Discovering New Light States at Neutrino Experiments
SSLAC-PUB-14197, FERMILAB-PUB-10-274-T
Discovering New Light States at Neutrino Experiments
Rouven Essig, ∗ Roni Harnik, † Jared Kaplan, ‡ and Natalia Toro § Theory Group, SLAC National Accelerator Laboratory, Menlo Park, CA 94025 Theoretical Physics Department, Fermilab, Batavia, IL60510, USA Theory Group, Stanford University, Stanford, CA 94305 (Dated: August 5, 2010)
Abstract
Experiments designed to measure neutrino oscillations also provide major opportunities for dis-covering very weakly coupled states. In order to produce neutrinos, experiments such as LSNDcollide thousands of Coulombs of protons into fixed targets, while MINOS and MiniBooNE alsofocus and then dump beams of muons. The neutrino detectors beyond these beam dumps aretherefore an excellent arena in which to look for long-lived pseudoscalars or for vector bosons thatkinetically mix with the photon. We show that these experiments have significant sensitivity be-yond previous beam dumps, and are able to partially close the gap between laboratory experimentsand supernovae constraints on pseudoscalars. Future upgrades to the NuMI beamline and ProjectX will lead to even greater opportunities for discovery. We also discuss thin target experimentswith muon beams, such as those available in COMPASS, and show that they constitute a powerfulprobe for leptophilic PNGBs. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] a r X i v : . [ h e p - ph ] A ug ontents I. Introduction
II. Constraints on PNGBs from Rare Meson Decays and Supernovas
III. Constraints from LSND on PNGBs and Dark Gauge Bosons
IV. Sensitivity of Modern Neutrino Experiments a -sstrahlung 22 V. Muon Fixed-Target Experiments with Thin Targets – COMPASS VI. Conclusions Acknowledgements A. Pseudoscalar Production References I. INTRODUCTION
In the last decade, several experiments have explored neutrino masses and mixings, butthese high-luminosity laboratories are also sensitive to rare production of new metastableparticles. Neutrino beams such as the LAMPF Neutrino Source at Los Alamos and NuMI2nd BooNE at Fermilab are produced through two basic stages. First, a high-intensity protonbeam impinges on a target and produces a large number of pions (and other hadrons), whichdecay to muons and neutrinos. The muons are stopped in a thick layer of rock, while theneutrinos travel unimpeded through the rock to the detector. In fact, short-baseline neutrinodetectors are situated behind the most intense proton and muon beam-dumps to date. Thusthey are ideally configured to search for long-lived particles produced by rare proton-nucleusor muon-nucleus interactions.Two classes of new physics scenarios naturally give rise to light, feebly coupled particlesof this type. An approximate symmetry broken at a high mass scale F naturally gives riseto light pseudoscalars — pseudo-Nambu-Goldstone bosons (PNGBs, or “generic axions”)— with couplings of order m X /F to Standard Model matter X . Alternately, a new “dark” U (1) gauge boson can naturally have small kinetic mixing (cid:15) with the photon, giving riseto suppressed interactions with all electrically charged matter [1]. Either spin-1 or spin-0bosons can be radiated in energetic-particle interactions with matter, with a very small rateproportional to the square of their weak coupling. The luminosities achieved in fixed-targetexperiments are such that thousands of these particles could be produced, and they offer anew window into weakly coupled sectors. Once produced, the lightest particles in a hiddensector could decay only through their weak couplings to Standard Model particles, and sowould be quite long-lived and weakly interacting. While ordinary products of the collisionare stopped in the shielding upstream of a neutrino detector, exotics could penetrate theshielding and decay within the detector, yielding a distinctive signal.This note summarizes the several classes of exotic particles that naturally give rise toobservable late-decay signals, their experimental signatures and typical kinematics. In anygiven model, the production cross section and lifetimes of these exotica are both determinedby a single small coupling parameter and by the mass of the produced particle, so in par-ticular, we present specific estimates for the sensitivity achievable with late-decay searchesin MINOS/MINERvA, MiniBooNE, and LSND.PNGB’s coupled to hadrons were searched for extensively in both proton and electronbeam-dump experiments in the 1980s, most notably in CHARM [2] at CERN, E774 [3] atFermilab, and the SLAC experiments E137 [4] and E141 [5]. Many of these limits haverecently been re-interpreted ([6–8]) in the context of kinetically mixed gauge bosons andthe associated scalar bosons that give them mass through the Higgs mechanism. Kinetically3ixed gauge bosons have been a subject of considerable recent interest [9–21] and discussionsof other collider, accelerator, and direct and indirect astrophysical probes for them can befound in e.g. [22–39]. In particular, [7] discussed the sensitivity of neutrino experiments tohadronic production in the case of kinetic mixing and the potential importance of existingLSND data as a constraint on these models. Additional constraints on both classes of modelsfrom supernovaes, rare decays, and radiative corrections have also been extensively discussed[6, 40–44].Our aim in this note is to present a more complete analysis of the sensitivity of past,present, and future neutrino experiments to new weakly-coupled physics, with a particlarfocus on PNGB models. We hope such a unified summary will facilitate new analyses ofneutrino-detector data to discover or constrain new weakly-coupled particles. We also dis-cuss the potential reach for experiments with muons beams that strike a fixed thin target.In § II, we review constraints on generic pseudoscalars from other arenas, in particular su-pernova data, rare meson decays, and the anomalous muon magnetic moment. The rangeexplored by neutrino experiments is complementary to all of these. In § III, we consider theimplications of existing LSND analyses for both PNGB’s and dark gauge bosons. Due tothe large number of protons dumped in LSND, we find that these analyses provide strongerconstraints on PNGB’s than the CHARM experiment. Our results for dark gauge bosonsare consistent with [7], but we have tried to clarify the experimental sensitivity. In § IV, weconsider the sensitivity to PNGBs that could be achieved by analyses using modern neu-trino beamlines, such as BooNE and NuMI, and their near detectors (the complementaryanalysis for kinetically mixed gauge bosons was presented in [7]). We consider both thestandard production mode in proton-nucleus collisions and the production of very forwardPNGBs off the stopping muons. The second mode is enhanced by the magnetic focus-ing of pions, so experiments using focused neutrino beams are uniquely sensitive to purelyleptophilic PNGBs that do not couple to hadrons. For ordinary PNGBs coupled to bothquarks and leptons, searches in MINOS/MINERvA and MiniBooNE would have sensitiv-ity comparable to, or perhaps slightly better than, the CHARM beam-dump limit [2]. Afuture “Project X” could significantly extend this reach into new territory. For leptophilicPNGBs, MINOS/MINERvA has slightly better sensitivity than the constraint from E137.In § V, we discuss muon fixed-target experiments using thin targets. Such an experimentcould be possible at the COMPASS experiment at CERN. We find that a COMPASS-like4etup can probe new territory in the parameter space of leptophilic PNGBs, closing a gapbetween muon g − § VI, andan appendix discusses the details of PNGB production off muon beams.
