Scalar Sector Phenomenology of Three-Loop Radiative Neutrino Mass Models
aa r X i v : . [ h e p - ph ] A ug August 2015
Scalar Sector Phenomenology of Three-LoopRadiative Neutrino Mass Models
Amine Ahriche, a,b, Kristian L. McDonald c, and Salah Nasri d, a Department of Physics, University of Jijel, PB 98 Ouled Aissa, DZ-18000 Jijel, Algeria b The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11,I-34014, Trieste, Italy c ARC Centre of Excellence for Particle Physics at the Terascale,School of Physics, The University of Sydney, NSW 2006, Australia d Physics Department, UAE University, POB 17551, Al Ain, United Arab Emirates
Abstract
We perform a phenomenological study of the scalar sector of two models thatgenerate neutrino mass at the three-loop level and contain viable dark matter can-didates. Both models contain a charged singlet scalar and a larger scalar multiplet(triplet or quintuplet). We investigate the effect of the extra scalars on the Higgsmass and analyze the modifications to the triple Higgs coupling. The new scalarscan give observable changes to the Higgs decay channel h → γγ and, furthermore,we find that the electroweak phase transition becomes strongly first-order in largeregions of parameter space. PACS : 12.60.Fr, 12.60.-I, 95.35.+d. [email protected] [email protected] [email protected] Introduction
The Standard Model (SM) of particle physics is a spectacularly successful theory thatstands as one of the truly great scientific achievements. Despite this success, however, thetheory possesses a number of short-comings, suggesting it will likely require extensionsand/or modifications in the future. The most obvious motivation for extending the SMis the need to incorporate gravity. However, a lack of present-day experimental guidancemakes the pursuit of the theory of quantum gravity an incredibly challenging task.Additional evidence that the SM is incomplete comes from the experimental observa-tion of neutrino mixing and the need to explain the missing gravitating (i.e. dark) matter,required on galactic scales. These puzzles motivate the addition of new particle species tothe SM and there is much hope that such new species will manifest in future experiments(in particular at Run II of the LHC). The neutrino mass and dark matter (DM) problemshave stimulated much research and there are multiple candidate solutions one can pursue.The problems may have independent solutions, though it seems reasonable to ask whetherthe two puzzles could share a unified or common solution. Could the neutrino mass andDM problems be related?In 2002, Krauss, Nasri and Trodden (KNT) proposed a simple extension of the SMmodel that admits a relationship between the existence of DM and the origin of neutrinomass [1]. In this approach, one adds new fields to the SM, such that neutrino mass isgenerated radiatively at the three-loop level, with one of the particles propagating in themass diagram being a DM candidate. The model employs two charged singlet scalars, S +1 , , and three generations of gauge-singlet fermions N . A Z symmetry with action { S +2 , N } → {− S +2 , − N } is also imposed. This ensures stability of the lightest fermion N , thereby giving a DM candidate, and also prevents a coupling between SM neutrinosand N , which would otherwise generate tree-level neutrino masses. The result is a typeof unified description for the origin of neutrino mass and DM, with the removal of theDM candidate simultaneously turning off neutrino mass. In recent years, a number of basic generalizations of the KNT model have appeared.In one such model (hereafter ‘the triplet model’), the singlet fields S +2 and N are replacedby SU (2) L triplets [6]. This model retains the Z symmetry to ensure DM stability andprevent tree-level neutrino masses, and gives a viable alternative unified framework forthe DM and neutrino mass problems. A further generalization exchanges the singletfields S +2 and N for SU (2) L quintuplet fields [7]. This model (hereafter ‘the quintuplet For recent studies of the KNT model see Refs. [2, 3, 4, 5]. Z symmetry in order to prevent tree-level neutrino masses,and is a viable theory for radiative neutrino mass, independent of DM considerations.Interestingly, the most-general Lagrangian for the model contains a single Z -breakingcoupling λ . When taken small, λ ≪
