# Complete one-loop corrections to decays of charged and CP-even neutral Higgs bosons into sfermions

**ABSTRACT** 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 was carried out in the minimal supersymmetric extension of the Standard Model and we use the on-shell renormalization scheme. For the down-type sfermions, we use running fermion masses and the trilinear coupling Af as input. Furthermore, 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.

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**ABSTRACT:**We analyze neutral Higgs boson decays into squark pairs in the minimal supersymmetric extension of the Standard Model and improve previous analyses. In particular the treatment of potentially large higher-order corrections due to the soft SUSY breaking parameters A_b, the trilinear Higgs coupling to sbottoms, and mu, the Higgsino mass parameter, is investigated. The remaining theoretical uncertainties including the SUSY-QCD corrections are analyzed quantitatively.Physical review D: Particles and fields 03/2011; - SourceAvailable from: Jonas M. Lindert[Show abstract] [Hide abstract]

**ABSTRACT:**We present the analysis of the signature 2j + ETmiss (+X) via squark-squark production and direct decay into the lightest neutralino, pp -> squark squark -> j j chi_1^0 chi_1^0 (+X), in next-to-leading order QCD within the framework of the minimal supersymmetric standard model. In our approximation the produced squarks are treated on shell. Thus, the calculation of production and decay factorizes. In this way, we provide a consistent, fully differential calculation of NLO QCD factorizable corrections to the given processes. Clustering final states into partonic jets, we investigate the experimental inclusive signature 2j + ETmiss for several benchmark scenarios. We compare resulting differential distributions with leading-order approximations rescaled by a flat K-factor and examine a possible impact for cut-and-count searches for supersymmetry at the LHC.Journal of High Energy Physics 07/2012; 2013(3). · 6.22 Impact Factor - SourceAvailable from: Davide Pagani[Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we present an analysis at NLO of the contribution from squark-squark production to the experimental signature 2j + 2l + missing E_T (+X) with opposite-sign same flavor leptons, taking into account decays and experimental cuts. We consider the case in which one squark decays directly into the lightest neutralino chi^0_1 and the other one into the second lightest neutralino and subsequently into l^+l^-chi^0_1 via an intermediate slepton. On one hand we study effects of the NLO corrections on invariant mass distributions which can be used for future parameter determination. On the other hand we analyze the impact on predictions for cut-and-count searches using this experimental signature.European Physical Journal C 03/2013; 73(5). · 5.44 Impact Factor

Page 1

arXiv:hep-ph/0701134v2 19 Jan 2007

HEPHY-PUB 832/07

hep-ph/0701134

Complete one-loop corrections to decays of

charged and CP-even neutral Higgs bosons

into sfermions

C. Weber,K. Kovaˇ r´ ık,H. Eberl,W. Majerotto

Institut f¨ ur Hochenergiephysik der¨Osterreichischen Akademie der Wissenschaften,

A-1050 Vienna, Austria

Abstract

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

was carried out in the minimal supersymmetric extension of the Standard Model and

we use the on-shell renormalization scheme. For the down-type sfermions, we use

DR running fermion masses and the trilinear coupling Af as input. Furthermore,

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.

Page 2

1Introduction

The Higgs boson is the last not discovered particle of the Standard Model (SM) and so 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 ˜ χ0

sfermions˜fm). The new decay channels might substantially influence the branching ratios

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˜f2 →˜f1h0).

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

h0

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.

i, charginos ˜ χ+

kand

k→˜fi¯˜fjand H±→˜fi¯˜f′

jwhere

k= (h0,H0) and˜f = (˜ u,˜d, ˜ s,˜ c,˜b,˜t, ˜ e, ˜ µ, ˜ τ).

2

Page 3

2Tree-level result

The tree-level widths for a neutral Higgs h0

fermions, h0

{1,2}= {h0,H0} decaying into two scalar

k→˜fi¯˜fjwith i,j = (1,2), are given by

Γtree(h0

k→˜fi¯˜fj) =

Nf

Cκ(m2

h0

k,m2

˜fi,m2

˜fj)

16πm3

h0

k

|G

˜f

ijk|2

(1)

with κ(x,y,z) =

Nf

C= 1 for sleptons, respectively.

