**3.3 Simulations of the** *H*<sup>±</sup> → *tb* **decay mode**

Finally, for the decay chain *H*<sup>±</sup> → *tb*, recall that the interaction term of the charged Higgs with the *t* and *b* quarks in the 2HDM of type II, as given by Ref.[10], is:

$$\mathcal{L} = \frac{g(R\_{33}^{-1})^{-1}}{2\sqrt{2}\ m\_W} V\_{tb} H^+ \overline{t} \left( m\_t \cot \beta (1 - \gamma\_5) + m\_b \tan \beta (1 + \gamma\_5) \right) b + h.c. \tag{27}$$

For the hadroproduction process *gb* → *tH*<sup>±</sup> (see Fig.3) with the decay mechanism *H*<sup>±</sup> → *tb*, the cross section for *gb* → *tH*<sup>±</sup> can be written as:

$$
\sigma(gb \to tH^{\pm}) \propto (R\_{33}^{-1})^{-2} \left( m\_t^2 \cot^2 \beta + m\_b^2 \tan^2 \beta \right) \,. \tag{28}
$$

Therefore, the decay width of *<sup>H</sup>*<sup>−</sup> <sup>→</sup> ¯*tb* is given by:

$$\Gamma(H^{-} \to \overline{\text{fb}}) \simeq \frac{3}{8} \frac{m\_{H^{\pm}}(R\_{33}^{-1})^{-2}}{\pi \upsilon^{2}} \left[ \left( m\_{t}^{2} \cot^{2} \beta + m\_{b}^{2} \tan^{2} \beta \right) \left( 1 - \frac{m\_{t}^{2}}{m\_{H^{\pm}}^{2}} - \frac{m\_{b}^{2}}{m\_{H^{\pm}}^{2}} \right) - \frac{4m\_{t}^{2} m\_{b}^{2}}{m\_{H^{\pm}}^{2}} \right]$$

$$\times \left[ 1 - \left( \frac{m\_{t} + m\_{b}}{m\_{H^{\pm}}} \right)^{2} \right]^{1/2} \left[ 1 - \left( \frac{m\_{t} - m\_{b}}{m\_{H^{\pm}}} \right)^{2} \right]^{1/2},\tag{29}$$

where the factor 3 takes into account the number of colours. The final state of the hadroproduction process contains two top quarks, one of which we required to decay semi-leptonically to provide the trigger, *t* → *νb* ( = *e*, *μ*), and the other hadronically, ¯*<sup>t</sup>* <sup>→</sup> *jjb*. The main background comes from *<sup>t</sup>*¯*tb* and *<sup>t</sup>*¯*tq* production with *<sup>t</sup>*¯*<sup>t</sup>* <sup>→</sup> *WbWb* <sup>→</sup> *<sup>ν</sup>bjjb*.

As such, we have used the production channel *pp* → *tH*<sup>±</sup> for this decay, and have tried to reconstruct the charged Higgs mass. That is, we have the following decay chain:

$$pp \to tH^{\pm} \to t(tb) \to (\ell \nu\_{\ell} b)(\langle jb \rangle b \to \ell \langle \rangle bbb\nu \,. \tag{30}$$

The procedure we have used in reconstructing the masses is:

pp → tH±(→ τ ν) pp → tH±(→ tb)

Constraining the Couplings of a Charged Higgs to Heavy Quarks 43

R-133

 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

<sup>33</sup> and tan *β* with fixed *mH*<sup>±</sup> =300 GeV [11].

<sup>−</sup>0.49 (*stat*) <sup>+</sup>0.46

0.5


> 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Fig. 6. Contour plots of the cross-sections for the processes *pp* → *tH*±(→ *τν*) (left) and

*Br*(*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>*+*τ*−*ν*¯*τ*) = 0.86 <sup>±</sup> 0.24 <sup>±</sup> 0.11 <sup>±</sup> 0.06% [30], which is consistent, within experimental uncertainties, with the SM, and with Belle [31]. Note also that the inclusive *b* → *cτν* branching ratio was determined at the LEP experiments [32]. The *B* → *τν* process has a smaller

from *fB*, the *B* meson decay constant, which from lattice QCD is *fB* = 191 ± 13 MeV. As such, the measurement of these processes will be important targets in coming *B* factory experiments. In order to test for the charged Higgs fermion couplings, we now determine the charged Higgs contributions to tauonic *B* decays, where it is straightforward to write down the amplitudes

to square of the charm Yukawa couplings, and since the branching ratio can change only by at most a few percent, we shall neglect such contributions here. Also, as we shall work with

We can now calculate the charged Higgs effect on the *B* → *Dτν* branching ratio, by utilising the vector and scalar form factors of the *B* → *D* transition. These are obtained using the

where *G*<sup>S</sup> and *G*<sup>P</sup> are scalar and pseudo-scalar effective couplings. These couplings are given from Eqs.(10), (11) and the similarly derived effective Lagrangian for charged leptons:

*<sup>e</sup>* ]33(*Mb*[*R*<sup>ˆ</sup> <sup>−</sup><sup>1</sup>

*<sup>e</sup>* ]33(*Mb*[*R*<sup>ˆ</sup> <sup>−</sup><sup>1</sup>

*Vcbcγμ*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*5)*bτγμ*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*5)*ντ* <sup>+</sup> *<sup>G</sup>*S*cbτ*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*5)*ντ* <sup>+</sup> *<sup>G</sup>*P*cγ*5*bτ*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*5)*ντ*

<sup>−</sup>0.51 (*syst*)) <sup>×</sup> <sup>10</sup>−<sup>4</sup> [33], and at

<sup>−</sup>0.42) <sup>×</sup> <sup>10</sup>−<sup>4</sup> [35]). Note

<sup>22</sup> are proportional

+h.c. , (31)


4

tan

<sup>−</sup>0.61) <sup>×</sup> <sup>10</sup>−5, where theoretical uncertainties came

<sup>→</sup> *<sup>D</sup>*+*τ*−*ν*) and *<sup>B</sup>* <sup>→</sup> *τν* processes. We should first

*<sup>d</sup>* ]22*Vcb* <sup>+</sup> *McVcb* cot<sup>2</sup> *<sup>β</sup>*) , (32)

*<sup>d</sup>* ]22*Vcb* <sup>−</sup> *McVcb* cot<sup>2</sup> *<sup>β</sup>*) , (33)

30 35 40 45 50

1.2

2.5

tan

branching ratio, as measured by Belle at (1.79 <sup>+</sup>0.56

that the SM predicts *Br*(*<sup>B</sup>* <sup>→</sup> *τν*)=(7.57 <sup>+</sup>0.98

BaBar (1.2 <sup>±</sup> 0.4 <sup>±</sup> 0.3 <sup>±</sup> 0.2) <sup>×</sup> <sup>10</sup>−<sup>4</sup> [34] (giving an average of (1.41 <sup>+</sup>0.43

*<sup>τ</sup>*−*<sup>ν</sup>* or *<sup>B</sup>*<sup>0</sup>

large tan *β* values, cot *β* terms can be neglected in the Lagrangian.

