Charged lepton beams as a source of effective neutrinos
aa r X i v : . [ h e p - ph ] F e b Charged lepton beams as a source of effective neutrinos
I. Alikhanov (a)
Institute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, RussiaInstitute of Applied Mathematics and Automation KBSC RAS, 360000 Nalchik, Russia
PACS – Neutrino interactions
PACS – Leptons
PACS – W bosons
Abstract – Neutrinos are likely the most poorly understood basic constituents of the StandardModel. In order to investigate precisely their interactions one should be able to create highintensity and well-collimated neutrino beams with known flavor compositions. This is a challengingproblem for neutrino experiments. We propose a method of studying neutrino interactions basedon the fact that a charged lepton is able to manifest itself effectively, with a certain probability, asa neutrino. The effective neutrino method may provide an additional tool for probing neutrino-induced reactions at e + e − and ep colliders as well as at other facilities that use charged leptonbeams. We derive the distributions of the effective neutrinos in the charged leptons and giveexamples of application of the method to electron–positron collisions. Introduction. –
Neutrinos have fundamental impli-cations for various fields of physics: particle physics, as-trophysics and cosmology [1, 2]. They are the only knownparticles that have led to an extension of the StandardModel due to the violation of the law of conservation ofthe family lepton numbers [3–5] and still display proper-ties requiring explanations. In order to investigate pre-cisely neutrino interactions one should be able to createhigh intensity and well-collimated neutrino beams withknown flavor compositions. This is a challenging prob-lem for neutrino experiments. The currently exploited ordiscussed techniques are based on the idea of generationof high energy neutrino fluxes through in-flight decays ofmesons and boosted beta-radioactive ions [6, 7].On the other hand, there are well-established exam-ples of probing dynamical properties of particles thatare unavailable in free states at ordinary accelerator ex-periments. First of all, this is the quark–parton model(QPM) [8] according to which a hadron can manifest itselfas a quark or gluon (parton) carrying the given fraction ofthe parent hadron momentum. The QPM represents thecross sections for lepton–hadron or hadron–hadron colli-sions as sums of lepton–quark, quark–quark, gluon–quarkor gluon–gluon subprocess cross sections. The observablesbecome thus dependent on the peculiarities of the par-ton interactions. Another example is the possibility of (a)
E-mail: [email protected] studying two photon processes at electron–positron col-liders [9–11]. Though there is no real photon in the ini-tial state, the electrically charged electron (positron) actsas an effective photon beam according to the Weizsäcker–Williams approximation. The massive electroweak bosons, W and Z , can be treated as partons in the description ofvarious reactions as well [12, 13].We propose to extend the concept of the effective (equiv-alent) particle also to neutrinos. We show that the dis-tributions of the effective neutrinos in the charged lep-tons can be derived within a simple approach. The effec-tive neutrino approximation (ENA) may provide a frame-work for probing neutrino-induced reactions observationof which in laboratory conditions is considered by manyunrealistic or at least very difficult. For example, relyingon the ENA one can investigate ν e ¯ ν e annihilation into dif-ferent final states at e + e − colliders in the same way asthe quark–antiquark annihilation was effectively analyzedthrough Drell–Yan processes [14]. Neutrino distributions in charged leptons. –
