$B\to D^{(*)}\tau\nu_\tau$ in the 2HDM with an anomalous $\tau$ coupling

TL;DR

This paper investigates the $R(D^{(*)})$ anomalies in B meson decays within the two-Higgs-doublet model, introducing an anomalous tau coupling to better fit experimental data from BaBar, Belle, and LHCb.

Contribution

It proposes a modified 2HDM with an anomalous tau coupling to explain the $R(D^{(*)})$ puzzle, showing all model types fit data similarly and improving fit with normalized ratios.

Findings

All four types of the model yield similar minimum chi-squared values.

Normalized $R(D^{(*)})$ with $B o au u$ data reduces chi-squared significantly.

The anomalous tau coupling provides a better fit to experimental results.

Abstract

The puzzle of $R(D^{(*)})$ associated with $B\to D^{(*)}\tau\nu$ decay is addressed in the two-HIggs-doublet model. An anomalous coupling of $\tau$ to the charged Higgs is introduced to fit the data from BaBar, Belle, and LHCb. It is shown that all of the four types of the model yield similar values of the minimum $\chi^2$. Also shown is that the newly normalized $R(D^{(*)})$ with the branching ratio of $B\to\tau\nu$ decay exhibits much smaller minimum $\chi^2$.

Tables

Table 1. Table 1: ξ 𝜉 \xi s for each type of 2HDM.

	${\xi }_A^u$	${\xi }_A^d$	${\xi }_A^{\ell }$
Type-I	$\cot \beta$	$-\cot \beta$	$-\cot \beta$
Type-II	$\cot \beta$	$\tan \beta$	$\tan \beta$
Type-X	$\cot \beta$	$-\cot \beta$	$\tan \beta$
Type-Y	$\cot \beta$	$\tan \beta$	$-\cot \beta$

Table 2. Table 2: Experimental data for R ​ ( D ( ∗ ) ) 𝑅 superscript 𝐷 R({D^{(*)}}) and ℬ ​ ( B → τ ​ ν ) ℬ → 𝐵 𝜏 𝜈 {\cal B}(B\to\tau\nu) . For R ​ ( D ( ∗ ) ) 𝑅 superscript 𝐷 {R(D^{(*)})} measurements the uncertainties are ± plus-or-minus \pm (statistical) ± plus-or-minus \pm (systematic).

	$R\left(D\right)$	$R\left(D^{\ast }\right)$	$ℬ\left(B\to \tau \nu \right)$
BABAR	$0.440\pm 0.058\pm 0.042$	$0.332\pm 0.024\pm 0.018$ BaBar1	${1.83}_{-0.49}^{+0.53}\times {10}^{-4}$ BaBar2
Belle(2015)	$0.375\pm 0.064\pm 0.026$	$0.293\pm 0.038\pm 0.015$ Belle1	$\left(1.25\pm 0.28\right)\times {10}^{-4}$ Belle3
Belle(1607)	$-$	$0.302\pm 0.030\pm 0.011$ Belle1607	$-$
Belle(1612)	$-$	$0.270\pm {0.035}_{-0.025}^{+0.028}$ Belle1612	$-$
LHCb	$-$	$0.336\pm 0.027\pm 0.030$ LHCb1	$-$

Table 3. Table 3: R τ ​ ( D ( ∗ ) ) subscript 𝑅 𝜏 superscript 𝐷 R_{\tau}({D^{(*)}}) values for different experiments Nandi . The value of ”Belle(1612)” is a new one.

	$R_{\tau }\left(D\right)\times {10}^3$	$R_{\tau }\left(D^{\ast }\right)\times {10}^3$
BABAR	$2.404\pm 0.838$	$1.814\pm 0.582$
BABAR($\tau$ tag)	$5.96\pm 2.26$	$4.49\pm 3.54$
Belle(2015)	$3.0\pm 1.1$	$2.344\pm 0.799$
Belle($\tau$ tag)	$5.7\pm 3.3$	$4.49\pm 3.54$
Belle(1607)	$-$	$2.416\pm 0.794$
Belle(1612)	$-$	$2.160\pm 0.835$

Table 4. Table 4: The best-fit R ​ ( D ( ∗ ) ) 𝑅 superscript 𝐷 {R(D^{(*)})} values with free η 𝜂 \eta for different Types of the model.

