Light sfermion interplay in the 125 GeV MSSM Higgs production and decay at the LHC

We study the effects from light sfermions on the lightest Higgs boson production and decay at the Large Hadron Collider (LHC) within the Minimal Supersymmetric Standard Model (MSSM). We find that the scenario with light coloured sfermions -- stops or sbottoms -- has the potential to explain a non-universal alteration, as hinted by LHC data, of the gluon-gluon Fusion ($\mu_{ggF}$) with respect to the Vector Boson Fusion ($\mu_{VBF}$) event rates and, in particular, can predict $\frac{\mu_{VBF}}{\mu_{ggF}}>1$ for all Higgs boson decay channels in large areas of the parameter space. We also find that the scenario with a light stop/sbottom can be complemented by the scenario in which the total Higgs width, $\Gamma_{\rm tot}$, is reduced due to a suppressed Yukawa coupling $Y_b$. In this case, the reduction of the Higgs production rates in the $ggF$ process which occurs in the maximal mixing scenario is compensated by the reduction of the $H\to b\bar{b}$ partial decay width, the largest component of $\Gamma_{\rm tot}$. Furthermore, we highlight the fact that, in the light stop/sbottom scenario, event rates with the Higgs boson decaying to a $b\bar{b}$ final state are predicted to be essentially below unity. Therefore, during the future LHC runs, the measurement of $h\to b\bar{b}$ final states is a matter of special importance. Amongst all viable MSSM configurations that we study (including revisiting a light stau solution), we emphasise most the scenario with a light stop, as the latter is also motivated by Dark Matter and Electro-Weak baryogenesis. We also perform fits of the MSSM against the LHC data, emphasising the fact that in most cases these are better than for the SM.

The magnitude of the signal is usually expressed via the "signal strength" parameters µ, defined for either the entire combination of or the individual decay/production modes, relative to the SM. In our study we define individual µ XY for a given production (X) and decay (Y ) channel, in terms of production cross sections σ and decays widths Γ (in preference to Branching Ratios (BRs)): where, generally, X = ggF, V BF, V H, ttH and Y = γγ, W W , ZZ, bb, ττ , etc. Notice that, in the above equations, κ X and κ Y are equal to the respective ratios of the couplings squared while κ h is the ratio of the total Higgs boson width in the MSSM relative to the SM. For example, for gg → h → γγ, we have Notice that the combination of individual production and decay channels which has been done by experimental papers is a non-trivial procedure which takes into account the efficiency of the various channels determining in turn the corresponding weights in the overall combination. The respective results as reported by ATLAS are given by [3] µ(h → γγ) = 1.6 ± 0.3 (3) µ(h → ZZ ( * ) ) = 1.5 ± 0.4 (4) µ(h → W W ( * ) ) = 1.0 ± 0.3 (5) µ(h → bb) = −0.4 ± 1.0 (6) µ(h → ττ ) = 0.8 ± 0.7 (7) while from the CMS collaboration one has [4] µ(h → γγ) = 0.77 ± 0.27 (8) µ(h → ZZ) = 0.92 ± 0.28 (9) µ(h → W W ) = 0.68 ± 0.20 (10) µ(h → bb) = 1.15 ± 0.62 (11) µ(h → ττ ) = 1.10 ± 0.41.
It is not possible to perform this combination accurately in this phenomenological study as for this one needs to know all details on various experimental efficiencies for all production and decay channels which are not publicly available. Moreover, the overall signal strength µ for all production channels combined does not carry valuable information about possible new physics since in most BSM scenarios the main production channels ggF and V BF are non-universally altered in comparison to the SM.
Luckily, both experiments have produced results for the µ X,Y parameters for ggF and V BF separately, as presented in Fig. 1. Herein, such results for both collaborations are visualised as likelihood contours in terms of Confidence Level (CL) rates for the different final states mentioned above. One can see that these results, on the one hand, are consistent with the SM model at 95% CL while, on the other hand, there is still a lot of room to accommodate deviations from the SM, at least in the ±40% range at 95%CL. One should also notice that for the h → γγ measurement, the ATLAS result is about 2σ above the SM prediction for both ggF and V BF production processes, while the CMS result is approximately 1σ below the SM value for ggF and about 1σ above the SM for V BF , respectively. Thus, one can also see that there is some tension between the ATLAS and CMS results. From Fig. 1 one can nonetheless see the interesting general pattern (still within the 1-2σ error interval) that µ V BF,γγ is actually bigger than µ ggF,γγ for both ATLAS and CMS, noting that for the CMS collaboration µ ggF,γγ is essentially below one 2 . This trend has been quantified by the ATLAS collaboration, who have produced a best fit value of [7] µ V BF µ ggF +ttH = 1.4 +0.4 −0.3 (stat) +0.6 −0.4 (sys) (13) for a combination of the γγ, ZZ and W W data. On the basis of the pattern of measured µ X,Y , it is clear that BSM solutions to the LHC data ought to be investigated thoroughly. Herein, in particular, we discuss the case of the Minimal Supersymmetric Standard Model (MSSM), assess how genuine Supersymmetry (SUSY) effects can affect the Higgs production or decay dynamics (or indeed both) and draw a picture of the preferred SUSY parameter space in the light of the Higgs LHC data.
In our analysis, we concentrate on the V BF and ggF productions channels only, which are the leading ones, and limit the study of the decay signatures to the cases of h → γγ, W W, ZZ final states, as these are the production and decay modes with the most accurate experimental results. We should also remark that we carried out our investigation using renormalisation-group-improved diagrammatic calculations, including higher-order logarithmic and threshold corrections, using CPsuperH [8,9] (version 2.3).
