JHEP01(2021)125 GeV for GeV. We ) Springer ) 13 13 10 10 October 25, 2020 January 21, 2021 December 1, 2020 September 2, 2020 − : : : − : 9 10 (10 (10 Revised . Furthermore, these the- Accepted Received Published R v c , Published for SISSA by https://doi.org/10.1007/JHEP01(2021)125 to be in the range R v GeV. The allowed parameter space can be probed 9 gauge interactions produce right-handed neutrinos and Keisuke Harigaya 10 L a,b theories that incorporate the Higgs Parity mechanism, [email protected] − & , B L R − v B U(1) × U(1) R × . 3 R SU(2) Lawrence J. Hall 2007.12711 SU(2) The Authors. a,b Beyond Standard Model, Cosmology of Theories beyond the SM, Higgs × c
The standard model Higgs quartic coupling vanishes at L , [email protected] SU(2) School of Natural Sciences, InstitutePrinceton, New for Jersey Advanced Study, 08540, U.S.A. E-mail: [email protected] Department of Physics, UniversityBerkeley, of California California, 94720, U.S.A. Theoretical Physics Group, Lawrence BerkeleyBerkeley, National California Laboratory, 94720, U.S.A. b c a Open Access Article funded by SCOAP Keywords: Physics, Neutrino Physics ArXiv ePrint: and successful natural leptogenesisfreeze-out, in require remarkable agreementishes, with whereas the freeze-in scale requires whereby the the Higgs warmness quartic of couplingcoupling dark van- matter, by precise future determinations collidersmass of and hierarchy. the lattice top computations, quark mass and and measurement QCD of the neutrino via the freeze-out orwhere freeze-in the mechanisms. lightest right-handedcreates In neutrino the both is baryon cases, asymmetry darkstructed of we matter that the find naturally and universe accounts the the for via thedark parameter decay leptogenesis. lightness matter, and space of A while stability a of maintaining theory the sufficient heavier of right-handed baryon neutrino one flavor asymmetry. is The con- dark matter abundance Abstract: study where this becomes the scale ofories Left-Right solve symmetry the breaking, strong CPcosmologies problem where and predict three right-handed neutrinos. We introduce David Dunsky, Sterile neutrino dark matterLeft-Right and Higgs leptogenesis Parity in JHEP01(2021)125 22 18 8 12 26 12 5 17 7 1 N 32 15 35 29 24 24 19 6 8 – i – 31 22 lifetime 37 1 20 4 N 5 28 3 6 1 3 10 7.2.2 Corrections from charged fermion Yukawa couplings 7.2.1 Corrections from lepton Yukawas 8.3 Natural leptogenesis8.4 for freeze-out cosmology Natural leptogenesis for freeze-in cosmology 8.1 Models for8.2 enhanced asymmetry parameter Radiative corrections: 7.1 The UV7.2 completion: tree-level The UV completion: quantum corrections 5.2 Enhancing the5.3 lepton asymmetry parameter Restriction on neutrino masses in freeze-in cosmology 6.1 Conventional LR6.2 symmetric theories Left-right Higgs Parity 4.1 Relativistic freeze-out4.2 and dilution Freeze-in 5.1 The baryon asymmetry in freeze-out and freeze-in cosmologies 3.1 Neutrino masses 3.2 The lightest right-handed neutrino as dark matter 2.1 Vanishing quartic 2.2 Left-right Higgs2.3 Parity Strong CP2.4 problem Prediction for the Higgs Parity symmetry breaking scale 9 Conclusions and discussion 8 Natural leptogenesis 7 A UV completion yielding a light, long-lived 6 Naturalness and radiative corrections in the effective field theory 5 Leptogenesis from heavy right-handed neutrino decay 3 Right-handed neutrino dark matter 4 Cosmological production of right-handed neutrino dark matter Contents 1 Introduction 2 Higgs Parity JHEP01(2021)125 ] , . ], ]. L i ∼ × v − 2 L , 15 H B 10 = h 1 (1.1) , i , with 9 / NP is part L i 0 Λ 2) 0 U(1) , H SU(2) h H ] dominate h (1 × 0 gauge group R SU(2) H 17 , 16 1) + SU(2) , SU(2) can be realized, and (2 ]; a candidate for the × . This Higgs Parity is H L 19 2) , ]. The , 8 18 , and spontaneously broken , the Higgs quartic coupling SO(10) (1 0 2) SU(2) , H ]. NP ]. We introduce Higgs doublet Λ (1 14 . ↔ R 13 – H at tree-level. Non-zero contributions One possibility is that 1) 11 , 2 ↔ , which is presently highly uncertain ) GeV (2 R = 0 1) 13 v H ¯ θ , 10 (2 L − – 1 – 9 H . This theory has the same number of gauge R v (10 , yielding the SM as the low energy effective theory. ], and may be within the reach of current searches. i ]. 8 0 ∼ 7 – H R ]. Indeed, the Higgs Parity theory we study actually has h 4 v 15 [ , rather than the conventional case of weak triplets and a ecm [ and the Higgs sector is extended to by 0 2) , as reviewed in section 27 , L,R − NP (1 NP . For this theory, precision measurements at future colliders will H Λ 10 R R Λ v SU(2) H × , first introduced in the 1970s [ L 1) + , − B (2 SU(2) , so that the SM Higgs quartic coupling necessarily vanishes at this Left- may be the scale where new symmetries emerge, for example Peccei-Quinn L R ] or supersymmetry [ v GeV, and not the weak scale. In this case, a variety of precision measurements H 3 , is based on the simple extension of the SM electroweak gauge group to ) U(1) NP = 2 Λ 13 × i 10 R R multiplet. Higgs Parity is imposed, − It has been known for many years that spacetime parity can solve the strong CP The most economical version of Higgs Parity, which we study in this paper and review In this paper we study a Higgs Parity extension of the SM [ H h 9 2) , arise at thedipole two-loop moment level of and order areGiven estimated the to simplicity of typicallywhy does generate the the the parity solution solution neutronthe involving literature? of electric an the anomalous The Peccei-Quinn answer strong symmetry may CP [ be problem that proposed it requires in an [ axion [ This will test whether precisionwhether gauge proton coupling decay is unification within in reach of future searchesproblem, [ in particularbroken in solely the by doublets contextone of less the relevant parameter gauge than group the SM, since the QCD coupling and play a key roll in sharpening this prediction for by Right (LR) symmetrycouplings breaking and scale charged fermionthree parameters Yukawa rather couplings than asmass two; of the but the one right-handed SM. Higgs of boson.while these The the Another is determines third Higgs irrelevant the provides as electroweak potential a scale it correlation has only between determines the the Higgs boson mass, the top quark mass, in section SU(2) multiplets, (2 spontaneously broken at Remarkably, in the limit thatis the predicted weak to scale vanish at isof far a below mirror sector, withThis mirror yields matter a heavier highly than predictive ordinary scheme matter for by dark a matter factor composed of mirror electrons [ physics. symmetry [ is extended to a parity interchanging these Higgs multiplets, The discovery at thesuggests Large a Hadron new Collider paradigmStandard of for Model a particle (SM) physics: Higgs(10 boson is the with the mass mass scale scaleat of 125 where GeV colliders, new searches [ the for physics rare Higgs beyond processes, and the quartic cosmological observations coupling could reveal vanishes, this new 1 Introduction JHEP01(2021)125 , , ) ) ]. 1 1 35 1.3 1.1 M M HH [ , the (1.2) (1.3) j terms gauge R ` as the c i v j i ` L keV and N N − i parameter B ) N 300 R − , v U(1) ) 2 1 and × j M ∼ ]. Furthermore, the ` ( . HH i R 1 L ` j 27 ` – i H M . Large values of these ` L in the relative strengths R is possible in this theory, 20 v H SU(2) ]. (+ c 2 has a very small mass j ` N 32 1 i and ` GeV lies inside the range ( N 2 1 above unity requires fine-tuning 2 R ij ± v c M 2 M 10 c and a reduction in the range for for Higgs Parity, does not alter this. 10 operator could also be present, but in + R the Higgs doublet. Alternatively, right- ∼ v j = 1 H R N , v i HH j c j N ` N ) the allowed range of the i i 2 gauge symmetry necessarily contain ` ij – 2 – . In the effective theory below the scale N 2 of this paper, we show that the LR Higgs Parity N i dark matter produced via freeze-in. R ¯ M ` ] and, remarkably, in the case that its abundance 4 ij 1 ) with 2 GeV. Increasing + 35 M N 6 L 1.3 and can be probed using 21 cm cosmology. ). dark matter can be probed by future precision collider + 10 SU(2) H 3 1 1 j × < H 1.1 N M doublets j N L R ] it is taken to be subdominant.) A virtue of adding the i led to a lowering of N v ` R i c 31 ` ij – y ij provides the dark matter, and we investigate the extent to which SU(2) y 28 1 ⊃ SU(2) ⊃ N can be added to the theory together with the two operators LR are the lepton doublets and i L i ]. With an abundance set by freeze-out (and subsequent dilution by the N ` ) for the SM and ( 35 SM+N we show that leptogenesis from the decay of – L 1.2 5 33 GeV. Lowering . As noted above, without interactions for neutrino masses the LR Higgs , the allowed ranges within the triangle were roughly 2 ± dark matter can arise as in [ = 1 10 = 1 1 c N 10 c In section In the LR Higgs Parity theory, neutrino masses are generated by the operators of ( Theories containing The minimal description of neutrino masses is to add the dimension 5 operator ∼ R determined by the Higgs mass. data that tightens the range of ( at the same time that the resulting reduced range for with Parity theory has oneinteractions, fewer ( relevant parameter thanThus the SM; addingis the determined by neutrino freeze-out, mass the required scale With v with no parameter spacein for the theory, butparameters opens were up also regions consistent to with larger values of of these two terms.it If could the be lightestinteractions dark right-handed [ matter, neutrino produceddecay in of a the heavier right-handed earlyspace neutrino, universe was via found to be restricted to a triangle, with a location that depended on generic structure of the operators leading to neutrino masses is Even though there areare three identical, operators, although the there flavor is matrices a for model the dependent coefficient the seesaw mechanism [ right-handed neutrinos is that, iflead they to are the produced cosmological in baryon the asymmetry early via universe, leptogenesis their [ decaysneutral can member of the to the SM, where handed neutrinos involving two flavor flavor matrices. (The axion can be searched for inof many its ways parameter and range. will be Intheory probed sections also in contains the coming a decadeleading dark over to much matter constraints candidate and that tests can on be the produced theory. in the early universe, cosmological dark matter with plausible production mechanisms [ JHEP01(2021)125 0 , . is 1 v 2) 0 , M ≡ GeV (2.1) (2.2) H space (1 2 13 0 / − 1 H 9 and 10 symmetric. . . H SU(4) forms a fun- m/λ 2 v
0 ) , is then very 0 = H partner
i SM 0 SU(4) 2 λ 2 | ], as a model that H H,H , Z h 8 ( H at tree-level, the low because of the small | H 0 0 2 . The top contribution λ H . v √ 4 + / | 0 2 v ) becomes H | gauge interaction with a new 2
= 0 2.1 L symmetry. The dominant effect 0 , and the dark matter mass, λ i H 0 λ
we study radiative corrections to 2 down to R and the SM Higgs is understood as v H 0 6 + h i v 0 2 SU(2) 1 + | SU(4) = H , vacuum alignment in the 0 h H 0 | i λ is exchanged with its λ H 2 charges. The scalar potential of λ h − symmetry under which 1) ) , 2 0 | + – 3 – (2 H . The nearly vanishing quartic coupling can be | 0 2 H
v 0 2 SU(4) the potential in eq. ( 0 SU(2) v H , 1
0 L we introduce UV completions of these operators that . is much larger than the electroweak scale, λ are either determined by symmetry or constrained by + 7 9 i 2 | | m 0 is N ) = λ H | | . The quartic coupling of the Higgs (SU(2) H H 2 ( 0 obtains a large vacuum expectation value 2 0 /v LE m 2 V is obtained only if the quadratic term is small, which requires a H v − 0 , symmetry that exchanges the 2 v 2 ) = m ∼ − 0 Z 0 λ v H,H ( we study the naturalness of leptogenesis in these theories and find highly symmetry is spontaneously broken by V 8 interaction. The SM Higgs field 0 SU(4) At tree-level the potential still leads to With positive In theories of sterile neutrino dark matter, there are naturalness issues for the small a Nambu-Goldstone boson with vanishing potential. quartic coupling. However, foris extremely fixed small by quantumis corrections renormalization which violate group the running from energy scale The hierarchy small value of small at the symmetryunderstood breaking by scale an approximatedamental global representation. For The and Higgs Parity isenergy spontaneously effective broken. potential After of integrating out We assume that the mass scale 2.1 Vanishing quartic Higgs Parity is a SU(2) where the brackets show the 2 Higgs Parity We begin withsimultaneously a predicts brief a review nearlyand of solves vanishing Higgs the Higgs strong Parity, quartic CP first coupling problem. introduced at in a [ scale space for dark matter.greatly In improve the section naturalness ofIn the long-lived, section light right-handed neutrinorestricted dark ranges matter. for the LRConclusions symmetry are breaking drawn scale, in section mass and long lifetime oftheory, the as sterile the neutrino. interactions Thisthe is observed of especially neutrino true masses inthe and the mixings. mass LR and In symmetric section lifetimedimension 5 in operators. the These effective lead theory to where significant naturalness quark constraints and on lepton the masses parameter arise from JHEP01(2021)125 R ' ) with v , the (2.3) , and ( 0 † ) 0 ¯ q SU(2) SM > ] and are , and the λ . None of 0 v, v 2 0 ↔ / λ v 1 14 q , . The gauge , the vacuum ) 9 , 0 . † ) = ( R λ i ∼ ¯ ` 2 i 0 m/λ 0 around the scale / EM R . ,H H 1 H R ↔ ≡ H h R L h − 0 H , ` H v = U(1) symmetry and nearly i ( i 2 L H × and the matter content / h as SU(2) . H c ( , and H h R L ) L 0 × H − SU(4) H . To obtain a viable vacuum, ) . Thus, Yukawa couplings are L B , where ¯ e v ↔ ) L 0 SU(3) are computed in [ L H,H H 2 1 , and hence N, , v ) i ( 2 H . From the perspective of running 1 1 1 / 0 ( U(1) 0 L R Z v (0 SU(2) v ( H ≡ h × and −−−→ × and ¯ ` SM ¯ R R q L Y and λ v − SU(2) H ↔ ) B ¯ 0) d q , the minima are ↔ , 6 1 U(1) , 0 SU(2) 0 ¯ ` u, / v . In this case, quantum corrections must be L ¯ 3 1 × U(1) 2 – 4 – < × 0 ↔ − L = (¯ v 0 ` × L ¯ λ q ) = ( R = i SU(2) 0 2 2 SU(2) H representations of i 0 / h SU(2) 1 . For , has a non-zero but small 1 × SU(2) H minima. By switching-on small negative 3) i c − h × m 0 , ) × H 0 c (1 h + < L ( 6 , v 0