Types	I	II	X	Y
$R\left(D\right)$	$0.342$	$0.362$	$0.342$	$0.362$
$R\left(D^{\ast }\right)$	$0.255$	$0.253$	$0.255$	$0.254$
${\chi }_{\text{min}}^2/\text{d.o.f.}$	$2.881$	$2.813$	$2.881$	$2.861$

Table 5. Table 5: The best-fit R τ ​ ( D ( ∗ ) ) subscript 𝑅 𝜏 superscript 𝐷 {R_{\tau}(D^{(*)})} values with free η 𝜂 \eta for different Types of the model.

Types	I	II	X	Y
$R_{\tau }\left(D\right)\times {10}^{-3}$	$2.828$	$2.885$	$2.828$	$2.915$
$R_{\tau }\left(D^{\ast }\right)\times {10}^{-3}$	$2.223$	$2.188$	$2.223$	$2.223$
${\chi }_{\text{min}}^2/\text{d.o.f.}$	$0.623$	$0.614$	$0.623$	$0.615$

Equations

$$R(D^{(\ast )})\equiv \frac{\mathcal{B}(B\to D^{(\ast )}\tau \nu )}{\mathcal{B}(B\to D^{(\ast )}\ell \nu )},$$

$$R(D{)}_{\mathrm{SM}}$$

$$R(D^{\ast }{)}_{\mathrm{SM}}$$

$${\mathcal{L}}_{\mathrm{Yukawa}}$$

$${\xi }_A^{\tau }\to \eta {\xi }_A^{\tau }.$$

$$\mathcal{L}(b\to c\ell \nu )$$

$$g_s^c$$

$$f_s^{\ell }$$

$$\mathcal{L}(B\to \tau \nu )$$

$$g_s^u$$

$$f_s^{\tau }$$

$${\Gamma }^{D^{(\ast )}}={\Gamma }_{\mathrm{SM}}^{D^{(\ast )}}+{\Gamma }_{\mathrm{mix}}^{D^{(\ast )}}+{\Gamma }_{H^{\pm }}^{D^{(\ast )}}.$$

$$\frac{d{\Gamma }_{\mathrm{SM}}^D}{ds}$$

$$\frac{d{\Gamma }_{\mathrm{mix}}^D}{ds}$$

$$\frac{d{\Gamma }_{H^{\pm }}^D}{ds}$$

$$P_D\equiv \frac{\sqrt{s^2+m_B^4+m_D^4-2(s{m}_B^2+s{m}_D^2+m_B^2{m}_D^2)}}{2m_B},$$

$$F_0$$

$$F_1$$

$$V_1(w)$$

$$S_1(w)$$

$$w$$

$${\rho }_D^2$$

$$\frac{d{\Gamma }_{\mathrm{SM}}^{D^{\ast }}}{ds}$$

$$\frac{d{\Gamma }_{\mathrm{mix}}^{D^{\ast }}}{ds}$$

$$\frac{d{\Gamma }_{H^{\pm }}^{D^{\ast }}}{ds}$$

$$H_{\pm }(s)$$

$$H_0(s)$$

$$H_s(s)$$

$$A_1(w^{\ast })$$

$$A_0(w^{\ast })$$

$$A_2(w^{\ast })$$

$$V(w^{\ast })$$

$$w^{\ast }=\frac{m_B^2+m_{D^{\ast }}^2-s}{2m_B{m}_{D^{\ast }}},r_{D^{\ast }}=\frac{2\sqrt{m_B{m}_{D^{\ast }}}}{m_B+m_{D^{\ast }}},$$

$$h_{A_1}(w^{\ast })$$

$$R_0(w^{\ast })$$

$$R_1(w^{\ast })$$

$$R_2(w^{\ast })$$

$${\rho }_{D^{\ast }}^2$$

$$R_1(1)$$

$$\mathcal{B}(B\to \tau \nu )=\mathcal{B}(B\to \tau \nu {)}_{\mathrm{SM}}{\left(1+r_{H^{\pm }}\right)}^2,$$