Quite apart from the fact that current data shows a tendency for µ V BF,Y Y > µ ggF,Y Y , the LHC measurements also point to a rather light Higgs mass. While the possibility that the SM Higgs state had such a mass would be merely a coincidence (as its mass is a free parameter), in the MSSM, in contrast, the mass of the lightest Higgs boson with SM-like behaviour is naturally confined to be ≤ 135 GeV [10,11] by SUSY itself, which in essence relates trilinear Higgs and gauge couplings, so that the former are of the same size as the latter, in turn implying such a naturally small Higgs mass value. Therefore, to some extent, the Higgs boson mass which is measured at the LHC favours the MSSM (or some other low energy SUSY realisation) over the SM, so that it is of the utmost importance to test the validity of this SUSY hypothesis against the LHC Higgs data and to establish the viable parameter space.
As we know, the MSSM Higgs sector consists of five Higgs bosons: two CP-even neutral bosons, h, H (with masses such that m h < m H ) 3 , one CP-odd, A, and a pair which is charged, H ± . At tree level, the mixing between the two CP-even neutral Higgs bosons is defined by the mixing angle α, which is a derived quantity uniquely determined by two independent parameters which can be taken as the mass of any of the five physical states (hereafter we take m h ) and the ratio between the Vacuum Expectation Values (VEVs) of the two Higgs doublet fields pertaining to the MSSM, denoted by tan β. However, while performing an analysis in higher orders (or in the presence of loop diagrams at lowest order as is the case for the hgg, hγγ and hγZ effective couplings), one ought to account for the sparticle sector of the MSSM too, which in turn implies the introduction of additional parameters.
In our paper we re-examine the light stop, sbottom and stau scenarios, but also extend previous research by allowing any combination of MSSM quantum corrections, mixing effects and/or light MSSM fermions entering loops. In particular, we are the first to discuss how the MSSM could explain a non-universal alteration in µ V BF,Y Y versus µ ggF,Y Y from their SM values such that µV BF µggF = 1, and use these to examine the compatibility of the MSSM against LHC data. We also examine its ability to produce enhanced (with respect to the SM) rates in the di-photon channel, such that µ V BF > 1 and/or µ ggF > 1 and explore the effects of deviations entering all other measured Higgs boson couplings to SM particles.
The plan of the paper is as follows. In Section II we introduce the general setup and the MSSM parameter space that we explore, specific to the Higgs sector. In Section III(A) we study the possible generic MSSM effects on the relevant dynamics, namely, onto Higgs production, decay and total width. In Sections III(B) and III (C) we study the effects of stops and sbottoms, respectively, where we find that both can give rise to non-universal alterations in µ V BF versus µ ggF as both particles are able to affect the ggF fusion rate but not the V BF one. Section III(D) explores the stau contribution, where we find that it can only produce a universal increase in cross section in the di-photon channel, irrespectively of the production channel, as it only appears in the γγ (and Zγ) decay loops. In view of the Higgs data potentially indicating a non-universality in the production channels compared to SM predictions, in Section III(E) we look at the combined effects of these scenarios as well as perform a χ 2 fit of the MSSM parameter space with respect to LHC data. We draw our conclusions in Section IV.

II. MSSM SETUP AND THE PARAMETER SPACE
The MSSM is essentially a straightforward supersymmetrisation of the SM with the minimal number of new parameters. It is the most widely studied potentially realistic SUSY model. Furthermore, while different assumptions about the SUSY breaking dynamics can be made and these in turn lead to quite different phenomenological predictions, one can always assume an EW scale configuration and scan over the SUSY parameters relevant at that energy.
From this point of view, it becomes important to specify the MSSM spectrum (of masses and couplings). The particle content of the MSSM is three generations of (chiral) quark and lepton superfields, the (vector) superfields necessary to gauge the SU (3) C × SU (2) L × U (1) Y group of the SM and two (chiral) SU (2) Higgs doublet superfields. The introduction of a second Higgs doublet, with respect to the SM, is necessary in order to cancel the anomalies produced by the fermionic members of the first Higgs superfield and also to give masses to both up-and down-type fermions.
The interactions between Higgs and matter superfields are described by the superpotential Here Q L contains the SU (2) (s)quark doublets and U c L and D c L the corresponding singlets, while the (s)lepton doublets and singlets reside in L L and E c L , respectively. In addition, H u and H d denote Higgs superfields with hypercharge Y = ± 1 2 . The MSSM assumes certain soft SUSY breaking terms at a grand unification scale M GUT = 3 × 10 16 GeV. These soft SUSY breaking terms are categorised as trilinear scalar couplings A f ij , gaugino masses M a , sfermion mass-squared termsm f ij , and bilinear scalar coupling B. In the MSSM, the SM-like Higgs is the lightest CP-even Higgs, which is defined as with mixing angle α given by tan 2α = tan 2β The mass of the MSSM Higgs can be written, at one loop, as [53][54][55] where M 2 S = 1 2 (M 2 t1 + M 2 t2 ) and X t = A t − µ cot β. From this expression, one can easily show that the maximum value of m h is obtained at the maximal stop mixing, i.e., at X t = √ 6M S . Also, in order to have m h = 125 GeV, one should assume that at least one stop has a mass of O(1) TeV, while the other stop can be light. In addition, a quite large stop mixing is required, i.e., A t > ∼ 1 TeV.
In our analysis we are interested in the stop, sbottom and stau states as light particles. In general, one can write the squared mass matrices of these particles in the basis of the gauge eigenstates (f L ,f R ) as where where T 3f is the third component of the weak isospin and Q f is the electric charge. Thus, the sfermion physical masses are given by and the mixing angles are given by In this regard, one notices that the mixing in the stop sector is very strong, hence one of the stops,t 1 , can be very light. Also, with large tan β and |µ|, the mixing in the sbottom and stau sectors can also be very strong, therefore lightb 1 andτ 1 are further obtained. LHC constraints on SUSY masses are generally quoted as around 600-700 GeV for stops and sbottoms and in the region of 300 GeV for staus [3,4], depending on assumptions regarding the decay processes and the masses of decay products. However, these results all rely strongly on a sizeable mass splitting between these sparticles and the Lightest Supersymmetric Sparticle (LSP), a neutralino, to which they decay. These limits are drastically reduced in the region of low mass splittings: e.g., if mt ≈ m t + mχ0 , then the stop signal becomes difficult to distinguish from the tt background and the LHC data are unable to constrain the stop mass. A similar situation arises for other mass splitting scenarios, such as when the stop mass is close to the mass of the LSP mt ≈ m c + mχ0 . In this case the stop mass limit is reduced down to the LEP limit ∼ 95 GeV [56]. The limits for sbottom and stau masses can also be markedly reduced down to LEP limits (∼ 95 GeV for sbottoms and ∼ 85 GeV for staus [56]) for appropriate mass splittings.
