Light Scalars and Dark Photons in Borexino and LSND Experiments
LLight Scalars and Dark Photonsin Borexino and LSND Experiments
Maxim Pospelov
1, 2, ∗ and Yu-Dai Tsai † Department of Physics and Astronomy,University of Victoria, Victoria, BC V8P 5C2, Canada Perimeter Institute for Theoretical Physics, Waterloo, ON N2J 2W9, Canada Fermilab, P.O. Box 500, Batavia, IL 60510, USAFermi National Accelerator Laboratory, Batavia, IL 60510, USA
Bringing an external radioactive source close to a large underground detectorcan significantly advance sensitivity not only to sterile neutrinos but also to “dark”gauge bosons and scalars. Here we address in detail the sensitivity reach of theBorexino-SOX configuration, which will see a powerful (a few PBq) Ce − Prsource installed next to the Borexino detector, to light scalar particles coupled to theSM fermions. The mass reach of this configuration is limited by the energy release inthe radioactive γ -cascade, which in this particular case is 2.2 MeV. Within that reachone year of operations will achieve an unprecedented sensitivity to coupling constantsof such scalars, reaching down to g ∼ − levels and probing significant parts ofparameter space not excluded by either beam dump constraints or astrophysicalbounds. Should the current proton charge radius discrepancy be caused by theexchange of a MeV-mass scalar, then the simplest models will be decisively probedin this setup. We also update the beam dump constraints on light scalars and vectors,and in particular rule out dark photons with masses below 1 MeV, and kinetic mixingcouplings (cid:15) (cid:38) − . ∗ [email protected] † [email protected]; [email protected] a r X i v : . [ h e p - ph ] S e p I. INTRODUCTION
Search for light weakly coupled states undergoes a revival in recent years [1]. Therehas been increased interest in models that operate with light sterile neutrinos, axion-likeparticles, dark photons, and dark scalars that can be searched for in a variety of particlephysics experiments. For a representative but incomplete set of theoretical ideas see, e.g. [2–10]. With more emphasis placed on the intensity frontier in recent years, experimentalsearches of exotic light particles are poised to continue [11].Some of this interest is cosmology-driven, exploiting possible connection of light particlesto dark matter, or perhaps to a force that mediates interactions between Standard Modeland dark matter particles [4, 6, 9]. In many cases, the interest in light new states is motivatedby “anomalous” results from previous experiments. The representative anomalies in thatrespect are the discrepancy in muon g − Ce − Pr source in close proximity to the detector. The source produces a large numberof electron antineutrinos, and their signals inside the Borexino as a function of the distancefrom the source can reveal or constrain sterile neutrinos with commensurate oscillationlength. In addition, it has already been pointed out that the same configuration will besensitive to the emission of light scalar (or vector) particles in the transitions between thenuclear levels in the final point of the β -decay chain [25].This note revisits the question of sensitivity of Borexino-SOX to light particles, includingdark scalars and newly considered below-MeV dark photons, and updates several aspects of[25]. We significantly expand the sensitivity reach by taking into account the decays of lightparticles inside the Borexino detector. Only scalar scattering on electrons was taken intoaccount in the previous consideration. In addition, we update the current leading boundson dark scalars and dark photons by considering the LSND measurements of the elasticelectron-neutrino cross section [13, 28]. Dark photons with masses below 1 MeV can beruled out with kinetic mixing coupling (cid:15) (cid:38) − To have a more specific target in terms of the light particles, in section II, we introducea light scalar coupled to leptons and protons, which might be responsible for the resolutionof the r p discrepancy [29]. In section III, we calculate the production rate of the scalarsby relating it to the corresponding nuclear transition rate of Nd. Taking into accountthe decay and the Compton absorption of the scalars inside the detector we arrive at theexpected counting rate, and derive the sensitivity to coupling constants within the massreach of this setup. Existing constraints on such light scalars are considered in section IV.In section V, we study the sensitivity reach of the Borexino-SOX setup in probing lightdark photons between a few hundred keV to 1 MeV with a small kinetic mixing (in fullawareness of the fact that such light dark photons are disfavored by cosmology). We reachthe conclusions in section VI.
II. SIMPLIFIED MODEL OF A LIGHT SCALAR AND THE PROTON SIZEANOMALY
Following the rebirth of interest in dark photons, other models of light bosons have beenclosely investigated. In particular, scalar particles are quite interesting, not least becausethey are expected to couple differently to particles with different masses. While it is difficultto create a simple and elegant model of dark scalars with MeV range masses, some attemptshave been made in refs. [30, 31]. We will consider a simplified Lagrangian at low energy inthe following form, L φ = 12 ( ∂ µ φ ) − m φ φ + ( g p ¯ pp + g n ¯ nn + g e ¯ ee + g µ ¯ µµ + g τ ¯ τ τ ) φ. (1)In principle, such Lagrangian can be UV-completed in a variety of ways, although it isdifficult to maintain both sizable couplings and small scalar mass m φ . In this study, wewill not analyze constraints related to UV completion, concentrating instead only on thelow-energy physics induced by (1). This simplified Lagrangian with MeV/sub-MeV scalarswas proposed in Ref. [29] (see also [32–35]) to explain a 7 σ disagreement between themeasurements of the proton-charge radius using e − p systems and the more precise muonicHydrogen Lamb shift determination of r p .More recent data with the Lamb shift in muonic deuterium [36] show no additionalsignificant deviations associated with the neutron, so that the new physics interpretation ofthe anomaly prefers g n /g p (cid:28)
1. Therefore, we will limit our considerations to g n = 0 case,which will also remove all constraints associated with neutron-nucleus scattering [37]. Ofcourse, the real origin of the r p discrepancy is a hotly debated subject, and new physics isperhaps a solution of “last resort”.Introducing the product of couplings, (cid:15) ≡ g e g p /e , one can easily calculate correctionsto the energy levels of muonic atoms due to the scalar exchange. When interpreted as aneffective correction to the extracted proton radius from the hydrogen and muonic hydrogen,this scalar exchange gives∆ r p | e H = − (cid:15) m φ , ∆ r p | µ H = − (cid:15) ( g µ /g e ) m φ f ( am φ ) (2) f = x (1 + x ) − and a ≡ ( αm µ m p ) − ( m µ + m p ) is the µ H Bohr radius. For the modificationsof the deuterium energy levels, one should make m p → m D substitution.The observed difference [16] is∆ r p | eH − ∆ r p | µ H = − . ± .
