Terrestrial and Solar Limits on Long-Lived Particles in a Dark Sector
SSLAC-PUB-13807SU-ITP-09/43
Terrestrial and Solar Limits on Long-Lived Particles in a Dark Sector
Philip Schuster, Natalia Toro, and Itay Yavin Theory Group, SLAC National Accelerator Laboratory, Menlo Park, CA 94025 Theory Group, Stanford University, Stanford, CA 94305 Center for Cosmology and Particle Physics, New York University, New York, NY 10003 (Dated: November 5, 2018)Dark matter charged under a new gauge sector, as motivated by recent data, suggests a richGeV-scale “dark sector” weakly coupled to the Standard Model by gauge kinetic mixing. The newgauge bosons can decay to Standard Model leptons, but this mode is suppressed if decays into lighter“dark sector” particles are kinematically allowed. These particles in turn typically have macroscopicdecay lifetimes that are constrained by two classes of experiments, which we discuss. Lifetimes of10 cm (cid:46) cτ (cid:46) cm are constrained by existing terrestrial beam-dump experiments. If, inaddition, dark matter captured in the Sun (or Earth) annihilates into these particles, lifetimes up to ∼ cm are constrained by solar observations. These bounds span fourteen orders of magnitudein lifetime, but they are not exhaustive. Accordingly, we identify promising new directions forexperiments including searches for displaced di-muons in B-factories, studies at high-energy and-intensity proton beam dumps, precision gamma-ray and electronic measurements of the Sun, andmilli-charge searches re-analyzed in this new context. I. LIGHT LONG-LIVED PARTICLES FROMNEW GAUGE SECTORS
Recent astrophysical anomalies have motivated models[1, 2, 3, 4, 5] of TeV-scale dark matter interacting withnew states at the GeV scale. A compelling candidate is anew gauge force, mediated by a GeV-mass vector-bosonthat mixes kinetically with hypercharge: L = L SM + i ¯ χ (cid:0) /∂ + ig /A (cid:48) (cid:1) χ + M χ ¯ χχ (1) − F (cid:48) ,µν F (cid:48) µν + (cid:15) θ W F (cid:48) µν B µν + L Dark . A program of searches in high-luminosity e + e − colliderdata [6, 7, 8], new fixed-target experiments [9, 10], andhigh-energy colliders [11, 12] can search quite exhaus-tively for the new gauge boson A (cid:48) if it decays into a leptonpair through the (cid:15) -suppressed kinetic mixing term.The A (cid:48) can, however, be accompanied by “dark sector”particles (other vectors, Higgs-like scalars, pseudoscalars,or fermions) to which it couples without (cid:15) -suppression,so that the decays of A (cid:48) into dark sector particles domi-nate. These particles may decay to Standard Model mat-ter (and indeed some must , if dark matter annihilationis to explain the astrophysical anomalies), but unlike the A (cid:48) their lifetimes are typically macroscopic — we will callsuch particles “long-lived particles” (LLP’s).In this paper, we discuss observational constraints onthe decays of the LLPs which cover lifetimes spanningfourteen orders of magnitude, 10 cm (cid:46) cτ (cid:46) cm. Aschematic diagram illustrating the type of experimentswe consider is shown in Fig. 1. The constraints fall intotwo categories: Terrestrial beam dump experiments (Sec. II): Ex-periments at proton and electron beam dumps
FIG. 1: A pictorial summary of the two similar approachesdiscussed in this paper: A strong source, either a beam dumpor dark matter annihilation in the Sun, produces dark sectorparticles. LLP’s can penetrate the shield, either the Sun’sinterior or the beam dump shielding. Decays downstreamover a baseline, L Baseline, can be detected. search for decays of long-lived particles in an in-strumented region downstream of a thick shield.Though A (cid:48) production scales as (cid:15) , the integratedluminosities ∼ ab − allow searches for LLP’swith mass < ∼ few GeV and 10 (cid:46) cτ (cid:46) cm. Annihilation of Dark Matter in the Sun (Sec.III): Dark matter captured in the Sun through A (cid:48) -mediated scattering annihilates efficiently, witha rate scaling as (cid:15) . Annihilations into LLP’slighter than the τ threshold result in e , µ , andlight hadrons, which can be detected when theLLP decays outside the Sun but before passing theEarth. In this case, neutrino, electron, and gammaray