The decays $h^0 \to b \bar{b}$ and $h^0 \to c \bar{c}$ in the light of the MSSM with quark flavour violation

We calculate the decay width of h0 ->b bbar in the Minimal Supersymmetric Standard Model (MSSM) with quark flavour violation (QFV) at full one-loop level. We study the effect of scharm-stop mixing and sstrange-sbottom mixing taking into account the constraints from the B meson data. We discuss and compare in detail the decays h0 ->c cbar and h0 ->b bbar within the framework of the perturbative mass insertion technique using the Flavour Expansion Theorem. The deviation of both decay widths from the Standard Model values can be quite large. Whereas in h0 ->c cbar it is almost entirely due to the flavour violating part of the MSSM, in h0 ->b bbar it is mainly due to the flavour conserving part. Nevertheless, the QFV contribution to Gamma(h0 ->b bbar) due to scharm-stop mixing and chargino exchange can go up to about 8%.


Introduction
In the Standard Model (SM) the Higgs mechanism is responsible for the mass of the fermions. Therefore, it is necessary to measure the Yukawa couplings very precisely. Since the Yukawa coupling is proportional to the fermion mass, the largest decay branching ratio of the Higgs boson, discovered by CMS and ATLAS at LHC [1,2] with a mass of approximately 125 GeV, is that of h 0 → bb. Within the SM this branching ratio is B(h 0 → bb) = 0.577 +3.2% −3.3% [3]. Although the Higgs boson properties measured so far are consistent with the SM, deviations from the SM are not yet excluded and could point to "New Physics".
An important extension of the SM is provided by Supersymmetry (SUSY), in particular by the Minimal Supersymmetric Standard Model (MSSM). In the MSSM, the discovered Higgs boson could be the lightest neutral Higgs boson h 0 . Quark flavour conservation (QFC) is usually assumed (apart from the quark flavour violation (QFV) induced by the Cabibbo-Kobayashi-Maskawa (CKM) matrix). However, SUSY QFV terms could be present in the mass mixing matrix of the squarks, especially mixing terms between the 2nd and the 3rd squark generations.
In a previous paper [4] we studied the impact ofc L,R −t L,R mixing on the decay h 0 → cc. We showed that the deviation from the SM width Γ(h 0 → cc) can go up to ±35%, due to QFV effects at one-loop level. In the present paper, we study the influence of this mixing in the decay h 0 → bb. (For completeness we have also studieds L,R −b L,R mixing effects, but they have turned out to be very small.) There are, however, constraints on the mixing between the 2 nd and the 3 rd generations of squarks from B-physics measurements (∆M Bs , B(b → sγ), B(b → sl + l − ), B(B s → µ + µ − ), B(B + → τ + ν)), as well as from m h 0 measurements and SUSY particle searches. We take into account all these constraints.
First, in our calculation of Γ(h 0 → bb) at full one-loop level, we will largely proceed analogously to the case of h 0 → cc [4][5][6][7][8], except for the particular features characteristic of the decays into bottom quarks, as the large tan β enhancement and resummation of the bottom Yukawa coupling.
The main new feature in this paper is the additional adoption of the perturbative mass insertion technique using the Flavour Expansion Theorem [9]. We will discuss it both in the h 0 → cc and h 0 → bb case. It gives systematic insight into the various QFV contributions. In particular, we show that due to the fact that the product T U 32 M U 23 is apriori unbounded by experiment, the correction to the width of h 0 → cc can become large so that perturbation theory breaks down. (For the definitions of T U and M U see eqs. (2.1), (2.2) and (4.10) below.) In the h 0 → bb case this is not possible.
In this paper we focus on thec R −t L ,c L −t R ,c R −t R , andc L −t L mixing which is described by the QFV parameters δ uRL 23 , δ uLR 23 ≡ (δ uRL 32 ) * , δ uRR 23 , and δ LL 23 , respectively. Thẽ t R −t L mixing is described by the QFC parameter δ uRL 33 . We also allows −b mixing. All parameters are assumed to be real, i.e. no CP-violation is considered. In principle, there might be in addition also trilinear non-holomorphic interactions, see eq. (1.5) in [12]. These interactions are not taken into account in this study.
3 h 0 → bb at one-loop level with flavour violation We write the decay width of h 0 → bb including the one-loop contributions as with the tree-level decay width where N C = 3, m h 0 is the on-shell mass of h 0 and the tree-level coupling s b 1 is

