**3. The** *H*± **decay channels at the LHC**

With the theory for a charged Higgs coupling to heavy quarks now developed, we shall now consider the case where the charged Higgs boson is heavier than the top quark mass. Our reasoning for doing this, in this illustrative example, is that experimental searches have already placed a lower limit on the mass of a charged Higgs, including LEP, which set a limit of *mH*<sup>±</sup> *>* 78.6 GeV [15]. Note that within the MSSM, the charged Higgs mass is constrained by the pseudo-scalar Higgs mass and *W*-boson mass at tree level, with only moderate higher-order corrections, resulting in *mH*<sup>±</sup> 120 GeV. Furthermore, the Tevatron constrains (in several different MSSM scenarios) *mH*<sup>±</sup> 150 GeV [16], and at the LHC ATLAS has so far found (for tan *β >* 22) *mH*<sup>±</sup> *>* 140 GeV [17] and CMS *mH*<sup>±</sup> 160 GeV [18].

As such, with *mH*<sup>±</sup> *mt*, the production mechanism at the LHC shall be the associated production *pp* → *tbH*<sup>±</sup> + *X* (the main production mechanisms are then *gg* → *tbH*±, *gb* → *tH*<sup>±</sup> and the parton level processes, as shown in Fig.3[19]), with alternative production mechanisms like quark-antiquark annihilation, *qq*¯ <sup>→</sup> *<sup>H</sup>*+*H*−[20] and *<sup>H</sup>*±<sup>+</sup> jet production, associated production with a *W* boson, *qq*¯ → *H*±*W*∓[21], or Higgs pair production having suppressed rates. Note that some of the above production processes may be enhanced in models with non-MFV, which we shall not consider here.

Once produced, it is expected that the decay channel *<sup>H</sup>*<sup>+</sup> <sup>→</sup> *τν* shall be the primary discovery channel for the charged Higgs boson. Recall that we shall consider the large tan *β* region, where the branching ratios of charged decays into SM particles is given in Fig.4[10]. For tan *<sup>β</sup>* <sup>=</sup> 40 the branching ratio for *<sup>H</sup>*<sup>+</sup> <sup>→</sup> *tb* is also quite high, we shall therefore consider both decay channels here. Note that we have assumed a heavy SUSY spectrum, such that the charged Higgs will decay only into SM particles for the maximal stop mixing scenario. For low values of tan *β*, below the top quark mass, the main decay channels are *H*<sup>±</sup> → *τ*±*ντ*, *cs*¯, *Wh*<sup>0</sup> and *t* ∗*b*.

As such we shall now simulate the charged Higgs boson in the LHC environment with as much care as is possible, where we have included QCD corrections, as well as fully analysing the *<sup>H</sup>*<sup>+</sup> <sup>→</sup> *tb* mode. We should note though that of the main production mechanisms in Fig.3, there will be a partial overlap when the *gb* → *tH*<sup>±</sup> is obtained from the *gg* → *tbH*<sup>±</sup> by a gluon splitting into a *b*-quark pair. The summing of both contributions must be done with care, so as to avoid double counting, as we shall now discuss in greater detail.

#### **3.1 The resolution of double-counting and the normalisation of the cross-section**

From the associated production *pp* → *tbH*<sup>±</sup> + *X*, two different mechanisms can be employed to calculate the production cross-section. The first is the four flavour scheme with no *b* quarks 8 Will-be-set-by-IN-TECH

g

t

H<sup>+</sup>

gb <sup>→</sup> tH<sup>+</sup>

b

t

H<sup>+</sup>

is the sum of these contributions, after the subtraction of common terms.

Fig. 3. The charged Higgs production at the LHC through the *gg* → *tbH*<sup>±</sup> process, the *gb* → *tH*<sup>±</sup> process, and there will also be parton level processes. The inclusive cross-section

has so far found (for tan *β >* 22) *mH*<sup>±</sup> *>* 140 GeV [17] and CMS *mH*<sup>±</sup> 160 GeV [18].

