PHYSICAL REVIEW D 81, 054004 (2010)

Invisible Higgs boson decay with B ! K constraint
C. S. Kim,1,2,* Seong Chan Park,2,† Kai Wang,2,‡ and Guohuai Zhu3,x1

Department of Physics, Yonsei University, Seoul 120-479, Korea
Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568, Japan
3
Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China(Received 18 November 2009; published 9 March 2010)2

If the Higgs boson were the only particle within the LHC accessible range, precision measurement of
the Higgs’s properties would play a unique role in studying electroweak symmetry breaking as well as
possible new physics. We try to use low energy experiments such as rare B decay to constrain achallenging decay mode of Higgs, in which a Higgs decays to a pair of light (  1  2 GeV) SM singlet
S and becomes invisible. By using the current experimental bound of rare decay B ! K and computing
the contribution of B ! KSS to (the) B ! K þ E
6 , we obtain an upper bound on the Higgs coupling tosuch light singlet. It is interesting that the partial width of the invisible decay mode h ! SS by taking theupper bound value of coupling is at a comparable level with h ! WW=ZZ or WW ðÞ decay modes,
making the Higgs identifiable but with a different predicted decay branching ratio from [the] standard
model Higgs decay. It will then have an impact on precision measurement of the Higgs’s properties. We
also study the implication for cosmology from such a light singlet and propose a solution to the potential
problem.
DOI: 10.1103/PhysRevD.81.054004

PACS numbers: 12.60.Fr, 13.20.He, 14.80.Ec

I. INTRODUCTION
Searching the Higgs boson, the last missing piece in thestandard model (SM) of particle physics, is one of theessential goals of the CERN Large Hadron Collider(LHC). The minimal Higgs boson model is the simplest
solution to electroweak symmetry breaking and also themost economic one to be consistent with existing precision
measurements. However, theoretical considerations suggest that the minimal Higgs boson model may not becomplete. Being a fundamental scalar, the Higgs boson
receives quantum corrections of quadratic divergence. To
solve this, there have been many theoretical proposals
which predict various new physics at OðTeVÞ. Direct
evidences of new resonances at the LHC can determine
what is the new physics model. However, if the Higgs
boson were the only particle at the LHC accessible range,
we will have to rely on precision measurements. The
precision measurement of Higgs boson properties canplay an important role to confirm electroweak symmetry
breaking mechanism [1] and test new physics [2]. For
instance, measurement on top quark Yukawa coupling yt
is crucial to probe the origin of fermion mass generation
while gg ! h production due to the top quark loop directly
depends on the coupling yt . On the other hand, the contribution from new physics may also change the gg ! hproduction rate significantly. One interesting scenario willbe that at the LHC one does discover the conventional
Higgs search channels, confirm it is the Higgs and measure*cskim@yonsei.ac.kr†
seongchan.park@ipmu.jp‡
kai.wang@ipmu.jp
xzhugh@zju.edu.cn

1550-7998= 2010=81(5)=054004(8)its mass but the observed event number is much smaller
than what we expect for the SM Higgs of that mass.
However, a new decay mode of Higgs boson that cannot
be easily identified will lead to the same consequence when
the new decay width is comparable with the conventional
SM Higgs width at the same mass [3]. For instance, if
Higgs decay has an invisible mode, it is impossible to fullyreconstruct such resonance and is very challenging to
identify at the hadron colliders [4].
In this paper, we want to consider the invisible decay ofHiggs to a pair of hidden sector scalar (S) particles in theminimal extension of the SM [5–8]. As the scalar particle isa singlet of the SM interactions it can only directly couple
to the Higgs by the interaction Lagrangian y 2 ^ y 2
H HS  H HS ;
2v0
2