A. Models of Weakly Coupled Light Exotics
1. Pseudo-Goldstone Bosons
Light pseudoscalars can arise as pseudo-Goldstone bosons in a large variety of well-motivated theories, such as multiple higgs doublet models, theories with an R-axion [45](from spontaneous and explicit R-symmetry breaking in a supersymmetric theory), axionmodels [46], the Next-to-Minimal Supersymmetric Standard model [47] (NMSSM), and re-cent dark matter models with a scalar portal to the dark sector [48]. The most importantpoint is that these particles are naturally light if there is an approximate shift symmetry,and that their interactions are proportional to the inverse of some symmetry breaking scale F . Using fermion equations of motion, the derivative coupling of a PNGB a to a fermionbilinear turns into the operator (in Weyl notation) L ⊃ m χ F aχχ, (1)which is the coupling we assume for the leptons and/or quarks, χ , of the Standard Model.A phenomenologically interesting sub-class of PNGB models are those where the PNGB isleptophilic, i.e. it couples preferentially (or only) to leptons; this scenario could arise if thelepton sector has its own higgs mechanism separate from that of the quark sector. LeptophilicPNGB are essentially unconstrained by searches for rare meson decays and proton fixedtarget experiments, so experiments that have muon beams, such as MINOS/MINERvA,MiniBooNE, and COMPASS, can easily be the most sensitive probes of these particles.The coupling of a PNGB with mass m a to a fermion with mass m (cid:96) induces a PGNBpartial width Γ (cid:96) = m a π (cid:16) m (cid:96) F (cid:17) (cid:113) − (4 m (cid:96) /m a ) (2)and the total width is well approximated by Γ e + Γ µ for m a (cid:46)
400 MeV (for larger masses,hadronic decays can also become important but we use the leptonic widths for masses up5 (cid:45) (cid:45) m a (cid:72) GeV (cid:76) F (cid:72) G e V (cid:76) PNGB Decay Length c Τ Prompt (cid:72) (cid:60) Μ m (cid:76) Displaced (cid:72) (cid:60) (cid:76)
Displaced (cid:72) (cid:62) (cid:76)
Invisible (cid:72) (cid:62)
100 cm (cid:76)
Invisible (cid:72) (cid:62)
100 m (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m A ' (cid:72) GeV (cid:76) Ε A' Decay Length c Τ Prompt (cid:72) (cid:60) Μ m (cid:76) Displaced (cid:72) (cid:60) (cid:76)
Displaced (cid:72) (cid:62) (cid:76)
Invisible (cid:72) (cid:62)
100 cm (cid:76)
Invisible (cid:72) (cid:62)
100 m (cid:76)
FIG. 1:
Left:
The rest-frame lifetime of a pseudo-Nambu-Goldstone boson (PNGB) as a functionof its mass m a and decay constant F . Right:
The lifetime of a dark photon A (cid:48) as a functionof its mass m A (cid:48) and (cid:15) , the strength of its mixing with the Standard Model hypercharge gaugeboson. In both plots, the black lines correspond to different decay lengths ( cτ ): 10 µm (solid), 1cm (dot-dashed), 1 m (dashed), and 100 m (dotted). In the blue, purple, red, green, and whiteshaded regions the decays are prompt ( < µ m), displaced with < > >
100 cm, or “invisible” with > E beam / mass. The feature in the left plot at 2 m µ occurs since PNGB’s coupling to a StandardModel particle is proportional to that particle’s mass, and at this point decays to two muons areallowed. The dip in the right plot near 0.7 GeV is due to the ρ -resonance. The lifetime for boththe PNGB and the A (cid:48) is calculated assuming decays directly into Standard Model particles. to 1 GeV). Thus, for example, proper lifetimes of 1 mm are obtained with F ≈
70 GeV( m a = 100 MeV) or F ≈
20 TeV ( m a = 300 MeV). Fig. 1 (left plot) shows the decay length( cτ ) of a PNGB as a function of its mass m a and decay constant F . Note that the decaylength is very different above and below the muon threshold, due to the much strongercoupling to muons as compared to electrons. We see that for F (cid:46) GeV, they decaypromptly and colliders should be able to set the best constraints. For larger F , collidersearches that look for displaced vertices or missing energy can still set limits, but searchesin beam dump experiments (with a large shield) become relevant.We ignore decays of PNGBs to two photons, since this is always subdominant in the massrange we consider in this paper. 6 . Kinetically Mixed Gauge Bosons Another class of light particles that has received significant interest in the last few yearsis a light, weakly coupled “sequestered sector” lying alongside the Standard Model. Newstates at the MeV–GeV scale are not in conflict with data, because the gauge and globalsymmetries of the Standard Model greatly restrict the couplings of ordinary matter to thesestates. It is, however, quite natural for the few allowed interactions to be suppressed by loopfactors.In particular, we consider a new MeV–GeV scale Abelian gauge boson A (cid:48) coupled toelectrically charged Standard Model particles ψδ L = (cid:15)eA (cid:48) µ ¯ ψγ µ ψ. (3)Such a coupling can generically originate from the kinetic mixing between the field strengthsof the Standard Model hypercharge and a hidden sector U (1) [1], δ L = (cid:15) Y F (cid:48) µν F µνY , (4)where F (cid:48) µν = ∂ µ A (cid:48) ν − ∂ ν A (cid:48) µ is the field strength of the A (cid:48) gauge boson, and similarly F µνY isthe hypercharge field strength. This mixing, assuming the hidden U (1) is broken so thatthe A (cid:48) is massive, is equivalent in low-energy interactions to assigning a hidden charge (cid:15)eq i to Standard Model particles of electromagnetic charge q i , where (cid:15) = (cid:15) Y / (cos θ W ) and θ W isthe Weinberg mixing angle (Eq. (3) if q ψ = 1) . Such a mixing can be generated in manyways, e.g. through loops of new heavy particles that couple to both the A (cid:48) and StandardModel hypercharge. We refer the reader to e.g. [27] for more detailed discussions.In Fig. 1 (right), we show the decay length of a vector boson A (cid:48) as a function of its mass m A (cid:48) and the parameter (cid:15) which sets the strength of its kinetic mixing with hypercharge.Assuming the A (cid:48) decays into Standard Model particles rather than exotics, its lifetime is γcτ (cid:39) N eff m A (cid:48) α(cid:15) (cid:39) . N eff (cid:18) E (cid:19)(cid:18) − (cid:15) (cid:19) (cid:18)
100 MeV m A (cid:48) (cid:19) , (5)where we have neglected phase-space corrections, and N eff counts the number of availabledecay products. If the A (cid:48) mixes kinetically with the photon, then N eff = 1 for m A (cid:48) < m µ (cid:40) µ ( m > m µ )2 e, γ ( m < m µ ) Proton-Nucleus( a/π -mixing) Same as π spectrum” ” Muon a -strahlung muon spectrum (avg.over material)Dark gauge boson 2 e , 2 µ , hadronicmodes Proton-Nucleus( π decay) 1 / π momentum” ” Muon-Nucleus( A (cid:48) -sstrahlung) muon spectrum (avg.over material) TABLE I: Summary of Signal Properties for Light Exotics considered in this paper. The PNGBdecay to two photons is never important for the mass range we consider, and can be ignored. Morecomplicated signals are possible if the PNGB or A (cid:48) can decay to other hidden sector particles beforedecaying to Standard Model particles, but we will not consider this possibility. when only A (cid:48) → e + e − decays are possible, and 2 + R ( m A (cid:48) ) for m A (cid:48) ≥ m µ , where R = σ ( e + e − → hadrons ; E = m A (cid:48) ) σ ( e + e − → µ + µ − ; E = m A (cid:48) ) [49].We summarize the possible production and decay mechanisms for both PNGBs and A (cid:48) ’sin Table I. In this paper, we only consider the case for which the PNGB or A (cid:48) decays directlyto Standard Model matter. More complicated signals are possible if they can decay to otherhidden sector particles before decaying to Standard Model particles. II. CONSTRAINTS ON PNGBS FROM RARE MESON DECAYS AND SUPER-NOVASA. Limits from Kaons and B-Meson Decays
