Update on very light CP-odd scalar in the Two-Higgs-Doublet Model
ABSTRACT In a previous work we have shown that a general two-Higgs-doublet model (THDM) with a very light CP-odd scalar can be compatible with electroweak precision data, such as the rho parameter, BR(b --> s gamma), R_b, A_b, BR(Upsilon --> A gamma), BR(eta --> A gamma), and (g-2) of muon. Prompted by the recent significant change in the theoretical status of the latter observable, we comment on the consequences for this model and update the allowed parameter region. It is found that the presence of a very light scalar with a mass of 0.2 GeV is still compatible with the new theoretical prediction of the muon anomalous magnetic moment. Comment: 10 pages, 11 figures, uses REVTEX 4, including updated figures and discussion considering the latest muon (g-2) data
arXiv:hep-ph/0205204v2 15 Aug 2002
Update on very light CP-odd scalar in the Two-Higgs-Doublet Model
F. Larios,1, ∗G. Tavares-Velasco,2, †and C.-P. Yuan3, ‡
1Departamento de F´ ısica Aplicada, CINVESTAV-M´ erida, A.P. 73, 97310 M´ erida, Yucat´ an, M´ exico
2Instituto de F´ ısica y Matem´ aticas, Universidad Michoacana de San Nicolas de Hidalgo,
Apdo. Postal 2-82, C.P. 58040, Morelia, Michoac´ an, M´ exico
3Department of Physics and Astronomy, Michigan State University, E. Lansing, MI 48824, USA
(Dated: February 1, 2008)
In a previous work we have shown that a general two-Higgs-doublet model (THDM) with a very
light CP-odd scalar can be compatible with electroweak precision data, such as the ρ parameter,
BR(b → sγ), Rb, Ab, BR(Υ → Aγ), BR(η → Aγ), and (g − 2) of muon. Prompted by the recent
significant change in the theoretical status of the latter observable, we comment on the consequences
for this model and update the allowed parameter region. It is found that the presence of a very light
scalar with a mass of 0.2 GeV is still compatible with the new theoretical prediction of the muon
anomalous magnetic moment.
PACS numbers: 12.60.Fr, 14.80.Cp, 12.15.Ji
The possibility of a Higgs boson decaying into a pair of light CP-odd scalars was considered in Ref. . Although
it is very unlikely that this particle can be accommodated in the minimal supersymmetric standard model (MSSM),
in the light of the restrictions imposed by the current low-energy data on the parameters of this model, a very light
CP-odd scalar A can still arise in some other extensions of the standard model (SM), such as the minimal composite
Higgs model , or the next-to-minimal supersymmetric model . Therefore, the existence of a very light CP-
odd scalar not only proves new physics but also casts the most commonly studied MSSM in doubt. Furthermore,
studying the couplings of the light CP-odd scalar to the SM fermions may help discriminating models of electroweak
symmetry breaking – either a weakly interacting model (e.g., the next-to-minimal supersymmetric model) or a strongly
interacting model (e.g., the minimal composite Higgs model). Apart from the above implications arising from the
existence of a very light CP-odd scalar, our main interest in studying this particle stems from the fact that its
phenomenology is indeed rather exciting: an interesting aspect of a light A is that if its mass MAis less than twice
that of the muon mµ, i.e. less than about 0.2GeV, it can only decay into a pair of electrons (A → e+e−) or photons
(A → γγ). Hence, the decay branching ratio BR(A → γγ) can be sizable. Consequently, A can behave like a
fermiophobic CP-odd scalar and predominantly decay into a photon pair, which would register in detectors of high
energy collider experiments as a single photon signature when the momentum of A is much larger than its mass .
