Neutrino Tridents at DUNE
Wolfgang Altmannshofer, Stefania Gori, Justo Martin-Albo, Alexandre Sousa, Michael Wallbank
FFERMILAB-PUB-19-062-LBNF-ND
Neutrino tridents at DUNE
Wolfgang Altmannshofer, ∗ Stefania Gori, † Justo Mart´ın-Albo, ‡ Alexandre Sousa, § and Michael Wallbank ¶ Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064, USA Department of Physics, Harvard University, Cambridge, MA 02138, USA Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA
Abstract
The DUNE near detector will collect an unprecedented large number of neutrino interactions, allowingthe precise measurement of rare processes such as neutrino trident production, i.e. the generation of alepton-antilepton pair through the scattering of a neutrino off a heavy nucleus. The event rate of this processis a powerful probe to a well-motivated parameter space of new physics beyond the Standard Model. In thispaper, we perform a detailed study of the sensitivity of the DUNE near detector to neutrino tridents. Weprovide predictions for the Standard Model cross sections and corresponding event rates at the near detectorfor the ν µ → ν µ µ + µ − , ν µ → ν µ e + e − and ν µ → ν e e + µ − trident interactions (and the corresponding anti-neutrino modes), discussing their uncertainties. We analyze all relevant backgrounds, utilize a Geant4-basedsimulation of the DUNE-near detector liquid argon TPC (the official DUNE simulation at the time of writingthis paper), and identify a set of selection cuts that would allow the DUNE near detector to measure the ν µ → ν µ µ + µ − cross section with a ∼
40% accuracy after running in neutrino and anti-neutrino modes for ∼ L µ − L τ that can explain the ( g − µ anomaly couldbe covered with large significance. As a byproduct, a new Monte Carlo tool to generate neutrino tridentevents is made publicly available. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] a r X i v : . [ h e p - ph ] J a n ONTENTS
I. Introduction 3II. Neutrino tridents in the Standard Model 4A. SM predictions for the neutrino trident cross section 41. Coherent scattering on nuclei 62. Incoherent scattering on individual nucleons 73. Results for the cross section and discussion of uncertainties 8B. Neutrino tridents at the DUNE near detector 101. The DUNE near detector 102. Expected event rates in the Standard Model 11III. Discovering SM muon tridents at DUNE 13A. Simulation 14B. Kinematic distributions and event selection 15C. Expected sensitivity 15IV. Neutrino tridents and new physics 18A. Model-independent discussion 19B. Z (cid:48) model based on gauged L µ − L τ L µ − L τ gauge boson 29References 302 . INTRODUCTION Neutrino trident production is a weak process by which a neutrino, scattering off the Coulombfield of a heavy nucleus, generates a pair of charged leptons [1–7]. Measurements of muonicneutrino tridents, ν µ → ν µ µ + µ − , were performed at the CHARM-II [8], CCFR [9] and NuTeV [10]experiments: σ ( ν µ → ν µ µ + µ − ) exp σ ( ν µ → ν µ µ + µ − ) SM = . ± .
64 (CHARM-II)0 . ± .
28 (CCFR)0 . +1 . − . (NuTeV)Both CHARM-II and CCFR found rates compatible with Standard Model (SM) expectations. Nosignal could be established at NuTeV. Future neutrino facilities, such as LBNF/DUNE [11–14], willoffer excellent prospects to improve these measurements [15–18]. A deviation from the event ratepredicted by the SM could be an indication of new interactions mediated by new gauge bosons [15].This could happen, for example, if neutrinos were charged under new gauge symmetries beyond theSM gauge group, SU (3) c × SU (2) L × U (1) Y .In this paper, we study in detail the prospects for measuring neutrino trident production atthe near detector of DUNE. As will be discussed below, the trident cross section is to a goodapproximation proportional to the charge squared ( Z ) of the target nuclei: Z = 18 for argon(DUNE), Z = 14 for silicon (CHARM II), and Z = 26 for iron (CCFR and NuTeV). As we willdemonstrate, despite the smaller Z compared to CCFR and NuTeV, the high-intensity muon-neutrino beam at the DUNE near detector leads to a sizable production rate of neutrino tridents.The main challenge to obtain a precise measurement of the trident cross section is to distinguishthe trident events from the copious backgrounds, mainly consisting of charged-current single-pionproduction events, ν µ N → µπN (cid:48) , as muon and pion tracks can be easily confused. Here, we identifya set of kinematic selection cuts that strongly suppress the background, allowing a measurement ofthe ν µ → ν µ µ + µ − cross section at DUNE.Our paper is organized as follows. In Section II, we compute the cross sections for severalneutrino-induced trident processes in the SM, and discuss the theoretical uncertainties in thecalculation. We also provide the predicted event rates at the DUNE near detector. Section IIIdescribes the sensitivity study. We analyze the kinematic distributions of signal and backgrounds,and determine the accuracy with which the ν µ → ν µ µ + µ − cross section can be measured at theDUNE near detector. In Section IV, we analyze the impact that such a measurement will have on3hysics beyond the SM, both model independently and in the context of a Z (cid:48) model with gauged L µ − L τ . We conclude in Section V. Details about nuclear and nucleon form factors and ourimplementation of the Borexino bound on the Z (cid:48) parameter space are given in the Appendices A,B, and C. Our neutrino trident Monte Carlo generator tool can be found as an ancillary file on the arXiv entry for this paper. II. NEUTRINO TRIDENTS IN THE STANDARD MODELA. SM predictions for the neutrino trident cross section
Lepton-pair production through the scattering of a neutrino in the Coulomb field of a nucleuscan proceed in the SM via the electro-weak interactions. Figure 1 shows example diagrams forvarious charged lepton flavor combinations that can be produced from a muon-neutrino in the initialstate: a µ + µ − pair can be generated by W and Z exchange (top, left and right diagrams); an e + e − pair can be generated by Z exchange (bottom left); an e + µ − pair can be generated by W exchange(bottom right). A muon neutrino cannot generate µ + e − in the SM. Analogous processes can beinduced by the other neutrino flavors and also by anti-neutrinos. The amplitude corresponding tothe diagrams shown in the figure has a first-order dependence on the Fermi constant. AdditionalSM diagrams where the lepton system interacts with the nucleus through W or Z boson exchangeinstead of photon exchange are suppressed by higher powers of the Fermi constant and are thereforenegligible.The weak gauge bosons of the SM are much heavier than the relevant momentum transfer inthe trident process. Therefore, the effect of the W and Z bosons is accurately described by a fourlepton contact interaction. After performing a Fierz transformation, the effective interaction can bewritten as H SMeff = G F √ (cid:88) i,j,k,l (cid:0) g Vijkl (¯ ν i γ α P L ν j )(¯ (cid:96) k γ α (cid:96) l ) + g Aijkl (¯ ν i γ α P L ν j )(¯ (cid:96) k γ α γ (cid:96) l ) (cid:1) , (1)with vector couplings g V and axial-vector couplings g A . The indexes i, j, k, l (= e, µ, τ ) denote theSM lepton flavors. The