(1)

where  is a dimension one coupling constant and v0 thevacuum expectation value (vev) of the Higgs boson. It is achallenge to identify such invisible Higgs at collider experiments and obtain any bound on invisible Higgs. The
only controlled experiments at this moment that can put
constraints on such decay mode are through low energy
processes such as rare B or K decays. In these processes the
Higgs is virtual, not interacting directly to B or K, but to
top quark and S. Therefore, the only difference is CKM
factor, for K it is about 105 smaller than B, so we would
need more than 1010 K’s. Therefore, we just focus on rare
B decays in this work.
In Table I, we show the theoretical estimates of branching ratios (BRs) within the SM [9–12] and their current
experimental bounds at B factories [13–15] for the decays
B ! M.
 The errors of the SM estimates in Table I are

054004-1

Ó 2010 The American Physical Society

C. S. KIM, SEONG CHAN PARK, KAI WANG, AND GUOHUAI ZHUTABLE I. Expected BRs in the SM and experimental bounds(90% C.L.) in units of 106 . The SM values for K, , K  include
the long distance contributions through intermediate on-shell ,which can be dominant for  case [9].
mode
B ! K
B ! B ! K  
B ! 

BRs in the SM [9–12]

Experimental bounds

5:1  0:8
9:7  2:1
8:4  1:4
0:49þ0:61
0:38<14 [13]<100 [14]<80 [15]<150 [13]good candidate of dark matter. The scalar particle could bein thermal equilibrium with the SM sector through interaction with Higgs boson in early universe and finally its
relic still may survive in the current universe in the form of
dark matter [5,18]. The relic density is determined byannihilation cross section of the scalar particle to the SMparticles as [19]
S h2 ’

mainly due to the hadronic transition form factors and the
CKM matrix elements, since those decay channels areamong the cleanest SM processes due to only involving
electroweak penguin diagrams [16], except for B ! [9]. Please note that by taking the ratios such as BrðB !
Þ=BrðB! lÞ, BrðB ! K  Þ=BrðB! lÞ, we
can reduce considerably the uncertainties related to thehadronic form factors [17]. For B ! K, similarly one
may consider the ratio BrðB ! KÞ=BrðB
! K‘þ ‘ Þ
where the uncertainties from the hadronic form factors
are canceled to a large extent [12].
Here we will focus on Bþ ! Kþ  decay as its experimental upper bound is closest to the SM prediction asshown in Table I. Using the SM expectation value
Br SM ðBþ ! Kþ Þ
 ¼ 5:1  0:8  106 ;(2)hS vrel i ¼

II. COSMOLOGICAL BOUND AND DECAY OF A
HIDDEN SECTOR SCALAR
If we assume the renormalizability of the theory andallow the mass term quartic self-interaction term and thequartic interaction term with the Higgs, the Lagrangian of
the scalar sector is written as^11
L scalar ¼ ð@SÞ2  m2S S2  S S4  S2 H y H:
2
2
2
4!

(4)

The Lagrangian respects the Z2 symmetry (S ! S) thus
S is a stable particle. Indeed this scalar particle can be a(5)^ 2 m2S
ðmS Þ:
m4h

(6)

P
The precise value of ðmS Þ ’ f x2f ð1  x2f Þ3=2 wherexf ¼ mf =mS depends on the actual mass of scalar particle
and the kinematically allowed channels. We found a stringent constraints on the annihilation cross section considering the WMAP data c h2 ¼ 0:1131  0:0034 [20] as
0:1 pb^ 2 m2S*) ^24
h jWMAP 5 yr
mh

2
 
1 GeV
mh
* 3:5 :
mS
150 GeV(3)

we can derive the corresponding constraint on Higgs invisible decay width.
To be kinematically allowed in B ! KSS, the singlet
scalar cannot be heavier than 1–2 GeV. Therefore, thescalar can be easily thermalized through the Higgs interactions in the early universe. We first discuss its cosmological bound in the next section. The third section is thediscussion on B decay. After taking into all the constraints,
we discuss its implication in Higgs in the Sec. IV andfinally present the conclusion in Sec. V.