In a previous work  we performed an extensive analysis within the framework of the two Higgs doublet model
(THDM) and found that a very light CP-odd scalar can still be compatible with precision data, such as the ρ
parameter, BR(b → sγ), Rb, Ab, BR(Υ → Aγ), and the muon anomalous magnetic moment aµ. We considered
different values for sin2(β − α) and found the constraints imposed on the remaining parameters of the model, which
we summarize in Table I, where MH (Mh) stands for the heavy (light) CP-even scalar and MH± for the charged
scalar. As for the soft breaking term µ12, it is not involved in any of the above observables, so they cannot be used to
constrain it. Since µ12has no relevance for the present discussion (the purpose of this work is to update the bounds
derived from the changes in the status of the theoretical value of the muon anomaly), we refer the reader to Ref. ,
where LEP-2 data were used to set bounds on this parameter. In obtaining the bounds shown in Table I we have
used the lower value of 110 GeV for Mh. We recall that the LEP-2 direct search bound requires Mh> 114.1GeV at
the 95%C.L. . However, in the presence of new physics such a bound can be substantially relaxed. As explained
in Ref. , the reason why the LEP-2 bound (Mh> 114.1 GeV) does not apply in our model is because this bound
is based on the SM specific value of BR(h → b¯b) . In the THDM, the new h → AA decay mode can significantly
reduce the h → b¯b branching ratio. This was clearly illustrated in the Fig. 9 of Ref.  for some allowed parameter
space of the model. In any case, the new decay channel (h → AA) registers as a di-photon signature (h → γγ) for
∗Electronic address: firstname.lastname@example.org
†Electronic address: email@example.com
‡Electronic address: firstname.lastname@example.org
TABLE I: Constraints from the low energy data for Type-I and Type-II THDMs, with MA = 0.2 GeV. The old calculation of
the hadronic light by light contribution to aµ, cf. Table II, was used together with the calculation of Ref.  for the hadronic
vacuum polarization. When sin2(β − α) = 1 there is no Mh dependence on ρ, otherwise, we assume Mh= 110 GeV.
(g − 2)µ
tanβ > 0.4
[tanβ > 1] b → sγMH+ > 100 GeV
[0.5 < tanβ < 1] b → sγ
[0.6 < tanβ < 1] Rb
MH+ > 200 − 600 GeV MH+ > 200 − 600 GeV
[sin2(β − α) = 1] ∆ρMH ∼ MH+
[sin2(β − α) = 0.8] ∆ρMH ∼ 1.2MH+
[sin2(β − α) = 0.5] ∆ρMH ∼ 1.7MH+
tanβ < 2.6
MH+ > 200 GeV
MH+ > 200 − 350 GeV
MH ∼ MH+
MH ∼ 1.2MH+
MH ∼ 1.7MH+
which LEP-2 has already set a lower bound. By taking both the AA and b¯b decay modes into consideration, a lower
bound for Mh> 103 GeV can be established in our light A scenario .
At this point we would like to emphasize that, given the recent measurements of aµ at Brookhaven National
Laboratory (BNL) , the bounds on new physics effects imposed by the muon (g − 2) data depend largely on the
theoretical value predicted by the SM for the nonperturbative hadronic contribution to aµ. In our analysis , we
followed a conservative approach and considered various predictions for the hadronic correction ahad
which in fact has been the source of debate recently [12, 13, 14]. For instance, the bounds shown in Table I were
obtained from the calculation presented in Ref. , which was the one allowing the largest parameter space.
After the completion of our work, it was evident that the latest precision measurement of aµ at BNL  along
with some theoretical predictions for ahad
disfavored the presence of a light A in the THDM. As is well known, the
BNL data opened the prospect for new physics as the experimental value of aµ appeared to be more than 2.6σ
above the theory prediction based on some calculations of the hadronic vacuum polarization. At the one-loop order,
a light CP-odd scalar can give a significant negative contribution to aµ, making it harder for this type of model to
be consistent with experiment. However, the two-loop calculation can yield a large correction to the one-loop result
as pointed out in . Although this fact seems to contradict perturbation theory, the unusual situation in which a
two-loop diagram can give a contribution of similar size or even larger than that from the one-loop diagrams within
a perturbative calculation was noted first by Bjorken and Weinberg when evaluating the Higgs scalar contribution
to the µ → eγ decay . It is straightforward to see that this situation also occurs in the calculation of the Higgs
scalar contribution to aµ. The reason is that the coupling of the Higgs scalar to the muon enters twice in the one-loop
diagram, whereas at the two-loop level there appears a diagram in which this coupling enters just once, together
with a line where the Higgs scalar couples to a heavy fermion pair (see Fig. 1). This gives rise to an enhancement
factor, due to the couplings, that compensates the suppression factor g2/(16π2), due to an additional loop. It turns
out that the diagram of Fig. 1 (c) gives by far the most dominant contribution at the two-loop level. Therefore, we
do not expect large uncertainties arising from unknown higher order terms. In our previous analysis, even when we
considered the two-loop calculation for the CP-odd scalar contribution to aµ, together with the hadronic correction
quoted in Ref. , i.e. that by Davier and H¨ ocker , we found that there was no allowed parameter space (in the
type-II THDM) in the tanβ vs. MAplane when MAwas below 3 GeV. Nevertheless, there were other SM calculations
yielding aµclose enough to the experimental value as to allow a very light A.