009 fm , (3)and can be ascribed to new physics, provided that it breaks lepton universality. In particular,it may originate from the g µ (cid:29) g e hierarchy, which would be expected from a scalar model.For simplicity, we will assume the mass-proportional coupling constants to the leptons andproton, thus g e = ( m e /m µ ) g µ , g τ = ( m τ /m µ ) g µ , g p = ( m p /m µ ) g µ , and plot the preferredparameter curve in Fig. 1 in green color on the (cid:15) − m φ plane.The best part of the new physics hypothesis is that it is ultimately testable with otherexperimental tools, of which there are many. The most direct way of discovering or limitingsuch particles is their productions in subatomic experiments with subsequent detection ofnew particle scattering or decay. The MeV-range masses suggested by the r p anomaly makenuclear physics tools preferable. Such light scalars can be produced in nuclear transitions,and in the next section, we calculate their production in the gamma decay of selected isotopesthat are going to be used in the search for sterile neutrinos. III. BOREXINO-SOX EXPERIMENT AS A PROBE OF SCALAR SECTOR
Here we consider the Borexino-SOX setup in which a radioactive Ce − Pr source willbe placed 8.25 meters away from the center of the Borexino detector.The decay of
Ce goes through Ce → βν − + Pr and then Pr → βν − + Nd( Nd ∗ ). A fraction of the decays results in the excited states of Nd ∗ that γ -decay tothe ground state. Then a small fraction of such decays will occur via an emission of a lightscalar, Nd ∗ → Nd + φ. (4)Small couplings of φ make it transparent to shielding and long-lived relative to the linearscale of the experiment. Nevertheless, very rare events caused by the scalar can still bedetected by the Borexino detector. The main processes via which such scalar can depositits energy are: eφ → eγ, Compton absorption φ → γγ , diphoton decay φ → e + e − , electron − positron decay (5)In what follows we put together an expected strength of such signal, starting from theprobability of the scalar emission. A. Emission of scalars in nuclear transitions
Let us find the probability of scalar particle emission in radioactive decays as a function ofits mass and coupling. About one percent of the Ce β -decays to the 2.185 MeV metastablestate of Nd. This excited state, Nd ∗ , then transitions to lower energy states via 1.485 MeVand 2.185 MeV gammas with approximately 30% and 70% branching ratios [22]. Ab initio calculation of a nuclear decay with an exotic particle in the final state could bea nontrivial task. Here, we benefit from the fact that the transition of interest ( Nd ∗ → Nd) are E1 and the scalar coupling to neutrons is zero, which allows us to link the emissionof the scalar to that of the γ -quanta and thus bypass complicated nuclear physics.In the multipole expansion, the relevant part of the interaction Hamiltonian with photonsis almost the same form as the corresponding counterpart of the scalar interaction, H int ,γ (cid:39) eωA (cid:88) p ( (cid:126)(cid:15) (cid:126)r p ); H int ,φ (cid:39) g p (cid:113) ω − m φ φ (cid:88) p ( (cid:126)n (cid:126)r p ) , (6)where A , φ are the amplitudes of the outgoing photon and scalar waves, (cid:126)(cid:15) and (cid:126)n are theunit vectors of photon polarization and the direction of the outgoing waves, and the sumis taken over the protons inside the nuclei. After squaring the amplitudes induced by theseHamiltonians, summing over polarizations and averaging over (cid:126)n , we arrive at both ratesbeing proportional to the the same square of the nuclear matrix element, (cid:104) (cid:80) p (cid:126)r p (cid:105) . In theratio of transition rates it cancels, leaving us with the desired relationΓ φ Γ γ, E = 12 (cid:16) g p e (cid:17) (cid:18) − m φ ω (cid:19) / . (7)All factors in this rate are very intuitive: besides the obvious ratio of couplings, the 1/2 factorreflects the ratio of independent polarizations for a photon and a scalar, while (1 − m φ /ω ) / takes into account the finite mass effect. B. Scalar decay and absorption
The Compton absorption e + φ → e + γ process leads to the energy deposition inside theBorexino detector. Since only the sum of the deposited energy is measured, we would need atotal cross section for this process. The differential cross section we derive is the same as Eq.(5) of [25] in the m φ (cid:28) E φ limit. But in this paper we do not take the limit and use the fullcross-section σ ( e + φ → e + γ ). The absorption length is then given by L abs = 1 / ( n e σ eφ → eγ ),where n e is the number density of electrons inside the Borexino detector. It is easy to seethat for the fiducial choice of parameters, the absorption length is much larger than thelinear size of the detector. The Compton absorption process dominates in the very low m φ regime, but the diphoton decays dominate in the medium and high mass range between afew hundred keV to 1.022 MeV (below pair production regime) as discussed below.The diphoton decay rate of the light scalar φ can be derived recasting the Higgs result[38], Γ( φ → γγ ) = α π m φ (cid:32) (cid:88) l = e, µ, τ g φll m l x l (cid:20) x l + ( x l −
