harvmac \#1[Ann. Phys. [**[\#1]{}**]{}]{} \#1[Comm. Math. Phys. [**[\#1]{}**]{}]{} \#1[Int. J. Mod. Phys. [**A[\#1]{}**]{}]{} \#1[J. Math. Phys. [**[\#1]{}**]{}]{} \#1[Mod. Phys. Lett. [**A[\#1]{}**]{}]{} \#1[Nucl. Phys. [**B[\#1]{}**]{}]{} \#1[Phys. Lett. [**B[\#1]{}**]{}]{} \#1[Phys. Rev. [**[\#1]{}**]{}]{} \#1[Phys. Rev. [**D[\#1]{}**]{}]{} \#1[Phys. Rep. [**C[\#1]{}**]{}]{} \#1[Phys. Rev. Lett. [**[\#1]{}**]{}]{} \#1[Prog. Theor. Phys. [**[\#1]{}**]{}]{} \#1[Rev. Mod. Phys. [**[\#1]{}**]{}]{} \#1[Suppl. Prog. Theor. Phys. [**[\#1]{}**]{}]{} \#1[Theor. Math. Phys. [**[\#1]{}**]{}]{}

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Kazunori Takenaga[^1] [E-mail address: f77498a@kyu-cc.cc.kyushu-u.ac.jp]{}

Department of Physics, Kyushu University

Fukuoka, 812 JAPAN

The oblique parameters $S$ and $T$ are studied in electroweak theory with two massless Higgs doublets. The effect of quadratic dependence on the charged Higgs mass in the parameter $T$ could be cancelled. The constraint on the charged Higgs mass from the parameter $T$ is not so stringent as it was expected. The parameter $S$ can be both positive and negative but it can’t take large negative value. We search allowed scalar mass regions for both $S$ and $T$ in the case of two different reference points.

The precise electroweak measurements at LEP have tested the standard model, and they have given constraint on the top quark mass $M_t\simeq
131^{+47}_{-28}$ GeV \[\]. Now the electroweak measurements reach a sensitivity at which they are probing effects of new physics beyond the standard model or constraining it. If new heavy particles exist and couple weakly only to light fermions, then their effect on low-energy electroweak phenomena appears through contributions to gauge-boson self-energies. This type of radiative correction is called an oblique correction. The oblique corrections are summarized by three parameters $S$, $T$ and $U$ \[\]. These parameters are defined in terms of the gauge-boson self-energies. The $S$, $T$ and $U$ are measures of the size of the radiative corrections arising from new physics beyond the standard model. The neutral current and many low-energy observables depend on $S$ and $T$, and the only measured weak interaction observable which depends on $U$ is $M_W$. $U$ is in general suppressed compared with $T$ by a factor $M_Z^2/M_{new}^2$, where $M_{new}$ is a typical mass scale of new physics. $U$ is often predicted to be very small. So one usually assumes $U\simeq 0$ and restricts $S$ and $T$ by the neutral current, $M_W$ and the other low-energy observables. Experimental limits on $S$ and $T$ differ depending on the values of $M_t$ and $M_H$ chosen as the reference standard model. It seems that negative values of both $S$ and $T$ are favored  though experimental and theoretical uncertainties in atomic parity violation are significant for $S$ \[\].

The standard model is working quite well, however, there are questions to be answered. The nature of the Higgs sector is one of the most mysterious problems. The symmetry-breaking mechanism is still unknown. The Higgs sector in the standard model is responsible for breaking the $SU(2)\times U(1)$ symmetry and for making gauge-bosons and fermions massive. A lot of models, multi-Higgs-doublet models, technicolor models, supersymmetric models $\cdots$, have been proposed with their own properties in the Higgs sector. None of new particles in these models, however, have been detected yet. Recently, by using the oblique parameter $S$ and $T$ in addition to the highly accurate low-energy measurements, it has become possible to impose some constraints on various kinds of models beyond the standard model. Originally, Peskin and Takeuchi estimated the contribution coming from QCD-like techicolor model to $S$ and $T$. They found that the predictions for $S$ in the model were outside experimentally allowed regions . Many authors performed estimations of the oblique parameters in various kinds of models . The oblique parameters are useful and transparent tools for probing or constraining new physics beyond the standard model.

