Neutron Decay to a Non-Abelian Dark Sector
NNeutron Decay to a Non-Abelian Dark Sector
Fatemeh Elahi and Mojtaba Mohammadi Najafabadi School of Particles and Accelerators,Institute for Research in Fundamental Sciences IPM, Tehran, Iran
Abstract
According to the Standard Model (SM), we expect to find a proton for each decaying neutron.However, the experiments counting the number of decayed neutrons and produced protons havea disagreement. This discrepancy suggests that neutrons might have an exotic decay to a DarkSector (DS). In this paper, we explore a scenario where neutrons decay to a dark Dirac fermion χ and a non-abelian dark gauge boson W (cid:48) . We discuss the cosmological implications of this scenarioassuming DS particles are produced via freeze-in. In our proposed scenario, DS has three portalswith the SM sector: (1) the fermion portal coming from the mixing of the neutron with χ , (2) ascalar portal, and (3) a non-renormalizable kinetic mixing between photon and dark gauge bosonswhich induces a vector portal between the two sectors. We show that neither the fermion portalnor the scalar portal should contribute to the production of the particles in the early universe.Specifically, we argue that the maximum temperature of the universe must be low enough toprevent the production of χ in the early universe. In this paper, we rely on the vector portal toconnect the two sectors, and we discuss the phenomenological bounds on the model. The mainconstraints come from ensuring the right relic abundance of dark matter and evading the neutronstar bounds. When dark gauge boson is very light, measurements of the Big Bang Nucleosynthesisimpose a stringent constraint as well. a r X i v : . [ h e p - ph ] M a y . INTRODUCTION Even though the Standard Model (SM) of particle physics can explain almost all observedphenomena, we are certain there exits physics beyond the SM. One of the most prominentquestions in the particle astrophysics community is the nature and origin of dark matter(DM). So far, we have not observed any unambiguous detection of DM. However, numer-ous experimental anomalies may be a hint of DM interaction with the SM. One of theseexperiments is the measurements of the neutron lifetime.Due to the importance of neutrons as one of the main building blocks of luminous matterand one of the key role players in the formation of light elements in the early universe,there have been several experiments that attempt to find the lifetime of the neutrons [1–10]. In the SM, we expect the branching ratio of a neutron to a proton, an electron, and aneutrino ( n → p + e + ¯ ν e ) to be 100%. In an experiment known as the bottle experiment [1–8], ultracold neutrons are stored for a time comparable to the neutron lifetime, then theremaining neutrons are counted. This experiment finds the total decay width or equivalentlythe lifetime of the neutrons. Their finding is τ bottle n = 879 . ± . s . In another experimentknown as the beam experiment [9, 10], the number of produced protons are counted, andtheir finding has been announced to be τ beam n → p + ... = 888 . ± . s . The lifetime of neutronsin these two experiments differ by 7 . s by 4 σ , which constitutes about 1% branching ratioof the neutron. The aforementioned discrepancy may be the result of an exotic decay ofneutrons to the dark sector. Due to the close mass of neutrons and protons and theirintimate structures, the easiest way to ensure an exotic decay of a neutron and the stabilityof protons is to assume the total mass of the exotic decay of neutron M f is greater thanthe mass of proton and electron: m n > M f > m p − m e . Furthermore, any baryon numberviolating process is severely constrained [11–25]. Therefore, we are led to consider scenarioswhere neutrons can decay to a new degree of freedom that has a baryonic charge.Numerous studies have explored different possibilities [26–43]. The most minimalisticscenario discussed in the literature is n → χγ , where χ is a fermionic DM that has abaryonic charge. If we assume m p < m χ < m n , we expect E γ < .
572 MeV [26]. However,experimental measurements disfavor this scenario [44, 45] for a photon with an energy in therange 0 .