1) arcsin ( √ x l ) (cid:21) θ (1 − x l )) (cid:33) . (8)where x l = m φ m l and θ ( x ) is the Heaviside step function. In principle, all charged particleswith couplings to φ will contribute to the rate. Here we take into account only the chargedleptons, while the inclusion of quarks would require additional information, beyond assuminga g p value. Therefore, this is an underestimation, with an actual Γ( φ → γγ ) being on thesame order but larger than (8). (One would need a proper UV-complete theory to make amore accurate prediction for the φ → γγ rate.)When the mass of the scalar m φ is larger than 2 m e , the electron-positron decay willdominate the diphoton and Compton absorption processes. We haveΓ( φ → e + e − ) = g e m φ π (cid:32) − m e m φ (cid:33) / . (9)The sum of these two rates determines the decay length, L dec = βγ (Γ( φ → e + e − ) + Γ( φ → γγ )) − , (10)where β is the velocity of the scalar, which depends on its mass and energy, β = (cid:113) − m φ /E ( c = 1 in our notations). The combination of absorption and decay, L dec , abs = ( L − + L − ) − , (11)is required for the total event rate.The decay/absorption length together with the geometric acceptance determines theprobability of energy deposition inside the detector per each emitted scalar particle, P deposit = (cid:90) d ( θ ) L dec , abs π π d cos θ = 1 L dec , abs (cid:90) √ − ( R/L ) (cid:112) R − L (1 − cos θ ) d cos θ = 1 L dec , abs × LR + ( L − R ) log (cid:0) LL + R − (cid:1) L , (12)where a spherical geometry of the detector is considered. Here R is the fiducial radius and L is the distance of the radiative source from the center of the detector. For our numericalresults we use R = 3 .
02 m and L = 8 .
25 m as proposed in the SOX project [39]. In the L (cid:29) R limit, the probability has a simple scaling with the total volume and the effectiveflux at the position of the detector, P deposit (cid:39) L dec , abs 43 πR πL , (13)but we use the complete expression (12) for the calculations below. C. Total event rate and sensitivity reach
Using formulae from the previous subsections, we can predict the signal strength as afunction of m φ and coupling constants. The excited state of Nd has two gamma tran-sitions, E = 2 .
185 MeV and E = 1 .
485 MeV, partitioned with Br = 0 . = 0 . φ of energy E i (2.185 MeV or 1.485 MeV as i = 0 or 1) in the Borexino detector is given by˙ N i = (cid:18) dNdt (cid:19) exp (cid:18) − tτ (cid:19) × Br Nd ∗ × Br i × (cid:16) g p e (cid:17) (cid:32) − (cid:18) m φ E i (cid:19) (cid:33) / × P deposit , i . (14)Here, (cid:0) dNdt (cid:1) is the initial source radioactivity in units of decays per time, and the projectedstrength is (cid:39) × decays per second. τ is the lifetime of Cr, τ = 285 days.Br Nd ∗ is the probability that the β -decay chain leads to the 2.185 MeV excited state of Nd, Br Nd ∗ (cid:39) .
01. Finally, P deposit , i is the probability of decay/absorption defined inthe previous subsection that depends on i via the dependence of the decay length and theabsorption rate on E i . Substituting relevant numbers we get the counting rate for the 2.185MeV energy as˙ N .
185 MeV (cid:20) countsday (cid:21) = 1 . × × exp (cid:18) − t [day]285 d (cid:19) × (cid:0) dNdt (cid:1) × (cid:16) g p e (cid:17) (cid:18) − (cid:16) m φ . (cid:17) (cid:19) / × P deposit , .
185 MeV (15)The resulting sensitivity reach of the three processes considered is plotted in the left panelof Fig. 1 as a blue curve. Here we assume the mass-proportional coupling strengths for φ to proton and leptons, and parametrize the coupling as (cid:15) = g p g e /e . The curve correspondsto a 3 σ sensitivity level with the assumption that the initial source strength is 5 PBq.For the derivation of the future sensitivity reach, we have followed the simplified proce-dure: For every point on the parameter space { m φ , (cid:15) } , we calculate the expected countingrate using Eq. (14). We then take an overall exposure of t exp = 365 days to arrive at anexpected number of signal events as a function of mass and coupling, N sig ( m φ , (cid:15) ). The back-ground is the total number of events in energy bins near E = 2 .
185 MeV and E = 1 .
485 MeV.The energy resolution at Borexino is 5% × (cid:112) /E. We use this as the bin size when weestimate the background rates at E = 2 .
185 MeV and E = 1 .
485 MeV. For the backgroundevent rate, we use the energy spectra shown in Fig. 2 in [39]. After all cuts, the backgroundrate is R backgr (cid:39)
200 counts/100t × E = 2 .