0:1 pb;hS vrel i

where S is the annihilation cross section of S to thestandard model particles through s-channel Higgs exchange diagrams and v is relative velocity between annihilating Ss. Since we are mainly interested in GeV scaleparticle, available channels are mainly to light leptons
ðe; ; ðÞÞ and quarks ðu; d; s; cð; bÞÞ and the cross section
is obtained as

and the current upper bound from BELLE [13] on this final
state asBr ðB ! K þ E
6 Þ < 14  106 ;PHYSICAL REVIEW D 81, 054004 (2010)(7)

If mh ’ 150ð115Þ GeV and mS ’ 1 GeV we get ^ *3:5ð1:2Þ, respectively, which is within the strong coupling
regime where the perturbative description of the model is
not available.
In Fig. 1, we presented the allowed parameter space in(^ ¼ =v0 , mS ) plane by the 5 yr WMAP data on the CDM
component with various values of Higgs mass (115, 150,
185) GeV taking threshold effects into account. Basically a
GeV scale mass range, only in which range B ! KSS isallowed, is not compatible with the cosmological observations.1 On the other hand, if the scalar is heavier (mS >2 GeV) even though the scalar cannot contribute to theB-decays but can be a successful dark matter candidate, if
the  coupling is properly chosen.
However, we can easily avoid this cosmological constraint provided that the singlet actually decays into light1
In Ref. [21], a scalar field in the mass range of 1 GeV has
been considered and the authors reached the same conclusion
with ours: a large coupling constant is required in order to avoid
overabundance. However, this large coupling constant is ruledout by the B ! K data, as we consider in Sec. III.

054004-2

INVISIBLE HIGGS BOSON DECAY WITH . . .
10

PHYSICAL REVIEW D 81, 054004 (2010)

Heff ¼

8

15
0

6
mh

mS GeV

115

18
5

Ge

V

4
2
Sh

2

0.2

0.11 Excluded

0.4

0.6

0.8

1.0

FIG. 1 (color online). Cosmological constraints for a stable S
from the relic abundance. Allowed parameter space in (^ ¼
=v0 , mS ) plane with mh ¼ 115, 150 and 185 GeV, respectively.

particles since only (absolutely) stable particles can significantly contribute to the dark matter density of thecurrent universe. As the longevity of the scalar particle isinherited by the Z2 symmetry, a mechanism of breaking Z2
symmetry leads to a natural way out. Indeed there is a very
promising source of the symmetry breaking. Quantum
gravity effect actually allows higher order operators andsome of them might break global symmetries such as Z2 .
For instance, the scalar particle may decay to a pair ofphotons or gluons through dimension five operators:SF F
SGa G
a
C1
þ C2
;




Vtb
Vts
Cs sð1 þ 5 ÞbSS:
2m2h

(9)

Intuitively, b ! sSS decay can be divided into two processes: first b quark decays to s quark plus a off-shell Higgs
boson h, and subsequently h decays to two light singlets.
From the interaction Lagrangian
term Hþ HS2 =2v0 , withp
ﬃﬃﬃ
Hþ ¼ ð  ; ðv0 þ h  i 0 Þ= 2Þ, it is easy to show thatthe Higgs boson decay h ! SS can proceed through atrilinear term hSS=2. But as we will see later, anotherterm  þ  S2 =2v0 is also crucial to guarantee the gauge
independence of the decay amplitude.
To evaluate the decay amplitude, the Wilson coefficientCs at scale b ¼ Oðmb Þ should be known, which can beobtained by matching the full theory to the effective theory
at scale around mW to obtain Cs ðmW Þ and then evolving
down to b . As the above operator does not mix with othereffective operators, the QCD running effects can be obtained straightforwardly with the calculation of the anomalous dimension of sð1 þ 5 Þb [22]:12=23s ðb ÞCs ðmW Þ:(10)
Cs ðb Þ ¼
s ðmW ÞCs ðmW Þ can be obtained by calculating the diagrams in
Fig. 2. Notice that the Higgs boson does not couple tos-quark by taking ms ¼ 0.
In Fig. 2, the first eight diagrams represent exactly theintuitive picture that first b ! sh, and then h ! SS. Since
the later one is a trivial tree level process, one may first
focus on the construction of an one-loop effective bsh
vertex V sð1 þ  Þbh
L bsh ¼ Cbsh Vtb
ts
5