782 MeV < E γ < .
664 MeV up to 2 . φ with mass m φ < .
572 MeV is also anotherpossibility. An important ingredient for both of these scenarios is a mixing between theneutron and χ , which has an effective Lagrangian of L eff = ¯ χ ( i /D − m χ ) χ − ¯ n ( i /∂ − m n + µ n σ µν F µν ) n − δm ¯ n R χ L . (1)To resolve the neutron lifetime discrepancy, δm is expected to be about 10 − MeV [26].A conversion of neutrons to χ with such strength has important consequences in the equationof states of Neutron Stars (NSs) [28]. That is the decay of neutrons in the NS cause theequation of state (pressure and energy density of the neutron star) to alter significantly, andthereby affect the mass and volume of NSs. Specifically, if neutron and χ are in chemicalequilibrium, due to the less interaction of χ comparatively, the conversion of neutrons to χ leads to a lower pressure. Integrating the Tolman-Oppenheimer-Volkoff equations [37, 39,40, 46–51], one can find the maximum mass of an NS as a function of its radius, and theupper limit is in contradiction with the properties of some of the neutron stars observed [28].The simplest solution is to consider DM scenarios that have repulsive self-interaction and arepulsive interaction with neutrons. That is to have a vector mediator, e.g. a dark photon.2n Ref. [28], the authors considered the decay of n → χA (cid:48) . To ensure the theory isconsistent with the observation of dense NSs with radius 2 M (cid:12) , we need m A (cid:48) /g D (cid:46) (45 −
60) MeV, where g D is the gauge coupling of the U (1) D . In this setup, we also have a mixingbetween the dark gauge boson and the SM photon: L = − (cid:15) F (cid:48) µν F µν , (2)which induces dark photon - electromagnetic current interaction with a coupling proportionalto (cid:15) . The authors of Ref. [28] did an extensive phenomenological study of this scenario, andshowed that the parameter space for m A (cid:48) < m e is severely constrained, while the casewhere m A (cid:48) > m e is slightly better. One of the main constraints comes from the era ofBig Bang Nucleosynthesis (BBN), which requires dark photon to decay early enough that itwon’t inject much energy during BBN. One way to escape this constraint is producing thedark sector (DS) particles via freeze-in. In this paper, we explore this possibility and showthat a χ that is a stable DM candidate and can justify the neutron decay experiments willnecessarily over-close the universe. Thereby, we must require the maximum temperatureof the universe to be low enough that χ is not produced in the early universe. To explainthe relic abundance of DM, we need to rely on the dark photon or the scalar, both ofwhich are unstable particles. Hence, in this paper, instead of a dark U (1) D , we considera dark non-abelian gauge SU (2) D , because even though it does not have any more freeparameters, the extra degrees of freedom helps with explaining the two observations of DMand neutron decay. Furthermore, for the freeze-in scenario to work, we should employ verysmall kinetic mixing, and this is more justified in the non-abelian kinetic mixing because ofits non-renormalizable nature. The main differences between our work and Ref. [28] are thefollowings:– In this paper, DS has a gauge SU (2) D rather than a gauge U (1) D .– We assume the relic abundance of DS particles is through freeze-in.– We turn off the scalar portal between the two sectors. More specifically, in the potentialterm λ φH | φ | | H | – with φ being the scalar responsible for the spontaneous breakingof the SU (2) D and H being the SM Higgs – we take λ φH = 0. We argue that theradiative correction is very suppressed, and thus our assumption is justifiable. Thischoice of λ φH has important consequences for the relic abundance of DM.The organization of the paper is as follows: In section II we explain the model andintroduce the degrees of freedom as well as the free parameters in the theory. Section III isdenoted to the phenomenology of the model including the constraints from NSs, the neutrondecay experiments, and the cosmological constraints which is discussed in Section III A.Direct Detection and Collider Constraints are explored in Section III C, and finally, theconcluding remarks are presented in the Conclusion IV. II. MODEL