185 MeV (For E = 1 .
485 MeV, the background rate is around 2300 counts/100t × E = 2 . N backgr (cid:39)
90. We then require N sig < (cid:112) N backgr that results in the sensitivity curve in Fig. 1. Based on our estimation,the inclusion of E = 1 .
485 MeV channel does not lead to a significant improvement: itallows one to increase the significance by roughly 0 . σ with respect to just considering themain 2.185 MeV channel. Should a strong signal be observed, however, the presence of twopeaks would be an unmistakable signature.In the above procedure, we have taken into account only the existing source-unrelatedbackgrounds. However, a question arises whether additional inverse beta decay (IBD) eventsin Borexino, p + ¯ ν → n + e + , which is the primary goal of the SOX project, may also affectthe search for E = 2 .
185 MeV abnormal energy deposition. If the location of IBD eventis inside the fiducial volume, then even the threshold IBD event creates 3.2 MeV energydeposition. (The positron at rest produces 1.0 MeV energy, and the neutron capture resultsin the additional 2.2 MeV). This is well outside the energy windows for the signal fromexotic scalars. Moreover, IBD events have a double structure in time, which can be usedto discriminate them. An interesting question arises whether the location of IBD eventsoutside the fiducial volume ( i.e. close to the edge of the detector) may lead to a loss ofpositron signal followed by the neutron capture inside the fiducial volume. For a neutronwith a typical kinetic energy of a keV would have to diffuse for at least 1m inside liquidscintillator to reach the fiducial volume. However, the estimates of Ref. [40] show that thetypical diffusion length is O (5 cm), which render the probability for such events to be small.Still, background events could occur when the neutron-proton capture takes place in thenon-scintillating buffer region at a radius R > .
25 m, if the 2.2 MeV capture gamma ray(with attenuation length ∼
90 cm) reach the fiducial volume at
R < R = 2 .
00 m) smaller than the R = 3 .
02 m used in the Borexinoanalysis [39]. We plot both the sensitivity reaches based on 2.00 m and 3.02 m fiducial radiiin Figure 1 and 2. One can regard the sensitivity reach with 2.00 m fiducial radius a moreconservative estimation. Furthermore, this gamma-ray background would have to appear ina radial dependent fashion in the detector, meaning that the background is stronger in theregime nearer to the buffer area. Such information on radius dependence can be applied tofurther subtract the background events. We leave the simulation to accurately determinethis background to future works.To be more inclusive, we also consider a variant of the scalar model when the couplingsto electrons and tauons are switched off (muonic scalar). In this case, the remaining energydepositing channel is the diphoton decay, and there is no gain in sensitivity for m φ > m e .We plot the corresponding sensitivity reach in the right panel of Fig. 1 also as a blue curve. IV. COMPARING TO EXISTING CONSTRAINTS
Here we reassess some limits on the couplings of very light scalars. The most significantones are from the beam dump experiments, meson decays and stellar energy losses. Theparticle physics constraints that rely on flavor changing processes are difficult to assess, asthey would necessarily involve couplings of φ to the heavy quarks. We leave them out asmodel-dependent constraints. A. Beam dump constraints
Among the beam dump experiments, the LSND is the leader given the number of particlesit has put on target. The LSND measurements of the elastic electron-neutrino cross section[13, 28] can be recast to put current-leading constraints on the parameter spaces of ourmodel, as well as models including light dark matter and millicharged particles [26, 41], andmodels with neutrino-heavy neutral lepton-photon dipole interactions [42]. Here we reviseprevious bounds discussing different production channels, and account for scalar decays andCompton absorptions inside the LSND volume.The collisions of primary protons with a target at LSND energies produce mostly pionsand electromagnetic radiations. Exotic particles, such as scalars φ can be produced in theprimary proton-nucleus collisions, as well as in the subsequent decays and absorptions ofpions. A detailed calculation of such processes would require a dedicated effort. It wouldalso require more knowledge about an actual model, beyond the naive Lagrangian (1). Inparticular, one would need to know how the scalars couple to pions and ∆-resonances, thatalongside nucleons are the most important players in the inelastic processes in the LSNDexperiment energy range. Here we resort to simple order-of-magnitude estimates, assumingthat the g p coupling is the largest, and drives the production of scalars φ .The important process for the pion production at LSND is the excitation of ∆ resonancein the collisions of incoming protons with nucleons inside the target. Assuming that thedecay of ∆’s saturates the pion production inside the target, we can estimate the associatedproduction of scalars in the ∆ → p + π + φ process. To that effect, we consider the followingtwo interaction terms, L int ∼ g p φ ¯ pp + g π ∆ p (∆ µ p ) ∂ µ π, (16)where ∆ µ is the Rarita-Schwinger spinor of ∆-resonance, g π ∆ p is the pion-delta-nucleoncoupling constant, and the isospin structure is suppressed. To estimate scalar production,we calculate the rates for ∆ → p + π , ∆ → p + π + φ and take the ratio finding N φ ∼ N π × Γ ∆ → pπφ Γ ∆ → pπ (cid:39) N π × . g p . (17)Notice that the decay rates are relatively large, being enhanced by the log( Q/m φ ), where Q is the energy release. The coefficient 0.04 is calculated for m φ = 1 MeV, and it varies from0.06 for m φ = 0 . m φ = 2 MeV.Depending on their charges, pions have very different histories inside the target. Thenegatively charged π − undergoes nuclear capture. In [25] the rate of the scalar productionin nuclear capture was overestimated, as it was linked to the production of photons in thecapture of π − by free protons via e → g p substitution. The radiative capture rate onprotons is about 40%. For the LSND target, however, the more relevant process is theradiative capture on nuclei with A ≥
