Implications of naturalness for the heavy Higgs bosons of supersymmetry
Kyu Jung Bae, Howard Baer, Vernon Barger, Dan Mickelson, Michael Savoy
IImplications of naturalness forthe heavy Higgs bosons of supersymmetry
Kyu Jung Bae ∗ , Howard Baer † , Vernon Barger ‡ ,Dan Mickelson § and Michael Savoy ¶ Dept. of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA Dept. of Physics, University of Wisconsin, Madison, WI 53706, USA
Abstract
Recently, it has been argued that various measures of SUSY naturalness– electroweak,Higgs mass and EENZ/BG– when applied consistently concur with one another and makevery specific predictions for natural supersymmetric spectra. Highly natural spectra arecharacterized by light higgsinos with mass not too far from m h and well-mixed but TeV-scale third generation squarks. We apply the unified naturalness measure to the caseof heavy Higgs bosons A , H and H ± . We find that their masses are bounded fromabove by naturalness depending on tan β : e.g. for 10% fine-tuning and tan β ∼
10, weexpect m A < ∼ . β as high as 50, then m A < ∼ H, A, H ± → W, Z, or h + (cid:54) E T + soft tracks sothat single heavy Higgs production is characterized by the presence of high p T W , Z or h bosons plus missing E T . These new heavy Higgs boson signatures seem to be challengingto extract from SM backgrounds. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] ¶ Email: [email protected] a r X i v : . [ h e p - ph ] J u l Introduction
The recent discovery of a Standard Model like Higgs boson with mass m h = 125 . ± . m h < ∼
135 GeV[3]. Such a large value of m h apparently requires TeV-scale top squarkswhich are highly mixed, i.e. a large trilinear soft SUSY breaking parameter A t [4]. Couplingthis result with recent SUSY search limits from LHC8[5, 6] (which require m ˜ g > ∼ . m ˜ g (cid:28) m ˜ q and m ˜ g > ∼ . m ˜ g ∼ m ˜ q ) imply, within the context of gravity-mediated SUSYbreaking models (SUGRA), a soft breaking scale characterized by a gravitino mass m / > ∼ m Z ∼ m h ∼
100 GeV. Thus, the Higgs mass and sparticlemass limits combine to sharpen the “Little Hierarchy”[8] typified by m h (cid:28) m / . The grow-ing Little Hierarchy has prompted several authors to question whether the MSSM is overlyfine-tuned, and either flatly wrong[9] or at least in need of additional features which sacrificeparsimony/minimality[10]. Before rushing to such drastic conclusions, it is prudent to ascertainif all SUSY spectra are fine-tuned or if some spectra are indeed natural. To proceed further one must adopt at least one of several quantitative naturalness measureswhich are available. We label these as • the electroweak measure ∆ EW [11, 12, 13, 14, 15], • the Higgs mass fine-tuning measure ∆ HS [16, 17] and • the traditional EENZ/BG measure ∆ BG [18, 19].Indeed, recently it has been shown that, if applied properly, then all three measures agreewith one another[20] and predict a very specific SUSY spectra with just ∼
10% fine-tuning. Ifapplied incorrectly– by not properly combining dependent quantities contributing to m Z or m h one with another– then overestimates[21] of fine-tuning can occur in ∆ HS and ∆ BG , often byorders of magnitude. ∆ EW The electroweak measure ∆ EW requires that there be no large/unnatural cancellations in de-riving the value of m Z from the weak scale scalar potential: m Z m H d + Σ dd ) − ( m H u + Σ uu ) tan β (tan β − − µ (cid:39) − m H u − µ (1)where m H u and m H d are the weak scale soft SUSY breaking Higgs masses, µ is the supersym-metric higgsino mass term and Σ uu and Σ dd contain an assortment of loop corrections to the1ffective potential. The ∆ EW measure asks for the largest contribution