Analogously, the decay width for the charged Higgs boson H+is given by

?

(x − y − z)2− 4yz and the colour factor Nf

C= 3 for squarks and

Γtree(H+→˜f↑

i

¯˜f↓

j) =

Nf

Cκ(m2

H+,m2

˜f↑

i,m2

˜f↓

j)

16πm3

H+

|G↑↓

ij1|2,

(2)

where˜f↑/↓stand for the up-type or down-type sfermions. The sfermion-Higgs boson

couplings G

G↑↓

i

˜f↓

The sfermion mass matrix is diagonalized by a real 2x2 rotation matrix R

rotation angle θ˜f[10, 11],

m2

RL

m2

RR

afmf

m2

˜f

ijkand G↑↓

jas well as all couplings needed in this paper, are given in [6].

ij1, defined by the interaction lagrangian Lint = G

˜f

ijkh0

k˜f∗

i˜fj+

ij1H+˜f↑∗

˜f

iαwith

M2

˜f=

m2

LL

m2

LR

=

m2

˜fL

afmf

˜fR

=

?

R

˜f?†

m2

˜f1

0

0

m2

˜f2

R

˜f,

(3)

which relates the mass eigenstates˜fi, i = 1,2, (m˜f1< m˜f2) to the gauge eigenstates˜fα,

α = L,R, by˜fi= R

the mass matrix are given by

˜f

iα˜fα. The left- and right-handed and the left-right mixing entries in

m2

˜fL

= M2

{˜ Q,˜L}+ (I3L

f−efsin2θW)cos2β m2

{˜U,˜ D,˜E}+ efsin2θWcos2β m2

Z+ m2

f,

(4)

m2

˜fR

= M2

Z+ m2

f,

(5)

af

= Af− µ(tanβ)−2I3L

f .

(6)

M˜Q, M˜L, M˜U, M˜Dand M˜Eare soft SUSY breaking masses, Af is the trilinear scalar

coupling parameter, µ the higgsino mass parameter, tanβ =v2

expectation values of the two neutral Higgs doublet states [10, 11], I3L

component of the weak isospin of the left-handed fermion f, ef the electric charge in

terms of the elementary charge e0, and θW is the Weinberg angle.

The mass eigenvalues and the mixing angle in terms of primary parameters are

v1is the ratio of the vacuum

denotes the third

f

m2

˜f1,2

=

1

2

?

m2

˜fL+ m2

−afmf

˜fL−m2

˜fR∓

?

(m2

˜fL−m2

˜fR)2+ 4a2

fm2

f

?

,

(7)

cosθ˜f

=

?(m2

˜f1)2+ a2

fm2

f

(0 ≤ θ˜f< π),

(8)

3

Page 4

and the trilinear soft breaking parameter Afcan be written as

1

2mf

The mass of the sneutrino ˜ ντis given by m2

For the crossed channels,˜f2→˜f1h0

Af

=

?

m2

˜f1− m2

˜f2

?

sin2θ˜f+ µ(tanβ)−2I3L

f .

(9)

˜ ντ= M2

2→˜f↓

κ(m2

˜L+1

1H+, the decay widths are

˜f1,m2

16πm3

˜f2

κ(m2

˜f↑

˜f↓

16πm3

˜f↑

j

2m2

Zcos2β.

kand˜f↑

Γtree(˜f2→˜f1h0

k) =

h0

k,m2

˜f2)

|G

˜f

12k|2,

(10)

Γtree(˜f↑

j→˜f↓

iH+) =

H+,m2

j,m2

i)

|G↑↓

ij1|2.