effective Lagrangian for *b* → *cτν* operators as given by

*<sup>G</sup>*<sup>S</sup> <sup>≡</sup> tan2 *<sup>β</sup>M<sup>τ</sup>* 2*v*2*M*<sup>2</sup> *H*±

*<sup>G</sup>*<sup>P</sup> <sup>≡</sup> tan2 *<sup>β</sup>M<sup>τ</sup>* 2*v*2*M*<sup>2</sup> *H*±

like to note that the higgsino diagram contributions, see Fig.1(a), to the *R*−<sup>1</sup>

[*R*ˆ <sup>−</sup><sup>1</sup>

[*R*ˆ <sup>−</sup><sup>1</sup>

30 35 40 45 50

0.45

0.55

0.35

R-133

 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0.25

*pp* <sup>→</sup> *tH*±(<sup>→</sup> *tb*) (right) versus *<sup>R</sup>*−<sup>1</sup>

for the *<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν* (*B*<sup>−</sup> <sup>→</sup> *<sup>D</sup>*<sup>0</sup>

<sup>L</sup>eff <sup>=</sup> <sup>−</sup> *<sup>G</sup>*<sup>F</sup>

√2


$$\left(m\_{\vec{j}\vec{l}\vec{b}} - m\_{\mathfrak{t}}\right)^2 + \left(m\_{\ell\vec{\nu}\vec{b}} - m\_{\mathfrak{t}}\right)^2\,\_{\vec{s}}$$

• Finally, we retained the top quarks that satisfy |*mjjb* − *mt*| *<* 12GeV and |*m�ν<sup>b</sup>* − *mt*| *<* 12GeV. This leaves two top quarks and one *b*-jet. There can be two possible combinations, where we retained both. It should be noted that only one of the combinations is the true combination (the combination that emerged from a charged Higgs), the other combination being combinatorial backgrounds.

Using these techniques we can now generate the correlation plot of the two LHC processes considered here, see Fig.6. In these plots we have considered three different values of *R*−<sup>1</sup> <sup>33</sup> , where these lines of constant *R*−<sup>1</sup> <sup>33</sup> are generated from three values of tan *β* (that is, tan *β* = 30, 40 and 50). Note that though this mode has a much larger branching ratio than *H*<sup>±</sup> → *τν*, it has at least three *b*-jets in its final state. As such, the combinatorial backgrounds associated with this channel make it a challenging task to work with [10], and not the best discovery channel for a charged Higgs at the LHC.

## **4. Charged Higgs at** *B***-factories**

Having now reviewed how a massive charged Higgs may be detected at the LHC, we shall now place greater constraints on the charged Higgs parameters by utilising the successful *B* factory results from KEK and SLAC. Note that *B* physics shall be a particularly fertile ground to place constraints on a charged Higgs. For example, it is well known that limits from *b* → *sγ* can give stronger constraints in generic 2HDMs than in SUSY models [29]!

The *B* decays of most interest here are those including a final *τ* particle, namely *B* → *Dτν* and *B* → *τν*[8]. An important feature of these processes is that a charged Higgs boson can contribute to the decay amplitude at tree-level in models such as the 2HDM and the MSSM. From the experimental perspective, since at least two neutrinos are present in the final state (on the signal side), a full-reconstruction is required for the *B* decay on the opposite side. For the *B* → *Dτν* process, the branching fraction has been measured at BaBar with 14 Will-be-set-by-IN-TECH

• We initially searched for one isolated lepton (both electrons and muons) with at least three tagged *b*-jets (this is done in order to include processes like *gg* → *tbH*) and at least two non *b*-jets. Furthermore, we used the cuts, where for *b* and non-*b* jets we used the same *pT*

*<sup>T</sup> >* 30GeV and |*η*| *<* 2.5. • Next we tried to reconstruct the *W* mass (where the *W* originates from the top decay) in both leptonic (*W* → *�ν*) and hadronic (*W* → *jj*) decays. For the leptonic decay we attributed the missing *pT* to the emergence of neutrinos from the leptonic *W* decay. Using the actual *W* mass we then reconstructed the longitudinal neutrino momentum. This gives a two fold ambiguity, both corresponding to the actual *W* mass, and neglecting the event if it gives an unphysical solution. Choosing both solutions the second *W* is reconstructed in the jet mode. We constructed all possible combinations of non-*b* jets and have plotted the invariant mass of the jets (*mjj*), retaining only those combinations of jets which are consistent with |*mjj* − *mW*| *<* 10GeV. Note that the rescaling is done by scaling the four

• We then attempted to reconstruct the top quarks, where we have, at present, reconstructed two *W* bosons and three tagged *b* jets. There can be six different combinations of *W*'s and

(*mjjb* <sup>−</sup> *mt*)<sup>2</sup> + (*m�ν<sup>b</sup>* <sup>−</sup> *mt*)<sup>2</sup> .

• Finally, we retained the top quarks that satisfy |*mjjb* − *mt*| *<* 12GeV and |*m�ν<sup>b</sup>* − *mt*| *<* 12GeV. This leaves two top quarks and one *b*-jet. There can be two possible combinations, where we retained both. It should be noted that only one of the combinations is the true combination (the combination that emerged from a charged Higgs), the other combination

Using these techniques we can now generate the correlation plot of the two LHC processes considered here, see Fig.6. In these plots we have considered three different values of *R*−<sup>1</sup>

30, 40 and 50). Note that though this mode has a much larger branching ratio than *H*<sup>±</sup> → *τν*, it has at least three *b*-jets in its final state. As such, the combinatorial backgrounds associated with this channel make it a challenging task to work with [10], and not the best discovery

Having now reviewed how a massive charged Higgs may be detected at the LHC, we shall now place greater constraints on the charged Higgs parameters by utilising the successful *B* factory results from KEK and SLAC. Note that *B* physics shall be a particularly fertile ground to place constraints on a charged Higgs. For example, it is well known that limits from *b* → *sγ*

The *B* decays of most interest here are those including a final *τ* particle, namely *B* → *Dτν* and *B* → *τν*[8]. An important feature of these processes is that a charged Higgs boson can contribute to the decay amplitude at tree-level in models such as the 2HDM and the MSSM. From the experimental perspective, since at least two neutrinos are present in the final state (on the signal side), a full-reconstruction is required for the *B* decay on the opposite side. For the *B* → *Dτν* process, the branching fraction has been measured at BaBar with

can give stronger constraints in generic 2HDMs than in SUSY models [29]!

*b*-jets that can give top quarks. As such, we chose the top quarks which minimise

*<sup>j</sup>* = *pj* × *mW*/*mjj*.

<sup>33</sup> are generated from three values of tan *β* (that is, tan *β* =

<sup>33</sup> ,

cuts, *p<sup>e</sup>*

*<sup>T</sup> >* 20GeV, *p*

*μ*

*<sup>T</sup> >* 6GeV, *p*

momenta of the jets with the *W* mass, that is, *p*�

being combinatorial backgrounds.

channel for a charged Higgs at the LHC.