q ` 3 2 2 1 1 2 1 1 1 2 2 1 2 / 2 λ , and we accordingly relabel or SU(3) (0 1 R H . The gauge charges of quarks, leptons, SU(3) i
1) SU(2) R . Threshold corrections to , and L 0 c and L R H is slightly destabilized and we may obtain v − × h (3 0 . −−−→ ) ) B c SU(2) 0 , . The presence of the right-handed neutrinos is now required by the 0) , . , v 0 2) Table 1 1 , v ) 0 , for which the potential has an accidental v , λ > SU(3) 3 SU(2) SU(2) v 0 U(1) − (2 SU(3) , i.e., The symmetry breaking pattern is, 0 symmetry are dominantly broken by ' does not include spacetime parity, (10 1 ) = ( ) = ( ) = (0 0 L . 2 i i i O λ 0 0 0 † Z R H H H H SU(2) . There also is a physically equivalent minimum connected to this by Higgs Parity, h h h 0 , , , v SU(2) In contrast to conventional Left-Right symmetric models, we do not introduce scalar The vacuum alignment can be also understood in the following way. For i i i ↔ If the , and creates the minimum of the potential at 1 ; the Higgs Parity explanation for the vanishing quartic coupling holds only if L H H H 1 . R h h h multiplets in v and H In this work, we considerunder the case where onlygroup the right-handed of (SM) the fermions areis theory charged listed is in table gauge symmetry. Higgs Parity maps ( v ( 2.2 Left-right Higgs Parity we need degenerate vacua with taken into account toquark determine Yukawa coupling. the The minimum. Colemann-Weinberg potential given( The by the dominant top Yukawa makes effect is given by the top vanishes is the scale typically minima of the potentialmass are of Higgses arethe as minima large for as dominates over the gauge contribution0 and generates a positive quarticfrom coupling low to high energy scales, the scale at which the SM Higgs quartic coupling nearly JHEP01(2021)125 , ]) = 2) . , t . In 14 3 (2.4) (2.5) sym- m , 0011) 8 2 . (1) 0 Z O subgroup ± 1) + (1 parameter R R ] and show 2 , M M v v θ Z 41 (2 f ij ij b c 1181 . ≡ ≡ ). In their model, f ij ij , it must be a complex For the top quark, 2.4 0) ) = (0 , 2 . Z 2 ) to be Hermitian and , ., y . y neutral and the 2 m 0) c c . . , ( , 2.4 2 2 S (1 Z , ], which used α + h 2 + h is , . The embedding of the theory † R R ) 40 c , (1 R H H , the running of the Higgs quartic in eq. ( v † 15 L L R ( v v H H f ij j j SU(3) SM ¯ ¯ ` ` y λ i i ` ` boson mass e ij ij Z M c M b since the top Yukawa coupling is + − R † R ], with Higgs Parity arising from a v – 5 – R H 14 † L H for a range of values for the top quark mass , R 8 GeV. H 1 j H ¯ q j ]. Then spacetime parity forbids the QCD i ¯ ` 16) q 8 . i ¯ ` 0 d ij M c ], but can be below the experimental upper bound from the + ], Higgs Parity including space-time parity is spontaneously ± ]. 8 8 [ L + 39 18 – ¯ . θ H R L 36 H H L j ` H i j = (125 ` ¯ q i h q doublets. The origin of the neutrino masses are discussed in section m ij M u ij R c M 2 c is exactly the same as in the SM. We follow the computation in [ unification is achieved in [ GeV, QCD coupling constant at the = = . In the setup of [ SM d R 4) N . ]. The first realistic model was proposed in [ , SU(2) . λ v , we take some of the masses of the Dirac fermions to be small. In this case, the u is absent at both tree-level and at one-loop. Two-loop corrections to the quark ν, , 0 e 7 ¯ θ 37 ± , −L 0 SO(10) . Solving the strong CP problem by restoring space-time parity was first pointed out v −L 36 To obtain the up and down quark masses solely from the exchange of 2 SO(10) and Higgs mass scalar rather than abecause pseudo-real of scalar. the complexsupersymmetry, In vacuum are expectation this imposed value case, [ of the the strong complex CP scalar, unless problem extra cannot symmetries, be such solved as by parity 2.4 Prediction for theBetween Higgs the Parity electroweak scale symmetry and breaking thecoupling Left-Right scale scale the running in the left(173 panel of figure broken without soft breaking and predictsinto vanishing of neutron electric dipole moment. in [ Higgses and Dirac fermionsspace-time to parity generate is the assumed to Yukawa coupling bechy softly in broken eq. in ( the Higgs potential to obtain the hierar- metry includes space-time parity [ at tree-level and requiresthus the enjoy quark real mass eigenvalues.hence matrices The determinant ofmass the matrix quark give mass non-zero matrix is then real and than the 2.3 Strong CP problem Higgs Parity can also solve the strong CP problem if These can arise, e.g.,or from from exchanges of the massive exchangethe Dirac of Dirac fermions a mass (as massive must consideredsection be scalar in around with [ or acorresponding below charge SM right-handed fermions dominantly come from the Dirac fermions rather forbidden at the renormalizable level, and arise from dimension-5 operators, JHEP01(2021)125 σ , 1 2 as a ) = ± (3.1) (2.6) g Z √ R e v M changes with an ( ln S 4 R R g α v v 2 can be as low π 3 R 64 v + 4 0 g 2 − is shown in the right panel g 4 g , R v p j uncertainties, j ln ) Predictions for the scale ν N is not exactly zero because of the σ − 2 R 0 show how the prediction changes with v g ! Right ) 2 ) ∗ v Z + ( ij √ 2 ij deviations about its mean, neutrino mass matrix, e M g y ( 6 2 S q 2 R α ± × ln 6 GeV. With 2 /v ] on the general properties and constraints of v M 2 – 6 – 2 ji v 35 ) y 16) 2 ij . 0 0 g M ± +
]. 2 9 g 18 ( . i . Contours of ) leads to a t 2 N π . Colored contours show how the prediction in m t i 3 2.5 ν m 128 = (125 + h t e y m ], 9 ln 4 t y 2 3 π ) Running of the SM quartic coupling. ( . The thickness of each curve corresponds to the 1-sigma uncertainty in the . 8 scheme is assumed. The prediction for the scale h Left m 0011 as a function of ' − MS . .( ) 0 1 R ± GeV. Future measurements of SM parameters can pin down the scale v ( 9 The value of the SM quartic coupling at the scale 10 SM 1181 λ . 3.1 Neutrino masses The effective Lagrangian of ( 3 Right-handed neutrino darkIn matter this section, weright-handed review neutrino the dark results matter in of LR [ theories. when the QCD0 coupling constant variesmeasured by Higgs mass, as accuracy of a few tens of percent [ where the of figure deviation in threshold correction [ Figure 1 function of the topthe quark uncertainty in mass, the QCD couping constant. The thickness of each countour corresponds to JHEP01(2021)125 ], is 43 ij ]. 2) (3.5) (3.6) (3.2) (3.3) (3.4) c 42 . is open and ! 9 1 ν − θ − 2 e 10 2 . + e × sin . 5 ) → , it may be sufficiently
ss 1 ( ij 5 1 arise from two different N mixing angle is given by (Overproduction) (Decay) m N 1 decays mediated by gauge ν θ − 1 1 exchange when kinematically − M N R (5) i 1 1 keV sin 2 D W m N , × 1 2 ij | ] may probe an order of magnitude δ 8 − 1 . i 1 5 45 y ≡ − − | , . The i v sec 1 ij i δ Σ jk y 30 ). 1 1 i y q 10 M M M 1 k is produced by the Dodelson-Widrow mecha- 3.6 3 keV 3 keV 1 v hadron(s) via 1 × M M = – 7 – N 5 + . v ij ≡ ± 1 ik 9 is diagonal, the lepton flavor mixing arises entirely ` 1 M y − MeV, the tree-level decay θ ij 1 ' − y 10 i ]. This decay rate is given by: 1 is, constraints arise from > θ sin 2 × M 2 43 1 1 5 is similar to ( i 2 2 2 R y v v 1 M ] and e-ASTROGAM [ ' i sin ij y decays into 44 mixing controlled by δ 1 5 exp 1 4 ν 1 = N and may overproduce photons relative to observed diffuse photon v θ M . Without loss of generality, we can work in a basis where ]. For 2 − ij 4 2 1 νγ 46 m π /M N α 2 sin R DM may be overproduced via the Dodelson-Widrow mechanism [ 9 v GeV. The experimental constraints on ≤ 1 ij 8192 Even though there is no symmetry that stabilizes c 2 as the right-handed neutrino responsible for the dark matter (DM) density of N | 3 real and positive. Upon integrating out the three heavy states, we obtain a 174 1 ' 1 i = i y N | νγ ' ij M i is a possible dilution factor after decays via → Σ v 1 M 1 2 N 1 D 2 Regardless of how small N Γ v DM decays into Note that our numbering of SM neutrinos does not necessarily coincide with the neutrino numbering M 3 1 the resultant constraint on exchange. For example, commonly found in the literature. Here nism. The higher photometricscopes sensitivities such of as next ATHENAsmaller generation [ decay x-ray rate and gamma-ray [ tele- These two constraints are summarized by the experimental limit on the mixing angle [ processes: 1) N backgrounds and galaxy fluxes [ where 3.2 The lightest right-handedWe define neutrino as darkthe matter universe. long-lived to be a DM candidate. In this basis, andfrom in the the charged limit lepton that mass matrix. diagonal such that with all mass matrix for the three light neutrinos: where JHEP01(2021)125 , 1 1 1 2 − N N N . See (4.1) (4.2) sec below and/or 1 27 2 keV 1 − exchange N M 10 R ] and shown W 35 have a thermal are frozen-in from i 2 N N and , DM considered in this 1 1 N N DM abundance is produced slightly increase above 1 exchange. If the reheat tem- , RH 1 N 3 T N R / 4 10 MeV W after inflation, as long as inf inf ] for a recent review. RH long-lived so that entropy is produced RH GeV is R T T )). In such a scenario, the required dilution to 2 2 v 43 1 10 abundance is reduced by an appropriate N , M 4.7 N 1 10 300 GeV mixing also generates a radiative decay of N RH T . L – 8 – to have the observed DM abundance requires ]. The parameter region with large 1 2 after inflation, the are also produced by freeze-in, via W 1 `H GeV 43 M M 2 1 − N 8 inf RH N 3 4 R mix with each other by a top-bottom-loop, and M T 10 ] bounds; see [ 10 keV 6 W L . For & 56 W – 4 = 1 ' inf . RH 53 1 T 1 exchange. The s and N N DM can be obtained over a wide range of parameter space. 004 DM exchange. ρ . R Ω 1 Ω 0 R ]. The relic density of R W N W . The experimental upper bounds on these decay rates are about ' ⇒ W . 57 ν , 3 − is overproduced as DM (see eq. ( 34 ` ], which has a stronger experimental upper limit of about 1 + N ` therm 49 , ]. Furthermore, the Y ] and warmness [ 47 eV. Such light sterile neutrino DM, however, is excluded by the Tremaine- [ 48 , 52 1 is excluded by these gauge-induced decays as discussed more in [ ]: – DM is diminished, and hence the warmness constraints on 34 − [ 100 may be DM if their abundance is diluted. If another right-handed neutrino, 1 R 35 50 for the warmness constraints on a pure freeze-in cosmology without any dilution. 1 v N sec At low reheating temperatures by freeze-in via or via the Yukawa couplings with At sufficiently high reheatingabundances temperatures from amount to the DM abundanceupon by decaying. making ' νγ 3 N exchange, and 25 The analysis is this section is also applicable to lower • • 1 4 − N R W realize figure is sufficiently long-lived such thatand it produces comes entropy to when dominate itwarmness the decays, bounds energy it [ density can of dilute the the universe DM abundance and cool the right-handed neutrinos reachthermal thermal yield equilibrium andm subsequently decouple with a Gunn [ The right-handed neutrinos coupleperature to of the the SM universe bath after via inflation is sufficiently high, In these two scenarios, 4.1 Relativistic freeze-out and dilution graphically in figure 4 Cosmological production ofIn right-handed this neutrino section, dark wepaper matter review [ the two production mechanisms of may decay into 10 into due to the emissionsmall of a hard photon [ allowed. In addition, JHEP01(2021)125 . ) ], 3 ¯ d RH /Z u can 61 L T (4.5) (4.3) (4.4) – /M − ]. In 2 ) and is the 2 W 59 v N 64 s 2 33 )[ left ¯ ud, ` can decay y ( + is the decay 2 ]. The light ` − and/or small ( N 3 56 2 2 atm RH , 4 MeV – is below 4 MeV, also decays into → M v T . As discussed in m M ) > 2 53 ¯ 2 ν 2 R ∆ & RH N N √ /v T 2 RH 2 T ν, νν v M = − ` 2 for large = ( + ]. The green-shaded region ). We considered the cases m 1 3 exchange to efficiently dilute 60 N , m 3.3 can beta decay through . ], may be relaxed. R is the energy density, . When `ud, ` 2 59 Pl 2 W 63 i 1 when N y and → N M ]. In our case, , . ) 2 during BBN ( ρ must be long-lived enough. 