3)
α is the mixing angle of the two CP-even Higgs bosons, h 0 and H 0 [13].
In the calculation of δΓ 1loop (h 0 → bb) we proceed in a way analogously to the calculation of δΓ 1loop (h 0 → cc) in Ref. [4]. In addition to the diagrams that contribute within the SM, δΓ 1loop (h 0 → bb) receives contributions from the exchange of SUSY particles and Higgs bosons. The corresponding diagrams are shown in Fig. 2 of [4], replacing c by b quarks and u ↔d. The dominant SUSY contribution is due to gluino and chargino exchange. The gluino and the chargino contribute also to the self-energy of the b quark.
As in Ref. [4] we use the DR renormalisation scheme, where all input parameters in the Lagrangian (masses, fields and coupling parameters) are UV finite, defined at the scale Q = 1 TeV. In order to obtain the shifts from the DR masses and fields to the physical scaleindependent quantities, we use on-shell renormalisation conditions. Moreover, we include in our calculations the contributions from real hard gluon/photon radiation from the final b quarks.
The one-loop corrected width Γ(h 0 → bb) is therefore given by where Γ g,impr includes the tree-level and the gluon loop contribution, see eq.(55) in [4], δΓg is the gluino one-loop contribution and δΓ EW is the electroweak one-loop contribution. Moreover, we have considered the large tan β enhancement and the resummation of the bottom Yukawa coupling [14]. It turns out, however, that in our case with large m A 0 close to the decoupling limit, the resummation effect is very small (< 0.1%).

Mass Insertion technique
In this section, we want to apply to the decays h 0 → cc and h 0 → bb the mass insertion technique as well as the Flavour Expansion Theorem (FET) as developed by Dedes et al. in [9]. Let us consider the expression by using Einstein summation convention. The diagonal elements of the squared mass matrix are denoted by M ii , and the off-diagonal ones by the matrix M I with the restriction M I ii = 0. This formula and all following MI formulas in this section have been checked with the Mathematica package MassToMI [15]. The generalized b 0 functions [9], where the first argument shows how many insertions are done, can be written recursively as  [16]. The general formula for a number of degenerate arguments is useful [9], for k ≥ 1 and m ≤ k. The derivative of b 0 with respect to the second argument reads The derivative of b 0 with respect to the first argument can be written as By using eq. (4.5) we can write b 0 (1, a, {b, b}) as b 0 (0,1) (a, b).