Before proceeding to the discussion of the distributions ofthe effective neutrinos in the charged leptons, it is usefulto begin with the Weizsäcker–Williams equivalent photonapproximation (EPA). The probability density of findinga photon inside an electron with fraction x of the parentp-1. Alikhanov Figure 1: The effective electron in the parent electron (a). Theeffective electron neutrino in the parent electron (b). electron momentum can be written as [15] f γ/e ( x, Q ) = α π (cid:20) − x ) x ln (cid:18) Q max Q min (cid:19) + O (1) (cid:21) , (1)where α = e / (4 π ) is the fine structure constant, Q min and Q max are the minimum and maximum of the magnitudeof the four-momentum transfer in the given process. Thenon-logarithmic term is, in general, also a function of x and Q .It is obvious that in the limit of vanishing electron massthe outgoing electron, after the emission of the photon,carries the fraction y = 1 − x of the initial momentum(see Fig. 1). This means that (1) represents simultane-ously the probability density of finding an electron withmomentum fraction y . Therefore the distribution of theeffective (equivalent) electrons in the electron is given by f e/e ( y, Q ) = α π (cid:20) y − y ln (cid:18) Q max Q min (cid:19) + O (1) (cid:21) , (2)where ≤ y ≤ . A detailed discussion of these functionscan be found in [16].In analogy with the EPA, the distribution of the effec-tive W − bosons in the electron gives simultaneously thedistribution of the effective electron neutrinos since theemission of W − is always accompanied by ν e as shown inFig. 1. Hence, relying on the results of [17], we readilyfind the distribution of the effective electron neutrinos inthe electron to be f Tν/e ( y, Q ) = α π
14 sin θ W " (cid:0) y + m W /s (cid:1) − y − m W /s × ln (cid:18) Q max + m W Q min + m W (cid:19) − s (cid:0) Q max − Q min (cid:1)(cid:21) , (3)where the T superscript is to indicate that the accompa-nied boson is transversely polarized, θ W is the Weinbergangle, s is the total center-of-mass (cms) energy of the re-action squared. The non-logarithmic term in the squarebrackets does not exceed unity because Q max < s . Notethat ≤ y ≤ − m W /s as the emitted boson is nowmassive [17] and the validity of the condition √ s > m W is simultaneously assumed. Also note that, in the high energy limit, (3) transforms into the distribution of theeffective W − bosons in the electron with the replace-ment y ⇄ (1 − y ) , being consistent with the spectrumfrom [12, 13]. This is similar to the property that theWeizsäcker–Williams equivalent photon spectrum of theelectron can be obtained from (2) by simply exchangingthe electron and photon momenta [16]. One can see thatat m W → and / (4 sin θ W ) → , (3) reduces to (2).We can generalize (3) to the case of the off-shell W bosons of arbitrary mass q ≥ as follows: d f Tν/e d q = α π
14 sin θ W " (cid:0) y + q /s (cid:1) − y − q /s × ln (cid:18) Q max + q Q min + q (cid:19) − s (cid:0) Q max − Q min (cid:1)(cid:21) ρ ( q ) . (4)Here we have made use of the Breit–Wigner density func-tion corresponding to the W propagator: ρ ( q ) = 1 π p q Γ( q )( q − m W ) + q Γ ( q ) (5)with the W width being Γ( q ) = α sin θ W p q . (6)Again ≤ y ≤ − q /s and s > q . In the on-shelllimit, (5) gives ρ ( q ) = δ ( q − m W ) , so that (4) turnsinto (3), as it should be.In the same way as above, adopting the distribution ofthe effective longitudinally polarized bosons in the electronfrom [17], we derive the related effective neutrino distri-bution: d f Lν/e d q = α π sin θ W y + q /s − y − q /s ρ ( q ) . (7)The distribution of the effective electron antineutrinos inthe positron is also given by (4) and (7) due to CP invari-ance. The presented results apply to the distributions ofthe other neutrinos, ν µ in µ − and ν τ in τ − , provided thatthe mass of the given parent charged lepton is negligiblysmall compared to the reaction energies and the masses ofthe boson ( m µ,τ ≪ s , m µ,τ ≪ q ). Application to e + e − collisions . – The Glashow resonance.