2 . v 2 1 N 2 2 1 2 N 2 62 ], Γ M . , N MeV [ N ) ,M M N Γ 2 58 q 4 2 R R 2 keV R v to be sufficiently small. 60 v 4 . The other two mass eigenstates are very v M , / > /v 2 ( 1 i 2 2 sol 59 y 2 v during BBN [ m RH ∗ – 9 – m 2 g T ∆ 24 GeV 2 affecting large scale structure [ = ( 10 N ' π √ & 1 2 2 while for 2 N m M from efficiently diluting dark matter, = ∓ M L 1 2 1 is taken from [ W m N exchange into right-handed fermions, . As we will see later, efficient leptogenesis require that RH N ] shows that the lightest neutrino mass eigenstate is closely ± 2 6 T . R 35 is the observed cosmic relic abundance, and 1 M decays too quickly through W 2 25 . N 0 3 νh, νZ, ` ). In the orange shaded region, the required ' ], these decays require M can decay through the couplings → 35 2 and have masses . These decay channels are unavoidable as they are independent of the , as set by its total decay rate DM 2 , and prevent 2 N 3 − is heavy enough, 2 right and Ω is thus fixed as i N ` and has a mass ν ( N 2 y , we show the constraints on 2 + 2 1 ], we used the above results, together with the radiative stability bound on ` N ν 2 N 1 2 sol M 35 and N m & 2 . For this case, ref. [ ∆ ν 2 → 3 energy density. The non-trivial shape of the blue-shaded region is due to the √ 2 M 1 M N The blue line itself is an interesting region of parameter space, which does not require In figure In ref. [ To achieve the dilution of The reheating bound from hadronic decays of N = & , to derive constrains on the neutrino mass matrix of ( . In addition, 3 2 1 R the blue-shaded region, the dependent effective degrees of freedom. any tuning but simply corresponds to the limit where the dominant decay is set entirely m which is excluded by hadronicis decays of excluded due togreen-shaded the region warmness shows of the sensitivity of future observations of 21 cm lines [ aligned with close to The mass of more detail in ref. [ N with M and free-parameter v at tree-level via exchange and active-sterile mixing to SM fermions, trinos decouple and heatreducing up the only effective electrons number andneutrinos of photons, and neutrinos the relatively [ bound cooling from neutrinos the and CMB, always beta decay through requires that Low reheating temperatures can also affect the CMB since some decays occur after neu- entropy density, temperature of where the numerical factor JHEP01(2021)125 2 ). N . We red decay 1 2 M , N exchange, 1 N R Z mass with the at- and 2 ), decaying too early ν R DM from freeze-out in W 1 N orange has a thermal abundance. ), and hot DM bounds ( is in tension. CMB Stage IV 2 , and the mass of R N R v v green ). We fix the nor plane. 1 1 , and the solar neutrino mass difference, N green M left , − R 2 atm v m – 10 – ∆ , the allowed region of abundance has two contributions: from √ )), neither 2 of DM and is hot. The red-shaded region is excluded 1 = DM produced by relativistic freeze-out and dilution from 4.1 N ), warm DM bounds ( 2 1 10% m N exchange. blue ], which parameterize the energy density of this component R -cm cosmology (dashed 65 W [ 21 decaying after Big Bang Nucleosynthesis ( eff as well as the prior thermal abundance from relativistic decoupling. 2 dilution ( ν, − N 1 m ` ] can cover the light red-shaded region and probe the limit where N + . ` 1 abundance is determined by scattering via heavy 67 and , N 1 right 66 , N eff → N 2 sol 2 ∆ m . The parameter space of N exchange. In this limit, the part of the blue line is already excluded, and high ∆ √ R R In sum, as can be seen from figure v decay in terms of the Left-Right symmetry breaking scale, ), and warm DM from = W 2 2 4.2 Freeze-in When the reheatof temperature the of right-handed the neutrinosInstead, universe (see the is ( below the thermalization temperature dominantly decays via the LR theories forms a bounded triangle in the The former contribution makes up by the effect of thelimits hot of component on the CMBof and DM structure when formation, relativistic as andlow set when by it current has becomeexperiments non-relativistic [ matter, respectively. The m by through N show constraints from to provide sufficient In addition wered show prospects ofmospheric improved neutrino searches mass for difference, hot DM from CMB telescopes (dashed Figure 2 JHEP01(2021)125 1 ]. ∓ − π 33 R ± from (4.6) (4.7) ` 1 W i 1 y to N or R R interaction. W DM to arise affects large W 1 1 N N `NH . 3 ! . The horizontal dashed νγ comes from the difference mediated by GeV inf RH 2 1 . 7 T 3 , N contours. Constraints from small / 10 4 , are subdominant since
1 decays too rapidly via pl 4 mixing to red , giving a commensurately stronger 1 N interaction or the 3 ! M L N / R with active neutrinos. 3 1 → W ) 1 W 4 R − N GeV R inf v RH `H DM v R gauge boson exchange is considered in [ region T 10 may be also produced from beta decays of W L s/ρ 1 10 1 1 − shown by the N
M blue M – 11 – B inf RH 5 T . Here, the factor of − 1 . More concretely, the free-streaming length is larger therm DM produced by freeze-in. The observed relic abundance 1 2 Y M 10 1 M . In the N 150 keV × 2)( , with projections from future probes of small scale structure . 1 3 are produced by the plane. In the green region, the warmness of / 3 ) ' ' , is long-lived. green 2 (4 1 show the reheat temperature after inflation for 1 1 ) on the mixing angle of green from freeze-in are not diluted, they are warmer than N 3 s N N DM decays too rapidly via Ω ,M 1 ρ 3.6 Ω 1 R N v N ( ⇒ , whether 1 region ]. In the blue and pink regions, the decay of N 68 . These contributions, however, are always subdominant to the direct freeze-in pink 3 between the non-thermal freeze-in and the thermal freeze-out distributions, as . The parameter space for N i lines show the limit ( The contours of figure p/T and h 2 freeze-out and dilution, for aby fixed a factor of approximately warm DM bound compared toin figure discussed in [ is needed to ensureN that production of from freeze-in, in the scale structure. Since The production of sterile neutrino DMFreeze-in production by from other sources, such as which, being UV-dominated, depends on the reheating temperature after inflation, occurs in the unshaded regionscale for values structure of are shownusing in the 21 cm lineand in in dashed the blue Figure 3 JHEP01(2021)125 1 2 ]. θ N 2 47 2 (5.1) . For 2 sin is small . η ∗ 31 y DM freeze-in and = 1 mixing is set by N 13 ν increases, y ; − 1 1 23 abundance is thermal, M is the efficiency factor, y N 2 η , N or DM is produced by freeze- † L 1 33 H † y N ` makes up DM, the decays of . Also, if the reheat temperature (Freeze-Out + Dilution) (5.2) 2 1 or DM from freeze-in is weakly con- N sufficiently long-lived. In the next N L 1 1 labeling the purple dotted contours. `H , the decay via N N 1 B , is high, R θ is the fraction of decays that occur when v 2 inf f RH 2 T therm ]. sin f B, 69 are weak; although, as ). In the next two sub-sections we discuss the decay is incorporated into the ηY decay is † – 12 – L 1 2 decay into 2 = /s H 1 2 N M † N L ` N M decay. Although the initial DM Y ρ 2 → N DM from freeze-out and dilution, shown in figure 2 is therefore short-lived. 1 N 3 N 28 79 decays yield a baryon asymmetry N 2 , is below the weak scale, the baryon asymmetry is reduced via active-sterile mixing overproduces the observed galactic x- accounts for the conversion of the lepton asymmetry into the = . N 1 1 parameter space becomes more tightly constrained. RH to have a sizable Yukawa coupling ) + Br( 79 N N T ) . L / f B 3 1 R i N 28 y `H , v 2 1 RH M , which differs in the two cosmologies, and the quantities → T M 2 4 ( 3 is reduced by the dilution produced from 2 . Likewise, bounds on -mediated decay, which is fixed by N MD-era, η N R shows that the parameter space for could be as low as about 100 TeV, with the reheat temperature after inflation 2 W 3 R N Br( v 79 28 is the asymmetry created per ≡ 100 GeV = B The lepton asymmetry yield from Figure B -mixing overproduces the observed amount of galactic gamma-rays, respectively [ Y L where the factor of baryon asymmetry via sphaleron processesthe [ temperature of the universe isasymmetry above into the weak a scale baryon where sphalerons asymmetry. convert the The lepton fraction depends on whether the temperature after the because only the lepton numbersphaleron produced processes. above the The weak scale is converted to baryons by 5.1 The baryon asymmetryWhen in the freeze-out reheat and temperatures freeze-in afterout inflation, and cosmologies subsequent dilution from the efficiency where and abundance of can produce a baryonasymmetry asymmetry requires through leptogenesis. Producingdue to a the large longevity enough of lepton 5 Leptogenesis from heavy right-handedIn neutrino both decay the freeze-out and freeze-in cosmologies, where strained compared to thatexample, of below is constrained to becomesection extremely we find small that, to ifcosmology, keep leptogenesis the via Similarly, the decay of rays and gamma-rays forUnlike the the mixing angle the free parameters W JHEP01(2021)125 ]. ' B 71 gray (5.3) , Y 70 and the [ 2 N = 1 are required as f MD NA . RH < v < v < T < v < T is greater than its natural matter-dominated era: that decay at temperatures 2 MD NA RH 2 T T T v < T , the reheat temperature is exchange does not generate N N R R v dark matter. Likewise, larger values W , required to produce the observed , as indicated by the dashed gray 1 v N 2 / 1 ) through MD 2 ). For large N v/T 5.3 ( drops below – 13 – 2 2 4 ) ) ) RH /v /v /v T is below the weak scale, as indicated by the dashed RH RH RH T T T ( ( ( 1 in the freeze-out cosmology. Larger values of RH . As T 1 11 is suppressed. is the temperature at the start of the non-adiabatic matter- required to produce the observed baryon asymmetry, − M ' during a radiation-dominated or 5 ) 10 / plane. The contours zig-zag through the plane due to the era- v v 1 when ) ) × is the temperature at the start of the adiabatic matter-dominated 1 = R 8 , according to eq. ( 4 v RH f T ' T ,M ( t R B therm contours of the asymmetry parameter, 2 v MD Y Y ( N 2 T M = Γ 4 3 = ( always decays before the electroweak phase transition so that . f 2 Purple ∗ = depends solely on NA N , in the . T shows contours of 11 contour, the baryon asymmetry can only be realized when MD falls below unity and 4 − T f In addition, there is no efficiency lost due to cancellations between the lepton asym- are required at low 10 increases due to the greater dilution necessary to realize 1 × required line, metry generated during productionknown with as washout, the since lepton the asymmetry production generated of during decay, era, and dominated era, after which the radiationFigure is dominated by the decay8 products of dependent change in high and Here, maximum, of the universe falls below baryon asymmetry, M of line. In thisabove regime, the the weak baryon scale, asymmetrypurple where is electroweak generated sphalerons only are by operative. To the left of the dot-dashed Figure 4 JHEP01(2021)125 2 2 2 R , 1 N W M N Y , (5.7) (5.8) (5.9) (5.5) (5.6) is the 2 i ]. The . by y R and 2 2 3 2 72 1 W 3 Y M M ] M , is low, = 1 72 i inf y RH is [ T 2 N to be Hermitian, the , which occurs when , is too low to induce a ij y β, x , of 2 decays; this accounts for again for this production ] η (Weak Washout) (Strong Washout) therm exchange is identical, N 2 1 Y 74 1 R sin 2 N become degenerate. , . eV eV + 1 ' 0 α W 2 3 3 32 η 1 x 2 − − M . Furthermore, we introduce an by the Yukawa coupling . is [ via 10 10 iβ (Freeze-In) (5.4) . ) 2 e 2 /s ) sin | 1 < > x and decays out-of-equilibrium. N x ) log ( 23 2 2 ( M therm 3 2 g x DM /M y g e e Y 2 ρ m m N M v 2 and , ) 2 ≡ | ], is given in the limit that = | are disparate, near unity when 1 (1 + : : 2 22 eV i 73 2 N y 23 y 3 − /s 1 π | y 1 − M 8 + reaches – 14 – M N i , and the lepton asymmetry is washed-out until B. 10 ρ ). The efficiency factor, X + 1 33 2 2 16 therm . y Y and for freeze-in is about < 1 x ( ≡ Y M = 5.2 − 3 2 2 1 therm . The function − R ∼ e e m α M m are small, the wash-out effect is negligible. Finally, we 1 W ) = ) to obtain the final result. eV Y ηY x therm T eV 2 2 4 i 2 ( 2 e tan x y g ) and ( m − 4.2 exchange, where we have set e 28 79 ηY m − √ 2 ≡ 10 R | 5.4 = 10 ) ≡ 32 W 22 B 2 23 y ) | y Y 03 /y x . | ( + g 23 from therm Im( y 2 22 + 0 | Y 2 y R N 03 2 W . ) at the transition between the weak and strong washout regimes [ Y 0 22 space contains a single phase y eV into a lepton and an anti-lepton [ π 2 ' 8 + 3 2 − × defined by N 33 2 y 10 α ( therm Conversely, in the limit when the reheat temperature after inflation, ' = ηY 2 e angle and is much less thanare unity comparable, when and much greater than unity as Since the Higgs Parity solutionheavy to the strong CP problem requires the Yukawa interection is out-of-equilibrium,esis. strongly reducing The the maximum efficiencym possible of leptogen- leptogenesis CP asymmetry parameter,ratio defined of by the difference between the branching In the strong washoutis regime, in where thermal equilibrium when In the weak washout regime, when freeze-in yield of mechanism. Since the freeze-in abundanceis of simply where Note that without amatter-dominated thermal era, abundance, so the that freeze-inthe no difference yield entropy between of is eq. produced ( when use the DM abundance from ( abundances are frozen-in and the resultant baryon asymmetry is any lepton asymmetry. Since JHEP01(2021)125 . . ) is 2 ∼ 4 N 5.2 ), is DM R 33 that ) can v 1 (5.10) (5.11) m R 5.4 N v 5.4 becomes . In the ,( B 4 33 Y drops below y , it is apparent RH . We thus focus 4 ), can match the , the asymmetry T , the strong washout 33 is negligible, while 1 y 2 33 ]. In the freeze-in is maximized when 5.2 as y 22 ∼ . 