values for the coefficients g V and g A for a variety of trident processes in theSM are listed in Table I. These factors are the same as obtained in Ref. [16]. Using the effectiveinteractions, there are two Feynman diagrams that contribute to the trident processes. They areshown in Figure 2.Given the above effective interactions, the cross sections for the trident processes can be computedin a straightforward way. The dominant contributions arise from the coherent elastic scattering of4 ✖ (cid:0)✁ ✗✖(cid:0)✰✌❲ ν µ ν µ µ − µ + γZ ✗✖ ✗✖ ❡(cid:0)❡✰✌❩ ✗✖ (cid:0)✁ ✗❡✂✰✌❲ Figure 1. Example diagrams for muon-neutrino-induced trident processes in the Standard Model. A secondset of diagrams where the photon couples to the negatively charged leptons is not shown. Analogous diagramsexist for processes induced by different neutrino flavors and by anti-neutrinos.Table I. Effective Standard Model vector and axial-vector couplings, as defined in Eq. (1), for a variety ofneutrino trident processes.Process g V SM g A SM ν e → ν e e + e − θ W − ν e → ν e µ + µ − − θ W +1 ν e → ν µ µ + e − − ν µ → ν µ e + e − − θ W +1 ν µ → ν µ µ + µ − θ W − ν µ → ν e e + µ − − the leptonic system on the full nucleus. We will also consider incoherent contributions from elasticscattering on individual nucleons (referred to as diffractive scattering in Refs. [16, 18]).In addition to the elastic scattering on the full nucleus or on individual nucleons, also inelasticprocesses can contribute to trident production. Inelastic processes include events where the nucleusscatters into an excited state, the excitation of a nucleon resonance, and deep-inelastic scattering. As5 ❦✵♣✰♣(cid:0)P ✵P q ◆◆ ❵(cid:0)✁❵✰❧✗✐✗❥ ❦ ❦✵♣(cid:0)♣✰P ✵P q ◆◆ ❵✰❧❵(cid:0)✁✗✐✗❥ Figure 2. Diagrams for the ν j → ν i (cid:96) − k (cid:96) + l trident process using the effective interaction of Eq. (1). shown in [16], deep-inelastic scattering is negligible for trident production at the SHiP experiment.We expect inelastic processes to be negligible for the neutrino energies we consider.All our results shown below are based on a calculation of the full 2 →
1. Coherent scattering on nuclei
The differential cross section of the coherent scattering process on a nucleus of mass m N isenhanced by Z and can be expressed as [2, 6] (see also [17, 18])d σ coh. = Z α G F π m N E ν d k (cid:48) E k (cid:48) d p + E + d p − E − d P (cid:48) E P (cid:48) H αβN L αβ q δ (4) ( k − k (cid:48) − p + − p − + q ) , (2)where the momenta of the incoming and outgoing particles are defined in Fig. 2 and E ν is theenergy of the incoming neutrino. The leptonic tensor L αβ is given by L αβ = (cid:88) s,s (cid:48) ,s + ,s − A α A † β , with A α = (¯ u (cid:48) γ µ P L u ) (cid:18) ¯ u − (cid:20) γ α p/ − − q/ + m − ( p − − q ) − m − γ µ ( g Vijkl + g Aijkl γ ) − γ µ ( g Vijkl + g Aijkl γ ) p/ + − q/ + m + ( p + − q ) − m γ α (cid:21) v + (cid:19) , (3)where m ± , s ± , and v + , u − are the masses, spins and spinors of the positively and negatively chargedleptons and s , s (cid:48) and u , u (cid:48) are the spins and spinors of the incoming and outgoing neutrinos.The relevant part of the hadronic tensor for coherent scattering on a spin 0 nucleus is H αβN = 4 P α P β (cid:2) F N ( q ) (cid:3) , (4)6here F N ( q ) is the electric form factor of the nucleus, N , and P the initial momentum of thenucleus. We use nuclear form factors based on measured charge distributions of nuclei [19]. Detailsabout the nuclear form factors are given in the Appendix A.The experimental signature of the coherent scattering are two opposite sign leptons without anyadditional hadronic activity.
2. Incoherent scattering on individual nucleons
In addition to the coherent scattering on the nucleus, the leptonic system can also scatter onindividual nucleons inside the nucleus. The corresponding differential cross sections have a similarform and readd σ p ( n ) = α G F π m p ( n ) E ν d k (cid:48) E k (cid:48) d p + E + d p − E − d P (cid:48) E P (cid:48) H αβp ( n ) L αβ q δ (4) ( k − k (cid:48) − p + − p − + q ) . (5)The leptonic tensor is still given by Eq. (3). The relevant part of the hadronic tensor for scatteringon the spin 1/2 protons (neutrons) is H αβp ( n ) = 4 P α P β m p ( n ) (cid:2) G p ( n ) E ( q ) (cid:3) q + 4 m p ( n ) + q (cid:2) G p ( n ) M ( q ) (cid:3) q + 4 m p ( n ) + g αβ q (cid:2) G p ( n ) M ( q ) (cid:3) , (6)where G p ( n ) E ( q ) and G p ( n ) M ( q ) are the electric and magnetic form factor of the proton (neutron)and m p ( n ) is the proton (neutron) mass. In our numerical calculations, we use form factors froma fit to electron-proton and electron-nucleus scattering data [20]. Details about the nucleon formfactors are given in the Appendix B.The differential trident cross section corresponding to the incoherent processes isd σ incoh. = Θ( | (cid:126)q | ) (cid:0) Z d σ p + ( A − Z ) d σ n (cid:1) , (7)where Z and ( A − Z ) are the number of protons and neutrons inside the nucleus, respectively. Weinclude a Pauli blocking factor derived from the ideal Fermi gas model of the nucleus [2]Θ( | (cid:126)q | ) = | (cid:126)q | p F − | (cid:126)q | p F , for | (cid:126)q | < p F , for | (cid:126)q | > p F , (8)with the Fermi momentum p F = 235 MeV and (cid:126)q the spatial component of the momentum transferto the nucleus.In addition to the two opposite sign leptons, the final state now contains an additional proton(or neutron) that is kicked out from the nucleus during the scattering process.7 . Results for the cross section and discussion of uncertainties To obtain the total cross sections for the coherent and incoherent processes discussed above, weintegrate the four-particle phase space in (2) and (5) numerically. Using the optimized integrationvariables identified in [2], we find that the numerical integration converges reasonably fast. Wechecked explicitly that our numerical computation accurately reproduces the cross section tablesfor a set of fixed neutrino energies given in [2]. We estimate the uncertainty of our numericalintegration procedure to be around the per-mille level, which is negligible compared to the otheruncertainties discussed below.In Figure 3 we show the cross sections for the ν µ → ν µ µ + µ − , ν µ → ν µ e + e − , and ν µ → ν e e + µ − processes for scattering on argon (left) and iron (right) as a function of the energy of the incomingneutrino. We show both the coherent and incoherent components. From the figure, we make thefollowing observations: • The cross sections fall steeply at low neutrino energy, as it becomes more and more difficultto produce the lepton pair via scattering with a low q photon from the Coulomb field. • The ν µ → ν e µ + e − process has the largest cross section over a broad range of neutrino energiessince it arises from a W mediated diagram (see Fig. 1). The cross section for the ν µ → ν e e + e − process is smaller due to the smaller couplings of the Z boson with leptons and neutrinos.The ν µ → ν µ µ + µ − process typically leads to the smallest cross section at low energies, dueto destructive interference between the W and the Z contributions and the relatively heavydi-muon pair in the final state. • For processes involving electrons, the incoherent cross section is approximately 5% −