Let us assume Dark Sector has a gauge SU (2) D that is spontaneously broken by a doublet φ : φ = (cid:18) √ ( G φ + iG φ ) ϕ + v φ + iG φ (cid:19) , (3)3here G iφ are the Goldstone bosons, which become the longitudinal component of the gaugebosons, and v φ is vacuum expectation value ( vev ) of φ . To ensure φ indeed acquires vev , werequire its potential to have the following form: V ( φ, H ) = − µ φ | φ | + λ φ | φ | − µ H | H | + λ H | H | + λ φH | φ | | H | , (4)with µ φ >
0. Since neutron is a fermion, the decay of a neutron to DS particles compels us toinclude fermionic degrees of freedom. Thereby, we introduce a Dirac fermion χ transformingas a doublet under SU (2) D : χ T = ( χ , χ ). Since there are severe constraints on the baryonnumber violating models [11, 12], we assume χ has a baryon charge of +1. The effectiveLagrangian becomes L NP = − W (cid:48) aµν W (cid:48) a,µν + i ¯ χ ( /D + m χ ) χ + | D µ φ | + η ¯ χ L φn + h.c. + C Y φ † τ a φW (cid:48) aµν F µν + ˜ C Y φ † τ a φW (cid:48) aµν ˜ F µν + ˜ C Y φ † τ a φ ˜ W (cid:48) aµν F µν + h.c − V ( φ, H ) , (5)where W (cid:48) aµν is the field strength tensor of SU (2) D and ˜ W (cid:48) aµν = (cid:15) µναβ W (cid:48) a,αβ is the dual of thefield tensor, and D µ = ∂ µ − ig D τ a W aµ . Once φ acquires vev , a mixing between the neutronand χ is induced, which results in the conversion of neutrons to χ and other dark sectorparticles.The Lagrangian terms written in the second line of Eq. 5 are the non-abelian kineticmixing between SU (2) D gauge bosons’ and the photon’s field tensor. Due to the presence of φ in these terms the kinetic mixing with the CP-odd component is not a total derivative, andthus contribute to the action. For simplicity, we assume ˜ C Y ≡ ˜ C Y = ˜ C Y . Note that thekinetic mixing terms are dimension 6 operators and therefore have an inverse mass-squareddimension. The suppressed mass dimension means there is a small couplings of the darkgauge bosons with SM particles.Note that with this setup, another effective Lagrangian term ¯ χ L φπp can be written aswell. This term may lead to proton decay, χ decay, or neither depending on the massspectrum. We must assume m p < m χ < m n to ensure the stability of proton as the lightestfermion charged under the baryon symmetry U (1) B . If m χ > m p + m π , χ may decay.However, in this paper, we fix m χ such that χ is also a stable particle. After SU (2) D Spontaneous Symmetry Breaking (SSB), W (cid:48) and φ get a mass proportionalto v φ : m W (cid:48) = g D v φ m φ = (cid:112) λ φ v φ . At low energies there is a residual Z symmetry remaining from the broken SU (2) D . Underthe Z symmetry, W (cid:48) ± and χ are odd, and the rest of the particles are even. It has beenshown that to explain the neutron decay anomaly ( n → χW (cid:48) ± ) and yet be safe from the NSconstraint, we necessarily need to have m χ (cid:29) m W (cid:48) . Therefore, W (cid:48) ± are the lightest particlescharged under Z , and thus are stable. This is while W (cid:48) mixes with photon after φ gets a vev , and thus it can decay (e.g, W (cid:48) → e + e − if m W (cid:48) > m e and W (cid:48) → γ for lighter W (cid:48) .)Similarly, due to the φW (cid:48) W (cid:48) coupling we can have φ → γγ decay. It is worth mentioningthat W (cid:48) and/or φ could be long-lived DM candidates.The free parameters in this model are the following: Even though after φ acquires a vev, there is a slight mass splitting between χ and χ , this mass splittingis negligible compared with m χ . Therefore, for the rest of the paper, we will assume both χ and χ havemass m χ and we will use χ to refer to both of them. masses : m χ , m φ , v φ . • couplings : η, g D , λ φH , and C Y , ˜ C Y with dimensions proportional to [ M − ].To find the cosmological constraints on the model, we need to briefly discuss the UVcompletion of the model. This is very similar to the model suggested in Ref. [28]: two colortriplet scalars with hypercharge 1 / and Φ ) are introduced, where Φ is also a doubletof SU (2) D . Thus, the UV Lagrangian can simply be written as L UV = λ ¯ d a P Lχ Φ a + λ (cid:15) abc ¯ u ca P R d b Φ c + µ Φ a Φ ∗ a φ, (6)where in the effective theory η = βv φ λ λ m m , (7)with β being the factor derived from confinement of quarks to neutrons, and its value β = 0 .
