Effective Lagrangian for the $\chi^{+}_j \chi^{-}_kH^{0}_l$ interaction
  in the minimal supersymmetric standard model and neutral Higgs decays
arXiv:0704.1913v3  [hep-ph]  31 Oct 2007
Effective Lagrangian for the χ+j χ
l interaction in the
minimal supersymmetric standard model and neutral
Higgs decays
Tarek Ibrahim
Department of Physics, Northeastern University, Boston, MA 02115-5000, USA
Department of Physics, Faculty of Science, University of Alexandria, Alexandria,
Egypt 1
Abstract
We extend previous analyses of the supersymmetric loop correction to the neutral
Higgs couplings to include the coupling χ+j χ
l . The analysis completes the pre-
vious analyses where similar corrections were computed for the τ̄ τH0l , b̄bH
l , c̄cH
and for t̄tH0l couplings within the minimal supersymmetric standard model. The
effective one loop Lagrangian is then applied to the computation of the neutral
Higgs decays. The sizes of the supersymmetric loop corrections of the neutral
Higgs decay widths into χ+i χ
j (i = 1, 2; j = 1, 2) are investigated and the super-
symmetric loop correction is found to be in the range of 7 ∼ 15% in significant
regions of the parameter space. By including the loop corrections of the other
decay channels b̄b, t̄t, τ̄ τ , c̄c, and χ0iχ
j (i = 1 − 4; j = 1 − 4), the corrections to
branching ratios for H0l → χ+i χ−j can reach as high as 40%. The effects of CP
phases on the branching ratio are also investigated.
Permanent address.
http://arxiv.org/abs/0704.1913v3
1 INTRODUCTION
The neutral Higgs couplings to different fields are of great current interest as they
enter in a variety of phenomena which are testable in low energy processes [1]. It is
known that supersymmetric corrections can affect the neutral Higgs boson decays
into bb̄, τ τ̄ and cc̄. The decay properties of the lightest Higgs boson in MSSM would
be different from those of the Standard Model Higgs boson when these corrections
are taken into consideration. Specifically the ratio of the branching ratios to bb̄
and τ τ̄ of the Higgs boson is an important piece of evidence that might distinguish
between the lightest MSSM Higgs boson and the Standard Model one at colliders.
In MSSM there are also other modes for neutral Higgs decays that do not exist in
Standard Model such as charginos and neutralinos.
In this paper we compute the one loop corrected effective Lagrangian for the
neutral Higgs and chargino couplings. We then analyze the effects of the loop
corrections to the neutral Higgs decays H0l → χ+j χ−k . In the analysis we also
include the effect of CP phases arising from the soft SUSY breaking parameters.
It is well known that large CP phases can be made compatible [2, 3, 4] with
experimental constraints on the electric dipole moments (edms) of the electron [5],
of the neutron [6], and of the Hg199 [7]. Further, if the phases are large they could
affect the Higgs sector physics. It is well known that one loop contributions to
the Higgs masses from the stop, sbottom, the chargino and neutralino sectors can
lift the lightest Higgs mass above MZ . The inclusion of the CP violating phases
brings mixings between the CP even and the CP odd Higgs [8, 9, 10, 22, 23, 24].
The CP violating phases modifies the physics of dark matter [11], and of other
phenomena [12]. (For a review see Ref.[13].)
The current analysis of ∆LH0χ+χ− and neutral Higgs decay into charginos is
based on the effective Lagrangian method where the couplings of the electroweak
eigen states H11 and H
2 with charginos are radiatively corrected using the zero
external momentum approximation. The same technique has been used in calcu-
lating the effective Lagrangian and decays ofH0l into quarks and leptons [1, 15, 16].
It has been used also in the analysis of the effective Lagrangian of charged Higgs
with quarks [1, 17] and their decays into t̄b and νττ [18] and into chargino + neu-
tralino [19]. The neutral Higgs decays into charginos have been investigated before
in the CP conserving case [20, 21]. In these analyses, the wave function renormal-
ization and the counter terms for the mass matrix elements are calculated beside
the vertex corrections of the mass eigen states h0, H0 and A0 with charginos. In
the effective Lagrangian technique with zero external momentum approximation,
the radiative corrections of the processes considered here originate only from the
vertex contributions. Thus our analysis of the neutral Higgs decays into charginos
is a partial one. However, as mentioned before the above analyses were carried out
in the CP conserving scenario. As far as we know, the analysis for the neutral Higgs
decays into charginos, with one loop corrections, in the CP violating case where
the neutral Higgs sector is modified in couplings, spectrum and mixings, does not
exist. We evaluate the radiative corrections to the Higgs boson masses and mixngs
by using the effective potential approximation. We include the corrections from
the top and bottom quarks and squarks [22], from the chargino, the W and the
charged Higgs sector [23] and from the neutralino, Z boson, and the neutral Higgs
bosons [24]. It is important to notice that the corrections to the Higgs effective
potential from the different sectors mentioned above are all one-loop corrections.
The corrections of the interaction ∆LH0χ+χ− to be considered in this work are all
one-loop level ones. So the analysis presented here is a consistent one loop study.
