Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams
Wolfgang Altmannshofer, Stefania Gori, Maxim Pospelov, Itay Yavin
NNeutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams
Wolfgang Altmannshofer, Stefania Gori, Maxim Pospelov,
1, 2 and Itay Yavin
1, 3 Perimeter Institute for Theoretical Physics 31 Caroline St. N, Waterloo, Ontario, Canada N2L 2Y5. Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada Department of Physics & Astronomy, McMaster University 1280 Main St. W. Hamilton, Ontario, Canada, L8S 4L8.
The production of a µ + µ − pair from the scattering of a muon-neutrino off the Coulomb field of anucleus, known as neutrino trident production, is a sub-weak process that has been observed in onlya couple of experiments. As such, we show that it constitutes an exquisitely sensitive probe in thesearch for new neutral currents among leptons, putting the strongest constraints on well-motivatedand well-hidden extensions of the Standard Model gauge group, including the one coupled to thedifference of the lepton number between the muon and tau flavor, L µ − L τ . The new gauge boson, Z (cid:48) ,increases the rate of neutrino trident production by inducing additional (¯ µγ α µ )(¯ νγ α ν ) interactions,which interfere constructively with the Standard Model contribution. Existing experimental resultsput significant restrictions on the parameter space of any model coupled to muon number L µ ,and disfavor a putative resolution to the muon g − (cid:48) for any mass m Z (cid:48) (cid:38)
400 MeV. The reach to the models’ parameter space can be widened with future searches ofthe trident production at high-intensity neutrino facilities such as the LBNE.
PACS numbers: 12.60.Cn, 13.15+g, 25.30.Pt
Introduction.
The Standard Model (SM) gauge groupis one of its most important defining features, giving fullyadequate description to all electroweak and strong in-teraction phenomena. However, there is no reason tobelieve that the SU (3) × SU (2) × U (1) structure is fi-nal, and its extensions, both at some high-energy scale,and at low energy, have been discussed in the literature,and subjected to a multitude of experimental searches.New Abelian gauge groups, U (1) X , are of particular in-terest, as many top-down approaches predict their possi-ble existence [1]. The simplest possibility is a new gaugegroup coupled to the SM via a gauge invariant renormal-izable portal, known as kinetic mixing [2], while the SMfields maintain their complete neutrality with respect to U (1) X . There are also well-known possibilities in whichthe SM fields carry a charge under a new force. Therequirement that such theories are valid up to very high-energy scales singles out the anomaly-free combinationsof gauged X = yB − (cid:80) i x i L i number. Here B is thebaryon number, L i are individual lepton flavor numbers,and y, x i are the constants related by the anomaly-freerequirement, 3 y = x e + x µ + x τ . Models with y, x e (cid:54) = 0 aregenerally well-constrained by electron and proton collid-ers, as well as neutrino scattering experiments. Yet, thereis one combination, y = x e = 0; x µ = − x τ , resulting in anew force associated with muon number minus tau num-ber ( L µ − L τ ) [3, 4] that is difficult to probe since it wouldaffect only neutrinos and the unstable leptons.One robust consequence of a new force that couples tomuons via a vector portal, either L µ and/or kinetic mix-ing with the photon, is the additional positive contribu-tion to the muon anomalous magnetic moment a µ . Sincethere exists a long-standing discrepancy in the muon g − ∼ . σ level, a possible increase of a µ by ∼ × − , due to anew vector force, may solve this problem. Until recentlythe existing constraints were sufficiently weak to afford the possibility of a “dark force” resolution to the g − (cid:48) (also known as“dark photon”) has been subjected to a multitude of ex-perimental tests that almost entirely rule out the regionof parameter space relevant to muon g − L µ ?In this Letter , we show that any model based on gaugedmuon number, L µ , is significantly restricted by the rareSM process of neutrino trident production : the produc-tion of a µ + µ − pair from the scattering of a muon-neutrino with heavy nuclei. The observation of this pro-cess in neutrino beam experiments at levels consistentwith the SM strongly constrains contributions from a newforce [13]. Perhaps more importantly, we show that fu-ture neutrino beam facilities, such as LBNE, may be ableto search for such forces in yet unconstrained regions ofthe parameter space. Our present work extends and gen-eralizes the arguments given in [13] for a heavy Z (cid:48) , and inparticular rules out such force as a solution of the muon g − m Z (cid:48) (cid:38)
400 MeV. Also, given the im-portance of this process for new physics, we recalculatethe rate of the neutrino trident production in the SM.
Muonic tridents in the SM and beyond.