observations together constrain dark matterannihilation into LLP’s with 10 cm (cid:46) cτ (cid:46) cm. a r X i v : . [ h e p - ph ] N ov n Sections II and III, we describe the best available lim-its. In Section IV, we interpret these limits in a par-ticular illustrative case: a Higgs-like scalar in the darksector that decays into two Standard Model fermions.We conclude in Section V, and highlight the value ofdirect A (cid:48) → (cid:96) + (cid:96) − searches, searches for displaced di-muons in B-factories, high-energy and high-intensity pro-ton beam dumps, precision gamma-ray and electronicmeasurements of the Sun, and milli-charge searches inthe context of these models. A. Decay Lengths in Dark Sectors
Before proceeding, we summarize the parametric scal-ing that governs the decays of various particles in a darksector which is coupled to the Standard Model throughgauge kinetic mixing.In non-Abelian dark sectors, multiple gauge bosons canmix with one another [4, 12], so that each acquires aneffective kinetic mixing (cid:15) eff and lifetime ∝ (cid:15) − eff . Par-ticles χ of other spins can have three-body decays intoStandard Model charged matter ( χ → f + f − χ (cid:48) throughan off-shell A (cid:48) ), with lifetimes ∝ (cid:15) − but longer thanthose of vectors due to three-body phase space. For (cid:15) ∼ − − few × − and moderate mass hierarchiesthese lifetimes are less than 1 m, and multi-body finalstates can be observed in a collider detector. B-factorysearches in higher-multiplicity final states (e.g. [13]) canbe quite sensitive to these decay modes [6, 7].Much longer lifetimes can arise from (cid:15) − scaling or de-cays through higher-dimension operators — these are themodes of greatest interest for our discussion. In partic-ular, we consider a Higgs-like boson h D below the A (cid:48) mass. The A (cid:48) can decay into h D only in models withmultiple dark-sector Higgses, as the second decay prod-uct must be a pseudoscalar. Several different paramet-ric decay lifetimes are possible. The dark Higgs can de-cay through gauge kinetic mixing, with (cid:15) − scaling, orfaster if it mixes with Standard Model Higgs bosons (e.g.through a quartic term (cid:15) λ | h D | | h | ). If the Dark Higgsis stabilized by an accidental symmetry so that its decayis mediated by higher-dimension operators as noted in[14, 15], the lifetimes can be significantly longer.For suitable parameter choices, all three decay scenar-ios (kinetic mixing, higgs mixing, and higher-dimensionoperators) lead to lifetimes 10 cm (cid:46) cτ (cid:46) cm,so that beam-dump and solar limits apply. The ki-netic mixing decay is a simple and illustrative bench-mark, because the decay lifetime and terrestrial produc-tion cross-sections both depend on (cid:15) and the particlemasses, with no additional free parameters. After dis-cussing the model-independent limits, we will considerthis case explicitly in Section IV. (a) ¯ qq A ! h D a D (b) ¯ qq A !∗ h D A ! FIG. 2: Feynman diagrams for (a) A (cid:48) decay into a dark-sectorhiggs and (b) higgs (cid:48) -strahlung process. II. BEAM-DUMP EXPERIMENTS
Production of dark-sector states in proton beam-dumpexperiments was discussed in [10], which identified threeproduction modes (meson decays, “higgs (cid:48) -strahlung”,and resonant A (cid:48) production). For concreteness, we willassume that a dark sector scalar higgs h D is the LLPof interest but the results in this section are insensitiveto details of the model, except for (cid:15) characterizing theproduction cross-section, the A (cid:48) mass and the mass andlifetime cτ of the LLP. We will consider two reactionsfor producing scalar dark sector particles: The model-independent higgs (cid:48) -strahlung process (see Figure 2(b))where a scalar LLP is directly produced, and the signif-icantly larger resonant process shown in Figure 2(a), inwhich the A (cid:48) can decay into LLP’s. Data from past ex-periments such as the axion search at CHARM [16] havenot been interpreted in terms of these processes anywherein the literature.We note that electron beam-dump experiments like theSLAC experiments E137 [17] and E56 [18, 19] are notcompetitive because of their lower energy and resultingdecreased acceptance [53]. We also note that h D canbe directly produced from pion decay, or through piondecays into A (cid:48) s followed by decays into h D . These mech-anisms are important in proton beam dumps where pionproduction is tremendously large for masses beneath m π .We refer the reader to [10] for further analysis. A. Limits from CHARM Axion Search