The Standard Model in-cludes processes the existence of which is not yet con-vincingly proved in experiments. This is the case for the s -channel resonant production of the on-shell W − bosonin ¯ ν e e − → W − . (8)This channel was predicted by Glashow in 1959 [18] andis now usually referred to as the Glashow resonance.p-2harged lepton beams as a source of effective neutrinos Figure 2: The Glashow resonance in the reaction e + e − → W + W − . The strategy of observation of the resonance adopted to-day is to search for it in large volume water/ice neutrinodetectors by exploiting the flux of very high-energy cosmicantineutrinos [19]. The reaction ¯ ν e e − → W − in the con-text of the effective electron approximation has been alsoconsidered in [20–22]. Below we show that in the frame-work of the ENA the Glashow resonance can be probed atelectron–positron colliders.The resonance will be excited when the incident elec-tron annihilates on an effective electron antineutrino fromthe positron as illustrated in Fig. 2. The CP conjugateresonance, ν e e + → W + , will appear similarly. The wholeprocess will be observed in the following form: e + e − → W + W − . (9)Note that only the transversely polarized bosons con-tribute to (9). Then, in analogy to the QPM, the crosssection for e + e − → W + W − will be given as a convolution: σ ( s ) = 2 Z ( − m W / √ s ) f Tν/e ( y, Q ) σ νe → W ( ys )d y, (10)where σ νe → W ( s ) is the cross section for the subprocesses ¯ ν e e − → W − and ν e e + → W + . The factor 2 implies thatthe distributions for ν e in e − and ¯ ν e in e + are equal toeach other because we assume CP invariance. In order toevaluate the integral of (10) in the resonance region weuse the narrow width approximation: σ νe → W ( s ) = 2 π α sin θ W δ ( s − m W ) . (11)Substituting (3) and (11) into (10) and taking into accountthat at s → ∞ for reaction (9), ( Q max + m W ) / ( Q min + m W ) → s/m W , we obtain σ ( s ) = πα θ W s ln (cid:18) sm W (cid:19) . (12)Figure 4 shows the cross section as a function of the cmsenergy. It should be emphasized that (12) coincides with Figure 3: The ν e ¯ ν e → Z annihilation in e + e − collisions. the exact cross section derived within the electroweak the-ory in the asymptotic limit [23, 24]. This is because thelogarithmic part of the exact cross section comes fromthe terms, in the amplitude squared, behaving as ∼ /t (with t = − Q being the Mandelstam variable). One cancheck that each term of this type is related to the Feyn-man diagram containing the neutrino [24]. This is eitherthe square of the diagram itself or an interference termwith this diagram. Our model provides thus an interpre-tation of the nature of the reaction e + e − → W + W − : itis asymptotically a sum of the Glashow resonance and its CP conjugate. Already at the cms energy √ s = 500 GeVthe contribution of the non-resonant channels (with the γ and Z boson exchanges) to e + e − → W + W − dropsdown to the level of a few percent and continues de-creasing at TeV energies. Such energies can be achievedat future electron–positron colliders as the InternationalLinear Collider (ILC) [25] and the Compact Linear Col-lider (CLIC) [26]. For instance, the integrated luminosityof 4000 fb − expected at the ILC for √ s = 500 GeV [27]would allow to produce ∼ × Glashow resonanceevents. The CLIC can study this reaction at substantiallyhigher energies in its second and third stages of operationwith the cms energies 1.5 TeV and 3 TeV, respectively.The corresponding integrated luminosities are planned tobe 2.5 ab − and 5 ab − [28], so that the CLIC will becapable of producing more than events in each of thestages. Exploiting specific observation channels along withthe limitations of experimental apparatus will certainly re-duce these numbers. For example, in the relatively clearchannel with two uncorrelated oppositely charged muonsin the final state (due to the subsequent W ± → µ ± + ν decays), one would observe approximately the total eventrate times (Γ W + → µ + ν / Γ W + → all ) ≈ − . The Z -burst mechanism. Another example is annihi-lation of neutrino–antineutrino pairs into the neutral elec-troweak boson: ν ¯ ν → Z. (13)This process has an important implication for cosmologyp-3. Alikhanov σ ( e + e - → W + W - Z )/ α σ ( e + e - → W + W - ) s ( TeV ) T o t a l c r o sss e c t i on ( pb ) Figure 4: Total cross sections for the Glashow resonance ex-citation in e + e − → W + W − given by (12) (dotted curve) andfor the Z -burst channel in e + e − → W + W − Z , (16), divided by α (solid curve). because provides a tool to probe the relic neutrino back-ground [29] but still remains unidentified experimentally.Ultra-high-energy neutrinos scattering onto relic neutri-nos may excite the Z resonance which, subsequently de-caying, would produce a shower of observable secondaryparticles. It is sometimes called the Z -burst mechanism.The ν e ¯ ν e → Z channel of this scenario can be tested at e + e − colliders in the following form (see Fig. 3): e + e − → W + W − + Z. (14)Within the ENA the cross section for (14) is representedas σ Z ( s ) = Z Z f Tν/e ( x, Q ) f Tν/e ( y, Q ) σ ν ¯ ν → Z ( xys )d x d y, (15)where σ ν ¯ ν → Z ( s ) is the cross section for the subprocess ν e ¯ ν e → Z . The narrow width approximation gives σ ν ¯ ν → Z ( s ) = 8 π α δ ( s − m Z ) / sin θ W , so that (15)asymptotically behaves as σ Z ( s ) ∝ α cos θ W sin θ W s ln (cid:18) sm W (cid:19) . (16)This cross section as a function of the cms energy is plot-ted in Fig. 4. The CLIC in the third energy stage at √ s = 3 TeV is capable of producing more than Z -burst events.Due to the leptonic decay modes of the final bosons, auseful signature of the appearance of an off-shell Z in the ν e ¯ ν e channel is two charged lepton pairs in the final state: e + e − → W + W − + Z → l + l − + l + l − (17)with one of the lepton pairs being highly massive (seeFig. 