16, which is in the range of ∼
2% [43]. Therefore,one may use N φ ( π − ) ∼ . × N π − × (cid:0) g π e (cid:1) as an estimate for the production rate of scalarsfrom the π − capture. Notice this is the coupling of scalars to pions that mostly determinesthe capture rate. Moreover, the number of π − is smaller than the total pion production,and therefore we expect the production of φ in the π − capture to be subdominant to ∆decays (17). Unlike the case with negatively charged pions, most of π + stop in the targetand decay. The scalar particle is then produced in the three-body decay, π + → µ + νφ , andin the four-body decay of the stopped µ + , µ + → e + ννφ . The decays of π are instantaneous,and they could also lead to the production of light scalars in π → γγφ . Direct estimatesof the corresponding branching ratios give ∼ . g µ ( π ) ) , and again we find that this issubdominant to (17) estimate because of g µ ( π ) < g p .A conservative estimate of the number of pions produced in the experiment is N π ∼ (see, e.g. , [28]). We take 300 MeV as an estimate for the average energy of scalars. Nowwe can estimate the expected number of events N LSND , i.e. the number of light scalars thatdeposit their energies in the LSND detector: N LSND ∼ N π × . g p × P survive + deposit in LSND (cid:39) N π × . g p × (cid:34) exp (cid:32) − L LSND − d LSND L dec (cid:33) − exp (cid:32) − L LSND + d LSND L dec , abs (cid:33)(cid:35) (cid:18) A LSND πL (cid:19) . (18)Here we conservatively assume spatially isotropic distribution, take L LSND = 30 m as thedistance between the target and the center of the detector, d LSND = 8 . S o x . + . M e V S o l a r P r odu c t i on r p favored L S ND S t e ll a r C oo li ng - - - - - m ϕ ( MeV ) ϵ S o l a r P r odu c t i on L S ND S ox . + . M e V r p f avo r ed S t e ll a r E ne r g y Lo ss - - - - - - - m ϕ ( MeV ) ϵ Figure 1. Future sensitivity reach of the Borexino-SOX setup and existing constraints placedon the coupling constant-mass parameter space. We conduct the analysis in two fiducial radii,2.00 m and 3.02 m, for the Borexino-SOX sensitivity reaches, in regard of the background fromthe 2.2 MeV n-p capture gamma ray discussed in section III C.
Left panel:
The g i ∝ m i scalingis assumed and (cid:15) is defined as (cid:15) = g p g e /e . Right panel: g e = g τ = 0, a g i ∝ m i scaling for µ and p , while (cid:15) = ( m e /m µ ) × g p g µ /e . The green curve is the parameter space that can explainthe proton-size anomaly. The experimental reach ( > σ ) by the Borexino-SOX setup is the blueregime. The recast of LSND constraints [28] is shown in purple, while the gray area is constrainedby the stellar energy loss [45]. The solar production constraint [46] is the protruding pink areabetween (cid:15) = 10 − and 10 − . detector itself, and A LSND (cid:39)
25 m is the cross-section of the detector looking from the side[28, 44]. L dec , abs are the decaying and absorption length determined by the physical processesEq. (5). Notice that we no longer use the assumption that L dec , abs (cid:29) L LSND and d LSND since in the high (cid:15) regime these three lengths could be comparable. The number densityof electrons in the LSND detector is n e = 2 . × m − , and the absorption again plays asubdominant role in the energy deposition process.Based on Fig. 10 of [28] and Fig. 28 of [13] we estimate that there are less than 20decay-in-flight events above 140 MeV during the exposure. We then determine the LSNDconstraint on the parameter space of the φ scalar as plotted in Fig. 1 in purple color. Wereiterate a rather approximate nature of the estimates. B. Solar emission and stellar energy loss
Thermal production of scalars may lead to abnormal energy losses (or abnormal thermalconductivity) that would alter the time evolution of well known stellar populations. In theregime of m φ > T , the thermally averaged energy loss is proportional to g e exp( − m φ /T star ) . Given the extreme strength of stellar constraints [45], one can safely exclude m φ <
250 keVfor the whole range of coupling constants considered in this paper.In addition, the non-thermal emission of scalars in nuclear reaction rates in the Sun canalso be constrained. The light scalar φ can be produced in the Sun through the nuclearinteraction p +D → He + φ. This process generates a 5.5 MeV φ flux that was constrained1by the search conducted by the Borexino experiment. The flux can be estimated asΦ φ, solar (cid:39) ( g p /e ) Φ ppν P esc P surv . (19)Here Φ ppν = 6 . × cm − s − is the proton-proton neutrino flux. P esc is the probabilityof the light scalar escaping the Sun while P surv is the probability of the scalar particle notdecay before it reaches the Borexino detector. P esc = exp (cid:18) − (cid:90) R (cid:12) dr n (cid:12) σ eφ → eγ (cid:19) (20) P surv = exp (cid:18) − L (cid:12) L dec (cid:19) (21)where R (cid:12) and L (cid:12) are the radius of the Sun and Earth-Sun distance respectively, while n (cid:12) is the mean-solar electron density. L dec is again determined by the decay processes in Eq. 5. For m φ < m e the φ particlecan survive and reach the Borexino detector when (cid:15) < − , and deposit its energy throughprocesses in Eq. (5). For m φ > m e the P surv is highly suppressed due to rapid di-electrondecays and thus m φ = 2 m e is where the constraint ends.Notice that it is difficult to impose the supernovae (SN) constraints on this model, becauseof the uncertainties in the choices of some couplings. In general, we believe that the couplingof scalars to nucleons can be large enough so that they remain trapped in the explosion zone,therefore avoiding the SN constraint. V. SENSITIVITY TO DARK PHOTONS BELOW 1 MEV