1, the model gives a long-lived DM candidate, whileturning λ off completely, λ →
0, activates a Z symmetry and gives absolutely stable DM.Thus, the quintuplet model is a viable model of radiative neutrino mass, with or withoutDM.Due to the presence of larger multiplets with non-trivial SU (2) charges in both thetriplet and quintuplet models, the phenomenology of the models is rather rich. In thepresent work, we extend the analysis of Refs. [6, 7] and undertake a more extensive study ofthe phenomenology of both models. We investigate the effect of the triplet and quintupletscalars on both the Higgs mass and the triple Higgs coupling, showing that the latter canexperience sizable modifications. We also study the effect of the new multiplets on theHiggs decay channels h → γγ, γZ . Our work shows that, e.g. observable changes areexpected to B ( h → γγ ), with some regions of parameter space already excluded for thetriplet model. The effect of the enlarged scalar sector on the electroweak phase transitionis also analyzed, revealing a tendency for a strongly first-order phase transition in largeregions of parameter space.Before proceeding we note that further generalizations of the KNT model are possible.Ref. [8] presented colored generalizations and other related three-loop models that employslightly modified loop topologies. A septuplet generalization of the KNT model wasproposed in Ref. [9]. This had the interesting feature of automatically containing anabsolutely stable DM candidate, without requiring a new symmetry. A minimal scale-invariant implementation was also recently studied [10]. More generally, a number ofauthors have studied connections between radiative neutrino mass and DM in recentyears, see e.g. Refs. [11]-[15].This paper is structured as follows. In Section 2, we outline the triplet and quintupletmodels, describing some key features and elucidating some stability constraints on thescalar potentials. We study the influence of the new multiplets on the Higgs mass andthe triple Higgs coupling in Section 3. The electroweak phase transition is considered inSection 4 and we turn to the Higgs decay channels h → γγ and h → γZ in Section 5.Conclusions are presented in Section 6. 2 Three-Loop Radiative Neutrino Masses
The SM employs the gauge symmetry G SM = SU (3) c × SU (2) L × U (1) Y . In this work,we consider extensions of the SM that include the charged singlet scalar S ∼ (1 , , T ∼ (1 , n + 1 ,
2) and three generations of chiral beyond-SM fermions, F i ∼ (1 , n + 1 , i = 1 , , , labels generations and numbers in parenthesis denotecharges under G SM . We use the integer n = 0 , , , to label the distinct models. Thecase with n = 0 is the KNT model, for which all beyond-SM fields are SU (2) L singlets: S ≡ S +1 , T ≡ S +2 and F ≡ N . For n = 1 ( n = 2) the multiplets T and F are SU (2) L triplets (quintuplets) and one has the triplet (quintuplet) model. In all cases, the newmultiplets are subject to a discrete symmetry with action { T, F } → {− T, −F } . Thisensures a stable DM candidate, which should be taken as the lightest fermion F ≡ F DM . Detailed analysis of the DM annihilation channels appears in Refs. [6, 7].With the aforementioned particle content, the Lagrangian contains the following terms:
L ⊃ { f αβ L cα L β S + + g iα F i T e αR + H . c } − F ci M ij F j − V ( H, S, T ) . (1)Here, L α ∼ (1 , , −
1) are SM lepton doublets, e αR ∼ (1 , , −