In this section and in Table II, we briefly summarize constraints from meson decayson PNGBs with sub-GeV masses. These constraints are controlled by two factors: thepartial width for the rare meson decays into a PNGB, and the fraction of PNGB’s thatdecay promptly enough to be included in the data samples (or, in the case of invisible-decaysearches, the fraction that decay outside the detector). The combination of these searches issensitive to PNGBs with F (cid:46) −
100 TeV over a wide range of masses. We call these limits“exclusions” below in the interest of brevity, but it should be emphasized that several of8ecay mode B.R. limit Mass range Decay region B + → K + + inv. [50] 3 × − < m a < m ( ∗ ) µ > . K + → π + + inv. [51] 0 . − × − < m a < m ( ∗ ) µ > . K + → π + + X X → e + e − [52] 1 . − × −
10 MeV < m a <
120 MeV < ( † ) B + → K + e + e − [53] 6 . × −
30 MeV < m a < m ( ∗ ) µ < µ m ( † ) d Γ /dm ( B + → K + (cid:96) + (cid:96) − ) [54] 1 . × − < m a < < . TABLE II: Summary of constraints on PNGBs from various experiments (B.R. = BranchingRatio). For decay of meson X , a characteristic transverse energy m X / ( ∗ ) : Searches extend beyond 2 m µ but only imply relevant limits on PNGB models below 2 m µ . ( † ) : Length scales are guesses only, but overall exclusion is insensitive to cutoff because it overlapswith reach of invisible searches. the results have been re-interpreted by non-experts (the authors) in a context very differentfrom the original experimental design, from which significant inaccuracies in the boundariescould have resulted. We focus here on the most constraining searches; a more exhaustiveset of limits are considered in [43], but our treatment of the decay regions and the PNGBlifetime differ from theirs.Following [42], we consider two-higgs-doublet models as a generic framework for a newPNGB a that couples to the Standard Model (whereas [43] assumes NMSSM couplings).In this framework, rare meson decays to a lighter meson and a are mediated primarily bya top-quark loop, which receives contributions at all scales up to the highest energy atwhich the PNGB couples to the top quark (typically either the F scale or the electroweaksymmetry-breaking scale). Therefore, the meson decay rates depend on the detailed UVdynamics through which the a particle couples to Standard Model fields, as discussed in[42]. We adopt the approximate formulaΓ( B + → K + a ) ≈ G F | V ∗ tb V ts | √ π m t m B (cid:18) v F (cid:19) ( kinematic )[ f ( m a ) ] , (6)and likewise for K + → π + a , but with the product of CKM matrices V ∗ tb V ts instead givenby V ∗ ts V td . Here G F is the Fermi coupling constant, v = 174 GeV is the electroweak Higgsvacuum expectation value, m t is the top mass, and m B the B -meson mass. This formulais obtained from (A.1) of [42] by assuming β = 45 ◦ in the two-higgs-doublet model, replac-ing sin θ → v/ (2 F ), and setting the model-dependent combination | X + X | = 1. This9ombination depends on the physical Higgs mass but is typically in the range 1–10 so thatour choice is conservative (although note that there are special choices of the Higgs masswhere the sum of X ’s crosses zero, in which case limits from meson decay are weaker). Thekinematic factor is typically in the range 0 . f ≈ .
33 [55]. The formula above with our parameter choices yields branchingfractions about a factor of 10 smaller than those in [43].The second factor that comes into play is the decay length of the PNGB. For visiblemeson decay searches, the PNGB must decay within a small distance (typically of ordermm-cm) from the meson decay from which it originates, so that all of the tracks producedby the meson reconstruct a single vertex. Likewise, invisible-decay searches require that thePNGB decay outside the detector volume, with a typical size of about a meter. We haveattempted to take these effects into account more precisely than [43], which uses a uniformlength scale of 10 m for all limits.The PNGB partial width to (cid:96) + (cid:96) − is given in Eq. (2) and the lifetime is shown in Fig. 1.Naively, one would expect that PNGB’s with characteristic lifetimes between 1 mm and 1m might be poorly constrained by both visible-decay and invisible-decay searches. However,the fraction of events in which the PNGB decays within an atypically short distance fromthe meson vertex can still set significant constraints; moreover, the limits from experimentswith different energy scales ( m K and m B ) overlap to fill this intermediate-lifetime region. a. Invisible decay searches for B + → K + ν ¯ ν or K + → π + X , where X decays invisibly,peak in a specified mass range. Below the muon threshold, these are sensitive to the highest F ’s probed in accelerator experiments. • B + → K + + inv. : A search at Belle [50] set an upper limit of 1 . × − on therate of B + → K + ν ¯ ν . This search imposes an upper limit on the energy deposited inthe electromagnetic calorimeter (ECL), which extends to 1.65 m from the beamline.Therefore, for this limit we consider only a decays outside this radius, assuming a typ-ical transverse momentum of m B /m a . A tighter limit could likely be set by searchingfor a narrow invisible resonance in the B + decays. • K + → π + + inv. : The search for K + → π + ν ¯ ν at Brookhaven E787 [51] also set anexplicit limit on the decay K + → π + X where X is invisible. The branching fractionfor this mode must be below about 5 × − for X masses below about 100 MeV, and10bout 10 − for X masses between 150 and 250 MeV. No limit is set between 100 to150 MeV, where several K + → π + ν ¯ ν candidates were seen. The transverse size of thedetector is roughly 1.3 m, and the kaons are produced at rest so that m K / K + decay limit extends to higher F but the B + decay search is able toprobe lower F because the PNGBs produced in B decays are more boosted and thereforelonger-lived. There are additional constraints from CLEO and BaBar searches for Υ(1 s )and Υ(3 s ) decays to γa [56–58], but this region is largely contained in the two identifiedabove. b. Visible decays Again, we focus here only on the most powerful visible-decay searches: • K + → π + X, X → e + e − : This resonant decay mode was searched for in the K µ experiment at KEK [52], which excluded kaon branching fractions to below about1 . × − for PNGB masses between 10 and 80 MeV, and below about 5 × − for PNGB masses from 80 to 120 MeV. The experiment also set a limit above 140MeV, but it is not as constraining as the B + -decay limits discussed below, so we donot include the constraint in this region for this mode. K µ used stopped kaons, sowe assume an initial energy of m K /
