Dark Matter in the Type Ib Seesaw Model
DDark Matter in the Type Ib Seesaw Model
Marco Chianese,
1, 2, ∗ Bowen Fu, † and Stephen F. King ‡ Dipartimento di Fisica "Ettore Pancini", Università degli studi di Napoli "Federico II",Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy INFN - Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy Department of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, United Kingdom (Dated: February 17, 2021)We consider a minimal type Ib seesaw model where the effective neutrino mass operator involvestwo different Higgs doublets, and the two right-handed neutrinos form a heavy Dirac mass. Wepropose a minimal dark matter extension of this model, in which the Dirac heavy neutrino iscoupled to a dark Dirac fermion and a dark complex scalar field, both charged under a discrete Z symmetry, where the lighter of the two is a dark matter candidate. Focussing on the fermionic darkmatter case, we explore the parameter space of the seesaw Yukawa couplings, the neutrino portalcouplings and dark scalar to dark fermion mass ratio, where correct dark matter relic abundance canbe produced by the freeze-in mechanism. By considering the mixing between between the standardmodel neutrinos and the heavy neutrino, we build a connection between the dark matter productionand current laboratory experiments ranging from collider to lepton flavour violating experiments.For a GeV mass heavy neutrino, the parameters related to dark matter production are constrainedby the experimental results directly and can be further tested by future experiments such as SHiP. CONTENTS
I. Introduction 1II. Minimal type Ib seesaw model with dark matter 3A. µ → γe I. INTRODUCTION
The masses of neutrinos and their mixing, evidenced by the neutrino oscillation experiments [1], is one of the openquestions in particle physics and indicates the existence new physics beyond the Standard Model (SM). Theoristshave developed multiple theories to explain the origin of the neutrino masses, most of which are different realisationsof the dimension-five Weinberg operator [2]. Typical tree-level realisations of the Weinberg operator include the typeI [3–6], II [7–12] and III [13–16] seesaw models. However, large seesaw coupling and small right-handed (RH) neutrinomass cannot be simultaneously achieved in the traditional seesaw models, which makes them hard to test. For thisreason, much attention has been focussed also on low scale seesaw models such as the inverse seesaw model [17] or thelinear seesaw model [18, 19], where both sorts of model are based on extensions of the right-handed neutrino sector.Loop models of neutrino mass provide further low energy alternatives [20]. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] F e b Recently a new version of the type I seesaw mechanism has been proposed called the type Ib seesaw mechanism [21]which may be just as testable as the low scale seesaw models above while allowing just two right-handed neutrinos [22].It had been pointed out a long time ago that the traditional Weinberg operator is not the only pathway to neutrinomass in models with multiple Higgs doublets [23]. The two Higgs doublet models (2HDMs) have been classified bymany authors [24–26], and the type Ib seesaw model [21] is based on the so called type II 2HDM in which one Higgsdoublet couples to down type quarks and charged leptons, while the other couples to up type quarks. Usually whenthe type I seesaw mechanism is combined with the type II 2HDM, the Weinberg operator [2] would involve only theHiggs doublets that couple to up type quarks. The novel feature of the type Ib seesaw mechanism is that the effectiveneutrino mass operator requires both types of Higgs doublets, which couple to up and down type quarks, while thetwo right-handed neutrinos form a single Dirac mass in the minimal case [21], as shown in Fig.1. In contrast to thetraditional type I seesaw models, the type Ib seesaw model allows large seesaw couplings and relatively small valueof the heavy neutrino mass simultaneously and thus is more testable than the traditional type I seesaw model, whichwe shall refer to as type Ia to distinguish it from type Ib. In this case the type Ib seesaw model [21] shares manyof the general features of testability as the inverse seesaw or linear seesaw models mentioned above, however it isdistinguished by the simplicity of the two right-handed neutrino sector [22] which form a single heavy Dirac mass inthe minimal case, as mentioned above, rather than relying on extending the right-handed neutrino sector as in otherlow energy seesaw models. This makes the minimal type Ib seesaw model particularly well suited for studying darkmatter produced via a heavy neutrino portal, as we now discuss.Besides neutrino mass and mixing, the existence of dark matter (DM) accounting for about 25% of the energydensity of the universe [27] also provides an important clue of physics beyond the standard model (BSM). Therelation between these intriguing phenomena has been investigated in many works [28–101]. One of the interestingpossibilities