Improved full one--loop corrections to A^0 ->\sq_1 \sq_2 and \sq_2 ->\sq_1 A^0

We calculate the full electroweak one-loop corrections to the decay of the CP-odd Higgs boson A^0 into scalar quarks in the minimal supersymmetric extension of the Standard Model (MSSM). Due to the complex structure of the electroweak sector a proper renormalization of many parameters in the on-shell renormalization scheme is necessary. For the decay into sbottom quarks, especially for large tanb, the corrections can be very large in the on-shell renormalization scheme, which makes the perturbation series unreliable. We solve this problem by an appropriate definition of the tree-level coupling in terms of running quark masses and running trilinear couplings A_q. We also discuss the decay of heavy scalar quarks into light scalar quarks and A^0. We find that the corrections are significant and therefore cannot be neglected.


Introduction
The Minimal Supersymmetric Standard Model (MSSM) [1] requires five physical Higgs bosons: two neutral CP-even (h 0 and H 0 ), one heavy neutral CP-odd (A 0 ), and two charged ones (H ± ) [2,3]. The existence of a CP-odd neutral Higgs boson would provide a conclusive evidence of physics beyond the SM. Searching for Higgs bosons is one of the main goals of present and future collider experiments at TEVATRON, LHC or an e + e − Linear Collider.
In this paper, we consider the decay of the CP-odd Higgs boson A 0 into two scalar quarks, A 0 →q 1q2 . The decays into squarks can be the dominant decay modes of Higgs bosons in a large parameter region if the squarks are relatively light [4,5]. In particular, the third generation squarkst i andb i can be much lighter than the other squarks due to their large Yukawa couplings and their large left-right mixing. We will calculate the full electroweak corrections in the on-shell scheme and will implement the SUSY-QCD corrections which were calculated previously [6]. The challenge of this calculation is the necessity to renormalize almost all parameters in the electroweak sector in only one single process. Due to the numerous electroweak interacting particles and the complex coupling structure we have to compute a large number of graphs. In general, the Higgs-squarksquark couplings consist of F -and D-terms and SUSY breaking terms, all depending on the squark mixing angle θq. As a first step we consider the case A 0 →q 1q2 where only F -terms and SUSY breaking terms enter in the coupling. Since A 0 only couples toq L -q R and due to the CP nature of A 0 , A 0 →q iqi vanishes (with real parameters also beyond the tree-level!). Despite the complexity, we have performed the calculation in an analytic way. The explicit formulae will be given elsewhere. We will, however, show the most important results of the numerical analysis. Furthermore, the crossed channelq 2 →q 1 A 0 is studied.
In case of the decay into sbottom quarks the decay width can receive large corrections which makes the perturbation expansion unreliable, especially for large tan β. In some cases the width can even become negative using the on-shell renormalization scheme. We will show that this problem can be fixed by an appropriate choice of the tree-level coupling in terms of DR running quark masses and running A q .

Tree-level result
The squark mixing is described by the squark mass matrix in the left-right basis (q L ,q R ), and in the mass basis (q 1 ,q 2 ),q =t orb, where Rq iα is a 2 x 2 rotation matrix with rotation angle θq, which relates the mass eigenstatesq i , i = 1, 2, (mq 1 < mq 2 ) to the gauge eigenstatesq α , α = L, R, byq i = Rq iαqα and MQ, MŨ , and MD are soft SUSY breaking masses, A q is the trilinear scalar coupling parameter, µ the higgsino mass parameter, tan β = v 2 v 1 is the ratio of the vacuum expectation values of the two neutral Higgs doublet states [2,3], I 3L q denotes the third component of the weak isospin of the quark q, e q the electric charge in terms of the elementary charge e 0 , and θ W is the Weinberg angle. The mass eigenvalues and the mixing angle in terms of primary parameters are and the trilinear breaking parameter A q can be written as At tree-level the decay width of A 0 →q 1q 2 is given by with κ(x, y, z) = (x − y − z) 2 − 4yz and the A 0 -q * i -q j coupling Gq ij3 given in [6].

Full Electroweak Corrections
The one-loop corrected (renormalized) amplitude Gq ren 123 can be expressed as Gq ren 123 = Gq 123 + ∆Gq 123 = Gq 123 + δGq where δGq 123 the wave-function corrections (Fig. 2). Note that in addition to the one-particle irreducible vertex graphs also one-loop induced reducible graphs with A 0 -Z 0 mixing have to be included. All parameters in the tree-level coupling Gq 123 have to be renormalized due to the shift from the bare to the on-shell values. These corrections are denoted by δGq (c) 123 . The full one-loop corrected decay width is then given by Since there are diagrams with photon exchange we also have to consider corrections due to real photon emission to cancel the infrared divergences ( Fig. 1). Therefore the corrected (UV-and IR-convergent) decay width is Throughout the paper we use the SUSY invariant dimensional reduction (DR) as regularization scheme. For convenience we perform the calculation in the 't Hooft-Feynman gauge, ξ = 1.