In the present study, we performed a large scan of parameter space using CPsuperH to produce the data points, concentrating on those parameters with an important role in the masses and couplings of the stops, sbottoms and staus as well as the mass of the Higgs boson and its couplings to the bottom quark 4 . These masses and couplings are largely independent of the M 1 mass parameter (they vary only ∼ 0.1% for M 1 ranged from 0.1 TeV -100 TeV). However when M 1 ≪ (M 2 , µ), the lightest neutralino mass, mχ0 1 ≈ M 1 , so that for any point in our parameter scan its values can be chosen to give whichever LSP mass is required to be consistent with cosmological and LHC constraints, without otherwise altering our conclusions. In Tab. I, we list the range of parameters of this scan.  To increase the number of points in the parameter space of interest, three further localised scans were performed, in each case reducing the scanned range of one variable, with the other variable ranges remaining as described in Tab. I. The altered ranges in these additional scans were: 1. 100 GeV ≤ M U3 ≤ 300 GeV to produce light stops; 2. 100 GeV ≤ M D3 ≤ 400 GeV to produce light sbottoms; 3. 100 GeV ≤ M E3 ≤ 400 GeV and 100 GeV ≤ M L3 ≤ 400 GeV to produce light staus.
In order to avoid colour breaking minima of thet orb fields, we apply the |A t |, |µ| ≤ 1.5(M Q3 + M U3 ) constraints to all plots and numerical results unless otherwise stated [57][58][59]. These requirements are somewhat conservative, in the light of a very recent analysis in Ref. [60], yet we maintained them in order to simplify our study.

III. MSSM EFFECTS IN HIGGS PRODUCTION AND DECAY
In this Section, we discuss MSSM effects which alter the Higgs event rates at the LHC as compared to those of the SM. We start with a first Subsection, in turn divided in the three parts corresponding to introducing the structure of Higgs cross sections, di-photon decay and total width. The remaining four Subsections deal with stop, sbottom, stau and their combined effects, respectively, in either of these contexts.
A. The three contexts for MSSM effects

MSSM Higgs production
We start our discussion with MSSM Higgs boson production via the gluon-gluon fusion process, which is the dominant channel for Higgs searches at the LHC. In the SM, this mode is predominantly mediated by top quarks via a one-loop triangle diagram while the contribution from other quarks, even the bottom one, is only at the few percent level.
In the MSSM, however, strongly interacting superpartners of the SM quarks, i.e., the squarks, could provide a sizeable contribution to this triangle loop. The lowest order parton-level cross section can be written aŝ whereŝ is the center-of-mass energy at the partonic level and ∆(ŝ − m 2 h ) is the Breit-Wigner form of the Higgs boson propagator, which is given by and Γ h is the total Higgs boson decay width, while its partial decay width, Γ LO (h → gg), is given by where Y f and g hSS are the MSSM Higgs couplings to the respective (s)particle species for fermion (spin-1/2) and scalar (spin-0) particles, respectively, entering the triangle diagram. The loop functions F 1/2,0 can be found, for example, in [19]. Here, x i is defined as 4m 2 i /m 2 h , with m i being the mass running in the loop. In the decoupling (or quasi-decoupling) regime, as in the case of the SM limit of the MSSM, the top quark contribution is dominant among the quarks, since it has the largest Yukawa coupling, while the contribution from the other quarks (mainly coming from the bottom quark) is at the percent level, as intimated. The role of the bottom quark can be dramatically different though in the non-decoupling regime, when the hbb Yukawa coupling, sin α cos β is enhanced by sin α/ cos β ≃ tan β in comparison to the SM, enabling the bottom quark contribution to the triangle loop to increase and even dominate over the top quark for large values of tan β. However, this is not a realistic possibility, since LHC data do not indicate such significant deviations of the Higgs couplings from SM the values (they are within a 50% or so range from the latter), while data on the Higgs mass measurement indicate that, if the MSSM is realised in Nature, then the decoupling or quasi-decoupling regime should take place. In fact, the Higgs boson mass is close to the one reached in the decoupling limit, requiring From eq. (24) one can see that the g hSS coupling has dimension one, while it is more convenient to define a dimensionlessĝ hSS to be used hereafter: where G F is the Fermi constant. So Γ LO (h → gg) will have a form One should also note that the functions F 1/2 (x) and F 0 (x) reach a plateau very quickly for x > 1 and their values are about 1.4 and 0.4, respectively. This fact has important consequences, which we will discuss together with the Higgs decay into two photons, in the next Subsection. The specific effects of stop and sbottom loops will be discussed in Sections III(b) and III(c).

MSSM Higgs decay into di-photons
In the SM, the one-loop partial decay width of the h state into two photons is given by [61] [62] while in the MSSM the one-loop partial decay width of the h state into two photons also gets a contribution from scalar particles represented by sfermions and charged Higgs boson and is given by where V, f , and S stand for Vector, fermion and scalar particles respectively, entering the one-loop triangle diagram, g hW W is the MSSM Higgs coupling to W -boson, while Y f andĝ hSS are the MSSM couplings of Higgs boson to fermions and scalars defined in the previous Subsection. The genuine SUSY contributions to Γ(h → γγ) are mediated by charged Higgs, charginos and charged sfermions. The SM-like part is dominated by W -gauge bosons, for which F 1 (x W ) ≃ −8.3, whereas the top quark loop is subdominant and enters with opposite sign, , with all other fermions contributing negligibly. It is also worth mentioning that F 0 (x S ) ∼ 0.4, which is about a factor 20 smaller than F 1 (x W ) and approximately a factor 4 smaller than F 1/2 (x f ).