[8, 9, 10, 11],
FIG. 1: Contribution from the THDM to the anomalous magnetic moment of muon: (a) neutral Higgs bosons, (b) charged
Higgs boson, and (c) the leading two-loop contribution from the CP-odd scalar.
TABLE II: Contributions to the anomalous magnetic moment of muon in the SM , prior to the discovery of a wrong sign in
the pion pole correction to ahad
µ (l.b.l), which significantly changed the atheory
prediction. All values are given in units of 10−11.
aThis value has been found to be wrong in Ref. .
Since the publication of , there has been a lot of controversyregarding the theoretical value of the muon anomalous
magnetic moment. It is evident that before claiming the presence of any new physics effect, an extensive reexamination
of every contribution to aµis necessary [12, 13, 14]. Along these lines, a reevaluation  of the hadronic light by light
contribution to aµfound a sign error in earlier calculations  of this contribution, which has resulted in a significant
change of the aµprediction. Once the corrected value is taken into account, the discrepancy between experiment and
theory reduces down to the level of 1.6σ. In the light of this result, we believe it is worth revisiting our work and
reexamining our previous bounds.
II.ALLOWED PARAMETER RANGE FOR MA AND tanβ
The SM prediction of aµis composed of the following three parts :
where the electroweak corrections have been computed to a very good accuracy: they have a combined error of
the order of 5 × 10−11, which is already about one order of magnitude smaller than the ultimate goal of the E821
experiment . In contrast, the hadronic contribution ahad
and can be decomposed into three parts , namely the hadronic vacuum polarization contribution aµ(h.v.p.), the
hadronic light-by-light correction aµ(l.b.l), and other hadronic higher order terms aµ(h.o.):
contains the bulk of the theoretical error (∼ 70 × 10−11)
µ (h.v.p.) + ahad
µ (l.b.l) + ahad
In our previous analysis we used the values shown in Table II for each contribution to atheory
µ (h.v.p.) predictions to be discussed below. In the months following the publication of our work, a new situation
arose: the sign of the pion pole contribution to the hadronic light by light correction was found to be wrong . Very
interestingly, this contribution alone represents about 70% of the full ahad
mistake, the ahad
µ (l.b.l.) value gets significantly changed and even its sign gets flipped. As a result, the discrepancy
between the experiment and theory reduces down to the level of 1.6σ. Subsequent publications have confirmed this
finding [20, 21, 22]. In Table III we list the most recent evaluations of ahad
calculation that is based on chiral perturbation theory :
 together with the
µ (l.b.l.). It turns out that after correcting this
µ (l.b.l.). In addition, there is one more
µ (l.b.l.) =
whereˆC is an unknown low-energy constant that parametrizes some subdominant terms. We will not consider this
result here but only mention it as an example of a calculation that is still open to debate. For the purpose of this
work we will take an average of the top three results shown in Table III and study the consequences on the allowed
parameter space of the THDM.
After introducing the corrected value of ahad
µ (l.b.l.), the sum of all the contributions to atheory
µ (h.v.p.) = 116584845.3(17.1)× 10−11,(3)
TABLE III: The most recent evaluations of the hadronic light by light contribution to ahad
contrast with the wrong one shown in Table II.
µ (l.b.l.). These corrected values
µ (l.b.l.) × 1011
aThis value accounts only for the pion pole contribution.
TABLE IV: Some of the most recent calculations of ahad
discrepancy ∆aµ between experiment and theory. The last column represents the bounds on any new physics contribution aNP
at the 95% C.L. All of the values are given in units of 10−11.