35 ,( commensurately grows to y . 33 [ R 2 y v 2 } eV 2 in figure 3 2 33 N 2 3 v , except in the region not y GeV as shown by the dot- − α, β m {z . We avoid this route. 2 32 3 M 10 B y 12 | 33 , since otherwise Y < 05 eV 10 m . − eV by increasing 0 × 3 2 3 − 3 v . This leads to a maximal natural 2 2 2 33 M 10 ∼ leptogenesis requires an enhancement y /v R 2 Unlike the freeze-out cosmology, R − v may be non-negligible. v to achieve a sufficiently large asymmetry N 5 3 GeV R 33 , it is possible that β v 23 M 10 m 2 m 2 R everywhere in the unshaded region of figure determines the size of the seesaw contribution 10 v v 1 eV) ∼ and . – 15 – 11 33 = (0 − , and for large angles y 33 11 ) 1 , α O y − 2 10 . Hence, except for a very small region near . ss , since ( 33 . For the freeze-out cosmology, v ∼ 10 × 4 π /M m 8 ) 8 3 is sufficient. However, at lower result in baryogenesis. At the lowest values of ∼ leptogenesis in both the freeze-out and freeze-in ) with the contours of required x / − 1 eV) ' ( 5 . 2 2 is subject to the similar constraints as dark matter and v 2 4 R ) g 0 ,M − inf RH v B N 1 (5) 3 5.11 22 , 22 33 T 2 3 Y , as shown by the shaded purple region of figure − 10 π v 3 y N y y m in figure 8 m (0 M + ∼ . Although this appears to enhance ∗ = = 1 ≡ O 33 33 y ∗ , where ( 1 eV) and R m . , the baryon asymmetry in the freeze-out plus dilution cosmology ( R v 2 (0 v , but 3 O M is aligned with the neutrino mass eigenstate m decaying above is taken much greater than ) for successful 2 33 2 . By comparing ( N 5.8 ∗ m remains /M by a slightly higher power, so the net effect is a decrease in mass via 2 dark matter, an insufficient baryon asymmetry is generated even if GeV, simultaneous v B 3 33 in this subsection. The coupling Y ν 1 2 23 12 m y N 33 Avoiding a finely tuned cancellation between the two terms, It is possible to choose above 10 If y 5 × 3 of ensure reduces Using this value for is too small, except fordashed the contour very labeled highest values of not necessarily the two terms arevalue comparable, for giving the asymmetry parameter In the freeze-outMoreover, cosmology the lastcosmology, term we assume is that negligible thestrongly last due suppressed term from to is strong less the washout than effects. long lifetime of for the freeze-in cosmology on to the 5.2 Enhancing the leptonFor asymmetry comparable parameter parameter is of order fraction of give non-perturbative and case of freeze-in cosmology theresuccessfully is yield no the dilution, observed so asymmetryshown that everywhere at the in baryon very figure asymmetry low of ( per decay ( cosmologies. For freeze-out, the baryonobserved asymmetry baryon asymmetry generated by At larger values of the weak scale, larger values are needed, as shown by the purple contours, as only the JHEP01(2021)125 χ ), 2 22 33 by y β. m 2 5.11 . It is (5.15) (5.16) (5.17) (5.12) (5.13) (5.14) /v 1) R sin 2 ) occurs. of ( v . For the α ' ∗ 33 χ 2 x 5.10 m ), the baryon sin and 5.15 fB increases) and χ 33 ) y x near unity and small decay requires either ( (5) 3 . A similar conclusion 2 g x R m , N ) v for any realistic neutrino ), and ( χ β. 1 − . 5.11 2 ) are not independent; they are χ . )(1 ss sin 2 ( 33 4 ) χ ), ( x χ . α χ m 2 2 33 ). ) 5.2 − + 2 χ m (1 + 2 and 5.4 χ then is the same as the sign of are nearly degenerate (i.e. ) sin 2 χ grows (so that (5) 3 )(1 4 x 4 x 3 1 χ . 3 ( m 33 χ (1 + 0 g M χ 2 3 2 2 m M 3 = ) 2 , m m χ ) ) 1 + (1 + m 05 eV 0 . and increases) in the following manner: ss ss 2 4 2 2 R R – 16 – − 0 = ( 33 ( 33 χ v v 2 v v ) is always positive. In terms of → 2 2 2 m m 3 2 ss 33 2 33 M χ 2 ( 33 )(1 χ 4 − + M M (5) 2 χ m m m and hence increasingly degenerate so as to keep their m = (5) 3 (5) 3 = = 33 x 3 = m m GeV R (1 + 2 33 m v 2 y M ∗ 10 ≡ m decays is 10 χ so that a cancellation between the two terms of ( , so that = 2 = 3 , respectively. Combining ( | 3 N and that the sign of 33 M 22 . 1 . That is, as m 1 m χ m 33 M 2 keV freeze-out dark matter and leptogenesis from m may be large when and | 1 5 ) ) − 2 N x small ss ( ( 33 < χ < m g is dominated by the dimension 5 contribution to its mass, or . 1 = 10 2 1 , m − m 11 ' (5) 3 with − x m B 33 10 goes to zero, the fine-tuning between the dimension-five and see-saw masses increases Y m Since We focus on the freeze-out cosmology for the rest of this subsection and identify There are two possibilities for this enhancement. One is to take × χ near unity 8 , are seen to be mutually exclusive: if This is an importantrelated by result the since neutrino spectrum. itχ The shows two that choices forspectrum. enhancing Thus x and asymmetry produced by The observed baryonand/or asymmetry can be explained by the enhancement from small Note that freeze-out cosmology, the lepton asymmetry parameter can be written as difference equal to grows (even faster than As since each becomes larger than applies to leptogenesis with freeze-in dark matter, ( having Alternatively, useful to introduce that the enhancement must be very significant at lower values of JHEP01(2021)125 , . , χ 3 2 2 R are ) re- /v β. 3 (5.21) (5.18) (5.19) (5.20) > M < m 4.5 M 2 , with | 2 resulting 3 3 sin 2 M 3 M m m α | lifetime. m is not neces- 2 positive (and ' 1 , if of eq. ( 2 i 3 β. sin N 2 2 sol 33 23 y decay can occur R 2 atm and y m v , with m m 2 m 2 fB sin 2 ∆ N . 33 ∆ 2 y 4.2 and α x and on the = 2 2 1 > m 2 R − . Enhancements in 3 | 33 . 1 M may possess a substantial sin /v 3 y 4 2 3 m 01 eV (5) 3 ), giving . 22 m | 0 96) fB M or large , so that the observed baryon for the following reasons. For m . , we find the observed baryon m 1 1 2 3 2 , ' 5.17 , 2 2 2 , r M 2 2 33 M ) 015 y , and < m & 3 . m | 3 3 r 3 (5) 2 3 GeV 05 eV) R . /m m v M m m 2 ± | 10 30; 0 (0 , to be larger than in the normal hierarchy. 1 m 3 , 10 ( ) , giving N 2 χ 3 x 20 1 3 3 . ( is fixed from ( N g − (0 – 17 – ∼ χ 1 r, ) ' GeV R . For enhancement by tuning in x M ) − 2 ( v from 2 keV 10 g x ) ( M 10 g 5 ss ; 1 ( 22 = − ) r m 3 χ √ 10 / ) gives M − 1 1 2 × 2 1 is required to be negligible. Then the SM neutrino masses are M DM from freeze-out and leptogenesis from ± χ )(1 2 keV 5.17 2 4 1 1 ' χ , spoiling the direct relationship between N | N − ) 5 3 very close to unity, , need not be long-lived. Consequently, − ss r ( 22 x M 3 (1 + 2 , and the effects of radiative corrections from 10 √ m 2 N 1 − × M 2 − M 75 1 + 2 | . 0 2 ' ' ), and negative. These give values for the enhancement factor of ' 3 χ 1 χ must be also cancelled by − The enhancement of the asymmetry requires To avoid the strong wash-out and maximize the allowed parameter space, the see- We conclude that For the case of a cancellation of large contributions to the neutrino mass For the case of > m DM cosmology puts restrictions on the neutrino mass matrix. x 2 (5) 2 1 saw contribution from determined by the see-saw contribution from enhancement by degeneracy, m In the freeze-in cosmologysarily without small leptogenesis, since discussed incontribution section from quired for the freeze-out cosmology. However,N requiring efficient leptogenesis in the freeze-in from a cancellation between seesawstudy and whether dimension leptogenesis 5 contributions. canenhancement In be for the obtained next naturally, section considering we both the5.3 origin in the Restriction on neutrino masses in freeze-in cosmology simultaneously throughout the largerequired unshaded and region can of arise in figure two ways: near degeneracy of very small, we find thatasymmetry ( requires Using this result, forasymmetry the results normal for hierarchy with m We see that the inverse hierarchy requires where the first tworespectively, while cases the are last for two a cases normal are for hierarchy, the with inverse hierarchy with JHEP01(2021)125 . ) 1 ' in , 4.5 2 33 22 (6.1) (6.2) must must y gauge (5.22) (5.23) 1 m i 33 23 L 01eV y . , y − can result m 3 B , 1 2 = 0 M 2 / /M U(1) 1 1 and hence is also negligible. × M and 3 (5) 3 R 1 1eV) i . m . /M y 2 2 (0 v 2 ! / 2 23 1 SU(2) y transforms. However, since , GeV R 1 22 v 11 N . . A similar problem arises from m ¯ seems to be hard to understand. ` 2 001eV) 10 . / 1 3 to i
(0 . y 3 ) unless it is enhanced by a very high 2 < 1 N 1 6.1 v becomes much larger than the observed M R 2 v ), the smallness of 3 keV M v / doublets 2 R 3 keV 1 2 ), the only available production mechanism 1.2 R /v M 13 2 ) 22 2 6.1 v − / 13 – 18 – m 3 1 R − 10 v ) can be approximated by SU(2) M ' ), 2 10 × 2 3.6 may be produced by the 3 M 4.5 − M 001eV) 1 √ violates ( . . 12 N 1 1 − (0 i i ' , despite the right-handed neutrinos and the right-handed y y eV. 3 is required to be small to avoid strong wash-out, interactions of ( < e jk (10 y is large, requiring the cancellation with 05 2 2 R . via /M ' 23 N v ) v 0 2 y 1 3 v ss , 1 23 ( 33 − 2 N 1 y 33 i M is model-dependent. In LR Higgs Parity, taking the examples of ( m ]. y y M 01 3 ' . , 23 , the see-saw contribution from 75 2 0 2 y 2 not tuned to be small, v | must be as large as the observed neutrino masses. Suppose that it is M 3 ' /M 33 χ 1 | y ) & M (5) 2 ss M 23 3 ( 23 y m M m ' ) must be ) with to be dark matter, whether in the context of (SM+N) or of Left-Right symmetry, ) . However, 1 2 R ss ( 23 5.22 In LR symmetric theories, In (SM+N), with the Since N 5.14 /v m 2 3 lepton asymmetry [ interactions. However, the smallnessWe of need the a coupling hierarchy charged leptons coming from the same Quite generally, light sterile neutrino dark matter has afrom small an numbers problem. approximate globalfreeze-in symmetry production under of whichis only via neutrino oscillations, and this also violates ( The value of or ( small parameters must beFor introduced sufficient to cosmological limit stability, its ( mass and decay rate, which is not largeeq. enough ( to explain the SM neutrino masses.6 We conclude that Naturalness and radiativeFor corrections in the effective field theory M Moreover, negligible. To obtainbe the non-negligible. two Since observedcompensate it. non-zero neutrino Then mass eigenvalues, SM neutrino masses. We obtain a relation similar to eq. ( However, JHEP01(2021)125 ¯ ` 1 1 R × † H M M Φ L ` (6.6) (6.3) (6.4) (6.5) under as U(1) and Φ Y , nor and 1 3 SU(2) × i i 2 y L Φ dark matter , y H U(1) 2 i 1 y × N L U(1) × from ¯ ` 1 i . . The dominant effect SU(2) y ., Φ mass matrix is diagonal, c in the SM Higgs must be . triplet that spontaneously U(3) 1 2 i j u N R − × N + h H , ` u 2 2 H Φ mixing and the latter introduces i . Also, the operators : ` 2 SU(2) d d U(3) d e ij Λ H H from the quark loop, q. y 4 × , and the LR symmetry then implies Φ¯ − − ¯ i q Φ + q ` ,H u to an d j 10 and ¯ e y † H u d 2 1 ∼ U(3) + H , . H ΦΦ i . The observed large neutrino mixing angles ¯ c 2 q × . ` 3 † to be small, in this and the next section we seek , ), the fraction of q 2 e ij Φ 1 : – 19 – y q + h M 6.1 u u M 2 = y U(3) Φ j ¯ . In the basis where the ` = 2 ,H and 1 d Φ ) Λ 1 i L . This can be achieved by a non-zero charge of H i M d ` b 2 † d y y , which can be decomposed under e ij π ∗ y ,H t Φ `H u 16 y decay. The UV completion discussed in the next section will . Then no symmetry can distinguish and i = ¯ H ` 2 3 . To satisfy ( to i 2 L N − , y L ∼ N 2 i containing ∆ 10 Φ = ( y ). This gives significant naturalness constraints on ∼ H 6.2 τ bi-fundamental and point out the difficulty in guaranteeing the stability 1 y ) has a naturalness problem if the cut-off scale of those interactions are i to be stable, the SM Higgs must almost exclusively come from only one of R y , thereby splitting the masses of 1 2.5 R : in certain regions of parameter space, radiative contributions to N ) and ( R . v SU(2) . In fact, the charged lepton Yukawa coupling arises from 3 , d 6.1 bi-fundamental scalar × 2 SU(2) H ) and ( R L M . We then argue why the problem can be avoided in Left-Right Higgs Parity, deferring 1 We must also introduce up and down quark Yukawa couplings, To make a comparison, we first examine the conventional LR symmetric theory with an While one can simply choose or 2.4 is as large as N u 1 e i These terms necessarily breakcomes the from the aforementioned quantum symmetry correction of to the mass of must be suppressed, since thethe former Yukawa introduces coupling of some symmetry. with the SM Higgs y very small. This can bebreaks achieved by coupling In order for H also solve this problem. 