Table Of ContentHEPHY-PUB 832/07
hep-ph/0701134
Complete one-loop corrections to decays of
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0 charged and CP-even neutral Higgs bosons
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into sfermions
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C. Weber,K. Kovaˇr´ık,H. Eberl,W. Majerotto
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0 Institut fu¨r Hochenergiephysik der O¨sterreichischen Akademie der Wissenschaften,
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A-1050 Vienna, Austria
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X Abstract
r
a We present the full one-loop corrections to charged and CP-even neutral Higgs
boson decays into sfermions including also the crossed channels. The calculation
wascarriedoutintheminimalsupersymmetricextensionoftheStandardModeland
we use the on-shell renormalization scheme. For the down-type sfermions, we use
DR running fermion masses and the trilinear coupling A as input. Furthermore,
f
we present the first numerical analysis for decays according to the Supersymmetric
Parameter Analysis project. This requires the renormalization of the whole MSSM.
The corrections are found to be numerically stable and not negligible.
1 Introduction
The Higgsboson is the last not discovered particle of the Standard Model (SM) andso the
search for the Higgs boson is the prime objective of the LHC and other future colliders.
Apart from the SM, the Higgs boson is also predicted by its minimal supersymmetric
extension - the Minimal Supersymmetric Standard Model (MSSM). As opposed to the
SM, the MSSM has not only one neutral Higgs boson but it predicts the existence of two
neutral CP-even Higgs bosons (h0, H0), one neutral CP-odd Higgs boson (A0) and two
charged Higgs bosons (H±). The existence of a charged Higgs boson or a CP-odd neutral
one would be clear evidence for physics beyond the SM.
A further difference in the MSSM is the possibility for the Higgs bosons to decay not only
into SM particles. In case the supersymmetric (SUSY) partners are not too heavy, the
Higgs bosons can decay into SUSY particles as well (neutralinos χ˜0, charginos χ˜+ and
i k
sfermions f˜ ). The new decay channels might substantially influence the branching ratios
m
of the MSSM Higgs bosons.
At tree-level the decays into SUSY particles were studied in [1, 2] and one-loop effects of
the decays into charginos and neutralinos were analyzed in [3, 4] and were found not to
be negligible. For the case of the CP-odd Higgs boson also the full one-loop corrections
to the decay into sfermions were analyzed in [5, 6].
This paper is the continuation of the effort in [5, 6] and includes the decays of the remain-
ing Higgs bosons of the MSSM into sfermions (including crossed channels f˜ f˜h0).
2 1
→
It also extends the SUSY-QCD one-loop analysis of [7] by including all SUSY-QCD and
electroweak effects. The emphasis is put on the decay into 3rd generation sfermions as
their masses can be light due to large mixings. Nevertheless, analytical and numerical
results are presented for all generations of sfermions (i.e. h0 f˜f˜¯ and H± f˜f¯˜′ where
k → i j → i j
h0 = (h0,H0) and f˜= (u˜,d˜,s˜,c˜,˜b,t˜,e˜,µ˜,τ˜).
k
The full electroweak corrections are calculated in the on-shell scheme [8] in the MSSM
with real parameters. Due to the known problems of the on-shell scheme as demonstrated
in [6], the artificially large on-shell parameters are replaced by the corresponding DR
counterparts. The numerical analysis is made using the DR input defined by the Super-
symmetric Parameter Analysis Project (SPA) [9]. In contrast to [6], the actual calculation
uses an on-shell input set fully consistent with the SPA convention. In order to obtain
such a input set, the renormalization of the whole MSSM is required.
The paper is organized as follows. In section 2 the tree-level formulae are given for all de-
cays. Section 3 and 4 show the full electroweak corrections including the bremsstrahlung
using the analytic formulae from the appendices A and B. The numerical analysis is
presented in section 5 and section 6 summarizes our conclusions.