(11)

with the coefficient in t’Hooft-Feynman gauge as [23,24](8)

where C1  C2  Oð1Þ are (unknown) parameters. One
should notice that both operators respect gauge symmetry
but break Z2 symmetry. The life time of the scalar issuppressed by a large cutoff scale (  MPlanck ) but certainly much shorter than the age of universe so that we can
avoid the strong constraint from the relic density
measurements.
III. B ! KSS AND INVISIBLE HIGGSIn this section we study the constraint on the interaction
term between the Higgs boson and the SM singlet from B
decays. Specifically we will look at B ! KSS decay whichcurrently has the most stringent experimental upper bound14  106 [13].
The effective Hamiltonian for this decay can be expressed as

Cbsh ðmW Þ ¼

g2 mb xt
3 þ xhð4Þ2 8v0
ð3  xt Þð1  xt Þ þ 2xt ð2  xt Þ lnxt; (12)
ð1  xt Þ3

where xt  m2t =m2W , xh  m2h =m2W with the approximation
m2b =ðm2W ; m2t ; m2h Þ ’ 0. Notice that this expression isgauge-dependent as the Higgs boson is off shell.
Although the calculation itself is straightforward, the issues about gauge dependence and renormalization scheme
ambiguities are a bit subtle which were finally settled down
by several groups a few years later [25].
But for the decay amplitude b ! sSS to be gauge invariant, the last diagram in Fig. 2, i.e. Fig. 2(i), has to be
included which (surprisingly at first look) does not containvirtual Higgs boson exchange at all. Actually Fig. 2(i)arises from the interaction term  þ  S2 =2v0 .
Therefore strictly speaking, b ! sSS cannot be factorized
into b ! sh and h ! SS.

054004-3

C. S. KIM, SEONG CHAN PARK, KAI WANG, AND GUOHUAI ZHUPHYSICAL REVIEW D 81, 054004 (2010)

FIG. 2. b-quark decays to s-quark plus two light singlets. The internal quark lines represent up, charm or top quarks, while theinternal dashed lines denote Higgs boson (h) or unphysical charged Goldstone bosons ( ).

Finally, summing all the diagrams, we obtain2

Cs ðmW Þ ¼

g2 3mb xt
ð4Þ2 8v0(13)

The calculation details can be found in the appendix. Heremb should be evaluated at the scale mW , but interestingly
when combined with the QCD evolution effect of Eq. (10),one has3
mb ðmW Þs ðb Þ

12=23s ðmW Þ

¼ mb ðmb Þ:q 
hK js bjB i
mb

q
¼
f ðq2 Þðp þ lÞ þ ðf0 ðq2 Þmb þ
m2  m2 fþ ðq2 ÞÞ B 2 K q ;(15)
qhK  jsð1 þ 5 ÞbjB i ¼

with the light-cone sum rules (LCSR) estimation [26]
0:162
0:173
þ
2
2
1  q =5:41
ð1  q2 =5:412 Þ2
0:33
f0 ðq2 Þ ¼
1  q2 =37:46
fþ ðq2 Þ ¼(14)

Please also note that in [11] the authors considered b !
sSS in an effective theory approach, however, with Cs asmodel independent free parameters.
To get the decay amplitude, the hadronic matrix elementhK jsð1 þ 5 ÞbjB i is needed as input, which can be
related to the known form factors through equation of
motion,2This expression has been obtained in [7]. However, in thederivation, they divided the process b ! sSS into b ! sh and
h ! SS. They then evaluate the bsh vertex with the approximation of vanishing Higgs boson mass. But even with these
unrigorousness or approximations, they do obtain finally thecorrect expression which is due to the almost completely cancellation between Fig. 2(h) and 2(i) up to Oðm2b =m2h Þ. However,generally this kind of cancellation does not happen and thesummation should be of order m2h =m2W , as pointed out by
Botella and Lim in [25].
3
We thank the referee for pointing this out to us.