10% ofthe coherent cross section. For the ν µ → ν µ µ + µ − process, however, about 30% of the crosssection is coming from incoherent scattering. Scattering on individual protons and neutronsprovides photons with higher q , which makes it easier to produce the (relatively) heavydi-muon pair. • Among the incoherent processes, the cross section for scattering on protons is approximatelyone order of magnitude larger than for scattering on neutrons because neutrons are electricallyneutral.In Fig. 3, estimates of the 1 σ and 2 σ uncertainties of the cross sections are indicated by theshaded bands. We consider uncertainties from form factors, higher order QED corrections, higher8 - E ν [ GeV ] σ t r i d e n t [ f b ] ν μ →ν μ μ + μ - in Ar - E ν [ GeV ] σ t r i d e n t [ f b ] ν μ →ν μ μ + μ - in Fe - E ν [ GeV ] σ t r i d e n t [ f b ] ν μ →ν μ e + e - in Ar - E ν [ GeV ] σ t r i d e n t [ f b ] ν μ →ν μ e + e - in Fe - E ν [ GeV ] σ t r i d e n t [ f b ] ν μ →ν e e + μ - in Ar - E ν [ GeV ] σ t r i d e n t [ f b ] ν μ →ν e e + μ - in Fe Figure 3. Standard Model predictions for the cross sections of the trident processes ν µ → ν µ µ + µ − , ν µ → ν µ e + e − , and ν µ → ν e e + µ − for scattering on argon (left) and iron (right) as a function of the energy ofthe incoming neutrino. Shown are both the coherent component (solid) and incoherent components (dashedfor proton, dotted for neutron). The 1 σ and 2 σ cross-section uncertainties are indicated by the shaded bands. order weak corrections, and nuclear modeling.Form factor uncertainties for the coherent scattering appear to be well under control (seeAppendix A). In our numerical analysis, we use 1% uncertainty on the coherent cross section comingfrom form factors. For the incoherent scattering we find differences in the cross section of a few9ercent, using different nucleon form factors (see Appendix B). For our numerical analysis, weassign a 3% value to the uncertainty arising from form factors.Higher order QED effects might lead to non-negligible corrections. Naively, we estimate thatthe effects could be of order Zα em / π (cid:39)
1% for argon. Moreover, at tree level there is ambiguityabout which value of α em should be used in the cross section computation. The q of the photonfrom the Coulomb field is typically very low, suggesting that the zero momentum value should beappropriate. We use α em = 1 /
137 and conservatively assign a 3% uncertainty to the total crosssection from higher order QED effects.At lowest order in the weak interactions, the value for the weak mixing angle is ambiguous. Usingthe on-shell value sin θ W = 0 . θ W =0 . ∼
5% shift in the cross section of the ν µ → ν µ µ + µ − process.In our numerical analysis we use the MS value at the electro-weak scale sin θ W = 0 . O (20%) are found bycomparing the Fermi gas model with more sophisticated shell models. As additional effects likerescattering or absorption of the nucleon in the nucleus might further modify the cross section,we use 30% uncertainty on all incoherent cross sections to be conservative. A more sophisticatednuclear model would be required to obtain a more precise prediction of the incoherent cross sections.To determine the total uncertainty we add all individual uncertainties in quadrature. Thefinal uncertainties on the coherent cross sections that we find are approximately 6% and they aredominated by our estimate of possible higher order electro-weak corrections. For the incoherentscattering cross sections, the by far dominant uncertainty is due to the nuclear modeling. B. Neutrino tridents at the DUNE near detector
1. The DUNE near detector
The
Deep Underground Neutrino Experiment (DUNE) [11–14] is an international project forneutrino physics and nucleon-decay searches, currently in the design and planning stages. Oncebuilt, DUNE will consist of two detectors exposed to a megawatt-scale, wide-band muon-neutrinobeam produced at the Fermi National Accelerator Laboratory (Illinois, USA). One of the detectors10ill record neutrino interactions near the beginning of the beamline, while the other, much larger,detector, comprising four 10-kilotonne liquid argon time projection chambers (TPCs), will beinstalled at a depth of 1.5 km at the Sanford Underground Research Facility (South Dakota, USA),about 1300 kilometres away of the neutrino source. Among the primary scientific goals of DUNEare the precision measurement of the parameters that govern neutrino mixing — including thosestill unknown: the octant in which the θ mixing angle lies, the neutrino mass ordering and thevalue of the CP-violation phase —, as well as nucleon-decay searches and neutrino astrophysics.One of the main roles of the DUNE near detector (ND), which will be located 570 meters awayfrom the beamline production target, is the precise characterization of the neutrino beam energyand composition, as well as the measurement to unprecedented accuracy of the cross sections andparticle yields of the various neutrino scattering processes. Additionally, as the ND will be exposedto an intense flux of neutrinos, it will collect an extraordinarily large sample of neutrino interactions,allowing for an extended science program that includes searches for new physics (e.g. heavy sterileneutrinos or non-standard interactions).The DUNE ND is presently under design. The baseline detector concept consists of a liquidargon TPC (LArTPC) and a magnetized high-resolution tracker [24]. The latter, not considered forthe study discussed in this paper, will consist of a large high-pressure argon gas TPC surroundedby an electromagnetic sampling calorimeter. The design of the LArTPC will be based on theArgonCube concept [25], which places identical but separate TPC modules in a common bath ofliquid argon. Each module features a central cathode and two drift volumes with pixelized chargereadouts and light detection systems. Module walls are kept thin to provide transparency to thetracks and showers produced in neutrino interactions. This detector configuration will mitigate theeffects of event pile-up and allow for an optimal use of liquid argon by boasting a relatively largeactive volume. The dimensions presently considered for the LArTPC, imposed by requirementson event statistics and containment, are 7 m width, 3 m height and 5 m depth, corresponding toan argon mass of about 147 tonnes. The definitive DUNE ND configuration will be defined in anupcoming near detector Conceptual Design Report (CDR) and in a subsequent
Technical DesignReport (TDR).
2. Expected event rates in the Standard Model
In Table II we show the number of expected events of muon-neutrino-induced Standard Modeltrident events at the DUNE near detector per tonne of argon and year of operation in the neutrino-11 able II. Expected number of muon-neutrino-induced Standard Model trident events at the DUNE neardetector per tonne of argon and year of operation in neutrino mode (first four rows) or anti-neutrino mode(last four rows). The numbers in parenthesis correspond to the total statistics in the 147-tonne LArTPC fora run of 3 years. Coherent Incoherent ν µ → ν µ µ + µ − . ± .
