Full one-loop corrections to SUSY Higgs boson decays into charginos

We present the decay widths of the heavier Higgs bosons $H^0$ and $A^0$ into chargino pairs in the minimal supersymmetric standard model, including full one-loop corrections. All parameters for charginos are renormalized in the on-shell scheme. The importance of the corrections to the chargino mass matrix and mixing matrices is pointed out. The full corrections are typically of the order of 10 %.


Introduction
The Minimal Supersymmetric Standard Model (MSSM) [1] is considered the most attractive extension of the Standard Model. This model contains two Higgs scalar doublets, implying the existence of five physical Higgs bosons [2]; two CP-even neutral bosons (h 0 , H 0 ), one CP-odd boson A 0 , and two charged bosons H ± . For the verification of the MSSM, detection and precision studies of these Higgs bosons are necessary.
The decay modes of the heavier Higgs bosons (H 0 , A 0 ) are in general complicated [3,4], especially if tan β, the ratio of the vacuum expectation values of the two Higgs scalars, is not much larger than one. For example, they may decay into pairs of the SUSY particles [3] such as squarks, sleptons, charginos, and neutralinos. In this paper, we focus our attention on the decays into charginos, with i, j = (1,2). Existing numerical analyses [3,4,5] at tree-level have shown that the decays (1) have in general non negligible branching ratios. These decays are also interesting because they are generated by gaugino-higgsino-Higgs boson couplings [2] at tree-level and very sensitive to the components of charginos. Detailed studies of these decays would therefore provide useful information about the chargino sector, complementary to the pair production processes e + e − →χ + with κ(x, y, z) ≡ ((x−y −z) 2 −4yz) 1/2 . η k represents the CP eigenvalue of H 0 k ; η 1,2 = 1 for the (h 0 , H 0 ) decays and η 3 = −1 for the A 0 decays. We use the abbreviation m i ≡ mχ± i .
In this paper, we assume that the contributions of CP violation and generation mixings of the quarks and squarks are negligible.
The chargino-Higgs boson couplings gF ijk , defined by the interaction lagrangian with a = 1, 2, c = 3, 4, are given by [2] gF The would-be Nambu-Goldstone boson H 0 4 ≡ G 0 is included here for later convenience. The mixing matrices (U, V ) for the charginos are determined by diagonalizing the chargino mass matrix X as Here M and µ are the mass parameters of the SU(2) gaugino and higgsino states, respectively. We choose U and V to be real. The effect of the mixings of H 0 k is represented by e k and d k , which take the values We also show the widths of the decaysχ + 2 → χ + 1 H 0 k at the tree-level [14] Γ tree (χ