Dark photon is a massive “copy” of the regular SM photon, which couples to the elec-tromagnetic current with a strength proportional to a small mixing angle (cid:15) , realized as akinetic mixing operator. The low-energy Lagrangian for dark photons can be written as L d . ph . = − F (cid:48) µν F (cid:48) µν + 12 m A (cid:48) ( A (cid:48) µ ) + (cid:15)A (cid:48) µ J EMµ . (22)Here J EMµ is an operator of the electromagnetic current.This model is very well studied, and in many ways, it is more attractive than the model ofscalars in (1) mainly because it has a natural UV completion. Zooming in on the parameterspace relevant for the Borexino-SOX, we discover that above 2 m e the combination of all beamdump constraints put strong limits on the dark photon model. For m A (cid:48) < m e the mostchallenging constraint comes from cosmology, where the inclusion of three A (cid:48) polarizations,fully thermalized with electron-photon fluid, will reduce the effective number of neutrinospecies to an unacceptable level N eff < (cid:15) . An interesting feature of the dark photon model belowthe 2 m e threshold is that the main decay channel is 3 γ , and it is mediated by the electronloop. The decay rate is very suppressed, and the effective-field-theory type calculationperformed in the limit of very light A (cid:48) [48] was recently generalized to the m A (cid:48) ∼ m e [49]. We take this decay rate, and in addition, calculate separately the cross section of the2scattering process e + A (cid:48) → e + γ . Due to the strong suppression of the loop-induced decay,we find that the Compton-type scattering gives the main contribution to the signal rate inBorexino.For the dark photon A (cid:48) , the emission rate (the rate of the nuclear state decay to A (cid:48) ) isdetermined by Γ A (cid:48) Γ γ, E = v A (cid:48) (3 − v A (cid:48) )2 (cid:15) , (23)where v A (cid:48) = (1 − m A (cid:48) /ω ) / . In the limit of m A (cid:48) (cid:28) ω , the ratio of the two rates becomessimply (cid:15) . Substituting relevant numbers we get the counting rate for the 2.185 MeV energyas ˙ N A (cid:48) , .
185 MeV (cid:20) countsday (cid:21) = 1 . × × exp (cid:18) − t [day]285 d (cid:19) × (cid:0) dNdt (cid:1) × (cid:15) × v A (cid:48) (3 − v A (cid:48) ) × P deposit , .
185 MeV (24)For the background event rate, we use the energy spectra shown in Fig. 2 in [39]. Weuse the 4th/green event spectrum with the fiducial volume (FV) cut. The background rateis around 200 counts/100t × E = 2 .
185 MeV (For E = 1 .
485 MeV, the background rate is about 2300 counts/100t × t exp = 365 days and consider thecoupling (cid:15) for each mass that gives N sig < (cid:112) N backgr . The energy resolution at Borexinois 5% × (cid:112) /E. We use this as the bin size when we estimate the background rates at E = 2 .
185 MeV and E = 1 .
485 MeV. In regard of the background from the 2.2 MeV n-pcapture gamma ray discussed in section III C, we again conduct the analysis in two fiducialradii, 2.00 m and 3.02 m, for the Borexino-SOX sensitivity reaches.Even though the particle A (cid:48) cannot decay to e + e − in the kinematic range we consider, thedecays to photons and the Compton-like absorption will lead to the beam dump constraintsfor this model. The LSND production is easy to estimate, given that π will always have an A (cid:48) γ decay mode with Br π → A (cid:48) γ = 2 (cid:15) .A compilation of all the considerations above is shown in Fig. 2. We find that the sensitiv-ity reach of the Borexino-SOX experiment, (cid:15) ∼ − in probing the light dark photons, iscomparable but slightly above the bound from recasting the LSND data. Furthermore, thisLSND bound covers up a small triangular parameter space for 10 − ≤ (cid:15) ≤ − , m (cid:48) A ≤ m e that was not excluded by the cooling of Supernova 1987A [50, 51], and the precision mea-surement of electron anomalous magnetic moment (see Fig. 7 of [50]), independently fromthe cosmological scenarios. Note that here we plot the “robust” constraint from [50] in ourFig. 2, which is the intersection of bounds from different supernova profile models. Also,both [50, 51] use the trapping criterion, rather than the energy transport criterion (see, e.g. ,[52, 53]), to set the upper limits for the SN exclusion regions, with the trapping criterionbeing more conservative. VI. CONCLUSION
We have considered in detail how the search of the sterile neutrinos in the Borexino-SOXexperiment can also be turned into a search for extremely weakly interacting bosons. The3
LSND
SOX 1.49 + ( ) SOX 1.49 + ( ) S t e ll a r E ne r g y Lo ss S N A a e - - - - m A ' ( MeV ) ϵ Figure 2. Future sensitivity reach for the Borexino-SOX setup and various existing constraintsin coupling constant-mass parameter space for dark photons with a small mixing angle (cid:15) . Again,we conduct the analysis in two fiducial radii, 2.00 m and 3.02 m, for the Borexino-SOX sensitivityreaches, in regard of the background from the 2.2 MeV n-p capture gamma ray discussed in sectionIII C.
