Higgs Boson Mass and Complex Snuetrino Dark Matter in the Supersymmetric Inverse Seesaw Models
aa r X i v : . [ h e p - ph ] M a r Higgs Boson Mass and Complex Snuetrino Dark Matterin the Supersymmetric Inverse Seesaw Models
Jun Guo, ∗ Zhaofeng Kang, † Tianjun Li,
1, 3, ‡ and Yandong Liu § State Key Laboratory of Theoretical Physics and Kavli Institute forTheoretical Physics China (KITPC), Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, P. R. China Center for High-Energy Physics, Peking University, Beijing, 100871, P. R. China School of Physical Electronics, University of Electronic Scienceand Technology of China, Chengdu 610054, P. R. China (Dated: June 12, 2018)
Abstract
The discovery of a relatively heavy Standard Model (SM)-like Higgs boson challenges natural-ness of the minimal supersymmetric standard model (MSSM) from both Higgs and dark matter(DM) sectors. We study these two aspects in the MSSM extended by the low-scale inverse seesawmechanism. Firstly, it admits a sizable radiative contribution to the Higgs boson mass m h , up to ∼ Y ν LH u ν c and a large sneutrino mixing.Secondly, the lightest sneutrino, highly complex as expected, is a viable thermal DM candidate.Owing to the correct DM relic density and the XENON100 experimental constraints, two scenariossurvive: a Higgs-portal complex DM with mass lying around the Higgs pole or above W threshold,and a coannihilating DM with slim prospect of detection. Given an extra family of sneutrinos, bothscenarios naturally work when we attempt to suppress the DM left-handed sneutrino component,confronting with enhancing m h . PACS numbers: 12.60.Jv, 14.70.Pw, 95.35.+d ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] . INTRODUCTION AND MOTIVATIONS The CMS and ATLAS Collaborations discovered a new resonance around 125.5 GeV [1].From the latest full collected data announced at the Moriond 2013 conference, it is quiteStandard Model (SM)-like. If this is confirmed at the next run of the √ s = 14 TeV LHC, itwould complete the picture of SM. But we can never conclude that the discovery of a highlySM-like Higgs boson at the LHC indicates an end to the particle physics: on the theoreticalside, the SM suffers the notorious gauge hierarchy problem if the discovered resonance isindeed a fundamental spin-0 boson; On the phenomenological side, the SM can not explainthe tiny neutrino mass origin and has no candidate for dark matter (DM), both of whichare clear signals for new physics beyond the SM.Supersymmetry (SUSY) is still the most promising underlying theory to account forthese two sides simultaneously. The supersymmetric SMs (SSMs) are free of quadraticdivergences involving scalars, and provide a weakly interactive massive particle (WIMP)DM candidate if R -pariy is conserved, i.e. , the Lightest Supersymmetric Particle (LSP)such as the lightest neutralino [2]. Of course, to explain the neutrino masses and mixings,we may have to supersymmetrize the well studied models with seesaw mechanisms. Amongthem, the inverse seesaw (ISS) mechanism [3] has an obvious advantage: it is suited forthe TeV-scale seesaw mechanism without turning to tiny Yukawa couplings between theneutrinos and Higgs doublet: Y ν LH u ν c . This property is found to be capable of mitigatingthe great stress in the Minimal SSM (MSSM) which, to have the relatively heavy SM-likeHiggs boson mass, incurs a rather serious fine-tuning from generating both the weak scaleand LSP neutralino dark matter phenomenology [4, 5].To demonstrate the consequence of this property, we consider that the new (single familyof) Yukawa coupling develops an IR-fixed point, which predicts Y ν ≃ .