(11)

3One-loop Corrections

The one-loop corrected (renormalized) amplitudes G

˜f ren

ijk

and G↑↓,ren

ij1

can be expressed as

G

˜f,ren

ijk

= G

˜f

ijk+ ∆G

˜f

ijk= G

˜f

ijk+ δG

˜f(v)

ijk+ δG

˜f(w)

ijk + δG

˜f(c)

ijk,

(12)

G↑↓,ren

ij1

= G↑↓

ij1+ ∆G↑↓

ij1= G↑↓

ij1+ δG↑↓(v)

ij1

+ δG↑↓(w)

ij1

+ δG↑↓(c)

ij1 ,

(13)

where δG

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 δG

ij1

come from the diagrams listed in Figs. 15 and

16. The analytic formulae are given in Appendix B. The wave-function corrections δG

can be written as

=1

2

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]

˜f(v)

ijk,δG

˜f(w)

ijk

and δG

˜f(c)

ijk and the corresponding terms for the couplings to the

˜f(v)

ijk and δG↑↓(v)

˜f(w)

ijk

δG

˜f(w)

ijk

?

δZ

˜f

i′iG

˜f

i′jk+ δZ

˜f

j′jG

˜f

ij′k+ δZH

lkG

˜f

ijl

?

,

(14)

δZ

˜f

ii= − Re˙Π

˜f

ij =

m2

˜f

ii(m2

˜fi),i = 1,2,

δZ

2

˜fi−m2

˜fj

ReΠ

˜f

ij(m2

˜fj),i,j = (1,2), i ?= j,˜f ?= ˜ νe,µ,τ

(15)

δZH

kk= − Re˙ΠH

kk(m2

h0

k),k = 1,2,

δZH

kl=

2

m2

h0

k−m2

h0

l

ReΠH

kl(m2

h0

l),k,l = (1,2), k ?= l.

(16)

4

Page 5

The explicit forms of the off-diagonal Higgs boson and sfermion self-energies and their

derivatives, ΠH

The coupling counter-term corrections which come from the shifting of the parameters in

the lagrangian can be expressed as

?

kl,˙ΠH

kkand Π

˜f

ij,˙Π

˜f

iiare given in Appendix A and in [6].

δG

˜f(c)

ijk

=

δR

˜f· G

˜f

LR,k· (R

˜f)T+ R

˜f· δG

˜f

LR,k· (R

˜f)T+ R

˜f· G

˜f

LR,k· (δR

˜f)T

?

ij. (17)

The counter term for the sfermion mixing angle, δθ˜f, is determined such as to cancel the

anti-hermitian part of the sfermion wave-function corrections [12, 13]. Analogously we fix

the Higgs boson mixing angle α by means of

δα =

1

4

?

δZH

21− δZH

12

?

=

1

2(m2

H0−m2

h0)Re

?

ΠH

12(m2

H0) + ΠH

21(m2

h0)

?

.

(18)

Using the relations

δG

δα

˜f

ij1

= −G

˜f

ij2,

δG

δα

˜f

ij2

= G

˜f

ij1,

(19)

and absorbing the counter terms for the mixing angles of the outer particles, δα and δθ˜f,

into δG

ijk

yields the symmetric wave-function corrections

˜f(w)

δG

˜f(w,symm.)

ijk

=

1

4

?

δZ

˜f

ii′ + δZ

˜f

i′i

?

G

˜f

i′jk+1

4

?

δZ

˜f

jj′ + δZ

˜f

j′j

?

G

˜f

ij′k+1

4

?

δZH

kl+ δZH

lk

?

G

˜f

ijl.

(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

δG

˜f(w+c)

ijk

= δG

˜f(w,symm.)

ijk

+

?

R

˜f·ˆδG

˜f

LR,k· (R

˜f)T?

ij,

(21)

The explicit forms of the counter termsˆδG

˜f

LR,kfor k = 1,2 are given by

(ˆδG

˜f

LR,1)11

= −√2hfmfcα

?δhf

hf

+δmf

mf

?

− gZmZefδs2

?δgZ

gZ

mZ

Wsα+β

+gZmZ(I3L

f−efs2

W)sα+β

+δmZ

+

δβ

tα+β

?

,

(22)

(ˆδG

˜f

LR,1)12

=

δhf

hf

(G

˜f

LR,1)12−hf

√2(δAfcα+ δµsα),

?δhf

hf

mf

?δgZ

gZ

(23)

(ˆδG

˜f

LR,1)22

= −√2hfmfcα

+δmf

?

+gZmZefs2

Wsα+β

+δmZ

mZ

+δs2

W

s2

W

+

δβ

tα+β

?