(16)

As discussed in [26], the uncertainty of the q2 dependenceof the form factors have not been fully analyzed in LCSR
but likely to be smaller than that at q2 ¼ 0 which is about
12%. Thus as an rough error estimation we assign a universal 12% uncertainty to the above form factors.
Then, the branching ratio can then be obtainedBrðB ! KSSÞ ¼

 V j2Z ðmB mK Þ2
2 jVtb
ts
2
C
ðm
Þ
dq2
s
b
5123 m3B B m4h
4m2S
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 hK jsð1 þ 5 ÞbjB i2 q2  4m2S
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðm2B  q2  m2K Þ2(17)
 4m2K :
q2

Taking as illustration
mh ¼ 130 GeV;
and with the values [27]054004-4

mS ¼ 1 GeV;(18)

INVISIBLE HIGGS BOSON DECAY WITH . . .

mb ðmb Þ ¼ 4:2 GeV;PHYSICAL REVIEW D 81, 054004 (2010)

mt ¼ 171:3 GeV;CKM ¼ 0:2257

A ¼ 0:814;(19)BrðB ! KSSÞ ¼ 0:82  1010

 Brexp ðB ! KE
6 Þ  BrSM ðB ! KÞ
’ 8  106

and Vts ¼ A2CKM , we can obtain the branching ratio 1010 ;IV. INVISIBLE HIGGS
If there exists such light SM singlet scalar, the Higgs
decay can be significantly modified. For mS ¼ 1 GeV, wetake the upper bound on  derived from the B ! KE
6 as
1

-1

Brh decay
10

-2

-3

100

150



2
4m2S 1=2
ðh ! SSÞ ¼
;
1 2
32mh
mh

(20)

where only the form factor uncertainty has been includedin the error estimation.

10

200

250

300

350

400

mh (GeV)

FIG. 3 (color online). Higgs boson decay BR with Invisible
decay mode predicted from current upper bound of B ! K insolid lines. (For comparison, dashed lines are for SM Higgs
decay BR.)(22)

where  is the dimension one coupling and mh , mS are the
Higgs boson mass and hidden sector scalar mass, respectively. To illustrate the feature, we scan mh and plot in
Fig. 3 how the Higgs decay BR will be changed due to the
h ! SS decay. The partial width of h ! SS is obtained bytaking mS ¼ 1 GeV and the  upper bound value computed for that mh point.
If mh < 150 GeV, h ! SS completely dominates the
Higgs decay and Higgs is only invisible. Even though thetraditional invisible Higgs search can be applied to searchfor such modes, it is impossible to identify the resonancethrough invisible modes at the LHC.
When mh > 150 GeV, the partial width of h ! SS is
comparable to the partial widths of conventional channels,
such as h ! W þ W  or h ! ZZ. The multilepton searches
for Higgs resonance are still valid but the decay BRs
significantly decrease. If the measured event numbers of
h ! W þ W  or h ! ZZ are below the expected numbers.
There are several possibilities:(i) There are more than one Higgs boson responsible for
the W gauge boson mass MW . The vacuum expectation value for the lightest Higgs boson is muchsmaller than v0 so that the coupling W þ W  H is gv0 .
(ii) The production of Higgs boson is suppressed due to
new physics. For instance, gg ! H production isless due to the top quark partner in the triangle
loop and significantly cancels the top quark loop.

1

1

-1

10

-1

Brh decay

Brh decay

10

(21)

and compute the upper bound of partial width for h ! SS.
The partial width of Higgs decaying into two scalar is

2 


130 GeV 4BrðB ! KSSÞ ¼ ð0:82  0:20Þ 
1 GeV
mh


10

2 


130 GeV 4
1 GeV
mh



10

10

-2

10

-3

100

150

200

250 300
mh (GeV)

350

400

10

-2

-3

100

150

200

250 300
mh (GeV)

350

400

FIG. 4 (color online). Solid lines correspond to the Higgs BR with invisible decay mode predicted from the upper bound value formS ¼ 1 GeV and BrðB ! KSSÞ ¼ 1  106 or 1  107 respectively. Dashed lines are the standard SM Higgs decay BR.