The CHARM axion search [16] had a 4 × detectionarea, extending over a 35 m decay region, 2.5 m deepand located 65 m behind a shielded proton target. Aveto counter was placed in front of the detection volume,so that the search was sensitive only to exotic-particledecays within the detector volume. In all, N p = 2 . × protons at an energy of 400 GeV were dumped on acopper target. No candidate e + e − , µ + µ − , or γγ eventswere observed.We estimate event rates in the approximation that theyare dominated by collisions in the first nuclear interaction2ength X nuc of the material: N X ≈ ( σ n ( X ) × A ) N X nuc. A × / mol N p , = σ n ( X ) × . × pb − , (2)where σ n ( X ) is the per-nucleon cross-section for the pro-cess of interest, A = 63 . N is Avogadro’s constant.We have modeled production in MadGraph [20] andused a Monte Carlo to determine geometric and lifetimeacceptances. In the mass range 0 . < m A (cid:48) < A (cid:48) production for a400 GeV incident proton is well approximated by σ ( m A (cid:48) ) ∼
20 pb (cid:16) (cid:15) − (cid:17) (cid:16) m (cid:17) − (3)(for masses above 4 GeV, σ ( m A (cid:48) ) ∼ < σ (4 GeV) × ( m A (cid:48) / ). In addition, the angular acceptancevaries considerably with the A (cid:48) mass. The typical en-ergies of A (cid:48) produced resonantly were ≈ −
50 GeV(with 10–20% exceeding 100 GeV); this is sufficient fora significant fraction of decay products to travel in thedirection of the detector. Taking into account angularacceptance and decay probability, Fig. 3 shows the re-sulting limits on (cid:15) Br ( X → (cid:96) + (cid:96) − ) for (cid:96) = e, µ as afunction of LLP lifetime for various A (cid:48) masses. Theselimits were obtained assuming a decay A (cid:48) → h D a D with m h D = m a D = 0 . m A (cid:48) , but are not very sensitive to thedetailed decay mode away from mass thresholds. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) c Τ (cid:72) cm (cid:76) Ε B r (cid:72) X (cid:174) l (cid:43) l (cid:45) (cid:76) Τ (cid:72) s (cid:76) FIG. 3: Limits on (cid:15) Br ( X → (cid:96) + (cid:96) − ), as a function of decaylength assuming resonant production of a vector A (cid:48) , with sub-sequent decay to X . From bottom to top, curves correspondto ten events expected (none were observed) for m A (cid:48) = 0.6,1, 2, 3, 4 GeV and m h D = 0 . m A (cid:48) . The regions above thecurves are excluded. The gray band represents the expectedband for the scalar model of Section IV, with the upper andlower lines corresponding to ( m A (cid:48) , m h D ) = (0 . , .
24) GeVand (4 , .
6) GeV, respectively. The width of the band comespredominantly from the opening of kaon decay channel athigher masses.
For the higgs (cid:48) -strahlung reaction, the cross-section issmaller by 4–5 orders of magnitude due to the combined effects of a weak coupling, phase space, and decreasingparton luminosity. For our benchmark m h D = 0 . m A (cid:48) and 0 . < m A (cid:48) <
10 GeV, the cross-section can beparametrized as σ ( h D A (cid:48) ) ≈ (0 .
001 pb)( m/ GeV) − (cid:0) m/ (cid:1) − (4)for α D = α EM . Limits on (cid:15) Br ( X → (cid:96) + (cid:96) − ) for thisreaction are shown in Figure 4. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) c Τ (cid:72) cm (cid:76) Ε B r (cid:72) X (cid:174) l (cid:43) l (cid:45) (cid:76) Τ (cid:72) s (cid:76) FIG. 4: Limits on (cid:15) Br ( X → (cid:96) + (cid:96) − ) as a function of decaylength, assuming radiation of X off an off-shell A (cid:48) as in thehiggs (cid:48) -strahlung process of [7]. From bottom to top, curvescorrespond to ten events expected (none were observed) for m A (cid:48) = 0.6, 1, 2, 3, 4 GeV and m h D = 0 . m A (cid:48) . The regionsabove the curves are excluded. The gray band represents theexpected band for the scalar model of Section I A, with the up-per and lower lines corresponding to ( m A (cid:48) , m h D ) = (0 . , . , .