2
2 Tree-level result
The tree-level widths for a neutral Higgs h0 = h0,H0 decaying into two scalar
{1,2} { }
fermions, h0 f˜f¯˜ with i,j = (1,2), are given by
k → i j
Nf κ(m2 ,m2 ,m2 )
Γtree(h0 f˜f˜¯) = C h0k f˜i f˜j Gf˜ 2 (1)
k → i j 16πm3 | ijk|
h0
k
with κ(x,y,z) = (x y z)2 4yz and the colour factor Nf = 3 for squarks and
− − − C
Nf = 1 for sleptonqs, respectively.
C
Analogously, the decay width for the charged Higgs boson H+ is given by
Nf κ(m2 ,m2 ,m2 )
Γtree(H+ f˜↑f˜¯↓) = C H+ f˜i↑ f˜j↓ G↑↓ 2, (2)
→ i j 16πm3 | ij1|
H+
where f˜↑/↓ stand for the up-type or down-type sfermions. The sfermion-Higgs boson
couplings Gf˜ and G↑↓ , defined by the interaction lagrangian = Gf˜ h0f˜∗f˜ +
ijk ij1 Lint ijk k i j
G↑↓ H+f˜↑∗f˜↓ as well as all couplings needed in this paper, are given in [6].
ij1 i j
The sfermion mass matrix is diagonalized by a real 2x2 rotation matrix Rf˜ with
iα
rotation angle θ [10, 11],
f˜
m2 m2 m2 a m m2 0
2 = LL LR = f˜L f f = Rf˜ † f˜1 Rf˜, (3)
Mf˜ m2 m2 a m m2 0 m2
RL RR f f f˜R (cid:16) (cid:17) f˜2
which relates the mass eigenstates f˜, i = 1,2, (m < m ) to the gauge eigenstates f˜ ,
i f˜1 f˜2 α
α = L,R, by f˜ = Rf˜f˜ . The left- and right-handed and the left-right mixing entries in
i iα α
the mass matrix are given by
m2 = M2 +(I3L e sin2θ )cos2βm2 +m2 , (4)
f˜L {Q˜,L˜} f − f W Z f
m2 = M2 +e sin2θ cos2βm2 +m2 , (5)
f˜R {U˜,D˜,E˜} f W Z f
af = Af µ(tanβ)−2If3L. (6)
−
M , M , M , M and M are soft SUSY breaking masses, A is the trilinear scalar
Q˜ L˜ U˜ D˜ E˜ f
coupling parameter, µ the higgsino mass parameter, tanβ = v2 is the ratio of the vacuum
v1
expectation values of the two neutral Higgs doublet states [10, 11], I3L denotes the third
f
component of the weak isospin of the left-handed fermion f, e the electric charge in
f
terms of the elementary charge e , and θ is the Weinberg angle.
0 W
The mass eigenvalues and the mixing angle in terms of primary parameters are
1
m2 = m2 +m2 (m2 m2 )2 +4a2m2 , (7)
f˜1,2 2 f˜L f˜R ∓ f˜L− f˜R f f
(cid:16) a m q (cid:17)
cosθ = − f f (0 θ < π), (8)
f˜ (m2 m2 )2 +a2m2 ≤ f˜
f˜L− f˜1 f f
q
3
and the trilinear soft breaking parameter A can be written as
f
1
Af = 2m m2f˜1 −m2f˜2 sin2θf˜+µ(tanβ)−2If3L. (9)
f
(cid:16) (cid:17)
The mass of the sneutrino ν˜ is given by m2 = M2 + 1 m2 cos2β.