75. This new largeYukawa coupling involving H u at the low energy contributes to the lightest CP-even Higgsboson mass m h radiatively. Using the effective potential method [6, 7], we first analyticallycalculate such corrections in some simplified cases and then employ the full numerical anal-yses. Enhancement up to 4 GeV can be obtained in the case of a large sneutrino mixing.This helps to alleviate the tension between a relatively heavy SM-like Higgs boson and theweak-scale naturalness.The sneutrino LSP in the SSMs with low-scale seesaw mechanism may be a good alter-native of the neutralino LSP DM [8, 9], especially after the discovery of the SM-like Higgsboson and null results from the DM detection experiments like XENON100 [10]. Specifiedto the low scale supersymmetric ISS, the sneutrino LSP is expected to be complex. This re-stricts the sneutrino DM into two possibilities: (I) Essentially it belongs to the Higgs-portalcomplex DM, and its mass has to be around m h / W boson mass m W ; (II) It2s a coannihilating DM, for example, the sneutrino LSP and Higgsino coannihilation. Thisallows a rather weak coupling between sneutrino DM and visible particles, so it is hard tobe detected. Given an extra family of sneutrinos, both scenarios naturally work when weattempt to suppress the left-handed sneutrino DM component, confronting with enhancing m h .This paper is organized as follows. In Section II, we briefly introduce the model andthen calculate the radiative correction to Higgs boson mass from the neutrino Yukawa cou-pling at the IR-fixed point. In Section III, we discuss the complex sneutrino dark matterphenomenology and investigate how they are consistent with the requirement of enhancingHiggs boson mass. The Section IV includes discussions and conclusion. II. THE LIGHTEST CP-EVEN HIGGS BOSON MASS
The models equipped with a low scale seesaw mechanism receive special attentions, byvirtue of its potential to be tested within our near future experiments. Specified to the SSMs,after the discovery of a relatively heavy SM-like Higgs boson, models capable of enhancingthe Higgs boson mass m h gain further theoretical preference, as stressed in the introduction.In type-I and III seesaw mechanisms, where the small neutrino mass m ν ∼ Y ν v /M R with M R the seesaw scale, the enhancement is impossible [12] because a low scale M R is at theprice of a negligibly small Y ν [37]. By contrast, in the type-II and inverse seesaw mechanismsthe smallness of m ν has other origins, and then the Higgs doublets are allowed to have largeneutrino Yukawa couplings. For instance, the supersymmetric type-II seesaw mechanismcan even enhance m h at the tree level [14] (Actually, it can simultaneously enhance thedi-photon rate [14].). However, such models are difficult to be embedded into a pertubativeGrand Unified Theory (GUT) picture. The inverse seesaw models, where only singlets areinvolved, are potential to lift the Higgs boson mass (at one-loop level) without violatingGUT.In this Section we will first briefly review the minimal supersymmetric ISS mechanism,and then discuss one of the new Yukawa couplings with the IR-fixed point behavior and itsimplication to the correction on the SM-like Higgs boson mass. A. The ISS Model with an IR-Fixed Point
The minimal supersymmetric model with ISS is the MSSM extended by two extra singlets ν c and N (a single family for the time being), which carry lepton numbers − W ISS = ( Y ν ν c LH u + M R ν c N ) + 12 µ N N . (1)The model contains one dimensionless parameter Y ν , and two dimension-one mass parameter M R and µ N . The µ N term softly breaks the lepton number by two units and largely accountsfor the smallness of tiny neutrino mass [38].The model naturally gives rise to a low scale seesaw mechanism without turning to smallparameters except for the massive parameter µ N . In the basis ( ν , ν c , N ) the neutrino massmatrix is given by M F = Y ν v u Y ν v u M R M R µ N , (2)where v u is the Vacuum Expectation Value (VEV) of Higgs field H u , i.e. , h H u i = v u . Thismatrix leads to the following combination as the light Majorana neutrino ν ≈ sin θ ν L − cos θ ν c , (3)where the mixing angle is approximated to be sin θ ≈ M R / p m D + M R with m D ≡ Y ν v u the Dirac mass for ν L and ν c . The neutrino mass assumes a form of m ν ≃ m D M R µ N . (4)Notably, if µ N , for some reason, can be arbitrarily small, then the sub-eV neutrino massscale can be obtained without turning to the large suppression from extremely small Y ν or(and) large M R . This merit of the ISS model is the basic observation of our article. Butnote that owing to the non-unitary constraint, M R should be several times larger than m D .From Ref. [15, 16] we set a rough bound M R & m D , (5)so as to make θ ≃ π/
2, i.e., the neutrino is dominated by left-handed neutrino. Finally, ν c and N form a Dirac fermion with mass approximately given by M R + m D / M R . (6)For a reason discussed later, we are interested in how large Y ν is allowed by pertubativityup to the GUT scale. Interestingly, we find that there is an IR-fixed point structure predict-ing Y ν . .