6) GeV respectively. The width of the bandcomes predominantly from the opening of kaon decay channelat higher masses.
III. DARK MATTER ANNIHILATION IN THESUN AND EARTH
We now move on to consider DM capture and annihi-lations in the Sun [21, 22] and the Earth [23, 24], andthe corresponding limits on the production rate and life-times of LLPs. We begin by reviewing the capture pro-cess of DM in celestial objects and the expected anni-hilation rates in relation to model parameters. Directannihilation into SM particles can only be detected bylooking for high energy neutrinos coming from the Sunand such searches have been discussed in the literatureextensively. Annihilations into long-lived neutral parti-cles, however, may have very different signatures whichwe now consider. First, they may again be observed inneutrino detectors if the LLPs decay into muons for ex-ample. Second, they may be observed in electron de-tectors such as FERMI. Third, gamma-ray observations,such as EGRET, Milagro and FERMI can detect the ra-diation associated with electronic decays of the LLPs or3heir direct decay into photons. This wide range of pos-sible probes of LLP decays is particularly welcome sincethe precise nature of these particles, if they exist, is stilllargely unconstrained.
A. Production Rate from Capture andAnnihilation of DM in the Sun and the Earth
We consider several mechanisms for Solar and Earthcapture and use the model described by Eq. (1) as abenchmark. We first describe the well studied case ofelastic scattering of DM against matter and the associ-ated capture rate. We then examine inelastic scatteringand the corresponding modified capture rate. In equilib-rium, the annihilation rate of DM is directly related tothe capture rate. However, the fraction of annihilationinto LLPs is model-dependent and so we parametrize it.The capture rate depends on the scattering cross-section of DM against nuclei. In the models we consider,the scattering is mediated through A (cid:48) exchange and isgiven by [2], σ χn = 16 πZ αα d (cid:15) µ ne A m A (cid:48) (5)where Z and A are the atomic and mass number, respec-tively. Since the elastic scattering of DM against nucleiis constrained to be (cid:46) − cm by direct detection ex-periments [25, 26], the mixing is bounded to be (cid:15) (cid:46) − for m A (cid:48) ∼ GeV. These bounds are dramatically weak-ened if DM predominantly scatters inelastically off nu-clei as in [27]. For example, DM-nucleon cross-sections σ χn ∼ − cm can be compatible with direct detectionlimits and the DAMA modulation signal for splittings ∼
100 keV [28].
Elastic Capture
Elastic capture in celestial objects proceeds throughthe basic mechanism discussed in [29, 30]. The scatter-ing can be spin-independent and spin-dependent. Bothtypes of processes are constrained by direct-detection ex-periments, but the latter much less so.If spin-independent scattering cross-sections saturatebounds from direct-detection experiments [25, 26], thecapture rates are C SI (cid:12) = 1 . × s − (cid:18) TeV m χ (cid:19) / , (6) C SI ⊕ = 4 . × s − (cid:18) TeV m χ (cid:19) / , (7)in the Sun and Earth, respectively [31]. The spin-dependent cross-section for proton coupling is most strin-gently constrained by KIMS [32] and setting the cross- section below the experimental bound we find, C SD (cid:12) = 4 . × s − (cid:18)
100 GeV m χ (cid:19) / . (8) Inelastic Capture
Inelastic DM (iDM) was proposed in [27] as an expla-nation for the discrepancy between the annular modu-lation signal reported by the DAMA collaboration andthe null results reported by other experiments. The ba-sic premise of this scenario is that dark matter scattersdominantly off nuclei into an excited state with splittings, ∼
100 keV, similar to the kinetic energy of DM in thehalo. With this hypothesis, the DAMA data is nicely fitwith WIMP- nucleon cross-sections of σ nχ (cid:38) − cm (see [28] for details). While interactions with matter inthe Earth are kinematically forbidden, the Sun’s grav-ity provides the necessary kinetic energy to overcome theinelastic threshold and allow for capture of DM in theSun.The capture rate in the Sun is shown in Fig. 5 for δ = 125 keV for several values of the WIMP-nucleoncross-section (see [33] for more details). The parameterspace is rather limited because above δ ∼
500 keV thereis no element in the Sun against which DM can scatterinelastically. We note that the high capture rates shownin Fig. 5 as compared with Eq. (6) are mostly just aresult of the higher WIMP-nucleon cross-section allowedin iDM models.