To be specific,and to take the least constrained case, we concentrate ona Z (cid:48) boson coupled to L µ − L τ , L Z (cid:48) = − (Z (cid:48) ) αβ (Z (cid:48) ) αβ + m (cid:48) Z (cid:48) α Z (cid:48) α (1)+ g (cid:48) Z (cid:48) α (cid:16) ¯ (cid:96) γ α (cid:96) − ¯ (cid:96) γ α (cid:96) + ¯ µ R γ α µ R − ¯ τ R γ α τ R (cid:17) . Here, g (cid:48) is the U (1) X gauge coupling, the field-strengthis (Z (cid:48) ) αβ = ∂ α Z (cid:48) β − ∂ β Z (cid:48) α , the electroweak doublets asso-ciated with left-handed muons and taus are (cid:96) = ( ν µ , µ L )and (cid:96) = ( ν τ , τ L ), and the right-handed electroweak sin-glets are µ R and τ R . The origin of the vector boson mass a r X i v : . [ h e p - ph ] A ug N Nµ + µ − νν k k p + p − qk Z " γ FIG. 1. The leading order contribution of the Z (cid:48) to neutrinotrident production (another diagram with µ + and µ − reversedis not shown). Other contributions at the same order in g (cid:48) are further suppressed by the Fermi scale. is not directly relevant for our work, and thus we suppressany additional pieces in (1) related to the correspondingHiggs sector.This model contributes to the neutrino trident pro-duction at lowest order through the diagram shown inFig. 1. This contribution interferes with the SM contri-bution coming from W ± /Z exchange. In order to gaininsight into the different contributions, in what followswe provide analytical results using the equivalent pho-ton approximation (EPA) [14, 15]. Under the EPA, thefull cross-section of a muon-neutrino scattering with anucleus N is related to the cross-section of the neutrinoscattering with a real photon through, σ ( ν µ N → ν µ N µ + µ − ) = (cid:90) σ ( ν µ γ → ν µ µ + µ − ) P ( s, q ) . (2)Here, P ( q , s ) is the probability of creating a virtual pho-ton in the field of the nucleus N with virtuality q whichresults in the energy being √ s in the center-of-mass frameof the incoming neutrino and a real photon. This proba-bility is given by [16] P ( q , s ) = Z e π dss dq q F ( q ) , (3)where Ze and F ( q ) are the charge and the electromag-netic form-factor of the nucleus, respectively. The in-tegral over s is done from 4 m to 2 E ν q , with the muonmass m and the neutrino energy E ν . The q integral has alower limit of 4 m / (2 E ν ) and the upper limit is regulatedby the exponential form-factor. We thus concentrate onthe computation of the cross-section σ ( ν µ γ → ν µ µ + µ − ).Computations of the full ν µ N → ν µ N µ + µ − process havebeen performed in [17–22] in the context of the V-A the-ory and of the SM.We begin with the differential cross-section for the νγ → νµ + µ − sub-process associated with a pure V-Acharged interaction between neutrinos and muons. It isgiven symbolically by dσ = 12 s dPS (cid:88) pol | M M | G e , (4) where G F = √ g / (8 M W ) is the Fermi constant. The3-body phase-space (with correction of a typo in the cor-responding expression of ref. [23]) is given bydPS = 12 1(4 π ) dt s d(cid:96) π v d Ω (cid:48) π , (5)where (cid:96) = ( p + + p − ) is the square of the invariantmass of the µ + µ − pair, Ω (cid:48) is the solid angle with re-spect to the photon four-vector in the µ + µ − rest-frame, v = (cid:112) − m /(cid:96) is the velocity of each muon in thatframe, and t ≡ k · q . M and M in (4) are the neutrinoand the muon-pair blocks in the amplitude, that formthe total amplitude according to M = G F e √ M M . Thefactor of 1 / M , explicitly, and summing over spins and po-larizations, we get (in agreement with result of ref. [16])12 (cid:88) pol | M M | ≡ |M V − A | (cid:39) × (cid:32) (6)( k · p + )( q · k )( q · p − ) A + ( k · p − )( q · k )( q · p + ) B + 2( k · p + )( k · p − )( p + · p − ) AB − ( k · p − )( p + · p − )( q · k ) AB − ( k · p + )( p + · p − )( q · k ) AB − ( k · p + )( k · p − )( q · p − ) AB + ( k · p + )( k · p + )( q · p − ) AB + ( k · p − )( k · p − )( q · p + ) AB − ( k · p + )( k · p − )( q · p + ) AB (cid:33) , where A = ( p − − q ) − m and B = ( q − p + ) − m .The result for the full SM contribution together with theZ (cid:48) vector-boson exchange can be obtained from the V-Amatrix-element contribution, if we neglect terms propor-tional to the muon mass. The full square of the matrix-element is defined as in Eq. (6) but with,12 (cid:88) pol | M M | = 512 |M V − A | × (cid:32) C V + C A (7) − C V C (Z (cid:48) ) V m (cid:48) k − m (cid:48) + (cid:18) C (Z (cid:48) ) V m (cid:48) k − m (cid:48) (cid:19) (cid:33) . Here, k is the momentum of the exchanged Z (cid:48) and the SMcoefficients of