In this paper we shall study the oblique parameters in electroweak theory with two massless Higgs doublets. In a previous paper \[\] we investigated phase structure of this model by analyzing the effective potential at finite temperature. We found that the phase transition was of first order and that the critical temperatures were significantly low compared with those in the standard model for wide ranges of scalar masses. We also estimated the “critical” temperatures for bubble nucleation and they were very close to the critical temperatures. We discussed that this model had many attractive features for electroweak baryogenesis even though there is no new CP violation. In this model there are four kinds of scalar particles: charged Higgs $H^{\pm}$, CP odd Higgs $A$, neutral Higgs (CP even) $h$ and scalon $H$. Their masses are denoted by $M_{H^{\pm}}, M_A, M_h$ and $M_H$, respectively. All these masses are free parameter except for $M_H$. We will find some constraints on these scalar masses by studying the oblique parameters $S$ and $T$ in this model. This model has two Higgs doublets which are massless by assumption. The oblique parameters in general two-Higgs-doublet model were studied in and the full expressions for them were obtained. But numerical analyses for these parameters are very difficult because there are six free parameters appearing in the expressions for the oblique parameters . Then one usually considers some special cases such that there are presumable mass hierarchies among new scalar particles. In the model with two-massless-Higgs doublets the situation changes drastically. This is because symmetry breaking occurs by radiative corrections to yield a non-trivial mass relation between the masses of scalon and other particles. In order to break the $SU(2)\times U(1)$ symmetry to the $U(1)_{em}$, we have to impose flatness condition on the Higgs potential along a suitable direction in VEV space of the Higgs fields \[\]. The scalon, which is associated with scale invariance of this model, becomes massive after symmetry breaking. After rotating the Higgs doublets such that only one kind of the Higgs field takes the VEV, the scalar mass terms in the Higgs potential are automatically diagonalized. As a result there is no angle dependence in the oblique parameters. In general two-Higgs-doublet model we need two different angles to diagonalize the scalar mass terms, which are usually denoted by $\alpha$ for neutral(CP even) sector and $\beta$ for charged and neutral(CP odd) sector, so that the oblique parameters depend on these two angles . In our case these two angles are the same because of the special direction in the VEV space and a flatness condition mentioned above. Thus the oblique parameters in our model is written in terms of only two free parameters. The parameter $T$ has quadratic dependences on the charged Higgs mass $M_{H^{\pm}}$ . It is usually believed that the constraint on $M_{H^{\pm}}$ from $T$ is strict due to its quadratic dependence. But we find that large mass differences between $M_{H^{\pm}}$ and two scalar masses $M_A, M_h$ could cancel the $M_{H^{\pm}}^2$ contribution to $T$. Therefore $M_{H^{\pm}}$ is not so restricted from $T$ as it was expected. On the other hand the parameter $S$ can be both positive and negative. Heavy charged Higgs can shift $S$ to negative value but this is impossible when the neutral scalars are heavy because of the mass relation in this model mentioned above. We find that it is impossible to obtain a large negative $S$.

In the next section we briefly review the Higgs sector of electroweak theory with two massless Higgs doublets. In section 3 the full expressions for the oblique parameters in our model are presented and the behaviors of $S$ and $T$ are discussed briefly. In section 4 we show the results of numerical analyses of $S$ and $T$ for two different reference points and discuss allowed regions for the scalar masses.

