RGE of a Cold Dark Matter Two-Singlet Model
aa r X i v : . [ h e p - ph ] A p r RGE of a Cold Dark Matter Two-Singlet Model
Abdessamad Abada a , b ∗ and Salah Nasri b † a Laboratoire de Physique des Particules et Physique Statistique,Ecole Normale Sup´erieure, BP 92 Vieux Kouba, 16050 Alger, Algeria b Physics Department, United Arab Emirates University,POB 17551, Al Ain, United Arab Emirates (Dated: October 5, 2018)
Abstract
We study via the renormalization group equations at one-loop order the perturbativity andvacuum stability of a two-singlet model of cold dark matter (DM) that consists in extending theStandard Model with two real gauge-singlet scalar fields. We then investigate the regions in theparameter space in which the model is viable. For this, we require the model to reproduce theobserved DM relic density abundance, to comply with the measured XENON 100 direct-detectionupper bounds, and to be consistent with the RGE perturbativity and vacuum-stability criteria upto 40TeV. For small mixing angle θ between the physical Higgs h and auxiliary field, and DM- h mutual coupling constant λ (4)0 , we find that the auxiliary-field mass is confined to the interval116GeV − θ enriches the existing viability regions without relocating them, while increasing λ (4)0 shrinksthem with a tiny relocation. We show that the model is consistent with the recent Higgs boson-likediscovery by the ATLAS and CMS experiments, while very light dark matter (masses below 5GeV)is ruled out by the same experiments. PACS numbers: 95.35.+d; 98.80.-k; 12.15.-y; 11.30.Qc.Keywords: cold dark matter. light WIMP. extension of Standard Model. RGE. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Now that there is more and more compelling evidence that the discovery in the ATLASand CMS experiments at the LHC is a Higgs particle with a mass m h ≃ − Z symmetry. Based on the DM relic-density and WIMPdirect detection studies, we have concluded that this two-singlet model is capable of bearingdark matter in a large region of the parameter space. Further constraints on the modelas well as some of its phenomenological implications have been studied in [15] where raremeson decays and Higgs production channels have been considered.2n the present work, we further the study of the two-singlet model and ask how high inthe energy scale it is computationally reliable. A standard treatment is the investigation ofthe running of the coupling constants in terms of the mass scale Λ via the renorrmalizationgroup equations (RGE). We believe one-loop calculations are amply sufficient for the presenttask; higher loops could be considered if the situation changes.The two standard issues to monitor are the perturbativity of the scalar coupling constantsand the vacuum stability of the theory. These issues were studied in [16] for the complexscalar singlet extension of the SM, and it was shown that the vacuum-stability requirementcan affect the DM relic density. Specific results from such studies depend on the cutoff scaleΛ m of the theory. Reversely, imposing perturbativity and vacuum stability may indicate atwhat Λ m the two-singlet extension is valid. In the early parts of this work, the second pointof view is adopted, whereas in the later part, the first is taken. Furthermore, up to onlyrecently, it has been anticipated that new physics such as supersymmetry would appear atthe LHC at the scale Λ ∼ m may be higher. As we shalldiscuss, we find that it can be ∼ m has to be made. Section V attempts atfinding the regions in the parameter space in which the model is predictive. In addition tothe DM relic-density constraint and the perturbativity-stability criteria deduced from theprevious two sections, we impose on the model to be within the current direct-detectionexperimental bounds. Section VI is devoted to concluding remarks.3 I. THE TWO-SINGLET MODEL
The model is obtained by adding to the Standard Model two real, spinless, and Z -symmetric SM-gauge-singlet fields. One is the dark matter field S with unbroken Z sym-metry, and the other an auxiliary field χ with spontaneously broken Z symmetry. Bothfields interact with the SM particles via the Higgs doublet H . Using the same notation asin [14], the potential function that involves S , H and χ is: U = ˜ m S − µ H † H − µ χ + η S + λ (cid:0) H † H (cid:1) + η χ + λ S H † H + η S χ + λ H † Hχ , (2.1)where ˜ m , µ and µ and all the coupling constants are real positive numbers .We are interested in monitoring the running of the scalar coupling constants. A one-looprenormalization-group calculation yields the following β -functions: β η = 316 π (cid:0) η + η + 4 λ (cid:1) ; β η = 316 π (cid:0) η + η + 4 λ (cid:1) ; β λ = 316 π (cid:18) λ + λ + λ − λ t + 8 λλ t − λg − λg ′ + 32 g g ′ + 94 g (cid:19) ; β η = 116 π (cid:0) η + η η + η η + 4 λ λ (cid:1) ; β λ = 116 π (cid:18) λ + λ η + 2 λ λ + η λ + 12 λ λ t − λ g − λ g ′ (cid:19) ; β λ = 116 π (cid:18) λ + λ η + 2 λ λ + η λ + 12 λ λ t − λ g − λ g ′ (cid:19) . (2.2)As usual, β g ≡ dg/d ln Λ where Λ is the running mass scale, starting from Λ = 100GeV.The constants g , g ′ and g s are the SM and strong gauge couplings, known [17] and given toone-loop order by the expression: G (Λ) = G (Λ ) r − a G G (Λ ) ln (cid:16) ΛΛ (cid:17) , (2.3)where a G = − π , π , − π , and G (Λ ) = 0 .