(24)

5

Page 6

for the sfermion couplings to the Higgs boson h0and

(ˆδG

˜f

LR,2)11

= −√2hfmfsα

?δhf

hf

+δmf

mf

?

+ gZmZefδs2

Wcα+β

−gZmZ(I3L

f−efs2

W)cα+β

?δgZ

gZ

+δmZ

mZ

− tα+βδβ

?

,

(25)

(ˆδG

˜f

LR,2)12

=

δhf

hf

(G

˜f

LR,2)12−hf

√2(δAfsα− δµcα),

?δhf

hf

mf

?δgZ

gZ

(26)

(ˆδG

˜f

LR,2)22

= −√2hfmfsα

+δmf

?

−gZmZefs2

Wcα+β

+δmZ

mZ

+δs2

W

s2

W

− tα+βδβ

?

(27)

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

?

with the symmetrized wave-function corrections

δG↑↓(w+c)

ij1

= δG↑↓(w,symm.)

ij1

+

R

˜f↑· δG↑↓

LR,1· (R

˜f↓)T?

ij+ δG↑↓(w,HW+HG)

ij1

(28)

δG↑↓(w,symm.)

ij1

=1

4(δZ

˜f↑

ii′ + δZ

˜f↑

i′i)G↑↓

i′j1+1

4(δZ

˜f↓

jj′ + δZ

˜f↓

j′j)G↑↓

ij′1+1

2δZH+

11G↑↓

ij1.

(29)

The single elements of the matrix corresponding to the variation with respect to the

couplings, δG↑↓

LR,1, are given explicitly as follows:

(δG↑↓

LR,1)11

= hf↓mf↓sβ

?δhf↓

hf↓

+δmf↓

mf↓

?δg

g

+δsβ

sβ

?

+ hf↑mf↑cβ

?δhf↑

hf↑

?

+δmf↑

mf↑

+δcβ

cβ

?

−gmW

√2

sin2β

+δmW

mW

+ cos2βδ tanβ

tanβ

(30)

(δG↑↓

LR,1)12

=

δhf↓

hf↓

(G↑↓

LR,1)12+ hf↓(δAf↓sβ+ Af↓δsβ+ δµcβ+ µδcβ)(31)

(δG↑↓

LR,1)21

=

δhf↑

hf↑

(G↑↓

LR,1)21+ hf↑(δAf↑cβ+ Af↑δcβ+ δµsβ+ µδsβ)(32)

(δG↑↓

LR,1)22

= hf↑mf↓cβ

?δhf↑

hf↑

+δmf↓

mf↓

+δcβ

cβ

?

+ hf↓mf↑sβ

?δhf↓

hf↓

+δmf↑

mf↑

+δsβ

sβ

?

(33)

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 δhfand δg

?−cos2β

sin2β

δhf

hf

=

δg

g

+δmf

mf

−δmW

mW

+

?δ tanβ

tanβ

,

δg

g

=δe

e−δ sinθW

sinθW

, (34)

6

Page 7

for

For the remaining counter terms we use the standard renormalization conditions. The fix-

ing of the angle β is performed using the condition that the renormalized A0-Z0transition

vanishes at p2= m2

?

up

down

?-type sfermions.

A0 as in [14], which gives the counter term

δ tanβ

tanβ

=

1

mZsin2βImΠA0Z0(m2

A0).

(35)

The higgsino mass parameter µ is fixed in the chargino sector by the chargino mass matrix,

δµ ≡ δX22, as explained in detail in [15, 16].

The counter term to the Standard Model parameter sinθW is determined using the on-

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 Af. The explicit form of

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 Afare taken DR without

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

width of H+(p) →˜f↑

k→˜fi¯˜fjγ (h0

j(−k2) + γ(k3) (Fig. 1) is given by

k= {h0,H0}) and H+→˜f↑

i

¯˜f↓

jγ. The decay

i(k1) +¯˜f↓

?a?

H

?

?

f

?

i

?

f

?

j

?

p

k

?

k

?

k

?

?b?

H

?

?

f

?

i

?

?

f

?

j

?c?