2. No vertex requirement is explicitly mentionedin the analysis, but common vertex requirements of order a few cm are frequentlyimposed in spectrometer analyses typical in these experiments; in any case, at muchlarger distances, the mass resolution would likely be degraded. We have used an adhoc but conservative estimate of 1 cm to produce the limit in Fig. 2 but even if thevertex requirement is much tighter (as tight as 0 . B + → K + + inv. search. • B + → K + (cid:96) + (cid:96) − : BaBar [53] and Belle [54, 59] have both measured the rate of the raredecay B + → K + (cid:96) + (cid:96) − for (cid:96) = e, µ . The observed branching fractions, (3–6) × − ,are consistent with Standard Model predictions, and can be translated into rough(conservative) limits on B + → K + a by requiring that this exotic decay not exceedthe total measured rate (we focus on the decays to K ± rather than K ∗ because theobserved rates are lower). The BaBar measurement includes lower-mass electron pairs,down to 30 MeV (compared to 140 MeV at Belle). The most recent Belle analysis[54] bins events by invariant mass, so that we can obtain a tighter limit ( ≈ − ) on11 + → K + a in the region of interest, 140-1440 MeV. In all cases we take the limit tobe the central value plus 2 σ . The BaBar measurement required that the (cid:96) + (cid:96) − pairoriginate from the same vertex as the K + . To set a conservative limit we requirethat the a decay within 100 µ m, the scale of BaBar’s vertex resolution. For the Belleanalysis we require only that the PNGB decay within 0.5 cm, which would pass therequirement of [59]. • Similar but slightly weaker limits are obtained from measurements of K → π(cid:96) + (cid:96) − ,e.g. [60–62]. We refer the reader to the original results and to [43] for details.It is worth emphasizing that the crude limits we have obtained are far weaker than thetightest limits that could be obtained by a directed analysis of BaBar or Belle data. Firstly,much tighter limits could be obtained by binning the 20–100 events in each sample morefinely, and accounting for the detection efficiency as a function of mass. Further improvementcould be obtained by including more displaced a decays, at the edge of the inner tracker (3cm) or even beyond. This direction is particularly worthy of exploration for the B → Kµ + µ − mode, for which there are no complementary searches near the high- F boundary of the Belle-excluded region, and for which the muon system gives an additional handle for studying highlydisplaced decays. B. Limits from Supernova SN 1987a
For completeness we also include the constraints on PNGBs from SN 1987a [63]. We adaptthe analysis of [64] to our setup, and obtain limits based on the assumption that PNGBsmust not be the dominant mechanism of energy loss from the supernova. The temperatureof the supernova core is conservatively estimated at T ∼
30 MeV [64], so PNGBs withmass significantly greater than these energies cannot be produced by the supernova, and aretherefore unconstrained. The flux of PNGBs from the core of the supernova is approximately[64] dN a dE a ∼ (cid:18) F (cid:19) e − E a /T . (7)However, if these PNGBs decay or are re-absorbed, then they will not escape from thesupernova and so they will not carry away any energy. Scattering and re-absorption dominate12 (cid:45) (cid:45) m a (cid:72) GeV (cid:76) F (cid:72) G e V (cid:76) (cid:45) (cid:45) m a (cid:72) GeV (cid:76) F (cid:72) G e V (cid:76) FIG. 2:
Left:
Constraints on pseudo-Nambu-Goldstone bosons as a function of their decayconstants F and their mass m a from various meson decays: K + → anything + e + e − (green), K + → π + + invisible (blue), B + → K + (cid:96) + (cid:96) − (yellow) ( (cid:96) = e, µ ), and B + → K + + invisible (red).Constraints from Υ(1 S ) or Υ(3 S ) → γa → γ + invisible and K + → π + (cid:96) + (cid:96) − decays are weakerthan those from B + → K + + invisible and B + → K + (cid:96) + (cid:96) − , respectively, and thus not shown.Details are in § II A.
Right:
Gray shaded background region is the combined exclusion region frommeson decays in the left figure. In the green exclusion region, the proton beam dump experimentCHARM at CERN would have seen at least five events (this exclusion region agrees roughly withthat in [2]) – see § IV. Here the PNGB is produced directly in the proton dump by a small mixingwith the pion. For m a < m µ , the PNGB decays to an electron pair, while in the “bubble” for m a > m µ the PNGB decays predominantly to a muon pair. The blue region is the limit from thesupernova SN 1987a (see § II B). The light red region is the constraint from the muon anomalousmagnetic moment and fills the gap for low m a and F left by the meson constraints (see § II C). Theregion excluded by the Fermilab E137 dump lies mostly within the CHARM excluded region andis not shown (it is instead shown in Fig. 5). over PNGB decay for the relevant region of parameter space, giving a mean free path of λ mfp ∼
10 m (cid:18) F GeV (cid:19) (8)for PNGBs with F < GeV. We see that for F significantly smaller than 10 GeV,the PNGB mean free path is much less than the estimated core size of 10 km, so for thesesmaller values of F , SN 1987a does not constrain the PNGB. The exclusion contours we haveplotted in Fig. 2 (right) correspond to requiring that the PNGBs carry away less energy thanneutrinos, meaning that the total integrated PNGB emission must be less than about 10 T . C. Limits from the anomalous muon magnetic moment
PNGB’s contribute to the anomalous magnetic moment of the muon, a µ , at the looplevel. For the mass range of interest in this paper ( (cid:46) a aµ = − π m µ F (cid:90) dx x x + (1 − x ) m a m µ . (9)While the experimental measurement of a µ is rather precise [66], the Standard Model pre-diction involves a hadronic contribution that must be estimated from experiments, whichdo not all agree. Using data from e + e − annihilation to hadrons, the theoretical value of a µ is smaller than the measured value by (316 ± × − [67], which is a 4 σ discrepancy.However, estimates from τ ’s give a smaller disagreement, with [68] finding a difference of(157 ± × − , which is a 1.9 σ discrepancy.Since the contribution from PNGB’s is negative, a very conservative limit is obtained byusing the 5 σ lower bound in [68], i.e. a aµ ≥ (157 − × × − , i.e. a aµ ≥ − × − . (10)This constraint is included in Fig. 2 (right) and 5. III. CONSTRAINTS FROM LSND ON PNGBS AND DARK GAUGE BOSONS
The Liquid Scintillator Neutrino Detector (LSND) experiment ran at the Los AlamosNeutron Scattering Center (LANSCE) in the 1990’s [69], and dumped O (10 ) 800 MeVprotons on a predominantly water-copper target. This produced ∼ pions, a very largenumber that allows LSND to be sensitive in principle to very weakly coupled PNGBs orgauge bosons. A detector of length 8.3 m and a diameter of 5.7 m was located 29.7 m awayfrom the target, 12 ◦ off-axis [84], and filled with dilute liquid scintillator (there was no open14 beam (GeV) N p X t (m) X d (m) n π (cid:15) geo ¯ E a (GeV)CHARM [2] 400 2 . ×
480 515 0.12 25LSND [69–71] 0 . ∼ . ×
541 553 0.002 2.7
TABLE III: Shown are the total number of incident protons N p , the distance from the target tothe open decay region in front of the detector (i.e. the thickness of the shield) X t , the distance fromthe target to the end of the detector X d , the geometric acceptance times the number of pions perincident proton n π (cid:15) geo , and the median PNGB energy E a . These numbers were used to calculatethe sensitivity of CHARM, MINOS/MINERvA (we always use the larger MINOS detector forestimates), and MiniBooNE to PNGBs produced directly in the proton dump. For LSND, we useda more involved procedure described in the text. decay region in front of the detector). Although the LSND collaboration did not searchfor signals that could originate from decays of long-lived exotics, approximate limits can beextracted from two published LSND analyses.We begin by reviewing the production of PNGB’s in proton-nucleus collisions (whichwill re-appear in our discussion of more recent neutrino experiments in § IV), then discussthe implications of two specific LSND analyses [70, 71] for PNGBs that decay to e + e − pairs or µ + µ − pairs, respectively. The calculation involves considerable uncertainties andassumptions, and should be taken only as a sensitivity estimate — a dedicated analysisby the LSND collaboration is required to obtain a reliable limit. However, this analysisdemonstrates that a dedicated analysis by the LSND collaboration would set some of thetightest constraints on PNGBs that couple to both quarks and leptons.We next discuss the production of dark gauge bosons in pion decay, and the implicationsof the same two analyses for these models. Our results are consistent with [7], but we haveconsiderably elaborated the discussion of the experimental sensitivity. The LSND sensitivityfor this model overlaps closely with that of the SLAC electron beam-dump experiment E137[4], as computed in [6]. A. Production of PNGBs at LSND and other Proton Beams