on the right-hand-sideto be comparable to m Z / m Z = 91 . | µ | ∼ m Z and also that m H u is driven radiatively tosmall, and not large, negative values[11, 13]. Also, the top squark contributions Σ uu (˜ t , ) areminimized for TeV-scale highly mixed top squarks, which also lift the Higgs mass to m h ∼ ∆ HS The Higgs mass fine-tuning measure ∆ HS asks that the radiative correction δm H u to the Higgsmass m h (cid:39) µ + m H u (Λ) + δm H u (2)be comparable to m h . This contribution is usually written as δm H u | rad ∼ − f t π ( m Q + m U + A t ) ln (Λ /m SUSY ) which is used to claim that third generation squarks m ˜ t , , ˜ b be approxi-mately less than 500 GeV and A t be small for natural SUSY. However, several approximationsare necessary to derive this result, the worst of which is to neglect that the value of m H u itselfcontributes to δm H u . By combining dependent contributions, then instead one requires thatthe two terms on the RHS of m h = µ + (cid:16) m H u (Λ) + δm H u (cid:17) (3)be comparable to m h . The recombination in Eq. 3 leads back to the EW measure since m H u (Λ) + δm H u = m H u ( weak ). ∆ BG The EENZ/BG measure[18, 19] (hereafter denoted simply by BG) is given by∆ BG ≡ max i [ c i ] where c i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ln m Z ∂ ln p i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p i m Z ∂m Z ∂p i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4)where the p i constitute the fundamental parameters of the model. Thus, ∆ BG measures thefractional change in m Z due to fractional variation in (high scale) parameters p i . The c i are known as sensitivity co-efficients [19]. For the pMSSM (MSSM defined only at the weakscale), then explicit evaluation gives ∆ BG (cid:39) ∆ EW . For models defined in terms of high scaleparameters, the BG measure can be evaluated by expanding the terms on the RHS of Eq.1 using semi-analytic RG solutions in terms of fundamental high scale parameters[22]: fortan β = 10 and taking Λ = m GUT , then one finds[23, 24] m Z (cid:39) − . µ + 3 . M − . M A t − . m H u − . m H d + 0 . m Q + 0 . m U + · · · (5) It is sometimes claimed that by using this method, then the SM would not be fine-tuned for large cutoffscales Λ (cid:29) Z -mass squared: e.g. c m Q = 0 . · ( m Q /m Z ). If one allows m Q ∼ m h ) then one obtains c m Q ∼
800 and so∆ BG ≥ e.g. M (Λ) = a M m / , A t = a A t m / , m Q = a Q m / etc. where the a i are just numbers. (For example, in stringtheory with dilaton-dominated SUSY breaking[26, 27], then we expect m = m / with m / = − A = √ m / ). The reason one scans multiple SUSY model soft term parameters is toaccount for a wide variety of possible hidden sectors. But this doesn’t mean each soft term isindependent from the others. By writing the soft terms in Eq. 5 as suitable multiples of m / ,then large positive and negative contributions can be combined/cancelled and one arrives atthe simpler expression[20]: m Z = − . µ (Λ) + a · m / . (6)The value of a is just some number which is the sum of all the coefficients of the terms ∝ m / . Using the BG measure applied to Eq. 6, then it is found that naturalness requires µ ∼ m Z andalso that am / ∼ m Z . The first requirement is the same as in ∆ EW . The second requirement isfulfilled either by m / ∼ m Z [19] (which seems unlikely in light of LHC Higgs mass measurementand sparticle mass bounds) or that m / is large but the co-efficient a is small[20]: i.e. thereare large cancellations in Eq. 5. Since µ (Λ) (cid:39) µ ( weak ), then also am / (cid:39) m H u ( weak ) and soa low value of ∆ BG also requires a low value of m H u : i.e. m H u is driven radiatively to smallnegative values. This latter situation is known as radiatively-driven natural supersymmetry , orRNS. The natural SUSY spectra is typified by a spectra of low-lying Higgsinos (cid:102) W ± , (cid:101) Z , with mass ∼ −