6.1 Conventional LRIn symmetric the theories conventional LR symmetricSU(2) theories, the SM Higgs is embedded into an the presentation of aof UV completion ( to thefar next above section. We showviolate that ( the lepton sector and on leptogenesis from hierarchies of parameters by breaking by appropriate symmetry breakingprotection fields. mentioned above, However, quantum because corrections of may destabilize the the absence hierarchies. ofSU(2) symmetry of imply no large symmetrythere distinction are between none between the the from an explanation for their suppression. At the tree-level, it is possible to obtain the desired the hierarchies JHEP01(2021)125 (6.8) (6.7) . This 5 . 0 , the cutoff and charged c ∼ 3 , Λ 2 is the standard τI i M U , I , 2 6 L R − up to c , which is violated for from the lepton sector R H H 10 1 Λ v , and j R mixing and the Yukawa 1 ν v > M to be µ 9 d M d 1 i − 2 H 2 H y Λ 10 m ¯ e Uγ − and 6 ' u 1 − for any i to be sufficiently large. Since the 2 y H 10 13 s for quark and lepton Yukawa cou- 33 c − R Φ ∼ y Λ v 10 e ij q q is broken by the charged lepton Yukawa y . 1 3 u , and the charged lepton Yukawas generate 2 ∗ 2 1 τI 2 H i Λ U y ). N m 33 . In the following we take i i 4 y τI 6.8 – 20 – − N U 10 from τ L R y 1 b decay requires ∼ y H H N can be suppressed. t 2 from charged leptons and quarks in the EFT. Loop momenta ij y 1 . In the next section, we present such a setup and show ), requires 1 N i i 3 , lead to ( R y c y v 2 , y 3.6 Λ ) partner of j 2 ,( appears in the charged current π 1 1 should also receive quantum corrections from HN 1 LR N i U (16 ` ). M ij . Two further diagrams involve the same vertices with different ∼ y 6.1 1 6 i = i j . We estimate this radiative correction to y ` ` L 5 ∆ receives quantum correction also in Left-Right Higgs Parity. The quantum ij , y . Similarly, N 1 i is the cut-off of the theory. This introduces y . Radiative corrections to . The dimension-five operators may be, however, UV-completed by introduction R Λ ). The stability of The Feynman diagrams for quantum corrections to Successful leptogenesis from This problem can be avoided by using different 2.4 > v c non-zero lepton Yukawa couplings. are shown inconnections figure of the Higgs lines. They are quadratically divergent for loop momenta above Λ of particles with massesthat below the quantum correction to flavor symmetry that distinguishes couplings, quantum corrections involving where the PMNS matrix PDG numbering for the correction is quadraticallyof divergent, the for effective loop theoryof momenta with ( above the dimension-five operators for the charged fermion masses correction from the quarkdiagram and in charged figure lepton Yukawa couplings is given by the Feynman plings and/or introducing supersymmetry, but we do not6.2 pursue this direction further. Left-right Higgs Parity The coupling violating the bound ( Figure 5 near the quark EFT cutoff scale, where coupling of JHEP01(2021)125 33 y and dark (6.9) ) and (6.10) (6.11) (6.12) (6.13) 1 may be for the = 2 2.5 N 3 , ij 2 j 2 y , 2 M R from /v 10 ) Λ 1 ) from the radiative R ) ) between to be j M H ` 6.1 2 1 . ( ( 6.9 i / i 3 y 2 . L ` . 2 , 4 H 4 R 1 Λ v R Λ R M v Λ R v 3 keV 1 ) requires /v 10 for the freeze-in cosmology. ∗ 1 τI contributions. Thus, for 2 5 Λ 6.2 ∗ 3 ) τI U − 2 τ j 3 lead to radiative corrections to the U R y , 2 j τI 10 6 H τI U 25 . . U 2 τ = 2 2 y j (0 2 τ in figure 3 j ` GeV y R , ij given in ( 2 v j (parenthesis) in the EFT. Loop momenta near the R y R Λ 12 3 v v 3 j , , not exceed the limit of ( M – 21 – M 1 2 R 10 2 1 4 =2 X M i ) ) H j 2 y /M 2 2 τ 2 ) 2 1 y π 1 2 π 1 ) ). L and 3 2 π M ` 25 π 1 1 ), . (16 H (16 i 3 keV 3 0 6.11 y and assumed no cancellation in ( (16 , ∼ ∼ (16 & dark matter, whether by freeze-out or freeze-in, 1 1 25 just allows the entire triangular regions of figure j . = 2 ∼ 1 1 0 1exp M R 1 M θ ) and ( i M N v ∆ y ∼ i, j ∆ v ( 6.9 j ∆ sin 2 ). We estimate this radiative correction to ij τI mass matrix leads to a radiative correction to 1 y U 1 1 2.4 M N ` Λ = 10 ∗ τI = U bounds max operator , the cutoff of the effective field theory described by the Lagrangian ( 1 y Λ R N . Radiative corrections to . H contributions. For ij R y Similarly, diagrams such as the one in figure H up to j ¯ ` = 3 R 1 ¯ where we assumedmatter, no cancellation a between cutoff freeze-out cosmology but limits very large For this not to exceed the value of ` Diagonalizing the where we used j chosen small enough to satisfyand this we bound. discuss However, this leptogenesis below. requires a significant Requiring this radiative correctiondecay to of v the third term of ( Figure 6 EFT cutoff scale lead to ( JHEP01(2021)125 ) 1 i × y ` 2.5 (7.1) (6.14) (6.15) ., for the greatly against c U(3) . 1 1 1 i M y N ) and ( + h b ¯ 2.4 S and a near unity and, is very light and 1 S i x 1 y N ¯ . We assume flavor S,ab to satisfy the bound β. 3 , and at the most, not S , 2 33 5 M y M sin 2 + 2 1 α a 2 S N a 1 R , only two pairs have significant for freeze-out dark matter and S This requires ¯ , the neutrino sector has M S /v 10 4 ) sin ¯ from the tau Yukawa may couple S S,a 6 Λ x . ( ) places a naturalness constraint on 1 λ M g and and 1 2 4 . and . 6.9 S 3 ( i 8 + S ) plane to natural leptogenesis. R 1 a , y x i , ) ¯ 2 ( S y can be obtained by introducing singlet fields i /v 10 g R a y Λ ¯ GeV R . In this section, we present a UV completion R ¯ flavor symmetry. Hierarchical breaking of the S S,ba , v v S H 1 6 10 R in section – 22 – ¯ shown in figure 3 S,a L H M M 1 10 1 . Requiring this value of ( M i ¯ 2 `H = y for the correction from figure 1 2 1 ` 2 ) /v / U(1) R ; we may instead start from the theory where only two M x + 3 v R ( 3 keV S ¯ S,ab g × v R and ∗ S 3 M L H 1 R . Thus, in the EFT the quadratic divergence of m 12 a H H − M S ,M 7 that naturally allows successful leptogenesis. i 3 keV . With three pairs of ∼ R ¯ ` ¯ ) 10 S S,a ), on leptogenesis U(1) ia R 33 ¯ `H × ¯ λ ¯ y ` M × , v 5.8 3 , 1 and + ¯ ), S = L . L M S H ( ¯ H L S,a 5.17 a U(3) are present. This suppresses the quantum correction to ¯ S i ¯ × ``H S ` ,M for the correction from figure S ) for freeze-in cosmology, unless ia ∗ ia λ λ and/or small masses R with the following couplings and masses, ) and ( and v 5.4 λ a = = U(3) ¯ S S , reopening large regions of the ) leads to the naturalness constraint 1 L ia 5.14 × ¯ λ ¯ ` M In the next section we give a UV completion of the lepton and quark sector. This is The quadratically divergent correction to and therefore, via ( 6.10 We will discuss a natural origin for and 6 a 33 symmetry breaking such that amongcoupling three pairs of pairs of following reason. Although the vertex corrections to and integrating out U(3) symmetry can explain the hierarchy 7.1 The UVThe completion: operators tree-level S quantum corrections, theshould UV occur at completion a of massfar scale the below above dimension-5 operatorsand show ( that the quantum corrections can be sufficiently suppressed. and 7 A UV completionAs yielding we a have light, seen long-lived in the previous section, to naturally protect the stability of shown by blue lines inlimits the figure range of important for two reasons:long-lived, first it and provides second an it understanding allows for a why very large reduction in the radiative corrections for given in ( from ( of ( y This is far below the required values of JHEP01(2021)125 1 N , is 2 (7.2) /v region, R v 33 blue m , do not obtain ¯ S symmetry may be is sufficiently light is much lighter and ¯ ` 1 1 and N N S U(3) DM in the freeze-out (left) These operators give , × 1 DM can be realized without 7 R ` . In the hatched . N 1 H ia 5 N . We may redefine the linear L ¯ λ . The charged fermion masses are ¯ S U(3) is obtained from the mass term R ¯ `H must be unnaturally tuned to keep H ` R 1 and L i ¯ to clarify the charge structure. H S y , the ∗ S L ia ¯ ∗ `H ¯ S 2 ` cut λ ¯ `H + . ¯ λM M ` for leptogenesis, approximately λ R 3 and 1 H + S R ' and R 2 ¯ `H – 23 – . However, to show that 2 ) H ¯ ` 3 2 , R 2 + does not couple to /M N L ¯ `H 3 i ¯ ` ¯ ` H M L ∗ ( S 2 cut M ≡ ``H 2 M x ¯ λ is the UV cutoff. The lower blue contour shows the same region if Λ L ∼ . to avoid the radiative correction of figure R = 1 v R , which is light. The operator 1 /v ¯ ` is a real parameter, we put the superscript required to set Λ being typical entries in the matrices symmetry is anarchically broken, the following higher-dimensional operators S 33 ¯ ` ¯ M λ y , but the massless combinations, which do not couple to i . The unhatched shaded regions are constraints solely on ¯ ` . The parameter space where the mass and stability of U(3) . This gives rise to Yukawa couplings between the massive linear combinations of and , one linear combination of ¯ S S × = 10 λ , e.g. the Planck scale (and, in the next subsection, from radiative corrections). If the S ` If there are (effectively) only two pairs of ¯ R Although stable when S to 7 and of S cut 1 /v i 1 ¯ with and stable, we must studyM higher-dimensional operators from the cutoffU(3) scale of the theory are allowed: ` Yukawa couplings. anarchically broken in the neutrino sector.has This a model smaller explains why Yukawa coupling than and freeze-in (right) cosmologies, as in figures ` combination as M fine tuning in the effectiveUV theory completed below the value of sufficiently large that theN tree and loopΛ contributions to Figure 7 JHEP01(2021)125 . , R . 2 v 1 i (7.4) (7.5) (7.6) (7.3) , but y S Pl cut M . ). In this M are larger M and Φ do not enter We delay a 1 1 i 7.2 dominantly Φ 2 8 y M 8 but, by the ap- S R replaced with S v M ) and ( M R 4 H 7.1 . still has a small (zero) 9 ) of interest. , . i R and c ) is replaced by can change the mass and . in ( N 0) R H λ , v S R GeV R 7.2 2 ) with v , + h M H 11 R 2 i v † ¯ R ` , = 2.5 10 1 H ¯ ` (1 R and charged lepton Yukawa cou- † × L ), among which the tau Yukawa is ). H b in eq. ( , 3 3 . It is possible to reduce the size of H ∗ 3 i R † ¯ 2.4 y S λ 33 7.3 , we obtain the Yukawa coupling b . S Φ H y , 2 ∗ 3 . R renormalizes 2 ij ) R A λ i M x H a y cut S H /v ∗ 3 , when − i 3 ), and R ¯ x R ` R 2 Φ M j M ) for the range of 1 a ¯ v ` 4 v M 3 with charge ¯ 1 i ` ( m 8.3 − Av ` ν x M 6.2 Φ – 24 – 2 2 Φ S 10 m = 4 R ij A v 4 Φ M > x e ij , and gives quantum contributions to ). y S,b , i.e. m 3 3 keV , R 2 ij takes the form of eq. ( v + ( ) and ( M 7.2 ¯ M ` ' y 2 2 2 S | 2 τ 3 ) ) 6.1 y 2 2 Φ smaller than M | M , π π 1 and are still given by eq. ( 2 Φ S 25 with values from 2 . 33 i 1 , shown in eq. ( m 0 i 1 y M (16 (16 ¯ ¯ ` y ), unless S M − . Only corrections below the scale S `H 2 S cut 2 cut ' L and inserting the vev of 7.3 = M M M ). Let us first consider the case where the charged lepton Yukawas M L' and `H Φ L 1 ' 2.4 have Yukawa interactions in eq. ( ) can satisfy ( ' M i R ¯ ` . The two-loop diagram shown in the left panel of figure 2 v R 1 7.3 ¯ v S 2 N cut S S 2 cut M M M , generating M 2 2 1 λ λ M ' ' ) that discriminates couples exclusively to leptons, not quarks, so that potential CP violating phases of 1 1 i The quantum correction above the scale The quantum corrections depend on the UV model that generates the dimension-5 In the model without 7.1 y In fact, further suppression results if supersymmetry existsΦ in the UV, since holomorphy of the super- M 8 9 ∆ ∆ potential can forbid the operators in eq. ( into the quark sector. Consequently, the strong CP problem remains solved when introducing decay rate of corrects proximate (accidental) symmetry, one linearmass combination and of coupling the to interactions in eq. ( arise from the exchange of a heavy scalar After integrating out We first discuss theplings. quantum All corrections three from the largest. The tauof Yukawa ( necessarily breaks the approximate or accidental symmetry the corrections to 7.2 The UV7.2.1 completion: quantum corrections Corrections from lepton Yukawas these corrections by taking case the effective theoryIt below is clear that ( discussion of the implications ofthan these the tree results result as of the ( quantum corrections to where we take the largest a mass and a coupling to JHEP01(2021)125 ` . ) 3 ' τ 1 y N i S N (7.9) (7.7) (7.8) τI (7.10) (7.11) (7.12) , when 1 U /M 1 2 and N obtains a λ τI L 1 H ¯ U S R N . H = ( R R , and , we obtain the v v 4 Φ R , and the Yukawa R S ¯ & . E in the left panel of H R /v R /m H a E E 3 H 2 ¯ E ` A , and the charged lepton M M . a M a S ¯ ` ∗ 3 τ ¯ E x y and R a a 1 H 25 1 E R . . x R − . 3 0 by a similar amount. H ) 3 ` E,a i H a M 1 ∗ 3 y 3 i ' . This term, after L ¯ 4 M ` z y 2 H 2 R 1 a R ¯ ` 1 . The diagrams are UV completions to + 10 , S , z /v S R Φ /v † ( † 1 R R o 3 v e aj a M v 2 ) `H R M N z H E ) a M 2 H a ∗ 3 τ to z 2 M y Ea R E 1 ' ¯ a `
U y