5). This is similar to the observation of massive µ + µ − pairs in pp collisions through Drell–Yan processes. Note Figure 5: The 4-lepton signal ( l + l − + l + l − ) for the ν e ¯ ν e → Z annihilation in e + e − collisions. that the leptons from the W ± boson decays are not neces-sarily from the same generation, so that a distinctive signalmay be, for example, of the form e ± µ ∓ + l + l − , as depictedin Figs. 6(a) and (b). Thus the signal in a detector will bevisible as an uncorrelated electron and a muon with longi-tudinal momenta of opposite signs (from the W ± → l ± + ν decays) and two leptons of the same generation with bal-anced transverse momenta (from the Z → l + l − decays).In contrast, the background events, shown in Figs. 6(c)and (d), will have a different signature and may be there-fore reduced. This is because in a background event, at √ s ≫ m W , both W + and W − will be emitted either by anelectron or by a positron predominantly along the beamaxis and decaying will give the charged leptons with longi-tudinal momenta of the same sign. The background fromthe photon conversion into charged fermion pairs, Fig. 7,will have the same behavior. The latter must be taken intoaccount provided the given experiment cannot distinguishbetween leptons of different generations. The quark de-cay modes of the bosons can also be used to identify the Z -burst events. In this case the signal will contain twojets with balanced transverse momenta plus two jets withuncorrelated momenta. The states with jets and chargedleptons may also be combined. The background can beisolated again due to the peculiar distribution of the lon-gitudinal components of the jet momenta. The Feynman- x variable, x F = 2 p ∗|| / √ s , should be useful in this regard ( p ∗|| is the longitudinal component of the particle momentumin the cms frame).It is notable that models with the so-called "secret neu-trino interactions" with hypothetical new mediators cou-pling to neutrinos [30] can be similarly tested at high mo-mentum transfers in electron–positron collisions. Conclusions . –
Neutrinos are likely the most poorlyunderstood basic constituents of the Standard Model. Itis a challenging problem to create high intensity and wellcollimated neutrino beams with known flavor compositionsfor related experiments. In this article we have proposedthe effective neutrino method as a framework for studyingneutrino interactions. A charged lepton is able to manifestp-4harged lepton beams as a source of effective neutrinos (a)(b) (d)(c)
Figure 6: An example of a 4-lepton signal for the ν e ¯ ν e → Z annihilation in e + e − collisions (a) and its signature in a detector (b).The diagram of the corresponding background event and its signature in the detector are shown by (c) and (d), respectively.In (b) and (d), azimuthal symmetry around the beam axis, denoted by the dotted line, is assumed.Figure 7: An example of a diagram for a process with pho-ton conversion into a fermion–antifermion pair that may alsocontribute to the background for the ν e ¯ ν e → Z annihilationevents. Its signature in a detector will be similar to that inFig. 6(d). itself, with a certain probability, as a neutrino. We havederived the distributions of the effective neutrinos in thecharged leptons. This is analogous to the parton densi-ties in hadrons introduced in order to investigate dynam-ical properties of quarks and gluons unavailable in freestates. The method may provide an additional tool forprobing neutrino-induced reactions at e + e − and ep collid-ers as well as at other facilities that use charged leptonbeams. For example, we have shown that i) the reac-tion e + e − → W + W − asymptotically is determined bythe Glashow resonance and its CP conjugate; and ii)the ν ¯ ν → Z annihilation can be observed in the chan- nel e + e − → W + W − Z . Both the Glashow resonance andthe Z -burst mechanism remain so far unidentified in tra-ditional experiments exploiting ultra-high-energy neutrinocomponent of cosmic rays, however can be accessed, ac-cording to our study, at electron–positron colliders oper-ating at reasonable center-of-mass energies.The proposed approach is simple and can be readilyapplied to other neutrino types as well as to neutrino–quark interactions. For example, high massive final statesin ν e q and ν µ q scattering processes in ep and µp collisionscan be treated in similar way. In closing, we point out thatthe effective neutrino method has at least the followingadvantages:1) Various neutrino-induced processes, observation ofwhich in the laboratory conditions is considered bymany unrealistic or at least very difficult, become ac-cessible with ordinary collider experiments.2) Techniques of charged particle beam acceleration arewell established.3) The collision energy can be varied in a wide range.4) High luminosities can be achieved.5) The interaction region is localized.p-5. Alikhanov ∗ ∗ ∗ I would like to thank Professor E. A.