75. We present the Yukawa running in this model and focus on the new Yukawa4 H Q (cid:144) GeV L Y Ν FIG. 1: Renormalization group equation running trajectory for Y ν in the inverse seesaw model,showing that the Yukawa coupling has an infrared-stable quasi-fixed point for large Y ν . Here, m t = 173 . β = 10 are chosen. term which is illustrated in Fig. 1. For numerical calculation, we choose the SUSY-breakingscale m S = 800 GeV, new sterile neutrino scale M R = 1000 GeV, and we simply consider onegeneration new sterile neutrinos. The new Yukawa coupling has infrared quasi-fixed pointbehavior, which restrict how large it can be at the TeV-scale while maintaining consistentwith perturbative unification. Here for fixed-point trajectory we adopt the same definitionlike in [17] which request the new Yukawa couplings is less than or equal to 3. B. Stau-sterile neutrino contribution to the Higgs boson mass
It is well known that at tree level the SM-like Higgs boson mass m h is predicted to belighter than M Z in the MSSM. This necessitates an significant radiative correction from thestop-top sector to lift the Higgs boson mass near 125 GeV, which is discovered by the recentCMS and ATLAS Collaborations [1]. However, such large correction typically requires stopsat the TeV scale and then renders the MSSM highly fine-tuned [4]. Therefore, how to liftthe Higgs boson mass at the less price of fine-tuning is a very interesting question.The supersymmetric ISS model with the IR-fixed point just provides a new source ofenhancement via the stau-sterile neutrino correction. We calculate the corrections followingthe effective potential method [6, 7] from the Colenman-Weinberg potential [18] of the Higgs5eld via a general formula ∆ V ( H u ) =2 X i (cid:2) F ( M S i ) − F ( M F i ) (cid:3) ,F ( M ) = M π (cid:2) ln( M /Q ) − / (cid:3) , (7)with Q the renormalization scale. Index i runs over all the scalar and fermion couplingsto Higgs doublets. Restricted to the neutrino sector, the fermion spectrum contains a lightMajorana neutrino, whose contribution to the effective potential is proportional to ( µ N /M R ) and thus can be safely dropped. While the Dirac sterile neutrinos have a large mass given byEq. (6). We postpone the discussion on the scalar spectrum to the next paragraph. Now, inthe decoupling region, i.e. , m A ≫ m h , the effective potential gives the following correctionto m h ∆ m h = (cid:26) sin β (cid:20) ∂ ∂v u − v u ∂∂v u (cid:21) + cos β (cid:20) ∂ ∂v d − v d ∂∂v d (cid:21) + sin β cos β ∂ ∂v u ∂v d (cid:27) ∆ V. (8)We now turn our attention to the scalar spectrum. First, the supersymmetric ISS modelintroduces new soft terms −L soft = −L MSSM soft + m e ν c | ˜ ν c | + m e N | ˜ N | + (cid:18) Y ν A Y ν ˜ L ˜ ν c H u + B M R ˜ ν c ˜ N + 12 B µ N ˜ N + c.c. (cid:19) , (9) where L MSSM soft is the MSSM SUSY breaking soft term. B M R ∼ m / M R with m / thegravitino mass measuring the scale of soft breaking parameters. It is noticed that in ourscenario the lepton number violating soft breaking parameter B µ N ∼ m / µ N is irrelevantlysmall (we will come back to this point later), and consequently three sneutrinos are highlycomplex scalars. Now, in the basis ( e ν L , ( e ν c ) ∗ , e N ) the sneutrino mass squared matrix takes aform of M S = m D + m e L + m Z cos 2 β m D ( A Y ν − µ cot β ) m D M R m D + M R + m e ν c B M R M R + m e N . (10)Because the analytical eigenvalues are extremely lengthy, in the ensuing discussion we willconsider some solvable limits to demonstrate which parameters can lift the Higgs bosonmass.There are two mixing terms which depend on H u and thus contribute to m h in termsof Eq. (8), one is the soft trilinear term X ν = A Y ν v u − µ cot β , while the other one comesfrom the F-term of ν c , i.e. , Y ν v u M R = m D M R (In light of the effective potential method,any origin of correction to m h can be traced back to the matrix entries depending on m D ).Turning off the mixing A -term will lead to a solvable matrix. As a warm up, we first consider6 m S M R X Ν M R FIG. 2: X ν M R versus m S M R for the contour of ∆ m h , which is defined as q (123 GeV) + ∆ m h −
123 GeVthroughout this work. the case by turning off the A ν and B M R mixing terms, setting the soft SUSY-breaking masssquares equal to m S and neglecting the small electroweak D-term contribution. Then in themass eigenstates the three sneutrino mass squares are given by m S , m D + M R + m S , m D + M R + m S . (11)The eigenstate corresponding to m S , which is independent on H u , does not contribute tothe effective potential. But it is merely a result of the approximations which we have taken.Using Eq. (8) we get ∆ m h = 14 π sin βY ν m D ln M R + m S M R . (12)It is similar to the non-mixing stop case but quantitatively less important due to a colorfactor and the smaller Yukawa coupling which leads to a suppression ( Y ν /h t ) . Even worseis that, from the neutrino side we have M R ∼ m S is constrained around the TeV scale by naturalness.Thus, the correction from sneutrino is less than 10 percents of that from the stop case, seethe left panel of Fig. 3.Next we take into account the soft trilinear term X ν , which would bring much differencein lifting m h . If other approximations are the same as the non-mixing case, we are still ableto find an analytical expression for the correction, in spite of somewhat complication∆ m h = 14 π sin βY ν m D (cid:18) ln m S + M R M R + X ν + 2 X ν M R M R − X ν m S + X ν M R + 4 X ν m S M R M R ln m S + M R m S (cid:19) . (13)7