is to connect the dark sector and the Standard Model through the RH neutrinos that realise the type Iseesaw, which is usually named the neutrino portal scenario [102–112]. Some recent research [102–104] shows thatdark matter particles can be dominantly produced through the neutrino Yukawa interactions in the seesaw sectornon-thermally by the so-called "freeze-in" mechanism [113, 114]. In those studies, the classical type I seesaw modelis adopted and the right-handed neutrinos are superheavy in order to realise the leptogenesis [115], which makes sucha model hard to be constrained and tested by the relevant experiments [116–123]. Moreover, within the frameworkof the traditional type I seesaw model, even if the leptogenesis is not considered, the right-handed neutrinos are stillrequired to be superheavy, otherwise the seesaw Yukawa coupling is too small to play a non-negligible role in darkmatter production and the connection between neutrino physics and dark matter is lost. This motivates studies ofDM in the type Ib seesaw model where the minimal 2RHN sector allows the simplest possible portal couplings, sincethis case has not been considered so far in the literature.In this paper, we consider a minimal version of the type Ib seesaw model with 2RHNs which form a single heavyDirac neutrino, within a type II 2HDM, where all fields transform under a Z symmetry in such as way as to requiretwo different Higgs doublets in the seesaw mechanism as shown in Fig.1. We then discuss a simple dark matterextension of this model, in which the Dirac heavy neutrino is coupled to a dark Dirac fermion and a dark complexscalar field, both odd under a discrete Z symmetry, where the lighter of the two is a dark matter candidate. Toreduce the number of free parameters, we derive analytical formulae which show that the dark matter productiondoes not depend on the individual mass of the dark scalar or dark fermion, but depends on the ratio of them, whichagrees with the numerical result in previous works [102, 103]. Focussing on the fermionic dark matter case, andconsidering the freeze-in production of dark matter, we investigate the parameter space of type Ib seesaw Yukawacouplings, neutrino portal couplings and the ratio of dark particle masses which give the correct dark matter relicabundance. By considering the mixing between between the standard model neutrinos and the heavy neutrino, webuild a connection between the dark matter production and current laboratory experiments ranging from colliderto lepton flavour violating experiments. For a GeV scale heavy neutrino, the parameters related to dark matterproduction are constrained by the experimental results directly and can be further tested by future experiments suchas SHiP.The paper is organised as follows. In Sec.II, we briefly introduce the model studied in this paper and discuss itsproperty and possible experimental constraints. In Sec.III, we derive the Boltzmann equations and provides someanalytical solutions. In Sec.IV, we present the numerical results from dark matter production and compare them tothe existing experimental constraints and future experimental sensitivities. Finally, we summarise and conclude inSec.V In the original type Ib seesaw model [21], a U (1) (cid:48) symmetry controlled the neutrino sector, and with this symmetry all Yukawa couplingswere forbidden. However, in the present proposal, the U (1) (cid:48) is replaced by a Z symmetry, and the standard model fermions transformunder Z in such as way as to allow their renormalisable couplings to the two Higgs doublets. FIG. 1. The type Ib seesaw mechanism involves two different Higgs doublets Φ and Φ . The minimal model involves tworight-handed neutrinos N R1 and N R2 which form a Dirac mass M N . Q α u Rβ d Rβ L α e Rβ Φ Φ N R1 N R2 φ χ L,R SU (2) L U (1) Y
16 23 − − − − − Z ω ω ω ω ω ω ω ω ω Z + + + + + + + + + − − TABLE I. Irreducible representations of the fields of the model under the electroweak SU (2) L × U (1) Y gauge symmetry, thediscrete Z symmetry (where we write ω = e i π/ ) and the unbroken Z dark symmetry. The fields Q α , L α are left-handed SMdoublets while u Rβ , d Rβ , e Rβ are right-handed SM singlets where α, β = 1 , , label the three families of quarks and leptons.The fields N R1 , are the two right-handed neutrinos, while φ and χ L,R are a dark complex scalar and dark Dirac fermion,respectively.
II. MINIMAL TYPE IB SEESAW MODEL WITH DARK MATTER