ADH 7011(94) 116591856.3 (95.54) 166.7(179.53)
DH  6924(62)116591769.3 (64.31) 253.7(164.92)
J 6974(105)116591833.3 (112.3) 189.7(188.8)
N 7031(77) 116591876.3 (78.88) 146.7(171.25)
TY 6952(64)116591797.3 (66.25) 225.7(165.81)
µ (h.v.p.) together with the respective theory prediction atheory
Allowed range for aNP
where the errors have been composed quadratically.
source of renewed interest lately.
some representative evaluations of aµ(h.v.p). In the second column of Table IV we show some of the most recent
results, which were compiled in , whereas in the third column we show the full theory prediction, which is obtained
after adding up each value in the second column to Eq. (3).
As for the experimental value aexp
µ , the data obtained during the 1999 running period combined with previous
measurements give 
1As for the aµ(h.v.p) term, its evaluation has also been the
2As in our previous work, here we will use a conservative approach and consider
= 116592023(152)× 10−11. (4)
One thus can obtain the discrepancy between experiment and theory ∆aµ= aexp
of aµ(h.v.p), as shown in the fourth column of Table IV. Finally, if we assume that the discrepancy between theory
and experiment is to be ascribed to new physics effects, we can obtain the bounds shown in the last column of the
same Table for the new physics contribution to the anomalous magnetic moment at the 95 % C.L., which is denoted
should be compared to those used in our previous analysis, cf. Eq. (2) in Ref. .
Given the new bounds on aNP
µ, we update the constraint imposed by it on the tanβ−MAplane within the THDM.
The analytical expressions for the contribution of either a CP-even or a CP-odd scalar (Fig. 1) can be found in
Appendix A of Ref. . We will use the two-loop calculation for the contribution from the CP-odd scalar . In
order to satisfy the bounds shown in Table I, we are assuming that the remaining four Higgs scalars are much heavier
than the CP-odd scalar A, so their contribution to aµturns out to be negligibly small as compared to that coming
from the latter. Also, we are considering that sin2(β −α) = 1. The reason why we make this choice is because in our
scenario with a very light CP-odd scalar the most convenient way to meet the constraint imposed by the ρ parameter
is to have MH and MH+ nearly degenerate and sin2(β − α) close to 1 . For comparison purposes, we will analyze
the bounds arisen from the theoretical predictions based on the DH , J , N  and TY  calculations of
µ (h.v.p.), which are the most representative and recent ones. We would like to note that, as observed through Fig.
2 to Fig. 6, the bounds from the J  and N  calculations are almost indistinguishable.
In Figs. 2-4 we show the allowed regions in the tanβ −MAplane for both types of THDMs. In Fig. 2, which shows
the low MAregime, it can be seen clearly that even if one considers the DH calculation of ahad
the smallest error, there is still the possibility of having a CP-odd scalar with a mass of the order of 0.2 GeV in the
type-II THDM as long as tanβ < 1.43, whereas for a type-I THDM tanβ has to be greater than 0.87. This is a very
significant change with respect to the results obtained when using the old (uncorrected) value of atheory
for each different evaluation
µ. Those bounds on aNP
µ , which is the one with
. In that case,
1Throughout this work we will systematically compose the errors in quadrature.
2For a summary of the most recent evaluations of ahad
, see Refs.  and .
the DH calculation did not allow for a light CP-odd scalar in either type of THDM, though other calculations did
allow such a possibility.
0 0.1 0.2
FIG. 2: The regions (above the curves for type-I and below the curves for type-II THDM) in the tanβ versus MA plane allowed
by the aµ data at the 95% CL. Four different curves are displayed depending whether the SM prediction is obtained from the
DH, J, N or TY calculation of ahad
µ (h.v.p.). The two-loop contribution from the light A has been used.
J or N
FIG. 3: The region (above the curves) in the tanβ versus MA plane of a type-I THDM allowed by the aµ data at the 95%
CL. The allowed regions based on the DH, J, N and TY calculations are above the curves. The two-loop contribution from the
light A has been used.