20 50 100 200 500 1000 20000.010.1110100 M Χ (cid:72) GeV (cid:76) C a p t u r e R a t e (cid:72) s ec (cid:45) (cid:76) FIG. 5: The inelastic capture rate of DM in the Sun againstthe DM mass. The black (solid) curve correspond to an in-elastic model with δ = 125 keV and σ nχ = 10 − cm , theblue (dashed) curve to σ nχ = 10 − cm , and the purple(dash-dot) curve to σ nχ = 10 − cm . nnihilation Rate In equilibrium the annihilation rate of DM is exactlyhalf its capture rate. More generally it is given by [34],Γ A = 12 C tanh ( t/τ eq ) , (9)where τ eq is the equilibrium time and t ≈ s is the dy-namical time of the system. In the elastic scattering case,DM can typically scatter in the Sun on a much shortertimescale than it annihilates. Hence, it thermalizes withthe rest of the matter in the Sun and concentrates in theinner core as it approaches its equilibrium configuration.This is not the case for the Earth where t/τ eq (cid:28) σ n (cid:38) − cm ,there are enough elastic collisions to ensure that equi-librium is reached. However, if σ n (cid:28) − cm , thecollision rate becomes so low that the WIMPs do notthermalize in the Sun. In Ref. [33] it was shown thatequilibrium is nevertheless obtained, and we will there-fore equate the annihilation rate with half the capturerate in this case as well. B. Limits from the Sun and the Earth
Bounds from Neutrino Detectors
If the LLP decays into a µ + µ − pair anywhere betweenthe Sun’s surface and the Earth, these muons can beobserved in underground neutrino detectors (Super-K,BAKSAL) as well as the neutrino telescopes (Ice-Cube,Antares) in either of two ways. If the decay happensvery close to the detector ( (cid:46) km) the muon pair maybe observed directly through their Cherenkov radiationas nearby tracks. Alternatively, if the LLP decays intoa µ + µ − pair far from the detector then the muons willyield high-energy neutrinos which can be detected as theyconvert back into a muon in the rock or ice near thedetector.The strongest bounds on the flux of upward go-ing muons come from Super-Kamiokande and are O (10 ) km − yr − [36]. Fig. 6 depicts the bound onthe annihilation rate of DM into LLPs that subsequentlydecay into muons, and the expected improvement in sen-sitivity with Ice-Cube and Antares. Some LLP’s scatteror decay inside the Sun, while others do not decay beforereaching Earth, or decay into modes that do not pro-duce neutrinos. Thus, the fraction of decays that can be detected near the Earth is, f decay = Br( LLP → µ (cid:48) s)e − R (cid:12) /(cid:96) int (cid:0) e − R (cid:12) /γcτ − e − AU/γcτ (cid:1) (10)where γ is the LLP’s boost factor, and (cid:96) int = ( n (cid:12) σ int ) − ≈ n − (cid:12) m A (cid:48) παα D (cid:15) (cid:39) R (cid:12) (cid:16) α D α (cid:17) (cid:18) − (cid:15) (cid:19) (cid:16) m A (cid:48)
100 MeV (cid:17) (11)is the LLP mean free path in the Sun. The suppression e − R (cid:12) /(cid:96) int ≈ (cid:15) and low m A (cid:48) , to which theCHARM analysis is sensitive. When the lifetime becomesmuch smaller than the Sun’s radius, the muons neverescape and no useful limits can be derived.LLPs produced in the Earth must decay near (¡1 km)the detector, or else the muons are stopped in the Earth.In this case, the muons can be observed directly, althoughone must take into account the special nature of suchevents. Since the LLPs are very boosted, their decayproducts (namely the muons) are highly collimated withan opening angle of about ∼ m LLP /m χ . The muons aretherefore no more than about a meter apart when theyreach the detector. We leave it for future work to inves-tigate the efficiency of observing such muon pairs, butthere is no obvious reason to suspect that they should beconsiderably more difficult to see than ordinary events.Hence, the limits from Super-Kamiokande can be used toconstrain the annihilation rate of DM in the Earth’s coreas shown in Fig. 7. Future results from Ice-Cube may im-prove on these limits by about two orders of magnitude.We note that such improvements will allow capture ratesin the Earth (Eq. (7)) to be probed despite the out-of-equilibrium suppression in Eq. (9).