We work in the usual $SU(2)\times U(1)$ gauge theory of electroweak model with two massless Higgs doublets $\Phi_1,\Phi_2$ and with 3 generations of quarks and leptons. There exist dangerous tree-level flavour-changing neutral current interactions mediated by scalar exchanges if we allow the most general Yukawa couplings. In order to avoid it, we restrict allowed Yukawa couplings by imposing a discrete symmetry\[\]: $\Phi_2 \rightarrow -\Phi_2, \quad u_{RA} \rightarrow -u_{RA}$. Then we have the following Yukawa couplings: where ${\tilde\Phi}_2=i{\tau}^2 \Phi^*_2$ , with $
\tau^a (a=1,2,3)$ being Pauli matrices. $Q_{LA}, l_{LA}$ are $SU(2)$ doublets of quarks and leptons respectively. A subscript $A$ distinguishes generations. The discrete symmetry must also be imposed on the Higgs self-interactions, otherwise the Yukawa couplings   lose its meaning \[\]. Then the Higgs potential is written as In our model the Higgs potential is a homogeneous polynomial of Higgs fields so that spontaneous symmetry breaking does not occur at the tree-level. We must consider at least the one-loop potential. In the previous paper we discussed how to obtain one-loop effective potential in this model. Calculating the contribution from one-loop order to a potential in a model with more than one Higgs doublets is not so straightforward as it appears. Discussions are given in . Here we quote only results. The direction in the VEV space of the Higgs fields, which is consistent with the final $U(1)_{em}$ invariance, is determined as where $n_{01}$ and $n_{02}$ are defined by $$n_{01}^2\equiv {\sqrt \lambda_2 \over{\sqrt \lambda_1 + \sqrt
\lambda_2}},\quad n_{02}^2\equiv {\sqrt \lambda_1 \over{\sqrt \lambda_1 +
\sqrt
\lambda_2}}.$$ The parameter $\rho$ is fixed by $v^2=1/{{\sqrt 2}G_F}$ after spontaneous symmetry breaking. Here $G_F$ is the Fermi coupling constant. The flatness condition on the Higgs potential is satisfied if The condition  guarantees the minimum value of the tree-level potential to be zero. Now we perform a rotation of the Higgs fields such that only one Higgs field takes VEV: Here $\beta$ is defined by $$\tan \beta \equiv{{\rho n_{01}}\over{\rho n_{02}}}
\equiv {v_1 \over v_2}, \qquad {\rm with}\quad v_1^2+v_2^2=v^2.$$ $C_{\beta}$ and $S_{\beta}$ are abbreviations for $\cos\beta$ and $\sin\beta$ respectively. It is clear that $\ph_2$ plays the same role of the Higgs in the standard model and $\ph_1$ represents an independent extra doublet. We expand $\ph_1$ and $\ph_2$ around the VEV as follows: Inserting  into the Higgs potential, we find that the scalar mass terms are given by where $g$, $g'$ and $\xi$ are $SU(2),U(1)$ and gauge-parameter respectively. The scalar masses in  are given as $$M_{H^{\pm}}^2(v) \equiv{1\over
2}({\sqrt{\lambda_1\lambda_2}}+\lambda_3)v^2,\quad
M_A^2(v)  \equiv -\lambda_5 v^2,\quad
M_h\equiv {\sqrt {\lambda_1 \lambda_2}}v^2.$$ The scalon denoted by $H$ becomes massive after spontaneous symmetry breaking. We identify the scalon as the usual Higgs boson in the standard model. Its mass is defined by the inverse propagator evaluated at zero momentum and is given by Here we have ignored fermion masses except for the top quark mass. $V_1$ is the effective potential of our model in one-loop approximation. Detailed results are given in .

The oblique parameters $S$, $T$ and $U$ are defined in terms of gauge-boson self-energies: where we adopt the notation of Peskin and Takeuchi , and $e$ is the electromagnetic charge, $\alpha_{em}$ is the electromagnetic fine structure constant, and $s^2\equiv
\sin^2\theta_W, c^2\equiv \cos^2\theta_W$. The indices $1 \sim 3$ in the vacuum polarization amplitudes refer to $SU(2)_L$ currents, while $Q$ refers to the $U_{em}(1)$ current. Note that the $S$, $T$ and $U$ do not include the standard model contributions. The oblique parameters defined above describe the effects of radiative corrections from new physics beyond the standard model. In our model the new physics contributions to  are isolated from those of the standard model. Only new scalar particles $M_{H^{\pm}}, M_A$ and $M_h$ contribute to . It is straightforward to calculate the contributions to . The results are where we have introduced functions $f$ and $g$ defined by Note that $f$ and $g$ vanish for $a=b$ because both functions are proportional to square of mass difference $(a-b)^2$ when $|a-b|\ll a,b$. We see from that $T$ has quadratic dependence on the charged Higgs mass $M_{H^{\pm}}$:

The second and the third terms in square brackets in  are important for smearing the leading quadratic mass dependence when mass differences between $M_A$ and $M_{H^{\pm}}$ or $M_h$ and $M_{H^{\pm}}$ become large. This is because when $a\gg b$, the second term in the function $f$ behaves as The right hand side in  becomes large enough if $b$ takes appropriate large value. So the second and the third terms could drive $T$ to negative value and smear out the leading quadratic dependence on $M_{H^{\pm}}$. The value of $M_{H^{\pm}}$ in this case must be relatively small, otherwise the term $M_{H^{\pm}}$ would become dominant contribution to $T$. In our model, however, relatively small charged Higgs mass can be obtained automatically when the neutral Higgs scalars become heavy. From  we see that the charged Higgs boson $M_{H^{\pm}}^2$ is essentially written in terms of two neutral scalars $M_A, M_h$: Then as $M_A, M_h$ become heavier, $M_{H^{\pm}}$ becomes lighter. Moreover, inserting  into $T$ we can reduce free parameters to two, [*i.e.*]{}, $M_A$ and $M_h$. So it is easy to perform numerical analyses in terms of two free parameters and one can simultaneously obtain the charged Higgs mass by  each time $M_h,M_A$ are fixed. The fourth term in a square bracket contributes to $T$ additively when mass difference $M_A-M_h$ is large.