07 0 . ± . ±
31) (216 ± ν µ → ν µ e + e − . ± .
17 0 . ± . ±
75) (79 ± ν µ → ν e e + µ − . ± . . ± . ± ± ν µ → ν e µ + e − ν µ → ¯ ν µ µ + µ − . ± .
04 0 . ± . ±
18) (141 ± ν µ → ¯ ν µ e + e − . ± .
13 0 . ± . ±
57) (57 ± ν µ → ¯ ν e e + µ − ν µ → ¯ ν e µ + e − . ± . . ± . ± ± beam (first four rows) or antineutrino-beam (last four rows) configurations. Note that the numberof events for the incoherent process is mainly coming from the scattering with protons. As discussedin Sec. II A 2, the neutron contribution (included as well in the table) is much smaller and amountsto only ∼
10% of the total incoherent cross section. In parenthesis, we also show the number ofexpected events for 147 tonnes of argon and a run of 3 years.The normalized neutrino beam energy spectra are shown in Fig. 4. The relevant integratedflux in neutrino mode is F ν µ = 1 . × − m − POT − and in antineutrino mode ¯ F ¯ ν µ = 0 . × − m − POT − [26]. We assume 1 . × POT per year.The numbers of events for the coherent process in Tab. II are then obtained via N trident = F σ N Ar N POT = F σ M D M Ar N POT , (9)where F is the relevant integrated neutrino flux as given above, σ is the neutrino trident cross12 .5 1 5 105. × - E ν ( GeV ) / Φ t o t × d Φ / d E ν ( G e V - ) neutrino mode ν μ ν μ ν e × - E ν ( GeV ) / Φ t o t × d Φ / d E ν ( G e V - ) anti - neutrino mode ν μ ν μ ν e Figure 4. Normalized energy spectra of the neutrino species at the DUNE near detector for the neutrino-beam(left) and antineutrino-beam (right) modes of operation. section (convoluted with the corresponding normalized energy distribution), N Ar the number ofargon nuclei in the detector, M D the mass of the detector, M Ar the mass of argon (39 . u ), and N POT is the number of protons on target. Similarly, we computed the number of events for theincoherent processes.In our calculation of the number of trident events we neglect all flux components but the ν µ component in neutrino mode and the ¯ ν µ component in anti-neutrino mode. Taking into accountthe scattering of the other components will increase the expected event numbers by a few percent,which is within the given uncertainties. The rates for the antineutrino-beam mode are smaller byapproximately 30%, mainly due to the lower flux. III. DISCOVERING SM MUON TRIDENTS AT DUNE
In this section, we discuss the prospects for detecting muon trident events, ν µ → ν µ µ + µ − , atthe DUNE near detector. As we will discuss in Section IV, this process is particularly relevantto test new light gauge bosons that couple to second generation leptons. A detailed discussion ofelectron tridents and electron-muon tridents at the DUNE near detector is left for future work.13 . Simulation The study presented here makes use of Monte-Carlo datasets generated with the official (at thetime of writing of the paper) DUNE Geant4 [27] simulation of the ND LArTPC. Each simulatedevent represents a different neutrino-argon interaction in the active volume of the detector. Allfinal-state particles produced in the interactions are propagated by Geant4 through the detectorgeometry until they deposit all their energy or leave its boundaries. In this process, additionalparticles (which are tracked as well) may be generated via scattering or decay. The trajectoriesand associated energy deposits left by charged particles in the active volume of the LArTPC arerecorded and written to an output file.For simplicity, charge collection and readout are not simulated, but their effect on the datais taken into account in our study with the introduction of the typical detection thresholds andresolutions expected from the ND LArTPC. Given that state-of-the-art TPCs have achievedvery high reconstruction efficiency ( > C++ source code of the event generator ispublicly available as an ancillary file on the arXiv.Several SM processes can constitute background for the muon trident process. In our simulation,we generate 10 neutrino interactions using the GENIE Monte Carlo generator [29, 30]. By far, themost important background is due to the mis-identification of charged-pion tracks. Roughly 38% ofthe events have a charged lepton and a charged pion in the final state, leading to two muon-likecharged tracks, as in our trident signal. We find that di-muon events from charged current charmproduction only represent less than one percent of the total background.14 . Kinematic distributions and event selection We identify a set of optimal kinematic variables that help discriminating between signal andbackground. Particularly, we use the number of tracks, the angle between tracks, the length of thetracks, and the total energy deposited within 10 cm of the neutrino interaction vertex ( E ).Figures 5 and 6 show the distribution for signal (coherent in red and incoherent in blue) andbackground (green) events of these kinematic variables. All distributions are area-normalized.Particularly, in the upper left panel of Fig. 5, we present the distribution for the number of tracks, N tracks , where we have considered a threshold of 100 MeV in energy deposited in the LAr for thedefinition of a track. The other panels have been evaluated considering only events that containtwo and only two tracks. We consider the distributions for the angle between the two tracks ( angle ,upper right plot), the length of the shortest track ( L min , lower left plot), and the difference inlength between the two tracks ( L max − L min ), lower right plot). Finally, in Fig. 6 we show the totalenergy deposited within 10 cm ( E ) of the neutrino interaction vertex. This includes the sum ofthe energies deposited by any charged particle (even those that deposit less than 100 MeV andthat, therefore, would not be classified as tracks) in a sphere of 10 cm radius around the interactionvertex.As expected, the background events tend to contain a larger number of tracks than the signal.The other distributions also show a clear discriminating power: the angle between the two tracksis typically much smaller in the signal than in the background. Moreover, the signal tracks (twomuons) tend to be longer than tracks in the background events (consisting typically in one muonplus one pion). Finally, the energy deposited in the vicinity of the interaction vertex for thecoherent signal events is compatible with the expectation from a pair of minimum ionization tracks,(d E/ d x ) mip ≈ . / cm. In contrast, both the incoherent signal and the background have, onaverage, more energy deposited around the vertex due to the hadronic activity generated in theinteraction. C. Expected sensitivity
The 147-tonne LArTPC at the DUNE near detector will record, in the neutrino-beam mode,close to 3 . × neutrino interactions per year, out of which only a couple hundred events willcorrespond to the trident process. Our event selection, therefore, has to achieve a backgroundsuppression of at least 6 orders of magnitude. 15 ��������������������������� � � � � ������������������ � ������ � � � � � � ��� � � ���������������������������� ��� ��� ��� ��� ����� - � ���������������� ����� ( ��� ) � � � � � � ��� � � ���������������������������� � ��� ��� ��� ��� ����� - � ��������������� � ��� ( �� ) � � � � � � ��� � � ���������������������������� � �� �� �� �� ���������������������������� � ��� - � ��� ( �� ) � � � � � � ��� � � Figure 5. Kinematic distributions for the coherent signal (in red), incoherent signal (in blue), and background(in green) used in our event selection: number of tracks (upper left plot), angle between the two selectedtracks (upper right plot), length of the shortest track (lower left plot), and difference in length between thetwo tracks (lower right plot). For the last three panels, we have only used events containing two and only twotracks. The dashed, black vertical lines indicate the optimized cut used in our analysis (see text for details).