As such, with *mH*<sup>±</sup> *mt*, the production mechanism at the LHC shall be the associated production *pp* → *tbH*<sup>±</sup> + *X* (the main production mechanisms are then *gg* → *tbH*±, *gb* → *tH*<sup>±</sup> and the parton level processes, as shown in Fig.3[19]), with alternative production mechanisms like quark-antiquark annihilation, *qq*¯ <sup>→</sup> *<sup>H</sup>*+*H*−[20] and *<sup>H</sup>*±<sup>+</sup> jet production, associated production with a *W* boson, *qq*¯ → *H*±*W*∓[21], or Higgs pair production having suppressed rates. Note that some of the above production processes may be enhanced in

Once produced, it is expected that the decay channel *<sup>H</sup>*<sup>+</sup> <sup>→</sup> *τν* shall be the primary discovery channel for the charged Higgs boson. Recall that we shall consider the large tan *β* region, where the branching ratios of charged decays into SM particles is given in Fig.4[10]. For tan *<sup>β</sup>* <sup>=</sup> 40 the branching ratio for *<sup>H</sup>*<sup>+</sup> <sup>→</sup> *tb* is also quite high, we shall therefore consider both decay channels here. Note that we have assumed a heavy SUSY spectrum, such that the charged Higgs will decay only into SM particles for the maximal stop mixing scenario. For low values of tan *β*, below the top quark mass, the main decay channels are *H*<sup>±</sup> → *τ*±*ντ*, *cs*¯,

As such we shall now simulate the charged Higgs boson in the LHC environment with as much care as is possible, where we have included QCD corrections, as well as fully analysing the *<sup>H</sup>*<sup>+</sup> <sup>→</sup> *tb* mode. We should note though that of the main production mechanisms in Fig.3, there will be a partial overlap when the *gb* → *tH*<sup>±</sup> is obtained from the *gg* → *tbH*<sup>±</sup> by a gluon splitting into a *b*-quark pair. The summing of both contributions must be done with care, so

From the associated production *pp* → *tbH*<sup>±</sup> + *X*, two different mechanisms can be employed to calculate the production cross-section. The first is the four flavour scheme with no *b* quarks

as to avoid double counting, as we shall now discuss in greater detail.

**3.1 The resolution of double-counting and the normalisation of the cross-section**

With the theory for a charged Higgs coupling to heavy quarks now developed, we shall now consider the case where the charged Higgs boson is heavier than the top quark mass. Our reasoning for doing this, in this illustrative example, is that experimental searches have already placed a lower limit on the mass of a charged Higgs, including LEP, which set a limit of *mH*<sup>±</sup> *>* 78.6 GeV [15]. Note that within the MSSM, the charged Higgs mass is constrained by the pseudo-scalar Higgs mass and *W*-boson mass at tree level, with only moderate higher-order corrections, resulting in *mH*<sup>±</sup> 120 GeV. Furthermore, the Tevatron constrains (in several different MSSM scenarios) *mH*<sup>±</sup> 150 GeV [16], and at the LHC ATLAS

g

**3. The** *H*± **decay channels at the LHC**

g b

models with non-MFV, which we shall not consider here.

*Wh*<sup>0</sup> and *t*

∗*b*.

gg <sup>→</sup> tH<sup>+</sup><sup>b</sup>

Fig. 4. The branching ratios of charged decays into SM particles as a function of *mH*<sup>±</sup> , for tan *β* = 5 (left panel), and tan *β* = 40 (right panel)[10].

in the initial state, the lowest order QCD production processes are gluon-gluon fusion and quark-antiquark annihilation, *gg* → *tbH*<sup>±</sup> and *qq*¯ → *tbH*<sup>±</sup> respectively. Note that potentially large logarithms ∝ ln(*μF*/*mb*), arising from the splitting of incoming gluons into nearly collinear *b*¯ *b* pairs, can be summed to all orders in perturbation theory by introducing bottom parton densities. This then defines the five flavour scheme. The use of bottom distribution functions is based on the approximation that the outgoing *b* quark is at small transverse momentum and massless, and the virtual *b* quark is quasi on-shell. In this scheme, the leading order process for the inclusive *tbH*<sup>±</sup> cross-section is gluon-bottom fusion, *gb* → *tH*±. The corrections to *gb* → *tH*<sup>±</sup> and tree-level processes *gg* → *tbH*<sup>±</sup> and *qq*¯ → *tbH*±. To all orders in perturbation theory the four and five flavour schemes are identical, but the way of ordering the perturbative expansion is different, and the results do now match exactly at finite order.