the vector and axial-vector currents in theinteraction of muon-neutrinos with muons are C V = +2 sin θ W , C A = , with θ W being the weak mixing angle.The second line in Eq. (7) features the Z (cid:48) contributionwith the vector-current coefficient defined as, C (Z (cid:48) ) V = 4 M W m (cid:48) g (cid:48) g = v v Z (cid:48) , (8)where v SM = 246 GeV is the SM Higgs vacuum expecta-tion value and v Z (cid:48) = m Z (cid:48) /g (cid:48) .Next we consider the phase-space integration. The to-tal cross-section is obtained by integrating over the entiresolid angle Ω (cid:48) , (cid:96) < t < s , and 4 m < (cid:96) < s . The inte-gration over phase-space is best done first over the solidangle, then over t and (cid:96) (see also ref. [23]). Keeping onlyleading log terms in the muon mass we find the followingexpression for the inclusive SM cross-section, σ (SM) (cid:39) (cid:0) C V + C A (cid:1) G α s π (cid:18) log (cid:16) sm (cid:17) − (cid:19) . (9)The destructive interference between the charged andneutral vector-boson contributions leads to a reductionof about 40% of the SM cross-section compared to thepure V-A theory. Our results corrects a missing factor of2 in the corresponding expression in ref. [16].In general we can write σ (SM+Z (cid:48) ) = σ (SM) + σ (inter) + σ (Z (cid:48) ) , (10)where the second term is the interference between theSM and the Z (cid:48) contributions. In the heavy mass limit, m Z (cid:48) (cid:29) √ s this can be expressed concisely as [13] σ (SM+Z (cid:48) ) σ (SM) (cid:39) (cid:16) θ W + 2 v /v Z (cid:48) (cid:17) (cid:0) θ W (cid:1) . (11)This expression also holds for the differential cross-section in this limit, up to muon mass corrections.In the limit of light Z (cid:48) , m Z (cid:48) (cid:28) √ s the expression ismore complex. In the leading log approximation, theinterference term is given by σ (inter) (cid:39) G F √ g (cid:48) C V α π log (cid:16) sm (cid:17) . (12)The Z (cid:48) contribution alone, for m (cid:28) m Z (cid:48) (cid:28) √ s , is σ (Z (cid:48) ) (cid:39) m (cid:48) g (cid:48) α π log (cid:18) m (cid:48) m (cid:19) , (13)while for m Z (cid:48) (cid:28) m (cid:28) √ s it is σ (Z (cid:48) ) (cid:39) m g (cid:48) α π log (cid:18) m m (cid:48) (cid:19) . (14)As can be expected, at high m Z (cid:48) the Z (cid:48) contribution is ad-ditive with respect to the SM one (as shown in Eq. (11))and decouples as m − (cid:48) . For light Z (cid:48) , on the other hand,the cross-section is only log sensitive to m Z (cid:48) and the cen-ter of mass energy of the event.To get the total ν µ N → ν µ N µ + µ − cross-section, thereal-photon contribution can be easily integrated againstthe Weizs¨acker-Williams probability distribution func-tion, Eq. (2), in 4 m < s < E ν q and 4 m / (2 E ν ) LHC FIG. 2. Parameter space for the Z (cid:48) gauge boson. The light-grey area is excluded at 95% C.L. by the CCFR measurementof the neutrino trident cross-section. The grey region withthe dotted contour is excluded by measurements of the SM Z boson decay to four leptons at the LHC [24, 25]. Thepurple (dark-grey) region is favored by the discrepancy in themuon g-2 and corresponds to an additional contribution of∆ a µ = (2 . ± . × − to the theoretical value [26]. cross sections reported in [19, 22], for V-A theory andfor the SM, for various neutrino energies, using both theEPA and the numerical calculation. For large m Z (cid:48) therelative size of the Z (cid:48) contribution is independent of theneutrino energy. For low m Z (cid:48) on the other hand, lowerneutrino energies lead to an enhanced sensitivity to theZ (cid:48) . Since the experimental searches employed a varietyof kinematical cuts, in determining the sensitivity to the { g (cid:48) , m Z (cid:48) } parameter space we use full numerical resultsfor the phase-space integration rather than analytic ap-proximations and keep the full dependence on the muonmass.Neutrino trident production has been searched for inseveral neutrino beam experiments. Both the CHARM-II collaboration [27] (using a neutrino beam with meanenergy of E ν ∼ 20 GeV and a glass target) and the CCFRcollaboration [28] (using a neutrino beam with mean en-ergy of E ν ∼ 160 GeV and an iron target) reported detec-tion of trident events and quoted cross-sections in goodagreement with the SM predictions, σ CHARM − II /σ SM = 1 . ± . , (15) σ CCFR /σ SM = 0 . ± . . (16)(Corresponding results from NuTeV can also be used al-beit with some caution due to a rather large differencein the background treatment between the initial