Fine-tuning in GGM and the 126 GeV Higgs particle
aa r X i v : . [ h e p - ph ] F e b Fine-tuning in GGM and the 126 GeV Higgs particle
Zygmunt Lalak ∗ and Marek Lewicki † Institute of Theoretical Physics, Faculty of Physics, University of Warsawul. Ho˙za 69, Warsaw, Poland
Abstract
In this paper we reanalyze the issue of fine-tuning in supersymmetric models which featureGeneralized Gauge Mediation (GGM) in the light of recent measurement of the mass of the lightHiggs particle and taking into account available data on the value of the muon magnetic moment g µ −
2. We consider GGM models with 3, 5 and 6 input parameters and reduce the fine-tuningby assuming simple relations between them at the high scale. We are able to find solutions whichgive the correct value of the light Higgs mass and are less fine-tuned than models with standardgauge mediation (and with gravity mediation), however one never finds fine-tung measure lowerthan about 10 if one neglects the data on g µ − g µ − g µ − µ and B µ , then in the case with neglected g µ − g µ − g µ − The discovery of the Higgs boson at LHC with mass of about 126GeV seems to favour MSSMwhich predicts that the lightest Higgs boson can’t be much heavier than Z boson. However Higgsmass this far from Z mass requires large radiative corrections which have to come from heavysupersymmetric particles. Such heavy sparticles reintroduce some fine-tuning in MSSM [1] becauselarge supersymmetric parameters also have to cancel out to secure electroweak symmetry breaking atthe right energy scale, thus threatening the motivation of SUSY as solution to naturalness problem.In MSSM large fine-tuning originates from requiring sparticles heavy enough to generate observedHiggs mass. In mSUGRA models the simplest way of increasing Higgs mass is to get maximal stopmixing which increases dominant stop correction, but requires large negative A-terms. In gaugemediated models [2] however, usually only negligible A-terms are generated at the SUSY breakingscale. So the Higgs mass can be increased only using non-universality of scalars and fermions throughsubdominant corrections. ∗ [email protected] † [email protected]
Neutral part of the scalar potential in MSSM takes the form V = ( µ + m H u ) | H u | + ( µ + m H d ) | H d | + ( bH u H d + h.c)+ 18 ( g + g ′ ) (cid:0) | H u | − | H d | (cid:1) . (2.1)Naturalness problem appears in MSSM when we require that the above potential gives correctelectroweak symmetry breaking, which gives us Z boson mass in terms of superymmetric parameters M Z = tan 2 β (cid:0) m H u tan β − m H d cot β (cid:1) − µ . (2.2)Pushing light Higgs mass to the observed value of 126GeV requires large radiative corrections, thebiggest one comes from top-stop loop [8] δm h = 3 g m t π m W (cid:20) log (cid:18) M S m t (cid:19) + X t M S (cid:18) − X t M S (cid:19)(cid:21) , (2.3)where M S = (cid:16) m t + m t (cid:17) is the average of stop masses, and X t = m t ( A t − µ cot β ) is an offdiagonal element of stop mass matrix. Parameters in (2.2) also receive top-stop loop corrections δm H u | stop = − Y t π (cid:0) m Q + m U + | A t | (cid:1) log (cid:18) M u TeV (cid:19) , (2.4)where m Q m U and A t are supersymmetric parameters that predict the stop mass, and M u is ascale at which soft masses are generated.So requiring correct Higgs mass gives large corrections that have to cancel out on the right handside of (2.2) to give the correct M Z .We define fine-tuning measure with respect to parameter a as follows [9]∆ a = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ln M Z ∂ ln a (cid:12)(cid:12)(cid:12)(cid:12) . (2.5)Fine-tuning connected with a set of independent parameters a i is then∆ = max a i ∆ a i . (2.6)2emembering that fine-tuning in the Standard Model actually appeared in the Higgs boson masswe can define fine-tuning with respect to Higgs mass in MSSM∆ h a = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ln m h ∂ ln a (cid:12)(cid:12)(cid:12)(cid:12) ; ∆ h = max a i ∆ h a i , (2.7)and calculate it numerically similarly to fine-tuning with respect to Z mass. In our Figures we plotseveral regions of allowed solutions corresponding to different models on top of each other, becauseonly the borders of these regions (corresponding to the minimal fine-tuning) are actually important.Figure 1 shows that as expected fine-tuning from Higgs boson mass turns out to be similar to the10020050010002000500010000120 122 124 126 128 130 ∆ m h [GeV] ∆ m h ∆Figure 1: Fine-tuning from Higgs mass and Z mass in mSUGRA model with soft terms generatedat scale M u = 2 . × GeV and with tan β = 40.one obtained from Z boson mass and usually is a few percent lower.3 Gravity vs Gauge mediation