Vertex and wave-function corrections
The relations between the unrenormalized (bare) and renormalized (physical) fields and couplings are with the notation H 0 Thus the renormalized Lagrangian is given by (up to the first order) The explicit form of the vertex corrections δGq (v) 123 will be given elsewhere. Due to the antisymmetry of the tree-level coupling, Gq ij3 = −Gq ji3 , the total wave-function correction reads δGq (w) 123 For the wave-function renormalization constants we use the conventional on-shell renormalization conditions [7] which lead to with the diagonal parts of the Higgs and squark self-energiesΠ ii (k 2 ).   The off-diagonal Higgs wave-function corrections can be combined with the contribution to δGq (v) 123 which come from A 0 -Z 0 mixing. First we show that the sum of the parts coming from the propagators of Z 0 and G 0 outside the loops is independent of the gauge The amplitudes of the two graphs of Fig. 3 in a general R ξ gauge are Contracting the Lorentz indices in M Z , and eliminating Π AG in favor of Π AZ by using the Slavnov-Taylor identity [8] we find the sum Finally we use the identity to obtain the result δGq (Z+G) 123 The gauge dependence of the propagators of the Z 0 and G 0 in Fig. 3 is completely removed. However, there still remain gauge dependences from vector particles and Goldstone bosons in the loops of Π AZ which cancel against their counter parts in the vertex, wave-function and counter term corrections.

Counter terms
Since all parameters in the tree-level coupling Gq 123 have to be renormalized, we get for up down -type squarks. The Yukawa coupling counter term can be decomposed into corrections to the electroweak coupling g, the masses of the quark q and the gauge boson W and the mixing angle β, For the trilinear coupling we get with eq. (7) In the on-shell scheme the renormalization condition for the electroweak gauge boson sector reads [9] δg with m W and m Z fixed as well as the quark and squark masses as the physical (pole) masses.

Renormalization of the electric charge e
Since we use as input parameter for α the MS value at the Z-pole, α ≡ α(m Z )| MS = e 2 /(4π), we get the counter term [10] δe e = 1 (4π) 2 with x f = m Z ∀ m f < m Z and x t = m t . N f C is the colour factor, N f C = 1, 3 for (s)leptons and (s)quarks, respectively. ∆ denotes the UV divergence factor, ∆ = 2/ǫ − γ + log 4π.

Renormalization of tan β
For tan β we use the condition [11] ImΠ A 0 Z 0 (m 2 A ) = 0 which gives the counter term δ tan β tan β = 1 m Z sin 2β Renormalization of µ The higgsino mass parameter µ is renormalized in the chargino sector [12,13] where it enters in the 22-element of the chargino mass matrix X,

Renormalization of θq
The counter term of the squark mixing angle, δθq, is fixed such that it cancels the antihermitian part of the squark wave-function corrections [14,15],

Infrared divergences
The infrared divergences in eq. (10) are cancelled by the inclusion of real photon emission, see the last two Feynman diagrams of Fig. 1. The decay width of A 0 (p) →q 1 (k 1 )+q 2 (k 2 )+ γ(k 3 ) can be written as with the phase-space integrals I n and I mn defined as [16] . (33) The corrected (UV-and IR-convergent) decay width is then given by (see eq. (11))

Improvement of One-loop Corrections
In the on-shell renormalization scheme, in case of the decay into sbottom quarks, especially for large tan β, the decay width can receive large corrections which makes the perturbation expansion unreliable. In some cases the corrected width can even become negative. It has been pointed out [17,18] that the source of these large corrections are mainly the counter terms for m b and the trilinear coupling A b . We show that this problem can be fixed by absorbing these large counter terms into the A 0 -squark-squark tree-level coupling and expanding the perturbation series around the new tree-level. The technical details will be given in a forthcoming paper.