00 1000 1200 1400 1600 1800 20000.51.01.52.0 m S D m h X t D m h FIG. 3: Left panel: ∆ m h versus m S with vanishing soft trilinear term. Right panel: ∆ m h versus X ν for M R = 1000 GeV, m S = 800 GeV, and tan β = 15. The red lines correspond to the fullnumerical results and green lines the approximated results. We would like to comment on the origins of various terms in the above equation [39]. The firstlogarithmic term exactly reproduces the result given in Eq. (12), extracting the correctionto the Higgs quartic coupling λ h , which is encoded in the Renormalization Group Equation(RGE) running from the supersymmetry breaking scale defined by the heavy sneutrino massscale to the Dirac sterile neutrino mass scale M R . While the second term stands for themixing effect after integrating out the sneutrinos, included as a shift to the boundary λ h (In the explicit Feymann diagram calculations, it can be obtained from the triangle and boxdiagrams.). The last term has an obscure dependence on the logarithmic log(( m S + M R ) /m S ),stemming from the different mass scales of e ν L and e ν c as shown in Eq. (10). Such a hybrid ofthe mixing and logarithmic terms is absent in the stop system, and noticeably it is negative.We now quantitatively analyze the corrections given in Eq. (13). m D has been fixed byvirtue of the IR-fixed of Y ν , and then it is not difficult to find that ∆ m h actually dependson two dimensionless parameters, x s = m S /M R and x a = X ν /M R . Denoting the functionin the brackets as f ( x s , x a ), it consists of three parts: the first and second parts are positivewhile the third part is negative. Hence the maximal mixing scenario here is more subtlethan the stop case. It is illustrative to consider two limits of x s : (1) If x s ≫
1, i.e., the largeSUSY breaking soft mass terms, we have a simple expression for f ( x s , x a ) → log x s − x a / x s .Thereby the mixing effect is negative but suppressed by large x s . (2) Oppositely, if x s ≪ f ( x s , x a ) → x a (1 + log √ x s ) + 2 x a . Thus, a moderate x s is needed to maximize thecorrections. Explicitly, we present a contour plot of ∆ m h in Fig. 2.We examine the difference between the approximately analytical and full numerical treat-ments. Fig. 3 shows that they give rise to almost the identical results, in the trivial caseof non-mixing (left panel) and the case with A ν mixing only (right panel). This indicatesthat the expression in Eq. (13) works well for universal soft masses. In Fig. 3, for simplicity8
000 1500 2000 2500 3000 35001.01.52.02.53.03.54.0 X t D m h X t D m h FIG. 4: Left: ∆ m h versus m S for the full numerical results with B M R = 0 . m S ; Right: ∆ m h versus m S for the full numerical results with B M R = m S . We have fixed M R = 1000 GeV, m S = 1200GeV, and tan β = 15. we set B M R ∼
0. However, a non-zero B M R may lead to an appreciable change. Althoughit is still possible to develop an analytical expression for ∆ m h , it is too lengthy to conveyany useful information. Therefore, we only display the change numerically in Fig. 4, withcolor codes of lines as before. It is clearly seen that, as | B M R | / approaches the sneutrinomass scale, the discrepancy between the full result and the approximation Eq. (13) becomesrather significant. III. A COMPLEX SNEUTRINO LSP IN THE ISS-MSSM
The presence of a DM candidate, i.e. , the neutral LSP, is one of the major attrac-tions of the SSMs. The lightest neutralino receives the most intensive attention. How-ever, the heaviness of the soft spectrum, owing to a relatively heavy Higgs boson and nullsparticle searches, along with the stringent bounds from direct detection experiments suchas XENON100 [10] [40], now threaten the viable neutralino DM. Another neutral LSPcandidate, the lightest sneutrino, has been investigated by many authors in many con-texts [8, 9, 19–21], and now may show some advantages.Roughly speaking, the sneutrino LSPs can be classified into the following three types
Complex sneutrino
As in the MSSM, the left-handed sneutrino e ν L , just the original pro-posal [8], is a complex scalar due to the conservation of global lepton number U (1) L .But the Z − boson mediated DM-nucleon spin-independent (SI) scattering has a verylarge cross section σ SI [19], which excludes the sneutrino LSP in the MSSM. Beyond it,a complex sneutrino LSP dominated by the SM singlets may be realized in the modelswhich conserve the lepton number with a high degree [24, 25].9 eal sneutrino Taking into account for generating tiny neutrino masses via the seesawmechanisms, U (1) L should be broken and then induces the mass splitting between theCP-odd and even components of complex sneutrino. If splitting is large, we actuallyhave a real sneutrino LSP [20]. Consequently, σ SI from Z − boson exchange will bezero. Pseudo-complex sneutrino
When the splitting is small, which is the usual case be-cause of suppression from small neutrino mass, we will get a pseudo-complex sneu-trino [20, 22, 23]. In this case, the DM-nucleon scattering mediated by Z − bosonbecomes inelastic and may be kinematically forbidden.In the low-scale ISS-MSSM, U (1) L is broken by µ N N / B µ N e N / m SUSY µ N e N /