Here we introduce the minimal version of the type Ib seesaw model with 2RHNs [21], where all fields transformunder a Z symmetry in such as way as to require two different Higgs doublets in the seesaw mechanism, as shown inFig.1. The two right-handed neutrinos N R1 and N R2 form a single heavy Dirac neutrino N . We also consider a darkmatter extension of this model to include a Z -odd dark sector containing a singlet Dirac fermion χ and a singletcomplex scalar φ . The fields of the model are summarized in Tab. I. The Z symmetry ensures that the couplingbetween the Higgs doublets and SM fermions follows the type II 2HDM pattern: The masses of the charged leptonsand − / charged quarks are generated by the spontaneous symmetry breaking (SSB) of the first Higgs doublet Φ ,while the / charged quarks gain masses from Φ . The full Lagrangian can be separated into parts as L = L + L seesawIb + L DS + L N R portal . (1)The first term is the Lagrangian of the 2HDM which includes couplings between charged fermions and Higgs doublets L ⊃ − Y uαβ Q α Φ u Rβ − Y dαβ Q α ˜Φ d Rβ − Y eαβ L α ˜Φ e Rβ + h . c . (2)The remaining terms are the type Ib seesaw Lagrangian, dark sector (DS) Lagrangian and the neutrino portal, whichtake the form L seesawIb = − Y α L α Φ N R − Y α L α Φ N R − M N N cR N R + h . c . , (3) L DS = χ (cid:0) i /∂ − m χ (cid:1) χ + | ∂ µ φ | − m φ | φ | + V ( φ ) , (4) L N R portal = y φ χ R N cR + y φ χ L N R + h . c . , (5)The two “right-handed” Weyl neutrinos can actually form a four component Dirac spinor N = ( N cR , N R ) with aDirac mass M N . Moreover, if the two portal couplings are equal y = y = y , the neutrino portal has a CP invariantform after N is introduced. The type Ib seesaw Lagrangian and the neutrino portal can be rewrite as L seesawIb = − Y ∗ α L cα Φ ∗ N L − Y α L α Φ N R − M N N L N R + h . c . , (6) L ParityN R portal = yφ χ N + h . c . , (7)where χ = ( χ L , χ R ) . Eq.(4) defines the kinetic and mass terms of the dark particles as well as a general potential ofthe dark scalar φ . The vacuum is required to appear at zero in the potential so that the Z symmetry is preserved.The neutrino portal in Eq.(7) includes a Yukawa-like coupling between the heavy neutrino in the visible sector anddark particles. Although the couplings between the Higgs doublets and the dark scalar is also not forbidden by thediscrete symmetries, we assume those couplings are negligible and in this study and we focus on the neutrino portal.In the type Ib seesaw Lagrangian, the heavy neutrino can be integrated out to generate a set of effective oper-ators [124], which leads an effective field theory for the low energy phenomenology. The dimension five effectiveoperators are Weinberg-type operators involving two different Higgs doublets [21] δ L d =5 = c d =5 αβ (cid:16) ( L cα Φ ∗ )(Φ † L β ) + ( L cβ Φ ∗ )(Φ † L α ) (cid:17) + (cid:0) c d =5 αβ (cid:1) ∗ (cid:0)(cid:0) L β Φ (cid:1) (cid:0) Φ T L cα (cid:1) + (cid:0) L α Φ (cid:1) (cid:0) Φ T L cβ (cid:1)(cid:1) . (8)Different from the type Ia seesaw model, the standard Weinberg operator with two Φ or two Φ is forbidden bythe Z symmetry and that only the new Weinberg-type operator that mixes the two Higgs doublets is allowed inthe model. When the Higgs doublets develop VEVs as (cid:104) Φ i (cid:105) = (cid:0) v i / √ , (cid:1) , the new Weinberg-type operator inducesMajorana mass terms m αβ ν α ν β for the light SM neutrinos, where m αβ = v v c d =5 αβ = v v M N (cid:0) Y ∗ α Y ∗ β + Y ∗ β Y ∗ α (cid:1) , (9)The smallness of the light neutrino masses may stem not only from the suppression of M N , but also from the Yukawacouplings. Since there are two different Yukawa couplings, one of them can be sizeable if the other one is small enough,allowing a low scale seesaw model. This enables the seesaw mechanism to play a role in dark matter even for GeVmass heavy neutrinos.It has been shown that, similar to the simplest minimal flavor violating type-I seesaw model [125], the Yukawacouplings Y α , Y α in type Ib seesaw model can be determined by the elements of the PMNS mixing matrix U PMNS the two mass squared splittings ∆ m sol and ∆ m atm , up to overall factors Y and Y [21]. In this minimal scenario, onlytwo neutrinos get masses and the lightest neutrino remains massless. On the other hand, considering the hierarchy ofthe neutrinos is undetermined, there are two distinguishable possibilities for the Yukawa couplings. For the case of anormal hierarchy (NH), the Yukawa couplings in the flavour basis, where the charged lepton mass matrix is diagonal,read Y α = Y √ (cid:16)(cid:112) ρ ( U PMNS ) α − (cid:112) − ρ ( U PMNS ) α (cid:17) , (10) Y α = Y √ (cid:16)(cid:112) ρ ( U PMNS ) α + (cid:112) − ρ ( U PMNS ) α (cid:17) , (11)where Y , Y are real numbers and ρ = ( √ r − √ r ) / ( √ r + √ r ) with r ≡ | ∆ m | / | ∆ m | . The neutrino massesin the NH are m = 0 , | m | = Y Y v v M N (1 − ρ ) , | m | = Y Y v v M N (1 + ρ ) . (12)For an inverted hierarchy (IH), the Yukawa couplings in the flavour basis are given by Y α = Y √ (cid:16)(cid:112) ρ ( U PMNS ) α − (cid:112) − ρ ( U PMNS ) α (cid:17) , (13) Y α = Y √ (cid:16)(cid:112) ρ ( U PMNS ) α + (cid:112) − ρ ( U PMNS ) α (cid:17) , (14)where ρ = ( √ r − / ( √ r + 1) with r ≡ | ∆ m | / | ∆ m | . The neutrino masses in the IH are m = 0 , | m | = Y Y v v M N (1 − ρ ) , | m | = Y Y v v M N (1 + ρ ) . (15)Since only the overall factors Y and Y are unfixed, we refer to Y and Y as Yukawa couplings in the rest of thepaper. With the central values of oscillation parameters [126] and setting δ CP = π , a combined value is fixed as Y Y v v M N = (cid:40) . × − GeV for NH . × − GeV for IH . (16) The Higgs portal couplings and constraints are discussed in the Appendix.