report[29] and the publication [30].) These results stronglyconstrain the gauged L µ − L τ model, and more gen-erally any new force that couples to both muons and (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) m Z ' (cid:72) GeV (cid:76) g ' (cid:37) (cid:37) CC F R (cid:72) g (cid:45) (cid:76) Μ (cid:177) Σ C H AR M (cid:45) II FIG. 3. Same as Fig. 2 but focusing on the low mass region.Constraints from CHARM-II and CCFR, Eqs. (15) and (16)are shown separately. We do not attempt a statistical com-bination of the results. The dashed lines show the expectedlimit if the trident cross-section could be measured with 10%or 30% accuracy using 5 GeV neutrinos scattering on Argon. muon-neutrinos. Implementing the phase space integra-tions that correspond to the signal selection criteria ofCCFR and CHARM-II, we arrive to the sensitivity plotsin Figs. 2 and 3. Our results show that the parameterspace favored by the muon g − m Z (cid:48) (cid:38) 400 MeV, proving the importanceof neutrino trident production for tests of physics beyondthe SM. Other constraints and future possibilities. As can beseen from Fig. 2, the region between 5 (cid:46) m Z (cid:48) (cid:46) 50 GeVis independently constrained by searches for the SM Z decay to four leptons at the LHC [24, 25]. The boundobtained by recasting the ATLAS search [25], based onthe full 7+8 TeV data set, extends down to g (cid:48) ∼ − at m Z (cid:48) ∼ 10 GeV. However, the sensitivity diminishesat low m Z (cid:48) because of the cuts employed in this specificLHC search, and in particular on the invariant mass ofsame flavor opposite sign leptons. The clear sensitivityof high-energy colliders to this region of parameter spacemotivates a dedicated search targeting the specific topol-ogy of an on-shell Z (cid:48) emitted from the muonic decay ofthe Z vector-boson and consequently decaying into a pairof muons. At quite low m Z (cid:48) a complication arises as theZ (cid:48) becomes more boosted and the muons originating fromits decay are more tightly collimated, forming a so-called“lepton-jet” [31]. Thus, low-mass leptonic Z (cid:48) points toan interesting prospect of a search for events with twoopposite-sign muons in addition to one muon-jet, alto-gether reconstructing the Z boson.Searches at B-factories for four lepton events can alsobe sensitive to the low m Z (cid:48) region. A search by BaBarlooked at the pair production of two narrow resonances, m Z ' (cid:72) GeV (cid:76) t r i d e n t e v e n t s (cid:144) t ono f A r (cid:144) P O T CC F R (cid:72) g (cid:45) (cid:76) Μ (cid:177) Σ SM FIG. 4. Expected number of trident events per ton of Argonand per 10 POT at the LBNE near detector for a neutrinoenergy of E ν = 5 GeV as a function of the Z (cid:48) mass. Thehorizontal line shows the SM prediction. The purple (darkgrey) region corresponds to Z (cid:48) masses and couplings that yielda contribution to the muon g-2 in the range ∆ a µ = (2 . ± . × − . The light grey region is excluded by CCFR. each decaying into a µ + µ − (or e + e − ) pair [32]. Whilethat search was optimized to an underlying two-bodyevent topology, with two equal masses, rather than oneresonance, we can use it to gain insight on the poten-tial sensitivity of a dedicated search of Z (cid:48) . Requiring theZ (cid:48) to contribute less than 10 events in each, 100 MeVwide, bin of the µ + µ − invariant mass distribution shownin ref. [32], we estimate a sensitivity to a coupling atthe level of g (cid:48) ∼ × − for Z (cid:48) masses in the range0 . (cid:46) m Z (cid:48) (cid:46) × − over a widekinematic window of m Z (cid:48) , open for direct Z (cid:48) productionwith subsequent decay to muon pairs.Perhaps even more interestingly, the low m Z (cid:48) regioncan be efficiently explored at the planned neutrino facil-ity LBNE, with its lower energy and higher luminosity, ascompared to past neutrino beam experiments. In Fig. 4we show an estimate for the expected number of tridentevents per ton of Argon and per 10 protons-on-target(POT) at the near detector at a LBNE-like run wherefor simplicity we set the neutrino energy to E ν = 5 GeV.For our estimate we use the expected charged currentrates from [33] and the charged current cross sectionsfrom [34]. With about one year of data (corresponding to ∼ × POT [35]) and a ∼ 18 ton Argon near detectorsetup [36], we expect O (100) trident events in the regionof parameter space favored by the muon g-2 anomalywith ∼ − Acknowledgments. We would like to thank T. 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