300 GeV, the closer to m Z the better, along with TeV-scale but highly mixed top-squarks ˜ t , .[11, 13] The gluino mass can range between current LHC8 limits and about 4 TeV,and may well lie beyond LHC14 reach[31]. First/second generation matter scalars may well liein the 5 −
30 TeV range, thus supplying at least a partial decoupling solution to the SUSYflavor, CP, proton decay and gravitino problem . In addition, it should be clear from Eq. 1that m H d / tan β ∼ m Z (a point mentioned previously in Ref. [32]). For m H d large, then oneexpects m A ∼ m H d . Requiring the term containing m H d in Eq. 1 to be comparable to m Z / If µ is also computed as µ = a µ m / as in the Giudice-Masiero mechanism[28], then m Z = const. × m / and ∆ BG ≡ µ -problem, such as Kim-Nilles[30], then µ is instead related to the Peccei-Quinn breaking scale and is expected to be independent. Inthe former case, then the responsibility is to find a suitable hidden sector which would actually generate m Z atits measured value. We are aware of no such models which even come close to that. Since m ˜ q, ˜ (cid:96) ∼ m / , then we would expect m / also at the 5 −
30 TeV level. then implies m A ∼ (cid:12)(cid:12)(cid:12) m H d (cid:12)(cid:12)(cid:12) < ∼ | µ | tan β . (7)Thus, for | µ | <
300 GeV, we would expect for tan β = 10 that m A < ∼ β ashigh as 50, we expect m A < ∼
15 TeV without becoming too unnatural.In this paper, we explore the implications of SUSY naturalness for the heavy Higgs bosonsof the MSSM: A , H and H ± . This topic has also been addressed in the recent paper [33].In Ref. [33], using several different naturalness measures along with a low mediation scaleΛ ∼ −
100 TeV and hard SUSY breaking contributions to the scalar potential, the authorsconclude that heavy Higgs bosons should lie around the 1 TeV scale, and that since the heavyHiggs bosons are less susceptible to having hidden decay modes, their search should be animportant component of the search for natural SUSY.In this paper, we will arrive at quite different conclusions. In Sec. 2, using the unifiednaturalness criteria, as embodied in ∆ EW , we will find that SUSY models which are valid allthe way up to Λ = m GUT (cid:39) × GeV can be found with fine-tuning at the ∆ EW ∼ − m A < ∼ β < ∼
15 while m A < ∼ β values ranging as high as 50 −
60. While the region m A < ∼ A , H and H ± branching fractions as a function of massfor a benchmark case with radiatively-driven naturalness. Since for naturalness µ ∼ − W , Z or h plus higgsinos, then the qualitatively new decay modes arise: A , H , H ± → W , Z or h plus missing E T ( (cid:54) E T ). These new decay modes– which are quite differentthan those expected in non-natural SUSY models with a bino-like LSP– offer new avenues forheavy Higgs searches at LHC. A simple mass bound from naturalness on heavy Higgs bosons can be directly read off fromEq. 1. The contribution to ∆ EW from the m H d term is given by C H d = m H d / (tan β − / ( m Z / . (8)Also the tree level value of m A is given by m A = m H u + m H d + 2 µ (cid:39) m H d − m H u ∼ m H d (9)4here the first partial equality holds when µ ∼ − m H u and the second arises when m H d (cid:29)− m H u . Combining these equations, then one expects roughly that m A < ∼ m Z tan β ∆ / EW ( max ) (10)where ∆ EW ( max ) is the maximal fine-tuning one is willing to tolerate. For ∆ − EW = 10%fine-tuning with tan β = 10, then one expects m A < ∼ uu (˜ t , ) and Σ uu (˜ b , )(complete expressions are provided in the appendix of Ref. [13]) can become large and arehighly tan β dependent.To evaluate the range of m A expected by naturalness, we will generate SUSY spectra usingIsajet[34, 35] in the 2-parameter non-universal Higgs model[36] (NUHM2) which allows for verylow values of ∆ EW <