65, 0 . , . G = g, g ′ , g s respectively. Thecoupling constant λ t is that between the Higgs field and the top quark. To one-loop order, The mutual couplings can be negative as discussed below, see (3.1).
4t runs according to [17]: β λ t = λ t π (cid:18) λ t − g s − g − g ′ (cid:19) , (2.4)with λ t (Λ ) = m t (Λ ) v = 0 .
7, where v is the Higgs vacuum expectation value (vev) and m t the top mass. Note that we are taking into consideration the fact that the top-quarkcontribution is dominant over that of the other fermions of the Standard Model.The model undergoes two spontaneous breakings of symmetry: one of the electroweak,with a vev v = 246 GeV, and one of the Z symmetry ( χ field), with a vev v we take inthis work equal to 150 GeV. Above v , the fields and parameters of the theory are thoseof (2.1). Below v , the (scalar) physical fields are S (DM), h (Higgs) and S (auxiliary),with parameters (masses and coupling constants) given in Eqs. (2.2–2.15) of [14]. We takethe values of the physical parameters at the mass scale Λ = 100 GeV. There are originallynine free physical parameters. The two vevs v and v are fixed, as well as the mass of thephysical Higgs field m h = 125 GeV [1, 2]. Also, the physical mutual coupling constant η (4)01 between S and S is determined by the DM relic-density constraint [18], which translatesinto the condition: h v σ ann i ≃ . × − GeV − , (2.5)where h v σ ann i is the thermally averaged annihilation cross-section of a pair of two DMparticles times their relative speed in the center-of-mass reference frame. This constraint isimposed throughout this work, together with the perturbativity restriction 0 ≤ η (4)01 ≤ √ π on its solution. The remaining free parameters of the model are the physical mutual couplingconstant λ (4)0 between h and S , the mixing angle θ between h and S , the DM mass m ,the mass m of the auxiliary physical field S , and the DM self-coupling constant η . Thislatter has so far been decoupled from the other coupling constants [14, 15], but not anymorein view of (2.2) now that the running is the focus. However, its initial value η (Λ ) isarbitrary and its β -function is always positive. This means η (Λ) will only increase asΛ increases, quickly if starting from a rather large initial value, slowly if not. Therefore,without loosing generality in the subsequent discussion, we fix η (Λ ) = 1. Hence, here toowe still effectively have four free parameters: λ (4)0 , θ , m , and m . The initial conditions for5he coupling constants in (2.1) in terms of these physical free parameters are as follows: η (Λ ) = 32 v (cid:20) m + m h + (cid:12)(cid:12) m − m h (cid:12)(cid:12) (cid:18) cos (2 θ ) + v v sin (2 θ ) (cid:19)(cid:21) ; λ (Λ ) = 32 v h m + m h − (cid:12)(cid:12) m − m h (cid:12)(cid:12) (cid:16) cos (2 θ ) − v v sin (2 θ ) (cid:17)i ; λ (Λ ) = sin (2 θ )2 vv (cid:12)(cid:12) m − m h (cid:12)(cid:12) ; η (Λ ) = 1cos (2 θ ) h η (4)01 cos θ − λ (4)0 sin θ i ; λ (Λ ) = 1cos (2 θ ) h λ (4)0 cos θ − η (4)01 sin θ i . (2.6)Note that normally, as we go down the mass scale, we should seam quantities in steps: at v , v , and Λ . However, the corrections to (2.6) are of one-loop order times ln vv or ln v Λ ,small enough for our present purposes to neglect. III. SCALARS ONLY
To see the effects of the scalar couplings only and how up in the mass scale the modelcan go, we switch off the non-Higgs SM couplings in (2.2). The perturbativity constraintwe impose on all dimensionless scalar coupling constants is G (Λ) ≤ √ π . Vacuum stabilitymeans that G (Λ) ≥ η , λ , and η , and the conditions: − p η λ ≤ λ ≤ √ π ; − √ η η ≤ η ≤ √ π ; − p η λ ≤ λ ≤ √ π (3.1)for the mutual couplings λ , η , and λ . Also, as a start, we let the masses m and m varyin the interval 1GeV − = 10 GeVup to 10 GeV. As is expected for scalars only, all coupling constants are increasing functionsof the scale Λ, with different but increasing rates. Also, the larger value the coupling