As stated above, so far our results have been derived from the two-loop contribution from the CP-odd scalar to aNP
It is also interesting to repeat the above analysis using only the one-loop calculation for aNP
in Figs. 5 and 6. The old (uncorrected) theory prediction based on the DH calculation required any new physics
contribution to aµto be positive. However, the one-loop contribution from a light CP-odd scalar is always negative.
Therefore, the old SM theory prediction for aµcombined with the THDM one-loop correction strongly disfavored the
existence of a very light CP-odd scalar. This is to be contrasted with the conclusion drawn from the corrected value
. In that case, there is indeed an allowed region of tanβ when MA∼ 0.2 GeV, though this region is smaller
than the one allowed by the two-loop calculation of aNP
(cf. Figs. 3 and 5, and Figs. 4 and 6). As shown in Fig. 4,
there is an interesting feature in the tanβ versus MAplane of a type-II THDM when MAis around 2.6GeV. It is
because for MA∼ 2.6GeV, the two-loop contribution from a light CP-odd scalar becomes as large as the respective
one-loop contribution but with an opposite sign, so the total effect cancels.
µ. Its result is depicted
A.Bounds on tanβ from meson decays
For completeness we now turn to analyze the bounds obtained on THDMs with a very light CP-odd scalar from
meson decays. A very light Higgs scalar (CP-odd or CP-even) can be a decay product of some hadrons, like the η
and Υ mesons. For the latter, a measured upper bound to the X + γ decay channel has been set  that can be
J or N
FIG. 4: The regions (below the curves) in the tanβ versus MA plane of a type-II THDM allowed by the aµ data at the 95%
CL. The allowed regions based on the DH, J, N and TY calculations are below the curves. The two-loop contribution from the
light A has been used.
J or N
FIG. 5: Same as Fig. 3, but only the one-loop contribution from the light A is considered.
0.01 0.11 10100
J or N
FIG. 6: Same as Fig. 4, but only the one-loop contribution from the light A is considered.
used to constrain the A¯bb coupling. Denote the Yukawa coupling of A¯bb to be kdmb/v, with kd= tanβ (cotβ) in
the type-II (type-I) model. Then, the data of the meson decay Υ → γ + X requires kd< 1. (We refer the reader to
Refs. [3, 4] for a detailed discussion.)
As shown in Ref. , there is another decay process that can strongly constrain tanβ, namely η → πS, where S
is a very light CP-even scalar. Those results can be translated into the case of a CP-odd scalar. In particular, the
experimental upper limit
BR(η → π0e+e−) ≤ 5 × 10−5
can be used to obtain the following constraint on a THDM CP-odd scalar with mass MAlying in the range 2me≤
≤ 1.5 (6)
where ku is cotβ for either type-I or type-II THDM and kd has been defined above. The function λ is given by
λ2(a,b,c) = a2+ b2+ c2− 2ab − 2ac − 2bc. From here we can conclude that cotβ ≥ 0.65 for type-I THDM and
0.55 ≤ tanβ ≤ 1.8 for type-II THDM. We thus can confirm that the hadron decay data together with the muon
(g − 2) measurement require tanβ to be of order 1 if there exists a very light pseudoscalar with a mass smaller than
III.OVERALL DESCRIPTION OF THE GENERAL THDM WITH A LIGHT A
Once the allowed parameter range for tanβ and MA has been updated, there remains five other parameters to
consider: the CP-even neutral Higgs mixing angle α, the soft breaking term µ12and the three other Higgs masses:
Mh, MH and MH+. Since we already know that tanβ has to be of order 1 we can address the status of the charged
Higgs mass MH+ independently of the other parameters. It turns out that both the b → sγ and the Rbdata require
H+to be considerably heavy [4, 29]:
∼350 GeV . (7)
Such a high lower bound for the H+mass affects the allowed values of the mixing angle α. In Ref.  we show
that the ρ parameter requires MH and MH+ to be very correlated depending on the value of sin2(β − α). In fact,
if a very light CP-odd scalar is to be allowed, the easiest way to satisfy the bound imposed by ρ ∼ 1 is to have
MH and MH+ degenerate and sin2(β − α) = 1. With this choice, Mhis not restricted since it does not contribute
to the ρ parameter. As we consider values of sin2(β − α) smaller than 1, it turns out that ρ is very sensitive to the
masses of H and H+. For instance, if sin2(β − α) = 0.5, MHmust be at least of the order of 500 GeV . Generally
speaking, our conclusion on the bounds on a very light CP-odd scalar in the THDM does not change significantly for
0.5 < sin2(β − α) < 1 as long as the other Higgs bosons in the model are heavy enough. For a very small value of
sin2(β − α), much less than 0.5, the ρ-parameter data would have required the mass of H to be at the TeV order.