054004-5

C. S. KIM, SEONG CHAN PARK, KAI WANG, AND GUOHUAI ZHU(iii) There exists unknown Higgs decay mode whichcannot be easily identified. The invisible Higgs
mode that we discuss here falls into this category.
Another example is the h ! N decay in some TeV
neutrino models [3].
We expect the SuperB or SuperBelle will improve themeasurement significantly and reduce the allowed regionof BrðB ! KSSÞ ¼ Brexp ðB ! KE
6 Þ  BrSM ðB ! KÞ.
In Fig. 4, we plot how the Higgs decay BR will change
accordingly for mS ¼ 1 GeV and improved bound onBrðB ! KSSÞ. As can be seen, if the value of BrðB !
KSSÞ becomes smaller than 2  106 , it will only change
the Higgs decay before on-shell WW threshold and will notsignificantly change the heavy Higgs decay.
V. CONCLUSION
We have studied the contribution of virtual Higgs in
B ! KE
6 by assuming Higgs coupling to a light SM singlet
scalar S, B ! KSS. For MS ¼ 1 GeV,2BrðB ! KSSÞ ¼ ð0:82  0:20Þ 
1 GeV

4
130 GeV 1010 :mhGiven the current experimental bound and subtracting the
known SM contribution,
Br exp ðB ! KE
6 Þ  BrSM ðB ! KÞ
 ’ 8  106 ;we obtain an upper bound on the coupling between the
Higgs and singlet scalar S. We take the upper bound valueof this coupling and compute the h ! SS decay partial
width. It is interesting that the partial width of h ! SSdecay is at comparable level when the Higgs mass is close
to the WW threshold. Consequently, Higgs may still be
discovered via the conventional Higgs search channels butwith a smaller event number. This will have some impact
on precision measurement of Higgs property. We expectthat the SuperB or SuperBelle experiments can improve the
B ! KE
6 measurement and put a stringent bound on possible invisible Higgs decay.
We have also studied the possible implication in cosmology from this scalar. It turns out that for the interestingregion of couplings between h and S, such light scalar may
not have enough annihilation cross section and will thenover close the universe if it is a stable particle. We propose
a scenario where S is not stable in the cosmological scale
but only a stable particle in B decay or collider
environments.

PHYSICAL REVIEW D 81, 054004 (2010)

part by WPI Initiative, MEXT, Japan. S. C. was supportedby the World Premier International Research Center
Initiative (WPI initiative) by MEXT and also supportedby the Grant-in-Aid for scientific research (Young
Scientists (B) 21740172) from JSPS, Japan. K. W. is supported by the World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan. G. Z. is supported in part by the National Science Foundation ofChina under Grant No. 10705024 and No. 10425525, andby the Scientific Research Foundation for the Returned
Overseas Chinese Scholars, State Education Ministry.
APPENDIX: CALCULATION DETAILS ON b ! sSS
In the calculations, we use the t’Hooft-Feynman gauge¼ 1. p, l and q denote the momentum of b-quark,s-quark and virtual Higgs boson, respectively. We havetaken the approximation q2  m2h ’ m2h and dropped acommon factor =m2h in the following expressions. We get
Fig : 2ðaÞ ¼

ig2 Vtb
VtsmsðlÞð1 þ 5 ÞbðpÞ b
4v0
ð4Þ2

xt ðx2t  1  2xt lnxt Þðxt  1Þ3

with xt  m2t =m2W .
Fig: 2ðbÞ ¼

V
ig2 Vtb
xt mb 1ts
sðlÞð1
þ

ÞbðpÞ

5
4v0
ð4Þ2

Z
1
 ðx; yÞdxdy ln 1 2
þ ln4   2
2

xþy1

2
mt y
þ
(A2)
1 ðx; yÞwith1 ðx; yÞ ’ ð1  x  yÞm2W þ ðx þ yÞm2t :The divergence of Fig. 2(b) can be canceled by that ofFig. 2(g):ig2 Vtb
Vts
xt mb 1
Fig: 2ðgÞ ¼