On the other hand, the function $g$ in $S$ increases monotonously as mass differences become large. The terms that could be negative in $S$ is the second term only in square brackets. In order to make this term negative, the argument in the logarithm must be smaller than one. For small value of the neutral scalar masses, this is possible by . But it is impossible for heavy neutral scalar masses. The contributions from new scalar particles to $S$ are all logarithmic so that a large changes of the value in $S$ can not be expected. $S$ also depends on only two free parameters by  as in $T$. It is necessary for us to make clear the reference point to define the parameters $S$ and $T$. We choose two different reference points: ${\rm  I} (M_t,M_H)=(140,100){\rm GeV},  {\rm  II} (M_t,M_H)=(140,300){\rm
GeV}$. At these reference points the experimental limits on $S$ and $T$ are given by The cesium data on atomic parity violation is included in both cases. We see that $S$ and $T$ are both negative. In analysis we assume $M_A\ge M_h$ without loss of generality because $f$ and $g$ are symmetric under $M_A \leftrightarrow M_h$. In order to make the formulae  valid, the scale of new physics must be larger than $M_Z$. We set the minimum value of $M_h$ and $M_A$ to be $140$ GeV for illustration. Perturbative triviality bounds on scalar masses in two-Higgs-doublet model are discussed in \[\], and are given as $M_{H^{\pm}}, M_A, M_h < 650
\sim 700$ GeV. We follow the results and set the upper bounds on the scalar masses to be $650$ GeV.

First, let us consider the case [I]{}. In this case the Higgs boson is relatively light so that it is natural to expect that other scalars are not extremely heavy. Actually, it is seen from  that extremely heavy neutral scalars give the imaginary charged Higgs mass. The maximum values of $M_A$ and $M_h$, that make the charged Higgs mass real, are $M_A=M_h \simeq 401$ GeV. So we assume that the upper bound on $M_A$ is $450$ GeV. This gives us a perturbative bound on $M_A$. In we display the allowed mass regions for $M_{H^{\pm}}$ and $M_h$.

[**Fig. 1**]{}

The minimum and maximum values for $M_h, M_A$ and $M_{H^{\pm}}$ in the allowed regions are $${\eqalign{
         M_h(min) &=140 {\rm {GeV}},            \cr
         M_A(min) &=369 {\rm {GeV}},            \cr
 M_{H^{\pm}}(min) &\simeq 223 (252) {\rm GeV}, \cr  }
\quad \eqalign{
          M_h(max) &=291 {\rm {GeV}},       \cr
          M_A(max) &=461 (450) {\rm {GeV}}, \cr
 M_{H^{\pm}}(max)  &\simeq 358 {\rm GeV}.   \cr}
}$$ Large mass regions for $M_{H^{\pm}}$ are allowed even though $T$ has the quadratic dependence on $M_{H^{\pm}}$. The charged Higgs boson mass is not so restricted as it was expected. The value in parenthesis corresponds to the case where we put the perturbative bound on $M_A$. In we show the allowed scalar mass regions from the experimental limit on $S$ at the reference point [I]{}.

[**Fig. 2**]{}

From fig. 2 we see that the allowed mass regions for $M_A$ and $M_h$ are very narrow and located in light mass regions. This is because if neutral scalars are heavy, then the charged Higgs boson is light by . So in this case there are no terms that give a negative value in $S$. We find that $${\eqalign{
M_h(min)&=140 {\rm {GeV}}, \cr M_A(min)&=140 {\rm {GeV}}, \cr} \quad
\eqalign{
M_h(max)&=155 {\rm {GeV}}, \cr
 M_A(max)&=170 {\rm {GeV}}. \cr}
}$$ The mass of the charged Higgs boson can not vary so much and its allowed mass region is almost a point: $M_{H^{\pm}} \simeq 400$ GeV. If we combine the constrains on the scalar masses from both $S$ and $T$, we see that there is no allowed mass regions of the scalar masses. The experimental limit on $S$ is very strict in this case if we believe the face value of $S$. But the cesium data on atomic parity violation, which is significant for $S$, is still ambiguous from both experimental and theoretical points of view. The contribution from atomic parity violation to $S$ is very sensitive and its data gives $S_{Q_W}=-2.7\pm 2.0\pm1.1$ by itself. Reducing the total uncertainties in $S$ is very important as mentioned in  not only for our model but also for other models. If we convert the allowed mass regions from $T$ into the values of $S$ we have $S$ is bounded above by a small positive value. This positive value is easily obtained due to the uncertainties in $S_{Q_W}$. At this stage the constrains on scalar masses from $T$ are more reliable than those from $S$. In and we show the changes of the values in $T_{min(max)}, S_{min(max)}$ against $M_h$ in order to clarify the possible values of $S$ and $T$ in our model.