10 (numerous other constrained models are evaluated in Ref. [20] andalways give much higher EW fine-tuning). The parameter space is given by m , m / , A , tan β, µ, m A , (NUHM2) . (11)The NUHM2 spectra and parameter spread versus ∆ EW were evaluated in Ref. [13] but withthe range of m A restricted to < . m A : m : 0 −
20 TeV ,m / : 0 . − , − < A /m < ,µ : 0 . − . , (12) m A : 0 . −
20 TeV , tan β : 3 − . We require of our solutions that: • electroweak symmetry be radiatively broken (REWSB), • the neutralino (cid:101) Z is the lightest MSSM particle, • the light chargino mass obeys the model independent LEP2 limit, m (cid:101) W > . • LHC search bounds on m ˜ g and m ˜ q are respected, • m h = 125 . ± . EW vs. m A . The dots arecolor-coded according to low, intermediate and high tan β values. From the plot, we see firstthat there is indeed an upper bound to m A given by naturalness. In fact, for tan β < EW <
10, then indeed m A < ∼ β >
15, we do not generate any solutions with ∆ EW <
10. For ∆ EW <
30 (dotted horizontal5igure 1: Plot of ∆ EW versus m A from a scan over NUHM2 parameter space.line), then we have m A < ∼ β <
15, and m A < ∼ β <
30 (60). Whilethese values provide upper bounds on m A from naturalness, we note that m A values as low as150-200 GeV can also be found. Since LHC14 searches for heavy Higgs are roughly sensitive to m A < ∼ m A and tan β , we nextadopt a proposed RNS benchmark point from Ref. [39]. This point has NUHM2 parametersgiven by m = 5 TeV , m / = 0 . , A = − . , tan β = 10 , with µ = 110 GeV and m A = 1 TeV . (13)The value of ∆ EW is found to be 13.8 . Here, we adopt this benchmark point, but now allow m A and tan β as free parameters and plot color-coded ranges of ∆ EW in the m A vs. tan β plane,as shown in Fig. 2.From Fig. 2, we see that indeed the region with lowest ∆ EW occurs around m A ∼ . − . β < ∼
10. The yellow colored regions have ∆ EW <
50. For these values, we find amore expansive region with m A < ∼ β < ∼
20. However, a second region with low∆ EW <
50 opens up at high tan β ∼ −
52 with m A < ∼ β ∼ − EW in the m A vs. tan β plane for the RNS benchmarkpoint Eq. 13. 7egion has greater fine-tuning, where the maximal contributions to ∆ EW we find arise from theradiative corrections Σ uu (˜ b ). In many studies of the prospects for heavy Higgs boson discovery at the LHC, it is assumed thatthe Standard Model decay modes of A , H and H ± are dominant. The prospects for discoveryare usually presented in the m A vs. tan β plane. At NLO in QCD, then the gluon fusionreactions gg → A, H are usually dominant out to m A,H < ∼ gg → A, H is thenthe
A, H → τ + τ − mode where the ditau mass can be reconstructed. Current search limitsfrom Atlas and CMS exclude m A < ∼ . β as high as 50. For lower tan β values, themass bounds are very much weaker[41] ( e.g. for tan β = 10, then m A >
400 GeV). Productionof heavy Higgs bosons in association with b -jets may aid the search[42]. In addition, the rarerdecays into dimuons may also be possible[43, 44], and recently dimuon signatures in associationwith b -jets have been explored[45, 46].The importance of heavy Higgs decay into SUSY modes was explored long ago[47] for thecase where the LSP was usually taken to be a bino. If SUSY decay modes of H or A are open,then the SM branching fractions diminish while the new SUSY modes offer novel detectionstrategies[48]. The unique feature of SUSY models with radiatively-driven naturalness is the presence of lighthiggsino states with mass ∼ −