Full text

I Introduction

One of the most interesting puzzles in flavor physics in recent years has been the excess of the semitaunic $B$ decays, $B\to{D^{(*)}}\tau\nu_{\tau}$. The excess is well expressed in terms of the ratio

[TABLE]

where ${\cal B}$ is the branching ratio. The standard model (SM) prediction is Na ; Fajfer

[TABLE]

The BaBar Collaboration has reported that the measured $R(D)$ exceeds the SM prediction by $2.0\sigma$, while $R(D^{*})$ exceeds the SM prediction by $2.7\sigma$, and the combined significance of the disagreement is $3.4\sigma$ BaBar1 ; BaBar_PRL . BaBar analyzed the possible effect of a charged Higgs boson in the Type-II two-Higgs-doublet model (2HDM), and excluded the model at the 99.8% confidence level.

The Belle measurements of ${R(D^{(*)})}$ are slightly smaller than those of BaBar, but still larger than the SM expectations Belle1 ; Belle1607 . Interestingly, Belle’s results are compatible with the Type-II 2HDM in the $\tan\beta/m_{H^{\pm}}$ region around $0.45c^{2}/\text{GeV}$ (where $\beta$ is the ratio of the two vacuum expectation values of the 2HDM) and zero Belle1 , and recent measurements of $R(D^{*})$ are consistent with the SM predictions Belle1612 On the other hand, LHCb reported that $R(D^{*})$ is larger than the SM predictions by $2.1\sigma$ LHCb1 .

In this paper we try to fit the global data on ${R(D^{(*)})}$ with the 2HDM of all types. The 2HDM is a natural extension of the SM Higgs sector, so it has been tested to fit the ${R(D^{(*)})}$ puzzle Andreas ; Fazio ; Cline ; Koerner . Out of all the types of 2HDM, the Type-II model is the most promising because the new physics (NP) effects are involved with the coupling of $\tan^{2}\beta$ while for other types the couplings are 1 or $1/\tan^{2}\beta$. As mentioned before, there is tension between BaBar and Belle regarding the compatibility of the Type-II 2HDM to the data. In this analysis we introduce an anomalous $\tau$ coupling to the charged Higgs Dhargyal . Since the NP effects are enhanced by new couplings and suppressed by the charged Higgs mass, the new couplings should be large enough to allow a heavy charged Higgs to fit the data. We also investigate possible roles of leptonic decay $B\to\tau\nu$ to solve the ${R(D^{(*)})}$ puzzle. It was suggested that the normalized ${R(D^{(*)})}$ with ${\cal B}(B\to\tau\nu)$, ${R_{\tau}(D^{(*)})}$ are consistent with the SM Nandi . We implement the global $\chi^{2}$ fitting to ${R(D^{(*)})}$ as well as ${R_{\tau}(D^{(*)})}$ with the anomalous $\tau$ coupling.

The paper is organized as follows. Section II introduces the 2HDM with the anomalous $\tau$ coupling to describe $B\to{D^{(*)}}\tau\nu$ and $B\to\tau\nu$ transitions. In Sec. III ${R(D^{(*)})}$ and ${R_{\tau}(D^{(*)})}$ are expressed in the 2HDM with the new coupling. Our results and discussions are given in Sec. IV, and conclusions follow in Sec. V.

II 2HDM with anomalous $\tau$ couplings

The Yukawa interaction in the 2HDM is given by Aoki

[TABLE]

where $v=\sqrt{v_{1}^{2}+v_{2}^{2}}=246$ GeV, $v_{1,2}$ are the vacuum expectation values (VEVs) of the scalar fields $\Phi_{1,2}$ of the 2HDM with $\tan\beta\equiv v_{2}/v_{1}$, and the $\xi$s are the couplings defined in Table 1.