2. Provided that there is no peculiar SUSY-breaking medi-ation such as in Ref. [25], the mass splitting between the CP-even and CP-odd parts of e N ,due to the soft bilinear term, is suppressed by the neutrino mass δm ∼ m SUSY µ N m e N ∼ m SUSY m e N ! (cid:18) M R m D (cid:19) m ν , (14)which is around 10 eV. Thereby, in this model the sneutrino LSP is expected to be com-plex [24]. Moreover, it is thermal since it is free of extreme suppression from Yukawa cou-plings with the visible sector. But note that enhancing U (1) L − violation by a abnormallylarge B µ N ∼ O (GeV ) can lead to an inelastic sneutrino DM, which actually is the case inmost references [23, 26, 27]. In this article, we insist on a normal SUSY breaking mediationmechanism to account for soft terms, and then we have a (highly) complex sneutrino LSP.Our purpose is to explore the viable sneutrino DM scenarios consistent with the enhancingHiggs boson mass. A. General Simplifications due to Complexity
We investigate what kind of mixture can lead to a good sneutrino DM, which has correctrelic density, is allowed by the XENON100 experiment and does not incur tremendous fine-tuning. In general, for the moment we do not restrict discussions to the setup made in theprevious Section. From the first glance, the sneutrino LSP is complicated because of threecomplex sneutrino system, but the complexity of the sneutrino LSP greatly simplifies thediscussions.Above all, we have to suppress the left-handed sneutrino fraction. The lightest sneutrinois a superposition of three sneutrinos e N = C e ν L e ν L + C e ν c ( e ν c ) ∗ + C e N e N , (15)10here the singlet fraction must dominate to suppress the Z − mediated DM-nucleon scatter-ing. It is justified to make an estimation on the doublet fraction C e ν L ≃ m D A ν m e ν L sin θ + m D M R m e ν L cos θ , (16)which is quite precise in the case of large mass splitting between e ν L and other two sneutrinos. θ ∈ ( − π/ ,
0) is the mixing angle of e ν c and e N from the 23-submatrix of Eq. (10)tan θ ≈ P − q P + 1 , P ≡ (cid:0) m D + m e ν c − m e N (cid:1) / B M R . (17)In Eq. (15) C e ν c ≈ − sin θ and C e N ≈ − cos θ . The DM-proton SI scattering cross sectionis σ p = µ p f p /π [2], with µ p the DM-proton reduced mass. For a DM with mass around 100GeV, XENON100 imposes the upper bound f p . . × − GeV − and in turn f p = C e ν L g /m Z . . × − GeV − ⇒ C e ν L . . . (18)A natural suppression needs a multi-TeV e ν L for Y ν ∼ O (0 . e ν L (but still heavier than e N and e ν c ) is allowed given a sufficiently small Y ν .With such a small C e ν L , the dynamics of the sneutrino LSP largely reduces to that of aHiggs-portal complex scalar DM. Concretely, e N annihilates into SM particles through fourways: (1) Contact interactions with H u from | F ν L | ; (2) h propagating in the s − channel; (3)Higgsinos/sneutrinos propagating in the t − channel; (4) Gauge interactions inheriting from e ν L . Decoupling e ν L means only case (1) left, giving rise to a Higgs-portal complex scalar DM | F ν L | → λ H | e N | | H u | , λ H ≡ (sin θ Y ν ) . (19)But cases (2) and (3) may cause deviations from an exact Higgs-port DM. In the first, theHiggsino-mediated processes might be important for light DM well below m W , since theircross sections are ∼ O ( λ H m e N / πM ), with further velocity/helicity suppressions [24].But owing to the bound indicated in Eq. (22), we find that it is far from enough to givethe correct relic density and thus can be ignored. Next, if C e ν L is not extremely suppressed,processes involving e ν L contributions to DM cross sections at higher order of C e ν L can beenhanced by large massive couplings. This is seen from the following terms, which areabsent in the ordinary Higgs-portal DM models, −L trilinear = | F ν c | + (cid:16) Y ν A ν e LH u e ν c + c.c. (cid:17) → − ( C e ν L m e ν L ) √ v u h | e N | − C e ν L m e ν L √ v u h e ν L e N ∗ + c.c. (20)The presence of extra C e ν L in the first and second terms are traced back to the fact thatthese trilinear couplings are also sources of the mixing terms between e ν L and singlets, seeEq. (10). 11e proceed to argue that terms in Eq. (20) can make no difference. The first term ofEq. (20) affects processes mediated by h . With it, the massive coupling constants of h | e N | , µ h , takes a form of µ h = √ v u λ H (cid:20) (cid:16) C e ν L m e ν L / p λ H v u (cid:17) (cid:21) . (21)The second term stands for the deviation from the exact Higgs-portal. In terms of Eq. (16),its order of magnitude is estimated to be at least ∼ O ( A ν / √ m e ν L ) , which is reliable barringthe large cancellation in Eq. (16). So, for a relatively large A ν but small m e ν L , which meansan appreciable C e ν L , the trilinear soft term dominates µ h . σ SI from h implies the bounds( C e ν L m e ν L ) / √ v u m DM , √ v u λ H /m DM . µ h /m DM . . × ( σ up p / − pb) / . (22)But the dynamics of the Higgs-portal DM with and without deformation (by trilinear softterms) are the same, provided that annihilation into a pair of h via the contact interaction inEq. (19) is irrelevant. Explicitly, it possesses σ chh v = (cid:0) √ λ H v u /m DM (cid:1) / πv u . . h , with a thermal averaged cross section σ thh v ≃ π (cid:18) C e ν L m e ν L √ v u m DM (cid:19) v u . (23)Even if the value in the bracket saturates its upper bound shown in Eq. (22), this cross sectionmerely gives 1 pb. Actually, it can never be saturated for m e ν L . m DM > m h . Insummary, the complex sneutrino DM is reduced to the Higgs-portal DM. B. The Sneutrino LSP Confronting with Enhancing Higgs Boson Mass