014 GeV is taken from Lattice QCD simulations [52]. Dijet searches at CMS [53]and ATLAS [54] push the masses of φ i to greater than 1 TeV.Since Φ i , with i = 1 , SU (3) c , we expect their number density in theearly universe to match that of photons (e.g., we expect them to be in thermal equilibriumwith thermal bath). Through their couplings with the dark sector, the production of χ andsubsequently φ and W (cid:48) should occur in abundance. As shown in [28], such set up leads tosevere constraints from CMB [55–57], BBN [58], and the Fermi -LAT observation of gammarays from dwarf spheroidal galaxies [59, 60]. The summary of these constraints in presentedhere:– Once χ becomes non-relativistic, it can only annihilate to W (cid:48) W (cid:48) and φφ efficiently.Therefore, if χ is in thermal equilibrium in the early universe, the abundant productionof W (cid:48) and φ becomes inevitable. On the other hand, W (cid:48) and φ can only decay after φ acquires a vev , which roughly occurs around 60 MeV , and it is extremely closeto BBN. The decay of W (cid:48) and φ near the BBN disturbs the Hubble rate and thus itsignificantly alters the production of light nuclei by diluting the baryon-photon ratioas well as causing photodissociation of the nuclei.– Similarly, decays near and during recombination will distort the CMB temperaturefluctuations and thus there are severe constraints on a model with light W (cid:48) from CMBas well.– For m W (cid:48) (cid:28) m χ , the annihilation of χχ to W (cid:48) W (cid:48) is Sommerfeld at low velocities, whichleads to an enhanced annihilation cross-sections in spheroidal galaxies and at the timeof recombination. Therefore, it is crucial that we do not have much χ in the universe.Thereby, in this paper, we explore another avenue. We assume dark sector particlesstart with zero abundance in the early universe and they get produced through freeze-inmechanism [61–63]. Consequently, in our set-up, we need the maximum temperature T max to be smaller than m Φ i so that they are not produced in the early universe. If the colormultiplets are not produced, then the production of χ is greatly reduced.The portals between the dark sector and the SM sector are via (1) the Higgs portal with astrength proportional to λ φH , (2) the kinetic mixing governed by C Y v φ , and (3) the effective This value is the maximum value allowed from the NS constraint χ and neutron which is ηv φ . For a successful freeze-in scenario, we need anextremely weak connection between the dark sector and the SM sector. Since C Y and η aredue to non-renormalizable interactions, we can justifiably assign them small values . Thisargument becomes more non-trivial for λ φH , which in general can take any value (cid:46)
1. Ifwe want to assign λ φH a small value, we must make sure that this choice is safe from loopcorrections. The radiative correction to λ φH comes from φφγγ vertex, which only opensup after φ acquires a vev and even then it is suppressed – the coupling is proportional to g D C Y v φ . Fig.1 shows one of the leading diagrams to radiative correction to λ φH , and as itis illustrated, in addition to the C Y v φ suppression, it is two loops suppressed. Therefore, ifthe value of λ φH is small at tree level, it does not get amplified much at loop levels. Forsimplicity, in this work, we assume λ φH = 0. ff '3 W '3 W gg – W – W – W HH f v Y C f v Y C Figure 1. The radiative corrections to λ φH . As illustrated this coupling is suppressed by twoloops as well as C Y v φ , and thus it is very small. Therefore, if we let λ φH = 0, we can easily makesure the radiative corrections to λ φH stay negligible. Having closed the Higgs portal, now we need to discuss the evolution of dark sectorparticles in the early universe, and how much they contribute to the relic abundance of thetotal DM. In the following section, we discuss the phenomenological constraints on each ofthese parameters including the ideal spot that explains the neutron decay anomaly and yieldsthe correct relic abundance of DM. Even though the number of free degrees of parametersis large, the numerous experimental and observations bounds on these parameters forces usto live in a small region of the parameter space.
III. PHENOMENOLOGY
One of the most important bounds on this model comes from NSs, where the conversionof the neutron to χ can have significant consequences. If neutrons and χ are in chemicalequilibrium, it is favorable for the neutrons to convert into χ , which due to its almostnon-interacting nature, results in a lower pressure in the neutron stars. By integrating theTolman-Oppenheimer-Volkoff equation, one finds the maximum mass with respect to theNS’s radius falls below the largest observed mass. Our scenario falls in the category thatthere is a repulsive interaction between χ and the neutrons and thus can be safe from thisconstraint as long as m W (cid:48) /g D <
60 MeV [28, 46]. The value of η is governed by the neutron decay anomaly. Since η is derived from non-renormalizableinteraction, we expect its value to be small. to χW (cid:48) and χφ areΓ n → χ W (cid:48) (cid:39) η ( m n − m χ ) πm n (cid:34) − (cid:18) m W (cid:48) m n − m χ (cid:19) (cid:35) / (8)Γ n → χφ (cid:39) η ( m n − m χ ) πm n (cid:34) − (cid:18) m φ m n − m χ (cid:19) (cid:35) / . (9)We have already discussed that the mass of χ should satisfy m p + m e < m χ < m n , inorder to both satisfy the neutron decay and yet be safe from stringent proton decay bounds.To have a decay that is kinematically allowed, we must have m χ + Min (cid:2) m W (cid:48) , m φ (cid:3) < m n .Therefore, let us consider the following benchmarks: • We consider two benchmarks where both W (cid:48) and φ are light enough that both decaysmentioned in Eq. 9 are allowed. For one of these benchmarks, we take m φ > m W (cid:48) : m χ = 937 .
992 MeV , m φ = 1 . , and m W (cid:48) = 0 . . Note that in this benchmark, φ decays to W (cid:48) , and thus it is not a DM candidate. Wewill denote this benchmark as A1 . To justify the neutron decay discrepancy, we need η (cid:39) . × − .Another benchmark we choose is when m φ ∼ m W (cid:48) : m χ = 937 .