The outline of the rest of the paper is as follows: In Sec. 2 we compute the
effective Lagrangian for the χ+j χ
l interaction. In Sec. 3 we give an analysis
of the decay widths of the neutral Higgs bosons into charginos using the effective
Lagrangian. In Sec. 4 we give a numerical analysis of the size of the loop effects
on the partial decay width and on the branching ratios. Conclusions are given in
Sec. 5.
2 LOOP CORRECTIONS TO NEUTRAL HIGGS
COUPLINGS
The tree-level Lagrangian for χ+j χ
0 interaction is
L = φjkχ+j PRχ+kH11 + ψjkχ+j PRχ+kH22 +H.c., (1)
where H11 andH
2 are the neutral states of the two Higgs isodoublets in the minimal
supersymmetric standard model (MSSM), i.e.,
(H1) =
, (H2) =
and the couplings φjk and ψjk are given by
φjk = −gUk2Vj1, ψjk = −gUk1Vj2 (3)
where U and V diagonalize the chargino mass matrix so that
U∗Mχ+V
−1 = diag(mχ+
, mχ+
) (4)
The loop corrections produce shifts in the couplings of Eq. (1) and the effective
Lagrangian with loop corrected couplings is given by
Leff = (φjk + δφjk)χ+j PRχ+kH11 +∆φjkχ+j PLχ+kH22 +
(ψjk + δψjk)χ
j PRχ
2 +∆ψjkχ
j PLχ
1 +H.c. (5)
In this work we calculate the loop correction to the χ+j χ
l using the zero external
momentum approximation.
2.1 Loop analysis of δφjk and ∆ψjk
Contributions to δφjk and ∆ψjk arise from the thirteen loop diagram of Fig. 1.
We note that the contribution from diagrams which have H+W+H0and H0Z0H0
vertices do not contribute in the effective Lagrangian with zero external momentum
approximation since these vertices are proportional to the external momentum. We
discuss now in detail the contribution of each of these diagrams in Fig. 1. We
begin with the loop diagram of Fig. 1i(a) which contributes to δφjk and ∆ψjk.
We calculate the corrections of the amplitude from Fig. 1i(a)
δM = iδφjkūjPRvk + i∆ψjkūjPLvk (6)
The idea is to extract, from the amplitude correction, the expressions for δφjk and
∆ψjk from those parts that are proportional to ūjPRvk and ūjPLvk respectively.
For this purpose we need b̃b̃H11 interaction which is given by
Lb̃b̃H1
= Hilb̃
1 +H.c. (7)
where Hil is given by
Hil = −
2 cos θW
sin2 θW )D
b1iDb1l −
sin2 θWD
b2iDb2l) cos β
gm2b√
2mW cos β
(D∗b1iDb1l +D
b2iDb2l)−
gmbAb√
2mW cos β
D∗b2iDb1l (8)
The matrix elements Dq are defined as
D+q M
q̃Dq = diag(m
) (9)
We need also the t̄χ+b̃ interaction which is given by
Lt̄χ+b̃ = −gχ̄
k [(U
− κbU∗k2D∗b2i)PL
−κtVk2D∗b1iPR]tb̃
i +H.c (10)
where κt,b are given by
2mW sin β
2mW cos β
For external momenta s, q and q − s the amplitude correction from loop 1i(a) is
given by
δM = −g2Hilū(q − s)[CLjlPL + CRjlPR]
(2π)4
[( 6s+ 6ℓ) +mt][C∗LkiPR + C
PL]v(s)
((s+ ℓ)2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)((ℓ+ q)
2 −m2
+ iǫ)
where CLjl and CRjl are given by
CLjl = U
− κbU∗j2D∗b2l
CRjl = −κtVj2D
The part in the numerator
[CLjlPL + CRjlPR][( 6s+ 6ℓ) +mt]
(C∗LkiPR + C
PL) (14)
could be written as
[CLjlC
PL + CRjlC
PR]( 6s+ 6ℓ)
+mt[CRjlC
PR + CLjlC
PL] (15)
by using the facts that γµPL = PRγ
µ, PLPR = 0, P
L = PL and P
R = PR. The
first term in Eq. (15) does not contribute to δφjk or ∆ψjk since it does not have
the same Lorentz structure. The second term of Eq. (15) contributes the part of
mtCRjlC
to δφjk and mtCLjlC
to ∆ψjk. Thus the loop corrections δφjk and
∆ψjk read
iδφjk = −g2HilmtCRjlC
i∆ψjk = −g2HilmtCLjlC
where
(2π)4
((s+ ℓ)2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)((ℓ+ q)
2 −m2
+ iǫ)
Now for zero external momentum approximation we set s = q = 0, and the integral
would read
(2π)4
(ℓ2 −m2t + iǫ)(ℓ2 −m2b̃l + iǫ)(ℓ
2 −m2
+ iǫ)
A detailed calculation of this integral is given in the appendix.