**4. Charged Higgs at** *B***-factories**

where these lines of constant *R*−<sup>1</sup>

*j*

Fig. 6. Contour plots of the cross-sections for the processes *pp* → *tH*±(→ *τν*) (left) and *pp* <sup>→</sup> *tH*±(<sup>→</sup> *tb*) (right) versus *<sup>R</sup>*−<sup>1</sup> <sup>33</sup> and tan *β* with fixed *mH*<sup>±</sup> =300 GeV [11].

*Br*(*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>*+*τ*−*ν*¯*τ*) = 0.86 <sup>±</sup> 0.24 <sup>±</sup> 0.11 <sup>±</sup> 0.06% [30], which is consistent, within experimental uncertainties, with the SM, and with Belle [31]. Note also that the inclusive *b* → *cτν* branching ratio was determined at the LEP experiments [32]. The *B* → *τν* process has a smaller branching ratio, as measured by Belle at (1.79 <sup>+</sup>0.56 <sup>−</sup>0.49 (*stat*) <sup>+</sup>0.46 <sup>−</sup>0.51 (*syst*)) <sup>×</sup> <sup>10</sup>−<sup>4</sup> [33], and at BaBar (1.2 <sup>±</sup> 0.4 <sup>±</sup> 0.3 <sup>±</sup> 0.2) <sup>×</sup> <sup>10</sup>−<sup>4</sup> [34] (giving an average of (1.41 <sup>+</sup>0.43 <sup>−</sup>0.42) <sup>×</sup> <sup>10</sup>−<sup>4</sup> [35]). Note that the SM predicts *Br*(*<sup>B</sup>* <sup>→</sup> *τν*)=(7.57 <sup>+</sup>0.98 <sup>−</sup>0.61) <sup>×</sup> <sup>10</sup>−5, where theoretical uncertainties came from *fB*, the *B* meson decay constant, which from lattice QCD is *fB* = 191 ± 13 MeV. As such, the measurement of these processes will be important targets in coming *B* factory experiments.

In order to test for the charged Higgs fermion couplings, we now determine the charged Higgs contributions to tauonic *B* decays, where it is straightforward to write down the amplitudes for the *<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν* (*B*<sup>−</sup> <sup>→</sup> *<sup>D</sup>*<sup>0</sup> *<sup>τ</sup>*−*<sup>ν</sup>* or *<sup>B</sup>*<sup>0</sup> <sup>→</sup> *<sup>D</sup>*+*τ*−*ν*) and *<sup>B</sup>* <sup>→</sup> *τν* processes. We should first like to note that the higgsino diagram contributions, see Fig.1(a), to the *R*−<sup>1</sup> <sup>22</sup> are proportional to square of the charm Yukawa couplings, and since the branching ratio can change only by at most a few percent, we shall neglect such contributions here. Also, as we shall work with large tan *β* values, cot *β* terms can be neglected in the Lagrangian.

We can now calculate the charged Higgs effect on the *B* → *Dτν* branching ratio, by utilising the vector and scalar form factors of the *B* → *D* transition. These are obtained using the effective Lagrangian for *b* → *cτν* operators as given by

$$\mathcal{L}\_{\rm eff} = -\frac{G\_{\rm F}}{\sqrt{2}} V\_{\rm cb} \overline{c} \gamma\_{\mu} (1 - \gamma\_{5}) b \overline{\tau} \gamma^{\mu} (1 - \gamma\_{5}) \nu\_{\tau} + G\_{\rm S} \overline{c} b \overline{\tau} (1 - \gamma\_{5}) \nu\_{\tau} + G\_{\rm P} \overline{c} \gamma\_{5} b \overline{\tau} (1 - \gamma\_{5}) \nu\_{\tau}$$
 
$$+ \text{h.c.} \quad (31)$$

where *G*<sup>S</sup> and *G*<sup>P</sup> are scalar and pseudo-scalar effective couplings. These couplings are given from Eqs.(10), (11) and the similarly derived effective Lagrangian for charged leptons:

$$\mathbf{G}\_{\mathbf{S}} \equiv \frac{\tan^2 \beta M\_{\mathbf{\bar{r}}}}{2v^2 M\_{H^{\pm}}^2} [\hat{\mathcal{R}}\_{\varepsilon}^{-1}]\_{33} (M\_b [\hat{\mathcal{R}}\_d^{-1}]\_{22} V\_{cb} + M\_{\mathbf{\bar{c}}} V\_{cb} \cot^2 \beta) \,\,\,\,\tag{32}$$

$$\mathcal{G}\_{\rm P} \equiv \frac{\tan^2 \beta M\_{\rm \tau}}{2v^2 M\_{H^{\pm}}^2} [\hat{\mathcal{R}}\_{\rm \varepsilon}^{-1}]\_{33} (M\_b[\hat{\mathcal{R}}\_d^{-1}]\_{22} V\_{cb} - M\_{\rm \varepsilon} V\_{cb} \cot^2 \beta) \, , \tag{33}$$

Using the matrix elements

by the following replacement.

*<sup>g</sup>* and *<sup>E</sup>*�

generalization of [*R*ˆ <sup>−</sup><sup>1</sup>

*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν* (*B*<sup>−</sup> <sup>→</sup> *<sup>D</sup>*<sup>0</sup>

seen from Eq.(36) where *R*−<sup>1</sup>

tan *<sup>β</sup>* and *<sup>R</sup>*<sup>−</sup><sup>1</sup>

*g*

where *E*(*i*)

the decay width of *B* → *τν* in the SM is given by:

<sup>Γ</sup>[*<sup>B</sup>* <sup>→</sup> *τντ*]*SM* <sup>=</sup> *<sup>G</sup>*<sup>2</sup>

[*R*ˆ <sup>−</sup><sup>1</sup> *<sup>d</sup>* ] 22 →

[*R*ˆ <sup>−</sup><sup>1</sup> *<sup>d</sup>* ] 11 →

the above equations are approximately the same because *E*�

contribution can be neglected in the evaluation with the [*R*ˆ <sup>−</sup><sup>1</sup>

*<sup>d</sup>* and *mH*<sup>±</sup> = 300GeV, have been given.

varied tan *β* in the range 30 *<* tan *β <* 50 for different values of *R*−<sup>1</sup>

ratios in Fig.7(b) gives the same line for different values of *R*−<sup>1</sup>

*<sup>d</sup>* ]<sup>11</sup> <sup>≈</sup> [*R*<sup>ˆ</sup> <sup>−</sup><sup>1</sup>

*<sup>τ</sup>*−*<sup>ν</sup>* or *<sup>B</sup>*<sup>0</sup>

**5. Determination of the effective couplings**

Γ[*B* → *τντ*]2*HDM* = Γ[*B* → *τντ*]*SM* ×

values for Super*B*, that is, we shall use *f*<sup>B</sup> = 200 ± 30MeV in our numerics.