Meade, Shih and Seiberg [3] defined gauge mediated models as those in which visible and hiddensectors decouple when gauge couplings vanish. They also have shown that in general such models canonly have six parameters determining the low energy sparticle spectrum. In this work we parametrisethe high energy soft SUSY breaking terms with three parameters corresponding to gaugino masses M = m Y , M = m w , M = m c and three parameters determining scalar masses Λ c , Λ w , Λ Y which give m f = 2 (cid:20) C ( f ) (cid:16) α π (cid:17) Λ c + C ( f ) (cid:16) α π (cid:17) Λ w + C ( f ) (cid:16) α π (cid:17) Λ Y (cid:21) , (3.1)where α i = g i / π and C ( f ) = 35 Y f C ( f ) = (cid:26) for f = Q, L, H u , H d f = U, D, E (3.2) C ( f ) = (cid:26) for f = Q, U, D f = E, L, H u , H d . A specific model of gauge mediation gives above quantities in terms of physical parameters presentin the model. As an example we use two of the models published in [10], the first of which (GGM1)is defined by the superpotential W = X i ( y i ¯ QQ + r i ¯ U U + s i ¯ EE ) , (3.3)with three independent parameters used to calculate soft massesΛ Q = y i F i y j X j Λ U = r i F i r j X j Λ E = s i F i s j X j . (3.4)In terms of which soft masses take the form m c = α π (2Λ Q + Λ U ) , m w = α π Q , m Y = α π (cid:18)
43 Λ Q + 83 Λ U + 2Λ E (cid:19) , Λ c = 2Λ Q + Λ U , Λ w = 3Λ Q , Λ Y = 43 Λ Q + 83 Λ Q + 2Λ E . (3.5)(3.6)The second model (GGM2) is defined by W = X i ( y i ¯ QQ + r i ¯ U U + s i ¯ EE + λ iq q ˜ q + λ il l ˜ l ) + F i X i , (3.7)with five independent parameters used to calculate soft massesΛ Q = y i F i y j X j Λ U = r i F i r j X j Λ E = s i F i s j X j Λ q = λ iq F i λ jq X j Λ l = λ il F i λ jl X j . (3.8)4gain we obtain soft masses of the form m c = α π (Λ q + 2Λ Q + Λ U ) , m w = α π (Λ l + 3Λ Q ) ,m Y = α π (cid:18)
23 Λ q + Λ l + 43 Λ Q + 83 Λ U + 2Λ E (cid:19) , (3.9)Λ c = Λ q + 2Λ Q + Λ U , Λ w = Λ l + 3Λ Q , Λ Y = 23 Λ q + Λ l + 43 Λ Q + 83 Λ U + 2Λ E . Main disadvantage of gauge mediation in respect to of fine-tuning comes from the fact thatonly negligible A -terms are generated at SUSY breaking scale. Large mixing in the sfermion massmatrices would increase its contribution to Higgs mass as in eq.(2.3), and make it easier to achievethe experimental result of Higgs boson mass. On the other hand prediction of nonuniversal gauginomasses makes it easier to avoid experimental bound on gluino mass. Nonuniversal scalar masseshelp avoiding bounds on masses of the first and second generation squraks. We use the followingbounds on sparticle masses [7] m g ≥ ,m u i , m d i , m c i , m s i ≥ i = 1 , ,m t i ≥ i = 1 , , (3.10) m b i ≥ i = 1 , ,m χ ≥ . We also assume M u = 10 GeV and tan β = 40. Figure 2 shows that generally models with largernumber of free parameters predict smaller fine-tuning because they allow to increase Higgs bosonmass with subdominant corrections.In a general model with 6 parameters, the biggest sources of fine-tuning are the gluon massparameter m c or contributions to scalar masses connected with color Λ c or weak interactions Λ w .The µ parameter contribution is small in solutions that minimize fine-tuning for a given Higgs mass,because µ can be decreased by increasing Λ Y and Λ w and decreasing Λ c which increases high scale m H u while keeping masses of coloured particles fixed. The walue of µ decreases with decreasingenergy scale and eventually runs negative to secure right elektroweak symmetry breaking, as we cansee from large tan β approximation of (2.2) m Z ≈ − m H u − | µ | . (3.11)As we can see increasing high scale m H u makes it run down towards smaller negative value and sodecreases µ required to obtain correct Z mass. Since colored particle masses that would increaseoverall fine-tuning aren’t changed we obtain a scenario with smaller µ parameter and similar fine-tuning. Meanwhile, increased Λ Y and Λ w give us larger sub dominant corrections to Higgs massdue to increased masses of non coloured particles.In model GGM2 squark and gluino masses obtain contributions from all parameters connectedwith color interactions Λ Q , Λ U , Λ q and fine-tuning coming from these masses is distributed amongthese fundamental parameters. The largest fine-tuning contribution turns out to come typically50020050010002000500010000120 122 124 126 128 130 ∆ m h [GeV] ∆ fullGGM ∆ GGM ∆ GGM Figure 2: Fine-tuning in models GGM1 and GGM2 as well as in the general six parameter case.from the µ parameter and can come from one of the parameters connected with color only if saidparameter is much larger than the other two.The same can be said about the model GGM1. The biggest source of fine-tuning is usually µ ,except cases where one of the other parameters is much larger than the other two. The simplest way of reducing fine-tuning is assuming we are considering a model that predicts theparameters which are not independet of each other, but instead are functions of some fundamental6arameters. For example, if gaugino masses M i are given functions of parameter M we obtain M i = f i ( M ) ,