Keeping this in mind, let us discuss possible sources of the enhancement of the h → γγ effective coupling which in the MSSM may come through one of the following possibilities: (a) by the induction of a large scalar contribution, due to the light stop or/and sbottom or/and stau, with negative couplingĝ hSS so that it interferes constructively with the dominant W -contribution; (b) via charged Higgs boson contributions; (c) via chargino contributions; (d) via modification of the Yukawa couplings of top and bottom quarks in the loop. In the decoupling or quasi-decoupling regimes which eventually take place, as discussed above, scenario (d) does not occur. As for case (b), then taking into account that the charged Higgs mass is limited to be above 200 GeV (see e.g. [31,63] and references there in), the fact that its loop contribution is suppressed by a factor of (M W /M H ± ) 2 and thatĝ hH + H − is of the order of the electroweak coupling (contrary to thê g hSS coupling for squarks and sleptons which can be large as we discuss below), we have found that the contribution from charged Higgs bosons is generally negligible. In case (c), the chargino contribution can be bigger than that of the charged Higgs, because of the ratio F 1/2 : F 0 ≃ 4 and because the chargino has a lower mass limit of approximately 100 GeV (coming from LEP2 [64]). We have found that the maximum chargino contribution is reached in the µ → M 2 , tan β → 1 limit (where µ is the Higgs mass parameter while M 2 is the gaugino soft breaking mass) and can enhance the SM h → γγ partial decay width by about 30%. This agrees with the recent results of [51]. The scenario with very light charginos is not the focus of our paper, where we assume charginos to have a mass of at least a few hundred GeVs, and for which the virtual chargino contribution to the h → γγ decay is negligible. Moreover, the effect from the light charginos which could alter only the h → γγ decay is qualitatively similar to the effect from the light staus, which quantitatively can be much larger [52], and which we consider in the current study in great detail together with the light sbottom and light stop scenarios. Therefore, in this study we concentrate on scenario (a) in which sizeable MSSM contributions via scalar loops are still possible.
It is worth stressing again two important details related to the scalar contribution to h → γγ. Firstly, the smallness of the loop function F 0 with respect to F 1/2 and with respect to F 1 too, and, secondly, the mass suppression factor, (M W /M S ) 2 , mentioned above. Therefore the only way to have a sizeable effect from the scalar loops is to be in a scenario with large couplingĝ hSS and light scalars. In such a scenario the scalar loop competing with the fermion loop has a larger relative contribution to Γ(h → gg) than to Γ(h → γγ) where it would also compete with the dominant vector boson loop. At the same time, the contribution from squarks is opposite for Higgs production via gluon-gluon fusion compared to di-photon decay: depending on the sign ofĝ hSS , they will destructively (constructively) interfere with top quarks in production loops and constructively (destructively) interfere with W -boson loops in Higgs boson decays. Therefore, any squark loop which causes an increase (decrease) in Γ(h → γγ) will cause a proportionally larger decrease (increase) in Γ(h → gg).

MSSM Higgs total decay width
The total Higgs decay width in the MSSM is given, similarly to the SM, by the sum of all the Higgs partial decay widths, i.e., Γ tot = Γ bb + Γ W W + Γ ZZ + Γ ττ . Other partial decay widths into SM particles are much smaller and can safely be neglected. As per decays into SUSY states, we assume that the lightest neutralino is heavy enough, so we do not have invisible decay channels with large rates. In the SM with a 125 GeV Higgs mass, these partial decay widths are given by Γ bb = 2.4 × 10 −3 GeV, Γ W W = 8.8 × 10 −4 GeV, Γ ZZ = 1.0 × 10 −4 GeV and Γ ττ = 2.4 × 10 −4 GeV.
In the MSSM, when m h ≈ 125 GeV, this width is dominated by the partial width to bb, Γ(h → bb), which is controlled by the bottom quark Yukawa coupling, Y b . In the SM, it is given by the expression At large tan β, sbottom-gluino and stop-chargino loops give corrections to this Yukawa, which can be approximated by [65] where and with α 3 , mg and A t being the SUSY-QCD constant, gluino mass and top quark trilinear parameter, respectively, and where the loop function I(a, b, c) is defined as I(a, b, c) is a positive definite function, therefore with positive m g , µ and A t , the correction ∆m b is positive, and Y b is reduced. In particular, we see that this correction is large for large values of µ.
As the total width of the Higgs is dominated by the partial width to bb, a reduction in Y b will lead to a reduction in both Γ(h → bb) and Γ tot , with a subsequent universal increase in all other BRs and µ X, irrespectively of the production channel. However, in the decoupling limit M A >> M Z , tan α → − cot β, therefore which along with sin α cos β → −1 means that Y b reduces to its SM value. Therefore, to have the possibility for some reduction of Y b , we also consider the parameter space with values of M A not too large (the quasi-decoupling regime), such that Y b can be reduced to be below its SM value. At the same time the M A values should be large enough such that M h ≈ 125 GeV is possible.
The results of our scan are presented in Fig. 2 where different values of κ bb are plotted in the (M A ,µ) and (A t , tan β) plane. From

B. Stop quark effects
As previously discussed, since F1 F0 ≈ −20, in order for stops to have a significant effect on (h → γγ), g ht1t1 is required to be very large. Furthermore, a positiveĝ ht1t1 coupling will decrease κ γγ whilst a negative coupling will increase κ γγ .
In the decoupling limit, the Higgs coupling to the lightest stop is given by [10] g ht1t1 = 1 2 cos 2β cos 2 θt − 4 3 where θt is the stop mixing angle defined by and X t is given in terms of the the Higgs-stop trilinear coupling as X t = A t − µ cot β. The first term in the equation is small compared to and so can be largely ignored. When X t is also Z > 0 will lead to a decrease of k γγ . For large X t , if mt 1 < mt 2 , it can be shown that sin 2θt ≃ −1 and the Higgs coupling to the lightest stop is strongly enhanced and negative. However, since m h ≈ 125 GeV, the scenario with light stops requires that the Higgs mixing should be near maximal, i.e., X t ≈ √ 6M SUSY , where M SUSY = 1 2 (mt 1 + mt 2 ). Hence, we are not free to consider very large values of X t as an independent parameter. In this case, one haŝ .