startsfrom at Λ , the faster it will go up. Fig. 2 shows the running of the mutual coupling constantsfor the same values of the parameters. For these values, the mutual coupling constants startwell below 1, and so run low; they are very much dominated by the self-couplings. Thissituation will stay for λ and λ in all regions, but not for η .The first coupling constant that leaves the perturbativity bound √ π is η , the self-coupling constant of the auxiliary scalar field χ , at about 1260 TeV for this set of values of6 T = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Η g = Η FIG. 1: The running of the self-couplings (scalars only). The self-coupling η of the auxiliary field χ dominates over the Higgs self-coupling λ . the parameters. This behavior is in fact typical. Indeed, η starts above 2 at Λ in all theparameter space, much higher than all the other coupling constants – only η can competewith it in some regions. As it intervenes squared in its own β -function, it will also moveup quicker. More precisely, from (2.6), we see that η (Λ ) depends on m and θ only. Theeffect of the mixing angle θ is small. As a function of m , starting from about 2, η (Λ )decreases slightly until m h and then picks up. It will pass the perturbativity bound √ π atabout m ≃ m > η (Λ) and evenin the case of the full RGE (see below), perturbativity puts a stricter upper bound on m ,irrespective of the other parameters of the model.In Fig. 1, the Higgs self-coupling λ starts just above 0.6 and does not pick up much whenrunning. This behavior is typical too. Indeed, λ (Λ ) is also a function of m and θ only,see (2.6). For a given θ , it will increase as a function of m to reach 3 ( m h /v ) at m = m h ,equal here to 0 .
77 for m h = 125GeV. Then it continues to increase, but with a smaller7 = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Λ g = Η FIG. 2: Running of the mutual couplings (scalars only). For these values of the parameters, allthree are well below the self-couplings. slope. The mixing angle θ enhances the behavior of λ (Λ ) as a function of m , but for, say θ = 15 o , λ (Λ ) will be less than 0 .
85 at m = 160 GeV. This situation implies that whenrunning as a function of the scale Λ, the Higgs self-coupling λ will increase, but will hardlyreach 1 before, say η , leaves the perturbativity bound.Increasing the dark-matter mass does affect the running of the couplings. Figs. 3 (self-couplings) and 4 (mutual couplings) display such effects for m = 100GeV. Among theself-couplings, η is still dominant, but tailed more closely by η this time. For both, thepositive acceleration is accentuated, something that makes η leave the perturbativity regionmuch earlier, at about 6 . λ staysflat.The major effect of increasing m is on the mutual coupling η , between the DM field S and the auxiliary field χ . Indeed, in Fig. 2 where m = 55GeV, η started and ran small We are implicitly confining the mixing angle θ to small values, a situation inferred from our work [15] onthe phenomenological implications of the model. This is discussed later in section V. = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Η g = Η FIG. 3: Running of the self-couplings (scalars only) for a larger DM mass. η is still dominant,even if η picks up faster behind it. like the other two mutual couplings. Here, whereas λ and λ (both Higgs related) stay closeto zero, η starts above 2.5 and runs up fast. In fact, for these values of the parameters, itleaves the perturbativity region earlier than η , at about 2 . m has a similar effect: it enhances the positive ac-celeration of the self-couplings η and η while leaving λ flat, and boosts up the mutualcoupling η away from λ and λ , which both remain not far from zero. It also makes η and η leave the perturbativity region earlier, without η necessarily taking over from η .Increasing λ (4)0 has also an effect. Figs. 5 (self) and 6 (mutual) show the running for λ (4)0 = 0 .