�0|*uγμγ*5*b*|*B*−� <sup>=</sup> *i f*<sup>B</sup> *<sup>p</sup><sup>μ</sup>* ,

Constraining the Couplings of a Charged Higgs to Heavy Quarks 45

�0|*uγ*5*b*|*B*−� = −*i f*<sup>B</sup>

<sup>2</sup> *f* <sup>2</sup> *Bm*<sup>2</sup> *<sup>τ</sup>mB* 

*F* <sup>8</sup>*<sup>π</sup>* <sup>|</sup>*Vub*<sup>|</sup>

which in the presence of a charged Higgs boson, is modified by a multiplicative factor to:

in the effective limits we have adopted. Note that our input parameters are the projected

Note that this link can be understood by recalling that in our generalized case of MFV, that is Eq.(16), the scalar and pseudo-scalar couplings, Eqs.(32), (33), (37), and (38) can be obtained

<sup>1</sup> + [*Eg*

<sup>1</sup> + [*Eg*

Using these results we have generated the contour plots in Fig.7(b), where a correlation of the

Collecting our numerical results from section 3 and the branching ratios calculated in the previous subsection, we have generated the plots in Figs.7 and 8. In these figures we can see correlations of the LHC cross-sections with the two *B* processes, where in these plots we have

shows the correlation of the LHC observables, whilst the correlation of *B*-decay branching

*ii* and tan *<sup>β</sup>* arise from the same combination (<sup>≡</sup> *<sup>R</sup>*−<sup>1</sup>

1

1

(3) <sup>−</sup> *<sup>E</sup>*� *g* (32)

(3) <sup>−</sup> *<sup>E</sup>*� *g* (31)

*M*<sup>2</sup> B *Mb* ,

<sup>1</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *B m*2 *H*±

] tan *β*

] tan *β*

*g*

*<sup>d</sup>* ]<sup>11</sup> and [*R*<sup>ˆ</sup> <sup>−</sup><sup>1</sup>

*<sup>d</sup>* ]22, which follows from fact that the higgsino diagram

<sup>→</sup> *<sup>D</sup>*+*τ*−*ν*) and *<sup>B</sup>* <sup>→</sup> *τν* branching ratios for various values of

(31) <sup>≈</sup> *<sup>E</sup>*�

*g* (32)

*<sup>d</sup>* ]22.

(*ij*) were defined in section 2.1. Notice that the right-handed sides of

<sup>1</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *τ m*2 *B* <sup>2</sup>

tan2 *β*

<sup>2</sup>

, (39)

, (40)

, (41)

, (42)

*ii* (*ii* = 11, 33). Fig.7(a)

*ii* tan<sup>2</sup> *<sup>β</sup>*) in

<sup>11</sup> . The reason for this can be

. This is the

where we shall now omit a prime from the fields in mass eigenstates. Recall that we shall neglect higgsino diagram contributions to the [*R*ˆ <sup>−</sup><sup>1</sup> *<sup>d</sup>* ]<sup>22</sup> proportional to the square of the charm Yukawa couplings, and also neglect the last terms in *G*<sup>S</sup> and *G*P.

In the heavy quark limit, these form factors can be parameterized by a unique function called the Isgur-Wise function. From the semi-leptonic decays *B* → *Dlν* and *B* → *D*∗*lν* (*l* = *e*, *μ*), the Isgur-Wise function is obtained in a one-parameter form, including the short distance and 1/*MQ* (*Q* = *b*, *c*) corrections. The short distance corrections for *B* → *Dτν* have also been calculated previously [36]. Here we adopt this Isgur-Wise function, but do not include the short distance and the 1/*MQ* corrections for simplicity.

Using the definitions,

$$x \equiv \frac{2p\_{\rm B^\*D}}{p\_{\rm B}^2} \; \; \; y \equiv \frac{2p\_{\rm B^\*o}}{p\_{\rm B}^2} \; \; \; r\_{\rm D} \equiv \frac{M\_{\rm D}^2}{M\_{\rm B}^2} \; \; \; r\_o \equiv \frac{M\_o^2}{M\_{\rm B}^2} \; \; \; \tag{34}$$

the differential decay width is given by

$$\frac{d^2\Gamma[B\to D\pi\nu]}{dxdy} = \frac{G\_\mathrm{F}^2|V\_{cb}|^2}{128\pi^3}M\_\mathrm{B}^5\rho\_\mathrm{D}(x,y) \; \; \; \tag{35}$$

where

$$\begin{split} \rho\_{\mathrm{D}}(\mathbf{x},y) & \equiv \left[ |f\_{+}|^{2}g\_{\mathrm{I}}(\mathbf{x},y) + 2\mathrm{Re}(f\_{+}f\_{-}^{\prime\prime})g\_{\mathrm{Z}}(\mathbf{x},y) + |f\_{-}^{\prime\prime}|^{2}g\_{\mathrm{3}}(\mathbf{x}) \right], \\ \rho\_{\mathrm{D}}(\mathbf{x},y) & \equiv (3-\mathbf{x}-2y-r\_{\mathrm{D}}+r\_{\mathrm{o}})(\mathbf{x}+2y-1-r\_{\mathrm{D}}-r\_{\mathrm{o}}) - (1+\mathbf{x}+r\_{\mathrm{D}})(1+r\_{\mathrm{D}}-r\_{\mathrm{o}}-\mathbf{x}) \,, \\ g\_{\mathrm{Z}}(\mathbf{x},y) & \equiv r\_{\mathrm{o}}(3-\mathbf{x}-2y-r\_{\mathrm{D}}+r\_{\mathrm{o}}) \; \\ g\_{\mathrm{Z}}(\mathbf{x}) & \equiv r\_{\mathrm{o}}(1+r\_{\mathrm{D}}-r\_{\mathrm{o}}-\mathbf{x}) \; \\ f\_{-}^{\prime} & \equiv \left\{ f\_{-}-\Delta\_{\mathrm{S}}[f\_{+}(1-r\_{\mathrm{D}})+f\_{-}(1+r\_{\mathrm{D}}-\mathbf{x})] \right\} \; \\ f\_{\pm} & \equiv \pm \frac{1\pm\sqrt{r\_{\mathrm{D}}}}{2\sqrt[4]{r\_{\mathrm{D}}}} \xi(w), \; \; \; \; \boldsymbol{w} = \frac{\mathbf{x}}{2\sqrt{r\_{\mathrm{D}}}} \; \; . \end{split}$$

Here Δ<sup>S</sup> ≡ <sup>√</sup>2*G*S*M*<sup>2</sup> B *<sup>G</sup>*F*VcbM<sup>τ</sup>* (*Mb*−*Mc* ). We use the following form of the Isgur-Wise function.

$$\begin{aligned} \xi(w) &= 1 - 8\rho\_1^2 z + (51\rho\_1^2 - 10)z^2 - (252\rho\_1^2 - 84)z^3 \\ z &= \frac{\sqrt{w+1} - \sqrt{2}}{\sqrt{w+1} + \sqrt{2}} \ . \end{aligned}$$

For the slope parameter we use *ρ*<sup>2</sup> <sup>1</sup> = 1.33 ± 0.22 [36, 37].