Correction to m b
If the Yukawa coupling h b is given at tree-level in terms of the pole mass m b , the oneloop corrections to the counter term δm b become very large due to gluon and gluino exchange contributions. We absorb these large counter terms and also the ones due to loops with electroweak interacting particles into the Higgs-squark-squark tree-level coupling by using the DR running massm b (Q = m A ). The large counter term due to the gluon loop is absorbed by using SM 2-loop renormalization group equations [18,19,20].
Thus we obtain the SM running bottomm b (Q) SM . For large tan β the counter term to m b can be very large due to the gluino-mediated graph [17,21,22]. Here we absorb the gluino contribution as well as the sizeable contributions from neutralino and chargino loops and the remaining electroweak self-energies into the Higgs-squark-squark tree-level coupling.
In such a way we obtain the full DR running bottom quark masŝ

Correction to A b
The second source of a very large correction (in the on-shell scheme) is the counter term for the trilinear coupling A b , eqs. (25, 26), especially the contribution of the left-right mixing elements of the squark mass matrix, m 2 LR = (m 2 q 1 − m 2 q 2 ) sin θq cos θq. As in the case of the large correction to m b we use DR runningÂ b (m A 0 ) in the Higgs-squark-squark tree-level coupling. Because of the fact that the counter term δA b (for large tan β) can become several orders of magnitude larger than the on-shell A b we useÂ b (m A 0 ) as input [18]. In order to be consistent we have to perform an iteration procedure to get the correct running and on-shell masses, mixing angles and other parameters.

Decays into stops:
In Fig. 4 we show the tree-level and the corrected width to A 0 →t 1t2 for tan β = 7 and {MQ, A, M, µ} = {300, −500, 120, −260} GeV as a function of the mass of the decaying Higgs boson, m A 0 . As can be seen for larger values of m A 0 , the electroweak corrections can be of the same size as the SUSY-QCD corrections.
In Fig. 5 the tree-level, the full electroweak and the full one-loop corrected (electroweak and SUSY-QCD) decay width of A 0 →t 1t2 are given as a function of A t . The electroweak corrections do not strongly depend on the parameter A t and are almost constant about 8%. As input parameters we have chosen the values given above as well as  Fig. 6 shows the tree-level, the full electroweak and the full one-loop corrected (electroweak and SUSY-QCD) decay width of A 0 →t 1t2 as a function of tan β with the same parameter set as above and m A 0 = 900 GeV. Again, in a large region of the parameter space the electroweak corrections are comparable to the SUSY-QCD ones.

Decays into sbottoms:
Here we illustrate the numerical improvement of the full one-loop corrections to A 0 →b 1b2 for large tan β.
In Fig. 7 we show two kinds of perturbation expansion for the input parameters the on-shell tree-level width (dotted line). The dashed and dash-dot-dotted line correspond to the on-shell electroweak and full (electroweak plus SUSY-QCD) one-loop width, respectively. For both corrections one can clearly see the invalidity of the on-shell perturbation expansion, in particular the electroweak corrections lead to an improper negative decay width. The second way of perturbation expansion is given by the dash-dotted and the solid line which correspond to the improved tree-level and improved full one-loop decay width, respectively. The smallness of the relative correction in this case shows that the improved tree-level is already a good approximation for A 0 →b 1b2 . The input parameters are the same as in the first case but now with running A b = −700 GeV. Squarks decays: Fig. 8 displays the decay widths of the crossed channelt 2 →t 1 A 0 as a function of A t . As can be seen, the electroweak corrections are as large as the SUSY-QCD ones in the considered region. The values of the input parameters are {tanβ, µ} = {35, −300} and {m A 0 , mg, MQ, A b , A τ } = {150, 1000, 300, −700, −700} GeV with the relations for the SUSY breaking masses given at the top of this section but with MŨ 3 = 500 GeV in order to get a quite acceptable mass splitting in the stop sector.    9 again demonstrates the numerical improvement in the large tanβ regime: The dotted and dash-dot-dotted lines correspond to the on-shell tree-level and on-shell oneloop decay widths ofb 2 →b 1 A 0 , whereas the dash-dotted and solid lines show the full improved tree-level and one-loop widths, respectively. The input parameters are the same as in Fig. 8 but with {MQ 3 , A} = {500, −700} GeV. In conclusion, we have calculated the full electroweak one-loop corrections to the decay widths A 0 →q 1q2 andq 2 →q 1 A 0 in the on-shell scheme. Moreover, we have included the SUSY-QCD corrections which were calculated in [6]. For the decay into sbottom quarks and large tan β an improvement of the on-shell perturbation expansion is necessary. This was done by an appropriate redefinition of the tree-level Higgs-squark-squark coupling. We find that the corrections are significant and in a wide range of the parameter space comparable to the SUSY-QCD corrections.