992 MeV , m φ = 1 . , and m W (cid:48) = 1 . , and we present this benchmark by A2 . The η that explain the neutron decay is η (cid:39) . × − . It is worth mentioning that φ , in this benchmark, can decay to twophotons. However, depending on C Y , φ can be a long lived DM candidate. • Another scenario is when φ is heavy such that the decay n → χφ is not kinematicallyallowed. In this case, we will also take two different benchmarks; one where m φ > m W (cid:48) : m χ = 937 .
992 MeV , m φ = 4 MeV , and m W (cid:48) = 1 . , which we use B1 to refer to this benchmark. Solving for the η that yields Br( n → χW (cid:48) ) (cid:39)
1% is η (cid:39) . × − .Another benchmark, satisfies m W (cid:48) < m φ < m W (cid:48) : m χ = 937 .
992 MeV , m φ = 2 . , and m W (cid:48) = 1 . . This benchmark is presented by B2 . Since the only parameter that has changed is m φ and n → χφ is forbidden, the desired η is still η (cid:39) . × − . • We also consider another case where m W (cid:48) is large enough that the decay of n → χW (cid:48) is not allowed. However, φ is light enough that allows the dark decay of neutrons: m χ = 937 .
992 MeV , m φ = 1 . , and m W (cid:48) = 2 . . This benchmark is referred by C . The desired η to justify the neutron decay anomalyis η (cid:39) . × − . The decay of n → χγ can occur via ¯ nσ µν φ † χF µν . However, because of the null search for monochromaticphoton [44] and exacerbating the tension of the axial coupling of the neutron [64], we expect this couplingto be very small and negligible in this study. . Cosmology In this section, we will discuss the cosmological constraints, and we will see satisfying therelic abundance and making sure DM candidates do not over-close the universe gives the moststringent bound for most of our benchmarks. Moreover, BBN, which strongly disfavors newdegrees of freedom injecting energy around the formation of nuclei in the early universe, putsimportant constraints on some of the benchmarks. As mentioned earlier, the observation oflarge neutron stars excludes part of the parameter space as well. The rest of the constraintsfrom various experiments and observations are also discussed in this section.
1. Relic Abundance
We are interested in a scenario where dark sector particles start with zero or negligibleabundance and then are slowly produced through their feeble interactions with SM particles.First, we will discuss the production of χ as it will be important to set the maximum temper-ature of the universe, then we will investigate the evolution of W (cid:48) and φ in the early universe. χ ProductionUp until φ acquires a vev , the production of χ is due to qq → qφχ , where q = u, d . TheBoltzmann equation describing the evolution of χ number is˙ n χ + 3 Hn χ = (cid:90) d Π q d Π q d Π q d Π φ d Π χ (2 π ) ( p i − p f ) |M| qq → qφχ f q f q , (10)where d Π i = d p i (2 π ) E i , and f q ∼ e − E q /T is the distribution function of the quarks in thermal bath,and s is the canonical Mandelstam variable. We can simplify Eq. 10 for any process that has threefinal state particles [62]:˙ n χ + 3 Hn χ = T (4 π ) (cid:90) d Ω (cid:90) ∞ ds s / |M| qq → qφχ K (cid:18) √ sT (cid:19) , (11)where K ( x ) denotes the modified Bessel function of the second kind. Eq. 11 is in the relativisticlimit where the masses of the particles involved are negligible to the temperature. The squaredMatrix Element (ME) of qq → qφχ in the relativistic limit is |M| qq → qφχ (cid:39) (cid:18) ηβ (cid:19) s . (12)The integral over s in Eq. 11 has a closed form (cid:90) ∞ ds s (2 n +1) / K (cid:18) √ sT (cid:19) = 4 n +1 T n +3 Γ( n + 1)Γ( n + 2) , (13)for n > −
1. Therefore, we can easily calculate the right hand side of Eq. 10. The left hand side we will assume that the temperature at which φ gets a vev is the value of vev itself T v φ ∼
60 MeV. an be converted to yield ( Y ≡ n/S ) with S being the entropy density: Y χ = (cid:90) T max dT (cid:26) − SHT (cid:20) T (4 π ) (cid:90) d Ω (cid:90) ∞ ds s / |M| qq → qφχ K (cid:18) √ sT (cid:19)(cid:21)(cid:27) (cid:39) ∗ . × π (cid:112) g ρ(cid:63) g S(cid:63) M P l (cid:18) ηβ (cid:19) T × θ ( T max − m χ ) (cid:39) × − A1 . × − A2 . × − B1 & B2 . C × (cid:18) T max GeV (cid:19) × θ ( T max − m χ ) (14)where the second equality is obtained by using the definitions S = π g S(cid:63) T and H = . √ g ρ(cid:63) T M Pl ,and θ ( x ) is the step function that ensures the universe has enough energy to produce χ . There isa constraint on the Y χ so that it does not over-close the universe: Y χ ≤ Ω total DM ρ c m χ s (cid:39) . × − × (cid:18) GeV m χ (cid:19) , (15)where for all of our benchmarks, the mass of χ is fixed to m χ = 937 .