Similarly, for the *B* → *τν* process, the relevant four fermion interactions are those of the *b* → *uτν* type [8]:

$$\mathcal{L}'\_{\rm eff} = -\frac{G\_{\rm F}}{\sqrt{2}} V\_{\rm ub} \overline{u} \gamma\_{\rm l} (1 - \gamma\_{5}) b \overline{\tau} \gamma^{\mu} (1 - \gamma\_{5}) \nu\_{\rm \tau} + G'\_{\rm S} \overline{u} b \overline{\tau} (1 - \gamma\_{5}) \nu\_{\rm \tau} + G'\_{\rm P} \overline{u} \gamma\_{5} b \overline{\tau} (1 - \gamma\_{5}) \nu\_{\rm \tau} \tag{36}$$
 
$$+ \text{h.c.} \tag{36}$$

$$\mathbf{G}'\_{\rm S} \equiv \frac{\tan^2 \beta M\_{\rm \tau}}{2v^2 M\_{H^{\pm}}^2} [\hat{\mathcal{R}}\_{\varepsilon}^{-1}]\_{33} (M\_b[\hat{\mathcal{R}}\_d^{-1}]\_{11} V\_{\rm ub} + M\_{\rm \mu} V\_{\rm ub} \cot^2 \beta) \; , \tag{37}$$

$$\mathbf{G}'\_{\mathbf{P}} \equiv \frac{\tan^2 \beta M\_{\mathbf{r}}}{2v^2 M\_{H^{\pm}}^2} [\hat{\mathbf{R}}\_{\varepsilon}^{-1}]\_{33} (M\_b[\hat{\mathbf{R}}\_d^{-1}]\_{11} V\_{\mathbf{u}b} - M\_{\mathbf{u}} V\_{\mathbf{u}b} \cot^2 \beta) \,. \tag{38}$$

Using the matrix elements

16 Will-be-set-by-IN-TECH

where we shall now omit a prime from the fields in mass eigenstates. Recall that we shall

In the heavy quark limit, these form factors can be parameterized by a unique function called the Isgur-Wise function. From the semi-leptonic decays *B* → *Dlν* and *B* → *D*∗*lν* (*l* = *e*, *μ*), the Isgur-Wise function is obtained in a one-parameter form, including the short distance and 1/*MQ* (*Q* = *b*, *c*) corrections. The short distance corrections for *B* → *Dτν* have also been calculated previously [36]. Here we adopt this Isgur-Wise function, but do not include the

, *<sup>r</sup>*<sup>D</sup> <sup>≡</sup> *<sup>M</sup>*<sup>2</sup>

<sup>F</sup>|*Vcb*| 2 <sup>128</sup>*π*<sup>3</sup> *<sup>M</sup>*<sup>5</sup>

<sup>−</sup> )*g*2(*x*, *<sup>y</sup>*) + | *<sup>f</sup>* �

*g*1(*x*, *y*) ≡ (3 − *x* − 2*y* − *r*<sup>D</sup> + *r*ø)(*x* + 2*y* − 1 − *r*<sup>D</sup> − *r*ø) − (1 + *x* + *r*D)(1 + *r*<sup>D</sup> − *r*<sup>ø</sup> − *x*) ,

*<sup>G</sup>*F*VcbM<sup>τ</sup>* (*Mb*−*Mc* ). We use the following form of the Isgur-Wise function.

<sup>1</sup> = 1.33 ± 0.22 [36, 37]. Similarly, for the *B* → *τν* process, the relevant four fermion interactions are those of the

D *M*<sup>2</sup> B

−|

<sup>1</sup> <sup>−</sup> <sup>10</sup>)*z*<sup>2</sup> <sup>−</sup> (252*ρ*<sup>2</sup>

<sup>1</sup> <sup>−</sup> <sup>84</sup>)*z*<sup>3</sup> ,

<sup>S</sup>*ubτ*(1 − *γ*5)*ντ* + *G*�

*<sup>d</sup>* ]11*Vub* <sup>+</sup> *MuVub* cot2 *<sup>β</sup>*) , (37)

*<sup>d</sup>* ]11*Vub* <sup>−</sup> *MuVub* cot2 *<sup>β</sup>*) . (38)

<sup>2</sup>*g*3(*x*)] ,

, *<sup>r</sup>*<sup>ø</sup> <sup>≡</sup> *<sup>M</sup>*<sup>2</sup> ø *M*<sup>2</sup> B

*<sup>d</sup>* ]<sup>22</sup> proportional to the square of the charm

, (34)

<sup>P</sup>*uγ*5*bτ*(1 − *γ*5)*ντ*

+h.c. , (36)

<sup>B</sup>*ρ*D(*x*, *y*) , (35)

neglect higgsino diagram contributions to the [*R*ˆ <sup>−</sup><sup>1</sup>

short distance and the 1/*MQ* corrections for simplicity.

*<sup>x</sup>* <sup>≡</sup> <sup>2</sup>*p*B·<sup>D</sup> *p*2 B

<sup>2</sup>*g*1(*x*, *y*) + 2Re(*f*<sup>+</sup> *f* �∗

<sup>−</sup> ≡ { *<sup>f</sup>*<sup>−</sup> − <sup>Δ</sup>S[ *<sup>f</sup>*+(<sup>1</sup> − *<sup>r</sup>*D) + *<sup>f</sup>*−(<sup>1</sup> + *<sup>r</sup>*<sup>D</sup> − *<sup>x</sup>*)]} ,

*<sup>ξ</sup>*(*w*), (*<sup>w</sup>* <sup>=</sup> *<sup>x</sup>*

*<sup>ξ</sup>*(*w*) = <sup>1</sup> <sup>−</sup> <sup>8</sup>*ρ*<sup>2</sup>

*z* =

[*R*ˆ <sup>−</sup><sup>1</sup>

[*R*ˆ <sup>−</sup><sup>1</sup>

2 <sup>√</sup>*r*<sup>D</sup> ) .

<sup>√</sup>*<sup>w</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> <sup>√</sup><sup>2</sup> <sup>√</sup>*<sup>w</sup>* <sup>+</sup> <sup>1</sup> <sup>+</sup> <sup>√</sup><sup>2</sup> .

*Vubuγμ*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*5)*bτγμ*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*5)*ντ* <sup>+</sup> *<sup>G</sup>*�

*<sup>e</sup>* ]33(*Mb*[*R*<sup>ˆ</sup> <sup>−</sup><sup>1</sup>

*<sup>e</sup>* ]33(*Mb*[*R*<sup>ˆ</sup> <sup>−</sup><sup>1</sup>

<sup>1</sup>*<sup>z</sup>* + (51*ρ*<sup>2</sup>

the differential decay width is given by

*g*2(*x*, *y*) ≡ *r*ø(3 − *x* − 2*y* − *r*<sup>D</sup> + *r*ø) , *g*3(*x*) ≡ *r*ø(1 + *r*<sup>D</sup> − *r*<sup>ø</sup> − *x*) ,

> <sup>1</sup> <sup>±</sup> <sup>√</sup>*r*<sup>D</sup> 2 <sup>√</sup><sup>4</sup> *<sup>r</sup>*<sup>D</sup>

<sup>√</sup>2*G*S*M*<sup>2</sup> B

For the slope parameter we use *ρ*<sup>2</sup>

, *<sup>y</sup>* <sup>≡</sup> <sup>2</sup>*p*B·<sup>ø</sup> *p*2 B

*dxdy* <sup>=</sup> *<sup>G</sup>*<sup>2</sup>

*<sup>d</sup>*2Γ[*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν*]