Left panel:
The experimental reach ( > σ ) by the Borexino-SOX setup is the blue curve.The constraint recasting the LSND data [28] is slightly stronger than the Borexino-Sox reach, andexcludes all the parameter space above the purple curve. Supernova cooling constrains the wholeregime below the dark blue curve on the upper-right corner [50, 51], while the gray area is againthe stellar energy loss bound [45]. reach of the experiment to the parameters of exotic scalars is limited by the energy release inradioactive cascades. It has to be less than 2.185 MeV for the radioactive source to be usedin SOX. However, in terms of the coupling constants, the reach of this experiment will bemuch farther down than even the most sensitive among the particle beam dump experiments.We find that with the proposed setup, coupling constants as low as (cid:15) ∼ − will beprobed. The improved analysis in this work includes particle decays inside the detector asthe main energy-deposition channel. It is the dominant process that significantly exceedsthe scalar Compton absorption above the hundred-keV mass regime. Similar revisions willapply to searches proposed in Ref. [25] that suggest using proton accelerators to populatenuclear metastable states. In addition, we study the sensitivity reach of the Borexino-SOXexperiment in probing a light dark photon below 1 MeV. The reach (cid:15) ∼ − is comparable,but slightly weaker than the bound already imposed by the existing LSND neutrino-electronscattering data. Combining this constraint with the supernova bound we completely ruleout the possibility of having a light dark photon below 1 MeV in this coupling range.In conclusion, one should not regard the SOX project as exclusively a search for sterile4neutrinos (motivated mostly by experimental anomalies), but a generic search for dark sectorparticles. The scalar case considered in this paper can be motivated by the proton chargeradius anomaly, and the SOX project provides tremendous sensitivity to this type of models.We encourage the Borexino collaboration to perform its own study of the sensitivity to newbosons using more detailed information about background and efficiencies. ACKNOWLEDGMENTS
We thank Drs. P. deNiverville and S. Zavatarelli for useful correspondence. We alsothank Drs. J. Dror, R. Lasenby and B. Safdi for helpful discussions. Research at thePerimeter Institute is supported in part by the Government of Canada through NSERC andby the Province of Ontario through MEDT. YT was supported by the Visiting GraduateFellow program at Perimeter Institute, U.S. National Science Foundation through grantPHY-1719877, and Cornell graduate fellowship, while parts of this work were completed. [1]
Fundamental Physics at the Intensity Frontier (2012) arXiv:1205.2671 [hep-ex].[2] V. Silveira and A. Zee, Phys. Lett.
B161 , 136 (1985).[3] B. Holdom, Phys. Lett.
B166 , 196 (1986).[4] C. Boehm and P. Fayet, Nucl. Phys.
B683 , 219 (2004), arXiv:hep-ph/0305261 [hep-ph].[5] A. Boyarsky, O. Ruchayskiy, and M. Shaposhnikov, Ann. Rev. Nucl. Part. Sci. , 191 (2009),arXiv:0901.0011 [hep-ph].[6] M. Pospelov, A. Ritz, and M. B. Voloshin, Phys. Lett. B662 , 53 (2008), arXiv:0711.4866[hep-ph].[7] S. N. Gninenko, Phys. Rev.
D83 , 015015 (2011), arXiv:1009.5536 [hep-ph].[8] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell, Phys. Rev.
D81 , 123530 (2010), arXiv:0905.4720 [hep-th].[9] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner, Phys. Rev.
D79 , 015014(2009), arXiv:0810.0713 [hep-ph].[10] J. Jaeckel and A. Ringwald, Ann. Rev. Nucl. Part. Sci. , 405 (2010), arXiv:1002.0329 [hep-ph].[11] J. Alexander et al. (2016) arXiv:1608.08632 [hep-ph].[12] G. W. Bennett et al. (Muon g-2), Phys. Rev. D73 , 072003 (2006), arXiv:hep-ex/0602035[hep-ex].[13] A. Aguilar-Arevalo et al. (LSND), Phys. Rev.
D64 , 112007 (2001), arXiv:hep-ex/0104049[hep-ex].[14] A. A. Aguilar-Arevalo et al. (MiniBooNE), Phys. Rev. Lett. , 161801 (2013),arXiv:1303.2588 [hep-ex].[15] J. M. Conrad and M. H. Shaevitz, (2016), arXiv:1609.07803 [hep-ex].[16] R. Pohl et al. , Nature , 213 (2010).[17] R. Pohl, R. Gilman, G. A. Miller, and K. Pachucki, Ann. Rev. Nucl. Part. Sci. , 175 (2013),arXiv:1301.0905 [physics.atom-ph].[18] A. Gando et al. (KamLAND-Zen), Phys. Rev. Lett. , 082503 (2016), [Addendum: Phys.Rev. Lett.117,no.10,109903(2016)], arXiv:1605.02889 [hep-ex]. [19] S. Andringa et al. (SNO+), Adv. High Energy Phys. , 6194250 (2016), arXiv:1508.05759[physics.ins-det].[20] G. Bellini et al. (Borexino), JHEP , 038 (2013), arXiv:1304.7721 [physics.ins-det].[21] C. Gustavino (LUNA), Proceedings, 37th International Conference on High Energy Physics(ICHEP 2014): Valencia, Spain, July 2-9, 2014 , Nucl. Part. Phys. Proc. , 1807(2016).[22] M. Nakahata et al. (Super-Kamiokande), Nucl. Instrum. Meth.