τ ν˜τ L˜ 2 Z
For the crossed channels, f˜ f˜h0 and f˜↑ f˜↓H+, the decay widths are
2 → 1 k 2 → 1
κ(m2 ,m2 ,m2 )
Γtree(f˜ f˜h0) = h0k f˜1 f˜2 Gf˜ 2, (10)
2 → 1 k 16πm3 | 12k|
f˜2
κ(m2 ,m2 ,m2 )
Γtree(f˜↑ f˜↓H+) = H+ f˜j↑ f˜i↓ G↑↓ 2. (11)
j → i 16πm3 | ij1|
f˜↑
j
3 One-loop Corrections
The one-loop corrected (renormalized) amplitudes Gf˜ren and G↑↓,ren can be expressed as
ijk ij1
Gf˜,ren = Gf˜ +∆Gf˜ = Gf˜ +δGf˜(v) +δGf˜(w) +δGf˜(c), (12)
ijk ijk ijk ijk ijk ijk ijk
G↑↓,ren = G↑↓ +∆G↑↓ = G↑↓ +δG↑↓(v) +δG↑↓(w) +δG↑↓(c), (13)
ij1 ij1 ij1 ij1 ij1 ij1 ij1
where δGf˜(v),δGf˜(w) and δGf˜(c) and the corresponding terms for the couplings to the
ijk ijk ijk
charged Higgs boson stand for the vertex corrections, the wave-function corrections and
the coupling counter-term corrections due to the shifts from the bare to the on-shell val-
ues, respectively.
Throughout the paper we use the SUSY invariant dimensional reduction (DR) as a reg-
ularization scheme. For convenience we perform the calculation in the ’t Hooft-Feynman
gauge, ξ = 1.
The vertex corrections δGf˜(v) and δG↑↓(v) come from the diagrams listed in Figs. 15 and
ijk ij1
16. The analytic formulae are given in Appendix B. The wave-function corrections δGf˜(w)
ijk
can be written as
δGf˜(w) = 1 δZf˜Gf˜ +δZf˜ Gf˜ +δZH Gf˜ , (14)
ijk 2 i′i i′jk j′j ij′k lk ijl
(cid:20) (cid:21)
with the implicit summations over i′,j′,l = 1,2. The wave-function renormalization
constants are determined by imposing the on-shell renormalization conditions [8]
δZf˜ = ReΠ˙f˜(m2 ), i = 1,2,
ii − ii f˜i
2 (15)
δZf˜ = ReΠf˜(m2 ), i,j = (1,2), i = j, f˜= ν˜
ij m2 m2 ij f˜j 6 6 e,µ,τ
f˜i− f˜j
δZH = ReΠ˙H(m2 ), k = 1,2,
kk − kk h0k
2 (16)
δZH = ReΠH(m2 ), k,l = (1,2), k = l.
kl m2h0k−m2h0l kl h0l 6
4
The explicit forms of the off-diagonal Higgs boson and sfermion self-energies and their
derivatives, ΠH, Π˙H and Πf˜, Π˙f˜ are given in Appendix A and in [6].
kl kk ij ii
The coupling counter-term corrections which come from the shifting of the parameters in
the lagrangian can be expressed as
δGf˜(c) = δRf˜ Gf˜ (Rf˜)T +Rf˜ δGf˜ (Rf˜)T + Rf˜ Gf˜ (δRf˜)T .(17)
ijk · LR,k · · LR,k · · LR,k ·
(cid:20) (cid:21)ij
The counter term for the sfermion mixing angle, δθ , is determined such as to cancel the
f˜
anti-hermitian part of the sfermion wave-function corrections [12, 13]. Analogously we fix
the Higgs boson mixing angle α by means of
1 1
δα = δZH δZH = Re ΠH(m2 )+ΠH(m2 ) . (18)
4 21 − 12 2(m2 m2 ) 12 H0 21 h0
(cid:16) (cid:17) H0− h0 (cid:16) (cid:17)
Using the relations
δGf˜ δGf˜
ij1 = Gf˜ , ij2 = Gf˜ , (19)
δα − ij2 δα ij1
and absorbing the counter terms for the mixing angles of the outer particles, δα and δθ ,
f˜
into δGf˜(w) yields the symmetric wave-function corrections
ijk
δGf˜(w,symm.) = 1 δZf˜ +δZf˜ Gf˜ + 1 δZf˜ +δZf˜ Gf˜ + 1 δZH +δZH Gf˜ .
ijk 4 ii′ i′i i′jk 4 jj′ j′j ij′k 4 kl lk ijl
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(20)
Note that in this symmetrized form momentum-independent contributions from four-
scalar couplings and tadpole shifts cancel out.