2) are the SM chargedlepton singlets and f αβ = − f βα denote Yukawa couplings. Lepton flavors are labeled bylower-case Greek letters, α, β ∈ { e, µ, τ } . The singlet-leptons e αR couple to the exotics T and F through the Yukawa matrix g iα , and the superscript “ c ” is used to denotecharge conjugation. The Lagrangian shows that both T and F are sequestered from SMneutrinos. We denote the SM Higgs doublet as H ∼ (1 , , ∼ S ( T ∗ ) in thescalar potential V ( H, S, T ) (discussed below) explicitly breaks lepton-number symmetry.The models therefore generate radiative neutrino masses, which appear at the three-looplevel as shown in Figure 1. Due to the Z symmetry, the neutral components of theexotic fermions F do not mix with SM neutrinos at any order in perturbation theory,and similarly there is no mixing between charged leptons and F . In both the tripletand quintuplet models, the charged scalar S can be within reach of TeV scale colliderexperiments [6, 7]. For n = 0 the scalar T is charged, while for n = 1 and n = 2 the neutral component of the scalar T has non-trivial couplings to the Z boson and id therefore excluded as a DM candidate by direct-detectionconstraints. S ∼ (1 , ,
2) and T ∼ (1 , n + 1 , F ∼ (1 , n + 1 ,
0) is a beyond-SM fermion. The case with n = 0 corresponds to the KNT model [1], while n = 1 gives the triplet model [6] and n = 2 gives the quintuplet model [7]. The case with n = 1 gives the triplet model, for which F i ∼ (1 , ,
0) and T ∼ (1 , ,
2) are SU (2) L triplets. We write the triplet fields in symmetric-matrix notation as T = T ab and F = F ab , where a, b ∈ { , } are SU (2) L indices. The multiplets contain the followingcomponents [6] F = F + L , F = F = 1 √ F L , F = F − L ≡ ( F + R ) c ,T = T ++ , T = T = 1 √ T + , T = T , (2)while the triplet mass term gives − ( F ci ) ab M ij ( F j ) cd ǫ ac ǫ bd = −F + iR M ij F + jL −
12 ( F iL ) c M ij F jL . (3)The neutral-fermion mass terms are brought to the correct sign by defining the Majoranafermions F i = F i,L − ( F i,L ) c . Radiative corrections from SM gauge bosons lift the degen-eracy between the components of F , leaving F as the lightest component [16], thoughfor most purposes this small splitting can be neglected [6]. We work in the diagonal basiswith M ij = diag( M , M , M ), where M ≡ M DM is the lightest triplet-fermion mass.According to the analysis in Ref. [6], the DM mass should lie in the range 2 . − . V ( H, S, T ) = − µ | H | + λ | H | + µ S | S | + λ S | S | + µ T [( T ∗ ) ab T ab ] + η T ∗ ) ab T ab ] + η T ∗ ) ab T bc ( T ∗ ) cd T da + λ SH | S | | H | + (cid:8) ¯ λ ST | S | + ¯ λ HT | H | (cid:9) [( T ∗ ) ab T ab ] − λ HT ( H ∗ ) a T ab ( T ∗ ) bc H c + λ ST S − ) T ab T cd ǫ ac ǫ bd + λ ∗ ST S + ) ( T ∗ ) ab ( T ∗ ) cd ǫ ac ǫ bd . (4)Vacuum stability requires that the quantities λ, λ S , η + η , η + η , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ (cid:0) λ HT − ¯ λ HT (cid:1) (cid:0) λ HT − ¯ λ HT (cid:1) η + η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ S ¯ λ ST ¯ λ ST η + η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5)are taken strictly positive, with λ = min( λ , µ S and the scalar triplet squared masses as positive ensures the absence of spontaneouscharge symmetry breaking and guarantees that h T i = 0. The latter is is necessary topreserve the Z symmetry and retain a stable DM candidate. The quintuplet model corresponds to n = 2, in which case one has F i ∼ (1 , ,
0) and T ∼ (1 , ,
2) as SU (2) L quintuplets. In symmetric-matrix notation, the quintuplets arewritten as T abcd and F abcd , where [7] F = F ++ L , F = F + L √ , F = F L √ , F = ( F + R ) c √ , F = ( F ++ R ) c ,T = T +++ , T = T ++ √ , T = T + √ , T = T √ , T = T − . (6)Observe that T + and T − are distinct fields with T − = ( T + ) ∗ . The explicit expansion ofthe fermion mass term gives −
12 ( F ci ) abcd M ij ( F j ) efgh ǫ ae ǫ bf ǫ cg ǫ dh +H . c . = −F ++ i M ij F ++ j −F + i M ij F + j − F i M ij F j , (7)where F is a Majorana fermion and the other four components of F combine to give twocharged (Dirac) fermions: F ++ = F ++ L + F ++ R , F + = F + L − F + R , F = F L + ( F L ) c . (8)Without loss of generality, we again employ the basis with M ij = diag( M , M , M ),where M ≡ M DM . According to the analysis of Ref. [7], the DM mass is expected to lie5n the range 5 . − .