Using the definitions,

*ρ*D(*x*, *y*) ≡ [| *f*+|

*f* �

*b* → *uτν* type [8]:

eff <sup>=</sup> <sup>−</sup> *<sup>G</sup>*<sup>F</sup> √2

<sup>S</sup> <sup>≡</sup> tan<sup>2</sup> *<sup>β</sup>M<sup>τ</sup>* 2*v*2*M*<sup>2</sup> *H*±

<sup>P</sup> <sup>≡</sup> tan2 *<sup>β</sup>M<sup>τ</sup>* 2*v*2*M*<sup>2</sup> *H*±

L�

*G*�

*G*�

Here Δ<sup>S</sup> ≡

*f*<sup>±</sup> = ±

where

Yukawa couplings, and also neglect the last terms in *G*<sup>S</sup> and *G*P.

$$
\begin{aligned}
\langle 0|\overline{u}\gamma^{\mu}\gamma\_5 b|B^-\rangle &= if\_\mathsf{B} p^\mu \\
\langle 0|\overline{u}\gamma\_5 b|B^-\rangle &= -if\_\mathsf{B} \frac{M\_\mathsf{B}^2}{M\_b} \ .
\end{aligned}
$$

the decay width of *B* → *τν* in the SM is given by:

$$
\Gamma[B \to \pi \nu\_{\tau}]\_{SM} = \frac{G\_F^2}{8\pi} |V\_{ub}|^2 f\_B^2 m\_\tau^2 m\_B \left(1 - \frac{m\_\tau^2}{m\_B^2}\right)^2,\tag{39}
$$

which in the presence of a charged Higgs boson, is modified by a multiplicative factor to:

$$
\Gamma[B \to \tau \nu\_{\tau}]\_{2HDM} = \Gamma[B \to \tau \nu\_{\tau}]\_{SM} \times \left(1 - \frac{m\_B^2}{m\_{H^\pm}^2} \tan^2 \beta\right)^2,\tag{40}
$$

in the effective limits we have adopted. Note that our input parameters are the projected values for Super*B*, that is, we shall use *f*<sup>B</sup> = 200 ± 30MeV in our numerics.

Note that this link can be understood by recalling that in our generalized case of MFV, that is Eq.(16), the scalar and pseudo-scalar couplings, Eqs.(32), (33), (37), and (38) can be obtained by the following replacement.

$$\left[ [\hat{R}\_d^{-1}]\_{22} \rightarrow \frac{1}{1 + [E\_{\overline{\mathcal{S}}}(^{3}) - E\_{\overline{\mathcal{S}}}'(^{32})] \tan \beta} '\right] \tag{41}$$

$$\left[\hat{\mathsf{R}}\_{d}^{-1}\right]\_{11} \to \frac{1}{1 + \left[E\_{\overline{\mathcal{S}}}^{(3)} - E\_{\overline{\mathcal{S}}}^{\prime(31)}\right] \tan \beta} \,'\,\tag{42}$$

where *E*(*i*) *<sup>g</sup>* and *<sup>E</sup>*� *g* (*ij*) were defined in section 2.1. Notice that the right-handed sides of the above equations are approximately the same because *E*� *g* (31) <sup>≈</sup> *<sup>E</sup>*� *g* (32) . This is the generalization of [*R*ˆ <sup>−</sup><sup>1</sup> *<sup>d</sup>* ]<sup>11</sup> <sup>≈</sup> [*R*<sup>ˆ</sup> <sup>−</sup><sup>1</sup> *<sup>d</sup>* ]22, which follows from fact that the higgsino diagram contribution can be neglected in the evaluation with the [*R*ˆ <sup>−</sup><sup>1</sup> *<sup>d</sup>* ]<sup>11</sup> and [*R*<sup>ˆ</sup> <sup>−</sup><sup>1</sup> *<sup>d</sup>* ]22.

Using these results we have generated the contour plots in Fig.7(b), where a correlation of the *<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν* (*B*<sup>−</sup> <sup>→</sup> *<sup>D</sup>*<sup>0</sup> *<sup>τ</sup>*−*<sup>ν</sup>* or *<sup>B</sup>*<sup>0</sup> <sup>→</sup> *<sup>D</sup>*+*τ*−*ν*) and *<sup>B</sup>* <sup>→</sup> *τν* branching ratios for various values of tan *<sup>β</sup>* and *<sup>R</sup>*<sup>−</sup><sup>1</sup> *<sup>d</sup>* and *mH*<sup>±</sup> = 300GeV, have been given.

#### **5. Determination of the effective couplings**

Collecting our numerical results from section 3 and the branching ratios calculated in the previous subsection, we have generated the plots in Figs.7 and 8. In these figures we can see correlations of the LHC cross-sections with the two *B* processes, where in these plots we have varied tan *β* in the range 30 *<* tan *β <* 50 for different values of *R*−<sup>1</sup> *ii* (*ii* = 11, 33). Fig.7(a) shows the correlation of the LHC observables, whilst the correlation of *B*-decay branching ratios in Fig.7(b) gives the same line for different values of *R*−<sup>1</sup> <sup>11</sup> . The reason for this can be seen from Eq.(36) where *R*−<sup>1</sup> *ii* and tan *<sup>β</sup>* arise from the same combination (<sup>≡</sup> *<sup>R</sup>*−<sup>1</sup> *ii* tan<sup>2</sup> *<sup>β</sup>*) in

the results of the *B*-factory experiments, as demonstrated pictorially in Fig.8. Assuming the charged Higgs boson mass to be known (taken to be 300 GeV in our present analysis) we have obtained cross-section measurement uncertainties as given in table 1. As can be seen from

Constraining the Couplings of a Charged Higgs to Heavy Quarks 47

luminosity. Armed with this information about tan *β*, from the LHC measurements, it can then be taken as an input to the *B*-decay measurements, namely *B* → *τν* and *B* → *Dτν*. In Ref.[10] it was inferred that for large values of tan *β* (≥ 40), measurements to a precision of 6-7% for high luminosity LHC results are possible. Our results are consistent with these observations. Future Super-*B* factories are expected to measure the *B* → *τν* and *B* → *Dτν* to a precision of 4% and 2.5% respectively [38]. The present world average experimental results for tauonic *<sup>B</sup>*-decays are *BR*(*<sup>B</sup>* <sup>→</sup> *τν*)=(1.51 <sup>±</sup> 0.33) <sup>×</sup> <sup>10</sup>−<sup>4</sup> and *BR*(*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν*)/*BR*(*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>μν*) = (41.6 ± 11.7 ± 5.2)% [30, 38]. Presently if one uses UTfit prescription of |*Vub*| then there is substantial disagreement between experimental and SM estimates for the branching fractions of *B* → *τν*. Recently, proposals have been given in Ref.[39] to reduce this tension between experimental and theoretical SM values of *B* → *τν*. Transforming the improved projected theoretical information of these decays along with future Super-*B* factory measurements one

To summarise, we have tried to demonstrate that at the LHC alone it is possible to measure