H

?

?

f

?

i

?

?

f

?

j

Figure 1: Real Bremsstrahlung diagrams relevant to cancel the IR-divergences in H+(p) →

˜f↑

j(−k2) + γ(k3). The diagrams for the neutral Higgs decays are analogous.

i(k1) +¯˜f↓

Γ(H+→˜f↑

i

¯˜f↓

jγ) =

NC

16π3mH+|G↑↓

ij1|2(−e0)2?

m2

H+I00+ e2

tm2

iI11+ e2

bm2

jI22

7

Page 8

−eteb

?

(m2

H+ − m2

i− m2

j)I12− I2− I1

?

− et

?

+ eb

(m2

j− m2

?

H+ − m2

i)I01− I1− I0

?

(m2

i− m2

H+ − m2

j)I02− I2− I0

??

,

with the phase-space integrals Inand Imndefined as [18]

Ii1...in=

1

π2

?d3k1

2E1

d3k2

2E2

d3k3

2E3

δ4(p − k1− k2− k3)

1

(2k3ki1+ λ2)...(2k3kin+ λ2).

(36)

The full IR-finite one-loop corrected decay width for the physical value l = 0 is then given

by

Γcorr(H+→˜f↑

i

¯˜f↓

j) ≡ Γ(H+→˜f↑

i

¯˜f↓

j) + Γ(H+→˜f↑

i

¯˜f↓

jγ)(37)

Analogously, for the neutral Higgs boson decays and the crossed channels the photon

emission processes yield

Γ(h0

k→˜fi¯˜fjγ) = NC

(e0ef)2|G

16π3mh0

˜f

ijk|2

k

??

m2

h0

k−m2

˜fi−m2

˜fj

?

I12+ m2

˜fiI11+ m2

˜fjI22−I1−I2

?

,

Γ(˜f2→˜f1h0

kγ) =

(e0ef)2|G

16π3m˜f2

˜f

ijk|2

??

m2

h0

k−m2

˜f1−m2

˜f2

?

I01− m2

˜f1I11− m2

˜f2I00− I0− I1

?

(38)

,

where the IR-finite decay widths are

Γcorr(h0

k→˜fi¯˜fj) ≡ Γ(h0

Γcorr(˜f2→˜f1h0

k→˜fi¯˜fj) + Γ(h0

k) + Γ(˜f2→˜f1h0

k→˜fi¯˜fjγ),

(39)

k) ≡ Γ(˜f2→˜f1h0

kγ).

(40)

5Numerical 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:

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Page 9

1. The SPA input, i.e. the on-shell electroweak SM parameters, the strong coupling

constant and the masses of the light quarks defined in the MS scheme and the masses

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

g′

0.36354 M1

103.21

g 0.64804M2

193.29

gs

1.08412M3

572.33

Yτ

0.10349Aτ

−445.4

−532.3

−938.9

10.0

Yt

0.89840At

Yb

0.13548Ab

µ 401.63tanβ

ML1

1.8121 · 102

1.1572 · 102

5.2628 · 102

5.0767 · 102

5.0549 · 102

2.5605 · 104

ML3

1.7945 · 102

1.1002 · 102

4.7091 · 102

3.8532 · 102

5.0137 · 102

−14.725 · 104

ME1

ME3

MQ1

MQ3

MU1

MU3

MD1

MD3

M2

H1

M2

H2

On-shell SUSY Parameters

g′

0.35565M1

100.31

g0.66547M2

197.01

gs

1.08419M3

612.81

Yτ

0.10771Aτ

−394.2

−495.0

1197.8

Yt

1.04638At

Yb

0.20481Ab

µ398.71tanβ10.31

ML1

1.8394 · 102

1.1784 · 102

5.6390 · 102

5.4540 · 102

5.4352 · 102

2.7220 · 104

ML3

1.8199 · 102

1.1172 · 102

5.0369 · 102

4.1021 · 102

5.3894 · 102

−15.726 · 104

ME1

ME3

MQ1

MQ3

MU1

MU3

MD1

MD3

M2

H1

M2

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 Abis large as there is a huge difference between the DR and the on-shell value

of the trilinear scalar coupling parameter. This is caused by fixing the of Abparameter

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

Page 10

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 Aband mbwhich take the original DR values. Although not

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 Abis applied for them as well.