12 ∆ M = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ln M Z ∂ ln M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M M z ∂M Z ∂M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M M z ∂M Z ∂M i ∂M i ∂M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.1)= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M M z M i M i f ′ i ( M ) ∂M Z ∂M i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M f ′ i ( M ) f i ( M ) M i M Z ∂M Z ∂M i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M f ′ i ( M ) f i ( M ) ∂ ln M Z ∂ ln M i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 c i ( M ) ∂ ln M Z ∂ ln M i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . If f i are simply proportional to M one finds∆ M = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 ∂ ln M Z ∂ ln M i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . If these functions were logarithms f i ( M ) = ˜ m ln M Q , (4.2) c i ( M ) = ˜ mM i . Keeping in mind that fine-tuning is proportional to soft terms ∆ M i ∝ M i , we obtain ∆ M ∝ ˜ m .We now check how the usual proportionality of nonuniversal soft masses and µ parameter helpsreducing fine-tuning in models GGM1 and GGM2 as well as in the general case. As one can seefrom Figure 3, simple proportionality of soft terms can greatly decrease fine-tuning in GGM butone still finds ∆ >
100 for m h = 126, even in the most general case. We have also checked that inmodels considered here (for example GGM1 in Figure 4) fine tuning coming only from the gaugemediated soft terms can cancel out very precisely if they are proportional to one another, as pointedout in [11]. g µ − In this section we check whether discussed models can accommodate the discrepancy between mea-sured muon magnetic moment and the standard model prediction [12] δa µ = a EXP µ − a SM µ = (2 . ± . − . (5.1)The simplest approximation of supersymmetric contribution to muon magnetic moment is obtainedby assuming that tan β is large and all masses in slepton sector are equall to M SUSY . This way oneobtains [13] δa SUSY µ ≈ (cid:18) g − g π + g π (cid:19) m µ M tan β, (5.2)70501002005001000120 122 124 126 128 130 ∆ m h [GeV]Λ i ∝ Λ j ∆ fullGGM ∆ GGM ∆ GGM ∆ m h [GeV]Λ i ∝ Λ j ∝ µ ∆ fullGGM ∆ GGM ∆ GGM Figure 3: Fine-tuning in models GGM1 and GGM2 as well as in the general six parameter case withparameters defining superparticle spectrum proportional to each other and to the µ parameter.which indicates a problem since Higgs boson mass depends on soft breaking terms only logarithmi-cally. We evaluate δa SUSY µ numerically using full 1-loop SUSY corrections and 2-loop QED logarith-mic corrections from [13]. From Figure 2 we can see that only the general case predicts δa µ within1 σ bound for m h = 126, while other models fall out of 2 σ bounds. Even in the most general case itis hard to increase δa µ because all slepton generations have the same mass at the high scale. The3rd generation gets negative contribution from large Yukawa coupling16 π ddt m L ⊃ | h τ | ( m H d + m L + m E + A τ ) (5.3)which can make stau tachionic before smuon is light enough to produce the required value of δa µ .Also requiering small masses in slepton sector means we can only increase Higgs mass withdominant squark corrections which increase fine-tuning. And we are left only with solutions withmuch higher fine-tuning than those that use sub dominant corrections to Higgs mass which wedescribed in previous chapters. We have reanalyzed the issue of fine-tuning in supersymmetric models which feature GeneralizedGauge Mediation (GGM) in the light of recent discovery of the 126 GeV Higgs particle and takinginto account available data on the value of the muon magnetic moment g µ −