To do this, we first require events with two and only two tracks, with an angle of at least 0.5degrees between them to ensure separation of the tracks. This requirement alone is able to suppressthe background by a factor of 2, while the signal is almost not affected ( ∼
90% efficiency). On topof this requirement, we optimize the cuts on the other variables shows in Fig. 5: angle , L min , and16 ������������������������������������� � ������ ���� ���� ���� ���� ���� ���������������������������������� � �� ( ��� ) � � � � � � ��� � � Figure 6. Total energy deposited within 10 cm from the neutrino interaction vertex. We show the distributionfor the coherent signal (in red), the incoherent signal (in blue), and for background events with and withoutthe requirement of exactly two tracks (lighter and darker green, respectively). The dashed black vertical lineindicates the cut used in our analysis. L max − L min . Particularly, we find the values of θ max , L M , ∆ L such that the requirements angle < θ max , L min > L M , L max − L min < ∆ L , (10)produces the largest S/ √ B per year. (The following discussion of the optimization of the cuts refersto the trident signal arising from the neutrino mode. The signal acceptance for the antineutrinomode is almost identical.) We perform a scan on the maximum angle between the two tracks, θ max ,the minimum length of the shortest track, L M , and the maximum difference between the lengthof the two tracks, ∆ L . We scan these cuts over a wide range: θ max ⊂ [0 , . L M ⊂ [100 , L ⊂ [0 , S/ √ B per year, after having asked for at least 10 background events in our generated sample. Theoptimized cuts that we find are given by θ max = 0 .
09 ( ∼ . L M = 375 cm, and ∆ L = 5cm. These cuts result in the following number of selected events: S coherent (cid:39) . , S proton (cid:39) . , S neutron (cid:39) . , B (cid:39) , (11)per year with S/ √ B ∼ . E <
50 MeV,17he total background (before applying any further cut) is suppressed by a factor of ∼
5, while thetotal signal acceptance is near 70% (this arises from a ∼
93% acceptance for the coherent signal, a ∼
7% for the incoherent-proton signal, and a ∼
85% for the incoherent-neutron signal). We findthat the cut on E is correlated with the other cuts we are employing in our analysis. In particular,if we first demand two and only two tracks and, on top of that, E <
50 MeV, the suppression ofthe background due to the E cut is reduced to a factor of ∼
3. This suppression factor is reducedto ∼
30% when we demand two tracks with a minimum length of 375 cm, as in the previouslyoptimized cuts. Potential systematic uncertainties impacting this vertex periphery cut on theenergy deposited have not been included in this analysis. Uncertainties at the level of 10%, arisingmainly from data/simulation discrepancies have been obtained by the MINERvA Collaborationusing a similar cut [31]. Over the next decade, as the DUNE analysis and simulation framework isdeveloped, such uncertainties should be further reduced. Note that the E distribution shown inFig. 6 can be affected by re-scattering processes that we neglect in our analysis.We re-run our cut optimization, having asked E <
50 MeV. We find that the optimal cuts areonly mildly modified to θ max = 0 .
08 ( ∼ . L M = 340 cm, and ∆ L = 4 cm. These cutslead to: S coherent (cid:39) . , S proton (cid:39) . , S neutron (cid:39) . , B (cid:39) , (12)per year with S/ √ B ∼ .
9. This shows that the requirement on the vertex activity does notsubstantially improve the accuracy of the measurement.These numbers show that a measurement of the SM di-muon trident production at the 40% levelcould be possibly obtained using ∼ ∼ ∼ S proton and S neutron , a separatemeasurement of the incoherent cross section appears to be very challenging. Note that our modelingof the kinematics of the nucleon in the incoherent processes might have sizable uncertainties (cf.discussion in section II A 3). However, we do not expect that a more detailed modeling wouldqualitatively change our conclusions with regards to the incoherent process. IV. NEUTRINO TRIDENTS AND NEW PHYSICS
Neutrino tridents are induced at the tree level by the electroweak interactions of the SM andthus can probe new interactions among neutrinos and charged leptons of electroweak strength. In18he following we discuss the sensitivity of neutrino tridents to heavy new physics parameterized ina model independent way by four fermion interactions (Sec. IV A), and in the context of a newphysics model with a light new Z (cid:48) gauge boson (Sec. IV B). A. Model-independent discussion
If the new physics is heavy compared to the relevant momentum transfer in the trident process,its effect is model-independently described by a modification of the effective four fermion interactionsintroduced in Eq. (1). Focusing on the case of muon-neutrinos interacting with muons, we write g Vµµµµ = 1 + 4 sin θ W + ∆ g Vµµµµ , g
Aµµµµ = − g Aµµµµ , (13)where ∆ g Vµµµµ and ∆ g Aµµµµ parameterize possible new physics contributions to the vector andaxial-vector couplings. Couplings involving other combinations of lepton flavors can be modifiedanalogously. Note, however, that for interactions that involve electrons, very strong constraints canbe derived from LEP bounds on electron contact interactions [32].The modified interactions of the muon-neutrinos with muons alter the cross section of the ν µ N → ν µ µ + µ − N trident process. We use the existing measurement of the trident cross section bythe CCFR experiment [9] and the expected sensitivities at the DUNE near detector discussed inSec. III C, to put bounds on ∆ g Vµµµµ and ∆ g Aµµµµ (see also [33]).Using the neutrino spectrum from the CCFR experiment (see [34]) and the spectrum at theDUNE near detector shown in Fig. 4, we find the cross sections σ CCFR (cid:39) ( g Vµµµµ ) × .
087 fb + ( g Aµµµµ ) × .
099 fb , (14) σ DUNE (cid:39) ( g Vµµµµ ) × . × − fb + ( g Aµµµµ ) × . × − fb , (15)where in both cases we only took into account coherent scattering. The CCFR trident measurementput a stringent cut on the hadronic energy at the event vertex region, which we expect to largelyeliminate incoherent trident events. Similarly, we anticipate that in a future DUNE measurementincoherent scattering events will be largely removed by cuts on the hadronic activity (see discussionin section III C).For the modifications relative to the SM cross sections we find σ CCFR σ SMCCFR (cid:39) (1 + 4 sin θ W + ∆ g Vµµµµ ) + 1 .
13 (1 − ∆ g Aµµµµ ) (1 + 4 sin θ W ) + 1 . , (16) σ DUNE σ SMDUNE (cid:39) (1 + 4 sin θ W + ∆ g Vµµµµ ) + 1 .