In this section, we will consider the production and experimental sensitivity to pseu-doscalars from proton beam dumps. We will focus on the LSND experiment because it can15et the most stringent limits, although the MINOS and MiniBooNE experiments can alsoset interesting limits. We will discuss them more extensively in § IV.Proton beam dumps can produce pseudoscalars directly through their mixing with pions.If the pseudoscalars couple to quarks they will interact via the operator m q F a ¯ qq = ⇒ c m π F π F aπ (11)where c is an O (1) parameter that depends on the up and down quark masses and anycoefficients in the pseudoscalar coupling to the quarks. Since the pseudoscalar mixes withthe pion, for every pion that is produced through a QCD process there is a probability ofapproximately (for c = 1) (cid:18) F π F (cid:19) (12)of instead producing a PNGB.We can estimate the production rate within the detector acceptance and the PNGBmomentum distribution by using the measured rates for π : N a = (cid:18) F π F (cid:19) n π N p (cid:15) geo . (13)Here, n π is the number of pions produced per incident proton, N p is the total number ofprotons, and (cid:15) geo is the geometric acceptance (the solid angle subtended by the detector atthe target divided by the solid angle of the beam). In practice, only the product n π N p (cid:15) geo is relevant but we have tabulated estimates for n π (cid:15) geo and N p in Table III for variousexperiments, for the sake of comparison. To determine the number of observable e + e − pairs,we must also account for the probability of decaying in the detectable region (either insidethe detector or in an open region upstream of the detector): N e = N a (cid:16) e − Xtγcτ − e − Xdγcτ (cid:17) (cid:39) N a X d − X t γcτ e − L γcτ , (14)where X t and X d are the minimum and maximum decay lengths (roughly, X t is the thicknessof the shield and X d is the distance from the target to the end of the detector), and in thesecond expression we have assumed that γcτ (cid:29) X d − X t .16 (cid:45) (cid:45) m a (cid:72) GeV (cid:76) F (cid:72) G e V (cid:76) FIG. 3: Sensitivity of various neutrino experiments to pseudo-Nambu-Goldstone bosons as a func-tion of their decay constants F and their mass m a . The thick (thin) black solid line correspondsto 10 (1000) events in LSND, the thick (thin) dashed blue line corresponds to 3 (1000) eventsin MiniBooNE, and the thick (thin) dot-dashed red line corresponds to 3 (1000) events in MI-NOS/MINERvA (in each case the inner regions correspond to more events than indicated by theline). Here the PNGB is produced directly in the proton dump by a small mixing with the pion.For m a < m µ , the PNGB decays to an electron pair, while in the “bubbles” for m a > m µ the PNGB decays predominantly to a muon pair. The gray shaded regions are the combined ex-isting constraints from other beam dump experiments, meson decays, anomalous muon magneticmoment, and SN 1987a shown in the right plot of Fig. 2. B. LSND Analyses Sensitive to PNGBs
We now focus on two LSND results and their implications for PNGB’s.
PNGBs decaying to e + e − : The analysis in [70] used ∼ . × protons on target andlooked for ν µ → ν e oscillations using ν µ from π + decay in flight [85]. The ν e are detectedvia the inclusive charged-current reaction ν e + C → e − + X . This analysis focused onidentifying electrons in the energy range 60 MeV to 200 MeV. Various cuts were used in theanalysis with an energy-dependent efficiency that is always near 10%. Clearly this analysis17hould be sensitive to PNGBs decaying to electrons inside the LSND detector, although it isimpossible to accurately estimate the efficiency without a dedicated analysis. One differencebetween our signal and the study in [70] is that in our case both an electron and a positronare produced in the detector, as opposed to just a single electron. However, it is impossibleto distinguish e + e − events from a single electron (or single photon) event, and it has beensuggested [75] that we should assume that the total e + e − pair energy (i.e. the PNGB energy)would have been measured as the energy of a “single-electron”. For simplicity, we will assumethat the detection efficiency for an e + e − pair is roughly the same as for the single electronanalysis, i.e. (cid:15) eff , ∼ . F π /F from the total number of pionsincident on the detector, (8 . ± . × cm − . This should be an excellent approximationfor m a ≈ m π ; it may be subject to additional O (1) uncertainties for m a (cid:28) m π and especiallyfor m a (cid:29) m π , which we neglect. In addition to the reconstruction efficiency, we must alsoaccount for the fraction of PNGBs with kinetic energy between 60 − m a and 200 − m a , whichfor small m a and the assumed distribution is approximately 25%. We note that the meanof the pion kinetic energy distribution is at ∼
275 MeV with a root-mean- square spreadof ∼
130 MeV, so an analysis including higher-energy electrons would be significantly moresensitive.We show the number of e + e − events obtained from PNGB decays in the F versus m a parameter space in Fig. 3, where the solid black thick and thin lines for m a < m µ show10 and 1000 e + e − events, respectively. We have assumed that the number of PNGB decaysinside the detector is given by Eq. (14) integrated over an PNGB energy from 60 MeV − m a to 200 MeV − m a , and multiplied by the efficiency (cid:15) eff , . The analysis of [70] (see theirFig. 29) indicates that 10 PNGB events in a 20 MeV energy bin below 200 MeV would havebeen easily noticed. We note that the sensitivity could have been increased by increasingthe energy threshold for electrons beyond 200 MeV. PNGBs decaying to µ + µ − : We now turn our attention to the analysis in [71], which wouldhave been sensitive to µ + µ − events. The analysis in [71] considered ∼ . × protons on18arget and searched for π → ν µ ¯ ν µ . The neutrinos interact with the nuclei in the detectorto produce muons through the reactions ν µ + C → µ − + p + X and ¯ ν µ + C → µ + + n + X .This analysis focused on identifying high-energy, muon-like beam excess events in the energyrange 160 MeV to 600 MeV electron equivalent. The muon was required to decay insidethe detector. Various selection cuts were used in the analysis [71] with an overall efficiencyof ∼ .
15 for identifying neutrino interactions in the detector. This analysis should clearlyhave been sensitive to PNGBs decaying to µ + µ − pairs inside the LSND detector, althoughit is impossible to accurately calculate the efficiency without a dedicated analysis. Wesimply estimate the efficiency for µ + µ − pairs to be reconstructed as single-muon events tobe (cid:15) eff , ∼ .