A421 , 113 (1999), arXiv:hep-ex/9807027 [hep-ex].[23] C. Aberle et al. , in
Proceedings, Community Summer Study 2013: Snowmass on the Missis-sippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013 (2013) arXiv:1307.2949[physics.acc-ph].[24] M. Abs et al. , (2015), arXiv:1511.05130 [physics.acc-ph].[25] E. Izaguirre, G. Krnjaic, and M. Pospelov, Phys. Lett.
B740 , 61 (2015), arXiv:1405.4864[hep-ph].[26] Y. Kahn, G. Krnjaic, J. Thaler, and M. Toups, Phys. Rev.
D91 , 055006 (2015),arXiv:1411.1055 [hep-ph].[27] E. Izaguirre, G. Krnjaic, and M. Pospelov, Phys. Rev.
D92 , 095014 (2015), arXiv:1507.02681[hep-ph].[28] L. B. Auerbach et al. (LSND), Phys. Rev.
D63 , 112001 (2001), arXiv:hep-ex/0101039 [hep-ex].[29] D. Tucker-Smith and I. Yavin, Phys. Rev.
D83 , 101702 (2011), arXiv:1011.4922 [hep-ph].[30] C.-Y. Chen, H. Davoudiasl, W. J. Marciano, and C. Zhang, Phys. Rev.
D93 , 035006 (2016),arXiv:1511.04715 [hep-ph].[31] B. Batell, N. Lange, D. McKeen, M. Pospelov, and A. Ritz, (2016), arXiv:1606.04943 [hep-ph].[32] V. Barger, C.-W. Chiang, W.-Y. Keung, and D. Marfatia, Phys. Rev. Lett. , 153001(2011), arXiv:1011.3519 [hep-ph].[33] B. Batell, D. McKeen, and M. Pospelov, Phys. Rev. Lett. , 011803 (2011), arXiv:1103.0721[hep-ph].[34] S. G. Karshenboim, D. McKeen, and M. Pospelov, Phys. Rev.
D90 , 073004 (2014), [Adden-dum: Phys. Rev.D90,no.7,079905(2014)], arXiv:1401.6154 [hep-ph].[35] Y.-S. Liu, D. McKeen, and G. A. Miller, Phys. Rev. Lett. , 101801 (2016),arXiv:1605.04612 [hep-ph].[36] R. Pohl et al. (CREMA), Science , 669 (2016).[37] R. Barbieri and T. E. O. Ericson, Phys. Lett. , 270 (1975).[38] A. Djouadi, Phys. Rept. , 1 (2008), arXiv:hep-ph/0503172 [hep-ph].[39] G. Bellini et al. (Borexino), Phys. Rev.
D88 , 072010 (2013), arXiv:1311.5347 [hep-ex].[40] P. Vogel and J. F. Beacom, Phys. Rev.
D60 , 053003 (1999), arXiv:hep-ph/9903554 [hep-ph].[41] G. Magill, R. Plestid, M. Pospelov, and Y.-D. Tsai, (2018), arXiv:1806.03310 [hep-ph].[42] G. Magill, R. Plestid, M. Pospelov, and Y.-D. Tsai, (2018), arXiv:1803.03262 [hep-ph].[43] J. E. Amaro, A. M. Lallena, and J. Nieves, Nucl. Phys.
A623 , 529 (1997), arXiv:nucl-th/9704022 [nucl-th].[44] C. Athanassopoulos et al. (LSND), Nucl. Instrum. Meth.
A388 , 149 (1997), arXiv:nucl-ex/9605002 [nucl-ex].[45] G. Raffelt and A. Weiss, Phys. Rev.
D51 , 1495 (1995), arXiv:hep-ph/9410205 [hep-ph].[46] O. Yu. Smirnov et al. (Borexino),
Proceedings, 2nd International Workshop on Prospects ofParticle Physics: Neutrino Physics and Astrophysics: Valday, Russia, February 1-8, 2015 , Phys. Part. Nucl. , 995 (2016), arXiv:1507.02432 [hep-ex].[47] K. M. Nollett and G. Steigman, Phys. Rev. D89 , 083508 (2014), arXiv:1312.5725 [astro-ph.CO].[48] M. Pospelov, A. Ritz, and M. B. Voloshin, Phys. Rev.
D78 , 115012 (2008), arXiv:0807.3279[hep-ph].[49] S. D. McDermott, H. H. Patel, and H. Ramani, (2017), arXiv:1705.00619 [hep-ph].[50] J. H. Chang, R. Essig, and S. D. McDermott, JHEP , 107 (2017), arXiv:1611.03864 [hep-ph].[51] E. Hardy and R. Lasenby, JHEP , 033 (2017), arXiv:1611.05852 [hep-ph].[52] A. Burrows, M. T. Ressell, and M. S. Turner, Phys. Rev. D42 , 3297 (1990).[53] W. Keil, H. T. Janka, and G. Raffelt, Phys. Rev.