One-loop corrections
We calculate the full one-loop corrections to the decay widths (3 The one-loop correction to the coupling F ijk is expressed as where δF ijk , and δF (c) ijk are the vertex correction, the wave function correction, and the counter terms for the parameters in Eq. (5), respectively.
The vertex correction δF (v) ijk comes from the diagrams listed in Fig. 1. In this paper we do not show the analytic forms of these diagrams.
The wave-function correction δF with the implicit summations over l = 1, 2 for k = 1 or 2, l = 3, 4 for k = 3, and i ′ , j ′ = (1, 2). The correction terms δZ +(L,R) for the chargino wave-functions are given by where p = i and are the self-energies of the charginos. δZ +R are obtained from Eqs. (11,12) by the exchange L ↔ R. The CP symmetry relation ReΠχ S,L ii = ReΠχ S,R ii is used in Eq. (11). The corrections δZ H 0 for the Higgs bosons are The Higgs boson self-energies Π H 0 (k 2 ) in Eqs. (14,15,16) include momentum-independent contributions from the tadpole shifts [15,16] and leading higher-order corrections. The latter contribution is relevant for the corrections to (m h 0 , m H 0 , α). For the A 0 decays, Eq. (16) already includes the contribution from the A 0 − Z 0 mixing in addition to the A 0 − G 0 mixing, using the Slavnov-Taylor identity, . The explicit forms of the self energies Πχ(p 2 ), Π H 0 ab (p 2 ), and Π AZ (m 2 A 0 ) are shown, for example, in Refs. [17,18].
To obtain ultraviolet finite corrections, we further need the counter term contribution F (c) ijk from the renormalization of the parameters in the tree-level couplings Eq. (5). The chargino mixing matrices (U, V ) are renormalized in the on-shell scheme, as described in Refs. [11,13]. In this scheme, extending Ref. [19] for quark and lepton mixings, the counter terms for (U, V ) are determined such as to cancel the antihermitian parts of the chargino wave-function corrections Eq. (12). As a result, after including (δV , δU) into Eq. (12) The counter term of β for A 0 decays is fixed by the condition [15,16] Inclusion of this counter term δβ cancels the half of δZ H 0 43 in Eq. (16). As usual, we use the pole mass m A 0 and on-shell tan β as inputs for the Higgs boson sector.
Since the zero-momentum contribution Π H 0 kl (0) to the masses and mixing angle of (h 0 , H 0 ) are often very large, we calculate (m h 0 , m H 0 ) and the effective mixing angle α eff , which is defined to cancel the zero-momentum part of Π H 0 ab (p 2 ) in Eq. (15), by FeynHiggs [20], which includes the leading higher-order corrections, and use these values both for the tree-level and corrected widths. After the inclusion of the corresponding counterterm δα, Eq. (15) is modified as with the DR renormalization scale Q = m Z for Π H 0 ab (p 2 ). Our calculation is performed in the ξ = 1 gauge. Although the on-shell mixing matrices generally depend on the gauge parameter [21,22], our (U, V ) may be understood as the ones improved by the pinch technique [23,24]. We ignore here very small differences of the on-shell β between the ξ = 1 results and improved ones by the pinch technique (see Refs. [24,25] for the case of CP-even Higgs bosons).
For the renormalization of the SU(2) gauge coupling g in Eq. (5), two schemes are used. In both the W-and Z-pole masses m W and m Z are input parameters. The Weinberg angle is defined by cos θ W = m W /m Z [26], and therefore In the α(m Z ) scheme we use as input the MS running electromagnetic coupling α(m Z ) (= e 2 (m Z )/(4π)). We have with δe given e. g. in [27,28], δm Z and δm W in [18].
In the other scheme, called here the G F scheme, the Fermi constant G F for the muon decay is input parameter, δZ e is the renormalization constant for the electric charge in the Thomson limit [29]. The term ∆r includes the full one-loop MSSM correction [30] and the leading two-loop QCD corrections [31].
The corrected widths are The process H 0 k →χ + iχ − j γ with real photon emission is included to cancel the infrared divergence by virtual photon loops.
One has to be careful in using the on-shell mixing matrices (U, V ) and masses m i (i = 1, 2) in the numerical analysis. When the gauge and Higgs boson sectors are fixed, the chargino sector is fixed by two independent parameters. Here we follow the method proposed in Refs. [11,13]: We fix the chargino sector by taking M ≡ X 11 and µ ≡ X 22 , where the onshell mass matrix X is defined to give the on-shell masses m i and on-shell mixing matrices (U, V ) by diagonalization. Note that, for given values of the on-shell M and µ, the oneloop corrected on-shell masses m i and mixing matrices (U, V ) are shifted [11,13] from the values obtained by the tree-level mass matrix X tree composed by the input parameters, the on-shell M, µ, tan β, and the pole mass m W . This is due to the shift of the off-diagonal elements of X from their tree-level values and related to the deviation of the gaugino couplings from the corresponding gauge couplings by SUSY-breaking loop corrections [32]. These shifts of m i and (U, V ), in addition to the "conventional" corrections shown in Eq. (21), have to be taken into account for a proper treatment of the loop corrections. (A slightly different scheme for the chargino sector was proposed in Ref. [12]. Apart from the different definition of the renormalized M and µ, their method is equivalent to ours.) The full one-loop corrections were calculated using the packages FeynArts, FormCalc, and LoopTools [33]. For the contributions of the quarks, leptons, and their superpartners, we also checked the consistency with Ref. [10], both analytically and numerically.