Using the above one finds for δφjk the contribution:
jk = κt
HilVj2D
(Uk1Db1i − κbUk2Db2i)f(m2t , m2b̃l, m
) (19)
where
f(x, y, z) =
(x− y)(x− z)(z − y)
× (zxln
+ xyln
+ yzln
), (20)
f(x, y, y) =
(y − x)2
× (xln
+ x− y) (21)
Similarly one finds for the correction ∆ψjk from the same loop the following con-
tribution
jk = κt
k2Db1i(U
− κbU∗j2D∗b2l)f(m
t , m
) (22)
Next for the loop Fig. 1ii(a) we find
jk = 0
jk = 0 (23)
For the loop of Fig. 1i(b) we find
jk = κb
FliUk2D
(Vj1Dt1l − κtVj2Dt2l)f(m
b , m
, m2t̃l)
jk = κb
j2Dt1l(V
− κtV ∗k2D∗t2i)f(m
b , m
, m2t̃l) (24)
where Fli is given by
Fli = −
2 cos θW
sin2 θW )D
t1lDt1i +
sin2 θWD
t2lDt2i) cos β
gmtµ√
2mW sin β
D∗t1lDt2i (25)
For the loop of Fig. 1ii(b) we find
jk = 0
jk = −κb
g2m2b
U∗j2Dt1i(V
− κtV ∗k2D∗t2i)f(m
b , m
b , m
) (26)
For loop of Fig. 1ii(c) we find
jk = 2g
ik sin βǫ
lj cos β
f(m2χ0
, m2χ0
, m2H+)
jk = 0 (27)
where ǫ
and ǫ are given by
ǫji = −gX4jV ∗i1 −
tan θWX1jV
ji = −gX∗3jUi1 +
X∗2jUi2 +
tan θWX
1jUi2 (28)
The parameters Q
ij are defined as:
[X∗3i(X
2j − tan θWX∗1j)] (29)
The matrix elements X are defined as
XTMχ0X = diag(mχ0
, mχ0
, mχ0
, mχ0
) (30)
For loop of Fig. 1i(c) we find
gmW cos β
[1 + 2 sin2 β − cos 2β tan2 θW ]
ik sin βǫ
ij cos β
f(m2χ0
, m2H+ , m
gmW cos β
[1 + 2 sin2 β − cos 2β tan2 θW ]
ǫik cos βǫ
ij sin β
f(m2χ0
, m2H+ , m
H+) (31)
For loop of Fig. 1i(d) we find
jk = g
3 mZ cos β
2 cos θW
((Ym1 − iYm3 sin β)(3Yl1 + iYl3 sin β)
−2(Ym2 − iYm3 cos β)(Yl2 + iYl3 cos β)− 4Ym2(Yl1 − iYl3 sin β) tanβ)
(Qki(Yl1 + iYl3 sin β) + Ski(Yl2 + iYl3 cos β))
(Qij(Ym1 + iYm3 sin β) + Sij(Ym2 + iYm3 cos β))
, m2H0m , m
jk = g
3 mZ cos β
2 cos θW
((Ym1 − iYm3 sin β)(3Yl1 + iYl3 sin β)
−2(Ym2 − iYm3 cos β)(Yl2 + iYl3 cos β)− 4Ym2(Yl1 − iYl3 sin β) tanβ)
(Q∗ik(Yl1 − iYl3 sin β) + S∗ik(Yl2 − iYl3 cos β))
(Q∗ji(Ym1 − iYm3 sin β) + S∗ji(Ym2 − iYm3 cos β))
, m2H0m , m
) (32)
where Qji = − 1√2gφij and Sji =
ψij , and the matrix elements Y are defined as
YM2HiggsY
T = diag(m2
For loop of Fig. 1ii(d) we find
jk = −g2
(Qli(Ym1 + iYm3 sin β) + Slj(Ym2 + iYm3 cos β))(Qki(Ym1 + iYm3 sin β)
+Ski(Ym2 + iYm3 cos β))
, m2H0m , m
jk = 0 (33)
For loop of Fig. 1ii(e) we find
jk = 0
cos2 θW
, m2Z0 , m
) (34)
The parameters L′ and R′ are defined by
ij = −Vi1V ∗j1 −
j2 + δij sin
ij = −U∗i1Uj1 −
U∗i2Uj2 + δij sin
2 θW (35)
For loop of Fig. 1i(e) we find
jk = −
2g3mZ cos β
cos3 θW
L′jiR
, m2Z0 , m
jk = −
2g3mZ cos β
cos3 θW
R′jiL
, m2Z0 , m
) (36)
For loop of Fig. 1ii(f) we find
jk = 0
jk = −4
Q”ilR
ljLik
f(m2χ0
, m2W+, m
) (37)
where L, R and Q” are defined as
Lij = −
X∗4iV
j2 +X
Rij =
X3iUj2 +X2iUj1
gQ” =
(X∗3i(gX
2j − g′X∗1j) + (i↔ j)) (38)
For loop of Fig. 1i(f) we find
jk = −
4g3mW cos β√
L∗ijRik
f(m2χ0
, m2W+, m
jk = −
4g3mW cos β√
R∗ijLik
f(m2χ0
, m2W+, m
W+) (39)
For loop of Fig. 1ii(g) we find
jk = 0
jk = −g2hτκτU∗j2V ∗k1
f(m2τ , m
τ , m
) (40)
where
2mW cos β
The loop corrections for δφjk and ∆ψjk are given by
δφjk =
∆ψjk =
jk (42)
2.2 Loop analysis of ∆φjk and δψjk
We do the same analysis of Fig. 2 as for Fig. 1. We write down here the final
results for both corrections from the thirteen loops together. The corrections are
written in the same order of the loops in Fig. 2.