The sum of wave-function and counter-term corrections then reads
δGf˜(w+c) = δGf˜(w,symm.) + Rf˜ δˆGf˜ (Rf˜)T , (21)
ijk ijk · LR,k · ij
h i
The explicit forms of the counter terms δˆGf˜ for k = 1,2 are given by
LR,k
δh δm
(δˆGf˜ ) = √2h m c f + f g m e δs2 s
LR,1 11 − f f α h m − Z Z f W α+β
(cid:18) f f (cid:19)
δg δm δβ
+g m (I3L e s2 )s Z + Z + , (22)
Z Z f − f W α+β g m t
(cid:18) Z Z α+β(cid:19)
δh h
(δˆGf˜ ) = f(Gf˜ ) f (δA c +δµs ), (23)
LR,1 12 h LR,1 12 − √2 f α α
f
δh δm
(δˆGf˜ ) = √2h m c f + f
LR,1 22 − f f α h m
(cid:18) f f (cid:19)
δg δm δs2 δβ
+g m e s2 s Z + Z + W + (24)
Z Z f W α+β g m s2 t
(cid:18) Z Z W α+β(cid:19)
5
for the sfermion couplings to the Higgs boson h0 and
δh δm
(δˆGf˜ ) = √2h m s f + f +g m e δs2 c
LR,2 11 − f f α h m Z Z f W α+β
(cid:18) f f (cid:19)
δg δm
g m (I3L e s2 )c Z + Z t δβ , (25)
− Z Z f − f W α+β g m − α+β
(cid:18) Z Z (cid:19)
δh h
(δˆGf˜ ) = f(Gf˜ ) f (δA s δµc ), (26)
LR,2 12 h LR,2 12 − √2 f α − α
f
δh δm
(δˆGf˜ ) = √2h m s f + f
LR,2 22 − f f α h m
(cid:18) f f (cid:19)
δg δm δs2
g m e s2 c Z + Z + W t δβ (27)
− Z Z f W α+β g m s2 − α+β
(cid:18) Z Z W (cid:19)
for the couplings to H0.
Analogously to the decays of the CP-even Higgs bosons, the sum of the wave-function
and counter-term corrections of the charged Higgs boson can be expressed as
δG↑↓(w+c) = δG↑↓(w,symm.) + Rf˜↑ δG↑↓ (Rf˜↓)T +δG↑↓(w,HW+HG) (28)
ij1 ij1 · LR,1 · ij ij1
h i
with the symmetrized wave-function corrections
1 1 1
δG↑↓(w,symm.) = (δZf˜↑ +δZf˜↑)G↑↓ + (δZf˜↓ +δZf˜↓)G↑↓ + δZH+G↑↓ . (29)
ij1 4 ii′ i′i i′j1 4 jj′ j′j ij′1 2 11 ij1
The single elements of the matrix corresponding to the variation with respect to the
couplings, δG↑↓ , are given explicitly as follows:
LR,1
δh δm δs δh δm δc
(δG↑↓ ) = h m s f↓ + f↓ + β +h m c f↑ + f↑ + β
LR,1 11 f↓ f↓ β h m s f↑ f↑ β h m c
(cid:18) f↓ f↓ β (cid:19) (cid:18) f↑ f↑ β (cid:19)
gm δg δm δtanβ
W sin2β + W +cos2β (30)
− √2 g m tanβ
(cid:18) W (cid:19)
δh
(δG↑↓ ) = f↓(G↑↓ ) +h (δA s +A δs +δµc +µδc ) (31)
LR,1 12 h LR,1 12 f↓ f↓ β f↓ β β β
f↓
δh
(δG↑↓ ) = f↑(G↑↓ ) +h (δA c +A δc +δµs +µδs ) (32)
LR,1 21 h LR,1 21 f↑ f↑ β f↑ β β β
f↑
δh δm δc δh δm δs
(δG↑↓ ) = h m c f↑ + f↓ + β +h m s f↓ + f↑ + β (33)
LR,1 22 f↑ f↓ β h m c f↓ f↑ β h m s
(cid:18) f↑ f↓ β (cid:19) (cid:18) f↓ f↑ β (cid:19)
The counter terms appearing in eqs. (22-33) can be fixed in the following manner. Some
of them can be decomposed further as is the case for δh and δg
f
δh δg δm δm cos2β δtanβ δg δe δsinθ
f = + f W + − , = W , (34)
hf g mf − mW ( sin2β) tanβ g e − sinθW
6
for up -type sfermions.