95 TeV, and all quintuplet members should have masses that exceed M DM . Consequently the quintuplet scalars cannot be produced directly at the LHC.The full scalar potential for the quintuplet model that respects the global symmetry Z is given by V ( H, S, T ) = − µ | H | + λ | H | + µ S | S | + λ S | S | + λ SH | S | | H | + µ T [( T ∗ ) abcd T abcd ]+ η T ∗ ) abcd T abcd ] + η T ∗ ) abcd T cdef ( T ∗ ) eflm T lmab ] + η T ∗ ) abcd T bcde ( T ∗ ) eflm T aflm ]+ (cid:8) λ ST | S | + λ HT | H | (cid:9) [( T ∗ ) abcd T abcd ] + λ HT ( T ∗ ) abcd T ebcd ( H ∗ ) e H a + κ S − ) T abcd T efgh ǫ ae ǫ bf ǫ cg ǫ dh + H . c . (9)Vacuum stability requires that the following quantities λ, λ S , η + η + η , η + η + η , η + η + η > , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ (cid:0) λ HT + λ HT (cid:1) (cid:0) λ HT + λ HT (cid:1) η + η + η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ S λ ST λ ST λ ST η + η + η (cid:0) η + η + η (cid:1) λ ST (cid:0) η + η + η (cid:1) η + η + η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (10)be strictly positive. Similar to the triplet case, spontaneous charge symmetry breakingis avoided and the neutral quintuplet remains VEV-less by taking the squared masses ofthe charged scalar and the quintuplet to be positive. This preserves the Z symmetry.In the numerical scans performed below, we impose the above mentioned conditionssuch as vacuum stability, charge non-breaking and h T i = 0, and also require the Higgsmass to be within the range reported by ATLAS and CMS, m h = 125 . ∓ .
21 GeV[17]. We restrict our attention to the perturbativity domain, demanding that the physicalvertices in Eqs. (4) and (9) be less than 3. For both the triplet [6] and quintuplet [7]models, we consider the charged singlet scalar mass between 100 GeV and 1 TeV. Themasses for the scalar multiplet members should be larger than the dark matter mass, i.e., M T > .
35 TeV and M T > .
65 TeV, for the triplet and quintuplet models, respectively.For the numerical analysis we consider 20,000 sets of benchmark points for both the tripletand quintuplet models. The benchmarks reproduce the observed DM relic density whilealso achieving viable neutrino masses and avoiding lepton flavor violating constraints (seeRefs. [6, 7] for discussion on constraints). The degeneracy between neutral and charged components of F is again lifted by radiative corrections. Higgs Mass and Triple Higgs Coupling
In order to estimate the Higgs mass and the triple Higgs coupling at one-loop, it isnecessary to properly define the effective potential, with the Higgs mass being its secondderivative and the triple Higgs coupling given by the third derivative: m h = ∂ ∂h V T =0 eff ( h ) (cid:12)(cid:12)(cid:12)(cid:12) h = υ ,λ hhh = ∂ ∂h V T =0 eff ( h ) (cid:12)(cid:12)(cid:12)(cid:12) h = υ . (11)Here h is the real part of the neutral component in the doublet, υ is its VEV, and V T =0 eff ( h )is the zero temperature one-loop Higgs effective potential. In this work we employ the DR ′ scheme, for which the effective potential is given by [18] V T =0 eff ( h ) = − µ h + λ h + X i n i m i ( h )64 π (cid:18) log m i ( h )Λ − (cid:19) , (12)where n i is the field multiplicity and Λ is the renormalization scale, which we take asthe measured value of the Higgs mass, Λ = 125 .
09 GeV [17]. The quantities m i ( h )are the field-dependent squared masses (presented in the appendix for both triplet andquintuplet models). In this class of models, h is the only scalar with a non-zero VEV, soall field-dependent masses can be written as m i ( h ) = µ i + α i h /
2, for constant α i .At tree-level, the parameter µ in the potential is given by µ = λυ . After includingone-loop corrections, the parameter µ is corrected as µ = λυ + 132 π X i n i α i m i (cid:18) ln m i Λ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) h = υ,µ ≡ µ + δµ , (13)in order to ensure the one-loop VEV value remains as υ = 246 GeV. The term δµ represents the radiative corrections to the µ -term, due to all fields, and is expected to bedominated by contributions from the new heavy fields. The Higgs mass at one-loop canbe similarly defined by m h = 2 λυ + υ π X i n i α i log m i Λ , (14)where the radiative corrections (i.e., second term in (14)) are also expected to be domi-nated by contributions from heavy new fields. Although m h is determined by experimentalobservations, the doublet quartic coupling λ can still be (very) small, relative to the SMvalue, while reproducing the observed Higgs mass. According to the size and sign of the7ne-loop contribution in (14), the quartic coupling λ must be smaller (larger) than thetree-level value 3 m h /υ , for a positive (negative) loop contribution.The triple Higgs coupling is the third derivative of (12), which can be simplified as λ hhh = 6 λυ + υ π X i n i α i (cid:18) α i υ m i + 3 log m i Λ (cid:19) . (15)Using (13) and (14), the triple Higgs coupling (15) can be simplified as λ hhh = 3 m h υ υ π m h X i n i α i m i ! . (16)The size of the radiative effects can be parameterized by the following dimensionlessquantities: δµ = µ − λυ µ , δm h = m h − λυ m h , δλ hhh = λ hhh − λυλ hhh . (17)These measure the relative strength of the radiative contributions to the Higgs bare mass-squared, the physical mass-squared µ , and the triple Higgs coupling, respectively. Usingthe previously mentioned benchmark points, in Fig. 2 we plot the triple Higgs couplingversus the mass-squared parameter µ , for both triplet and quintuplet models. We alsoshow the relative strength of the radiative contributions to the Higgs mass, triple Higgscoupling and the parameter µ , as defined in Eq. (17).One notices from Fig. 2-Top that the mass-squared parameter µ can be large, evenup to 100 (500) times the Higgs VEV-squared υ for the triplet (quintuplet) models. Thelarger values are required in order to balance the radiatively induced mass term in theLagrangian, i.e., the second term on the left-hand side of (13). The radiative correctionscan also be negative, depending on the value of the Higgs quartic coupling; i.e., for λ & .