10%. Combining this information from the LHC with improved *B*-factory measurements, one can measure all four observables indicated in the introduction. These observables represent effective couplings of a charged Higgs boson to the bottom quark and the three generations of up-type quarks, thus demonstrating that it is possible to test the charged Higgs boson couplings to quarks by the combination of low energy measurements at future Super-*B* factories and charged Higgs boson production at the LHC. Something which shall be realisable in the very near future as results from the LHC are already starting to emerge

[1] P. Fayet and S. Ferrara, Phys. Rept. 32, 249 (1977); H.P. Nilles, Phys. Rep. 110, 1 (1984);

[7] W.S. Hou, Phys. Lett. B296, 179 (1992); D. Chang, W.S. Hou and W.Y. Keung, Phys. Rev. D48, 217 (1993); D. Atwood, L. Reina and A. Soni, Phys. Rev. D55, 3156 (1997). [8] H. Itoh, S. Komine and Y. Okada, Prog. Theor. Phys. 114, 179 (2005)

[10] K. A. Assamagan, Y. Coadou and A. Deandrea, Eur. Phys. J. directC 4, 9 (2002)

[11] A. S. Cornell, A. Deandrea, N. Gaur, H. Itoh, M. Klasen, Y. Okada, Phys. Rev. D81,

<sup>22</sup> to a fairly high precision.

[17, 18], and which will require more refined analyses in the near future.

[4] H.E. Haber, G.L. Kane and T. Sterling, Nucl. Phys. B161, 493 (1979).

[9] L. J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D 50, 7048 (1994).

[12] M. Gorbahn, S. Jäger, U. Nierste and S. Trine, arXiv:0901.2065 [hep-ph].

H.E. Haber and G. L. Kane, Phys. Rep. 117, 75 (1985). [2] S.L. Glashow and S. Weinberg, Phys. Rev. D15, 1958 (1977). [3] H. Georgi and D.V. Nanopoulos, Phys. Lett. 82B, 95 (1979).

[5] L.J. Hall and M.B. Wise, Nucl. Phys. B187, 397 (1981). [6] J.F. Donoghue and L.F. Li, Phys. Rev. D19, 945 (1979).

[arXiv:hep-ph/0409228].

[arXiv:hep-ph/0203121].

115008 (2010).

the charged Higgs boson couplings, namely tan *β* and *R*−<sup>1</sup>

<sup>33</sup> and tan *β* with an accuracy of about 10% at high

<sup>33</sup> , to an accuracy of less than

this, it might be possible to measure *R*−<sup>1</sup>

<sup>11</sup> and *<sup>R</sup>*−<sup>1</sup>

can measure *R*−<sup>1</sup>

**6. References**

Fig. 7. Correlation plots of the cross-sections for the processes *pp* → *t*(*b*)*H*±(→ *τν*) and *pp* <sup>→</sup> *<sup>t</sup>*(*b*)*H*±(<sup>→</sup> *tb*) for three values of *<sup>R</sup>*−<sup>1</sup> <sup>33</sup> and tan *β* (left) and of the branching ratios for *<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν* and *<sup>B</sup>* <sup>→</sup> *τν* (right) for various values of tan *<sup>β</sup>* and *<sup>R</sup>*<sup>−</sup><sup>1</sup> *<sup>d</sup>* with fixed *mH*<sup>±</sup> = 300 GeV [11].

Fig. 8. Contour plots of the *B* → *Dτν* branching ratio correlated with the cross-section *σ*(*pp* → *t*(*b*)*H*±(→ *τν*) (*a*) left) and *σ*(*pp* → *t*(*b*)*H*±(→ *tb*) (*a*) right), and the *B* → *τν* branching ratio correlated with the cross-section *σ*(*pp* → *t*(*b*)*H*±(→ *τν*) (*b*) left) and *<sup>σ</sup>*(*pp* <sup>→</sup> *<sup>t</sup>*(*b*)*H*±(<sup>→</sup> *tb*) (*b*) right), for various values of tan *<sup>β</sup>* and *<sup>R</sup>*−<sup>1</sup> (the bracketed numbers in the key refer to the appropriate *R*−<sup>1</sup> for each process being considered)[11].

the tauonic *B*-decays considered in this work. Hence the measurement of these two *B*-decays will only give an estimate of the product of *R*−<sup>1</sup> <sup>11</sup> and tan *β*. However, by considering the correlations of the *B*-decay observables with LHC observables, as shown in Fig.8, one can remove this degeneracy. So in principle it is possible to measure the four parameters (tan *β* and *R*−<sup>1</sup> *ii* with *ii* = 11, 22, 33) using the six correlation plots shown in Figs.7 and 8.

The primary question to be answered in this effective test of the charged Higgs couplings is "to what precision can we test *R*−1?". From our simulations we can safely assume that the LHC shall determine, to some level of precision, values for *mH*<sup>±</sup> and/or tan *β*. These values can then be converted into a value for *R*−<sup>1</sup> with all the precision afforded to us from the results of the *B*-factory experiments, as demonstrated pictorially in Fig.8. Assuming the charged Higgs boson mass to be known (taken to be 300 GeV in our present analysis) we have obtained cross-section measurement uncertainties as given in table 1. As can be seen from this, it might be possible to measure *R*−<sup>1</sup> <sup>33</sup> and tan *β* with an accuracy of about 10% at high luminosity. Armed with this information about tan *β*, from the LHC measurements, it can then be taken as an input to the *B*-decay measurements, namely *B* → *τν* and *B* → *Dτν*. In Ref.[10] it was inferred that for large values of tan *β* (≥ 40), measurements to a precision of 6-7% for high luminosity LHC results are possible. Our results are consistent with these observations. Future Super-*B* factories are expected to measure the *B* → *τν* and *B* → *Dτν* to a precision of 4% and 2.5% respectively [38]. The present world average experimental results for tauonic *<sup>B</sup>*-decays are *BR*(*<sup>B</sup>* <sup>→</sup> *τν*)=(1.51 <sup>±</sup> 0.33) <sup>×</sup> <sup>10</sup>−<sup>4</sup> and *BR*(*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν*)/*BR*(*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>μν*) = (41.6 ± 11.7 ± 5.2)% [30, 38]. Presently if one uses UTfit prescription of |*Vub*| then there is substantial disagreement between experimental and SM estimates for the branching fractions of *B* → *τν*. Recently, proposals have been given in Ref.[39] to reduce this tension between experimental and theoretical SM values of *B* → *τν*. Transforming the improved projected theoretical information of these decays along with future Super-*B* factory measurements one can measure *R*−<sup>1</sup> <sup>11</sup> and *<sup>R</sup>*−<sup>1</sup> <sup>22</sup> to a fairly high precision.

To summarise, we have tried to demonstrate that at the LHC alone it is possible to measure the charged Higgs boson couplings, namely tan *β* and *R*−<sup>1</sup> <sup>33</sup> , to an accuracy of less than 10%. Combining this information from the LHC with improved *B*-factory measurements, one can measure all four observables indicated in the introduction. These observables represent effective couplings of a charged Higgs boson to the bottom quark and the three generations of up-type quarks, thus demonstrating that it is possible to test the charged Higgs boson couplings to quarks by the combination of low energy measurements at future Super-*B* factories and charged Higgs boson production at the LHC. Something which shall be realisable in the very near future as results from the LHC are already starting to emerge [17, 18], and which will require more refined analyses in the near future.