In 2HDM, it is common to define the ratio of Higgs VEVs as tan β = v /v , where v and v are the VEVs of Φ and Φ respectively. Assuming there is no complex relative phase between the VEVs, the Higgs VEVs follow the relation (cid:112) v + v = v = 246 GeV and Eq.(16) can be simplified Y Y sin 2 βM N = (cid:40) . × − GeV − for NH . × − GeV − for IH . (17)In summary, there are four free parameters constrained by one relation Eq.(17) in the minimal type Ib seesaw model.As will be shown later, the quantity that matters in dark matter production is the sum of the squared Yukawa couplingsinstead of their product. Therefore it is useful to derive a lower limit to the sum of squared Yukawa couplings fromEq.(17) using the inequality of arithmetic and geometric means (AM–GM inequality) Y + Y (cid:38) . × − M N GeV β for NH . × − M N GeV β for IH . (18)For the simplicity, we focus on the normal hierarchy of neutrino mass ordering from now on and discuss the experi-mental constraints on the model. A. µ → γe The first experimental constraint on type Ib seesaw model is from the µ → γe decay. In the framework of type Ibseesaw model, the total × mass matrix of neutrinos is given in the flavour basis, where the charged lepton massmatrix is diagonal, by [21] ν e ν µ ν τ N L N cR M ν = ν ce ν cµ ν cτ N cL N R Y ∗ v √ Y ∗ v √
20 0 0 Y ∗ v √ Y ∗ v √
20 0 0 Y ∗ v √ Y ∗ v √ Y ∗ v √ Y ∗ v √ Y ∗ v √ M N Y ∗ v √ Y ∗ v √ Y ∗ v √ M N ≡ (cid:18) m TD m D M (cid:19) . (19) M ν is a symmetric complex matrix and thus can be diagonalised by a unitary transformation U of the form U T M ν U .In the flavour basis, the × unitary matrix U takes the approximate expression [21] U (cid:39) I × − ΘΘ † − Θ † I × − Θ † Θ2 (cid:18) U PMNS I × (cid:19) (20)where Θ = m † D M − is a × matrix in this model, while and I × , I × are unit matrices of the specified dimension.The eµ element of the hermitian matrix η defined by η = ΘΘ † / is constrained by µ → γe through the neutrinomixing [21] | η eµ | = (cid:12)(cid:12) Y e Y ∗ µ v + Y e Y ∗ µ v (cid:12)(cid:12) M N (cid:46) . × − . (21)Besides Y and Y , it is clear from Eq.(10) and Eq.(11) that the mixing between ν e and ν µ also depends on theunconstrained relative Majorana phase δ M in the PMNS mixing matrix. B. Neutrino mixing
For sub-TeV heavy neutrino masses M N , the mixing between the SM neutrinos and the heavy neutrino is alsoconstrained by existing collider data [116–118] as well as future experiments [119–121] like the SHiP experiment [122]and FCC- ee [123]. The strength of the mixing between SM neutrinos and the heavy neutrino is represented by thequantity U α = (cid:88) i = L,R | U αi | , α = e, µ, τ, (22)where U is the × unitary matrix in Eq.(20) and in the above expression we have summed over the two heavyneutrino indices N cL , N R for each light neutrino flavour ν e , ν µ , ν τ . More specifically, for the SM neutrinos ν e , ν µ , ν τ inthe flavour basis Eqs. 10, 11, 19, 20, 22 give for central values of oscillation parameters [126] U e = (0 .
031 + 0 .
029 cos δ M ) v Y + (0 . − .
029 cos δ M ) v Y M N , (23a) U µ = (0 . − .
16 cos δ M ) v Y + (0 .
27 + 0 .
16 cos δ M ) v Y M N , (23b) U τ = (0 .
20 + 0 .
13 cos δ M ) v Y + (0 . − .
13 cos δ M ) v Y M N , (23c)where δ M is the unmeasured relative Majorana phase. If only one of the seesaw Yukawa couplings dominates, thequantity U α is proportional to v i Y i /M N , where i = 1 , depending on which Yukawa coupling is dominating. UsingEq.(16), the dependence on one of the Yukawa couplings can be removed and lower limits of U α can be obtained. Forexample, removing Y leads to simplification of Eq.(23) as U e = (3 . × − ) (0 .
031 + 0 .
029 cos δ M ) (cid:18) GeV v Y (cid:19) + (0 . − .
029 cos δ M ) (cid:18) v Y M N (cid:19) ≥ . × − (cid:112) . − . δ M GeV M N , (24a) U µ = (3 . × − ) (0 . − .
16 cos δ M ) (cid:18) GeV v Y (cid:19) + (0 .
27 + 0 .
16 cos δ M ) (cid:18) v Y M N (cid:19) ≥ . × − (cid:112) . − . δ M GeV M N , (24b) U τ = (3 . × − ) (0 .
20 + 0 .
13 cos δ M ) (cid:18) GeV v Y (cid:19) + (0 . − .