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

mA0 and the relevant soft supersymmetry breaking terms M{˜ Q,˜U,˜D,˜L,˜E}. These parameters

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 Afand mfare taken DR (dash-dotted curve

denotes SPA tree-level and blue solid curve stands for the full one loop decay 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 final one-loop decay width. For comparison 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˜D3= 150 GeV, m2

A0 = 106GeV}

510 15 2025

?0.15

?0.05

0.05

0.15

tanβ

Γ(H0→˜b1¯˜b1) [GeV]

Figure 2: On-shell tree-level (dotted line), full on-shell one-loop decay width (dashed line),

tree-level (dash-dotted line) and full one-loop corrected width (solid line) of H0→˜b1¯˜b1

as a function of tanβ according to the SPA convention.

10

Page 11

Figs. 2, 3 and 4 show the dependence of the decay width on tanβ and clearly demonstrate

the known fact [7, 5] that the counter term for Abgrows with tanβ. As mentioned above

such a large counter term causes the perturbation series to break down. The full one-loop

results in the on-shell scheme (red dashed curve) differ from those where DR parameters

are used (blue solid) only by higher orders. Nevertheless, due to the perturbation series

breakdown, the higher order corrections are no longer suppressed by the coupling con-

stant. This is the reason why the results using the DR parameters should be viewed as

the final one-loop corrected decay widths for the processes calculated in this paper.

{M˜D1= 150 GeV, m2

A0 = 106GeV}

5 10 15

tanβ

20 25

?1

0

1

2

3

Γ(H0→˜d1¯˜d2) [MeV]

Figure 3: On-shell tree-level (dotted line), full on-shell one-loop decay width (dashed line),

tree-level (dash-dotted line) and full one-loop corrected width (solid line) of H0→˜d1¯˜d2

as a function of tanβ according to the SPA convention.

5 1015

tanβ

20 25

0

0.2

0.4

0.6

Γ(H0→ ˜ τ1˜ τ1) [GeV]

Figure 4: On-shell tree-level (dotted line), full on-shell one-loop decay width (dashed line),

tree-level (dash-dotted line) and full one-loop corrected width (solid line) of H0→ ˜ τ1˜ τ1

as a function of tanβ according to the SPA convention.

11

Page 12

The next class of plots shows the dependence on the superpotential parameter µ (see

Figs. 6, 7, 8 and 5). The typical behaviour of the tree-level is governed by the square

of the coupling G

all one-loop corrections in this case are factorizable this dependence is preserved in the

full one-loop result. In case down-type sfermions are involved, the on-shell curves are

deformed by the large difference in the Af parameter. The corrections can reach up to

40% for some areas of parameter space and are therefore not negligible.

˜f

ijk. It implies that the tree-level is a quadratic function of µ and as

{M˜D3= 150 GeV, m2

A0 = 106GeV}

5 10152025

?1

0

1

2

3

tanβ

Γ(H+→˜t1¯˜b2) [GeV]

Figure 5: On-shell tree-level (dotted line), full on-shell one-loop decay width (dashed line),

tree-level (dash-dotted line) and full one-loop corrected width (solid line) of H+→˜t1¯˜b2

as a function of tanβ according to the SPA convention.

{M˜D2= 150 GeV, m2

A0 = 106GeV}

?1000

?5000 5001000

?0.5

0

0.5

1

1.5

µ [GeV]

Γ(H0→ ˜ s1¯˜ s2) [MeV]

Figure 6: On-shell tree-level (dotted line), full on-shell one-loop decay width (dashed line),

tree-level (dash-dotted line) and full one-loop corrected width (solid line) of H0→ ˜ s1¯˜ s2

as a function of µ according to the SPA convention.

12

Page 13

Furthermore, the pseudothreshold in Fig. 7 comes from the sbottom contribution to the

Higgs wave-function correction.