08. Due to the fact that the extra fields in the quintuplet model are much heavierthan those of the triplet model, their radiative contributions are larger and therefore the µ parameter values are larger, as it is evident from the figures. From Fig. 2-Bottom,one notices that the relative radiative-contributions to the Higgs mass and triple couplingare proportional, i.e. when the Higgs mass is completely generated radiatively, the tripleHiggs coupling is also dominated by radiative effects. We also observe that for most of thebenchmark points, in both the triplet and quintuplet models, the mass-squared parameter µ is fully radiative, as shown in the palette.The relevant quantity for collider phenomenology is the relative enhancement in thetriple Higgs coupling, with respect to the SM value, which is defined as∆ = λ hhh − λ SMhhh λ SMhhh . (18)8 λ hhh / υ µ / υ Triplet Model 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15-300 -200 -100 0 100 200 300 400 500 600 λ hhh / υ µ / υ Quintuplet Model 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16-0.2 0 0.2 0.4 0.6 0.8 110 -5 -4 -3 -2 -1 δ λ hhh δ m h2 Triplet Model -1.5-1-0.5 0 0.5 1 1.5 2 -0.2 0 0.2 0.4 0.6 0.8 110 -5 -4 -3 -2 -1 δ λ hhh δ m h2 Quintuplet Model -1.5-1-0.5 0 0.5 1 1.5 2
Figure 2:
Top: the triple Higgs coupling versus the mass parameter parameter µ , both inunits of the Higgs VEV, for the triplet (left) and quintuplet (right) models. The palettegives the Higgs quartic coupling λ , and the red line shows the SM triple Higgs couplingvalue. Bottom: the relative radiative contribution to the triple Higgs coupling versus therelative radiative contribution to the Higgs mass. The palette gives the relative radiativecontribution to the Higgs mass parameter δµ . According to Eq. (16), the relative enhancement of the triple Higgs coupling is given by∆ = P i = SM n i α i m i π m h υ + P i = all n i α i m i . (19)In Fig. 3, we show the relative enhancement of the triple Higgs coupling, Eq. (19), for thebenchmark sets used previously. The figure shows that the relative enhancement of thetriple Higgs coupling, with respect to the SM, are larger for large values of the quarticcoupling λ , and smaller for small values of the charged scalar mass. Also, one notices thatthe relative enhancement in the triple Higgs coupling, with respect to the SM value, isalways positive, contrary to other models [19, 20], and furthermore it can exceed 35% for9 -5 -4 -3 -2 -1 ∆ = ( λ hhh - λ hhh S M ) / λ hhh S M λ Triplet Model 100 200 300 400 500 600 700 800 900 1000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510 -5 -4 -3 -2 -1 ∆ = ( λ hhh - λ hhh S M ) / λ hhh S M λ Quintuplet Model 100 200 300 400 500 600 700 800 900 1000
Figure 3:
The relative enhancement in the triple Higgs coupling with respect to the SM, ∆ , versus the Higgs quartic coupling. The palette shows the mass of the charged scalar S + in GeV. The black point at ( λ = λ SM , ) refers to the SM. both the triplet and quintuplet models. The SM cannot successfully explain baryogenesis [21] for two reasons: (1) the CP violatingsource in the CKM matrix is too small and (2) the electroweak phase transition (EWPT)is not strongly first order. The latter is required to suppress the B+L violating processesin the broken phase, inside the bubble, when its wall is expanding during the transition.In the SM, the criterion for a strongly first-order EWPT [22], υ ( T c ) /T c > , (20)is not fulfilled since the ratio is given by υ c /T c ∼ λ , which would require a Higgs massbelow 42 GeV [23]. Here T c is the critical temperature at which the effective potentialexhibits two degenerate minima, one at zero and the other at υ ( T c ). Both T c and υ ( T c )are determined using the full effective potential at finite temperature, which is given by[24] V eff ( h, T ) = V T =0 eff ( h ) + T π X i n i J B,F (cid:0) m i /T (cid:1) + V ring ( h, T ); (21) J B,F ( α ) = Z ∞ x log(1 ∓ exp( −√ x + α )) . (22)In the above, we include an important leading term from the higher-order loop corrections,which can play an important role during the