down
Fornthe reomaining counter terms we use the standard renormalization conditions. The fix-
ing of the angle β is performed using the condition that the renormalized A0-Z0 transition
vanishes at p2 = m2 as in [14], which gives the counter term
A0
δtanβ 1
= ImΠ (m2 ). (35)
tanβ m sin2β A0Z0 A0
Z
Thehiggsinomassparameterµisfixedinthecharginosectorbythecharginomassmatrix,
δµ δX , as explained in detail in [15, 16].
22
≡
The counter term to the Standard Model parameter sinθ is determined using the on-
W
shell masses of the gauge bosons as in [17]. To avoid the problems with light quarks in
the fine structure constant α, we use the MS value at the Z-pole with the counter term
given in [5, 21].
The on-shell counter term that has the biggest influence and also poses a serious problem
is the counter term to the trilinear scalar coupling parameter A . The explicit form of
f
the counter term was already given in [5] and it was shown in [5, 6] that this counter term
becomes very large for large values of tanβ. One of the aims of this paper is to show that
this problem is present in all Higgs decays into sfermions. The solution takes advantage of
the fact that the SPA convention which we use here, defines the SUSY parameters in the
DR scheme. Therefore, the trilinear scalar coupling parameters A are taken DR without
f
the use of the large on-shell counter term.
4 Infrared divergences
To cancel infrared divergences we introduce a small photon mass l and include the real
photon emission processes h0 f˜f˜¯γ (h0 = h0,H0 ) and H+ f˜↑f˜¯↓γ. The decay
k → i j k { } → i j
width of H+(p) f˜↑(k ) +f˜¯↓( k )+γ(k ) (Fig. 1) is given by
→ i 1 j − 2 3
~" ~" ~"
(a) fi (b) fi (c) fi
k1
(cid:13)
p
+ + +
H H H
k3 (cid:13) (cid:13)
k2
~# ~# ~#
fj fj fj
Figure1: RealBremsstrahlungdiagramsrelevanttocanceltheIR-divergencesinH+(p)
f˜↑(k ) +f˜¯↓( k )+γ(k ). The diagrams for the neutral Higgs decays are analogous. →
i 1 j − 2 3
Γ(H+ f˜↑f˜¯↓γ) = NC G↑↓ 2( e )2 m2 I +e2m2I +e2m2I
→ i j 16π3m | ij1| − 0 H+ 00 t i 11 b j 22
H+
h
7
e e (m2 m2 m2)I I I e (m2 m2 m2)I I I
− t b H+ − i − j 12 − 2 − 1 − t j − H+ − i 01 − 1 − 0
(cid:16) (cid:17) (cid:16) (cid:17)
+e (m2 m2 m2)I I I ,
b i − H+ − j 02 − 2 − 0
(cid:16) (cid:17)i
with the phase-space integrals I and I defined as [18]
n mn
1 d3k d3k d3k 1
I = 1 2 3 δ4(p k k k ) . (36)
i1...in π2 2E 2E 2E − 1 − 2 − 3 (2k k +λ2)...(2k k +λ2)
Z 1 2 3 3 i1 3 in
The full IR-finite one-loop corrected decay width for the physical value l = 0 is then given
by
Γcorr(H+ f˜↑f˜¯↓) Γ(H+ f˜↑f˜¯↓)+Γ(H+ f˜↑f˜¯↓γ) (37)
→ i j ≡ → i j → i j