13 cos δ M ) (cid:18) v Y M N (cid:19) ≥ . × − (cid:112) . − . δ M GeV M N , (24c)where the inequalities are the application of the AM–GM inequality. It can be deduced from Eq.(24) that the lowestallowed value of U α is achieved when cos δ M = 1 . For each neutrino flavour, the minimum is achieved when Y = 1 . × − (cid:112) M N /v or Y = 2 . × − (cid:112) M N /v for e neutrino, (25a) Y = 3 . × − (cid:112) M N /v or Y = 7 . × − (cid:112) M N /v for µ neutrino, (25b) Y = 8 . × − (cid:112) M N /v or Y = 3 . × − (cid:112) M N /v for τ neutrino, (25c)for the cases v Y (cid:28) v Y and v Y (cid:29) v Y , respectively. The derivation of such a lower limit of U α relies onthe particular relation of the seesaw couplings Eq.(16) in the framework of type Ib seesaw model and thus it isdistinguishable from the minimum U α required in other types of seesaw models. III. DARK MATTER PRODUCTION
In principle, both the dark scalar and dark fermion can be dark matter candidate, depending on their masses. Forsimplicity, we focus on the mass case where the dark scalar is heavier than the dark fermion and we require the dark N φ ∗ χν α , (cid:96) + α φ , φ − yY α N φ ∗ χν α , (cid:96) − α φ , φ +2 yY ∗ α (a) Neutrino Yukawa processes χ φφ ∗ NN yy φ χχ NN yy (b) Dark sector processes independent of neutrino Yukawa couplings FIG. 2. Processes responsible for the dark matter production considered in this study. scalar is heavy enough to decay into the dark fermion and heavy neutrinos, i.e. m φ > m χ + M N , to keep a single DMscenario. In general, both the freeze-out and freeze-in mechanism can produced the correct dark matter relic density.In this work, we focus on the freeze-in and assume neglectable comoving number density of dark particles at the endof reheating.The Feynman diagrams for processes that are relevant to dark matter production are shown in Fig.2. There aretwo classes of processes named as neutrino Yukawa processes and dark sector processes, respectively. The neutrinoYukawa processes are the scattering between SM particles into one dark scalar and one dark fermion, mediated bythe heavy neutrino, while the dark sector processes are the scattering of two heavy neutrinos into two dark scalars ortwo dark fermions.The evolution of the dark particle number density follows the Boltzmann equation. Here we use a variation of theBoltzmann equation which shows the evolution of yield Y as a function of the photon temperature T . The yield Y isdefined as the ratio of the number density and the entropy density, Y ≡ n/ s . The Boltzmann equations for the darkparticles are given by H T (cid:18) T g s ∗ ( T ) dg s ∗ dT (cid:19) − dY φ dT = − s (cid:104) σ v (cid:105) DS φφ (cid:16) Y eq φ (cid:17) − s (cid:104) σ v (cid:105) ν − Yukawa χφ Y eq φ Y eq χ + (cid:104) Γ φ (cid:105) (cid:32) Y φ − Y eq φ Y eq χ Y χ (cid:33) , (26) H T (cid:18) T g s ∗ ( T ) dg s ∗ dT (cid:19) − dY χ dT = − s (cid:104) σ v (cid:105) DS χχ (cid:0) Y eq χ (cid:1) − s (cid:104) σ v (cid:105) ν − Yukawa χφ Y eq φ Y eq χ − (cid:104) Γ φ (cid:105) (cid:32) Y φ − Y eq φ Y eq χ Y χ (cid:33) , (27)where (cid:104) σ v (cid:105) is the thermal averaged cross section and (cid:104) Γ (cid:105) is the thermal averaged decay rate. The superscripts“DS” and “ ν -Yukawa” refer to the total contributions from the dark sector processes and the neutrino Yukawa ones,respectively.The heavy neutrino N is assumed to be in thermal equilibrium. The Boltzmann equation for total darkmatter yield is obtained by adding Eq.(26) and Eq.(27) together H T (cid:18) T g s ∗ ( T ) dg s ∗ dT (cid:19) − dY DM dT = − s (cid:104) σ v (cid:105) DS φφ (cid:16) Y eq φ (cid:17) − s (cid:104) σ v (cid:105) DS χχ (cid:0) Y eq χ (cid:1) − s (cid:104) σ v (cid:105) ν − Yukawa χφ Y eq φ Y eq χ . (28)The contribution from dark scalar decay is cancelled as it does not change the number of dark matter particles. Thefinal yield should meet the yield of dark matter today, which can be calculated with the observed relic abundance Ω DM h , entropy density s and critical density ρ crit /h Y DM , = Ω DM h ρ crit /h s m χ . (29)The observed relic abundance is provided by the Planck Collaboration at 68% C.L. [27]: Ω obsDM h = 0 . ± . . (30)In general, the Boltzmann equation can be solved numerically. However, we would like to derive some analyticalresults which can confirm and be confirmed by the numerical results later. In the limit m φ (cid:29) M N , m χ , the scatteringamplitude for the contribution from ν -Yukawa processes is (cid:88) internal d.o.f. (cid:90) |M| ν − Yukawa d Ω = 2 πy (cid:0) Y + Y (cid:1) (cid:32) − m φ s (cid:33) . (31)Then, with the approximation of Maxwell-Boltzmann distribution for all the particles, the thermal averaged crosssection is (cid:104) σ v (cid:105) ν − Yukawa χφ = g χ g φ n eq χ n eq φ y (cid:0) Y + Y (cid:1) T π (cid:90) ∞ m φ ds (cid:113) s − m φ K (cid:0) √ s/T (cid:1) (cid:32) − m φ s (cid:33) (32) = g χ g φ n eq χ n eq φ y (cid:0) Y + Y (cid:1) T π (cid:90) ∞ ( m φ /T ) ds (cid:48) (cid:113) s (cid:48) − ( m φ /T ) K (cid:16) √ s (cid:48) (cid:17) (cid:18) − ( m φ /T ) s (cid:48) (cid:19) , (33)where the second step is rescaling the variable from s → T s (cid:48) . The new variable is s (cid:48) is dimensionless as well as theintegral. Therefore the integral only depends on the dimensionless quantity m φ /T and can be replaced by a function F ( T /m φ ) defined as the antiderivative of the integrand (cid:104) σ v (cid:105) ν − Yukawa χφ = g χ g φ n eq χ n eq φ y (cid:0) Y + Y (cid:1) T π F ( T /m φ ) . (34)Then the yield of DM through ν -Yukawa processes is Y ν − YukawaDM = y (cid:0) Y + Y (cid:1) π (cid:90) T RH dT T H s F ( T /m φ ) = y (cid:0) Y + Y (cid:1) π m φ C (cid:90) T RH /m φ dT (cid:48) F ( T (cid:48) ) T (cid:48) (35)where C = T / ( H s ) (cid:39) . × GeV is a constant. In the second step, the variable