Here we introduce an anomalous factor $\eta$ to enhance $\xi_{A}^{\tau}$ Dhargyal . The motivation is that $\tau$ is screened from the second Higgs VEV $v_{2}$ and the neutral component of $\Phi_{2}$ by a factor of $\eta$. In this case the tau mass is $\sim Y_{\rm Yukawa}v_{2}/\eta$, effectively enhancing the Yukawa coupling of $\tau$ to $H^{\pm}$, while that of $\tau$ to neutral Higgses remains unchanged. This kind of model can be easily constructed within extra dimensions. For example, as in Refs. Archer ; Agashe , the overlappings between the wave functions of $\tau$ and the neutral component of $\Phi_{2}$ over the extra dimension would determine the strength of the $\tau$ coupling to the neutral part of $\Phi_{2}$. We could simply assume that the overlapping of $\tau$ and the neutral $\Phi_{2}$ is rather weak compared to other cases. The enhancement occurs for Type-I and Type-Y models because in these models leptons couple only to $\Phi_{2}$. The same thing could happen for $\tau$ and the neutral component of $\Phi_{1}$ to screen $\tau$ from $v_{1}$, resulting in $\eta$ enhancement for $\tau$-$H^{\pm}$ couplings in Type-II and X models. In this work we assume that phenomenologically $\tau$ couplings to $H^{\pm}$ are enhanced by a factor of $\eta$ for all types of the model,

[TABLE]

Now the effective Lagrangian for the $b\to c\ell\nu$ transition is

[TABLE]

where

[TABLE]

For $B\to\tau\nu$ decay,

[TABLE]

where

[TABLE]

Note that $\xi_{A}^{\tau}$ contains the enhancement factor $\eta$, $\xi_{A}^{\tau}=\eta\xi_{A}^{\ell=e,\mu}$.

III $B\to{D^{(*)}}\tau\nu$ and $B\to\tau\nu$ decays

The decay rates of $B\to{D^{(*)}}\ell\nu$ in the 2HDM can be expressed as

[TABLE]

The differential decay rates for $B\to D\ell\nu$ are given by

[TABLE]

where $s=(p_{B}-p_{D})^{2}$ is the momentum-transfer squared, and

[TABLE]

is the momentum of $D$ in the $B$ rest frame. The form factors $F_{0}$ and $F_{1}$ are given by

[TABLE]

where

[TABLE]

with

[TABLE]

For $B\to D^{*}\ell\nu$ decay,

[TABLE]

where $P_{D^{*}}=P_{D}(m_{D}\to m_{D^{*}})$. The form factors are given by

[TABLE]

where

[TABLE]

with

[TABLE]

and

[TABLE]

HereDhargyal

[TABLE]

For the leptonic two-body decay $B\to\tau\nu$, the branching ratio is

[TABLE]

where

[TABLE]

Here $f_{B}$ and $\tau_{B}$ are the decay constant and lifetime of $B$, respectively.

The experimental data are summarized in Table 2 Nandi .

At first we try to fit the data of Table 2 by minimizing $\chi^{2}$. BABAR results BaBar1 already ruled out the Type-II 2HDM. We introduce an anomalous $\tau$ coupling for all types of 2HDM, which will be shown to significantly reduce the $\chi^{2}$ minimum.

In addition, it was suggested that the ratio

[TABLE]

has some advantages in this analysis Nandi . First of all the $\tau$ detection systematics is canceled in the ratio. But it should be noted that the ratio of Eq. (43) introduces the theoretical error on $V_{ub}$.

We use the values of $R_{\tau}({D^{(*)}})$ in Table 3 for the fit.

IV Results and Discussions

In our analysis $\tan\beta$ and $M_{H^{\pm}}$ are by default the fitting parameters to minimize $\chi^{2}$, defined by

[TABLE]

where the $x_{i}$s are model predictions and the $(\mu_{i}\pm\delta\mu_{i})$s are experimental data. Figure 1 shows the $R(D)$ values vs $\chi^{2}$ with the anomalous $\tau$ coupling $\eta$. In Fig. 1(a), $\eta$ is set to be an additional fitting parameter, $-1000\leq\eta\leq 1000$. Plots for the Type-I and Type-X models are overlapped.