Gluino contribution to h 0 → cc
As a first example, we want to calculate the self-energy of the c-quark due tog andũ i in the loop. We decompose the charm self-energy Σ c defined by the Lagrangian L = −c Σ c c, We assume real input parameters, therefore Σ LR c = Σ RL c , and Allowing the squaredũ-mass matrix (eq. (2.1)) in the form GeV, and the QFV elements of the 3 × 3 matrices M 2 Q and M 2 U are written as M Q ij and M U ij , respectively. We neglect the terms proportional to µ/ tan β assuming that tan β is large. The matrix elements M 25 = M 52 =v 2 T U 22 are assumed to be zero, because T U 22 is strongly constrained by the colour-breaking condition being proportional to the squared charm-Yukawa coupling (see Appendix D of [4]). Using eq. (4.2) we get where the QFV contributions read with ρ = 0. The graphs corresponding to the terms T 2 and T 3 are given in Figs. 1(a) and 1(b) or 1(c) and 1(d), respectively. Note that there is no contribution with no mass insertion because we have a helicity flip, and also practically no contribution with only one insertion, because T U 22 is very small. Thus, all terms in eq. (4.12) are quark-flavour violating. The interactions related to the mass insertions are given by the effective Lagrangian Figure 1. Quark-flavour violating mass insertions to the charm quark self-energy with gluino, corresponding to T 2 and T 3 in eq. (4.12).
Neglecting the charm mass and m h 0 compared to the gluino andũ i masses, for the coefficients c v L and c v R we have with C 0 being the scalar Passarino-Veltman integral with three propagators, and the coupling c h 0ũ * iũ j is given by eq. (65) of [4], Assuming that T U 23 , T U 32 , T U 33 are non-zero and real, we can approximate c h 0ũ * iũ j by The mass insertion expansions for the coefficients c v L and c v R , are equal (for real input parameters), where In terms of b 0 -functions we have The graphs corresponding to the terms T v 1 and T v 2 are given in Figs. 2(a) and 2(b) or 2(c) to 2(f), respectively. Comparing the results for the charm self-energy, eqs. (4.11),(4.12), , we see that T 2 = T v 1v 2 . The same holds for the term proportional to we have a factor 3 in the term T v 2 compared to that in T 3 . This can also be seen by comparing Fig. 1(d) with Figs. 2(d) to 2(f). Thus we can deduce the result T v 3 from the term T 4 in eq. (4.12) by adding a prefactor of 3 for all the terms with three T U elements.
In a recent paper by A. Brignole [17] the width Γ(h 0 → bb) was also considered in a quark flavour changing scenario. There only the graphs of Figs. 2(d) to 2(f) were taken, which are, however, much suppressed compared to Figs. 2(a) to 2(c).
The leading term in the SUSY contribution to the DR m c is UV-finite and therefore scale independent. As M Q 23 is strongly constrained by B-physics observables, this term is nearly proportional to the product of the two insertions T U 32 and M U 23 , see Fig. 1. The resummed SM running charm mass m c | SM is ∼ 0.6 GeV. The SUSY DR running charm mass can be written then as m c ∼ 0.6 GeV + ∆mg c with In order to find bounds for T U 32 and M U 23 we also have studied the decay t → ch 0 , having written a numerical program for its decay width. However, the product T U 32 M U 23 cannot be directly constrained by this process. In principle, one could get individual bounds on T U 32 and M U 23 but the effects of these parameters on the width turn out to be numerically too small [12].
Neglecting the wave-function contributions, which are proportional to the tree-level coupling s 1 c we get the approximate result for the decay h 0 → cc, where Σ LR,g c is given in eq. (4.9) or in the MI approximation in eq. (4.11) with ρ = 1. Γ g,impr can be taken from eq. (55) and Γ tree from eq. (9) in [4].

Gluino and chargino contributions to h 0 → bb
We decompose the bottom self energy Σ b defined by the Lagrangian L = −b Σ b b as follows , as we assume real input parameters, and Allowing the squaredd-mass matrix in the form where the quark flavour conserving (FC) and quark flavour violating (FV) contributions read with ρ = 0. As in the charm sector, the vertex contribution can be directly deduced from the self energy,

with an additional factor 3 for some terms in
, and accordingly ρ = 1. The interactions related to the mass insertions are given by We will apply the mass insertion technique for the self-energy amplitude of the bottomquark and for the vertex amplitude with a chargino in the loop. The relevant term for the self energy calculation is proportional to c i6 . Using eq. (4.25) we get Neglecting the term proportional to the SU(2) coupling g and the bottom mass in the loop integrals, we get Concerning the mass insertions in theũ i line, we have the same structure as in eq. (4.26), but for theũ sector. We have M RL 33 →v 2 T U 33 , T D → T U , and M D 23 → M U 23 . Therefore, we can use the results for the bottom self energy with gluino in the loop. Using eq. (4.10) we obtain where with ρ = 0. The graphs corresponding to T 1 , T 2 and T 3 are shown in Figs.

The formula with linear mass insertion reads
Assuming m W M 2 , µ, the linear term vanishes, and X 22 f ((X 2 ) 22 ) ∼ µf (µ 2 ). We get the final approximate result with the terms T x i taken from eq. (4.34) with m 2 χ + m → µ 2 . Neglecting the wave-function renormalization contributions, which are proportional to the tree-level coupling s 1 b we get the approximate result for the decay h 0 → bb, where Σ LR,g b and Σ LR,χ + b are given in eq. (4.26) and eq. (4.31) or in the MI-approximation in eq. (4.28) and eq. (4.37), respectively, with ρ = 1. Γ g,impr is given by eq. (55) and Γ tree by eq. (9) in [4], with c → b.
In [18] the chirally enhanced corrections to Higgs vertices in the most general MSSM were discussed analytically by taking into account gluino-squark loops. We qualitatively agree with their results on h 0 → bb. A study including two-loop SUSY-QCD corrections was performed in [19].