∆φjk = κt
k2Db1i(U
− κbU∗j2D∗b2l)f(m
t , m
−κtht
g2m2t
V ∗k2Db1i(U
− κbU∗j2D∗b2i)f(m
t , m
t , m
j2Dt1l(V
− κtV ∗k2D∗t2i)f(m
b , m
, m2t̃l)
gmW sin β
[1 + 2 cos2 β + cos 2β tan2 θW ]
ǫik cos βǫ
ij sin β
f(m2χ0
, m2H+ , m
mZ cos β
2 cos θW
(tanβ(Yl2 − iYl3 cos β)(3Ym2 + iYm3 cos β)
−4Yl1(Ym2 − iYm3 cos β)− 2 tanβ(Ym1 − iYm3 sin β)(Yl1 + iYl3 sin β))
(Q∗ik(Yl1 − iYl3 sin β) + S∗ik(Yl2 − iYl3 cos β))
(Q∗ji(Ym1 − iYm3 sin β) + S∗ji(Ym2 − iYm3 cos β))
, m2H0m , m
cos2 θW
, m2Z0 , m
2g3mZ sin β
cos3 θW
R′jiL
, m2Z0 , m
S”ilR
ljLik
f(m2χ0
, m2W+, m
4g3mW sin β√
R∗ijLik
f(m2χ0
, m2W+, m
+0 (43)
where G and E are given by
Gij =
2 cos θW
sin2 θW )D
b1iDb1j −
sin2 θWD
b2iDb2j) sin β
gmbµ√
2mW cos β
D∗b1iDb2j
Eij =
2 cos θW
sin2 θW )D
t1iDt1j +
sin2 θWD
t2iDt2j) sin β
2mW sin β
(D∗t1iDt1j +D
t2iDt2j)−
gmtAt√
2mW sin β
D∗t2iDt2j (44)
and S” is given by
S”li = −
sin β
δli −Q”li cos β − R”li)
R”li =
∗X∗1lX
1i + m̃2
∗X∗2lX
2i − µ∗(X∗3lX∗4i +X∗4lX∗3i)) (45)
The corrections δψjk are given by
δψjk = κt
GilVj2D
(Uk1Db1i − κbUk2Db2i)f(m2t , m2b̃l , m
EliUk2D
(Vj1Dt1l − κtVj2Dt2l)f(m
b , m
, m2t̃l)
ik sin βǫ
lj cos β
f(m2χ0
, m2χ0
, m2H+)
gmW sin β
[1 + 2 cos2 β + cos 2β tan2 θW ]
ik sin βǫ
ij cos β
f(m2χ0
, m2H+ , m
mZ cos β
2 cos θW
(tanβ(Yl2 − iYl3 cos β)(3Ym2 + iYm3 cos β)
−4Yl1(Ym2 − iYm3 cos β)− 2 tanβ(Ym1 − iYm3 sin β)(Yl1 + iYl3 sin β))
(Qki(Yl1 + iYl3 sin β) + Ski(Yl2 + iYl3 cos β))
(Qij(Ym1 + iYm3 sin β) + Sij(Ym2 + iYm3 cos β))
, m2H0m , m
(Qlj(Ym1 + iYm3 sin β) + Slj(Ym2 + iYm3 cos β))(Qki(Ym1 + iYm3 sin β)
+Ski(Ym2 + iYm3 cos β))
, m2H0m , m
2g3mZ sin β
cos3 θW
L′jiR
, m2Z0 , m
3mW sin β√
L∗ijRik
f(m2χ0
, m2W+, m
+0 (46)
where S ′ is given by
[X∗4j(X
2i − tan θWX∗1i)] (47)
3 Neutral Higgs decays including loop effects
We summarize now the result of the analysis. Thus Leff of Eq.(5) may be written
as follows
Leff = H0l χ+j (αlSjk + γ5αlPjk)χ+k +H.c (48)
where
αlSjk =
((Yl1+iYl3 sin β)(φjk+δφjk+∆ψjk)+(Yl2+iYl3 cos β)(ψjk+δψjk+∆φjk))
and where
αlPjk =
((Yl1+iYl3 sin β)(φjk+δφjk−∆ψjk)+(Yl2+iYl3 cos β)(ψjk+δψjk−∆φjk))
Next we discuss the implications of the above result for the decay of the neutral
Higgs.
Γljk(H
l → χ+j χ−k ) =
)2 − 4m2
((|αlSjk|)2 + (|αlPjk |)2)(M2H0
((|αlSjk|)2 − (|αlPjk |)2)(2mχ+
)])(51)
There are many channels for H0l decays. The important channels for the decay
of the neutral Higgs boson are b̄b, t̄t, s̄s, c̄c, τ̄ τ , χ+i χ
j and χ
j . There is another
set of channels that neutral Higgs can also decay into: these are modes of decaying
into the other fermions of the SM, squarks, sleptons, other Higgs bosons, W and
Z boson pairs, one Higgs and a vector boson, γγ pairs and finally into the gluonic
decay i.e, H0l → gg. We neglect the lightest SM fermions for the smallness of their
couplings. We choose the region in the parameter space where we can ignore the
other channels which either are not allowed kinematically or suppressed by their
couplings. Thus in this work, squarks and sleptons are too heavy to be relevant
in neutral Higgs decay. The neutral Higgs decays into nonsupersymmetric final
states that involve gauge bosons and/or other Higgs bosons are ignored as well.