s
ðlÞð1
þ

ÞbðpÞ
5
4v0
ð4Þ2

Z1
xm2 þ ð1  xÞm2W
þ ln4 dx ln t: (A3)
2
0
The sum of Figs. 2(b) and 2(g) then gives (taking the scale
 ¼ mW )Fig : 2ðb þ gÞ ¼


ig2 Vtb
Vts
x2t mb

s
ðlÞð1
þ

ÞbðpÞ
5
4v0
ð4Þ2



ACKNOWLEDGMENTS
C. S. K. was supported in part by the Basic Science
Research Program through the NRF of Korea funded by
MOEST (2009-0088395), in part by KOSEF through the
Joint Research Program (F01-2009-000-10031-0), and in(A1)

ð3xt  5Þðxt  1Þ  2ðxt  2Þ lnxt:ðxt  1Þ3
(A4)

It is clear that for Figs. 2(a), 2(b), and 2(g), the internal up
and charm quarks contributions are suppressed at least by

054004-6

INVISIBLE HIGGS BOSON DECAY WITH . . .

PHYSICAL REVIEW D 81, 054004 (2010)

m2u;c =m2t compared to the virtual top quark contribution andwithcan be safely neglected.
For Fig. 2(c), the internal top quark contribution is
Fig : 2ðcÞt ¼

2 ðx; yÞ ’ xm2t þ ð1  xÞm2W :ig2 Vtb
VtsmsðlÞð1
þ 5 ÞbðpÞ b
2
4v0
ð4Þ



2x2t lnxt  ð3xt  1Þðxt  1Þ:
ðxt  1Þ3

The divergence here can be canceled when the contributions from the internal up and charm quarks are included,
then we get(A5)
Fig : 2ðdÞ ¼

But here the internal up and charm quarks contributions are
not suppressed, which can be obtained from the above
expression by taking the limit xt ! 0 and changing thecorresponding CKM factors. We then obtain using the
CKM unitarity condition,
Fig : 2ðcÞ ¼

ig2 Vtb
VtsmsðlÞð1 þ 5 ÞbðpÞ b
2
4v0
ð4Þ
 2

2xt lnxt  ð3xt  1Þðxt  1Þ
 1 : (A6)ðxt  1Þ3

ig3 Vtb
VtsmsðlÞð1 þ 5 ÞbðpÞ b
2
8mW
ð4Þ

Z
1
1
   þ ln4  
dxdy
2
xþy1


 ðx; yÞ ð1 þ x þ yÞm2t
 2 ln 2 2 þ
2 ðx; yÞ(A7)[1] M. Duhrssen, S. Heinemeyer, H. Logan, D. Rainwater, G.
Weiglein, and D. Zeppenfeld, Phys. Rev. D 70, 113009(2004).
[2] I. Low, R. Rattazzi, and A. Vichi, arXiv:0907.5413.
[3] S. C. Park, K. Wang, and T. T. Yanagida, arXiv:0909.2937.
[4] O. J. P. Eboli and D. Zeppenfeld, Phys. Lett. B 495, 147(2000).
[5] V. Silveira and A. Zee, Phys. Lett. B 161, 136 (1985).
[6] H. Davoudiasl, R. Kitano, T. Li, and H. Murayama, Phys.
Lett. B 609, 117 (2005).
[7] C. Bird, P. Jackson, R. V. Kowalewski, and M. Pospelov,
Phys. Rev. Lett. 93, 201803 (2004); C. Bird, R. V.
Kowalewski, and M. Pospelov, Mod. Phys. Lett. A 21,
457 (2006).
[8] H. Sung Cheon, S. K. Kang, and C. S. Kim, J. Cosmol.
Astropart. Phys. 05 (2008) 004; Phys. Lett. B 675, 203(2009).
[9] J. F. Kamenik and C. Smith, Phys. Lett. B 680, 471 (2009).
[10] J. H. Jeon, C. S. Kim, J. Lee, and C. Yu, Phys. Lett. B 636,
270 (2006).
[11] W. Altmannshofer, A. J. Buras, D. M. Straub, and M.
Wick, J. High Energy Phys. 04 (2009) 022.
[12] M. Bartsch, M. Beylich, G. Buchalla, and D. N. Gao, J.