In conclusion, with the recent correction to the SM prediction of aµ, the current muon (g − 2) data, together with
other precision data (cf. Table I), still allows a light (MA∼ 0.2GeV) CP-odd scalar boson in the THDM. Due to this
new development in the SM theory calculation of muon’s (g−2) , the allowed range of tanβ in the Type-I or Type-II
THDM is modified, and our result is summarized in Table V. It is interesting to note that the phenomenology at
high energy colliders predicted by the THDM with a light CP-odd Higgs boson is dramatically different from that
predicted by the usual THDM in which the mass of the CP-odd scalar is at the weak scale. A detailed discussion
on this point can be found in Ref. . In particular, various potential discovery modes were studied in there: it was
found that the Fermilab Tevatron, the CERN large hadron collider (LHC) and the planned e+e−linear collider (LC)
have a great potential to either detect or exclude a very light A in the THDM.
Finally, we note that while a light CP-odd scalar in THDM is still compatible with all the precision data, it has been
shown recently in Ref.  that a light CP-odd scalar in the MSSM will violate the constraint derived from the Zb¯b
coupling. This is because in the MSSM, the masses of the five Higgs bosons are related by the mass relations required
by supersymmetry. Hence, with a light CP-odd scalar, the mass of the other Higgs bosons cannot be arbitrary large,
and it is difficult to yield the decoupling limit when calculating low energy observables.
TABLE V: Constraints on tanβ from the muon (g − 2) data for Type-I and Type-II THDM, with MA = 0.2 GeV, based on
various SM theory predictions of ahad
Theory prediction Type-I THDM Type-II THDM
DH  tanβ > 0.87
J tanβ > 0.54
N tanβ > 0.53
TY tanβ > 0.73
µ (h.v.p). The two-loop contribution for the CP-odd scalar has been used.
tanβ < 1.43
tanβ < 2.19
tanβ < 2.24
tanβ < 1.67
During the review process of this manuscript, the muon (g − 2) collaboration announced a new result based on
data collected in the year 2000 , in which the experimental uncertainty has been reduced to one half that of
the previous measurement while the central value of aexp
11659204(7)(5) × 10−10
(0.7 ppm).] According to the latest experimental data, we have updated Figs. 2 to 6 in
this paper to Figs. 7 to 11. The new data suggests that a very light CP-odd scalar is not allowed in the type-I or
type-II THDM based on the SM calculation done by DH  and TY . However, based on the N  and J 
calculations, a very light CP-odd scalar is still possible though the allowed parameter space of the THDM has been
remains about the same. [The new data yields aexp
FIG. 7: Same as Fig. 2, but with the latest experimental data from the muon (g − 2) collaboration . There is no allowed
region in this range of parameters according to the DH  and TY  calculations.
FIG. 8: Same as Fig. 3, but with the latest experimental data from the muon (g − 2) collaboration . There is no allowed
region in this range of parameters according to the DH  and TY  calculations.
FIG. 9: Same as Fig. 4, but with the latest experimental data from the muon (g − 2) collaboration . The region allowed
by the DH  and TY  calculations is bounded by the respective lines.
0.01 0.11 10100
FIG. 10: Same as Fig. 5, but with the latest experimental data from the muon (g − 2) collaboration . There is no allowed
region in this range of parameters according to the DH  and TY  calculations.
FIG. 11: Same as Fig. 6, but with the latest experimental data from the muon (g − 2) collaboration considered . There is
no allowed region in this range of parameters according to the DH  and TY  calculations.
FL would like to thank Conacyt and SNI (M´ exico) for support. GTV acknowledges support from SEP-PROMEP.
The work of CPY was supported in part by NSF grant PHY-0100677.
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