In the region of large tan β, these decays typically contribute less than 1% of the
total Higgs decay rate [25]. Thus we can neglect these final states.
We calculate the radiative corrected partial decay widths of the important
channels mentioned above. In the case of CP violating case under investigation
we use for the radiatively corrected Γ of neutral Higgs into quarks and leptons the
analysis of [16], for the radiatively corrected partial widths into charginos we use
the current analysis, and for the radiatively corrected decay width into neutralino
we use [26]. We define
Γ(H0l → χ+i χ−j )− Γ0(H0l → χ+i χ−j )
Γ0(H0l → χ+i χ−j )
where the first term in the numerator is the decay width including the full loop
corrections and the second term is the decay width evaluated at the tree level.
Finally to quantify the size of the loop effects on the branching ratios of the
neutral Higgs decay we define the following quantity
Br(H0l → χ+i χ−j )− Br0(H0l → χ+i χ−j )
Br0(H0l → χ+i χ−j )
where the first term in the numerator is the branching ratio including the full loop
corrections and the second term is the branching ratio evaluated at the tree level.
The analysis of this section is utilized in Sec.(4) where we give a numerical analysis
of the size of the loop effects and discuss the effect of the loop corrections on decay
widths and branching ratios.
4 NUMERICAL ANALYSIS
In this section we discuss in a quantitative fashion the size of loop effects on the
partial decay width and the branching ratios of the neutral Higgs bosons into
charginos. The analysis of Sec. 2 is quite general and valid for the minimal su-
persymmetric standard model. For the sake of numerical analysis we will limit
the parameter space by working within the framework of the SUGRA model [14].
Specifically we will work within the framework of the the extended mSUGRA
model including CP phases. We take as our parameter space at the grand unifica-
tion scale to be the following: the universal scalar mass m0, the universal gaugino
mass m1/2, the universal trilinear coupling |A0|, the ratio of the Higgs vacuum ex-
pectation values tanβ =< H2 > / < H1 > where H2 gives mass to the up quarks
and H1 gives mass to the down quarks and the leptons. In addition, we take for CP
phases the following: the phase θµ of the Higgs mixing parameter µ, the phase αA0
of the trilinear coupling A0 and the phases ξi(i = 1, 2, 3) of the SU(3)C , SU(2)L
and U(1)Y gaugino masses. In this analysis the electroweak symmetry is broken by
radiative effects which allows one to determine the magnitude of µ by fixing MZ .
In the analysis we use one loop renormalization group (RGEs) equations for the
evolution of the soft susy breaking parameters and for the parameter µ, and two
loop RGEs for the gauge and Yukawa couplings. In the numerical analysis we com-
pute the loop corrections and also analyze their dependence on the phases. The
masses of particles involved in the analysis are ordered as follows: for charginos
< mχ+
and for the neutral Higgs (mH1 , mH2, mH3) → (mH , mh, mA) in the
limit of no CP mixing where mH is the heavy CP even Higgs, mh is the light CP
even Higgs, and mA is the CP odd Higgs.
We investigate the question of how large loop corrections are relative to the
tree values. We first discuss the magnitude of the loop corrections of the partial
decay width defined in Eq.(52). As we mentioned earlier the loop corrections to
the partial decay width of the chargino channel have been investigated before in
the CP conserving case [20, 21]. The correction in these analyses is of the order of
∼ 10% of the tree level value. Our analysis supports this conclusion. In Figs. (3)
and (4) we give a plot of ∆Γ
l (l = 1, 3) as a function of tan β for the specific set
of inputs given in the captions of these figures. We notice that the partial decay
width gets a change of 7 ∼ 15% of its tree level value. We also notice that the
CP violating phase θµ can affect the magnitude of this change. This effect has not
been addressed in the previous analyses as they are working in the CP conserving
scenario. To compare between our analysis and the previous ones we have to notice
that these analyses are using the general SUSY parameter space where they put
by hand all the parameters that control the analysis. In [20], the authors choose
the SUSY parameter set SPS1a of the Snowmass Points and Slopes as a reference
point. They choose for the trilinear couplings the values of At = −487 GeV,
Ab = −766 GeV and Aτ = −250 GeV. The values of the other parameters are:
M = 197.6 GeV, M ′ = 98 GeV, µ = 353.1 GeV, tan β = 10, mA0 = 393.6 GeV,
MQ̃1,2 = 558.9 GeV, MŨ1,2 = 540.5 GeV, MD̃1,2 = 538.5 GeV, ML̃1,2 = 197.9 GeV,
MẼ1,2 = 137.8 GeV, MQ̃3 = 512.2 GeV, MŨ3 = 432.8 GeV, MD̃3 = 536.5 GeV,
ML̃3 = 196.4 GeV and MẼ3 = 134.8 GeV. In all the figures of [20], these values
are used, if not specified otherwise. In our mSUGRA analysis the magnitude of all
these parameters and others are fixed by the five input parameters m0 = 100 GeV,
m1/2 = 250 GeV, tanβ = 10, A0 = −100 GeV and a positive sign of µ in the CP
conserving scenario [27]. These parameters are different from those of our Figs.