54 (1 − ∆ g Aµµµµ ) (1 + 4 sin θ W ) + 1 . . (17)19 C F R CCFR
DUNE
DUNE 25 % DUNE - - - - - - - Δ g μμμμ V Δ g μμμμ A ν μ N → ν μ μ + μ - N Figure 7. 95% CL. sensitivity of a 40% (blue hashed regions) and a 25% (dashed contours) uncertaintymeasurement of the ν µ N → ν µ µ + µ − N cross section at the DUNE near detector to modifications of thevector and axial-vector couplings of muon-neutrinos to muons. The gray regions are excluded at 95% CL. byexisting measurements of the cross section by the CCFR collaboration. The intersection of the thin blacklines indicates the SM point. In Fig. 7 we show the regions in the ∆ g Vµµµµ vs. ∆ g Aµµµµ plane that are excluded by the existingCCFR measurement σ CCFR /σ SMCCFR = 0 . ± .
28 [9] at the 95% C.L. in gray. The currentlyallowed region corresponds to the white ring including the SM point ∆ g Vµµµµ = ∆ g Aµµµµ = 0. In thecentral gray region the new physics interferes destructively with the SM and leads to a too smalltrident cross section. Outside the white ring, the trident cross section is significantly larger thanobserved. The result of our baseline analysis (corresponding to an expected measurement with40% uncertainty) does not extend the sensitivity into parameter space that is unconstrained by theCCFR measurement. However, it is likely that the use of a magnetized spectrometer, as it is beingconsidered for the DUNE ND, able to identify the charge signal of the trident final state, alongwith more sophisticated deep-learning based event selection, will significantly improve separationbetween neutrino trident interactions and backgrounds. Therefore, we also present the region thatcould be probed by a 25% measurement of the neutrino trident cross section at DUNE, which would20xtend the coverage of new physics parameter space substantially. B. Z (cid:48) model based on gauged L µ − L τ A class of example models that modify the trident cross section are models that contain anadditional neutral gauge boson, Z (cid:48) , that couples to neutrinos and charged leptons. A consistentway of introducing such a Z (cid:48) is to gauge an anomaly free global symmetry of the SM. Of particularinterest is the Z (cid:48) that is based on gauging the difference between muon-number and tau-number, L µ − L τ [35, 36]. Such a Z (cid:48) is relatively weakly constrained and can for example address thelongstanding discrepancy between SM prediction and measurement of the anomalous magneticmoment of the muon, ( g − µ [37, 38]. The L µ − L τ Z (cid:48) has also been used in models to explain B physics anomalies [39] and as a portal to dark matter [40, 41]. The ν µ N → ν µ µ + µ − N tridentprocess has been identified as important probe of gauged L µ − L τ models over a broad range of Z (cid:48) masses [15, 39].The interactions of the Z (cid:48) with leptons and neutrinos are given by L L µ − L τ = g (cid:48) Z (cid:48) α (cid:104) (¯ µγ α µ ) − (¯ τ γ α τ ) + (¯ ν µ γ α P L ν µ ) − (¯ ν τ γ α P L ν τ ) (cid:105) , (18)where g (cid:48) is the L µ − L τ gauge coupling. Note that the Z (cid:48) couples purely vectorially to muonsand taus. If the Z (cid:48) is heavy when compared to the momentum exchanged in the process, it canbe integrated out, and its effect on the ν µ N → ν µ µ + µ − N process is described by the effectivecouplings ∆ g Vµµµµ = ( g (cid:48) ) v m Z (cid:48) , ∆ g Aµµµµ = 0 , (19)where m Z (cid:48) is the Z (cid:48) mass and v (cid:39)
246 GeV is the electroweak breaking vacuum expectation value.Using the expression for the cross section in (16) we find the following bound from the existingCCFR measurement g (cid:48) (cid:46) . × (cid:16) m Z (cid:48)
100 GeV (cid:17) for m Z (cid:48) (cid:38) few GeV . (20)The bound is applicable as long as the Z (cid:48) mass is heavier than the average momentum transfer inthe trident reaction at CCFR, which – given the neutrino energy spectrum at CCFR – is around afew GeV. For lower m Z (cid:48) , the Z (cid:48) propagator is saturated by the momentum transfer and the CCFRbound on g (cid:48) improves only logarithmically. A measurement of the trident process at the DUNE neardetector has the potential to considerably improve the sensitivity for low-mass Z (cid:48) bosons. Because21f the much lower energy of the neutrino beam compared to CCFR, also the momentum transfer ismuch smaller and the scaling in eq. (20) extends to smaller Z (cid:48) masses.In Fig. 8 we show the existing CCFR constraint on the model parameter space in the m Z (cid:48) vs. g (cid:48) plane and compare it to the region of parameter space where the anomaly in ( g − µ = 2 a µ can beexplained. The green region shows the 1 σ and 2 σ preferred parameter space corresponding to ashift ∆ a µ = a exp µ − a SM µ = (2 . ± . × − [42] (see also [43]). In the figure, we also show theconstraints from LHC searches for the Z (cid:48) in the pp → µ + µ − Z (cid:48) → µ + µ − µ + µ − process [15, 44] (seealso [45]), direct searches for the Z (cid:48) at BaBar using the e + e − → µ + µ − Z (cid:48) → µ + µ − µ + µ − process [46],and constraints from LEP precision measurements of leptonic Z couplings [39, 47]. Also a Borexinobound on non-standard contributions to neutrino-electron scattering [48–50] has been used toconstrain the L µ − L τ gauge boson [51–53]. Our version of this constraint (see appendix C) is alsoshown. For very light Z (cid:48) masses of O (few MeV) and below, strong constraints from measurementsof the effective number of relativistic degrees of freedom during Big Bang Nucleosynthesis (BBN)apply [51, 54, 55] (see the vertical dot-dashed line in the figure). For m (cid:48) Z (cid:39)
10 MeV, the tension inthe Hubble parameter H can be ameliorated [55]. Taking into account all relevant constraints,the region of parameter space left that may explain ( g − µ lies below the di-muon threshold m Z (cid:48) (cid:46)
210 MeV.A measurement of the ν µ N → ν µ µ + µ − N cross section at the SM value with 40% uncertainty atthe DUNE near detector is sensitive to the region delimited by the blue contour. We find that theparameter space that is motivated by ( g − µ could be covered in its majority.Other proposals to cover the remaining region of parameter space favored by ( g − µ include LHCsearches for µ + µ − + E T / [56], searches for γ + E/ at Belle II [57], muon fixed-target experiments [58, 59],high-intensity electron fixed-target experiments [60], or searches for Z + E/ at future electron-positroncolliders [61]. V. CONCLUSIONS