Analogously, for the neutral Higgs boson decays and the crossed channels the photon
emission processes yield
(e e )2 Gf˜ 2
Γ(h0 f˜f˜¯ γ) = N 0 f | ijk| m2 m2 m2 I +m2 I +m2 I I I ,
k → i j C 16π3mh0k (cid:20)(cid:16) h0k− f˜i− f˜j(cid:17) 12 f˜i 11 f˜j 22− 1− 2(cid:21)
(e e )2 Gf˜ 2
Γ(f˜ f˜h0γ) = 0 f | ijk| m2 m2 m2 I m2 I m2 I I I ,
2 → 1 k 16π3mf˜2 (cid:20)(cid:16) h0k− f˜1− f˜2(cid:17) 01 − f˜1 11 − f˜2 00 − 0 − 1(cid:21)
(38)
where the IR-finite decay widths are
Γcorr(h0 f˜f¯˜) Γ(h0 f˜f˜¯) + Γ(h0 f˜f˜¯ γ), (39)
k → i j ≡ k → i j k → i j
Γcorr(f˜ f˜h0) Γ(f˜ f˜h0) + Γ(f˜ f˜h0γ). (40)
2 → 1 k ≡ 2 → 1 k 2 → 1 k
5 Numerical analysis
The numerical results presented in this section are based on the SPS1a’ benchmark point
as proposed by the Supersymmetric Parameter Analysis Project (SPA) [9]. A consistent
implementation of the SPA convention into the calculation of a decay width and the nu-
merical analysis is a non-trivial endeavor. As the electroweak one-loop calculations are
carried out in the on-shell scheme and the SPA project proposes the SUSY input set in
the DR scheme at the scale of 1 TeV, a conversion of the input values is necessary. This
conversion requires the renormalization of the whole MSSM in order to transform the
input parameters correctly. Moreover, the numerical analysis of a decay makes varying
fundamental SUSY parameters necessary. That is why the above mentioned transforma-
tion of parameters has to be carried out for every single parameter point. In our case
this is provided by the not-yet-public routine DRbar2OS which couples to the spectrum
calculator SPheno [19]. The transformation is performed in the following two steps:
8
1. The SPA input, i.e. the on-shell electroweak SM parameters, the strong coupling
constant andthemasses ofthelight quarksdefinedintheMSscheme andthemasses
of the leptons and the top quark defined as pole masses and the SUSY parameters
defined in the DR scheme at 1 TeV, is given to SPheno which transforms it to a
pure DR input set including also higher loop corrections.
2. The pure DR set is taken as input for the DRbar2OS routine which yields as output
the complete set in the on-shell scheme. An example of different sets of parameters
for the SPS1a’ benchmark point can be seen in Table 1.
All plots below show the dependence of the decay width on a DR parameter. By varying
a single DR parameter and transforming subsequently to the on-shell scheme, almost all
parameters are influenced. That means, not only the corresponding on-shell parameter
changes, but also the other parameters through loop effects.