EWPT dynamics, namely the so-called daisy10ontributions [25] V ring ( h, T ) = − T π X i n i (cid:8) ˜ m i ( h, T ) − m i ( h ) (cid:9) . (23)The summation is over scalar and longitudinal gauge degrees of freedom, with ˜ m i ( h, T ) = m i ( h ) + Π( T ) their thermal masses, and Π( T ) are the thermal parts of the self energy(given in the appendix). For our analysis we include the daisy contributions by followingan alternative approach to Eq. (23), i.e. by replacing the field dependent masses of thescalar and longitudinal gauge fields by their thermal masses ˜ m i ( h, T ) in the full effectivepotential (21). In order to account for all the (heavy and light) degrees of freedom, weevaluate the integrals (22) numerically.The strength of the EWPT can be improved when new bosonic degrees of freedomare present, as occurs in the present models. It is clear from (14) that for large valuesfor the couplings { λ SH , λ HT , ¯ λ HT for triplet model and λ SH , λ HT , for quintuplet model } and/or small mass-values for the extra (singlet and multiplet) scalars, the one-loop correc-tion to the Higgs mass can be significant, allowing the Higgs self-coupling to be smaller.Consequently one can fulfill the criterion (20) without conflicting with recent Higgs massmeasurements [17].Analyses of similar models [4, 19, 26] has shown that extra scalars can help to generatea strongly first order EWPT by: (a) relaxing the Higgs self-coupling λ to be as small as O (10 − ); and (b) enhancing the value of the effective potential at the wrong vacuum atthe critical temperature, without suppressing the ratio υ ( T c ) /T c , which relaxes the severebound on the mass of the SM Higgs. υ c / T c T c (GeV) Triplet Model 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.6 0.8 1 1.2 1.4 1.6 100 110 120 130 140 150 υ c / T c T c (GeV) Quintuplet Model 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Figure 4:
The ratio υ ( T c ) /T c versus the critical temperature for the (left) triplet and (right)quintuplet models. The palette shows the Higgs quartic coupling λ .
11n Fig. 4, we plot the ratio υ ( T c ) /T c verses the critical temperature T c , using the20,000 benchmark points for the triplet and quintuplet models. The figure shows that theEWPT is strongly first-order for a majority of the benchmark sets, with the ratio υ ( T c ) /T c predominantly taking values between 1.2 and 1.5 in both the triplet and quintuplet models.The transition temperature is a bit larger than the typical SM value ∼
100 GeV, and canbe as large as 150 GeV and 140 GeV for the triplet and quintuplet models, respectively.One can read from the palettes in Fig. 4 that, for fixed Higgs quartic coupling values, theratio υ ( T c ) /T c is inversely proportional to the critical temperature.Inspecting Fig. 4, one is lead to the conclusion that the increased EWPT strength isnot only a consequence of a small Higgs quartic coupling λ , but can also be due to thetransition dynamics; the existence of heavy scalars makes the Higgs VEV slowly decayingwith respect to the temperature. Consequently the evolving (increasing or decreasing)effective potential at the wrong vacuum makes the transition occurring at the mentionedtemperature values, therefore giving a large ratio υ ( T c ) /T c .An interesting issue, discussed in the literature [27], is a possible correlation betweenthe EWPT strength and the relative enhancement in the triple Higgs coupling ∆. InRef. [19] it was shown that such a correlation is not clear for a model with extra chargedand neutral scalars from two inert doublets. In Fig. 5, we plot the relative enhancementof the triple Higgs coupling, ∆, versus the ratio υ ( T c ) /T c , for the 20,000 benchmarks forthe triplet and quintuplet models. From Figure 5, it is not clear whether a correlationbetween the relative enhancement in the triple Higgs coupling and the EWPT strengthexists. This issue deserves a detailed and model-independent investigation. h → γγ and h → γZ In July 2012, the ATLAS [28] and CMS [29] collaborations announced the observation of ascalar particle with mass ≃