Paschos , for read-ing the manuscript and encouraging comments. This workwas partly supported by the Program of fundamental sci-entific research of the Presidium of the Russian Academyof Sciences "Physics of fundamental interactions and nu-clear technologies".
References[1]
King S. F., Merle A., Morisi S., Shimizu Y. and
M. Tanimoto , New J. Phys. , (2014) 045018.[2] Bilenky S. M. , Phys. Part. Nucl. , (2015) 475.[3] Fukuda Y. et al. [Super-Kamiokande Collabora-tion] , Phys. Rev. Lett. , (1999) 2644.[4] Ahmad Q. R. et al. [SNO Collaboration] , Phys. Rev.Lett. , (2001) 011301.[5] Eguchi K. et al. [KamLAND Collaboration] , Phys.Rev. Lett. , (2003) 021802.[6] Zucchelli P. , Phys. Lett. B , (2002) 166.[7] Volpe C. , J. Phys. G , (2004) L1.[8] Bjorken J. D. and
Paschos E. A. , Phys. Rev. , (1969) 1975.[9] Brodsky S. J., Kinoshita T. and
Terazawa H. , Phys.Rev. D , (1971) 1532.[10] Walsh T. F. and
Zerwas P. M. , Phys. Lett. B , (1973) 195.[11] Budnev V. M., Ginzburg I. F., Meledin G. V. and
Serbo V. G. , Phys. Rept. , (1975) 181. [12] Kane G. L., Repko W. W. and
Rolnick W. B. , Phys.Lett. B , (1984) 367.[13] Dawson S. , Nucl. Phys. B , (1985) 42.[14] Drell S. D. and
Yan T. M. , Phys. Rev. Lett. , (1970)316.[15] Frixione S., Mangano M. L., Nason P. and
RidolfiG. , Phys. Lett. B , (1993) 339.[16] Chen M. S. and
Zerwas P. M. , Phys. Rev. D , (1975)187.[17] Alikhanov I. , arXiv:1812.05578 preprint, 2018.[18]
Glashow S. L. , Phys. Rev. , (1960) 316.[19] Berezinsky V. S. and
Gazizov A. Z. , JETP Lett. , (1977) 254.[20] Alikhanov I. , Eur. Phys. J. C , (2008) 479.[21] Alikhanov I. , Phys. Lett. B , (2015) 295.[22] Alikhanov I. , Phys. Lett. B , (2016) 247.[23] Flambaum V. V., Khriplovich I. B. and
Sushkov O.P. , Sov. J. Nucl. Phys. , (1975) 537.[24] Alles W., Boyer C. and
Buras A. J. , Nucl. Phys. B , (1977) 125.[25] Fujii K. et al. , ILC-NOTE-2015-067, arXiv:1506.05992preprint, 2015.[26]
Burrows P. N. et al. [CLICdp and CLIC Collabo-rations] , CERN Yellow Rep. Monogr. , (2018) 1.[27] Barklow T., Brau J., Fujii K., Gao J., List J.,Walker N. and
Yokoya K. , arXiv:1506.07830 preprint,2015.[28]
Robson A. and
Roloff P. , CLICdp-Note-2018-002,arXiv:1812.01644 preprint, 2018.[29]
Weiler T. J. , Phys. Rev. Lett. , (1982) 234.[30] Kolb E. W. and
Turner M. S. , Phys. Rev. D , (1987)2895.(1987)2895.