With the above analyses, in this subsection we explore the viable scenarios for sneutrinoDM, taking into account the requirement to lift the Higgs boson mass. But a single familyof sneutrino fails, because of the contradiction between these two aspects. On the one hand,to significantly lift m h we need Y ν ∼ m D . M R ∼ O (1) TeV. On the otherhand, it renders sneutrinos heavy. Moreover, the mixings between e ν L and e ν c / e N are largeby virtue of the large off-diagonal entries, i.e. , m D X ν e ν † L e ν c and m D M R e ν † L e N , in the sneutrinomass matrix Eq. (10). Therefore, without large fine-tuning, the single family case can notlift m h sizably and at the same time has a sneutrino LSP around the weak scale with asufficiently small e ν L fraction. An extra family of sneutrino, which has Yukawa coupling Y ν for definiteness [41], is thus introduced. 12ow good scenarios can be accommodated. Because Y ν , the third family Yukawa coupling,has approached the IR-fixed point, Y ν should be much smaller than it. As a result, e ν c and e N can be around the weak scale. While m e ν L, is assumed to be properly heavier and hencethe lightest sneutrino e N is dominated by e N and/or e ν c . According to the analysis made inthe above subsection, a sufficient suppression of the doublet component can be realized viaa heavy m e ν L, for large Y ν, or a small Y ν, for light m e ν L, . Both are well motivated and willbe discussed respectively in the following. Hereafter, the subscript “2” will be dropped andsneutrinos refer to the second family unless otherwise specified.
1. Higgs-Portal Sneutrino DM Inspired by Natural SUSY
In the natural SUSY framework, the first and second families of sfermions are assumedto be much heavier than the third family, says lying above 3 TeV. Such a pattern may berelated to the SM fermion flavor structure [28]. While sterile neutrinos are SM singlets andhave different flavor structure, and hence their superpartners are allowed to be light. As aresult, we naturally get a light sneutrino LSP with negligible e ν L component.We are working in the Higgs-portal sneutrino DM, so its correct relic density, viewingfrom the stringent XENON100 experimental constraint, needs to be studied carefully. Asmentioned before, the cross section from DM annihilating into a pair of Higgs bosons viacontact interaction is not large enough. Consequently, annihilations of the viable sneutrinoLSP are completely specified by h in the s − channel. Correct relic density restricts the viablesneutrino LSP only to two possibilities (see the left panel of Fig. 5) • m e N ≃ m h /
2. When the DM mass lies below the W threshold and closes to the Higgspole, DM annihilations will benefit from the Higgs resonance enhancement. Thencorrect relic density can be got for a small µ h . In turn, the XENON100 bound can beevaded, see the right panel of Fig. 5. The Higgs invisible decay may impose an evenmore stringent constraint for m DM < m h /
2. In the right panel of Fig. 5 we label thepoints giving invisible Higgs decay to a pair of e N with branching ratio larger than10%, which, assumed to be the upper bound, excludes e N below 55 GeV. • m e N & m W . Bare in mind that the SM-like Higgs boson decay is always dominatedby the W W mode for m h &
160 GeV [29], thus e N will dominantly annihilate into W W once it is kinematically allowed. Increasing DM mass will decrease the DMannihilation cross section, but new accessible channels such as
ZZ/hh/t ¯ t can partiallycompensate the decrease. As a result, even without significantly increasing µ h , thesneutrino LSP with mass extending above a few times of m W still can acquire correctrelic density, see the left panel of Fig. 5. On the other hand, a heavier scalar DM13elps to reduce σ SI from h , so the XENON100 experimental constraint is satisfied forthe heavier e N , see Fig. 5. Actually, from it we see that the latest LUX result hasexcluded the sneutrino DM between 65 and 150 GeV.We modify MicrOMEGAs 3.2 [30] by including the ISS-MSSM and then using it to calculatethe sneutrino DM relic density and σ SI . Based on the previous semi-analytical analysis, wescan the following three-dimensional parameter space A ν = 200 GeV , Y ν ∈ [0 . , . ,m e N ∈ [ − M R , M R ] , m e ν c = km e N with k ∈ [ − . , . . (24)Additionally, we take M R = 9 m D and B M R = (100 GeV) M R , and fix the irrelevant MSSMparameters as the followingtan β = 20 , m e l = 10 TeV , µ = 400 GeV , (25)while all the other sparticles are decoupled for simplicity. Comments are in orders. First,the upper bound on Y ν covers the limit indicated by Eq. (22) and is consistent with the largeYukawa coupling of the third family. Second, we allow negative m e ν c and m e N such that theLSP, via a moderately large cancellation, can be light around m W even for the large Y ν andheavy M R case. Compared to the method turning back to a large B M R to get a light DM,this way allows a widely varying mixing angle θ . But they do not show essential difference.