The production of a pair of charged leptons through the scattering of a neutrino on a heavynucleus (i.e. neutrino trident production) is a powerful probe of new physics in the leptonic sector.In this paper we have studied the sensitivity to this process of the planned DUNE near detector.In the SM, neutrino trident production proceeds via the weak interaction, and thus the crosssection can be computed with good accuracy. Here, we have provide SM predictions for the crosssections and the expected rates at the DUNE near detector for a variety ν µ and ν µ of neutrino and22 .001 0.010 0.100 1 105. × - m Z ' ( GeV ) g' B o r e x i n o BaBar BB N LHC
CCFR ( g - ) μ DUNE
Figure 8. Existing constraints and DUNE sensitivity in the L µ − L τ parameter space. Shown in greenis the region where the ( g − µ anomaly can be explained at the 1 σ and 2 σ level. The parameterregions already probed by existing constraints are shaded in gray and correspond to a CMS search for pp → µ + µ − Z (cid:48) → µ + µ − µ + µ − [44] (“LHC”), a BaBar search for e + e − → µ + µ − Z (cid:48) → µ + µ − µ + µ − [46](“BaBar”), precision measurements of Z → (cid:96) + (cid:96) − and Z → ν ¯ ν couplings [39, 47] (“LEP”), a previousmeasurement of the trident cross section [9, 15] (“CCFR”), a measurement of the scattering rate of solarneutrinos on electrons [48–50] (“Borexino”), and bounds from Big Bang Nucleosynthesis [51, 54, 55] (“BBN”).The DUNE sensitivity shown by the solid blue line assumes 6.5 year running in neutrino mode, leading to ameasurement of the trident cross section with 40% precision. antineutrino-induced trident processes: ( − ) ν µ → ( − ) ν µ µ + µ − , ( − ) ν µ → ( − ) ν µ e + e − , and ( − ) ν µ → ( − ) ν e e ± µ ∓ .We estimate that the uncertainties of our predictions for the dominant coherent scattering process areapproximately 6%, mainly due to higher order electroweak corrections. Sub-dominant contributionsfrom incoherent scattering have larger uncertainties due to nuclear modeling.We find that at the DUNE near detector, one can expect ∼ ν µ → ν µ µ + µ − events per year, ∼ ν µ → ν µ e + e − events per year, and ∼ ν µ → ν e e + µ − events per year. This impliesfavorable conditions for performing precise measurements of the cross sections of such processes.In this paper, we performed a state-of-the-art analysis for the future sensitivity of DUNE tomuon neutrino tridents using a Geant4-based simulation of the DUNE near detector liquid argonTPC. Thanks to the very distinctive kinematical features of the signal, if compared to the muon23nclusive production background (two long tracks with a relatively small opening angle and a smallenergy deposited around the neutrino vertex interaction), the background rate can be reduced by ∼ O (10) signal event/year and O (20) background/year. The main source of background arises frompion-muon production, where both the pion and the muon produce long tracks, and originate fromthe first part of the liquid argon. A further suppression of the background might be obtained viathe magnetized spectrometer, whose sampling calorimeter should improve the separation betweenmuons and pions.We find that the ν µ → ν µ µ + µ − trident cross section can be measured with good precision at theDUNE near detector. Taking into account approximately three years running each in neutrino andantineutrino mode, we anticipate a measurement with an accuracy of ∼ Z (cid:48) model. We find that a measurement at DUNE couldsignificantly extend the coverage of new physics parameter space compared to the existing tridentmeasurement from the CCFR and CHARM II experiments. This is particularly the case for lightnew physics. As a benchmark new physics model we considered an extension of the SM by a new Z (cid:48) gauge boson that is based on gauging the difference of muon-number and tau-number, L µ − L τ .We provide a summary of existing constraints on the Z (cid:48) parameter space in Fig. 8. Interestinglyenough, there is viable parameter space where the Z (cid:48) can explain the long-standing discrepancy inthe anomalous magnetic moment of the muon, ( g − µ . We find that the parameter space thatis motivated by ( g − µ could be largely covered by a measurement of the ν µ → ν µ µ + µ − tridentcross section at DUNE. ACKNOWLEDGEMENTS
We thank Chris Ontko for the collaboration in the early stages of this paper. We thank theDUNE Collaboration for reviewing this manuscript and providing computing resources for thesimulation of neutrino interactions in the DUNE near detector. The research of WA is supported24y the National Science Foundation under Grant No. PHY-1912719. SG is supported by a NationalScience Foundation CAREER Grant No. PHY-1915852. The work of WA and SG was in partperformed at the Aspen Center for Physics, which is supported by National Science FoundationGrant PHY-1607611. The work of AS and MW was supported by the Office of High Energy Physicsat the Department of Energy through grant DE-SC011784 to the University of Cincinnati.
Appendix A: Nuclear form factors
In our predictions for the coherent neutrino trident process we use electric form factors based onnuclear charge density distributions that have been fitted to elastic electron scattering data [19].The form factors are expressed as F N ( q ) = (cid:90) dr r sin( qr ) qr ρ N ( r ) , (A1)where q = (cid:112) q and ρ N is a spherically symmetric charge density distribution of the nucleus N ,normalized as (cid:82) dr r ρ N ( r ) = 1, such that F N (0) = 1. The charge distributions ρ N can beparameterized in various different ways. In Fig. 9 we compare the form factors for argon and ironthat we obtain using various parameterizations that are available from [19]. We show the formfactors based on charge densities parameterized by a Fourier-Bessel series expansion in purple. Formfactors based on the three parameter Fermi charge distribution and the three parameter Gaussiancharge distribution are shown in red and orange, respectively.We also compare these form factors with other phenomenological parameterizations which aremuch less precise. In particular we consider a two parameter charge density distribution ρ N ( r ) = N { ( r − r ) /σ } , (A2)with r = 1 .
18 fm × A − .
48 fm and σ = 0 .
55 fm [2] (shown in solid gray in the plots of Fig. 9)and r = 1 .
126 fm × A and σ = 0 .