(3) and (4). By using these parameters and fixing some of them by hand when
needed to match their values in the analysis of [20], we were able to have a fair
agreement with their Figs. (2-9). As an example of this check we show in Table.1
a comparison of the two works. For the input of Fig. 2 of [20] with CP violating
phases are set to zero we can see that partial decay widths in both works have
the same behavior as functions of masses and their magnitudes are fairly close to
each other. However it seems that our loop corrected values of the partial widths
are different from those of Eberl et al. This could be understood since our loop
analysis of the effective lagrangian includes only the vertex corrections beside the
corrections in the Higgs potential.
case Γtreeeberl Γ
our Γ
eberl Γ
2.a mA0 = 700 GeV 0.95 GeV 0.94 GeV 0.85 GeV 0.80 GeV
2.a mA0 = 800 GeV 1.18 GeV 1.17 GeV 1.0 GeV 0.91 GeV
2.b mH0 = 800 GeV 0.7 GeV 0.69 GeV 0.63 GeV 0.58 GeV
2.b mH0 = 900 GeV 0.8 GeV 0.8 GeV 0.73 GeV 0.70 GeV
Table 1: A comparison between the current analysis and Eberl et al [20] for bench-
mark cases.
In the work of Ref. [21] only 8 out of 26 diagrams of the present analysis
are calculated and they correspond to the vertex corrections from Figs. (1,2ii(a)),
(1,2ii(b)), (1,2i(b)) and (1,2i(a)). By considering these diagrams only in the com-
parison, our analysis is in fair agreement with their Figs (2-4) and Figs. (6,8) for
their inputs.
Now we turn to address the question of how much loop corrections can affect
the branching ratios into charginos. The branching ratio of a decay mode is defined
to be the ratio between the partial decay rate of this mode and the total decay
rate. In the parameter space under investigation this total decay rate includes the
rates of decays into charginos, heavy quarks, taus and neutralinos. In Figs. (5)
and (6) we give a plot of ∆Br
l (l = 1, 3) defined by Eq.(53) as a function of tanβ
for the specific set of inputs given in the captions of these figures. Fig. (5) is for
the neutral Higgs H1 boson and Fig. (6) is for the neutral Higgs H3 boson. In all
regions of the parameter space investigated in this work, the decay of the lightest
Higgs boson H2 into charginos is forbidden kinematically, since we have in these
regions the fact that 2mχ−
> mH2 . The analysis of Figs. (5) and (6) shows that
the loop correction varies strongly with tanβ with the correction changing sign
for the case of H3 decay. Further, the analysis shows that the loop correction can
be as large as about −40% of the tree contribution for both H1 and H3 cases. We
also notice that the behavior of ∆Br
l (l = 1, 3) as a function of tan β changes
considerably by changing the phase of µ. So for some values of this phase we
find that this parameter increases as tanβ increases and for other values of θµ
we see that it decreases as tan β increases. As shown in the previous figures, the
parameter tan β is playing a strong role. This parameter is important at the tree
level through the diagonalizing mass matrices of the chargino and neutral Higgs
and their spectrum. At the loop level it has extra effect explicitly in α
jk and
implicitly through the radiatively corrected matrix elements Ylm and through the
corrections δφjk, ∆φjk, δψjk, ∆ψjk. The values of the branching ratios themselves
at tree and one loop levels are shown in Table.2.
θµ(rad) Br
0(H1) Br
loop(H1) Br
0(H3) Br
loop(H3)
0.5 6% 4.7% 18.2% 13.8%
1.0 8.4% 6.9% 21.3% 18.1%
1.5 9.2% 7.9% 23.4% 22.2%
Table 2: Values of branching ratios at tree and one-loop levels of neutral Higgs
into the channel χ+1 χ
1 at tan β = 24 for the input of Figs. (5) and (6)
We notice that their magnitudes are not negligible for the region of the param-
eter space investigated. These non negligible branching ratios for the decay of the
neutral Higgs into charginos suggest that these decay modes could be measurable
at the soon-to-operate LHC. However, one should also consider the production
rates for H1 and H3 bosons to assess whether the change in branching ratios could
be detectable at colliders. This analysis goes beyond the scope of the current work.
We also notice that the phase of the parameter µ affects the tree level branching
ratios as well. This comes mainly from the structure of the chargino matrix. The
more important channels in the region of the parameter space investigated are the
decay into bottom and top quarks. They have the highest values of branching
ratios. The radiative corrections of these channels are also more than those of
the charginos and neutralinos. These channels were studied before [1, 15, 16] as
mentioned above. However a 20% of branching ratio for the case of neutral Higgs
as shown in the above table is not very small and could justify carrying out the
current analysis.