T is rescale as T → m φ T (cid:48) .Again, T (cid:48) is dimensionless and the integral only depends on its upper limit T RH /m φ . If the reheating temperature ishigh enough ( T RH (cid:29) m φ ), the integral above is not sensitive to the upper limit and behaves like a constant. Noticethat the change in relativistic degrees of freedom caused by the decouple of dark particles is neglected since it onlycontributes few percents to the final results. After all these approximations are made, the yield of dark matter through ν -Yukawa processes is derived as Y ν − YukawaDM (cid:39) y (cid:0) Y + Y (cid:1) π m φ C × . . (36)The yield contributed by the ν -Yukawa processes depends on both the neutrino portal coupling y and the sum ofthe squared seesaw couplings. If the ν -Yukawa processes are dominating the DM production, the required relationbetween the seesaw Yukawa couplings and the neutrino portal coupling can be estimated by requiring the yield inEq.(36) equals the yield in Eq.(29). The result is y (cid:0) Y + Y (cid:1) (cid:39) . × − m φ m χ , (37)from which it can be inferred that the relation between the couplings only depends on the ratio of dark particlemasses. Although the dark matter production through the ν -Yukawa processes depends on the seesaw couplings, itis not affected by the VEVs of the 2HDM directly as in the cases of muon decay and neutrino mixing. The onlyinfluence from the 2HDM parameters is the minimum value of Y + Y in Eq.(18).For the dark sector processes, similar treatment can be applied and the yields follow Y φφ DM (cid:39) y π m φ C × . and Y χχ DM (cid:39) y π m φ C × . . (38)The ratio of yields from φφ and χχ process is around , which can be read from the spin-averaged amplitudes at highenergy limit |M| φφ ∗ → NN (cid:39) |M| χχ → NN (cid:39) y . In total, the contribution from the dark sector processes is Y DSDM (cid:39) y π m φ C × . . (39) As shown in [127], such an approximation may cause up to around 50% difference in the reaction rates for relativistic particles.
When the dark sector coupling is large enough, the dark sector processes can dominate the dark matter productionand the coupling is determined by y (cid:39) . × − m φ m χ . (40)With Eq.(36) and Eq.(39), the dominance of dark matter production can be obtained by evaluating the ratio of theyields r Y ≡ Y DSDM /Y ν − YukawaDM . If y/ (cid:112) Y + Y > . , r Y > and the dark sector processes dominate the dark matterproduction; if y/ (cid:112) Y + Y < . , r Y < and the ν -Yukawa processes dominate the dark matter production. When y/ (cid:112) Y + Y = 0 . , r Y = 1 and the contributions from dark sector and ν -Yukawa processes are equal. In the case r Y = 1 , the seesaw couplings satisfy (cid:0) Y + Y (cid:1) (cid:39) . × − m φ m χ . (41)As dark scalar decay is required for single dark matter scenario, the ratio of dark particles mass always satisfies m φ /m χ > and therefore Y + Y (cid:38) . × − . If only one of the seesaw couplings dominates, the minimum valueof the dominating coupling is around . × − . Notice that the quantity Y + Y is constrained by Eq.(18) and itcan be turned into a constraint on the dark particles mass ratio m φ /m χ m φ m χ (cid:38) . × − (cid:18) M N GeV (cid:19) β . (42)When the right side of Eq.(42) is larger than 1, the ν -Yukawa processes dominate the dark matter production definitelywhen m φ /m χ is below the threshold value. IV. RESULTS
In this section, we show the numerical result for relation between the seesaw couplings and the neutrino portalcoupling, and discuss the constraints from existing and future experiments. We use the open code MicrOmegas [128],with model generated by LanHEP [129], to compute the dark matter relic density. As the first step, we start with therelation between the couplings, which can be compared with the analytical calculations. The numerical results showthat y (cid:0) Y + Y (cid:1) (cid:39) . × − m φ m χ , (43)when the ν -Yukawa processes dominate the dark matter production and y (cid:39) . × − m φ m χ . (44)when the dark sector processes dominate. These numerical results are consistent with the analytical calculation inSec.III, within the errors from the approximations applied.Following a full numerical calculation, which does not use the above approximations, the required portal coupling y for the observed relic abundance is shown in Fig.3. The value of y only depends on the ratio of dark particle masses m φ /m χ and Y + Y if T RH (cid:29) m φ (cid:29) m χ , M N . Therefore the values of y in different panels of Fig.3 are roughlythe same for the same values of m φ /m χ and Y + Y . The red lines represent Y = Y , which depend on M N dueto Eq.(18). Y dominates the neutrino Yukawa production of DM on the left of the red lines while Y dominates onthe right. The figures are symmetric relative to the red lines since the dependence is on Y + Y . The grey areas areexcluded either by the muon decay or by the perturbativity limit of the neutrino portal coupling. The light shadowedareas in Fig.3(a) are constrained by the collider data while it is weaker than the constraint from muon decay inFig.3(b). The future SHiP sensitivity is also marked out in Fig.3(a) as dot-dashed lines.In Fig.3, the yellow lines mark where the ν -Yukawa processes and dark sector processes contribute equally to therelic abundance. The dark sector processes are dominating the dark matter production above the yellow lines while the ν -Yukawa processes are dominating under them. Along the yellow lines, the ratio y/ (cid:112) Y + Y has numerical valuesaround . , agreeing with the analytical result. In Fig.3(a), where the parameter space is highly constrained by thecollider data, the dark sector dominance is favoured: the ν -Yukawa dominance is fully constrained when Y dominatesthe seesaw couplings and only a small parameter space is allowed for ν -Yukawa dominance when Y dominates. InFig.3(c) and Fig.3(d), there are minimum values of m φ /m χ for the dark sector process to dominate. The reason isthat the relation in Eq.(41) cannot be satisfied for small m φ /m χ since the minimum value of Y + Y increases as0 (a) M N = 1 GeV (b) M N = 10 GeV(c) M N = 10 GeV (d) M N = 10 GeV