300 GeV, the closer to m Z the better. This fact means thatfor most of the mass range of m A,H , then SUSY decay modes should be open. Furthermore,the higgsino-like LSP implies that the SUSY decay modes will generally be quite different thanin earlier models where a bino-like LSP was considered.In Fig. 3, we show the branching fraction as calculated by Isajet[34] of the pseudoscalar A boson versus m A for the RNS benchmark point from Sec. 2, but now with m A taken asvariable, with tan β = 10. At low m A ∼
200 GeV, then SUSY decay modes are kinematicallyclosed and A → b ¯ b at ∼
85% as is typical when SM decay modes are considered and the t ¯ t modeis closed. As m A increases beyond 200 GeV, then already the A → higgsino pairs opens up,and the SM branching fractions diminish. For m A > ∼
700 GeV, then the mixed higgsino/winomode A → (cid:102) W (cid:102) W turns on and rapidly dominates the branching fraction. This is because theSUSY Higgs coupling to -inos involves a product of gaugino component of one -ino times thehiggsino components of the other -ino and in this case (cid:102) W is higgsino-like and (cid:102) W is wino-like.For m A > ∼ ∼
50% level. For m A > ∼ A → (cid:101) Z (cid:101) Z and (cid:101) Z (cid:101) Z are important. For TeV-scale values of m A , the SM decaymode A → b ¯ b drops to below the 10% level while A → τ ¯ τ has dropped to the percent level. Inthis case, then the search for heavy Higgs bosons utilizing SM decay modes will be much moredifficult. See p. 178-179 of [49]. A vs. m A for the RNS benchmark point Eq. 13 but withvariable m A . 9igure 4: Branching fraction of H vs. m H for the RNS benchmark point Eq. 13 but withvariable m A .In Fig. 4, we show the branching fractions of the heavy scalar Higgs H versus m H for thesame RNS benchmark point. The overall behavior is similar to the case of the pseudoscalar A :at low values of m H , then the SM decay modes are dominant, but once m H is heavy enough,the supersymmetric decay modes quickly open up and dominate the branching fractions. Atlarge m H , then the H → (cid:102) W (cid:102) W , (cid:101) Z (cid:101) Z and (cid:101) Z (cid:101) Z decay modes are dominant.In Fig. 5 we show the branching fractions of H + versus m H + for the same RNS benchmarkpoint. In this case, at low values of m H + , then H + → t ¯ b is dominant followed by H + → τ + ν τ .As m H + increases, then H + → (cid:102) W +1 (cid:101) Z turns on and later also (cid:102) W +2 (cid:101) Z , (cid:102) W +1 (cid:101) Z and (cid:102) W +2 (cid:101) Z allturn on. At m H + > ∼ H , A → W + (cid:54) E T We have seen that for m A,H > ∼ H, A → (cid:102) W ± (cid:102) W ∓ .Since the (cid:102) W is higgsino-like, it tends to have only a small mass gap with the LSP: m (cid:101) W − m (cid:101) Z ∼ H + vs. m H + for the RNS benchmark point Eq. 13 but withvariable m A . 11igure 6: Transverse mass distribution for e + plus (cid:54) E T events at LHC14 from W + , W + Z and A, H production from benchmark point Eq. 13.10 −
20 GeV. In this case, the visible energy from (cid:102) W → f ¯ f (cid:48) (cid:101) Z decay (where f denotes SMfermions) is quite soft– most of the energy goes into making up the (cid:101) Z rest mass– and so thehiggsinos are only quasi-visible. On the other hand, the branching fractions for (cid:102) W decay inthe RNS model have been plotted out in Ref. [31] and found to be: (cid:102) W → (cid:102) W Z , (cid:101) Z W , (cid:101) Z W each at about 30% with (cid:101) Z W accounting for the remainder. Thus, we expect s -channel H and A production to give rise to gg → H, A → W + (cid:54) E T → (cid:96) ± + (cid:54) E T (14)which is a rather unique signature for heavy Higgs boson production.The dominant backgrounds come from direct W production followed by W → (cid:96)ν (cid:96) decayand also W Z production followed by Z → ν ¯ ν and W → (cid:96)ν (cid:96) . In Fig. 6, we plot the e + + (cid:54) E T transverse mass distribution from the signal using the RNS benchmark point with m A = 1TeV along with SM backgrounds. The signal from A, H production with m A,H ∼ β = 10 and 30 is well below background. 