2. We consider GGM8.010.11101001000120 122 124 126 128 130 ∆ m h [GeV]∆ GGM without ∆ µ , ∆ B µ ∆ GGM Figure 4: Fine-tuning in model GGM1 with and without contribution to fine-tuning from µ and B µ parameters and with soft terms proportional to each other.models with 3, 5 and 6 input parameters and reduce the fine-tuning by assuming simple relationsbetween them at the high scale. We are able to find solutions which give the correct value of thelight Higgs mass and are less fine-tuned than models with standard gauge mediation, however onenever obtains fine-tung measure lower than about 10 if one neglects the data on g µ − g µ − g µ − µ and B µ , since itisn’t obvious that the origin of these two parameters has anything to do with gauge mediation. Itis interesting to note, that once this is done, then in the case with neglected g µ − g µ − g µ − σ σ σ σ σ σ
120 122 124 126 128 130 δ a M SS M µ m h [GeV]SUSY contribution to muon g-2 δa fullGGMµ δa GGM µ δa GGM µ ∆ m h [GeV]fine-tuning∆ fullGGM ∆ GGM ∆ GGM Figure 5: Regions of largest possible SUSY contribution to muon g-2 and corresponding fine-tuningTo sum up, in models featuring GGM one can naturally obtain fine-tuning smaller than that inmodels with gravity mediation, despite vanishing A-terms at the high scale. Moreover, consideringexclusively fine-tuning coming from gauge-mediated soft masses one can easily achieve arbitrar-ily small fine-tuning while staying with the correct value of the light Higgs mass. Imposing theagreement of the model with the g µ − σ band. Even a small decrease ofthe measured value of the Higgs mass would allow for much better agreement of GGM models withmeasured g µ − Acknowledgements
Authors thank S. Pokorski for very helpful discussions. ZL thanks D. Ghilencea for discussions.This work was supported by the Foundation for Polish Science International PhD Projects Pro-gramme co-financed by the EU European Regional Development Fund. This work has been partiallysupported by Polish Ministry for Science and Education under grant N N202 091839, by NationalScience Centre under research grant DEC-2011/01/M/ST2/02466 and by National Science Centreunder research grant DEC-2011/01/M/ST2/02466.
A Numerical procedure
The numerical procedure we used is similar to the ones used in existing codes like [6]. We work withquantities renormalized in DR and use renormalization group equations (RGE), to iteratively find10ow energy parameters for a given set of high energy set terms.calculate radiative corrections to couplings g i ( M Z ), h t ( M Z ), h b ( M Z ), h τ ( M Z ) (use SM valuesin the first run) ❄ RGE : M z → M u include soft breaking terms given at high scale M u ❄ RGE : M u → M EW SB iteratively calculate µ , B µ and the mass spectrum(in the first run find estimates for M EW SB , µ and B µ ) ❄ if µ convergedcalculate physical masses ✻ R G E : M E W S B → M Z Figure 6: Schematic of the algorithm we used. all the stepsare described in the appendix.
A.1 M Z Scale At M Z scale we include radiative corrections to couplings. We set Yukawa couplings using tree levelrelations h t = m t √ v sin β , h b = m b √ v cos β , h τ = m τ √ v cos β , (A.1)where m t , m b , m τ are fermion masses and v is the Higgs field void expectation value. At firstiteration we use physical masses and SM Higgs vev v ≈ ,
22. At next iterations above quantitiesare renormalized in DR scheme and one-loop corrections are included. To calculate top mass weuse 2-loop QCD corrections [15] and 1-loop corrections from super partners from the appendix of[16]. While calculating bottom mass we follow Les Houches Accord [17], starting from running mass11n
M S scheme in SM m bMSSM . Next applying procedure described in [18] we find DR mass at M Z ,from which we get MSSM value by including corrections described in appendix D of [16]. Whilecalculating tau mass we include only leading corrections from [16]. We calculate Higgs vev in MSSMusing v = 4 M Z + ℜ Π TZZ ( M Z ) g + 3 g / , (A.2)where we include Z self interactions described in appendix D of [16]. To calculate g , g i g in DR in MSSM we use procedure described in appendix C of [16]. A.2 RGE and M u scale after calculating coupling constants at M Z scale we numerically solve renormalization group equa-tions [14],[19], to find their values at M u scale, at which we include the soft breaking terms.Then we solve RGEs again to get sot terms, coupling constants, tan β and Higgs vev v at scale M EW SB = p m ¯ t ( M EW SB ) m ¯ t ( M EW SB ). At first iteration we take µ = sgn( µ )1GeV and B µ = 0and run to scale at which the above equation is fulfilled.Next using potential minimization conditionswe find new values of µ and B µ . A.3 Calculation of physical masses
To calculate physical masses we use only leading corrections described in [16] everywhere but theHiggs sector. In the Higgs masses calculation we use full one-loop corrections from [16] and leadingtwo-loop corrections described in [20].
A.4 Constraints imposed on the scalar potential
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