4. The self-coupling η dominates and leaves the perturbativity region at about251TeV. The mutual coupling η is raised above 1 at Λ and so can run high, while λ and λ stay here too just above zero. It leaves the perturbativity region at about 794GeV, wellbehind η . Higher values of λ (4)0 are more difficult to achieve as the relic-density constraint(2.5) may not be satisfied [14].Finally, changing the mixing angle θ has little effect on the self-coupling constants. It9 = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Λ g = Η FIG. 4: Running of the mutual couplings (scalars only) for a larger DM mass. η starts above 1,much higher than the two others, even higher than the self-coupling η . helps the mutual coupling constants η and λ start higher, but not by much: they staywith λ well below one. IV. THE FULL RGE
In the previous situation, ‘scalars only’, all running coupling constants were positive,and so there were no issues related to vacuum stability. We now reintroduce the other SMparticles and see their effects. Fig. 7 displays the behavior of the self-couplings under thefull RGE for the same values of the parameters as in Fig. 1 (scalars only). The dramaticeffect is on the Higgs self-coupling constant λ which quickly gets into negative territory, atabout 15TeV, thus rendering the theory unstable beyond this mass scale. This is betterdisplayed in Fig. 8 where the RG behavior of λ is shown by itself. Such a negative slope for λ is expected, given the negative contributions to β λ in (2.2). Here too η is dominant overthe other couplings and still controls perturbativity, leaving its region much later, at about1600 TeV, farther away from the situation ‘scalars only’. This looks to be a somewhat general10 = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Η g = Η FIG. 5: Running of the self-couplings (scalars only) with a larger physical Higgs self-coupling λ (4)0 .The self-coupling η is still dominant. trend: the non-Higgs SM particles seem to flatten the runnings of the scalar couplings.The runnings of the mutual coupling constants for the same set of parameters’ values isdisplayed in Fig. 9. They also get flattened by the other SM particles, but they stay positive.Here too they dwell well below the self-couplings, as in the ‘scalars only’ case. In fact, manyof the effects on the running coupling constants coming from varying the parameters aresimilar to those of the previous situation since the SM particles do not intervene in theinitial values of the couplings (self and mutual) at Λ . This means that increasing m and m will raise the mutual coupling η and not the two others, higher than η in some regions.For example, Fig. 10 shows the running of the self-couplings when m = 100GeV. Both η and η run faster but λ is little affected. Fig. 11 shows the running of the mutual couplingsfrom the full RGE also at m = 100GeV. As in the case ‘scalars only’, larger m boosts up η (Λ ), much higher than λ and λ , at about 2.2 here, which makes it run quickly high,leaving the perturbativity region before η , as in the case ‘scalars only’.Raising λ (4)0 will also make the self-couplings η and η run faster while affecting very11 = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Λ g = Η FIG. 6: Running of mutual couplings (scalars only). Larger λ (4)0 helps η rise well above λ and λ , but not enough to win over the self-coupling η . little λ . It will also make the mutual coupling η starts higher, and so demarked from λ and λ . By contrast, the effect of θ is not very dramatic: the self-couplings are not muchaffected and the mutuals only evolve differently, without any particular boosting of η . V. REGIONS OF VIABILITY
The foregoing discussion showed us how the scalar parameters of the two-singlet modelbehave as a function of the mass scale Λ. From the situation ‘scalars only’ we understoodthat the two couplings that control perturbativity are η and η . The full RGE brought instability: the change of sign of λ is the vacuum stability criterion to use. Equipped withthese indicators, we can try to investigate in a more systematic way the viability regions ofthe model, regions in the space of parameters in which the model is predictive. Rememberthat this model has four parameters: the dark-matter mass m , the physical auxiliary fieldmass m , the physical Higgs self-coupling λ (4)0 , and the mixing angle θ between the physicalHiggs and the auxiliary field. The way we proceed is to vary λ (4)0 and θ and try to find the12 = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Η g = Η FIG. 7: Running of the self-couplings (full RGE). η controls perturbativity and the Higgs coupling λ becomes negative quickly. regions of viability of the model in the ( m , m )-plane.We have by now a number