992 MeV. It is clear that if χ is produced, it quickly over-closes the universe. Thereby, we require T max < m χ to prevent theproduction of χ in the early universe. In other words, even though χ is a stable particle, it doesnot contribute to the relic abundance of the DM in the universe. Notice that these calculationsonly depend on the coupling of χ with quarks and thus the results are the same if we had assumea U (1) D instead of the SU (2) D . W (cid:48) and φ ProductionFor the case where λ φH = 0, the main mechanism for the production of φ and W (cid:48) µ is via thekinetic mixing term. The leading diagram of W (cid:48) /φ production for T > v φ is shown in Fig. 2,where J EM represents any particle (lepton or hadron) that has electromagnetic charge and has asignificant abundance at T < m χ . The Boltzmann equation governing the number density of W (cid:48) and φ is similar to Eq. 11, and the squared ME for the process of our interest J EM J EM → φφW (cid:48) isthe following: |M| J EM J EM → φφW (cid:48) = Q C Y (4 s − s ( t + u ) − tu )4 s , (16)with Q being charge of the initial state particles, and s, t, and u are the Mandelstam variables. Inthe limit where s (cid:29) t, u , we get |M| J EM J EM → φφW (cid:48) → Q C Y s . In Eq. 16, we have let m W (cid:48) = m φ = 0,because φ still has not acquired a vev . Note that W (cid:48) and φ do not get thermal corrections as well,since they live in a much colder sector. In the SM sector, however, for T (cid:29) m SM ( T = 0),the thermal correction to the mass of particles becomes important. For simplicity, we assume that m SM ( T ) ∼ T . The only exception is for proton, where for T < m p , we assume m p ( T ) = m p ( T = 0).Furthermore, we make the reasonable assumption that unstable particles decay more efficiently toSM particles than annihilate to W (cid:48) and φ . The yield of W (cid:48) and φ coming from Fig. 2 is Y UVi (cid:39) g i αM P l × C Y . × π (cid:112) g ρ(cid:63) g S(cid:63) T , (17)where i = W (cid:48) /φ and g i represents the number of φ and W (cid:48) produced.
2. CMB and BBN constraints
We know that W (cid:48) decays and if it injects energy during BBN, its energetic decay productsmight disturb the production of the light nuclei by diluting the ration of baryons to photons.Furthermore, the injection of energy may cause photodissociation which will affect the CosmicMicrowave Background (CMB) fluctuations. To avoid these effects, we follow the convention ofRef. [65] and require W (cid:48) to decay before it exceeds half of the energy density of the universe. Thetemperature at which this occurs is T dom ≈ m W (cid:48) Y W (cid:48) f , (19)where f = 1 / W (cid:48) . We requirethat the lifetime of the W (cid:48) is smaller than H − ( T dom ). The lifetime of W (cid:48) if m W (cid:48) > m e is τ heavy (cid:39) παC Y v φ m W (cid:48) (cid:32) − (cid:18) m e m W (cid:48) (cid:19) (cid:33) − / , (20) The small differences between Eq. 11 and Eq. 18 are due to the number of final state particles, and thefact that final state particles are massive after SU (2) D SSB.