#### **6. References**

18 Will-be-set-by-IN-TECH

4

0 5 10 15 20

Br (B ) 105

<sup>33</sup> and tan *β* (left) and of the branching ratios for

(1.4,1.4) (1,1) (0.6,0.6) (0.8,0.6) tan = 30 tan = 40 tan = 50

*<sup>d</sup>* with fixed *mH*<sup>±</sup> = 300 GeV

(1.4,1.4) (1,1) (0.6,0.6) (0.8,0.6) tan = 30 tan = 40 tan = 50

1 2 3 4 5<sup>0</sup>

( t b) ] [pb]

[p p t H-

<sup>11</sup> and tan *β*. However, by considering the

Br (B -) 105

R11 -1 = 1.4 R11 -1 = 1 R11 -1 = 0.8 R11 -1 = 0.6 tan- = 30 tan- = 40 tan-= 50 SM

4.5

5

Br (B

Fig. 7. Correlation plots of the cross-sections for the processes *pp* → *t*(*b*)*H*±(→ *τν*) and

4.5

Fig. 8. Contour plots of the *B* → *Dτν* branching ratio correlated with the cross-section *σ*(*pp* → *t*(*b*)*H*±(→ *τν*) (*a*) left) and *σ*(*pp* → *t*(*b*)*H*±(→ *tb*) (*a*) right), and the *B* → *τν* branching ratio correlated with the cross-section *σ*(*pp* → *t*(*b*)*H*±(→ *τν*) (*b*) left) and

in the key refer to the appropriate *R*−<sup>1</sup> for each process being considered)[11].

*<sup>σ</sup>*(*pp* <sup>→</sup> *<sup>t</sup>*(*b*)*H*±(<sup>→</sup> *tb*) (*b*) right), for various values of tan *<sup>β</sup>* and *<sup>R</sup>*−<sup>1</sup> (the bracketed numbers

the tauonic *B*-decays considered in this work. Hence the measurement of these two *B*-decays

correlations of the *B*-decay observables with LHC observables, as shown in Fig.8, one can remove this degeneracy. So in principle it is possible to measure the four parameters (tan *β*

The primary question to be answered in this effective test of the charged Higgs couplings is "to what precision can we test *R*−1?". From our simulations we can safely assume that the LHC shall determine, to some level of precision, values for *mH*<sup>±</sup> and/or tan *β*. These values can then be converted into a value for *R*−<sup>1</sup> with all the precision afforded to us from

*ii* with *ii* = 11, 22, 33) using the six correlation plots shown in Figs.7 and 8.

5

Br (B D -) 103

Br (B -) 105

0.2 0.4 0.6

 ( -) ] [pb]

[p p t H-

5.5

6

6.5

 D ) 103

5.5

6

6.5

7

0

*pp* <sup>→</sup> *<sup>t</sup>*(*b*)*H*±(<sup>→</sup> *tb*) for three values of *<sup>R</sup>*−<sup>1</sup>

0.1 0.2 0.3 0.4 0.5 0.6

*<sup>B</sup>* <sup>→</sup> *<sup>D</sup>τν* and *<sup>B</sup>* <sup>→</sup> *τν* (right) for various values of tan *<sup>β</sup>* and *<sup>R</sup>*<sup>−</sup><sup>1</sup>

(1.4,1.4) (1,1) (0.6,0.6) (0.8,0.6) tan = 30 tan = 40 tan = 50

1 2 3 4 5<sup>4</sup>

( t b) ] [pb]

[p p t H-

will only give an estimate of the product of *R*−<sup>1</sup>

(a) (b)

( ) ] [pb]

[p p t H-

(a) (b)

1

2

[p p

(1.4,1.4) (1,1) (0.6,0.6) (0.8,0.6) tan = 30 tan = 40 tan = 50

[11].

4

and *R*−<sup>1</sup>

0.2 0.4 0.6

 ( -) ] [pb]

[p p t H-

4.5

5

Br (B D -

) 103

5.5

6

6.5

t H-

( t b ) ] [pb]

3

4

5

6

R33 -1 = 1.4 R33 -1 = 1 R33 -1 = 0.6 tan- = 30 tan- = 40 tan-= 50


**1. Introduction**

by,

Symmetries, global or local, always play an important role in the conceptual aspects of physics be in broken or unbroken phase. Spontaneous breaking of the continuous symmetries always generates various excitations with varying mass spectra. Axion is one of that type, generated via spontaneous breaking of a global Chiral U(1) symmetry named after its discoverers, Peccei and Queen. This symmetry is usually denoted by U(1)*PQ*. To give a brief introduction to this particle and its origin we have to turn our attention to the development of the standard model of particle physics and its associated symmetries. The standard model of particle physics describes the strong, weak and electromagnetic interactions among elementary particles. The symmetry group for this model is, SUc(3) × SU(2) × U(1). The strong interaction ( Quantum Chromo Dynamics (QCD)) part of the Lagrangian has SU(3) color symmetry and it is given

**Introduction to Axion Photon Interaction in** 

**Particle Physics and Photon Dispersion in** 

<sup>L</sup> <sup>=</sup> <sup>−</sup> <sup>1</sup>

non-zero vacuum expectation values, i.e.,

<sup>2</sup>*g*<sup>2</sup> Tr F*<sup>a</sup>*

pion multiplets indicate that the isospin part is also conserved approximately.

*μν*F*μν*

It was realized long ago that, in the limit of vanishingly small quark masses (chiral limit), Strong interaction lagrangian has a global U(2)<sup>V</sup> × U(2)<sup>A</sup> symmetry. This symmetry group would further break spontaneously to produce the hadron multiplets. The vector part of the symmetry breaks to iso-spin times baryon number symmetry given by U(2)<sup>V</sup> = SU(2)<sup>I</sup> × U(1)B. In nature baryon number is seen to be conserved and the mass spectra of nucleon and

So one is left with the axial vector symmetry. QCD being a nonabelian gauge it is believed that this theory is confining in the infrared region. The confining property of the theory is likely to generate condensates of antiquark quark pairs. Thus *u*- and *d* quark condensates would have

and they would break the *U*(2)*<sup>A</sup>* symmetry. Now if the axial symmetry is broken, we would expect nearly four degenerate and massless pseudoscalar mesons. Interestingly enough, out of the four we observe three light pseudoscalar Nambu Goldstone (NG) Bosons in nature, i.e., the pions. They are light, m*<sup>π</sup>* � 0, but the other one (with approximately same mass) is not

*<sup>&</sup>lt;* <sup>0</sup>|*u*¯(0)*u*(0)|<sup>0</sup> *<sup>&</sup>gt;*=*<sup>&</sup>lt;* <sup>0</sup><sup>|</sup> ¯*d*(0)*d*(0)|<sup>0</sup> *<sup>&</sup>gt;*�<sup>=</sup> 0 . (2)

*<sup>a</sup>* + *q*¯(*i*/*D* − *m*)*q*. (1)

**Magnetized Media**

*Banaras Hindu University (MMV), Varanasi,* 

Avijit K. Ganguly

*India* 

**3**