In Figs. (7) and (8) we give a plot of ∆Br
l (l = 1, 3) as a function of |A0|
for the specific set of inputs given in the caption of these figures. The analysis of
these figures shows that the loop corrections are substantial and reaches the value
of −38% of the tree contribution for the case of H1 decay and the value of −43%
for the case of H3 decay.
Next we investigate the effects of CP violating phases on the loop corrections
of the neutral Higgs decays into charginos. In Figs. (9) and (10) we give a plot
of ∆Br
l (l = 1, 3) as a function of θµ for the specific set of inputs given in the
caption of these figures. The analysis of the figures shows that the loop correction
has a sharp dependence on θµ. Further, the correction is changing sign as θµ varies
from 0 to π for two cases of H3 decay. Thus θµ affects not only the magnitude of
l but also its sign depending on the value of θµ.
In Figs. (11) and (12) we give a plot of ∆Br
l (l = 1, 3) as a function of αA0
for the specific set of inputs given in the caption of these figures. Here also we find
a very substantial dependence of ∆Br
l on αA0. This dependence is very large in
the case of H3 decay and it exceeds −40% of the tree contribution.
In Figs. (13) and (14) we give a plot of ∆Br
l (l = 1, 3) as a function of ξ2 for
the specific set of inputs given in the caption of these figures. Here we find a small
effect of this phase on the loop corrections.
5 CONCLUSION
In this paper we have carried out an analysis of the supersymmetric loop correc-
tions to χ+j χ
l couplings within MSSM. In supersymmetry after spontaneous
breaking of electroweak symmetry one is left with three neutral Higgs bosons
which in the absence of CP phases consist of two CP even Higgs bosons and one
CP odd Higgs boson. In the absence of loop corrections, the lightest Higgs bo-
son mass satisfies the inequality mh < MZ and by including these corrections the
lightest Higgs mass can be lifted above MZ . With the inclusion of CP phases
the Higgs boson mass eigenstates are no longer CP even and CP odd states when
loop corrections to the Higgs boson mass matrix are included. Further, inclusion
of loop corrections to the couplings of charginos and neutral Higgs is in general
dependent on CP phases. Thus the decays of neutral Higgs into charginos can be
sensitive to the loop corrections and to the CP violating phases. The effect of the
supersymmetric loop corrections is found to to be in the range of 7 ∼ 15% for the
partial decay width. For the branching ratios it is found to be be rather large, as
much as 40% in some regions of the parameter space. The effect of CP phases on
the modifications of the partial decay width and the branching ratio is found to
be substantial in some regions of the MSSM parameter space.
Acknowledgments
I wish to acknowledge useful discussions with Professor Pran Nath. The support
of the Physics Department at Alexandria University is also acknowledged.
6 APPENDIX
The integral of import to this work is
(2π)4
(k2 −m21 + iǫ)(k2 −m22 + iǫ)(k2 −m23 + iǫ)
It could be written in the form
(2π)4
where
a = k2 −m21 + iǫ
b = k2 −m22 + iǫ
c = k2 −m23 + iǫ (56)
Using Feynman parametrization, 1
could be written as
∫ 1−x
[a + (b− a)x+ (c− a)z]3
The denominator in the above integral could be written in the form k2 +M2 + iǫ
where M2 = (m21 − m22)x + (m21 − m23)z −m21. Thus the integral J can take the
(2π)4
∫ 1−x
[k2 +M2 + iǫ]3
Now integrating over k and using the standard integral, for n ≥ 3
(2π)4
(k2 + Λ + iǫ)n
= iπ2
Γ(n− 2)
one can find that the integral J has the form
(4π)2
∫ 1−x
α + βz
where α = (m21 −m22)x−m21 and β = m21 −m23. Integrating over z one can get for
the integral J the form of
(4π)2
m21 −m23
dx ln(δ1x−m23)− ln(δ2x−m21) (61)
where δ1 = m
3 −m22 and δ2 = m21 −m22. Finally we integrate over x to get for J
the form of
(4π)2
f(m21, m
3) (62)
where
f(m21, m
m21 −m23
m23 −m22
m21 −m22
×[m22m23 ln(
) +m23m
1 ln(
) +m21m
2 ln(
)] (63)
This is the famous form factor that appears in the analysis of the radiative correc-
tions for the quark and lepton masses [28], the decay rates of neutral and charged
Higgs into quarks and leptons [1, 15, 16, 18] and in the b→ sγ process [17]. In the
latter process, the authors are using different form factor H(
). This form
factor could be easily converted to our f(m21, m
3) through the simple relation
m23f(m
3) = H(
) (64)
For the case where two of the masses are equal, m2 = m3, one can repeat the
same analysis with b = c = k2 −m23 + iǫ and a = k2 −m21 + iǫ. By doing so one
can get for the form factor J
(4π)2
(m23 −m21)2
[m21 ln(
) +m21 −m23] (65)
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s2, v2s1, v1
jf2f1
Figure 1: Set of diagrams contributing to radiative corrections δφjk and ∆ψjk. (i):
(a) s1 = b̃
i , s2 = b̃
l , f = t; (b) s1 = t̃i, s2 = t̃l, f = b̄; (c) s1 = H
+, s2 = H
f = χ0i ; (d) s1 = H
l , s2 = H
m, f = χ
i ; (e) v1 = Z
0, v2 = Z
0, f = χ+i ; (f)
v1 = W
+, v2 = W
+, f = χ0i . (ii): (a) f1 = t, f2 = t, s = b̃
i ; (b) f1 = b̄, f2 = b̄,
s = t̃i; (c)f1 = χ
i , f2 = χ
l , s = H
+; (d) f1 = χ
i , f2 = χ
l , s = H
m; (e) f1 = χ
f2 = χ
l , v = Z
0; (f) f1 = χ
i , f2 = χ
l , v = W ; (g) f1 = τ
+, f2 = τ
+, s = ν̃τ .