Numerical results
In this section we demonstrate the effects of QFV due toc −t mixing in the decays of h 0 to bb and cc in the MSSM. 1 In order to find an explicit scenario where both decay widths deviate appreciably from the SM values, we have performed two scans over wide parameter regions. In the first calculation we have scanned 8750000 parameter points. From them only 17% have satisfied the existing theoretical and experimental constraints (see Appendix B). The parameters involved and their variations are given as follows: A detailed study of the MSSM QFV parameter space has also been done in [20].
The results of the scans are summarised in Fig. 4, where the distributions of the deviation from the SM width for h 0 → bb and h 0 → cc are shown. We take Γ SM (h 0 → bb) = 2.35 MeV [21], Γ SM (h 0 → cc) = 0.118 MeV [3], m b (m b ) MS = 4.2 GeV, m c (m c ) MS = 1.275 GeV [22], and α s (m Z ) = 0.1185 [23]. The y-axis counts the number of survived parameter points for each bin of the deviation. It is seen that in the case of h 0 → bb ( Fig. 4(a)) the detailed variation of the elements M U and M Q can increase the effect and the deviation from the SM can go up to ∼ 30% at certain parameter points. In the case of h 0 → cc (Fig. 4(b)) a large deviation from the SM value due to large values of the product T U 32 M U 23 , discussed at the end of Section 3.2, is in principle possible. Since there exists no physical constraint on this product we will only show results with a deviation from the SM up to ∼ ±50%. Based on the results from the scans we have chosen a reference scenario with strongc−t mixing to demonstrate the effects of QFV in both h 0 to bb and cc decays. The corresponding MSSM parameters at Q = 1 TeV are given in Table 1. This scenario satisfies all present experimental and theoretical constraints, see Appendix B. The resulting physical masses of the particles are shown in Table 2. We also show the flavour decomposition of the up-type squarksũ i , i = 1, ..., 6 in Table 3. For the calculation of the masses and the mixing, as well as for the low-energy observables, especially those in the B meson sector (see Table 4), we use the public code SPheno v3.3.3 [24,25]. Both the widths Γ(h 0 → bb) and Γ(h 0 → cc) are calculated at full one-loop level in the MSSM with QFV using the packages FeynArts [26] and FormCalc [27]. We also use the packages SSP [28] and LoopTools [27]. For creating the Fortran code for the mass insertion formulas MassToMI [15] was very helpful. In the following unless specified otherwise we show various parameter dependences of Γ/Γ SM − 1 for Γ(h 0 → bb) and Γ(h 0 → cc) with all other parameters fixed as in Table 1.