FIG. 3. Required portal coupling y in order to achieve the correct dark matter relic abundance in the m φ /m χ − Y plane withdifferent heavy neutrino mass. The reheating temperature is set to be T RH = 10 GeV and tan β = 10 to avoid gravitationaleffects [104]. m φ is fixed to be GeV and m χ changes from − GeV. The relation between Y and Y is determined byEq.(17). The neutrino Yukawa process and dark sector process make equal contribution to the relic abundance along the yellowlines. the heavy neutrino becomes massive in Eq.(18). As a result, the dark matter production is definitely dominated by ν -Yukawa processes when m φ /m χ is below the threshold value determined by the heavy neutrino mass. In particular,the dark matter production cannot be dominated by the dark sector process when m φ /m χ is smaller than 28 and . × in Fig.3(c) and Fig.3(d), respectively, and the results agree with Eq.(42).Fig.4 shows the constraints and predicted dark matter dominance. As many experimental constraints are directlylinked to the neutrino mixing, the result is presented in M N - U α plane, where U α are defined by Eq (22) in theframework of Type Ib seesaw model. The black lines show the constraints on U α . Below the solid black line, theparameter space is the excluded by the structure of Yukawa couplings as shown in Eq.(24), which is determined bythe neutrino data, regardless of the Majorana phase δ M . The region above the dotted line is excluded by the muondecay. Although both the muon decay constraint and the expression of U depend on the heavy neutrino mass, thedependence is cancelled since both of them are proportional to v i Y i /M N when only one of the Yukawa couplingsdominates, and therefore muon decay constraint appears to be independent of M N . The shadowed region above thedashed line is excluded by the collider data from multiple experiments [119] and the one below the dash-dotted line is1 FIG. 4. Constraints and predicted dark matter dominance in the M N - U α plane of the minimal type Ib seesaw model in whichthere is only a single heavy Dirac neutrino of mass M N . The correct relic abundance of dark matter can be produced over theentire the white region. The left panels show the regions of definite dark sector dominance (meaning that the neutrino Yukawacouplings definitely do not play a role in dark matter production), which occurs in the white regions below the coloured dashedlines, for different values of tan β . The right panels show analogous regions for different ratio of dark particle masses. The blacklines mark the constraints on the quantity U α in the type Ib model. The red and orange lines stand for the future sensitivityof SHiP [122] and FCC- ee [123]. excluded by Big Bang Nucleosynthesis (BBN) data [130, 131]. Besides the existing constraints, the future experiment2sensitivity of SHiP [122] and FCC- ee [123] are also shown in Fig.4 as the red lines and the orange lines, respectively.The green, blue and purple lines in Fig.4 mark lowest values of U α that the ν -Yukawa process can dominate thedark matter production for different benchmarks, i.e. the dark matter production is definitely dominated by darksector process below these lines for the corresponding benchmarks. Along those lines, the Yukawa couplings are fixedto the threshold values corresponding to the selected dark particle mass ratios on the yellow lines in Fig.3.In the left panels, the dashed coloured lines are obtained in the limit m φ /m χ → which can never been actuallyreached due to the required mass ordering m φ > m χ + M N in the single dark matter scenario. In such a limit,the dominance of dark matter production switches when the dominant coupling is around . × − . The coloursgreen, blue and purple stand for tan β equals 10, 30 and 60, which cover the maximum value of tan β in mostrecent literatures [132–143]. These lines show interesting behaviours as tan β and M N change. To understand theirbehaviours, the first step is to notice that U α are always determined by their values when the dark matter productionis driven by Y . Suppose the value of Y ( Y ) when the dominance of dark matter production switches in the Y ( Y ) dominating region is Y ( Y ). Then Y = Y since the figures in Fig.3 are symmetric relative to the red lines Y = Y . And as shown in Eq. (23), U α are proportional to either v Y or v Y when their ratio is far from one, withthe same coefficients. However, in general, Y dominance in dark matter production does not mean v Y dominancein U α . Indeed, according to Eq. (16), v Y dominance requires Y > . × − GeV M N /v while Y dominancerequires Y > . × − GeV M N / ( v v ) . When M N is larger than ( Y ) v / (3 . × − GeV ) , v Y dominates U α while Y dominates the dark matter production. This situation does not appear when Y dominates the dark matterproduction as tan β > . Therefore the dark sector dominance and U α dominance regarding to the Yukawa couplingshave three scenarios: (1) Y and v Y (2) Y and v Y (3) Y and v Y . Y = Y in scenario (1) and (2) while Y = Y in scenario (3). From the discussion before, it is easy to conclude that the U α in scenario (1) is smaller than the U α in scenario (3). In scenario (2), U α is proportional to v Y which is smaller than v Y , and therefore smaller thanthe value of U α in scenario (3). In summary, U α are always determined by the scenario when Y dominates the darkmatter production.As a result, the dashed coloured lines move downwards as tan β increases when heavy neutrino is light, because U α are proportional to v Y and thus cos β in that region. It can be observed that those lines tends to touch thetype Ib limit as the heavy neutrino mass increases. The reason for such tendency is because the values of the Yukawacouplings when U α is minimised is proportional to √ M N as shown in Eq. (25). After Y become smaller than thevalue for minimum U α , the lines leave the type Ib limit. One may observe, especially in the case of ν e , that some ofthe lines approach the type Ib limit again after leaving it. This is because U α become proportional to v Y ratherthan v Y as the mass of the heavy neutrino grows. In the case of m φ /m χ → , this change happens when M N islarger than . × GeV cos β (around 47 and 12 GeV for tan =