Thus, if m 2 t2 ≈ m 2 t1 , it is possible to get a very large Higgs coupling to stops. However, with a light stop, such that m 2 t1 ≪ m 2 t2 , one findsĝ Thereforeĝ ht1t1 is both negative (making the overall stop loop contribution of the same sign as the W loop), thereby increasing k γγ , and fixed, which limits the overall contribution to (h → γγ) of a stop loop of a particular mass. Small deviations from this prediction should be expected as in practice we are only requiring near maximal mixing. Cuts have been applied such that only points with a Higgs mass within 2 GeV of the maximum value for each value of Xt MSUSY are kept. The pink-shaded window indicates the X t /M SUSY > 3 region, where the majority of points do not pass the colour breaking minima conditions. To isolate the influence of light stops, the following cuts are also applied to both plots: m H ± , m χ ± 1,2 , mτ 1,2 , mb 1,2 , mt 2 > 300 GeV. This is illustrated in Fig. 3(a) where we present results for κ γγ as a function of the lightest stop mass, mt 1 , where 124 GeV < m h < 126 GeV, for the scan described in Section II. Together with Fig. 3(b) which presents κ γγ versus X t /M SUSY as well as m h versus X t /M SUSY , these two figures provide a clear illustration of the argument discussed above: the effect of the stop is limited because of the correlation betweenĝ ht1t1 and the Higgs mass. As can be seen from Fig. 3(a), even mt 1 ∼ 120 GeV would lead to a modest increase of κ γγ ≈ 1.2. At the same time, in Fig. 3 (b) which presents κ γγ , κ gg and m h as functions of Xt MSUSY , we can see that the effect of light stops on κ γγ and κ gg could be much larger if m h was not limited to be in the 124-126 GeV mass window: outside of this window κ γγ could be as large as 1.8 for a stop quark mass of about 120 GeV, however the majority of points with X t /M SUSY > 3 do not pass the colour breaking minima conditions discussed in Section II, limiting the maximum κ γγ to around 1.5.
In this figure it is clear that, in order to get m h ≈ 125 GeV, X t /M SUSY has to be near its maximal mixing value of √ 6 ≈ 2.4. In this region there is little variation in k γγ and we are limited to 0.9 k γγ 1.2. , mτ 1,2 , mb 1,2 , mt 2 > 300 GeV.
Let us consider the overall effect of light stops on µ ggF,γγ = k gg × k γγ × k −1 h via its effects on k γγ and k gg in the parameter space where the total width is close to the SM one. From Fig. 3 (b) we would expect that, in general, either k γγ is increased with a relatively larger decrease in k gg , causing an overall decrease in µ ggF,γγ , or k γγ is decreased with a relatively larger increase in k gg , causing an overall increase in µ ggF,γγ . This is demonstrated in Fig. 4(a), where we see that (other than for a few points very near µ ggF,γγ = 1 where other factors such as small changes in the total width play a role) we have µ ggF,γγ > 1 when k gg > 1 (red) and vice versa (black). This means that if the total width of the Higgs boson is unchanged, then stop loops alone can produce a universal increase in all decay channels (µ ggF,Y > 1) via increasing the ggF production channel but will not produce an isolated increase in (h → γγ), as this will always be cancelled by a relatively larger decrease in ggF production.
We are naturally lead to consider the possibility of counteracting the effect of a reduced κ gg caused by light stops by reducing Γ bb as discussed in Section IIIA.3. This would mean that when the stop coupling is negative, producing an increase in k γγ and bigger relative decrease of κ gg , the BRs in all channels other than bb can be increased such that the overall value for µ ggF,Y remains ≈ 1. In this scenario, µ V BF,γγ > µ ggF,γγ as the V BF channel will be increased by both k γγ > 1 and the increased BR to photons from the reduced total width, without the reduced production rate of the ggF channel. This is demonstrated in Fig. 4 (b) where the effects of light stops and reduced Γ bb (via a reduction in the bottom Yukawa coupling) are combined together and the resulting µ V BF,γγ and µ ggF,γγ values along with current best fit CMS and ATLAS data are plotted. One can see in Fig. 4 (b)(i) that the smaller the κ bb values, the larger the universal µ V BF,γγ and µ ggF,γγ alterations it will cause, while Fig. 4 (b)(ii) clearly demonstrates how decreased stop quark masses lead to an increase of the non-universal alteration of these couplings, which can be expressed through the µ V BF,γγ /µ ggF,γγ ratio.
In this scenario, the same situation takes place for all other decay channels with the exception of the decay to bottoms, (µ V BF,W W/ZZ/τ τ /γγ > µ ggF,W W/ZZ/τ τ /γγ ), as will be discussed further in Section III.E. Furthermore, in this region, the di-photon decay channel can be increased by a factor of up to 1.2 relative to the other decay channels. Of note, the area of parameter space with light stops and a Higgs mass of ≈ 125 GeV is relatively small, because a heavy M SUSY is preferred to give large logarithmic parts to the radiative corrections to the Higgs Mass. The scenario where M U3 ∼ M Q3 300 GeV requires fine-tuning of the stop mixing parameter X t in order to achieve a lightest stop with mass ≤ 300 GeV. However, as X t is fixed by the near maximal mixing requirement (X t ≈ √ 6M SUSY ), this is not possible. Hence, the area of parameter space with a ≈ 125 GeV lightest Higgs mass and light stops (with mass ≤ 300 GeV) is where M U3 ≪ M Q3 , generally with M U3 ≤ 300 GeV and M Q3 2 TeV. (In this region it is easy to show that mt 1 ≈ M U3 and mt 2 ≈ M Q3 ). This explains the reason for choosing the reduced range of M U3 described in Section II for the additional scan. The relationship between M U3 , M Q3 and the lightest stop mass is shown in Fig. 5, exemplifying a strong correlation between mt 1 and M U3 as discussed.