[**Fig. 3**]{},

In this calculation the perturbative bound on $M_A$ is assumed and we stop the calculation at $M_A=M_h=401$ GeV in order to avoid the imaginary charged Higgs mass.

The same analyses are performed at the reference point [II]{}. As in the reference point [I]{} we find the allowed mass regions for $M_{H^{\pm}},M_h$ from $T$ which are displayed in .

[**Fig. 5**]{}

In this calculation we set the triviality bounds on the scalar masses discussed below eq. . We obtain $${\eqalign{
M_h(min)&=  140 {\rm  GeV}, \cr
M_A(min)&=  593 {\rm  GeV}, \cr
M_{H^{\pm}}(min)&\simeq  544 {\rm  GeV}, \cr}
\quad \eqalign{
M_h(max)&=  554 {\rm  GeV}, \cr
M_A(max)&= 650 {\rm  GeV},\cr
M_{H^{\pm}}(max)&\simeq 616 {\rm  GeV}.\cr}
}$$ As for $S$ at this reference value there is no allowed mass regions that explain the experimental limit on $S$. If we convert the allowed mass regions from $T$ into the value of $S$ we have This upper bound on $S$ is obtainable due to the uncertainties in atomic parity violation as in the case of [I]{}. Fig. 6 and fig. 7 correspond to fig. 3 and fig. 4 in different reference point.

[**Fig. 6**]{},

We have not assumed any mass hierarchies among the scalar particles. If we assume $M_{H^{\pm}}=M_A$ the custodial symmetry exists in the Higgs potential. In this case the contribution to $T$ is zero because of the property of the function $f$. On the other hand, $S$ has the non-trivial contributions from the scalar particles . The changes of the values in $S$ are shown in , for the reference values [I]{}, [II]{} respectively.

[**Fig. 8**]{},

The possible values of $S$ at the reference point [I]{} for $M_h=140\sim
450$ GeV are $$\eqalign{
-1.64\times 10^{-2} &\le S_{custodial} \le 1.97 \times 10^{-2},  \cr
 247{\rm GeV}&\le M_{H^{\pm}}=M_A \le 362 {\rm GeV}.
}$$ At the reference point [II]{} we have $$\eqalign{
-1.98\times 10^{-2} &\le S_{custodial} \le 0.52 \times 10^{-2},  \cr
541{\rm GeV}&\le M_{H^{\pm}}=M_A \le 616 {\rm GeV} \cr
}$$ for $M_h=140\sim 650$ GeV. Both positive and negative value are possible. The behavior of $S_{custodial}$ is similar with those of $S$ without the custodial symmetry so that it does not take large negative values.

We have studied the oblique parameters $S$ and $T$ in electroweak theory with two massless Higgs doublets at the two different reference points. We have performed numerical analyses of $S$ and $T$ by using the specific properties in this model. We have found that the effects of quadratic dependence on the charged Higgs mass in $T$ could be cancelled and the constraint on $M_{H^{\pm}}$ from the experimental limit on $T$ is not so stringent for both [I]{} and [II]{} as it was expected. On the other hand, the experimental limit on $S$ for both [I]{} and [II]{} are very severe for our model if we believe the face values . We can not say that there are no allowed mass regions for the scalar particles that satisfy the experimental limits on both $S$ and $T$ in this model because large uncertainties from atomic parity violation exist in $S$. It is interesting that the future experiments reduce the uncertainties to the accuracy such that we can conclude whether our model is excluded or not.

[**Acknowledgments**]{}

The author would like to thank A.Kakuto for careful reading of the manuscript. He also thanks K. Inoue and T. Inami for valuable discussions.

[^1]: $^\dagger$
