Future DUNE constraints on EFT

Journal of High Energy Physics, Apr 2018

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 \( \mathcal{O}(30) \) 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.

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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. Paris-Sud, Universite Paris-Saclay In the near future, fundamental interactions at high-energy scales may be most e ciently studied via precision measurements at low energies. A universal language to assemble and interpret precision measurements is the so-called SMEFT, which is an e ective eld 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 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 dimension-6 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, parity-violating scattering of electrons, and a plethora of meson, nuclear, and tau decay { 1 { processes. For certain dimension-6 operators the low-energy 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 low-energy measurements is worse, they often play a vital role in lifting at directions in the notoriously multi-dimensional 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 low-energy 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 core-collapse 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 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 o 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 SU(2) local symmetry of the SMEFT, we also include in our analysis other future experiments that do not involve neutrinos: parity-violating 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 higher-dimensional (non-renormalizable) 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 higher-dimensional SMEFT operators. This way, measurements of wEFT parameters in experiments with E mZ provide non-trivial 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 model-independent 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 tree-level map between the wEFT and SMEFT coe cients. 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 e ective 4-fermion 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 2-component 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 low-energy 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 high-energy 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 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 L = LSM + Pi vc2i OD=6, where LSM is the SM Lagrangian, v = ( 2GF ) 1=2 i ' 246 GeV, each OD=6 is a gauge-invariant 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 4-lepton 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 avor-diagonal interactions. Not all the vertex corrections above are independent, as in the dimension-6 SMEFT there is the relation g Z a L L gZea = gW ea . L The vertex corrections can be expressed by a combination of dimension-6 Wilson coe cients in any operator basis (see e.g. [10] for the map to the so-called 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 4-lepton 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 non-standard 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 dimension-6 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 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 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 right-handed quark and chirality-violating 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 anti-neutrino 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 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 de ned by LSMEFT we obtain the following relations between the parameters of the two theories: as zero when they multiply dimension-6 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 non-standard 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 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 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 CHARM-II [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 tree-level formula for the ratio of the trident cross section to its SM-predicted 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 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 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 center-of-mass 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], CHARM-II [23], and BNL-E734 [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 anti-neutrino 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 anti-neutrinos 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 well-known Llewellyn-Smith 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 Llewellyn-Smith 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 Llewellyn-Smith 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 100-t 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 4-fermion 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 LEP-1 [48] and LEP-2 [49] experiments. The LEP-1 data place per-mille level constraints on the vertex corrections gZf , f = q; `, while the LEP-2 data probe gLW f , and the 4-fermion 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 parity-violating M ller scattering, which probes the electron's axial self-coupling 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), (3.8-3.9) and (3.13-3.14). 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 per-mille 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, lepton-quark interactions can be probed in colliders. Existing LEP measurements [48, 49] of electron-quark interactions are compiled in [10] with the conclusion that, in combination with the low-energy measurements, chirality-conserving 2-electron2-quark operators are constrained in a model-independent way at a percent level. The sensitivity improves greatly when lepton pair Drell-Yan production at the LHC is taken into account, and in fact 2-lepton-2-quark operators can be accessed for all three lepton generations [60{62]. For eeqq and eedd operators the current sensitivity is at a per-mille level, and is expected to improve to an O(10 4) level after the high-luminosity run is completed [ 62 ]. That level of precision may be di cult to beat in DUNE. However, a full model-independent analysis of the LHC constraints has not been attempted yet, due to a large number of di erent SMEFT operators contributing to the Drell-Yan process. In this analysis we focus on low-energy 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 low-energy precision experiments. 5 Future evolution of SMEFT constraints In this section we translate the projected low-energy constraints discussed in sections 3 and 4 into the SMEFT language. The goal is to illustrate the place of DUNE in the landscape of low-energy 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 non-zero at a time. Then we compare the constraints on that parameter using the current and future low-energy 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 4-fermion 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 near-future 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 4-muon 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 low-energy precision measurements, assuming only one Wilson coe cient is non-zero 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 best-case 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 parity-violating 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 dimension-6 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 4-muon 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 non-zero 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 low-energy 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 non-zero Wilson coe cient at a time. Working in the Higgs basis of the SMEFT, the constraints on seven vertex corrections and on nine four-fermion 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 4-fermion 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 near-future 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 16/17041-0 and 16/02636-8. A.F is partially supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie 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. 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Adam Falkowski, Giovanni Grilli di Cortona, Zahra Tabrizi. Future DUNE constraints on EFT, Journal of High Energy Physics, 2018, 101, DOI: 10.1007/JHEP04(2018)101