Numerical results
We present numerical results for the tree-level and one-loop widths of the decays A 0 → χ + 1χ − 1 , H 0 →χ + 1χ − 1 , and χ + 2 → χ + 1 h 0 . The SUSY parameter set SPS1a of the Snowmass Points and Slopes in Ref. [34] is chosen as reference point; For the trilinear breaking terms A t , A b and A τ we use the DR running values given at the scale of the mass of the decaying particle, For the values of the Yukawa couplings of the third generation quarks (h t , h b ), we take the running ones at the scale of the decaying particle mass. In the G F scheme for the renormalization of g, we use G F = 1.16639×10 −5 GeV −2 instead of α(m Z ).
We compare three cases: the "naive" tree-level width Γ naive tree , the tree-level width already including the loop corrections to the chargino mass matrix Γ tree , and the full one-loop width Γ corr. .
In Fig. 2    A comparison of two renormalization schemes for fixing g, the α(m Z ) scheme and the G F scheme, is shown in Fig. 3 for the decay A 0 →χ + 1χ − 1 as functions of m A 0 . The difference between these two schemes is below 1%, scaling with the one-loop correction part, and mainly a higher order effect. Since the Higgs boson couplings to charginos are very sensitive to the gaugino-higgsino mixing, it is interesting to study the dependence of the decay widths on the gaugino and higgsino components ofχ ± 1 . Fig. 4 shows the tree-level and one-loop corrected widths of A 0 →χ + 1 +χ − 1 as functions of µ for fixed M. One can see that in the region where the light charginoχ + 1 becomes a pure wino the width gets very small. The correction grows from ∼ −1% for µ ∼ 120 GeV to 20% for µ ∼ 600 GeV. The µ dependence of the decay width H 0 →χ + 1 +χ − 1 is not shown because its behavior is similar to that shown in Fig. 4. Fig. 5 shows the tree-level and one-loop corrected widths of A 0 →χ + 1 +χ − 1 as functions of M for fixed µ. In the whole range of this figureχ + 1 is gaugino-like. In (a), for increasing M the decay widths decreases due to phase space. The correction, see (b), gets up to 30% near the threshold. Again, the H 0 →χ + 1 +χ − 1 is not shown because of a similar behavior. Fig. 6 shows the decay widths for A 0 →χ + 1 +χ − 1 in (a) and H 0 →χ + 1 +χ − 1 in (b) as functions of tan β. The correction is in the range of ∼ 10% and the dependence on tan β is small. We examined the difference of the renormalization scheme taking the DR value for tan β at the scale Q = 454.7 GeV as input parameter instead of the on-shell tan β. For these processes the difference is small, e.g. in the Fig. 6 (a) it is about 0.5% for low and 0.2% for large tan β, respectively.  2,3) are taken to be equal to mQ, while the other parameters are unchanged. The relative corrections Γ tree /Γ naive tree − 1 (dashed lines), stemming from the shift of the chargino mass matrix by the renormalization, are negative. The remaining conventional corrections shown in Eq. (21) (dotted lines) are positive. The total correction Γ corr. − Γ naive tree (solid lines) is positive and in the range of 6 − 11% in (a) and 4 − 7% in (b). The corrections become quite insensitive to mQ for large mQ. The total correction consists of the mQ dependent (s)fermion contribution and the remaining contribution, the latter of which is ∼ 7.8% for (a) and ∼ 9.6% for (b). Again, these two types of loop corrections are of comparable order.  Figure 7: Correction of the full one-loop corrected (solid), the tree-level (dashed), and the conventional one-loop corrected width (dotted) for the decays A 0 →χ + 1 +χ − 1 (a) and H 0 → χ + 1 +χ − 1 (b) relative to the naive tree-level width as a function of mQ. (Note that the treelevel already includes the correction due to the chargino mass matrix renormalization.) lines denote Γ tree /Γ naive tree − 1. They show the effect due to the chargino mass matrix renormalization. The solid lines show the total correction in terms of the naive tree level width, Γ corr. /Γ naive tree − 1. The dotted lines stand for Γ corr. /Γ tree − 1. This is the total correction in terms of the tree-level result, where the chargino mass matrix renormalization effect is already included. One sees that Γ tree /Γ naive tree −1 and Γ corr. /Γ naive tree −1 are much stronger dependent on A t compared to Γ corr. /Γ tree −1. This shows that the A t dependence of the corrected widths comes mainly from the shifts of the masses and mixing matrices of the charginos.

Conclusions
We have calculated the full one-loop corrections to the decays (H 0 , A 0 ) →χ + i +χ − j (i, j = 1, 2). All parameters in the chargino mass matrix X and mixing matrices (U, V ) are renormalized in the on-shell scheme. The importance of the corrections to these matrices, in addition to the conventional corrections (vertex and wave-function corrections with counter terms), was emphasized. We have studied the dependence of the corrections on the SUSY parameters. The corrections to the widths of the decays (H 0 , A 0 ) →χ + 1 +χ − 1 are of the order of 10%, but can be larger near the thresholds. The corrections from quarks, leptons, and their superpartners were shown to be of similar order of magnitude as the other loop corrections. We also showed that the correction to the decayχ + 2 →χ + 1 h 0 can be to ∼ −10%.