As can be seen from Eqs. (III) and (14), the 2HDM contributes to $R(D)$ as

[TABLE]

where the coefficients are omitted for simplicity. For free $\eta$, Types I and X behave similarly because $\xi_{A}^{u}$ and $\xi_{A}^{d}$ are the same (see Table 1). This is also true for Types II and Y. We also consider the case of fixed $\eta\equiv\tan^{2}\beta$ as in Ref. Dhargyal in Fig. 1(b). The dominant contribution to Eq. (45) comes from

[TABLE]

Since the first term is negative for Types I and II for $\eta\equiv\tan^{2}\beta>0$, the $\chi^{2}$ values are very poor compared to those for Types X and Y, as shown in Fig. 1(b). If we allow $\eta=-\tan^{4}\beta$ for the Type-I model, the $\chi^{2}$ distribution over $R(D)$ overlaps with that for the Type-X model. Similar things happen for the Type-II model with $\eta=-\tan^{2}\beta$ and the Type-Y model. In this case, Eq. (45) is not the same for Types II and Y; the sign of $\eta$ is more relevant to the $\chi^{2}$ distribution than the power of $\eta$. We can see that introducing the anomalous $\tau$ coupling improves the $\chi^{2}$ fitting, and any Type of 2HDM model is as good (or bad) as another. The best-fit values of ${R(D^{(*)})}$ and the corresponding minimum $\chi^{2}$ per degree of freedom (d.o.f.) are given in Table 4,

and the allowed region of $R(D)$ and $R(D^{*})$ at the $1\sigma$ level is given in Fig. 2

Figure 3 shows the allowed region of $m_{H^{\pm}}$ vs $\tan\beta$.

In Fig. 3 (a), $\eta$ is a free parameter within $-1000\leq\eta\leq 1000$. In this case $m_{H^{\pm}}$ cannot be large enough because the $R(D)_{H^{\pm}}$ term of Eq. (46) gets smaller and cannot fit the data. One exception is the Type-II model. As shown in Eq. (46), there is a $\tan^{2}\beta$ enhancement for $R(D)_{H^{\pm}}$, which allows $m_{H^{\pm}}$ to be large. If we require that the charged Higgs mass is $m_{H^{\pm}}\gtrsim 500$ GeV, only the Type-II model survives in Fig. 3(a). In Fig. 3 (b) we fix $\eta\equiv\pm\tan^{\alpha}\beta$ for some $\alpha$. For Types X and Y, the allowed stripe stretches to larger $m_{H^{\pm}}$ with smaller $\tan\beta$ as $\alpha$ goes from 2 to 3. This is because $R(D)_{H^{\pm}}\sim\eta m_{b}m_{\tau}/m_{H^{\pm}}^{2}+\left(\eta m_{b}m_{\tau}/m_{H^{\pm}}^{2}\right)^{2}$. Also shown in Fig. 3(b) are the Type-I model with $\eta=-\tan^{4}\beta$ and the Type-II model with $\eta=-\tan^{2}\beta$ for comparison. It would be expected from Eq. (46) that stripes for the Type-X and Y models are coincident up to $\sim{\cal O}(m_{c}/m_{b})$. They also overlap with the stripe of the Type-I model with $\eta=-\tan^{4}\beta$. The stripe for the Type-II model with $\eta=-\tan^{2}\beta$ lies in the lowest region of $\tan\beta$ since there is already a $\tan^{2}\beta$ term in $R(D)_{H^{\pm}}$.

Now we turn to the ${R_{\tau}(D^{(*)})}$. Figure 4 shows $R_{\tau}(D)$ vs $\chi^{2}$.

Note that the minimum $\chi^{2}$ reduces significantly compared to Fig. 1; $\chi^{2}_{\text{min}}/\text{d.o.f.}=$ 0.623 (Type-I, X), 0.614 (Type-II) 0.615 (Type-Y) for free $\eta$ in Fig. 4(a). As discussed in Ref. Nandi , ${R_{\tau}(D^{(*)})}$ values from the BABAR and Belle results are consistent with each other and not so far from the SM predictions Nandi ,

[TABLE]

In Fig. 4(b) we fix $\eta=\pm\tan^{\alpha}\beta$ for some $\alpha$. Any Type of the model is as good as another. $R_{\tau}(D^{*})$ vs $\chi^{2}$ shows similar behavior. The new contribution to ${\cal B}(B\to\tau\nu)$ is

[TABLE]

where terms of ${\cal O}(m_{u}/m_{b})$ are neglected. As in Eq. (46), only the combination of $\xi_{A}^{d}\xi_{A}^{\tau}$ is relevant, and thus the Type-I model with $\eta=-\tan^{4}\beta$ looks much like the Type-X models with $\eta=\tan^{2}\beta$, and so on.