60 MeV, the only electromagnetically chargedparticles that are still in the plasma and have not decayed are the electrons and protons. J EM
DM accumulating at the Galactic Center or near dwarf spheroidal galaxies, annihilates to W (cid:48) :(e.g, W (cid:48) + W (cid:48) − → W (cid:48) ( ∗ )3 W (cid:48) ). Depending on the mass of W (cid:48) , we may either have W (cid:48) → e + e − or W (cid:48) → γ . An excess emission of positron may be detected by Voyager [66] and the AMS-02 [67].As discussed in [28] and [68], any claim on the detection of DM from the excess positron suffersfrom large uncertainties and it is not reliable. he Fermi -LAT collaboration [69, 70] is searching for the excess in photons, and Ref. [28] has de-rived the constraint on DM coming from 6 years of
Fermi -LAT observations of 15 dwarf spheroidalgalaxies. Ref. [28] has shown that the region parameter space satisfying (cid:104) σv (cid:105) W (cid:48) + W (cid:48)−→ γW (cid:48) → γ > . × − cm / s , is excluded. We can approximate this annihilation as (cid:104) σv (cid:105) W (cid:48) + W (cid:48)−→ γW (cid:48) → γ (cid:39) α D C Y v φ π m W (cid:48) , (22)where α D = g D / (4 π ). As can be seen, this cross section is extremely small and does not provideany noteworthy bound on the parameter space. C. Direct Detection and Collider Constraints
The cross section for W (cid:48) ± to scatter on proton with W (cid:48) being the mediator is σ e W (cid:48) = 8 παα D C Y v φ µ e W (cid:48) ( m W (cid:48) + q ) , (23)where µ e W (cid:48) ≡ m W (cid:48) m e m W (cid:48) + m e is the reduced mass of the DM-electron system, and q is the momentumtransfer between the DM and electron. At electron ionization experiments like SENSEI [71] andXENON10[70], the targeted electrons are usually bound to atoms with typical velocity of a boundelectron being v e ∼ α . The minimum energy transferred required in these experiments to knock outthe bound electron and detect the DM-electron scattering is q ∼ αm e . For all of our benchmarks,we have m W (cid:48) (cid:29) αm e and thus the momentum transfer can be neglected. For such heavy W (cid:48) , thebounds are rather very mild and they do no provide any noticeable bound on our parameter space.
1. BaBar and SLAC
Another constraint on C Y comes from the direct production of W (cid:48) and γ at E137 [72] and BaBar [73] experiments. Ref. [28] has worked out this constraints and has found that C Y v φ < . × − , which means that C Y < × − GeV − which is much weaker bound than the ones wehave discussed so far. Since the coupling of W (cid:48) with W (cid:48) ± does not play any role, this constraintis oblivious to the value of g D . The BaBar bound is shown as shaded purple in Fig. 6.Yet, another important constraint comes from the electron beam dump experiment at
SLAC [74],which consists of a 20 GeV electron beam hitting upon a set of fixed aluminum plates. Throughthe W (cid:48) − γ , we can create a pair of DM candidates ( W (cid:48) ± ): eN → eN W (cid:48) ∗ → eN W (cid:48) + W (cid:48) − . TheDM would then travel through a 179 m hill, followed by 204 m of air and then would be detectedby an electromagnetic calorimeter. This process, however, is suppressed. That is because theproduction of on-shell W (cid:48) is favored, which then would decay back to either e + e − or 3 γ . The largeelectron-positron pair background coming from the SM photon overwhelms the signal. The SLAC experiment requires C Y < . − .
2. Electric Dipole Moment of neutrons
Since we can introduce a CP-odd kinetic mixing between the non-abelian fields strength andphoton (e.g., ˜ C Y φ † τ a φW (cid:48) µνa F µν ), we get a constraint on ˜ C Y from the contribution of this scenario n neutron EDM. The leading contribution is shown in Fig. 5: Doing the calculation, we get g W' n fc n h h i Y C~ Figure 5. The new contribution to the Electric Dipole Moment of neutron. Since this diagram issuppressed by η , its contribution is very small. d new n (cid:39) ˜ C Y i v φ η π log Λ m W (cid:48) (24)where Λ ∼ m Φ m Φ . Measurements exclude any contribution to the electric dipole moment thatexceeds d new n < − e.cm . Given the value of m W (cid:48) in our benchmarks, EDM measurements requireus to ˜ C Y < GeV − , and this constraint is much weaker than perturbativity. D. Astrophysical bounds
We have already summarized the importance of NS in constraining any model that discussesnon-standard neutron decay. Recall that to evade NS bounds we moved to models with dark vectormediators and we had to fix m W (cid:48) /g D ∼
60 MeV. This constraint is presented in Fig. 6 as shadedgreen.Another astrophysical bound comes from the cooling rate of Supernova1987A (
SN1987A ) [75].Through the mixing with photon, DM can be produced through the implosion of a newly born NS.Since DM does not interact with baryonic matter strongly, it can leaves the supernova, resulting ina faster cooling rate. If DM is produced in appreciable number, then the cooling rate can be fasterthan observed. For