Scan1 Scan2
In Fig. 5 the dependence on the QFV parameters δ uRL 23 and δ uLR 23 is shown. It is seen that in the case of bb (Fig. 5(a)) the variation due to correlatedc R −t L andc L −t R mixing can vary up to ∼ 6% in the region allowed by the constraints. Comparing Fig. 5(a) with Fig. 5(b) one can see that there exist regions where both widths considered simultaneously deviate from their SM prediction. Hence Γ(h 0 → bb) tends to depend more onc R −t L mixing, while Γ(h 0 → cc) depends more onc L −t R mixing. Table 1. Reference scenario: shown are the basic MSSM parameters at Q = 1 TeV, except for m A 0 which is the pole mass (i.e. the physical mass) of A 0 , with T U 33 = 1450 GeV (corresponding to δ uRL 33 = 0.1). All other squark parameters not shown here are zero.   This tendency can also be seen in Fig. 6. On the left hand side (Fig. 6(a)) the dependence of Γ/Γ SM (h 0 → bb) on the QFV parameters δ uRR 23 and δ uRL 23 is shown. The variation due toc R −t L andc R −t R mixing is ∼ 7%. In the same scenario, however, the variation of Γ/Γ SM (h 0 → cc) (not shown here) is only ∼ 3%. On the right hand side (Fig. 6(b)) Γ/Γ SM (h 0 → cc) is shown as a function of δ uRR 23 and δ uLR 23 . The variation is large and can go up to ∼ 30%, see also [4]. In the same scenario, however, Γ/Γ SM (h 0 → bb) varies only by less than one percent.
In Section 4.1, in agreement with our results in Ref. [4], we have shown that in the case   of cc the deviation from the SM is entirely due to QFV. However, it is known that in the MSSM Γ(h 0 → bb) can differ considerably from the SM due to QFC contributions [29]. In Fig. 7  chargino exchange which involves mixing in theũ sector. On the other hand, in the bb case the gluino exchange, which plays a major role in the cc case, involvesd quarks whose QFV mixing effect is strongly suppressed, and hence the QFV component of the gluino exchange contribution is very small. Therefore, it is not shown in this figure. It is also interesting that the "h 0 " contribution depends significantly on the QFV parameter δ uRL 23 . After all, the variation of Γ/Γ SM (h 0 → bb) in the shown QFV parameter range, which can be taken as QFV effect, can be as large as ∼ 7%. Fig.7(b) demonstrates the quality of our approximated result (4.38). By comparing numerically the different MI orders we realized that the MI formulas converge fast forg FC andχ + FC, but not forχ + FV. This can be seen by comparing Fig. 7(a) with Fig. 7(b). Thus, the difference between the dotted curve and the upper curve in Fig. 7(a) is mainly due to the relatively slow MI convergence of theχ + FV contribution.
Although the decay h 0 → bb is dominant, the measurement of its branching ratio and width at the LHC will be a big challenge. At LHC one always measures σ(pp → h 0 X)B(h 0 → bb). The largest Higgs boson production cross section is due to gluon gluon fusion. However, due to the huge QCD background it will be difficult to isolate the h 0 → bb mode. The other production modes (vector boson fusion, Higgs radiation from W ± Z, and associated tth 0 production) have smaller cross sections, but may have less background. In any case, high luminosity at LHC would be needed [30]. A model independent and precise measurement of B(h 0 → bb) and Γ(h 0 → bb) would be possible at a e + e − linear collider such as ILC [31].

Conclusions
In analogy to our previous paper [4], we have calculated the decay width of h 0 → bb in the MSSM with quark flavour violation at full one-loop level. We have studied the effects ofc−t mixing, taking into account all constraints on the QFV parameters from B-meson data. We have discussed in detail both the decays h 0 → cc and h 0 → bb within the perturbative mass insertion technique applying the Flavour Expansion Theorem [9]. There are cases, where the charm self-energy and consequently the correction to the width Γ(h 0 → cc) can become unacceptably large. This is due to the product M U 23 T U 32 , for which there exists no bound. In general, the deviation of Γ(h 0 → bb) from the SM can be large (up to 30%), mainly coming from the QFC part of the MSSM. The QFV contribution due toc L,R −t L,R mixing and chargino exchange is smaller but can nevertheless reach ∼ 7% at certain parameter points. The QFV part due to gluino exchange, which is due tos L,R −b L,R mixing, is very small.

A Interaction Lagrangian
• In the MSSM the interaction of the lightest neutral Higgs boson, h 0 , with two bottom quarks is given by with the tree-level coupling s b 1 given by eq. (3.3).
• The interaction of gluino, down-type squark and a bottom quark is given by where T α are the SU(3) colour group generators and summation over r, l = 1, 2, 3 and over α = 1, ..., 8 is understood. In our case the parameter M 3 = mge iφ 3 is taken as real, φ 3 = 0.
• The interaction of chargino, up-type squark and a bottom quark is given by Lχ+ m bũ i =b kũ im P L + lũ im P R χ + * mũ i +χ + * m kũ * im P R + lũ * im P L bũ * i , (A. 6) where the couplings kũ im and lũ im are given by U and V are unitary matrices that diagonalise the charging mass matrix U * XV † = diag(mχ± 2m W sin β(cos β) . The interaction Lagrangian for the h 0 → cc case is given in Ref. [4]. Table 4. Constraints on the MSSM parameters from the B-physics experiments relevant mainly for the mixing between the 2nd and the 3rd generations of squarks and from the data on the h 0 mass. The last column shows the constraints at 95% CL obtained by combining the experimental error quadratically with the theoretical uncertainty, except for m h 0 , see Ref. [4].

B Theoretical and experimental constraints
The experimental and theoretical constraints taken into account in the present note are discussed in detail in Ref. [4]. Here we only list the updated constraints from B-physics and those on the Higgs boson mass in Table 4.