Table 5 shows the best-fit values of ${R_{\tau}(D^{(*)})}$ and $\chi^{2}_{\text{min}}/\text{d.o.f}$,

and Fig. 5 shows the allowed region of $R_{\tau}(D)$ and $R_{\tau}(D^{*})$ at the $1\sigma$ level.

Figure 6 shows the allowed region in the $m_{H^{\pm}}$-$\tan\beta$ plane to fit the ${R_{\tau}(D^{(*)})}$.

In Fig. 6(a), $\eta$ is a free parameter. For Types X and Y, almost the entire region is allowed. The different behaviors of the Type I and II models are due to the factors of $\eta/\tan^{2}\beta$ (Type-I) and $\eta\tan^{2}\beta$ (Type-II) in Eq. (46).

In Fig. 6(b), $\eta\equiv\pm\tan^{\alpha}\beta$ for some $\alpha$. Compared to Fig. 3(b), each Type shows similar behavior, but with broader bands. The reason is that the ${R_{\tau}(D^{(*)})}$ values are more consistent with each other than ${R(D^{(*)})}$ ones, and thus more points in the $m_{H^{\pm}}$-$\tan\beta$ plane are allowed around $\chi^{2}_{\text{min}}$. And the bands for Types X and Y with $\eta=\tan^{2}\beta$ stretch to the region of $m_{H^{\pm}}=1000$ GeV.

V Conclusions

In this work we tried to solve the puzzle of ${R(D^{(*)})}$ in the 2HDM. We introduced $\eta$ as an anomalous $\tau$ coupling to $H^{+}$ to fit the data through minimizing $\chi^{2}$. To fit the excess of the data over the SM predictions it needs to enhance the charged Higgs contributions, which come in the form of ${R(D^{(*)})}_{H^{\pm}}\sim\eta m_{b}m_{\tau}/m_{H^{\pm}}^{2}+(\eta m_{b}m_{\tau}/m_{H^{\pm}}^{2})^{2}$. Thus, for small values of $\eta$, $m_{H^{\pm}}$ cannot be large enough to avoid detection. For the Type-II the situation is alleviated because there is already a factor of $\tan^{2}\beta$ (but with opposite sign) in ${R(D^{(*)})}_{H^{\pm}}$. As shown in Fig. 3(b), a large $m_{H^{\pm}}\sim 1000$ GeV is allowed if ${R(D^{(*)})}_{H^{\pm}}\sim(m_{b}m_{\tau}/m_{H^{\pm}}^{2})\tan^{3}\beta+(m_{b}m_{\tau}/m_{H^{\pm}}^{2})^{2}\tan^{6}\beta$ in any Type of 2HDM model.

The new ratios ${R_{\tau}(D^{(*)})}$ fit much better. Contributions of the form $\sim(m_{b}m_{\tau}/m_{H^{\pm}}^{2})\tan^{2}\beta+(m_{b}m_{\tau}/m_{H^{\pm}}^{2})^{2}\tan^{4}\beta$ allow a large $m_{H^{\pm}}\sim 1000$ GeV as in Fig. 6(b), which is not true for the ${R(D^{(*)})}$ fitting. In both cases of ${R(D^{(*)})}$ and ${R_{\tau}(D^{(*)})}$ fitting, any type of 2HDM is as good as another with an appropriate $\eta$. For a sufficiently large $m_{H^{\pm}}\gtrsim 1000$ GeV, new contributions of the form $\sim(m_{b}m_{\tau}/m_{H^{\pm}}^{2})\tan^{k}\beta$ with $k=2$ fit the data well for ${R_{\tau}(D^{(*)})}$, while $k\geq 3$ for ${R(D^{(*)})}$. It should be noted that the errors in ${R_{\tau}(D^{(*)})}$ are still large.