† 1 L H Max ' R = N H ' b a 1 , and 3 ∗ H 1 i a ) ¯ 3 ∗ 3 E obtains a vev, gives a mass mixing between λ y ¯ S i ` 2 R b z M ` 1 3 ∗ ∆ a R R ¯ ` . In the second equality we use 1 ∆ /v λ e ia H z 2 , τ a Φ z H . 3 ∗ 3 y ¯ E R z m 6 = S M a H R 25 1 . 2 L S H z 4 R 2 and (0 v S g M . The two-loop diagram with external , after integrating out e E ¯ 2 2 ` ' M , the SM right-handed charged leptons originate from R z ) ) 2 2 R v 2 R e v in the diagram, one of external π π 1 1 ' e /v R e L ∗ (16 (16 y ` ) > z W H < z τ y ' . Two-loop diagrams correcting the mass and decay rate of the dark matter, E E i generates a mass-mixing between . This term, after m L' τI m 9 2 R U We next consider the case where the charged lepton yukawas arise from the exchange 3 Without /v τI 3 10 U 1 ¯ In the second line,M we use figure yukawa coupling When coupling is of heavy fermions When The mass mixing also induces a coupling of The diagram in the right panel of figure vev, gives a mass mixing between the neutrino masses aremasses generated are by generated the bythe the exchange EFT exchange of diagrams of a of a heavy figure heavy singlet scalar, where we assume (
† L y L /v & U H S & H . Thus, with Λ a a 1 1 i S ¯ ¯ U D y i i M M q q . In the up, down or charged lepton sectors, if ∆ S ∆ u u d ia ia z z z . R 6 ∼ = = j H d ` ), from a UV completion with u u R y L L H 7.14 R replaced by , integrating out j W ` R Λ v , the correction is minimized for the largest u , so that the light fermion masses are “flipped” rather than “seesaw”, with ¯ q R ) and ( R , on the other hand, the SM right-handed up quarks dominantly come from v H > z R E , with Lagrangian v & . Two-loop diagrams correcting the decay rate and mass of the dark matter, ) with U 7.13 ¯ u D E M 6.9 D, m < z by a similar amount. We see that eqs. ( U 1 i i ` via a seesaw,m and similarly forrather the than down quark Yukawasthe by Yukawa coupling integrating out light mass is seesawed, while it becomes flipped as With We next consider theintroduce quantum a corrections UV from completion chargedand for fermion the Yukawa up couplings. and down We quark Yukawas by heavy fermions y or eqs. ( and ( greatly improved. When we take 7.2.2 Corrections from charged fermion Yukawa couplings Similarly, the mass mixing also induces a coupling of The two-loop diagram in the right panel of figure For correction is minimized for the smallest Figure 9 neutrino masses are generatedmasses by are the generated exchange by of thethe a exchange EFT heavy of diagrams singlet a of heavy figure fermion, JHEP01(2021)125 , L R ), Φ . 5 H H 7.3 E/ (7.18) (7.19) (7.17) GeV , easily D 9 , , ) 9 10 R (right), and ) involving v . e Φ q q 5 10 , z R U , when the charged , v exchange 1 exchange d . N E , z MeV/10 keV for (figure :Φ R 9 v 10 , u L R ): to be natural throughout or ∗ a < H H , z e 1 exchange exchange 1 1 8 , the cutoff scale of the EFT (left), or scalar, ) and visualized in figure z E R E R v e ia M M E E v τ z M 6.8 ,M : :Φ y ( Φ 3 τ τ . As a result, the quadratically D R y y 2 v ∗ R E v 18 14 1 . M e ,M i 2 − − M y U ) × R 10 10 M D 2 ), from the left panel of figure consistent with Higgs Parity, R /v × v M b R R ' 7.7 y i U j v , are less than t ` ` H in the UV complete theory are shown by the y GeV D M L 1 = max( 3 3 b 11 τ τ i – 27 – M H y y y y ∗ 1 , where we may integrate out the heavy fermions , 3 t decay rate from figure 10 ¯ ` y R i U M 1 ` as calculated in eq. ( v 2 × × ) ) for any M N ∗ 1 2 3 L R b i (3 d lc 3 π 1 y , where the SM right-handed fermions are dominantly y 2 u,d,e . We consider the correction from the third generation z H H / 3 t R d R lc 1 ). y v or z (16 H 2 > z ) )( D 2 2 & ∗ 7.18 u,d,e keV) u π 1 kb 1 / < m z i . A possible tree-level contribution from the Planck scale, ( 1 y R u kb (16 U,D,E < z q q . The correction is small enough for z M ∆ ( ( M R & < v , which generate the operator v R U,D,E D U . 1 i S 2 H ∼ y M M R 10 U,D,E M m ∗ U ∆ /v R M L R . The radiative correction of ( M S v M H H 2 M ) for ) 2 E GeV . Two-loop diagrams correcting the decay rate of the dark matter, π 1 6.2 13 , the quantum correction is bounded by (16 ¯ 10 E , respectively. Each diagram is a UV completion to the EFT diagram of figure Furthermore, corrections to the In summary, these UV completions easily allow small When the heavy fermion masses . ¯ D D, L' , R i j ¯ ` ` v satisfies ( is natural if lepton (charged fermion) yukawa couplings, can be made small enough in either cosmology which is even smaller than ( the allowed regions of figures where we take exchange. For U, fermions and their LRthird partners, generation. since the For smallestto possible obtain corrections the are dimension-5 largest operators, for the the quantum correction is bounded by where we assume generating the dimension-five quarkdivergent radiative masses corrections is to below are absent. The radiativediagrams corrections in to figure lepton masses are generated bythe the up-type exchange quark of and aU down-type heavy quark fermion, masses are generated by the exchange of heavy fermions, Figure 10 JHEP01(2021)125 . 1 Φ = N 33 E/ GeV y (7.21) (7.20) . Thus, can be 13 ) − can decay 3 ), the 12 IH 2 . This may . or ¯ 10 N E 7.14 ) 2 2 007( | . in the cosmology 1 0 , eI and R exchange is ) v U ) or ( | ¯ R D MeV/10 keV for to reach NH , , , where we took 7.8 W 4 | | 10 ¯ R U / | | 3 3 (3 + 22( v 3 3 3 . ) < 2 via m m 1 | | | and the charged component m m | | 2 1 2 . If the active neutrinos obey ¯ q , the SM fermion masses are < > | eI N /M M > < 3 . For ( R | | . The allowed parameter space U 3 = 0 | | | 2 2 v | R / 3 m 2 2 has charged lepton masses arising singlets, | v 2 5 m m we simply chose parameters of our 2 | 1 | | m 4 R m m . 1 | | | , , 5 R > u,d,e v M , , M eI can naturally be much larger. Hence, z | (30 keV 3 > 2 U around the upper bound, 1 | π | and m 2 < R ] M | 2 v SU(2) 4 m , R 76 | 1536 beta decay rate relaxes the blue bound that 3 /v 2 30 :023 NH : NH 30 :67 IH : IH 67 :023 NH : IH S = U,D,E ...... For – 28 – N 0 0 0 0 0 0 M − R M ` v . The decay rate of ' ' ' ' ' ' + ¯ ` ` 2 2 2 2 2 2 1 | | | | | | lifetime is natural for 1 3 2 3 2 1 N u,d,e ) shows that e e e e e e y . With “flipped” masses, 1 → and 2 U U U U U U 2 exchange is excluded for NH and allowed for IH. | | | | | | N exchange, and has “flipped” rather than “seesaw” charged for all cases. The bounds from warmness and BBN are = ¯ DM that is hot is q ) sufficiently less than N N 7.19 exchange, relaxing the upper bound on R E 1 R Φ R 11 v = = W N + Γ W 2 2 ,M ) | | ¯ d 1 2 ) is satisfied if D u,d,e u by eI eI z − 2 6.1 U U ,M | | N DM. U ¯ ud, ` 1 + M ` ( N ( → S ) shows that the 2 N M ; but the suppression of the Γ 2 ), the fraction of 7.18 that satisfy the constraints required for dark matter, and parameters that enhance exchange, rather than 7.20 2 DM is shown in figure , E 1 , typical for natural leptogenesis. For radiative corrections involving “seesaw” charged 1 The PMNS matrix elements are given by [ For sufficiently small Dirac masses obtain large masses 6 decaying dominantly via N N ¯ ` − 2 In this section we studyneed the for fine-tuning extent of to parameters. whichtheory In successful sections to leptogenesis obtain can a occurfor realistic without the light neutrino spectrum,leptogenesis decay to rates, realistic masses values. and interactions While this is certainly possible, in this section we study arises from insufficient dilution,From permitting ( the highest allowed N 8 Natural leptogenesis The suppression is mostan significant for IH, NH the with suppressionof is also strongest when as in figure with freeze-out and dilution by in only into the first generation of from fermion masses. Inmade such natural UV for completions, the entire parameter of“flipped” figures with right-handed statessuppress the dominantly decay of stability requirement ( 10 fermions, ( exchange; for “flipped” masses ( the UV completion with the largest natural range for by choosing JHEP01(2021)125 , 3 ` 33 2 atm y m ↔ ∆ 2 ). Such ` √ = 2.5 is light and . The shaded 2 be naturally R m 1 1 v N N so that a cancel- u,d,e in eq. ( 33 y to explain the mass proportional to y 0 1 ij i ' b y is suppressed, shifting the 2 23 c N parameter space by requiring and ) R , or discrete symmetries ) 33 beta decay rate decreases as the two 3 , v c , can be explained by introducing 1 3 2 , ` ' , or by increasing , are far lighter than 2 M M N 3 ` ( ( 22 N ' c for leptogenesis? U,D,E 2 and M 33 . The M y 2 R v – 29 – N contours show how the insufficient dilution boundary DM produced by relativistic freeze-out and dilution from . Can a sufficiently long lifetime for i 1 blue ν N and symmetry rotating 1 obey a normal (NH) or inverted hierarchy (IH). Bounds from hot , except that the beta decay rate of ). The becoming heavy, reducing the kinematically allowed decay channels N 3 ¯ ` 2 ν right SU(2) ( and and ¯ q 2 sol 2 ν m ∆ we have seen that sufficient leptogenesis typically requires an enhancement √ . The symmetry is explicitly broken in the coupling 5 = 2 ` 2 . The parameter space of m → − 2 ) insufficient dilution region to higher ` In section that can occur by near degeneracy of ) and decay when the masses of the heavy fermions, 2 blue left Highly degenerate right-handed neutrinos, an approximate flavor symmetrysymmetries ensuring include that an and splitting of the two heaviest SM neutrinos. Such a flavor structure — symmetry in the choices be made naturalIn by addition, introducing in approximate the symmetriesleading last to in section mixing the we between UV foundmaintained a completion? in radiative the correction presence to of an enhanced 8.1 Models for enhanced asymmetry parameter previous section thatsufficiently allows stable, us and to also limits start the with size of an radiativeof understanding corrections. of why lation between contributions to the light neutrino masses is required. Can these parameter depends on whether DM are discussed in the text. the extra naturalness constraintsa imposed natural on theory the without fine-tuning. We will use the UV completion described in the N regions are identical to( figure heaviest generations of and inducing suppressions from the( PMNS matrix. We show the allowed regions for Figure 11 JHEP01(2021)125 . R 1 of H and Φ (8.5) (8.3) (8.4) (8.1) (8.2) R given 3 give a | , which N L ¯ χ ¯ ` `H , so the | ¯ ` ) ` , H N 3 Φ L . x M c . and , − ``H 6 ν 2 − + h ), giving 3 M 10 , it is generated by 2( ). In summary, the ' 5.12 / ν 2 ¯ σ∂N 2 † 2 7.9 τ 2 π M y in ( 8 N ) . . ' 2 23 c ss & . ( − ) ., 2 δZ x c ) m 10 . ( + h ∗ g 23 2 ' + , , and drop generation indices. Let + h S and 3 δZ 3 ¯ ` S 2 ) + = 1 (5) M R R ¯ 1 2 σ∂N 23 x m and , v † 3 ¯ S `H 3 N + δZ N . ` ) 6 M + R M 33 + ( L 10 2 ) δZ – 30 – ¯ `SH min( `H ) can be explained in the following manner. Since , there is a quantum correction to . Only one linear combination of λ 33 R 33 v S + c δZ ln 2 5.10 L λ M 2 = − + (1 + 2 2 π . We obtain the same bound for the case when charged 2 is of order g 22 33 − 5 and couplings b 16 ) 10 λ`SH = x δZ S ( ( > ¯ σ∂N , we only consider g L × = † | . Near the resonance 2 ), one-loop quantum corrections from the coupling q 33 2 τ , while there is no corresponding quantum correction to 5 χ N L | y y ) with ) ' 7.4 is 12 | 22 & 3 ) , 2.5 | x δZ ∗ kj ( M 3 2 , ∗ kj g x 2 π gives the dimension-5 operator x − 8 ki , as in ( , obtains a Majorana mass and hence the SM neutrino remains massless. ki M 2 S x x 0) N | M , | = (1 + ' . This quantum correction upsets the cancellation, giving a lower bound 2 R , ij L 2 H , L δZ (1 ¯ `H ` Since there is no symmetry forbidding the Majorana mass of Cancellation between the SM neutrino mass contributions from the see-saw of The symmetry is also explicitly broken by the charged lepton Yukawa couplings. For and is dominantly This can be interpreted as a cancellation between quantum corrections. Below theby scale the diagram in figure Integrating out corresponding to eq. ( we are interested in large us introduce only one singlet maximum natural lepton masses are generatedmaximum natural by value heavy for fermion exchange, as inthe ( first dimension-5 operator of eq. ( where we use where we conservatively do not include a log-enhancement. This generates a mass splitting model since the masses and the couplings have differentexample, flavor when charges the from charged each lepton other. charge Yukawas arise from the exchangewave-function of renormalization, a heavy scalar masses and explicit breaking in the couplings — may be naturally obtained from a flavor JHEP01(2021)125 . , R R , v W (8.6) , the Y R is close /v ! 2 Λ ! ) Λ 2 in either cos- , solid), when /M 3 2 are as naturally Max N M ) ( (5) 3 x ( ≡ Freeze-In m Freeze-In g , natural leptogenesis x 1 , respectively. Because Strong Washout Weak Washout and
(Freeze-Out + Dilution) > ) , which is assumed for the = 10 R ss ( 3 `H R /v Y 16 is negligible compared to m . Λ 1 therm /v . The dashed contours above and Y − Λ `H ! L ) Y ` eV from = 1 H ss ) 4 eV ( 2 − ss and B 2 R ( 2 m Y − 1 10 /v m . 10 Λ 03 .