As such, in order to resolve the double-counting problem during event generation we use MATCHIG[22] as an external process to PYTHIA6.4.11[23]. In this program, when the *gb* → *tH*<sup>−</sup> (*g*¯ *<sup>b</sup>* <sup>→</sup> ¯*tH*+) process is generated, there will be an accompanying outgoing ¯ *b* (*b*) quark. For low transverse momenta of this accompanying *b* quark, this process, including initial state parton showers, describes the cross-section well. However, for large transverse momentum of the accompanying *<sup>b</sup>*-quark one instead uses the exact matrix element of the *gg* <sup>→</sup> *<sup>t</sup>*¯ *bH*<sup>−</sup> (*gg* → ¯*tbH*+) process. Whilst for low transverse momenta, this process can be described in terms of the gluon splitting to *b*¯ *b* times the matrix element of the *gb* → *tH*<sup>±</sup> process. As was shown in Ref.[24], for low transverse momenta ( 100GeV) the *gg* <sup>→</sup> *<sup>t</sup>*¯ *bH*± approach underestimates the differential cross-section. Therefore, when the accompanying *b*-quark is observed, it is necessary to use both the *g*¯ *<sup>b</sup>* <sup>→</sup> *tH*<sup>±</sup> and the *gg* <sup>→</sup> *<sup>t</sup>*¯ *bH*± processes together, appropriately matched to remove the double-counting.

To do this MATCHIG defines a double-counting term *<sup>σ</sup>*DC, given by the part of the *gg* <sup>→</sup> *<sup>t</sup>*¯ *bH*± process which is already included in the *g*¯ *b* → *tH*<sup>±</sup> process. This term is then subtracted from the sum of the cross-sections of the two processes. The double-counting term is given by the leading contribution of the *b* quark density as:

$$\sigma\_{\rm DC} = \int d\mathbf{x}\_1 d\mathbf{x}\_2 \left[ g(\mathbf{x}\_1, \mu\_F) b'(\mathbf{x}\_2, \mu\_F) \frac{d\hat{\sigma}\_{2 \to 2}}{d\mathbf{x}\_1 d\mathbf{x}\_2} (\mathbf{x}\_1, \mathbf{x}\_2) + \mathbf{x}\_1 \leftrightarrow \mathbf{x}\_2 \right] \tag{22}$$

sum is still normalised to the LO total cross-section, we renormalise it to NLO precision

Constraining the Couplings of a Charged Higgs to Heavy Quarks 39

computations given in Ref.[25, 26], which has been shown to be in good agreement with the one performed in Ref.[27]. For a Higgs boson mass of 300 GeV and in the tan *β* region of 30–50 considered here, the correction varies very little and can be well approximated with a constant

As has already been mentioned, the *τν* decay channel offers a high transverse momenta, *pT*, of the *τ* and a large missing energy signature that can be discovered at the LHC over a vast region of the parameter space, where constraints have already been determined [17, 18]. To simulate this the events were generated in PYTHIA using the *gb* → *tH*<sup>±</sup> process, explicitly using the mechanism *pp* → *t*(*b*)*H*<sup>±</sup> → *jjb*(*b*)*τν*. That is, the associated top quark is required to decay hadronically, *t* → *jjb*. The charged Higgs decays into a *τ* lepton, *H*<sup>±</sup> → *τ*±*ντ*, and the hadronic decays of the *τ* are considered. The backgrounds considered are QCD, *W*+ jets,

Γ(*H*<sup>±</sup> → *tb*) + Γ(*H*<sup>±</sup> → *τ*±*ντ*)

*<sup>t</sup>* cot<sup>2</sup> *<sup>β</sup>* + *<sup>m</sup>*<sup>2</sup>

If the decay *H*<sup>±</sup> → *tb* is kinematically allowed, comparing its width with Eq.(25) can give a

Note that a measurement of the signal rate in *H*<sup>±</sup> → *τ*±*ντ* can allow a determination of tan *β*.