2. Coannihilating Sneutrino LSP Inspired by the Semi-Constrained ISS
We now turn to a scenario inspired by the semi-constrained ISS-MSSM. In this scenario,the MSSM part is described by the CMSSM with free parameters m , M / and A , etc.The ISS-sector contains the universal SUSY breaking soft mass terms m e S , A ν and B M R .For the sake of enhancing Higgs boson mass, | A ν | typically is multi-TeV, says 5 TeV (Butthe enhancement is purely due to the mixing effect and thus is limited, typically around 2GeV.). Note that m e ν L , by naturalness, is favored to be at the sub-TeV scale. So we need arather small Y ν , typically at the order of 0 . C e ν L . The LSP annihilation rateis suppressed by small Y ν , and then the coannihilation effect [31] is needed to reduce itsnumber density.If the next-to-the LSP (NLSP) and the sneutrino LSP have sufficiently small mass dif-ference δm , coannihilation effect will play an important role in reducing number densityof the sneutrino LSP. The effective annihilation cross section σ eff , in terms of Ref. [31], isa weighted sum of the LSP-LSP, LSP-NLSP, and NLSP-NLSP annihilation cross sections,14 DM(GeV) m DM(GeV)0 0.05 0.1 0.15 0.2 0.25( Y ν C e ν c ) Ω h Y ν C e ν c ) Ω h σ p S I ( c m ) m e N (GeV)1e-501e-491e-481e-471e-461e-451e-441e-431e-421e-41 50 100 150 200 250 300 σ p S I ( c m ) m e N (GeV)Xenon100 2012 excluding lineLUX excluding line0 . < Ω h < . h < . . < Ω h < Br inv. > . < Ω h < . h < . . < Ω h < Br inv. > FIG. 5: Left: The DM relic density versus the effective coupling constant between the up-typeHiggs doublet and sneutrino LSP. Points in the narrow band between the two black lines haverelic density consistent with the measurement: 0 . < Ω h < .