. When L = 0 W is modified to remove the quadratic divergence R + 0 13 – 31 – therm R /v /s /s Y v 1 1 Λ in blue shading for the freeze-out cosmology and , dashed), and when is unable to reach the observed baryon asymmetry M M 03 DM DM lifetime . L ρ ρ decay rate depends on the fourth power of 0 B ` , dotted). 13 1 H Y 1 . ) = 1 ) N β min N ). x ss ( are as naturally degenerate as can be ( χ ( for the purpose of showing the theoretical maximum allowed is that, for a theory with g 2 m sin 2 at 6.10 M 13 α ) 2 therm x and ( Y g and 1 . (5) ) sin 3 0 m x ( in either cosmology, which is only possible when M g 1 2 max y are comparable ( ) is given in eq. ( π x : when 1 2 . A diagram contributing to a non-zero neutrino mass for the case with tree-level 8 ( ) g x M max ( 28 79 y g . A key result of figure The vertical gray lines show the asymmetry enhancement for three representative val- The parameter space where and B 3 Y degenerate as can be ( requires to unity. Thusfirst, there the are structure two of ways the to theory construct below natural theories of leptogenesis. In the the baryon asymmetry inand the the freeze-in orange cosmology region is extends identical down to to the match freeze-out theues cosmology blue of region. M the radiative correction toresults the are sensitive toThe this allowed parameter ratio; space natural withincosmology leptogenesis the due freeze-in becomes to cosmology implausible is themoment for greater additional to than saturate contribution the to freeze-out region of the freeze-in cosmology in figure where without tuning isorange shown shading in for figure the freeze-inbelow cosmology, show for the analagous regions for mology, cancellation between 8.2 Radiative corrections: Naturalness thus limits the maximum baryon asymmetry generated by Figure 12 JHEP01(2021)125 ) ) ) 2 1 ). x N M ( 4.5 2.4 g 2.5 replaced by . Λ , solid), when satisfy eqs. ( shaded region, 8.1 3 decay, in the (freeze- leptogenesis requires max must be to realize the ) m 2 2 x ) with 3 N . This is because the ( N of this fermion. g green M 6.9 we construct an explicit and are as naturally degenerate S 5.2 7 contours show the analagous 2 M and (5) 3 m 2 m . In the freeze-out cosmology, together in a way that ensures M ) and ( leptogenesis without fine-tuning. In 3 and 2 orange ) M 1 eV) N 6.12 . ss ( 3 . In section (0 and R m O v , as discussed in section 3 , and 2 . In the second, a symmetry is introduced ) require that N blue , respectively. In the DM and 7 1 ,M can be freely adjusted to generate a large solid, dashed, and dotted lines show representative 3.3 N R 2 – 32 – = 10 with , v , remain 2 R 3 /M gray 33 3 N y m /v is, the more degenerate Λ so large that fine-tuning is needed for sufficient stability of M ) and show that in this theory the radiative corrections R 33 and 2.5 y /v and , dashed), and when 2 Λ have the maximal natural degeneracy ( 1 . 3 m together with ( , we understand it to be the mass ), which are identical to ( ), relates M 2 R ) = 1 i = 0 ) and ( x y 7.8 ( 3.3 g R < v and can be less than one if the effective field theory described by ( 2.4 , dotted). /v 2 ) Λ and Λ R , this is not the case as is shown in section M 1 ; the greater min i ) and ( χ R 33 /v ) shaded regions, the observed baryon asymmetry from y . The upper and lower dashed v y R at (Λ 7.7 v & when ) x ) Λ ( x g orange . Parameter space for simultaneous ( Λ = , g for ) is generated by physics below the scale are comparable ( 1 ); such a theory is provided in section 3 2.5 > blue The ratio M 6.9 , the mass of the fermion which upon integrating out generates the operators of ( ) , when S x 1 ( Although it appears the massindependent ratio of neutrino mass matrix, ( the active neutrino masses, the smallness of M Thus, when we take 8.3 Natural leptogenesis for freeze-out cosmology to naturally yield near degeneracy of and ( model that generates ( are given by ( and as can be ( of ( N exclusion regions for DM is too warm.g In bothobserved freeze-out baryon or asymmetry. freeze-in The cosmologies, vertical values successful of Figure 13 the ( out, freeze-in) cosmology, requires JHEP01(2021)125 . . , , . 3 1 4 2 2 (5) 2 N m are M /v max m (8.7) ) 2 R . For (5) 3 x v 1 eV) 1 are far ( . ) and lifetime. m g 3 N 1 2 (0 3 ! 1 m M m M ' N − and ) (equivalently, 2 ) 3 2 m ss m , a positive definite ( 3 ( ), is greater than 2 2 33 − /M m y 2 3 m 8.7 , which is why this case 0 m M , which is not visible ' ( 2 > 2 3 33 m The first scenario gives a y GeV m , respectively. . The gold, red, and green 13 11 , otherwise 1 . 10 1 and 2 10 | , as set by ( ∼ 3 when incorporating leptogenesis > x 33 m affecting the stability of | β < ) , but equal to are smaller, so that slightly higher y 1 eV) , and 1 . R ; that is, where neutrino masses are 1 < 1 1 DM cosmology. The shaded regions i v i | , and hence in freeze-out cosmology ), and natural stability bounds for y , (0 2 max 2 y 1 when sin 2 1 ,M 1 . Within the unshaded region of each y , dashed), and when = 1 . m 3 R , dotted), which occurs for α | N x 8.7 ≈ v β 2 , (top left panel), Constraint from neutrino masses ( ) x 0 = 0 min )), is unable to match the observed baryon 3
> m ) = 1 sin χ 2 R < m x sin 2 are as naturally degenerate as can be ( 8.6 ( m 3 /v at g 2 α – 33 – − only when , but newly added is a hatched gold region where Λ 2 m 2 ) M , the variation in location of the naturally allowed 2 1 x m ( sin ( g 2 14 and (upper ( m > M 3 . 2 1 4 2 R M ) and . This is because the radiative corrections to N )). This is violated if v v 1 , so that if M 2 8.7 8.7 compared to the second, meaning leptogenesis can be realized in the top left panel compared to the top right panel. However, ∆ M 2 atm , R 2 33 x m 1 v are comparable ( m 12 y : when √ 2 must not only be less than q ≤ 3 13 M set by ( 2 33 means radiative corrections to R m y | ≈ does not show the parameter region where radiative corrections to the 3 33 ), exceed 2 33 /v . generated by − y and y DM remain from figure , we show the constraints on m Λ 2 14 3 B 14 1 , (top right panel), can be reached in the top right panel compared to the top left. Identical 6.12 Y ≈ | 0 14 N since it is determined solely by the positive-definite dimension five mass contribution, M 1 x m | 2 ,( ); that is, where neutrino masses are incompatible with a natural 2 > 1 √ ). For the solid and dashed triangles, M m ), so that m 3 N | , 6.10 m ' = that satisfies the neutrino mass relations, ( 6.10 14 5.10 2 ,( 2 Last, figure Among the four panels of figure In figure ), shown in figure 33 2 33 m A consistent neutrino mass spectrum requires ) y is not necessarily positive because it may have a non-negligible see-saw contribution with a negative sign. y x 11 12 3 ( max example, when m quantity, would be negative (seeis ( absent in figure reasoning explains the slight variationobey in a the normal bottom hierarchy. two panels when the active neutrinos mass of less constraining than the radiative corrections to and if relatively larger value of at slightly lower values of a lower value of values of taken in each panel.of This is because theDM, apex ( of each triangleWhen is the determined active by neutrinos the obeyfigure value an inverted hierarchy, as shown by the top two panels of incompatible with leptogenesis fortriangle, the natural specified leptogenesis iscontours show possible the for allowed regions when region can be understood by the differences in the values and relative signs of as naturally degenerate asThe right can side be of eachy ( triangle marks the regionThe where left sideregion of where the triangle,asymmetry i.e. with the boundary of the hatched gold region, marks the constraining natural leptogenesis is inconsistent withregion the reside observed neutrino three masses. trianglesg Within with the the allowed same representativesolid), values when of naturally and consistently within the freeze-out and ( JHEP01(2021)125 is 33 y 2 ). N . We fix Λ Bottom are as naturally (5) 3 DM from freeze-out m 1 N set by consistent neutrino and ) 33 y ss leptogenesis can naturally be ( 3 2 m N DM and in accord with the active 1 N set to its largest, natural value, and ) ) and Normal Hierarchy (NH, x DM and ( g 1 Top N , dashed) and ). The dashed and dotted contours show the same ) = 1 – 34 – x ( g green , , and places a strong upper bound on the cutoff R v red , dotted). Naturalness and neutrino mass consistency excludes , region indicates where the baryon asymmetry generated by is set to its largest, natural value, and min ) χ gold x ( ( gold at g ) 10 are comparable, ( x , ( 3 1 g , M 1 . are sufficiently large that they must be unnaturally tuned with tree contributions = 0 and 1 i 2 R y masses by the Inverted Hierarchy (IH, M . The hatched /v 3 ν Λ . The parameter space where frozen-out 2 DM stable when 1 N and 2 ν to keep masses for region when degenerate as can beareas ( with too lowthe or high values of realized without radiative corrections affectingneutrino the stability mass of spectrum. Theas shaded in (unhatched) figure regions solelyunable constrain to match the observedset baryon by asymmetry consistent with neutrino masses.corrections to The right, downward sloping contours mark where the radiative Figure 14 JHEP01(2021)125 R ). at so )). v . 1 2 2 (8.8) − 5.6 6.10 . Each ηY . After 10 1 ,( . When 1 eV) contract decreases ! . , as shown ) occurs: `H 5.3 max (0 2 Y 33 15 β < 10 for the same R y y . Within the 8.7 − 1 /v 4 ) Λ because sin 2 ∼ − = 1 33 α x 10 is defined in ( β 14 do affect regions of 2 m , is unable to match 2 1 − e sin m sin 2 M , dotted. The right side therm 22 α ∆ Y m 2 1 min ( . χ 0 remain of order the observed 22 sin | ), is greater than eV. in the freeze-out cosmology, so ' m at 33 2 05 ) 8.8 DM from freeze-in. The shaded are not more degenerate than the 33 . when leptogenesis is incorporated m x y 0 | ( ηY ) 1 ) and . For example, in the left panel of 22 g in freeze-in cosmology 1 − N demonstrates that naturally reaching , but newly added is a hatched gold 33 m 8.8 and and is at its largest when ,M 01 m 3 . 15 ) R Constraint from neutrino masses 22 2 33 v x and ( ( y
is vertical unlike figure m = 0 g and spanning six decades from 33 , as set by ( 2 , dashed; ) R set by ( 15 33 m 22 – 35 – R y , the constraints from /v 33 m is required to realize the observed baryon asymme- R Λ y so that v/v ) = 1 R ( /v . In the right panel, 2 due to the additional contribution from x 2 , v Λ ( 33 1 4 2 R g y M v v M assumes 1 eV) = 22 . , and the triangular region shrinks ( for freeze-out. In this case, leptogenesis can probe lower GeV, but only for parameter space already excluded by the in the freeze-in cosmology, as is shown in section 15 (0 22 13 x m 8.3 m for freeze-out. 33 , at the maximum possible ' , solid; , and the left side of the triangular regions of figure 10 therm 2 y √ DM allowed in the freeze-in cosmology requires DM remain from figure R ) Y are unknown quantities generally misaligned with the active neu- 14 N 1 < 1 W 33 1 max . and 33 ) Y N 0 . R 33 N eV to avoid strong wash-out, a similar relationship to ( m and x 1 v m ( i m ) ≤ − y g − x 2 001 ( : ∆ . 22 05 eV `H , we show the constraints on 2 g . 0 22 ηY and Y 0 m ( , , < 15 /M 3 . 22 . For larger values of x m 22 3 2 eV | − m generated by ˜ √ m 3 m M 33 14 10 , is much smaller, and larger − B m ' Y | 10 2 33 15 y 2 2 Since The left side of the triangle in figure In figure 33 6∼ y e m 2 e reasons discussed in section due to the slightthat enhancement in try. The right panelobserved of neutrino figure mass spectrum. If they are significantly more degenerate, trino masses, it is impossible toand know inverted the hierarchies. exact parameter Nevertheless, spaceneutrino associated since masses, with the the normal variations inwhen the allowed scanning parameter space over do possiblefigure not values change dramatically of its maximum is independentm of If to match those of figure the observed baryon asymmetryunshaded with region of each triangle,contour natural color corresponds leptognesis to is a possible different by for the legend at thethe bottom highest of masses the of figure. Figure masses. Within the allowed regionvalues reside of three triangles associatedof with each the triangle three marks familiar The the left region side where of thewhere triangle, i.e. the boundary of the hatched gold region, marks the region where naturally and consistentlyregions in constraining the cosmologyregion with where natural and consistent leptogenesis is inconsistent with the observed neutrino too do they tie requiring parameter space for constraints from 8.4 Natural leptogenesis forJust freeze-in as cosmology neutrino mass relations tie together on figure JHEP01(2021)125 , 2 N DM 1 violet N . Regions resembling Λ 2 sol m are as naturally is set to its largest (5) 3 . + ∆ ) is relatively large at . The hatched m x ( 7 = 1 2 33 2 atm g y x m and leptogenesis can naturally ) ∆ 2 DM and in accord with the and ss √ N ( 3 1 2 − N m = are sufficiently large that they must , resembling the Normal Hierarchy. 1 eV) 33 . 1 i is greater than the mass of the heavy m (0 y 2 atm 2 DM stable when m ' N and 1 ∆ ) N √ 33 is relatively small at , dashed) and 2 atm − m region, the baryon asymmetry generated by m 2 33 = − y ∆ ) = 1 22 33 √ – 36 – x DM from freeze-in and gold constrained by neutrino masses. The right, downward m ( m and = ( g 1 2 33 22 N , is unable to match the observed baryon asymmetry with , as occurs for the model of section 22 y R m and m , and places a strong upper bound on the cutoff R 1 eV) . 