1, a little bit less than the efficiency for the original analysis. We estimate thePNGB kinetic energy distribution as described above. To obtain the total number of pionsincident on the detector, we rescale the value we used above for the a → e + e − analysis to1 . × cm − , since this analysis uses a factor of ∼ − m a + 2 m µ to 600 MeV − m a + 2 m µ , we show the numberof µ + µ − events obtained from PNGB decays in the F versus m a parameter space in Fig. 3,where the solid black thick and thin lines for the region m a > m µ show 10 and 1000 µ + µ − events, respectively. The analysis in [71] indicates that 10 PNGB events in a given energybin would have been noticed, and so we believe that this is a reasonable number of eventswith which to set a tentative limit. C. LSND Limits on Light Vector Bosons
In [7] it was shown that LSND may be sensitive to a new light vector boson A (cid:48) thatmixes with U (1) hypercharge; however, the precise sensitivity was unclear due to uncertain-ties in the LSND experiment. We believe that we have clarified these uncertainties usinginformation from [70, 71, 75].With approximately 1 . × protons on target at 800 MeV, the LSND experimentproduced about 1 . × π ’s. The vast majority of these π ’s decay to γγ , so with akinetic mixing parameter (cid:15) we expect a branching fraction of 2 (cid:15) for π → γA (cid:48) . We will only19onsider the simplest scenario, where the A (cid:48) decays to e + e − with a lifetime τ A (cid:48) = 3 α(cid:15) m A (cid:48) (cid:113) − m e m A (cid:48) (cid:16) m e m A (cid:48) (cid:17) (15)that is also set by the kinetic mixing. Due to the huge number of pions produced at LSND,the A (cid:48) may be visible even for very small (cid:15) .To set a reliable limit, we use the results of the π → ν ¯ ν search discussed previously [71].They performed a Monte Carlo simulation to determine the flux of neutrinos in the LSNDdetector from this decay and plotted it in their Fig. 1 [71]. In our case we want to consider π → γA (cid:48) , but in the limit that m A (cid:48) (cid:28) m π the neutrino results should give a very goodapproximation to the distribution of A (cid:48) particles; beyond this limit we will continue to usethe neutrino results with the caveat that we expect an O (1) uncertainty in the rates. Giventhis distribution of A (cid:48) particles, we computed the number of A (cid:48) s that would have decayedinside the LSND detector using the lifetime formula above, and the 2 (cid:15) branching ratio of π → γA (cid:48) .The only reliable data that can exclude an A (cid:48) decaying to e + e − inside the LSND detectoris the analysis of [70], which we discussed above in order to determine the LSND sensitivityto PNGBs. From the study of π backgrounds in [70] and from discussions with LSNDcollaborators [75], we expect that an A (cid:48) decay to e + e − would have been interpreted as asingle-electron event in the LSND detector. The data on these types of events has only beenpublished for energy depositions below 200 MeV, and the analysis in question had an overallreconstruction efficiency of about 10%. Thus we use the Monte Carlo data [70, 75] belowthis energy to set our tentative limits, which are displayed in Fig. 4. Our limits are basedon the conservative assumption that a signal of 50 or more events after the 10% efficiencycut should have been visible. IV. SENSITIVITY OF MODERN NEUTRINO EXPERIMENTSA. Production from Proton-Nucleus Collisions
We have discussed this production mode already in the context of LSND, in § III A. Asimilar analysis was also carried out by the CHARM experiment at CERN [2], a proton20 .01 0.1 110 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m A ' (cid:72) GeV (cid:76) Ε E137E141E774 a Μ a e (cid:85) (cid:72) (cid:76) SN FIG. 4: The sensitivity of various beam dump, collider, and astrophysical probes of light vectorbosons that kinetically mix with the standard model hypercharge, as a function of the kineticmixing parameter (cid:15) and the vector boson mass m A (cid:48) (from [6]). The thick (thin) solid black linecorresponds to 10 (1000) signal events in LSND. The region enclosed by the thick line can be viewedas a very rough exclusion limit from the LSND experiment. A re-analysis of the LSND data by theLSND collaboration could further extend the sensitivity of that experiment [7]. Further details aredescribed in the text. beam dump with a 400 GeV beam, a 35 m decay region, and a 3 × F π /F ) . However,the production rate of very weakly coupled PGNBs could actually be significantly greaterthan this scaled rate because PGNBs produced within the target will virtually always escapefrom the target, whereas many of the pions produced by proton interactions might not makeit out of the beam dump. This issue may merit further exploration.For the CHARM experiment, we take the experimental parameters from [2]. Our calcu-21ation of the PNGB exclusion region is shown in Fig. 2 (right) and roughly agrees with theresults in [2].The MiniBooNE estimate is shown as a thick (thin) blue dashed line in Fig. 3, corre-sponding to 3 (1000) events. It is obtained as follows. We take the π production crosssection to be the average of the π + and π − production cross sections found in [74]. Thesecross sections are given as a function of angle and momentum, and one can show that thefraction of π ’s produced in the target that point towards the 12 m detector at a distanceof 541 m away is roughly 0 . π + and π − produced per “particle-producing reaction”, which we take to be per incident proton ontarget – we average these to find about 0.89 π ’s per incident proton.The MINOS/MINERvA estimate is shown as a thick (thin) red dot-dashed line in Fig. 3,again corresponding to 3 (1000) events. This estimate for MINOS was obtained by using dataon their π + momentum distributions from Fig. 9.3 of [72]. Based on the assumption that themomentum distribution for a general PGNB would be similar, we estimate from that figurethat a fraction 0 . .
18 pions produced perproton on target, which allows us to estimate the total number of PGNBs produced.Searches in either MiniBooNE and MINOS/MINERvA would have significant overlapwith the existing CHARM result, perhaps slightly extending the region probed. However,the proposed “Project X” upgrades to NuMI, which would increase the beam intensity bya factor of ∼ F . B. Production through Muon a -sstrahlung A more unique search opportunity at MINOS/MINERvA and MiniBooNE arises fromtheir magnetic focusing of charged pions, which in turn focuses the neutrinos and muonsfrom their decay towards the detectors. As a result of this focusing, approximately one in athousand protons on target produces a muon that points toward the neutrino detector. Asthese muons stop in the rock upstream of the detector, they can radiate very forward PNGB’sin a process that is analogous to ordinary bremsstrahlung. Unlike photon bremsstrahlung,however, the typical PNGB produced by this process carries a large fraction of the incident22 beam (GeV) N (cid:96) ¯ E (cid:96) (GeV) X t (m) X d (m) W (m) E thresh (GeV) E137 [4] 20.0 N e = 1 . ×
20 180 380 3 2.0
MINOS/MINERvA [76] 120.0 N µ = 2 . ×
20 240 270 3 1.0MiniBooNE [77] 8.9 N µ = 2 . × TABLE IV: Shown are the proton beam energy E beam , the total number of incident electrons (forE137) or muons (for MINOS/MINERvA and MiniBooNE) N (cid:96) and their average energy ¯ E (cid:96) , thedistance from the start of the electron or muon dump to the open decay region (if any) in front ofthe detector or to the detector itself (i.e. the thickness of the shield) X t , the distance from the startof the muon dump to the end of the detector X d , the diameter of the detector W , and the thresholdenergy E thresh to detect an electron or muon that originates from an PNGB decay. These numberswere used to calculate the sensitivity of E137 [4], MINOS/MINERvA [76] (we always use the largerMINOS detector for estimates), and MiniBooNE [77] to PNGBs produced by bremsstrahlung offan electron beam (in the case of E137) or off a muon beam that is produced in a proton dump. muon’s energy. The production rate and kinematics can be reliably calculated using theWeizs¨acker-Williams approximation [78], where the nuclei in the fixed target provide aneffective photon beam. We have relegated the details to Appendix A, so in this section wewill simply give the results along with their physical motivation.We will be interested in a wide range of values for m a ; with this in mind we note thatproduction is dominated by emission angles θ a (cid:46) max (cid:18) m a E , m µ E (cid:19) (16)for E a ∼ E . This is very useful for estimating the angular acceptance, which we computeby taking the ratio of the solid angle subtended by the detector to the solid angle within θ a of the beam direction. We also considered acceptance issues associated with the PNGBdecay, but because we require the PNGB to decay within the MiniBooNE detector or in therelatively small open decay region in front of the MINOS/MINERvA detector, this is notimportant, and we ignore it.The cross section is always peaked near E a ≈ E , even when m a (cid:28) m µ . This is in starkcontrast to the analogous formula for photon bremsstrahlung, where the rate is proportionalto the inverse of the photon energy. The difference is due to the contrasting soft emissionbehavior