ig3 VtbVtsxmsðlÞð1
þ 5 ÞbðpÞ t b
2
32mW
ð4Þ



2xt ð5xt  6Þ lnxt  ð9xt  11Þðxt  1Þ:
ðxt  1Þ3
(A9)For Fig. 2(e), we have
Fig : 2ðeÞ ¼

ig3 Vtb
VtsxmsðlÞð1 þ 5 ÞbðpÞ t b
2
32mW
ð4Þ



The virtual top quark contribution to Fig. 2(d) is
Fig: 2ðdÞt ¼(A8)

2xt ð3xt  2Þ lnxt  ð7xt  5Þðxt  1Þ:
ðxt  1Þ3
(A10)

Here the internal up and charm quarks contributions areagain negligibly small due to the Oðm2u;c =m2t Þ suppression.
It is easy to show that the contribution of Fig. 2(f) vanishes
by using the equation of motion sðlÞ=l6 ¼ 0. The cancellation between Fig. 2(h) and 2(i) is obvious by approximating the Higgs boson propagator i=ðq2  m2h Þ ’ i=m2h .

High Energy Phys. 11 (2009) 011.
[13] K. F. Chen et al. (Belle Collaboration), Phys. Rev. Lett. 99,
221802 (2007).
[14] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett.
94, 101801 (2005).
[15] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D 78,
072007 (2008).
[16] G. Buchalla and A. J. Buras, Nucl. Phys. B400, 225(1993); Y. Grossman, Z. Ligeti, and E. Nardi, Nucl.
Phys. B465, 369 (1996); B480, 753(E) (1996).
[17] T. M. Aliev and C. S. Kim, Phys. Rev. D 58, 013003(1998).
[18] C. P. Burgess, M. Pospelov, and T. ter Veldhuis, Nucl.
Phys. B619, 709 (2001).
[19] G. Bertone, D. Hooper, and J. Silk, Phys. Rep. 405, 279(2005).
[20] E. Komatsu et al. (WMAP Collaboration), Astrophys. J.
Suppl. Ser. 180, 330 (2009).
[21] M. C. Bento, O. Bertolami, and R. Rosenfeld, Phys. Lett.
B 518, 276 (2001).
[22] Y. B. Dai, C. S. Huang, and H. W. Huang, Phys. Lett. B
390, 257 (1997); 513, 429(E) (2001).
[23] R. S. Willey and H. L. Yu, Phys. Rev. D 26, 3086 (1982).

054004-7

C. S. KIM, SEONG CHAN PARK, KAI WANG, AND GUOHUAI ZHU[24] B. Grzadkowski and P. Krawczyk, Z. Phys. C 18, 43(1983).
[25] F. J. Botella and C. S. Lim, Phys. Rev. Lett. 56, 1651
(1986); Phys. Rev. D 34, 301 (1986); B. Grinstein, L. J.
Hall, and L. Randall, Phys. Lett. B 211, 363 (1988); A. A.
Johansen, V. A. Khoze, and N. G. Uraltsev, Yad. Fiz. 49,
1174 (1989); [Sov. J. Nucl. Phys. 49, 727 (1989)]; J. G.

PHYSICAL REVIEW D 81, 054004 (2010)

Korner, N. Nasrallah, and K. Schilcher, Phys. Rev. D 41,
888 (1990); R. Ferrari, A. Le Yaouanc, L. Oliver, and J. C.
Raynal, Phys. Rev. D 52, 3036 (1995).
[26] P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005).
[27] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1(2008) and 2009 partial update for the 2010 edition (http://
pdg.lbl.gov).

054004-8