2 sol therm v < v , dotted). Naturalness and neutrino mass consistency excludes Y (0 m Λ 1 . ∆ 0 min We fix χ √ ) ' 33 2 = at . In the hatched Left: m ) are comparable, ( 3 ηY 22 x − ( 3 set by consistent neutrino masses. Each contour corresponds to a specific m g M 22 33 m y ( and 22 We fix are only allowed if 2 m 1 M M . The parameter space where Right: . , as shown by the legend at the bottom. The dashed and dotted contours show the same set to its largest natural value, and R ) = 1 x /v ( Consequently, areas with too low orwith larger high values of region shows the inconsistentfermion region that where generates the it. massthe of Inverted Hierarchy. Consequently, x sloping contours indicate where thebe radiative unnaturally corrections to tuned withnatural tree value, contributions to and keep Λ region when degenerate as can be ( Figure 15 be realized without radiativeactive corrections neutrino affecting mass the spectrum.from stability freeze-in of The as unhatched inat shaded figure the regions maximum possible are constraintsg solely on JHEP01(2021)125 . The 4 , is below inf RH T plane of figure ) 1 ,M R GeV. In extensions of the SM v ( ) GeV; remarkably, this coincides is greater than the mass of the 13 13 2 10 . Finally, the allowed region where 10 N R are overproduced, but are diluted by − − v 1 9 8 , the allowed region is slightly enlarged, N R are weaker than the radiative corrections v (10 is produced via new gauge boson exchange 1 = 10 for the same reasons discussed for the freeze- decays again produce the baryon asymmetry. 1 , or are already excluded by other means. M 1 2 R ), successful dark matter and baryogenesis can N v N 15 M – 37 – , may be dark matter and the decay of a heavier 2.5 1 are produced by the new gauge boson exchange and N 2 N ) and ( decays also create the baryon asymmetry. In the freeze-in 2.4 2 lies between the triangular regions in the left and right panels N 2 . 2 ), are greater than N 1 eV) . , that generates it, which is inconsistent. This region is always more 6.12 (0 does not show the region of parameter space where radiative corrections S ,( . 1 M 15 ) N is required to generate the observed baryon asymmetry. Consequently, the 33 R . By combining Left-Right Higgs Parity with space-time parity, the absence . m later decouple from the thermal bath; symmetry breaking scale, and hence precise measurements of SM parameters v R symmetry called Higgs Parity, the spontaneous breaking of Higgs Parity yields the i − 2 v 2 15 , may create the baryon asymmetry of the universe through leptogenesis. N Z 22 2 Z and either do not show up on figure N i m The freeze-out cosmology is tightly constrained. With quark and lepton masses gener- We studied two cosmological histories of the universe. In the freeze-out cosmology, In this paper, we identified Higgs Parity with Left-Right symmetry, which is broken Last, figure Within the hatched violet region, the mass of and leptogenesis becomes challenging. We do not analyze this region in this work. 1 ( y 2 22 be achieved simultaneously in thesymmetry unshaded breaking regions scale is of predicted the with to be the window predictedcan from be SM probed parameters by 21 andthe cm Higgs effective line theory cosmology Parity. has and The a by parameter precise UV measurements completion space of below SM parameters. If cosmology, the reheating temperaturethermalized, is but low, an appropriate so amountaround that of the the completion of right-handed reheating. neutrinosby Yukawa are couplings to not SM particles. The ated by the effective theory of ( the reheating temperature ofinitially the thermalized universe via is exchange high ofmetry. enough additional that gauge right-handed bosons neutrinos requiredthe are by late-time Left-Right decay sym- of at scale of CP violation inright-handed neutrinos. strong interactions The lightest, isone, explained. Left-Right symmetry predicts three with a SM as a low energyat effective the theory. The SMcan Higgs narrow quartic down coupling the is symmetry predictedsymmetry to breaking breaking vanish scale. scale are Observable correlated quantities with correlated SM with parameters. the 9 Conclusions and discussion The discovery of thecoupling Higgs nearly with vanishes at a a mass high of energy scale 125 GeV has revealed that the Higgs quartic constraining than the regionM where the reheat temperature after inflation, to the mass of out cosmology: the radiativeto corrections to naturally allowed triangular regionm shifts to higher of figure heavy fermion, and large JHEP01(2021)125 . . ) R ). 1 − R v 1 v 2.5 M 10 10 , where depends R ∼ v ; quantum doublet to R R 1 v v R N and, together R v (see e.g. figure from SU(2) h , allow 3 m , including the entire ) N 14 1 . In some of the param- is allowed, but discovery 1 R ,M to the SM lepton doublets N v R 1 v plane for a fixed Higgs mass ( N , and hence SM parameters. If ) 1 R . Second, the v ), that generate charged fermion R ,M DM is not protected by any symme- v t 2.4 1 m ( N and the Higgs mass, ) will hone in on the scale Z acts as a direct substitute for the scale h t . M m ( R m S – 38 – v , α h which can forbid its decay is explicitly broken by a m , and keV for the normal (inverted) hierarchy of SM neutri- 1 GeV and around a few keV, respectively. This param- ) N Z 12 and M 10 are nearly degenerate, or the see-saw contribution to the ( ) S 3 , Z is nearly cancelled by a contribution from dimension-5 oper- 100(30) α 2 3 , M − N . t ( N , has Yukawa couplings to generate the charged lepton Yukawa 2 S 3 m 1 and tau Yukawa couplings destabilize ¯ ` α decay too fast. We identified two types of quantum corrections. ∼ . However, the near degeneracy or cancellation may be destabilized 3 1 1 N GeV is required, giving precise predictions for future measurements of . The freeze-in cosmology, on the other hand, is consistent with simul- N M 8.1 12 are small and significant enhancement of the CP asymmetry is required. 11 , and to a lesser extent, 3 t GeV. For a normal hierarchy, a wider range of 10 , , we recast the constraints on the 2 m 11 < N 16 10 R v GeV and is embedded, . Also, observations of cosmic 21 cm line radiation will discover DM to be warm, must have significant Yukawa couplings for efficient leptogenesis, while the tau ∼ s ) 1 3 are required to be above α R 12 v N 1 N M In figure Constraints on the freeze-out cosmology, summarised in figure In most of parameter space, sufficient baryon asymmetry requires either the two heav- Naturalness of the scheme further constraints the parameter space as well as the origin (10 and 13 t Consequently, for fixed The allowed parameter space is inFuture remarkable measurements agreement with of the observed topwith quark determination mass. of theThis neutrino can mass then hierarchy, will be narrow confirmed the or allowed excluded range by of 21 cm line cosmology. Here we assume that the CP asymmetry of leptogenesisand is not enhanced by eithereter degeneracy region or cancellation, can be probed by measurements of SMand parameters several and values the of warmnessdominantly a on of DM. strong coupling constant. In Higgs Parity, the scale of SM neutrinos canconfirmed, probe this parameter space.m For example, if anunless inverted hierarchy is or constraints on the warmness of DM will narrow down by quantum corrections, limiting the enhancement.the This masses excludes of lower values of 10 nos, respectively. Measurements of SM parameters, the warmness of DM, and the hierarchy ier right-handed neutrinos SM neutrino masses from ators. These two features canpresented in be section explained naturally by UV models of the neutrino sector which couplings. The chiral symmetrycombination of of this Yukawa andcorrections, the the quark UV Yukawas. completionmasses, To of requires suppress fields the the with resulting operators masses quantum of below ( and Higgs, making First, Yukawa coupling explicitly breaks anycorrections symmetry that involving distinguishes eter space, to suppressshould these be UV-completed quantum by corrections, fields the with neutrino a mass mass operators below of ( taneous dark matter andunshaded baryogenesis region over a of figure wide range of of the fermion masses in thetry. model. Quantum The corrections stability may of induce Yukawa couplings of as shown in figure JHEP01(2021)125 and 2 ν . The 10% ]. . Remarkably, . If the Dirac t 80 R m – v . This is a very − 77 ` + ` at its central value, and 1 ) N for given SM parameters Z , a component of hot dark is at most only R M R → ( v R S W 2 v α N at its central value throughout, since , the running is slightly altered. If h R v m , this corresponds to an increase in the MeV, which is smaller than the expected . The branching ratio of the decay into , there exists a set of new particles with R , is sharpened, corresponding to the right- t v R 20 Yukawa couplings are sufficiently small. In 16 v m 2 , and the mass of the top quark, N 1 , the increase in u,d,e – 39 – R y MeV. If the Dirac masses of fermions generating M v , 1 values. We fix decays dominantly via , for a normal neutrino mass hierarchy too much hot DM from freeze-out, natural leptogenesis, and consistent 150 N σ 2 7 1 2 . The center triangle fixes N N ) are smaller than ± 7.15 4 GeV , or equivalently . 0 R v the Inverted Hierarchy (IH) in accordance with the top left panel of ± 0 . ) and ( Left 7.9 by the Normal Hierarchy (NH), in accordance with the bottom right panel of = 173 t within its uncertainty do not appreciably change the parameter space. The m . Even for this extreme case, the prediction for h ), this hot component provides 10% of dark matter. However, in the case of R Right v m 2.4 . The parameter space of u,d,e and . y 14 14 In the freeze-out cosmology, if DM, natural leptogenesis, and the observed neutrino masses are consistent with the current , which creates lepton asymmetry, is less than unity, but this can be compensated by the masses are fixed by: 1 3 hand blue side of`H the allowed regions inenhancement figure of the CP asymmetry.theory When of charged ( fermion massesUV arise completions from discussed the in effective section uncertainty of top quark mass measurements at future lepton collidersmatter [ is predicted duenatural to possibility, occurring the whenever subdominant the this decay case the mode prediction for masses is increased only byprediction a for factor the of top two. quarkthe mass first For by fixed generation Yukawascorresponding are increase above in the top quark mass is figure the running of gauge couplingmass constants terms is in that eqs. of ( all the of SM the up Dirac to the masses scale are smaller than N measurement of the triangles to thevariations left in and right at ν figure Figure 16 neutrino masses in terms of the mass of JHEP01(2021)125 ] (see, 83 ]. , , especially R 82 , v must be above SPIRE 70 R IN v ][ ; ] if the Higgs Parity (2012) 1 81 15 716 , while for the inverted hierarchy R arXiv:1207.7235 W [ Phys. Lett. B , branching ratios can be computed because 2 (2012) 30 N – 40 – 716 is the complex conjugate of the PMNS matrix. R GeV, constraining the parameters. W 12 10 ), which permits any use, distribution and reproduction in ]. decays dominantly via 2 . The relevant Phys. Lett. B N ; the constraint is typically stronger since the maximal temper- , Observation of a new particle in the search for the Standard Model 7% SPIRE . R 0 v IN Observation of a New Boson at a Mass of 125 GeV with the CMS ][ CC-BY 4.0 This article is distributed under the terms of the Creative Commons collaboration, ]). As we have shown in this paper, the baryon asymmetry can be produced collaboration, 84 is required to be above R v arXiv:1207.7214 ATLAS Higgs boson with the[ ATLAS detector at the LHC CMS Experiment at the LHC We conclude the paper by stressing the importance of cosmology and precise mea- Theories of Higgs Parity suffer from the domain wall problem [ The freeze-in cosmology is also constrained, as shown in figure GeV. If the CP asymmetry of leptogenesis is not enhanced by degeneracy or cancella- 9 [1] [2] Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. This work was supportedergy in and part Nuclear by Physics,05CH11231 the of (LJH) Director, the and US Office DE-SC0009988grants Department of (KH), PHY-1316783 of Science, by and Energy the Office PHY-1521446 underSackler National of (LJH), Foundation Contracts Science Fund High as DE-AC02- (KH). Foundation En- well under as by the Raymond and Beverly parameters by future collidersneutrino and hierarchy. lattice computations, and by the measurement ofAcknowledgments the experiments will be difficult inerations the on near the future. early universe, In cosmological(including testing observations, and such those predictions theories, of of theoretical SM neutrinos) consid- parameters tion play of key dark roles. matterThe and In theory baryon this can densities paper, be in we in a investigated fact Left-Right the symmetric probed produc- Higgs by Parity the theory. warmness of DM, precise determination of SM in the freeze-in cosmology, safely avoiding the domain wallsurements problem. for Higgs Parity.particles New physics are scales heavy in and/orthese theories very theories of weakly by Higgs coupled Parity discovery to are of high. 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