523 fm [16, 18] (shown in dashed gray), where A is the massnumber of the nucleus. Finally, in dotted gray, we show a simple exponential form factor [6] F N ( q ) = exp (cid:26) − a q (cid:27) , with a = 1 . × A . (A3)This is the form factor that was used in [15].We observe that the form factors that are based on the fitted nuclear charge distributionsfrom [19] (purple, orange, and red lines in the figure) agree very well over a broad range of relevantmomentum transfer. Our predictions for the trident cross sections differ by less than 1% using25 .0 0.1 0.2 0.3 0.4 0.5 0.6 0.710 (cid:45) q (cid:72) GeV (cid:76) (cid:200) F A r (cid:72) q (cid:76) (cid:200) Ar (cid:45) q (cid:72) GeV (cid:76) (cid:200) F F e (cid:72) q (cid:76) (cid:200) Fe Figure 9. Electric form factors, F N ( q ), of argon (left panel) and iron (right panel) based on variousparameterizations of the nuclear charge density distributions. See text for details. these parameterizations. As default for our numerical calculations we choose the Fourier-Besselseries expansion for argon and iron ρ N ( r ) = N (cid:80) n a ( n ) N j ( nπr/R N ) for r < R N , r > R N , (A4)where j is the spherical Bessel function of 0-th order. (These correspond to the purple lines inFig. 9). The coefficients a ( n ) N and R N are given in Table III.The trident cross sections that we obtain with the two parameter form factors used in [2] differat most by few % from the results using our default form factors. We find that the form factorsused in [16, 18] (dashed gray in the figure) tend to be somewhat smaller than the others andunderestimate the trident cross sections by approximately 10% for both argon and iron. Thesimple exponential form factors used in [15] (dotted gray in the figure) give cross sections thatagree reasonably well in the case of iron, but tend to overestimate the cross sections for argon by5% − Appendix B: Nucleon form factors
We use proton and neutron form factors from [20] (see also [62] for a recent re-evaluation ofnucleon form factors). The form factors of the proton were obtained from fits to measurements of26 rgon iron argon iron R a (9) . − . a (1) . . a (10) − . . a (2) . . a (11) − . − . a (3) . . a (12) . . a (4) − . − . a (13) − . − . a (5) − . − . a (14) . . a (6) − . . a (15) − . − . a (7) . . a (16) – 0 . a (8) − . − . a (17) – − . the electron-proton elastic scattering cross section and polarization transfer measurements. Theform factors of the neutron were obtained from fits to electron-nucleus (mainly deuterium and He) scattering data. The following parameterizations are used for the electric form factor G pE andmagnetic form factor G pM of the proton G pE ( q ) = 1 + a Ep τ b Ep, τ + b Ep, τ + b Ep, τ , (B1) G pM ( q ) µ p = 1 + a Mp τ b Mp, τ + b Mp, τ + b Mp, τ , (B2)where τ = q / (4 m p ) and the magnetic moment of the proton is µ p (cid:39) . G nM ( q ) µ n = 1 + a Mn τ b Mn, τ + b Mn, τ + b Mn, τ , (B3)where τ = q / (4 m n ) and the magnetic moment of the neutron is µ n (cid:39) − . a and b parameters are collected in Table IV.Finally, the electric form factor of the neutron is parametrized in the following way G nE ( q ) = Aτ Bτ G D ( q ) , (B4)where A = 1 . B = 3 .
63, and the standard dipole form factor is G D ( q ) = (1 + q /m V ) − , with m V = 0 .
71 GeV . 27 roton neutron proton a M a E -0.19 b M b E b M b E b M b E q (cid:72) GeV (cid:76) (cid:200) G E p (cid:72) q (cid:76) (cid:200) , (cid:200) G M p (cid:72) q (cid:76) (cid:200) proton q (cid:72) GeV (cid:76) (cid:200) G E n (cid:72) q (cid:76) (cid:200) , (cid:200) G M n (cid:72) q (cid:76) (cid:200) neutron Figure 10. Electric (red) and magnetic (orange) form factors of the proton and neutron from [20]. Forcomparison, simple dipole form factors are shown with the dashed lines (in this case, the electric form factorof the neutron vanishes) .
Fig. 10 shows the electric (red) and magnetic (orange) form factors of the proton and neutron(B1) - (B4). For comparison, the standard dipole form factors used in [2, 18] are shown withthe dashed lines. We see that the different sets of form factors do not differ appreciably at lowmomentum transfer. To estimate form factor uncertainties we computed the incoherent tridentcross sections also with the standard dipole form factors and found few % differences with respectto the calculation using the form factors in (B1) - (B4). In view of the other uncertainties fromnuclear modeling discussed in Sec. II A, this difference is insignificant.28 ppendix C: Borexino bound on the L µ − L τ gauge boson In this appendix we detail our treatment of the Borexino constraint shown in Fig. 8. TheBorexino experiment measures the rate of low energy solar neutrinos that scatter elastically onelectrons [49, 50]. The most precise measurement is obtained for Be neutrinos which have an energyof E ν = 862 keV. The good agreement of the measured scattering rate with the expectations fromthe Standard Model allows one to put bounds on non-standard contributions to the neutrino-electronscattering cross section.The neutrino scattering rate at Borexino is proportional to the neutrino-electron scattering crosssection of the three neutrino flavors, weighted by their respective fluxes at the earth. For Be solarneutrinos, the flux ratios at the earth are approximately φ ν e : φ ν µ : φ ν τ (cid:39)
54% : 20% : 26% , (C1)where we used the expressions from [63] and the latest neutrino mixing parameters from [64]. Thedifferential scattering cross sections can be written as [65] ddy σ ( ν i e → ν i e ) SM = G F m e E ν π (cid:104)(cid:0) g Viiee − g Aiiee (cid:1) + (cid:0) g Viiee + g Aiiee (cid:1) (1 − y ) − (cid:0) ( g Viiee ) − ( g Aiiee ) (cid:1) m e E ν y (cid:21) , (C2)where y is the electron recoil energy E R normalized to the energy of the incoming neutrino E ν , y = E R E ν , and the relevant couplings are given in the SM by (cf. Tab. I) g Veeee = 1 + 4 sin θ W , g Aeeee = − g Vµµee = g Vττee = − θ W , and g Aµµee = g Aττee = 1. Integrating over y between theminimal recoil energy considered at Borexino y min (cid:39) .
22 [50] and the maximal value allowed bykinematics y max = E ν E ν + m e (cid:39) .
77 we find σ ( ν e e → ν e e ) SM : σ ( ν µ e → ν µ e ) SM : σ ( ν τ e → ν τ e ) SM (cid:39) . . (C3)The L µ − L τ gauge boson can contribute to the ν µ e → ν µ e and ν τ e → ν τ e scattering processes atthe 1-loop level, through kinetic mixing between the Z (cid:48) and the SM photon. The kinetic mixingbecomes relevant if the momentum transfer is small compared to the muon and tau masses, as isthe case in the low energy neutrino scattering at Borexino. We find that the Z (cid:48) contributions canbe easily incorporated by making the following replacements in Eq. (C2) g Vµµee → g Vµµee − e ( g (cid:48) ) π log (cid:18) m τ m µ (cid:19) v m Z (cid:48) + 2 m e yE ν , (C4) g Vττee → g Vττee + e ( g (cid:48) ) π log (cid:18) m τ m µ (cid:19) v m Z (cid:48) + 2 m e yE ν . (C5)29ontributions to the ν e e → ν e e process arise first at 2-loop and require Z (cid:48) mixing with the SM Z boson. They are therefore negligible. Note that the Z (cid:48) contributions to the muon-neutrino andtau-neutrino scattering differ by a relative minus sign. The new physics interferes constructively in ν µ e → ν µ e and destructively in ν τ e → ν τ e , resulting in a partial cancellation of the new physicseffect.The change in the neutrino scattering rate at Borexino due to the presence of the Z (cid:48) can bedetermined as σ Borexino σ SMBorexino = σ ( ν e e → ν e e ) φ ν e + σ ( ν µ e → ν µ e ) φ ν µ + σ ( ν τ e → ν τ e ) φ ν τ σ ( ν e e → ν e e ) SM φ ν e + σ ( ν µ e → ν µ e ) SM φ ν µ + σ ( ν τ e → ν τ e ) SM φ ν τ . (C6)Standard Model predictions for the scattering rate depend on the solar model, in particularon the assumed metallicity of the sun. Combining the predictions from [66] with the Borexinomeasurement in [50], we find the following range of allowed values for the scattering rate at the 2 σ level: 0 . < σ Borexino /σ SMBorexino < .
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