500 1000 1500 200010 (cid:45) (cid:45) M Χ (cid:72) GeV (cid:76) (cid:71) A (cid:72) s (cid:45) (cid:76) (cid:137) f d eca y FIG. 6: The purple curves show the current bounds fromSuper-K on the annihilation rate of DM in the Sun into LLPsassuming 100% branching ratio into 1-step (solid) or 2-step(dashed) cascades which result in muons. The red (orange)curves are the expected bounds from Antares (Ice-Cube). .1 10 100010 (cid:45) Γ Τ (cid:72) sec (cid:76) (cid:71) a nn (cid:72) s (cid:45) (cid:76) (cid:137) f d eca y FIG. 7: The purple curves show the constraints from Super-K on the annihilation rate of LLP in the Earth as a functionof their lifetime for m χ = TeV (solid) and m χ = 0 . Bounds from Electron/Positron Detectors
Injection of highly energetic leptons between the Sunand Earth can also be observed in dedicated satellitemissions that look for electrons/positrons in cosmic rayssuch as PAMELA [37] and FERMI-LAT [38]. The en-ergy spectrum of such electrons is determined by theirproduction mechanism only and does not suffer from theuncertainties present in galactic propagation models. Weconsider 1-step and 2-step processes, where the LLP’sdecay directly into electron/positron pairs or do so viaan additional intermediate state, respectively. To obtainthe energy spectrum for each type of cascade we assumedthe different mass scales involved in the cascade are wellseparated and that all particles decay isotropically. Therelevant formulae can be found in the appendix of [39].Decay of the LLP into a pair of muons is very similar andis in fact well approximated as a 2-step process.For such short cascades, the electron energy spectrumis much harder than the cosmic-ray background. There-fore, the strongest constraints are derived from the high-est energy bins available. It is then straightforward toderive the bounds on the annihilation rate in the Sunby demanding that the resulting flux is lower than thehighest energy bins in FERMI/LAT,Γ A × f decay (cid:46) . × s − (cid:18) m χ (cid:19) (12)Γ A × f decay (cid:46) . × s − (cid:18) m χ (cid:19) (13)for the 1-step and 2-step cascades, respectively. We notethat these are about an order of magnitude stronger thanthe neutrino constraints from Super-K. γ Ray Constraints
Electronic decays of the LLP will usually lead to finalstate radiation (FSR) of photons unless the LLP is rightabove the electron pair threshold. This radiation canbe detected by observations of the γ -ray spectrum of theSun. Both EGRET and FERMI have looked at the Sun’semission spectrum for E γ (cid:38)
100 MeV and these agreewell with the predicted spectrum from inverse Comptonscattering of the Sun’s light against cosmic rays [40, 41].We can therefore derive bounds on the electronic de-cays of LLPs by demanding that the resulting FSR issmaller than the observed gamma ray spectrum. As inthe eletronic flux, the FSR energy spectrum is harderthan that observed and so the highest energy measure-ments ( E γ (cid:38)
100 200 500 1000 2000 5000 1 (cid:180) (cid:45) (cid:45) M Χ (cid:72) GeV (cid:76) (cid:71) A (cid:72) s (cid:45) (cid:76) (cid:137) f d eca y FIG. 8: FSR constraints on Γ ann × f decay from Milagro(green), EGRET (purple) and Fermi solar data (blue) for one-step (solid) and two-step (dashed) cascade decays, with decayefficiency f decay defined in (10). The bounds were derived asexplained in the text. IV. EXAMPLE: DARK-SECTOR SCALARS
In this Section we discuss the implications of the previ-ously identified limits for a particular class of long-livedparticles: Higgs-like dark scalars h D that decay throughloop diagrams with a rate controlled by the gauge ki-netic mixing (cid:15) . We then discuss the dependence of thesecontours on variation of model assumptions.In a single-Higgs model the width of the light scalar6 D to each lepton species is,Γ h D → (cid:96) + (cid:96) − = 2 α D α (cid:15) π I ( m (cid:96) , m A (cid:48) , m h D ) m (cid:96) m V m h , (14)where I is a loop integral defined in [7], and very nearlyequal to 2 β , where β = (1 − m (cid:96) /m h D ) / . This formulamust be corrected by mixing angles in a model with twoor more Higgses in the dark sector, but given our com-plete ignorance of the dark sector’s structure we ignorethese and take (14) as a benchmark. For decays abovethe muon threshold, this gives cτ ≈ (2 × cm) × Br ( h