In Tab. II, we give three different benchmark points for scenarios where light stops give rise to µV BF µggF > 1, where we have also included a value for the minimum fine-tuning for each of the benchmark points.
Our fine-tuning parameter is based on the electroweak fine-tuning parameter (see e.g. [66] for details). This value is derived by noting that the minimisation condition for the Higgs potential gives rise to the equation for the Z-boson mass, where Σ u u and Σ d d are the radiative corrections to m 2 Hu and m 2 H d . The electroweak fine-tuning parameter is then defined as where C Hu = | − m 2 Hu tan 2 β/(tan 2 β − 1)|, with analogous definitions for C H d , C µ , C Σ u u and C Σ d d . If tan β is moderate or large, we have  Taking into account that Σ u u is not defined if our starting point is the theory at the EW scale (rather than the GUT scale) the measure of fine-tuning gives a minimum value for ∆ EW , which could be larger if there is a large cancellation between m 2 Hu and Σ u u as discussed by Baer et. al. in [66]. Keeping this in mind we will be using this definition of fine-tuning in our paper. We see that for Benchmark points 1 and 3, ∆ −1 ∼ 2 − 5%, as there is no requirement for a large µ. However Benchmark 2 has a reduced k bb which requires a large ∆m b and hence a large µ, leading to a larger minimum fine-tuning, with ∆ −1 ∼ 0.1%.
We should also mention here that the light stop scenario has been discussed recently in connection to EWBG [67,68]. The latter requires A t M Q3 /2 (or X t /M SUSY 1 √ 2 ) in order to achieve a strong phase transition and therefore is not realised in the maximal mixing scenario which we consider in our paper. Remarkably, for these values of X t /M SUSY , the 125 GeV Higgs mass can be only achieved for extremely large values of M Q3 ≃ 10 6 TeV, as shown in [68]. At the same time, X t /M SUSY 1 √ 2 leads toĝ ht1t1 > 0 and therefore to a constructive interference of the light stops inside ggF production. It was suggested in [68] that the overall enhancement for ggF production coming from the light stop can be compensated by a significant invisible Higgs boson decay into light neutralinos.
We would also like to note that the scenario with an altered Y b coupling that we consider in this paper suggests an alternative solution to the problem of how to compensate the enhancement of ggF production due to light stops. Analogously to the parameter space region where Y b is decreased, there is another where Y b is increased, which is realised in the µ < 0 region. In this case, the enhancement of ggF production is compensated by the respective decrease of the γγ/ZZ/W W/τ τ decay rates while the BR(h → bb) will be increased. Since, at the moment, the LHC is not quite sensitive to the h → bb signature, this scenario is perfectly viable.

C. Sbottom quark effects
Similarly to stops, light sbottoms loops may also alter Higgs production via ggF and decay to di-photons. However, there are some important differences between the sbottom loop contribution and the stop loops.
In the decoupling limit, the Higgs coupling to sbottoms is given bŷ where θb is the sbottom mixing angle defined by and The first major difference with respect to the stop case is that this coupling does not have any dependence on X t and, hence, is not constrained by the requirement of m h ≈ 125 GeV. In particular, when mb 1 < mb 2 with a large positive µ and tan β, leading to a large negative X b , it can be shown than sin 2θb ∼ 1. The last term in eq. (42) therefore dominates, givingĝ and leading to a large negative coupling due to the negative X b . As the Higgs-sbottom coupling ultimately depends on X 2 b , via the sin 2θb term, it is also possible to get a large negative coupling if µ is large and negative such that X b is large and positive. However, as we are interested in the parameter space where Y b has the possibility of being small, which requires a positive µ (see Section III.A), we have only considered positive µ.
The second important difference with respect to the stop case is that N c,b Q 2 b = 1/3 versus N c,t Q 2 t = 4/3 for stops. This will not affect gluon fusion, but means that the sbottom mass will need to be 1 2 that of a stop mass with the same coupling strength to Higgs bosons to produce the same alteration in decay to photons.
The result of the two factors discussed above is that, firstly, due to the opposing effects of the sbottom coupling having a larger maximum magnitude compared to stops, but the sbottom loop effects of decay to di-photons being suppressed by a factor of compared to stops, the relative effects of sbottom loops on κ γγ compared to stop loops is difficult to predict. Secondly, the maximum effect of sbottom loops on gluon fusion can be larger than that of a stop with the same mass, because the coupling isn't constrained by the Higgs mass and can become larger in magnitude than the stop coupling, while the loop contribution isn't constrained by a charge factor as is the case for di-photon decay.
Both of these effects can be observed in Fig. 6(a). We see that the largest possible increase in κ γγ (black) for a given sbottom mass is smaller than that for a stop of the same mass, and only very light sbottoms ) for 120 GeV ≤ mb 1 ≤ 150 GeV. To isolate the influence of light sbottoms, the following cuts are also applied: m H ± , m χ ± 1,2 , mt 1,2 , mτ 1,2 , mb 2 > 300 GeV.
≃ 80 GeV are able to produce κ γγ ≈ 1.2 (compare to mt 1 ≈ 120 GeV). Also, as expected, κ gg (green) has a larger reduction for sbottoms compared to stops of similar mass, with κ gg as low as 0.7, for mb 1 ≈ 250 GeV. Fig. 6(b) shows how κ γγ (black) and κ gg (green) depend on for 120 GeV < mb 1 < 150 GeV, confirming that the largest deviations from the SM occur for large and negative X b , and that the effect on κ gg is much larger than for κ γγ .
The combined effect of the sbottom loops can be seen in Fig. 7(a), where the lightest sbottom mass is plotted against µ ggF,γγ . We see that for the majority of parameter space, µ ggF,γγ is suppressed, other than a small region where it is increased due an increased κ gg . This small region of increased κ gg occurs for very small values of X b where the final term of eq. (42) does not dominate and the coupling is small and positive.