30 and 60). As tan β grows, the region for definitedark sector dominance becomes small, which means the ν -Yukawa dominance is less constrained.In the right panels, the green, blue and purple lines stand for the mass ratio of dark particles 1, and , with tan β = 10 . As U α are proportional to v Y and thus (cid:112) m φ /m χ when Y dominates the dark matter productionaccording to Eq. (41), the lower limit of U α for ν -Yukawa dominance move up as the mass ratio of dark particlesincreases and the intervals between two adjacent lines are roughly two orders of magnitude. In the case M N = 1 GeV,the ν -Yukawa dominance is almost forbidden due to collider constraint in ν µ mixing for m φ /m χ = 10 and totallyexcluded for m φ /m χ = 10 , which is consistent with the result in Fig.3(a). The larger mass ratio m φ /m χ is, theeasier the ν -Yukawa dominance is to be tested by the upcoming SHiP and FCC- ee results. V. CONCLUSION
In this paper, we have proposed a minimal type Ib seesaw model, based on type II 2HDM, where a Z symmetryensures that the effective neutrino mass operator involves two different Higgs doublets, and the two right-handedneutrinos form a single heavy Dirac mass, providing an extremely simple seesaw model with low scale testability.For example the whole of neutrino mass and mixing may be accounted for by a single heavy Dirac neutrino of massaround the GeV scale with large couplings to SM fermions, making it eminently discoverable at colliders and SHiP. Weemphasise that there are no other heavy neutrinos required, since only one Dirac neutrino is needed in the minimalmodel. To explain dark matter, we have discussed a minimal extension of this model, in which the single heavy Diracneutrino of mass M N is coupled to a dark Dirac fermion and a dark complex scalar field, both charged under a discrete Z symmetry, where the lighter of the two, assumed to be the fermion, is a dark matter candidate. It is remarkablethat such a single heavy Dirac neutrino of mass around the GeV scale can not only account for neutrino mass andmixing but can also act as a portal for dark matter.We have studied analytically and numerically the dark matter production in the simple dark matter extension ofthe minimal type Ib seesaw model, with the heavy Dirac neutrino portal to a dark scalar and a dark fermion. Dueto the special structure of the type Ib seesaw model, the parameters in the model are highly constrained by the3oscillation data and the dark matter production has an interesting dependence on the seesaw Yukawa couplings. Ithas been proved analytically and confirmed numerically that the dark matter production only depends on the ratio ofdark particle masses in the case of non-degenerate masses. We have shown the required neutrino portal coupling fordifferent values of seesaw couplings and dark particle mass ratio and highlighted the regions for different productionmechanism. Since the dark matter production has a symmetric dependence on the seesaw couplings, the requiredvalue of portal coupling is symmetric with respect to the two seesaw couplings. Although dark matter productioninvolving the type Ib seesaw Yukawa interaction is favoured when the mass of the heavy neutrino is large, it is stillconstrained for the GeV mass heavy neutrino accessible to low energy experiments.We have presented the regions of parameter space where the dark matter can be produced through the type Ib seesawYukawa interaction with the neutrino mixing characterised by the quantities U α which are relevant for experiment.The regions where the type Ib seesaw Yukawa interaction can affect dark matter production are shown for differentbenchmark values of tan β and dark particle mass ratios. In all allowed regions dark matter may be produced throughthe Dirac neutrino portal. The neutrino mixing is seen to be constrained by the current experimental results and aretestable at future experiments such as SHiP and FCC- ee , especially when tan β is small and the dark particle massratio is large. The discovery of the single heavy Dirac neutrino of the minimal type Ib seesaw model would not onlyunlock the secret of the origin of neutrino mass but could also provide important information on the mechanism ofdark matter production. ACKNOWLEDGMENTS
MC acknowledges partial support from the research grant number 2017W4HA7S “NAT-NET: Neutrino andAstroparticle Theory Network” under the program PRIN 2017 funded by the Italian Ministero dell’Universitàe della Ricerca (MUR) and from the research project TAsP (Theoretical Astroparticle Physics) funded by theIstituto Nazionale di Fisica Nucleare (INFN). BF acknowledges the Chinese Scholarship Council (CSC) GrantNo. 201809210011 under agreements [2018]3101 and [2019]536. SFK acknowledges the STFC Consolidated GrantST/L000296/1 and the European Union’s Horizon 2020 Research and Innovation programme under Marie Sklodowska-Curie grant agreement HIDDeN European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN).
Appendix A: Constraints on the Higgs portal couplings
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