125 GeV, with roughly 5 σ confidence level. Subsequently thisvalue was updated to m h = 125 . ± .
21 GeV [17]. An important question is whether ornot this really is the SM Higgs or an alternative Higgs-like state with different properties.Indeed, a fit to the data, performed by both the ATLAS and CMS collaborations, seemsto (almost) show agreement with the SM, with the reported values being 1 . ∓ .
27 [30]and 1 . ± .
24 [31], respectively.Defining R γγ as the branching ratio of h → γγ scaled by the SM value, we find that12 ∆ = ( λ hhh - λ hhh S M ) / λ hhh S M υ c /T c TripletQunituplet
Figure 5:
The relative enhancement in the triple Higgs coupling ∆ versus the ratio υ ( T c ) /T c for the benchmark points used previously. the present models give R γγ = B ( h → γγ ) B SM ( h → γγ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) υ P i ϑ i m Xi A γγ ( τ i ) A γγ ( τ W ) + N c Q t A γγ / ( τ t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (24)where i stands for all charged scalars X i , τ i = m h / m X i , with m X i being the mass ofthe charged particle X i running inside the loop, N c = 3 is the color number, and Q t isthe top quark electric charge in units of | e | . The parameters ϑ i are given for triplet andquintuplet members in Table 1. In the above, the loop amplitudes A γγk for spin 0, spin1 / A γγ ( x ) = − x − [ x − f ( x )] ,A γγ / ( x ) = 2 x − [ x + ( x − f ( x )] ,A γγ ( x ) = − x − (cid:2) x + 3 x + 3 (2 x − f ( x ) (cid:3) , (25) f ( x ) = arcsin ( √ x ) x ≤ − h log √ − x − −√ − x − − iπ i x > . (26)Another important Higgs decay channel that can be modified by the presence of extracharged scalars is h → γZ . This channel is similarly parameterized as R γZ = B ( h → γZ ) B SM ( h → γZ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s w c w υ P i κ i m Xi A γZ ( τ i , ζ i ) c A γZ ( τ W , ζ W ) + 2 (1 − s / A γZ / ( τ t , ζ t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (27)13odel Charged fields ϑ i κ i S + λ SH s w c w λ SH Triplet T + 2 λ HT − ¯ λ HT s w c w λ HT − ¯ λ HT T ++ λ HT − − s w s w c w λ HT S + λ SH s w c w λ SH T − λ HT + λ HT s w s w c w ( λ HT + λ HT )Quintuplet T + 2 λ HT + λ HT s w c w λ HT + λ HT T ++ λ HT + λ HT − − s w s w c w λ HT + λ HT T +++ λ HT − − s w s w c w λ HT Table 1: The parameters ϑ i and κ i , which are relevant for the Higgs decay channels h → γγ and h → γZ .where ζ i = m Z / m X i , and the functions A γZk are given by [32] A γZ ( x, y ) = I ( x, y ) ,A γZ / ( x, y ) = I ( x, y ) − I ( x, y ) ,A γZ ( x, y ) = (cid:2) (1 + 2 x ) s /c − (5 + 2 x ) (cid:3) I ( x, y ) + 4 (cid:0) − s /c (cid:1) I ( x, y ) , (28) I ( x, y ) = − x − y ) + f ( x ) − f ( y )2( x − y ) + y [ g ( x ) − g ( y )]( x − y ) , I ( x, y ) = f ( x ) − f ( y )2( x − y ) ,g ( x ) = √ x − − √ x ) x ≤ √ − x − h log √ − x − −√ − x − − iπ i x > . (29)The parameters κ i are shown in Table 1 for both the triplet and quintuplet models.The deviation of the channels h → γγ, γZ from their SM values is sensitive to themass of the scalars and the strength with which they couple to the Higgs doublet, i.e. onthe parameters m S , m , m , λ SH , λ HT and ¯ λ HT for the triplet model, and on m S , m , m , m , λ SH and λ HT , for the quintuplet model. Depending on the relative signof the couplings to the Higgs doublet, the new contributions can strengthen or weakenthe deviation of B ( h → γγ ) from its SM value. In Fig. 6, we present R γγ versus R γZ forthe considered 20,000 sets of benchmark parameters for both the triplet and quintupletmodels.We remark that some benchmarks in the triplet model are already excluded by therecent measurements of ATLAS [17] and CMS [31], while more precise measurementsare required to probe benchmarks for the quintuplet model. In contrast to other modelswith extra singlets [4] and doublets [19], the decay channel h → γZ can be significantly14 R γγ R γ Z ATLASCMS