of tools at our disposal. First the DM relic-density constraint(2.5), which has been applied throughout and will continue so. We have the RGE analysisof this work. We will require both η (Λ) and η (Λ) to be smaller than √ π , and λ (Λ) tobe positive.There is one important issue to address though before we proceed, and that is how farwe want the model to be perturbatively predictive and stable. The maximum value Λ m forthe mass scale Λ should not be very high for two reasons. One, more conceptual, is that wewant to recognize and allow the model to be intermediary between the Standard Model andsome possible higher structure. The second reason, more practical, is that a too-high Λ m is too restrictive for the parameters themselves. For example, for the parameters we usedin the previous sections, in particular m = 55GeV and m = 110 GeV, we have seen that λ gets negative already for Λ ≃ η leaves the perturbativity region muchlater, for Λ ≃ m = 67GeV,13 - - - = Log H L (cid:144) L L Λ H T L Λ H L = Θ = ° , m = m = FIG. 8: The running of the Higgs self-coupling λ (full RGE). It gets negative at about 15TeV forthis set of parameters’ values. m = 135GeV, and θ = 15 o , λ can live positive until about 400TeV whereas η leavesperturbativity at about 50TeV. In this section, we set Λ m ≃ σ det = m N (cid:0) m N − m B (cid:1) π ( m N + m ) v " λ (3)0 cos θm h − η (3)01 sin θm . (5.1)In this relation, m N is the nucleon mass and m B the baryon mass in the chiral limit. Thequantities λ (3)0 and η (3)01 are coupling constants of cubic terms in the theory after spontaneousbreaking of the two symmetries [14]: λ (3)0 = λ cos θ + η v sin θ ; η (3)01 = η v cos θ − λ v sin θ. (5.2)The condition we impose is that σ det be within the XENON 100 upper-bounds [11].In work [15], we studied phenomenological implications of the model and constraints onit, using rare meson decays and Higgs production. A number of inferences were deduced, but14 T = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Λ g = Η FIG. 9: Running of the mutual couplings (full RGE). The inclusion of the other SM particlesflattens the runnings. we will prefer to retain only two. One is that the mixing angle θ is to be chosen small. Thisis emphasized in view of the mounting evidence of a SM Higgs particle found by ATLASand CMS at the LHC [1, 2]. The other is that the physical self-coupling λ (4)0 is to be smalltoo. This was already observed in [14], where the relic-density constraint has the tendencyof ‘shutting down’ high values of λ (4)0 . At the end of the next section, we will comment onpossible larger values for λ (4)0 .In this section, the display range of m and m is from 1GeV to 160GeV. Indeed, thereis no reliable data to discuss regarding a dark-matter mass below the GeV, and in view ofthe behavior of η at Λ as a function of m , taking this latter beyond 160GeV is outside theperturbativity region. In practice, m was taken up to 200GeV, with no additional featuresto report.Let us start with λ (4)0 and θ both very small. Fig. 12 displays the regions (blue) for whichthe model is viable up to Λ m ≃ λ (4)0 = 0 .
01 and θ = 1 o . We see that themass m is confined to the interval 116GeV − = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Η g = Η FIG. 10: Running of the self-couplings (full RGE) with m larger. η is more closely tailed by η and λ decreases and turns negative at about 10TeV. mainly to the region above 118GeV, the left boundary of which having a positive slope as m increases. The DM mass m has also a small showing in the narrow interval 57GeV − θ is to enrich the existing regions withoutrelocating them. This is displayed in Figs. 13 and 14 for which θ is increased to 5 o and 15 o respectively. We see that, as θ increases, the region between the narrow band and the largerone to the right gets populated. This means more dark-matter masses above 60GeV areallowed, but m stays in the same interval, roughly 116GeV − λ (4)0 has the opposite effect, that of shrink-ing existing viability regions. Indeed, compare Fig. 15 for which λ (4)0 = 0 . θ = 15 o (Λ m ≃ λ (4)0 raises η (Λ ) wellenough above 1 so that this latter will leave the perturbativity region sooner. Increasingit is also caught up by the relic-density constraint, which tends to shut down such largervalues of λ (4)0 when the dark-matter mass m is large. The direct-detection constraint has16 = Log H L (cid:144) L L g H T L Λ H L = Θ = ° , m = m = g = Λ g = Λ g = Η FIG. 11: Running of the mutual couplings (full RGE) with m larger. η starts well above λ and λ and leaves the perturbativity region before the self-coupling η . also a similar effect. VI. CONCLUDING REMARKS