The one-loop width of H+→ ˜ ντ¯˜ τ2 (see Fig. 8) is unexpectedly sensitive to the large

difference of the on-shell and DR Abparameters above µ = 600 GeV although there is

no such parameter at tree-level. It is caused by the enhanced contribution of the vertex

diagram with a 4-sfermion coupling. This diagram contains the coupling of the charged

Higgs boson and a stop-sbottom pair where the Abparameter appears.

{M˜U3= 150 GeV, m2

A0 = 106GeV}

?1000

?5000500 1000

0

2.5

5

7.5

10

12.5

µ [GeV]

Γ(H0→˜t1¯˜t1) [GeV]

Figure 7: On-shell tree-level (dotted line), full on-shell one-loop decay width (dashed line),

tree-level (dash-dotted line) and full one-loop corrected width (solid line) of H0→˜t1¯˜t2as

a function of µ according to the SPA convention.

?2000 200 4006008001000

0

1

2

3

4

µ [GeV]

Γ(H+→ ˜ ντ¯˜ τ2) [GeV]

Figure 8: On-shell tree-level (dotted line), full on-shell one-loop decay width (dashed line),

tree-level (dash-dotted line) and full one-loop corrected width (solid line) of H+→ ˜ ντ¯˜ τ2

as a function of µ according to the SPA convention.

13

Page 14

Fig. 9 illustrates the aforementioned problems of the perturbation series in case of using

the on-shell Abparameter. As one can see there is no obvious divergence of the decay

width in the on-shell scheme. Nevertheless, the full one-loop widths in the on-shell scheme

and the one with Abtaken DR are far apart. This separation is a pure two loop effect

coming from using different Abvalues when calculating the δAbcounter term.

In this particular case the electroweak corrections interfere destructively with the QCD

corrections reducing them by half.

{M˜D3= 150 GeV}

?900 ?600 ?3000 300600 900

0

0.05

0.1

0.15

0.2

0.25

µ [GeV]

Γ(˜b2→˜b1h0) [GeV]

Figure 9: On-shell tree-level (dotted line), full on-shell one-loop decay width (dashed line),

tree-level (dash-dotted line) and full one-loop corrected width (solid line) of˜b2→˜b1h0as

a function of µ according to the SPA convention.

?1000

?5000 5001000

0

0.2

0.4

0.6

0.8

1

At[GeV]

Γ(˜t2→˜t1h0) [GeV]

Figure 10: On-shell tree-level (dotted line), tree-level (dash-dotted line) and full one-

loop corrected width (solid line) of˜t2→˜t1h0as a function of Ataccording to the SPA

convention.

14

Page 15

The only plot over the trilinear coupling Af is shown in Fig. 10 which is for the decay

˜t2→˜t1h0. The region At= (−120 GeV,320 GeV) is kinematically forbidden. Although

at some regions of parameter space (e.g. At= (700 GeV,900 GeV)) the correction to the

SPA tree-level is large, one can see there is also a large difference between SPA tree-level

and the tree-level used in the perturbation expansion denoted by the dotted line.

6Conclusions

We have calculated the full electroweak one-loop corrections to the charged and CP-even

neutral Higgs boson decays to sfermions including the crossed channels. We have also

included the SUSY-QCD corrections which were calculated in [7]. Similar to [5] and [6],

the on-shell parameters Aband mb(and the corresponding down-type parameters for the

first two generations) were replaced by their DR values to avoid the numerically large

counter term. Furthermore, we have presented the first consistent numerical analysis for

a one-loop decay width based on the Supersymmetric Parameter Analysis project [9].

This required the renormalization of the whole MSSM in a way that allows to carry out a

transformation between the on-shell and the DR scheme for every single parameter point.

The corrections were shown not to be negligible and were comparable in the magnitude

to the QCD in some regions of parameter space.

Acknowledgements

We thank W. Porod for his generous support in including SPheno in our numerical calcu-

lations and for useful discussions. The authors acknowledge support from EU under the

MRTN-CT-2006-035505 network programme and from the ”Acciones Integradas 2005-

2006”, project No. 13/2005. This work is supported by the ”Fonds zur F¨ orderung der

wissenschaftlichen Forschung” of Austria, project No. P18959-N16.

15

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