and
given the future experiments we take into account in our analysis). An O(10 4) relative
precision can be achieved for some parameters such as the Z boson coupling to muons or
light quarks or certain 4fermion LLQQ operators. This translates into O(30) (O(300))
TeV indirect reach for new BSM particles, assuming they are weakly (strongly) coupled to
the SM muons and light quarks. Those scales are well beyond the direct reach of current and
nearfuture colliders, and also surpass the indirect reach of the past electroweak precision
tests in the LEP e+e
collider. Even for the more realistic scenario about systematic
errors the improvement in the reach for new physics is spectacular in some cases. On
the other hand, in the pessimistic scenario the impact of DUNE on the precision program
is marginal, except in the 4muon sector where the existing loose trident constraints can
be improved by an order of magnitude. These considerations highlight the importance of
precision measurements in DUNE and the e orts to reduce experimental and theoretical
sources of systematic errors.
Coe cient
(current)
(no sys.)
(0:1% sys.)
(1% sys.)
(w/o DUNE)
gW e
L
g
Z
L
gZu
L
gZu
R
gZd
L
gZd
R
gW q1
R
in units of 10 4 on SMEFT Wilson coe cient from current and future
lowenergy precision measurements, assuming only one Wilson coe cient is nonzero at a time. We
display only those coe cients in the Higgs basis for which the constraints can be improved by at
least a factor of ve in the bestcase scenario. The current uncertainties are extracted from ref. [10],
while the future ones also include projections from
scattering on electrons and nucleons and
trident production in DUNE, as well from APV and parityviolating electron scattering experiments
discussed in section 4. The future constraints are shown under three di erent hypotheses about
the size of systematic errors in DUNE. The nal column shows the future projections without the
DUNE input.
Technically there is no problem including the future projections into a fully global
SMEFT analysis akin to that in ref. [10] where all dimension6 operators are present
simultaneously. Such a step will be crucial once the real data are available, as only a global
analysis is basis independent and contains the complete information about the constraints.
However, in the present case such an exercise is not very illuminating and we do not show
global t projections here. The reason is that the DUNE and other observable we consider
in this paper constrain only a limited number of linear combinations of the Wilson
coefcients, leaving many weakly constrained directions in the SMEFT parameter space. As
a result, the improvement in sensitivity is hardly visible in our global t projection, again
with the exception of the 4muon operators.
HJEP04(218)
4
1
4
6
Current
Future w/o DUNE
Future w/ DUNE
Future w/ DUNE+syst.
δgLZμ
δgLZu
[cle]2211
precision measurements, assuming only one Wilson coe cient in the Higgs basis is nonzero at a
time. The current constraints are represented by blue error bars. We future constraints are shown
with the DUNE input assuming only statistical errors (red) or 0:1% systematic errors (dotted pink),
and also without the DUNE input (orange).
6
Conclusions
In this work, we investigated the precision reach in the determination of the SMEFT Wilson
coe cients relevant for lowenergy experiments. We studied observables related to trident
production, neutrino scattering o
electrons and neutrino scattering o
nuclei at DUNE
ND, while leaving for a future work an estimate of the sensitivity of oscillation processes
to the SMEFT parameters. Moreover, information from parity violating M ller scattering,
APV, and polarization asymmetry experiments is also included, as they are sensitive to
the same SMEFT parameters as the ones probed by DUNE. Our projections are combined
and compared with the current constraints on the SMEFT parameters.
Our main results are summarized in table 6 and illustrated in gure 4. There we assume
the presence of only one nonzero Wilson coe cient at a time. Working in the Higgs basis of
the SMEFT, the constraints on seven vertex corrections and on nine fourfermion operators
are improved by at least a factor of ve in the best case scenario. With the optimistic
assumption that the DUNE errors will be dominated by statistics, one can reach an O(10 4)
relative precision for Z boson couplings to muons and light quarks, and for some 4fermion
LLQQ operators. This could probe the new physics scale
> O(30)(O(300)) TeV, for
weakly (strongly) coupled new physics to the SM muons or light quarks. This is beyond
the direct reach of the LHC or nearfuture colliders, as well as beyond the indirect reach
of electroweak precision measurements at LEP. Without the DUNE input, the expected
precision is typically an order of magnitude worse. The projections are degraded with less
optimistic assumptions about the systematic errors achievable in DUNE, which encourages
future e orts to reduce experimental and theoretical sources of these errors.
Acknowledgments
GGdC and ZT are supported by Fundac~ao de Amparo a Pesquisa do Estado de S~ao Paulo
(FAPESP) under contracts 16/170410 and 16/026368. A.F is partially supported by
the European Union's Horizon 2020 research and innovation programme under the Marie
SklodowskaCurie grant agreements No 690575 and No 674896. ZT is particularly grateful
to the hospitality of LPT Orsay where this work was initiated and acknowledges useful
discussions with Yuber F. Perez. We also thank Paride Paradisi for pointing out a typo in
an earlier version of eq. (3.11).
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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