If we consider the possibility of counteracting the effect of a reduced κ gg by reducing Γ bb as we did for the stops, we find that we would expect sbottom loops to have a similar effect to stop loops in the (µ V BF +V H , µ ggF +ttH ) plane. As in the stop case, V BF production channels will be unaffected by the gluon fusion rate, but the decays to all particles (other than sbottoms) will still be increased by the reduction in Γ bb , such that µ V BF,γγ > µ ggF,γγ . This is demonstrated in Fig. 7(b)(i), which is analogous to Fig. 4(b)(i) for stops, where we have plotted µ V BF,γγ versus µ ggF,γγ for different values of κ bb , for 120 GeV < mb 1 < 300 GeV. In the case of sbottoms, as the arrows indicate, their main effect is to reduce κ gg , reducing µ ggF,γγ , with a much smaller effect on κ γγ , producing only a small increase in µ V BF,γγ . The reduced Yukawa coupling to bottoms then reduces the total width, causing a universal increase in µ γγ irrespective of the production channel. Overall, we see that, in this situation, light sbottoms can produce quite large non-universal alterations, which can be measured by the µV BF,γγ µggF,γγ ratio which can be even larger than in the light stop scenario. Fig. 7(b)(ii) is similar to Fig. 7(b)(i), but with the colours indicating the sbottom mass in each case. We see that the largest effects are produced by the lightest sbottoms, as expected, but that a significant effect giving , mt 1,2 , mτ 1,2 , mb 2 > 300 GeV.

D. Stau effects
In addition to light stops and sbottoms, the lightest stau may give important contributions that in particular could enhance κ γγ . For staus, N c Q 2 = 1, a factor of 3 larger than sbottoms, and since the Higgsstau coupling like the Higgs-sbottom coupling also does not depend on X t , and hence is not constrained by the Higgs mass, light stau effects on Γ(h → γγ) could be more significant than sbottom effects, with the caveat of a different (running) bottom mass versus the tau mass.
The Higgs coupling to the lightest stau, normalised by v/ √ 2 = M W /g, with v the SM Higgs VEV, is given byĝ with Similarly to sbottoms, for a large positive µ, with large tan β, X τ is large and negative, and we find that which is large and negative. Thus the stau contribution may enhance Γ(h → γγ) in a large tan β scenario with large and positive µ. (As in the sbottom case, a large negative µ would also give rise to a large negative coupling, but we only consider positive µ, as required such that Y b may be reduced). As intimated, since N c Q 2 is 3 times larger for the stau than the sbottom, the minimum mass at which its contribution to κ γγ can become large is approximately a factor of √ 3 times heavier than for the sbottom. This is demonstrated in Fig. 8(a), where κ γγ (black) is plotted against the stau mass, indeed showing that we can have κ γγ > 1.2 when m τ 180 GeV. It also shows that light staus have no effect on κ gg as ) for 120 GeV ≤ mt 1 ≤ 140 GeV. To isolate the influence of light staus, the following cuts are also applied: m H ± , m χ ± 1,2 , mt 1,2 , mb 1,2 , mτ 2 > 300 GeV.
expected. (The points with a slight reduction in κ gg have sbottoms or stop masses ∼ 300 GeV, just above the mass cut applied for these particles). In Fig. 8(b), κ γγ and κ gg are plotted against , showing that as for the sbottom, the coupling becomes largest for large and negative X τ . , mt 1,2 , mb 1,2 , mτ 2 > 300 GeV.
As staus are colourless and do not affect the gluon-gluon fusion production channel, µ ggF,γγ follows a very similar pattern as κ γγ . This is illustrated in Fig.9(a) where we see that for mτ 1 180 GeV, the value of µ ggF,γγ can become > 1.2.
In Fig.9(b(i,ii)), we see that the effect of the light staus on the (µ V BF +V H , µ ggF +ttH ) plane is as expected, causing a universal increase in decay to di-photons irrespective of production channel, magnifying also universal effects which may be caused by a reduction in Γ hbb .

E. Combined effect and fit of the LHC data
In Fig. 10 we present results for µ V BF versus µ ggF for the γγ, W W , ZZ, τ τ and bb decay channels in the (µ V BF , µ ggF ) plane, where we have included all points from our scan for which any or all of the scenarios discussed in the previous Subsections are realised. We see that, for each final state, the majority of parameter space has µ V BF > µ ggF , and comparing with experimental measurements, we see that 6 out of 8 measurements have µ V BF > µ ggF for their best fit values. Therefore one can expect that the MSSM will provide a better fit to the data and, in general, that the light stop and sbottom scenarios would be able to explain a non-universal alteration of µ V BF > µ ggF , if this is confirmed at the upgraded LHC starting in 2015. In Fig. 10 we have also stratified by shading according to the values of the fine-tuning parameter ∆, as described in section III B. We see that the points with a large universal increase in µ ggF and µ V BF have a larger fine tuning in general. This is expected, as for these points µ, on which ∆ depends, is required to be large to reduce Y b as discussed in section III A.
To quantify how well the scenarios we have discussed fit current LHC data, we have calculated the χ 2 for each scenario and compared to the best fit values for µ ggF and µ V BF in Fig. 1. For each collaboration (ATLAS, CMS) the systematic errors on the values of µ ggF and µ V BF for a single decay mode are correlated. To take this correlation into account, we actually calculate a "profiled log likelyhood ratio" test statistics. However, under the assumption that the data is distributed as a multivariate Gaussian (valid to a good approximation), this reduces to a χ 2 statistics with a non-diagonal covariance matrix. This is discussed fully in [69]. There are 6 degrees of freedom from the ATLAS data and 10 from the CMS data, giving 16 degrees of freedom overall (3 channels for ATLAS (γγ, W W and ZZ) and 5 from CMS (γγ, W W , ZZ, τ τ , bb), each with ggF and V BF channels). The regions of interest for which a χ 2 was calculated were defined as; 1) light stops only, 2) light sbottoms only, 3) light staus only, 4) lights stops and/or sbottoms and/or staus. In each case a "light" mass was defined as between 120 and 300 GeV.