12igure 7: Dilepton cluster transverse mass distribution for e + e − plus (cid:54) E T events at LHC14 from ZZ and A, H production from benchmark point Eq. 13. H , A → Z + (cid:54) E T As mentioned above, (cid:102) W → (cid:102) W Z at about 30-35% in radiatively-driven natural SUSY. Thus,an alternative signature comes from gg → H, A → Z + (cid:54) E T → (cid:96) + (cid:96) − + (cid:54) E T . (15)The background to this process comes from ZZ production where one Z → ν ¯ ν whilst theother goes as Z → (cid:96) + (cid:96) − . In Fig. 7 we plot the distribution in cluster transverse mass[51] m T ( (cid:96) + (cid:96) − , (cid:54) E T ) from heavy Higgs H , A production followed by their decays to Z ( → (cid:96) + (cid:96) − )+ (cid:54) E T from the RNS benchmark point for m A = 1 TeV along with ZZ background. Here we see thatsignal from A, H production with m A,H ∼ β = 10. If we increase tan β to 30, then signal and BG become comparable at very large m T ( (cid:96) + (cid:96) − , (cid:54) E T ) although in this range the event rate is quite limited. H , A → h + (cid:54) E T A third possible signature consists of
A, H → (cid:101) Z , (cid:101) Z , where (cid:101) Z , → (cid:101) Z , h resulting in a( h → b ¯ b )+ (cid:54) E T signature. We expect such a signal to lie well below backgrounds from Zh and13 Z production. In this paper we have examined the implications of SUSY naturalness for the heavy Higgs bosonsector. We use the ∆ EW measure of naturalness, although we show that– properly applied– theHiggs mass fine-tuning and also the EENZ/BG fine-tuning would give similar results since∆ HS (cid:39) ∆ BG (cid:39) ∆ EW (16)so long as dependent terms are properly combined before evaluating naturalness.Using the ∆ EW measure, then we find upper bounds on the heavy Higgs masses: for 10%fine-tuning and tan β ∼
10, we expect m A < ∼ . β ashigh as 50, then m A < ∼ (cid:102) W ± and (cid:101) Z , are expected to have mass ∼ − m Z the more natural), then almost always there will be supersymmetricdecays modes open to the heavy SUSY Higgs states. We evaluated these branching fractionsand find that they can in fact be the dominant decay modes, especially if m A,H > m (cid:101) W + m (cid:101) W ,in which case this decay mode tends to dominate. The supersymmetric decay modes diminishthe SM decay modes of H , A and H ± making standard search techniques more difficult fora specified heavy Higgs mass. However, qualitatively new heavy Higgs search modes appearthanks to the supersymmetric decay modes. Foremost among these are the decays H, A → (cid:102) W ± (cid:102) W ∓ which results in final states characterized by W , Z or h plus (cid:54) E T . These new signaturesseem to be rather challenging to extract from SM backgrounds which occur at much higherrates. It may well be that forward b -jet tagging in bg → bA or bH production or gb → tH + production followed by A, H, H ± → SU SY decays could ameliorate the situation.
Acknowledgments
We thank X. Tata for discussions. This work was supported in part by the US Department ofEnergy, Office of High Energy Physics.
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