• We first searched for events having one *τ* jet, two light non-*τ* jets and at least one (or two)

• A *W*-boson from the top quark decay was first reconstructed using a light jet pair. Note

• In this case, due to the presence of missing energy (the neutrino) in the charged Higgs decay, we can not reconstruct the charged Higgs mass. Instead we constructed the

then rescaled the four momenta of such jets in order to arrive at the correct *W*-boson mass. • We then reconstructed the top quark by pairing the above constructed *W*-boson with the bottom quarks. Choosing the combination which minimises *<sup>χ</sup>*<sup>2</sup> = (*mjjb* <sup>−</sup> *mt*)2, we only

that we retained all the combinations of light jets that satisfy |*mjj* − *mW*|

<sup>=</sup> *<sup>m</sup>*<sup>2</sup>

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

*<sup>τ</sup>* tan<sup>2</sup> *<sup>β</sup>*

*<sup>b</sup>* tan2 *<sup>β</sup>*) + *<sup>m</sup>*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *τ m*2 *H*± . (25)

*<sup>τ</sup>* tan<sup>2</sup> *<sup>β</sup>* . (26)

<sup>2</sup> *<* 25GeV. We

single top production *Wt*, and *<sup>t</sup>*¯*t*, with one *<sup>W</sup>* <sup>→</sup> *jj* and the other *<sup>W</sup>*<sup>±</sup> <sup>→</sup> *<sup>τ</sup>*±*ντ*.

 *m*2 *<sup>τ</sup>* tan2 *<sup>β</sup>*

8*πν*<sup>2</sup>

*Br*(*H*<sup>±</sup> <sup>→</sup> *<sup>τ</sup>*±*ντ*) � <sup>Γ</sup>(*H*<sup>±</sup> <sup>→</sup> *<sup>τ</sup>*±*ντ*)

3(*R*−<sup>1</sup> *<sup>t</sup>* )2(*m*<sup>2</sup>

*b*-jets. There is no isolated hard lepton in this configuration.

retained the events that satisfied |*mjjb* − *mt*| *<* 25GeV.

Note that we were required to impose additional cuts, namely:

*nf* =5

MS <sup>=</sup> 226 MeV in the

using CTEQ6M parton densities and the corresponding value of *λ*

**3.2 Simulations of the** *H*<sup>±</sup> → *τν* **decay mode**

The width of the process *H*<sup>±</sup> → *τ*±*ντ* is:

<sup>Γ</sup>(*H*<sup>−</sup> <sup>→</sup> *<sup>τ</sup>*−*ντ*) � *mH*<sup>±</sup>

rough estimate of the *H*<sup>±</sup> → *τ*±*ντ* branching ratio:

Our approach for this process is as follows:

transverse mass of the charged Higgs.

factor of 1.2.

Fig. 5. Plots of the transverse mass of the charged Higgs in *H* → *τν* for a luminosity of 300fb−<sup>1</sup> scaled to 30fb−1. The three lines in each plot correspond to positive events (the dotted red lines), negative events (dotted and blue) and matched events (shaded portion and black). The three graphs corresponds to three different values of *R*−<sup>1</sup> as indicated in each plot.

where *b*� (*x*, *μ*<sup>2</sup> *<sup>F</sup>*) is the leading order *b*-quark density given by [22]:

$$b'(x, \mu\_F^2) \approx \frac{a\_s}{2\pi} \log \frac{\mu\_F^2}{m\_b^2} \int \frac{dz}{z} P\_{\theta \mathcal{g}}(z) \text{ g}\left(\frac{x}{z}, \mu\_F^2\right) \,, \tag{23}$$

with *Pqg* the *<sup>g</sup>* <sup>→</sup> *qq*¯ splitting function, *<sup>g</sup>*(*x*, *<sup>μ</sup>*<sup>2</sup> *<sup>F</sup>*) the gluon density function, *μ<sup>F</sup>* the factorization scale and *z* the longitudinal gluon momentum fraction taken by the *b*-quark.