D → µ + µ − ) αα D × (cid:16) − (cid:15) (cid:17) (cid:0) m A (cid:48) (cid:1) (cid:16) m hD (cid:17) . (15)We take the ratio of hadronic to muonic widths from aspectator model, as in [43] (we use the parameter choicesof [44] with r = 1) – the µ + µ − branching fraction variesbetween 20 and 50% below the muon threshold, and fallsto 1–2% above the kaon threshold.The terrestrial direct production limits and those fromDM annihilation in the Sun (assuming annihilation prod-ucts include the A (cid:48) and/or h D ) are summarized in Figure9, where for definiteness we have assumed m h D = m a D =0 . m A (cid:48) , and α D = α .The most constraining terrestrial experiment is theCHARM beam dump, which searched for e + e − and µ + µ − events and found no candidates in 2 . × pro-tons dumped. These results imply that (cid:15) (cid:38) − are ex-cluded for gauge boson masses up to a few GeV. Larger (cid:15) ∼ . − g − m =100 GeV–1 TeV) an-nihilates through the A (cid:48) , smaller (cid:15) can be probed bymeasurements of the solar electron, photon, and neu-trino flux. For m h D < m τ , the photon flux from finalstate radiation of dark Higgses that escape the Sun ismost constraining, with sensitivity to annihilation rates ∼ − / s, four to five orders of magnitude belowthe expected capture rate for σ χn = 10 − cm . Weassume the capture rate is determined by the scatteringcross-section (5) from exchange of an A (cid:48) , with no mixing-angle suppression. In this case, the solar production rate,like the terrestrial beam-dump production rate, scales as (cid:15) , leading to a very similar shape of exclusion region(blue region for m DM = 100 GeV, dark blue dashed linefor m DM = 1 TeV).Unlike muons, pions, and kaons, the τ decayspromptly, producing neutrinos that can escape the Suneven if the dark-sector Higgs decays promptly. Thusdark matter scenarios above this mass threshold are con-strained (green vertical band in Fig. 9), with little sen-sitivity to dark matter mass.As we have noted, the decay lifetimes of Higgs-likescalars are quite model-dependent, and can be signifi-cantly longer or shorter than those quoted. This wouldqualitatively affect both beam-dump (red) and solar FSR (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) h D (cid:76) (cid:72) GeV (cid:76) Ε CHARM
Τ(cid:174)Ν (cid:72) g (cid:45) (cid:76) Μ solar Γ 's (cid:72) filled: m Χ (cid:61)
100 GeV,dashed: m Χ (cid:61) (cid:76) IIIIII
FIG. 9: Combined limits on a higgs-like scalar h D producedin A (cid:48) decays, with leptonic width (14), m A (cid:48) = 2 . m h D , and α D = α . The gray region in the upper left is the model-independent limit on (cid:15) for given A (cid:48) mass from ( g − µ (see[45]). The red region corresponds to the limit from direct A (cid:48) production in CHARM, with h D decays into electrons ormuons (see Sec II). Where this limit is obscured by others,the outline of the CHARM exclusion is drawn in dark red.The blue and green regions are solar limits (Sec. III B) rele-vant if weak-scale dark matter couples to the A (cid:48) , and kineticmixing mediates its scattering off ordinary matter. The blueand green regions are excluded by photon and neutrino fluxobservations, respectively, for a benchmark dark matter mass m χ = 100 GeV. The dashed blue line outlines the analogousexclusion from photon flux when m χ = 1 TeV. The neutrinolimits are controlled by the τ threshold and are insensitiveto the dark matter mass. Possible strategies for exploringregions I, II, and III are discussed in Section V. (blue) constraints. The upper limits on (cid:15) correspond totypical γcτ comparable to the thickness of the beam-dump shielding (or R (cid:12) ), with only logarithmic sensi-tivity to production rate. These shift up (down) forlonger (shorter) lifetimes. The lower limits are deter-mined equally by the lifetime (which typically exceedsthe size of the experiment) and production rate, so it issomewhat less affected by variation of the decay length.The solar constraints also depend on various assump-tions about dark matter. The dependence on dark mat-ter mass, shown in the Fig. 9 (dashed blue line), arisesmostly from the resulting boost of the late-decaying Hig-gses. In models with a reduced DM-matter scatteringcross-section, the luminosity of the Sun in dark matterannihilation is reduced and