12. Right: The Direct detectionexclusion limits on the sneutrino LSP in the σ p SI − m DM plane. Assuming the observed DM relicdensity can be realized from non-thermal production and dilution mechanism, we also show thepoints with smaller and larger relic densities, labeled as the red and green points, respectively.Black circles denote the points with branching ratio of Higgs invisible decay ≥ denoted as σ , σ , and σ , respectively. Viewing from the sneutrino system in the sce-narios under consideration, a large σ eff should be ascribed to a large σ rather than σ .The natural candidates for the NLSP include the Higgsino and doublet-like snerutrino whichhave full SU (2) L interactions. While the colored sparticles, especially the light stop, shouldbe moderately heavy owing to the LHC bounds. So we do not consider them in this paper.We focus on the Higgsino NLSP case. This is a reasonable choice because by naturalnessthe µ − term should be small, and thus light Higgsinos. We now make a numerical analysis.In terms of the above arguments, we scan a slice of the parameter space as the following A ν = 5 . , Y ν ∈ [0 . , . , µ ∈ [100 , ,m e ν c = m e N ∈ [ µ − , µ + 30] , m e l = [400 , . (26)The other parameters are chosen as before. In this scenario the bounds from direct detectionsare weak, see the right panel of Fig. 6 where most of the region is untouched, except for thosewith a larger C e ν L . This is not surprising. Because σ is allowed to be small, the sneutrinoLSP can couple to Higgs, more widely, the visible sector, weakly. As a consequence, it maybecome deeply dark, even for the indirect detections. Instead, we may have to count on thepossible hints if a light non-LSP Higgsino is discovered at colliders.15 e N (GeV) m e N (GeV)0 5 10 15 20 m e χ − m e N (GeV)1e-041e-031e-021e-011e+001e+011e+021e+03 Ω h m e χ − m e N (GeV)1e-041e-031e-021e-011e+001e+011e+021e+03 Ω h C e ν L C e ν L LUX excluding lineXenon100 2012 excluding line100 150 200 250 300 350 400 m e N (GeV)1e-501e-481e-461e-441e-421e-401e-38 σ p S I ( c m ) m e N (GeV)1e-501e-481e-461e-441e-421e-401e-38 σ p S I ( c m ) FIG. 6: Left: The DM relic density versus the sneutrino-Higgsino mass difference. Right: Thedirect detection exclusion limits on the sneutrino LSP in the σ p SI − m DM plane. The heavier LSPmay have a larger C e ν L , so it is more sensitive to direct detections. IV. DISCUSSIONS AND CONCLUSION
The discovery of a relatively heavy SM-like Higgs boson is a good news to SUSY but notto the MSSM, whose little hierarchy problem is exacerbated. Moreover, the viable parameterspace for the neutralino LSP dark matter is greatly constrained. In this work we studied thetwo aspects of the MSSM extended by the inverse seesaw mechanism, which is an elegantmechanism to realize the low-scale seesaw mechanism without turning to very small Yukawacouplings. This feature makes the model contribute to a sizable radiative correction to m h , up to ∼ Y ν LH u ν c and a largesneutrino mixing. Thus, the little hierarchy problem can be alleviated. Furthermore, itmakes the lightest (highly complex) sneutrino be a viable thermal DM candidate. Owing tothe stringent constraints from the correct DM relic density and XENON100 experiment, wefound that there are only two viable scenarios for the LSP sneutrino • Its dynamics is reduced to that of the Higgs-portal complex DM, with mass around m h / m W . The upcoming experiments such as XENON1T can excludethem, especially the latter. Additionally, we may observe a hint of DM with resonantenhancement from Higgs invisible decay. • It is a coannihilating DM, likely with Higgsinos. Because now a fairly weak couplingbetween the sneutrino DM and visible particles is allowed, it is hard to be excluded.Taking into account the requirement of enhancing m h which needs large sneutrino mixing,we should introduce an extra family of sneutrinos to account for sneutrino DM. And both16cenarios naturally work when we attempted to suppress the DM left-handed sneutrinocomponent.In this article we did not consider the supersymmetric ISS models extended by gaugegroups [23, 32, 33]. Actually, such supersymmetric models have even more significant effectsto enhance the SM-like Higgs boson mass [32, 33]. The LHC detection in the presence of asinglet-like sneutrino DM possesses some special signatures when the ordinary LSP is a stausneutrino [34]. Consequently, the colored sparticle like stop decay is characterized by a longdecay chain and the presence of leptons in the final state, which may weaken the ATLASMET plus jets constraint. Other collider phenomeplogy consequences of this model are alsostudied [35]. Acknowledgement
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1) but at the same time a very largesoft mass squared for the right-handed sneutrino, unfortunately, the correction is negative.[38] Based on the next-to-MSSM, the Dirac mass M R can be dynamically generated [24, 36]. Inparticular, Ref. [24] observed that, in the presence of a singlet with a TeV-scale VEV, thesmall µ N − term can be related to a dimension-five operator suppressed by the Planck scale.[39] We note that our approximation expression is different from that in Ref. [36] which is consistentwith that in Ref. [17]. However, from our understanding, the particle contents and interactionsin their inverse seesaw models are not the same as the extended MSSM with extra vector-likeparticles.[40] After the completion of this work, the LUX Collaboration announced their new results,which gave the even more stringent constraint, one magnitude of order stronger than theXENON100 experiment. But this does not qualitatively affect our discussions, so we onlymention XENON100 in the text and show LUX data in figures.[41] As a matter of fact, to produce the realistic neutrino masses and mixings, we need extrafamilies as well.term can be related to a dimension-five operator suppressed by the Planck scale.[39] We note that our approximation expression is different from that in Ref. [36] which is consistentwith that in Ref. [17]. However, from our understanding, the particle contents and interactionsin their inverse seesaw models are not the same as the extended MSSM with extra vector-likeparticles.[40] After the completion of this work, the LUX Collaboration announced their new results,which gave the even more stringent constraint, one magnitude of order stronger than theXENON100 experiment. But this does not qualitatively affect our discussions, so we onlymention XENON100 in the text and show LUX data in figures.[41] As a matter of fact, to produce the realistic neutrino masses and mixings, we need extrafamilies as well.