of gauge bosons and goldstone bosons – the former have soft singularities, whileemission of the latter vanishes in the soft limit due to their derivative couplings. The23 (cid:45) (cid:45) m a (cid:72) GeV (cid:76) F (cid:72) G e V (cid:76) FIG. 5: Sensitivity and constraints of various experiments to leptophilic pseudo-Nambu-Goldstonebosons as a function of their decay constants F and their mass m a . Here the PNGB is produced bybremsstrahlung off an incident muon or electron beam. Thick (thin) lines show rough sensitivityregions and correspond to 3 (1000) displaced e + e − pairs in MINOS/MINERvA (red dot-dashedlines), MiniBooNE (blue dashed lines), and in a thin target experiment using the COMPASS muonbeam (green dotted line, see § V). The thick (thin) dotted black lines correspond to S/ √ B = 3 (10),where S ( B ) are the number of prompt µ + µ − signal (radiative background) events in COMPASS(see § V) (we have ignored the Bethe-Heitler background and the finite acceptance, so these linesshould not be viewed as real significance lines but only as very rough estimates of what could beprobed). Inside the gray shaded region, E137 would have seen at least one event – since they sawnone, this region gives their approximate constraint. Details are described in the text. The lightred region is the constraint from the muon anomalous magnetic moment (see § II C). peaking toward high energy fraction E a /E is further enhanced by a larger phase spacewhen m a (cid:29) m µ . The total cross section for pseudoscalar production from a muon beam hasthe parametric form σ ≈ m µ F α max( m µ , m a ) (17)Note that the formula in the case of an electron beam dump would be identical except with24 µ → m e , so we see that muons are a much more efficient source of PNGBs as long as m a (cid:29) m e .To get a rough idea of our experimental reach, it is useful to have approximate formulasfor the total production rates. For a thin target, the yield is N a ∼ N µ m µ αF T e m e max( m µ , m a ) (18)where T e is the number of electron radiation lengths of material. In the case of a thicktarget, where the beam of muons is completely dumped, it is more difficult to give a simpleparametric formula for the pseudoscalar yield because muons stop due to minimum ionizationinteractions, as opposed to bremsstrahlung. However the equation above with T e ∼ e + e − pairs in MINOS/MINERvA (red dot-dashed lines) andMiniBooNE (blue dashed lines) as a function of the PNGB decay constant F and the PNGBmass m a . The thick (thin) lines correspond to 3 events (1000 events). The gray shaded regioncorresponds to the approximate constraint from E137: inside the region, more than 1 e + e − event would have been seen (note that we calculated this region using the procedure in [6],but changing the couplings from an A (cid:48) to those relevant for PNGBs). Our estimate forthe E137 constrained region agrees with that in [4]. We see that MINOS/MINERvA canextend the E137 region, although the MiniBooNE region is contained within the E137 regionsince many PNGBs are produced in the dump with a large enough angle causing them tomiss the detector (this is because the MiniBooNE muon beam has a lower energy and theirdetector is further away compared to MINOS/MINERvA). Note that these lines have beencalculated with the full formula as detailed in the appendix. The NOVA experiment and“Project X” upgrades to the NuMI beamline will have a factor of 5–10 more protons andso will significantly extend the reach. The light red region is the constraint from the muonanomalous magnetic moment, while the other lines will be discussed in § V.25 . MUON FIXED-TARGET EXPERIMENTS WITH THIN TARGETS – COM-PASS
In this note, we have argued that new light states may be produced by bremsstrahlungoff a dumped muon beam. Here we will briefly comment on the potential of using a fixedtarget setup to constrain weakly coupled light states. Compared to a dump experiment,the target will be much thinner, and we can thus search for particles with a much shorterlifetime.Electron fixed target experiments have been used to probe new light states [6, 26, 37, 79],but muon fixed-target setups are different in several respects. The intensity of muon beamsis obviously lower, but this may be partly compensated for by a much thicker target —a muon beam can easily traverse a meter of material leaving a relatively quiet off-beamenvironment. Furthermore, muon beams will have an obvious advantage over electrons inPNGB searches, as their couplings are proportional to particle mass. For example, it isinstructive to consider the COMPASS experiment at CERN [80], where a 160 GeV muonbeam strikes a low- Z polarized target (they have also used higher- Z , but much thinner,targets). A detailed description of their polarized target is given in [80], and we approximateit as a 130 cm long target consisting of Lithium, with a packing factor of 0.5 (i.e. a columndensity of 34.7 g/cm , or 0.42 radiation lengths). In total, we estimate that this experimenthas collided about ∼ muons. Here we will illustrate the rough reach of this COMPASS-like setup assuming the Lithium target. We have checked that a higher Z target, such asTungsten, of a similar thickness would probe a somewhat larger parameter space, includinghigh- F regions above the muon threshold. We will not show this in any figures, focusinginstead on the data set already collected by COMPASS.There are two regions of parameter space to consider, one in which F is low and the PNGBdecays promptly, and one in which F is high and the PNGB decays with a displaced vertex.In the case of a prompt decay, one must search for a peak in the di-muon invariant mass ontop of a sizable standard model background (decays to electrons may not be searched forthis way because their interactions in the target considerably degrade the mass resolution).The signal to background ratio (where the background comes from di-lepton production via26n off-shell photon) for such a search is roughly [6] SB ∼ m µ F α m a δm , (19)where δm is the resolution-limited mass window used in the search. Based on the angularand momentum resolutions for COMPASS [81], we infer a mass resolution σ m = 11 MeV,making a window width δm ≈ . σ m = 27 MeV appropriate for estimating sensitivity.Additional two-photon diagrams (which we refer to as the “Bethe-Heitler” background) andtheir interference have been neglected; these actually dominate over the “radiative” dileptonproduction assumed in 19, but can be reduced to an order-1 fraction of the background bykinematic selection.We can combine Eq. (19) with the expected number of signal events using the formulasin Appendix A to get a very rough estimate for the sensitivity of such a search. Theseestimates omit several factors, all of which degrade sensitivity: the finite detector acceptance,the additional Bethe-Heitler background, and the finite acceptafinite of kinematic selectionneeded to veto the dominant Bethe-Heitler background. Assuming 10 muons on targetand a mass window δm = 27 MeV [81], we show the “3 σ ” and “10 σ ” estimated reach inFig. 5 (the neglected factors are expected to reduce the sensitivity each point in parameterspace by a factor of 2–4). Such a search in existing COMPASS data (or that of a similarexperiment) would be sensitive to leptophilic PNGBs with F near the weak scale and massesbetween the dimuon threshold and a few GeV. Higher F ’s can be probed by increasing thetarget thickness in radiation lengths, for example by using a high- Z target of comparablelength.Another parameter range that can be studied by muon fixed-target experiments is thatwhere the PNGB decays are significantly displaced. If the vertex can be reconstructed tobe downstream of the target region, beyond the tails of Standard Model backgrounds, a fewevents could be enough to claim a discovery. The transverse vertex resolution of COMPASSis about 0.1 mm [80], so that the vertex resolution in the direction of the beam is about σ z = (0 . × E beam /m a . In Fig. 5, we show the area of parameter space that yields 3or 1000 events in the region between 5 σ z and 10 meters behind the 130 cm Lithium target(for this estimate we assume that the PNGB is produced in the middle of the target). Notethat the vertex resolution is worse for lower-mass PNGBs, so that for m a (cid:46) M eV , we27ave 5 σ z (cid:38)
10 m, and there is no good search region (which is why the plot cuts off for low m a ). An analysis of the data in COMPASS, or a similar experiment of this type, can covernew ground with respect to E137 or the neutrino experiments at lower F , and can close thewindow between the latter searches and the muon g − Z targetsuch as Tungsten, more PNGBs would be produced so that some region of parameter spaceat F ∼ GeV above the muon threshold may also give rise to observably large rates ofdisplaced decays.In summary, the very rough estimates above suggest that a fixed target experiment with afocused muon beam may be able to probe unconstrained regions in PNGB parameter space.In fact, the existing data set of the COMPASS experiment may already be able to set someinteresting new limits for leptophilic PNGBs. It may also be worthwhile to consider usingmore diffuse muon “beams” such as those in neutrino factories in a thin target setup due totheir higher intensity. We leave this for future thought.