In this work, we have studied the effects and consequences of the renormalization groupequations at one-loop order on a two-singlet model of cold dark matter that consists inextending the Standard Model with two real gauge-singlet scalar fields. The two issueswe monitored are perturbativity and vacuum stability. The former is controlled by theauxiliary-field self-coupling η and the mutual coupling η between the dark matter and theauxiliary fields. The latter is controlled by the Higgs self-coupling λ . When the non-HiggsSM coupling constants are switched off, all scalar couplings are positive increasing functionsof the scale Λ. Reintroducing them flattens the rates for all the scalar couplings and makesthe Higgs coupling λ turn negative at some scale. The mutual couplings λ (DM-Higgs) and λ (Higgs-auxiliary) stay always well below one, whereas η boosts up for larger m (DMmass) and/or m (auxiliary-field mass), dominating over η in some regions.17
50 100 160050100160 0 50 100 160 050100160 m H GeV L m H G e V L Λ H L = Θ = FIG. 12: Regions of viability of the two-singlet model (in blue). Physical Higgs self-coupling λ (4)0 and mixing angle θ very small. We then have investigated the regions in the space of parameters in which the model is vi-able. We have plotted these regions in the ( m , m )-plane while varying the physical mutualcoupling λ (4)0 between the dark matter S and the physical Higgs h , and the mixing angle θ between h and the physical auxiliary field. We have required that the model reproducesthe DM relic density abundance, and that it complies with the measured direct-detectionupper bounds – those of the XENON 100 experiment. We have also imposed the RGEperturbativity and vacuum-stability criteria that we deduced from this work together witha maximum cutoff Λ m ≃ λ (4)0 and θ , the auxiliary-field mass m is confined to the interval 116GeV − m is confined mainly to the region above 118GeV, with a small showing18
50 100 160050100160 0 50 100 160 050100160 m H GeV L m H G e V L Λ H L = Θ = FIG. 13: Regions of viability (blue) of the model. λ (4)0 still very small, but θ larger. The region isricher, but not relocated. in the narrow interval 57GeV − θ enriches the existing viability regionswithout relocating them, while increasing λ (4)0 has the opposite effect, that of shrinking themwithout substantial relocation.It is pertinent at this stage to comment on the implications of the Higgs discovery at theLHC on the possibility of having a light dark matter WIMP S with a mass m . h → S S becomes open, and therefore will lower the number ofHiggs decays into SM particles. On the other hand, The ATLAS and CMS published dataon Higgs boson searches seem to indicate that the observed boson is SM-like, and so, oneexpects to have stringent constraints on the parameter space when it comes to light dark-matter masses. In [19], a global fit to the Higgs boson data that includes those presented19
50 100 160050100160 0 50 100 160 050100160 m H GeV L m H G e V L Λ H L = Θ =
FIG. 14: The region of viability (blue) is even richer for larger mixing angle θ . at the Moriond 2013 conference by the ATLAS and CMS collaborations [20, 21] has beenperformed; see [22] for earlier fits. It has been found that any extra invisible Higgs bosondecay must be bounded by the following condition on the corresponding branching ratio:Br( h → invisible) < . (6.1)It turns out that in our two-singlet model, the branching fraction of the invisible width ofthe Higgs boson is smaller than the bound above for m . m = 55GeV used frequently in this work, the ratio Γ( h → S S ) / Γ( h → b ¯ b ) is lessthan 17%, quite consistent with the above current bound. Therefore, we conclude that thetwo-singlet model is consistent with the current available data regarding the Higgs bosonsearches.Finally, we ask whether the model allows for very light cold dark matter. Below 5GeV,20
50 100 160050100160 0 50 100 160 050100160 m H GeV L m H G e V L Λ H L = Θ =
FIG. 15: The physical Higgs self-coupling λ (4)0 shrinks the viability region (blue) as it increases. direct detection puts no experimental bound on the total cross section σ det for non-relativisticelastic scattering of a dark matter WIMP off a nucleon target. Such a situation allows forvery small m regions of viability, but only when λ (4)0 is quite large ( ∼ θ not too small ( ∼ o and above). However, for such values of the parameters, the branchingfraction of the invisible Higgs decay is larger than 25%, which is excluded by the currentLHC available data. [1] G. Aad et al . [ATLAS Collaboration], Phys. Lett. B , 1 (2012).[2] S. Chatrchyan et al . [CMS Collaboration], Phys. Lett. B , 30 (2012).[3] P.A.R. Ade et al. [Planck Collaboration], arXiv:1303.5062 [astro-ph.CO] .[4] V. Silveira and A. Zee, Phys. Lett. B , 136 (1985).
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