Figure 6:
The modified Higgs decay rates B ( h → γγ ) vs B ( h → γZ ) , scaled by their SMvalues, due to the extra charged scalars, for 20,000 randomly chosen sets of benchmarkparameters for the triplet (red) and quintuplet (green) models. The intervals between themagenta (green) lines represent the ATLAS (CMS) recent measurements on the h → γγ channel, and the blue point represents the SM. modified, with respect to the SM value, particularly in the quintuplet model. This can beunderstood from the large κ i coupling values for the scalar multiplet members in Table 1. Models of radiative neutrino mass with DM candidates can explain some of the short-comings of the SM while generating observable experimental signals. In this work, weperformed a detailed study of the scalar-sector phenomenology for a pair of three-loopneutrino mass models with DM candidates. The models, referred to as the triplet [6] andquintuplet [7] models, generate neutrino mass via a diagram with the same topology as theKNT model. We investigated the effect of the extra scalars on the Higgs mass, the tripleHiggs coupling, and the Higgs decay channels h → γγ, γZ . We also studied the strengthof the electroweak phase transition. In both models, it was shown that the beyond-SMmultiplets can modify the triple Higgs coupling and the Higgs decay channels away fromtheir SM values. The electroweak phase transition was found to be strongly first-order insignificant regions of parameter space. Measurements of the Higgs decay channels alreadyexclude some regions of parameter space for the triplet model, and future improvementswill further explore the parameter space for both models.15 cknowledgments AA thanks the ICTP for the hospitality during the last stage of this work. AA is supportedby the Algerian Ministry of Higher Education and Scientific Research under the CNEPRUProject No D01720130042. KM is supported by the Australian Research Council.
A Field Dependent Masses
The charged scalar and SM field-dependent masses are given by: m χ = − µ + λh + Π H , m h = − µ + 3 λh + Π H , m t = y t h , m W = m W W = g h + Π W ,m BB = g h + Π B , m W B = g g h , m S = µ S + λ SH h + Π S , (30)with Π H = (cid:0) λ + 9 g + 3 g ′ + 3 y t + 2 λ SH (cid:1) T , Π S = (cid:0) λ SH + 4 λ S + 3 g ′ (cid:1) T , Π LW = Π LW = 116 g T , Π LB = 2716 g ′ T , Π TW = Π TB = 0 . (31)Here, we ignored the triplet and quintuplet contributions since they decouple from thethermal plasma due to their large masses, relative to the relevant typical temperature of O (100 GeV).The triplet members field dependant masses are given by: m = µ T + λ HT − ¯ λ HT h + Π T , m = µ T + 2 λ HT − ¯ λ HT h + Π T ,m = µ T + λ HT h + Π T , Π T = (cid:0) g + 3 g ′ + 2¯ λ HT + 4 λ HT + 2¯ λ ST (cid:1) T , (32)and the quintuplet members field dependant masses are given by: m − = µ T + λ HT + λ HT h + Π T , m = µ T + 4 λ HT + 3 λ HT h + Π T ,m = µ T + 2 λ HT + λ HT h + Π T , m = µ T + 4 λ HT + λ HT h + Π T ,m = µ T + λ HT h + Π T , Π T = (cid:0) g + 3 g ′ + 2 λ ST + 4 λ HT + 2 λ HT (cid:1) T . (33) References [1] L. M. Krauss, S. Nasri and M. Trodden, Phys. Rev. D , 085002 (2003)[hep-ph/0210389]. 162] E. A. Baltz and L. Bergstrom, Phys. Rev. D , 043516 (2003) [hep-ph/0211325].[3] K. Cheung and O. Seto, Phys. Rev. D , 113009 (2004) [hep-ph/0403003].[4] A. Ahriche and S. 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