For each case, we have calculated the χ 2 for each point in the parameter space from our scan that matched the aforementioned particle mass criteria. The results are presented in Fig. 11 as the χ 2 per degree of freedom (χ 2 /N DOF ), when compared to (a) just the ATLAS results, (b) just the CMS results, (c) both ATLAS and CMS results.
First of all, one can see that the SM already fits the data well, the χ 2 /N DOF is about 1 for ATLAS ( Fig. 11(a)) and is about 0.6 for CMS ( Fig. 11(b)), while for the ATLAS+CMS combination we have got χ 2 /N DOF ∼ 0.75 (Fig. 11(c)). One should note that the χ 2 /N DOF is lower for CMS, mainly due to the best fit point for h → ZZ being very close to the SM value, and for h → bb being reasonably close to the SM with very large error bars. It is hard to judge which scenario is preferred due to the large experimental errors, but one can see that the combined scenario as well as the scenario with light staus give the lowest value of χ 2 /N DOF .

IV. CONCLUSIONS
In this paper we have explored the effect from light sfermions on the production and decay of the lightest Higgs boson within the MSSM culminating with a fit of the relevant parameter space to LHC data.
FIG. 10: µ V BF vs µ ggF for the di-photon, WW, ZZ, τ τ and bb decay channels where the lightest stop and/or lightest stau and/or lightest sbottom has a mass between 120 GeV -300 GeV. ATLAS (circle) and CMS (square) best fit results for each channel are also plotted. The 68% Confidence Level (CL) for the experimental results are included. ATLAS results were not available for the bb and τ τ channels. Colour gradients denote different values of the fine-tuning parameter ∆ −1 × 100 as described in subsection III B.
We have found that the scenario with light coloured sfermions, namely stops and sbottoms, has the potential to explain a non-universal alteration of the two most relevant Higgs production channels, i.e., µ V BF = µ ggF , and predicts µV BF µggF > 1 in all Higgs boson decay channels in the majority of the parameter space. These light stop and light sbottom scenarios are realised in specific regions of the parameter space in terms of soft-breaking mass terms, namely where M Q3 >> M U3 and M Q3 >> M D3 , respectively.
The specific feature of the scenario with a light stop is thatĝ ht1t1 is negative (whenever one is near maximal stop mixing), which makes the overall stop loop contribution of the opposite sign as the top quark contribution and of same sign as the W loop. As a result, one obtains a decreased k gg and an increased k γγ couplings. It is important to notice that the relative decrease of k gg is bigger than the relative increase of k γγ , so the overall effect from light stops per se would lead to a decrease of the Higgs production via gluon-gluon fusion decaying to di-photons (µ ggF,γγ ) as well as a reduction in µ ggF compared to the SM for all decay channels (µ SM ggF = 1 by definition). This scenario would be somewhat consistent with CMS data, where µ ggF < 1 for all decay channels.
However, this prediction is in tension with the ATLAS data where µ ggF ≈ 1.5 for both the γγ and ZZ decay channels. Therefore, we also consider the scenario where we have both light stops and a suppressed hbb coupling. In this case, the reduced h → bb partial width (and related κ bb parameter) causes an enhancement of the BRs and hence signal strengths of the other decay channels, which can compensate for the reduced production via ggF . In this case, depending on the degree of suppression of the hbb coupling, the ggF signal strength in all channels can be increased, either to match the SM level, or greater, e.g., to µ ggF ≈ 1.5, in order to be more consistent with the ATLAS data. The exceptions being µ ggF,γγ , which can be slightly enhanced, and µ ggF,bb , which would be reduced compared to the other channels. This reduction of the hbb coupling was achieved with a large µ (1-5 TeV), intermediate M A (300-800 GeV) and intermediate-to-large tan β (20-50).
One should also note that, in the light stop scenario,ĝ ht1t1 is fixed (in the maximal mixing scenario) to , which limits the maximal contribution of the stop to production via ggF and to the h → γγ decay. We have also found that the effect from light sbottoms on the gluon fusion rate can be potentially larger (since Higgs-sbottom-sbottom coupling is not correlated with the Higgs boson mass), which in its turn requires a bigger decrease for κ bb to satisfy the experimental data. One can see that, in the light stop/sbottom scenarios, µ ggF,bb and µ V BF,bb are predicted to be essentially below unity, especially the µ ggF,bb value, which is doubly suppressed both via ggF production (due to the negative interference from stop/sbottom loops) and from decay (due to κ bb suppression). Therefore, in the future LHC runs, the measurement of µ ggF,bb and µ V BF,bb is particularly important, as this will enable one to exploit an additional constraint in order to pin down possible MSSM effects in the Higgs sector.
In contrast, light staus were found to be able to only universally increase the signal strengths, irrespective of the production channel, generally complementing the effect of a reduced hbb coupling.
Furthermore, we showed that the non-universal solutions (µ ggF = µ V BF ) had a fairly low minimum fine-tuning measure, as low as ∼ 5%, while in regions where a universal increase in signal strength (µ ggF ∼ µ V BF > 1) is caused due to a suppressed Y b , the fine-tuning was much larger due to the requirement of a large µ parameter.
Finally, we performed a χ 2 fit for these MSSM scenarios which showed that they all fit data better than the SM.
To conclude, we have found that the MSSM with light stops or sbottoms has a good potential to explain a non-universal alteration of the Higgs production rates in different channels as compared to the SM and that this can be complemented by the fact that a reduced Γ h could increase the relevant signal strengths to be equal to or greater than the SM prediction.
As an outlook, we should mention the fact that, among the light squark solutions considered, the light stop one is also attractive from a cosmological point of view. In fact, the scenario where the neutralino (the LSP) is degenerate with the lightest stop in the 100-300 GeV range (to satisfy the experimental data on stop quark searches) predicts a plausibly low amount of DM (via the stop-neutralino co-annihilation channel). At the same time the light stop scenario can provide a crucial link to EWBG, specifically in case of very light stops. In this connection, we also suggest an alternative solution of compensating the enhancement of ggF production due to light stops (which takes place away from maximal mixing scenario), through an increase of Y b and a respective decrease of the γγ/ZZ/W W/τ τ decay rates.