SN1987A , the energy loss per unit mass should be smaller than 10 erg / g / s atthe temperature of the plasma, which equates roughly 10 MeV. The shaded red region in Fig. 6illustrates the constraint coming from SN1987A . IV. CONCLUSION
In this paper, we presented a model that can explain the discrepancy between the total decaywidth of the neutron and its decay width to protons. In the Standard Model (SM), we expectthe branching ratio of n → pe − ¯ v e to be 100%. However, the two bottle experiment and beamexperiment which tried calculating the decay width of the neutron, one by counting the remainingneutron and another by counting the produced protons, show a discrepancy in their results. One if we had not turned off λ φH coupling, we could have an arguably more important contribution to neutronEDM. - - - - - - l og [ C Y ⇥ G e V ]
Supernova
BaBar
Supernova
BaBar
Supernova
BaBar [73] are also shown in Redand Purple, respectively. The rest of the bounds discussed are much weaker than these bounds andare not presented here. The constraints are very sensitive to the value of m W (cid:48) , but the value of m φ does not change the bounds in a visible way. Thereby, we presented the bounds for benchmarks A2 , B1 , and B2 together.potential answer could be that neutrons decay to dark sector (DS) particles with a branchingratio of 1%. The observation of large neutron stars with radius of two solar radius leads us toonly consider DS models with vector mediators to ensure a repulsive interaction between darkmatter (DM) candidates as well as between DM and neutrons. A dark U (1) D gauge has alreadybeen discussed in details and has been shown the resulting free parameter space is very small.The important constraints on this scenario comes from the measurements of Cosmic MicrowaveBackground (CMB) and Big Bang Nucleosynthesis (BBN) which strongly disfavor the existence ofa light degree of freedom in large abundance at late times. To avoid these constraints, we consideredthe production of DS through freeze-in mechanism. Even with freeze-in, however, we showed thatthe region of the parameter space that explains the neutron decay anomaly will necessarily leadto the over production of χ – the fermionic DM candidate in our theory. Thereby, we considered alow T max ( e.g, T max ∼ m χ ). This temperature is valid according to the current constraints on thereheat temperature of the universe.Since χ in this model cannot account for the relic abundance of DM in the early universe, weconsidered a DS with gauged SU (2) D . The extra degrees of freedom in this model can successfullyaccount for the observed relic abundance of DM. Yet, the number of free parameters in this modelis very much like gauged U (1) D , due to the intricate relationship between the particles of DS.One important advantage of DS scenarios that attempt to explain another theoretical or ex-perimental anomaly is that the freedom over the new parameter space becomes much smaller. Inthis paper, we only had a few free parameters we could play with: the kinetic mixing coupling, thedark gauge coupling and m W (cid:48) which could vary over a small region. For m W (cid:48) > m e , satisfyingthe right relic abundance gave the best bound on the kinetic mixing between the two sectors. forlighter W (cid:48) ( m W (cid:48) < m e ), BBN constraints became much more significant. The main constrainton g D is from making sure the self interaction of DM, as well as the interaction between DM and eutrons are repulsive enough that they do not change the equation of state of large neutron starssignificantly. ACKNOWLEDGMENTS
We would like to thank H. Mehrabpour and J. Unwin for numerous useful conversations. FE isalso thankful to CERN theory division and Mainz Cluster of Excellence for their hospitality.
Appendix A: The squared Matrix Elements of the processes that produce W (cid:48) and φ for T < v φ . The exact Matrix Element of the processes presented in Fig. 3 and Fig. 4 are the following: |M| J EM J EM → φW (cid:48) = Q C Y v φ g D v φ + √ s ) m W (cid:48) (4 s + 5 s ( t + u ) − tu ) + 5 s ( s − t ) m W (cid:48) s |M| γ J EM → J EM W (cid:48) = Q C Y v φ m W (cid:48) s (2 s − t + u ) + s (2 s − t − u )) m W (cid:48) ( s − t ) |M| J EM J EM → W (cid:48) + W (cid:48)− = g D Q C Y v φ m W (cid:48) (cid:16) s − m W (cid:48) (cid:17) (cid:16) m SM ( m W (cid:48) s − m W (cid:48) )+ 2 m SM ( s − m W (cid:48) )( s + 4 m W (cid:48) ( s + 4( t + u )) − m W (cid:48) ) − m W (cid:48) − m W (cid:48) ( s − t + u )) − m W (cid:48) ( s + 8 s ( t + u ) + 11 t + 10 tu + 11 u )+4 m W (cid:48) s (3 t + 2 tu + 3 u ) + s ( s − ( t − u ) ) (cid:17) , where the Mandeslestam ( s, t, u ) variables are defined as usual: s = 4( T + m SM ) t = − T + m SM ) + 2 T (cid:113) T + m SM − m W (cid:48) (cos θ ) + m SM + m W (cid:48) u = − T + m SM ) − T (cid:113) T + m SM − m W (cid:48) (cos θ ) + m SM + m W (cid:48) . 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