VI. CONCLUSIONS
We have explored the sensitivity of neutrino oscillation experiments to three types of newlight states – vector bosons that kinetically mix with the photon, pseudoscalars that coupleto quarks and leptons, and pseudoscalars that couple preferentially to leptons. The first twoare strongly constrained by rare decays, fixed target experiments, and supernova 1987a (allof which we have reviewed), whereas there are fewer tests of the third class.The sensitivity of the LSND experiment to vector bosons was discussed in [7], but thedetails of the LSND detector and analyses were not considered. We have shown that theanalyses of [70, 71] would have been sensitive to vector bosons with mass below 2 m µ anda large range of coupling strengths. This sensitivity has significant overlap with the E137experiment [4], but LSND does probe a new region at very weak coupling, and in any case theLSND results serve as an important cross-check. Because LSND dumped a larger number ofprotons, other neutrino experiments are not as sensitive to the light vector boson scenario.We also considered the sensitivity of various neutrino experiments to pseudo-scalars thatcouple to quarks, so that they can be produced in proton beam dumps. LSND remains themost sensitive experiment for pseudoscalars with mass below 2 m µ , nearly closing the gapbetween fixed target experiments and supernova constraints. However, other experiments28re more sensitive to heavier pseudoscalars. The MiniBooNE and MINOS experiments arecurrently competitive with the best limits on these particles. However, the estimates thatwe derive for these experiments may be too conservative because we estimate the rate ofPGNB production by scaling the pion production rate, which may be an underestimate ifmany pions get stuck in the beam dump. This issue merits further study.The MINOS and MiniBooNE experiments produce neutrinos from focused muon beams;the requisite muon beam dumps provide a unique opportunity to search for pseudoscalarsthat couple preferentially to leptons, since we expect these particles to couple far more tomuons than to electrons. The MINOS experiment is sensitive to leptophilic pseudoscalarswith decay constants almost an order of magnitude greater than any previous experiment(in particular E137), while MiniBooNE can probe a region that is contained within that ofE137.A thin target experiment with a muon beam, such as that available in COMPASS, offersa unique probe for leptophilic PNGBs. An analysis using the existing COMPASS data setand looking for either e + e − pairs originating from displaced vertices behind their (Lithium)target or for a spike in the µ + µ − invariant mass spectrum of muons coming from the targetshould be sensitive to new regions of parameter space. A similar experiment using a higher- Z target would have even more sensitivity.Upgrades to the NuMi beamline followed by the proposed “Project X” experiment willexplore new parameter space for both standard and leptophilic pseudoscalars. It is our hopethat in the future these experiments will perform dedicated analyses to explore and constrainnew weakly coupled low-mass particles. Acknowledgements
We thank Philip Schuster for collaboration in early stages of this work. We also thankJames Bjorken, Joe Lykken, Chris Polly, Geoff Mills, Simona Murgia, Michael Peskin,Ronald Ransome, Brian Rebel, Byron Roe, David Schmitz, Tomer Volansky, Jay Wacker,Hywel White, and Geralyn Zeller for very useful discussions. We especially want to thankWilliam Louie for extensive discussions and correspondence about the LSND detector andpublications. RE and JK are supported by the US DOE under contract number DE-AC02-76SF00515. Fermilab is operated by Fermi Research Alliance, LLC, under Contract DE-29C02-07CH11359 with the United States Department of Energy. We acknowledge the hos-pitality of the Aspen Center for Physics where part of this work was done.
Appendix A: Pseudoscalar Production
We are interested in the pseudoscalar production rate from muons braking in a fixedtarget. This process is analogous to ordinary bremsstrahlung, and it can be reliably calcu-lated using the Weizs¨acker-Williams approximation [78], where the nuclei in the fixed targetprovide an effective photon beam. When the incoming muon has energy E , the differentialcross section to produce a pseudoscalar of mass m a with energy E a = xE is dσdx d cos θ a = m µ F α E xU (cid:20) x − m a x (1 − x ) U + 2 m a U ( m a (1 − x ) + m µ x (1 − x )) (cid:21) χ (A1)where U = E θ a x + m µ x + m a − xx (A2)is the virtuality of the intermediate muon in initial state bremsstrahlung, and χ is a formfactor that can be found in [78]. We will be interested in a wide range of values for m a ; withthis in mind we see that production is dominated by θ a (cid:46) max (cid:18) m a E , m µ E (cid:19) (A3)for x ∼
1. For angles larger than this, the differential cross section falls off rapidly, as 1 /θ a , sothis angular scale sets the width of the PNGB beam for the purposes of angular acceptance.We compute the angular acceptance by taking the ratio of the solid angle subtended by thedetector to the solid angle within θ a of the beam direction.Integrating Eq. (A1) over the angle θ a , we find dσdx = m µ F α m µ x (cid:20) f (1 + f ) χ ( Z ) + (cid:18) f (1 + f ) log(1 + f ) − f + 2 f f (1 + f ) (cid:19) χ ( Z ) (cid:21) (A4)where f = m a (1 − x ) m µ x (A5)30nd we have now included the form factors χ ( Z ) = Z ln(184 Z − / ) + Z ln(1194 Z − / ) (A6) χ ( Z ) = Z + Z. (A7)It is important to note that the cross section is always peaked near x ∼
1, even when m a (cid:28) m µ . This is in stark contrast to the analogous formula for bremsstrahlung, which isproportional to 1 /x . The difference is due to the different soft emission behavior of gaugebosons and goldstone bosons – the former have soft singularities, while emission of the lattervanishes in the soft limit. The cross section is dominated for x ∼ σ ≈ m µ F α max( m µ , m a ) (A8)The max( m µ , m a ) factor comes from the presence of the function f ∼ m a /m µ in the de-nominator for larger m a /m µ . The formula in the case of electrons would be identical with m µ → m e , so we see that as claimed, muons are a much more efficient source of PNGBs aslong as m a (cid:29) m e .To use these formulae we must account for the way that the muon slows in a beam dump.The number of PNGBs produced per incident muon is dYdx = N A X e A (cid:90) E E a dE (cid:90) T e dt e I µ ( E , E , t e ) dσdx (cid:48) (A9)where E is the energy of the original incident muons, E a is the energy of the producedPNGBs, x (cid:48) = E a /E , x = E a /E , N A is Avogadro’s number, X e is the unit (electron)radiation length in g/cm , A is the atomic number of the material, and T e is the total numberof (electron) radiation lengths in the target or beam dump. The function I µ ( E , E , t e ) isthe distribution of muon energies after the muons have traversed t e radiation lengths.It is essential to remember that muons stop primarily through minimum ionizing interac-tions, and not through radiation, so the number of radiation lengths is not directly relatedto the muon energy loss. We therefore use electron radiation lengths in our computations,so that t e and T e can be much greater than 1 without completely depleting the energy ofthe muon beam. We estimate the function I µ using the relevant material properties (if theintervening material is earth, we use Silicon).31o compute the number of electron or muon pairs from PNGB decays in a detector, wesimply integrate Eq. (A9) times the probability that an PNGB with this energy decays in(or in some cases in front of) the relevant detector (see Eq. (14)). [1] B. Holdom, Phys. Lett. B166 , 196 (1986).[2] F. Bergsma et al. (CHARM), Phys. Lett.
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