Including kinematic constraints due to finite center of mass energy (CM) and finite *b* quark mass, the resulting expression for the double-counting term can be written as [24]:

$$
\sigma\_{\rm DC} = \int\_{\tau\_{\rm min}}^{1} \frac{d\tau}{\tau} \int\_{\frac{1}{2}\log\tau}^{-\frac{1}{2}\log\tau} dy^\* \frac{\pi}{\hat{\mathcal{S}}} \int\_{-1}^{1} \frac{\beta\_{34}}{2} d(\cos\hat{\theta}) \left| \mathcal{M}\_{2\to 2} \right|^2 \frac{a\_s(\mu\_R^2)}{2\pi}
$$

$$
\times \left[ \int\_{x\_1}^{z\_{\rm max}} dz P\_{q\xi}(z) \int\_{\Omega\_{\rm min}^2}^{\Omega\_{\rm max}} \frac{d(Q^2)}{Q^2 + m\_b^2} \frac{x\_1}{z} g\left(\frac{x\_1}{z}, \mu\_F^2\right) x\_2 \mathbf{g}(\mathbf{x}\_2, \mu\_F^2) + \mathbf{x}\_1 \leftrightarrow \mathbf{x}\_2 \right]. \tag{24}
$$

Here <sup>M</sup>2→<sup>2</sup> is the matrix element for the *<sup>g</sup>*¯ *b* → *tH*<sup>±</sup> process, *μ<sup>F</sup>* and *μ<sup>R</sup>* are the factorization and renormalization scales as in the *gg* <sup>→</sup> *<sup>t</sup>*¯ *bH*± process, and the kinematical variables are *<sup>τ</sup>* <sup>=</sup> *<sup>x</sup>*1*x*2, *<sup>x</sup>*1,2 <sup>=</sup> <sup>√</sup>*τe*±*y*<sup>∗</sup> , *<sup>s</sup>* <sup>=</sup> *<sup>τ</sup>s*. *<sup>θ</sup>* is the polar angle of the *t*-quark in the CM system of the *g*¯ *<sup>b</sup>* <sup>→</sup> *tH*<sup>±</sup> scattering, and *<sup>β</sup>*<sup>34</sup> <sup>=</sup> *<sup>s</sup>*−<sup>1</sup> (*<sup>s</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *<sup>t</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup> *<sup>H</sup>*<sup>±</sup> )<sup>2</sup> <sup>−</sup> <sup>4</sup>*m*<sup>2</sup> *<sup>t</sup> <sup>m</sup>*<sup>2</sup> *<sup>H</sup>*<sup>±</sup> . *<sup>Q</sup>*<sup>2</sup> is the virtuality of the incoming *b*-quark and *z* is identified with the ratio of the CM energies of the *gb* system and the *gg* system.

Note that since the double-counting contribution should be subtracted from the sum of the positive processes, this weight is negative for double-counting events. This means that if all three processes are run simultaneously in PYTHIA, the total cross-section will be correctly matched.

With use of MATCHIG, issues of double-counting in our event generator are resolved. However, we shall not use the Monte-Carlo event generator, PYTHIA, to calculate the precise normalisation of the cross-sections, for though it gives an accurate description of the simulated data in both the low and high transverse momenta regions (with the inclusion of the external process MATCHIG), we can more accurately determine these by taking the leading order cross-section multiplied by an appropriate *k*-factor. The reason for this is that the matched sum is still normalised to the LO total cross-section, we renormalise it to NLO precision using CTEQ6M parton densities and the corresponding value of *λ nf* =5 MS <sup>=</sup> 226 MeV in the computations given in Ref.[25, 26], which has been shown to be in good agreement with the one performed in Ref.[27]. For a Higgs boson mass of 300 GeV and in the tan *β* region of 30–50 considered here, the correction varies very little and can be well approximated with a constant factor of 1.2.