s2, v2s1, v1
jf2f1
Figure 2: Set of diagrams contributing to radiative corrections ∆φjk and δψjk. (i):
(a) s1 = b̃
i , s2 = b̃
l , f = t; (b) s1 = t̃i, s2 = t̃l, f = b̄; (c) s1 = H
+, s2 = H
f = χ0i ; (d) s1 = H
l , s2 = H
m, f = χ
i ; (e) v1 = Z
0, v2 = Z
0, f = χ+i ; (f)
v1 = W
+, v2 = W
+, f = χ0i . (ii): (a) f1 = t, f2 = t, s = b̃
i ; (b) f1 = b̄, f2 = b̄,
s = t̃i; (c)f1 = χ
i , f2 = χ
l , s = H
+; (d) f1 = χ
i , f2 = χ
l , s = H
m; (e) f1 = χ
f2 = χ
l , v = Z
0; (f) f1 = χ
i , f2 = χ
l , v = W ; (g) f1 = τ
+, f2 = τ
+, s = ν̃τ .
Figure 3: tanβ dependence of ∆Γ1 → χ+1 χ−1 . The curves in ascending order
correspond to θµ = 0.2, 0.4, 0.6 (rad). The input is m0 = 350 GeV, m1/2 = 180
GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.8 (rad) and |A0| = 250
Figure 4: tanβ dependence of ∆Γ3 → χ+1 χ−1 . The curves in ascending order
correspond to θµ = 0.2, 0.4, 0.6 (rad). The input is m0 = 350 GeV, m1/2 = 180
GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), αA0 = 0.8 (rad) and |A0| = 250
∆ Br1
Figure 5: tan β dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order
at tanβ = 40 correspond to θµ = 0.5, 0.1, 1.0, 1.5 and 2.0 (rad). The input is
m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad),
αA0 = 0.3 (rad) and |A0| = 250 GeV.
∆ Br3
Figure 6: tan β dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order
at tanβ = 40 correspond to θµ = 0.5, 0.1, 1.0, 1.5 and 2.0(rad). The input is
m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad),
αA0 = 0.3 (rad) and |A0| = 250 GeV.
∆ Br1
Figure 7: |A0| dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at
|A0| = 0 correspond to tanβ = 40, 35, 30, 25 and 20. The input is m0 = 500 GeV,
m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), θµ = 0.7 (rad) and
αA0 = 0.1 (rad).
∆ Br3
Figure 8: |A0| dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at
|A0| = 0 correspond to tanβ = 40, 35, 30, 25 and 20. The input is m0 = 500 GeV,
m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad), ξ3 = 0.6 (rad), θµ = 0.7 (rad) and
αA0 = 0.1 (rad).
∆ Br1
Figure 9: θµ dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at
θµ = 2.0 (rad) correspond to |A0| = 100, 250, 500, 750 and 900 GeV. The input
is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad),
ξ3 = 0.6 (rad) and αA0 = 0.2 (rad).
0-505
∆ Br3
Figure 10: θµ dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at
θµ = π (rad) correspond to |A0| = 100, 250, 500, 750 and 900 GeV. The input is
tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad),
ξ3 = 0.6 (rad) and αA0 = 0.2 (rad).
∆ Br1
Figure 11: α0 dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at
αA0 = 2.2 (rad) correspond to |A0| = 500, 450, 400, 100 and 200 GeV. The input
is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad),
ξ3 = 0.6 (rad) and θµ = 0.1 (rad).
∆ Br3
Figure 12: α0 dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at
αA0 = 2.2 (rad) correspond to |A0| = 500, 450, 400, 100 and 200 GeV. The input
is tanβ = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ2 = 0.5 (rad),
ξ3 = 0.6 (rad) and θµ = 0.1 (rad).
∆ Br1
Figure 13: ξ2 dependence of ∆Br1 → χ+1 χ−1 . The curves in ascending order at
ξ2 = 0.75 (rad) correspond to |A0| = 50, 100, 150, 200 and 250 GeV. The input is
tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ3 = 0.6 (rad) and
θµ = 0.2 (rad) and αA0 = 0.3 (rad).
∆ Br3
Figure 14: ξ2 dependence of ∆Br3 → χ+1 χ−1 . The curves in ascending order at
ξ2 = 0.75 (rad) correspond to |A0| = 50, 100, 150, 200 and 250 GeV. The input is
tan β = 20.0, m0 = 500 GeV, m1/2 = 150 GeV, ξ1 = 0.4 (rad), ξ3 = 0.6 (rad) and
θµ = 0.2 (rad) and αA0 = 0.3 (rad).