the resulting FSR limits areweakened. We note that a mixing angle weakens thelower- (cid:15) limit significantly, but has only logarithmic im-pact on the high- (cid:15) sensitivity limit.7 . DISCUSSION Dark sectors coupled to the Standard Model throughgauge kinetic mixing may contain an array of macroscop-ically long-lived particles (LLP’s) in addition to the ki-netically mixed gauge boson. Gauge bosons can escapedetection in accelerator experiments if they decay intoLLP’s with lifetimes between 10 cm and 10 cm, ratherthan to Standard Model fermions. LLPs with lifetimesin this range would escape collider detectors unobserved,but are sufficiently short-lived that BBN constraints arelargely irrelevant.Remarkably, two classes of data constrain these sce-narios across these fourteen decades in lifetime. The pro-ton beam dump experiment CHARM places strong con-straints on scalar LLP’s with 10 (cid:46) cτ (cid:46) cm decayinginto muons or electrons. If dark matter annihilates intodark sector LLP’s, then Solar capture of dark matter re-sults in limits on LLP’s with 10 cm (cid:46) cτ (cid:46) cm forsimilar decay modes.These limits each depend on different assumptions andproduction mechanisms, so they cannot be combined intoa robust exclusion. Nonetheless, the case of a Higgs-like boson in the dark sector decaying through kineticmixing allows an illustrative comparison. The sensitiv-ity of solar limits — when applicable — nearly matcheson to the parameter region excluded by CHARM. To-gether, these constraints disfavor A (cid:48) decays to scalars,pseudoscalars and other long-lived particles in the param-eter region best suited to explain the DAMA modulationsignal through iDM [14, 15, 46]. In particular, if LLP’sdecay through higher-dimension operators, lowering themass scale of these operators to evade BBN constraintsmay bring them into conflict with experimental data.Given these limits and the remaining unconstrainedparameter regions, our analysis suggests several promis-ing searches with existing data and small-scale experi-ments: Prompt Vector Decays to (cid:96) + (cid:96) − : Remarkably, for (cid:15) ∼ − − − , A (cid:48) decays into LLPs are much more constrained than decays directlyinto Standard Model states. This observationcompounds the motivation for a sharply definedsearch program to constrain direct A (cid:48) decays to (cid:96) + (cid:96) − [8, 9]. Searches in e + e − Collider Data:
LLP lifetimes tooshort to be observed in a beam-dump experiment( γcτ (cid:46)
10m at E LLP ∼
50 GeV, e.g. region Iof Fig. 9) give rise to observable displaced decayswithin the detectors of e + e − collider experimentssuch as Babar, Belle, KLOE, and BES-III. More-over, such lifetimes arise from (cid:15) (cid:38) − for whichmany thousands of events are possible in γ + A (cid:48) and h D A (cid:48) production. The most visible such decaymay be into µ + µ − (or π + π − ) pairs reconstructinga displaced vertex, which may be accompanied bya photon, invisible particles, and/or additional e , µ , or π pairs. High-Energy and High Intensity Beam Dumps:
High-energy beam dumps with cumulative chargeson target exceeding a coulomb may extend sensitiv-ity into region II of Fig. 9. Neutrino factories withnear detectors such as MINOS have geometrieswell-suited for beam-dump LLP searches [10, 47],and integrated luminosity ∼ larger than thatof CHARM. Solar Gamma Ray and Electron Measurements:
LLP lifetimes ∼ − seconds (region III inFig. 9) are too short to disrupt light elementabundances, but only a fraction of LLP’s candecay between the Sun and the Earth, so verysensitive solar measurements are needed to probethese scenarios. Solar gamma-ray observationsabove 10 GeV from ACTs and FERMI can explorethis region, as can searches for electronic signalscorrelated with the Sun [48]. Milli-Charged Particle Searches:
Electron beam-dump experiments searching for milli-chargedparticles (e.g. [49, 50]) may be sensitive to LLPscattering off a nuclear target N in the activedetector volume, such as h D + N → A (cid:48) + N .Searches in existing data may be sensitive to (cid:15) (cid:38) − (100 MeV /m A (cid:48) ) and m LLP (cid:46)
100 MeV.
Note added: While this work was brought to completiontwo related works [51], and [52] appeared.
Acknowledgments
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