DR SUSY Parameters On-shell SUSY Parameters
g′ 0.36354 M 103.21 g′ 0.35565 M 100.31
1 1
g 0.64804 M 193.29 g 0.66547 M 197.01
2 2
g 1.08412 M 572.33 g 1.08419 M 612.81
s 3 s 3
Y 0.10349 A 445.4 Y 0.10771 A 394.2
τ τ τ τ
− −
Y 0.89840 A 532.3 Y 1.04638 A 495.0
t t t t
− −
Y 0.13548 A 938.9 Y 0.20481 A 1197.8
b b b b
−
µ 401.63 tanβ 10.0 µ 398.71 tanβ 10.31
M 1.8121 102 M 1.7945 102 M 1.8394 102 M 1.8199 102
L1 · L3 · L1 · L3 ·
M 1.1572 102 M 1.1002 102 M 1.1784 102 M 1.1172 102
E1 · E3 · E1 · E3 ·
M 5.2628 102 M 4.7091 102 M 5.6390 102 M 5.0369 102
Q1 · Q3 · Q1 · Q3 ·
M 5.0767 102 M 3.8532 102 M 5.4540 102 M 4.1021 102
U1 · U3 · U1 · U3 ·
M 5.0549 102 M 5.0137 102 M 5.4352 102 M 5.3894 102
D1 · D3 · D1 · D3 ·
M2 2.5605 104 M2 14.725 104 M2 2.7220 104 M2 15.726 104
H1 · H2 − · H1 · H2 − ·
Table 1: The input parameters for the SPS1a’ point according to the SPA project.
Left: DR input values at Q = 1 TeV, Right: on-shell values used in the calculation.
Comparing the parameter sets in Table 1 one can easily see that the on-shell counter
term for A is large as there is a huge difference between the DR and the on-shell value
b
of the trilinear scalar coupling parameter. This is caused by fixing the of A parameter
b
in the sfermion sector [5]. This fixing was shown to lead to a numerically large counter
term which should be avoided. The decays of Higgs bosons into sfermions (and the cor-
responding crossed channels) are the only 2-body decays that are affected directly as the
9
trilinear coupling parameter appears at tree-level. In this case, a very large counter term
makes the perturbative expansion unreliable. Here we make use of the fact that the in-
put parameters are given in the DR scheme. That means our calculation uses on-shell
parameters except for the A and m which take the original DR values. Although not
b b
shown in the Table 1, this behaviour is common to all down-type trilinear scalar coupling
parameters, and the same strategy as described for the A is applied for them as well.
b
A distinct feature of all decay modes involving down-type sfermions is a large difference
between the on-shell and the SPA tree-level. The origin of this difference is again the
large counter term for the trilinear scalar couplings.
Keeping the numerical analysis strictly confined to the SPS1a’ benchmark scenario would
mean that most of the possible decays are kinematically not allowed. The vertical red line
denotes the position of the SPS1a’ parameter point for the kinematically allowed decay
modes. For the other decays we slightly deviate from the SPS1a’ point adjusting mainly
m and the relevant soft supersymmetry breaking terms M . These parameters
A0 {Q˜,U˜,D˜,L˜,E˜}
only influence the kinematics and have usually no effect on the couplings.
In general, we always show the results using the on-shell parameters (dotted curve for on-
shell tree-level and red dashed curve for on-shell full one loop decay width) as well as the
improved decay widths where the parameters A andm aretakenDR (dash-dottedcurve
f f
denotes SPA tree-level and blue solid curve stands for the full one loopdecay width). This
convention does not apply in cases where there is no down-type trilinear scalar coupling
entering the tree-level. There we show only the on-shell and SPA tree-level together with
the finalone-loopdecay width. Forcomparison with other calculations, the SPA tree-level
is shown as defined in [9] taking all parameters in the couplings in the DR scheme and
using the proper masses for the kinematics.
M = 150 GeV, m2 = 106 GeV
{ D˜3 A0 }
] 0.15
V
e
G
[
) 0.05
¯˜b1
1
˜b
→-0.05
0
H
(
Γ -0.15
5 10 15 20 25
tanβ
Figure2: On-shelltree-level (dottedline), fullon-shellone-loopdecay width(dashedline),
tree-level (dash-dotted line) and full one-loop corrected width (solid line) of H0 ˜b ˜b¯
1 1
→
as a function of tanβ according to the SPA convention.
10
