**1. Introduction**

28 Will-be-set-by-IN-TECH

28 Particle Physics

Squires, E.J. (1981), Some Comments on the Three-Fermion Composite Quark and Lepton

t'Hooft, G. (1971a), Renormalization of Massless Yang-Mills Fields, *Nuclear Physics B*, Vol. 33,

t'Hooft, G. (1971b), Renormalizable Lagrangians for Massive Yang-Mills Fields, *Nuclear*

Veltman, M. (2003), *Facts and Mysteries in Elementary Particle Physics*, (World Scientific

Weinberg, S. (1967), A Model of Leptons, *Physical Review Letters*, Vol. 19, No. 21, pp. 1264-1266.

Wu, C.S. *et al.* (1957), Experimental Test of Parity Conservation in Beta Decay, *Physical Review*,

Wilczek, F. (2005) In Search of Symmetry Lost, *Nature*, Vol. 433, No. 3, pp. 239-247.

Model, *Journal of Physics G*, Vol. 7, No. 4, pp. L47-L49.

No. 1, pp. 173-199.

*Physics B*, Vol. 35, No. 1, pp. 167-188.

Publishing Company, Singapore).

Vol. 105, No. 4, pp. 1413-1415.

The Standard Model (SM) of particle physics has been an incredibly successful theory which has been confirmed experimentally many times, however, it still has some short-comings. As such physicists continue to search for models beyond the SM which might explain issues such as naturalness (the hierarchy problem). Among the possible discoveries that would signal the existence of these new physics models (among several) would be the discovery of a charged Higgs boson.

Recall that in the SM we have a single complex Higgs doublet, which through the Higgs mechanism, is responsible for breaking the Electroweak (EW) symmetry and endowing our particles with their mass. As a result we expect one neutral scalar particle (known as the Higgs boson) to emerge. Now whilst physicists have become comfortable with this idea, we have not yet detected this illusive Higgs boson. Furthermore, this approach leads to the hierarchy problem, where extreme fine-tuning is required to stabilise the Higgs mass against quadratic divergences. As such a simple extension to the SM, which is trivially consistent with all available data, is to consider the addition of extra *SU*(2) singlets and/or doublets to the spectrum of the Higgs sector. One such extension shall be our focus here, that where we have two complex Higgs doublets, the so-called Two-Higgs Doublet Models (2HDMs). Such models, after EW symmetry breaking, will give rise to a charged Higgs boson in the physical spectrum. Note also that by having these two complex Higgs doublets we can significantly modify the Flavour Changing Neutral Current (FCNC) Higgs interactions in the large tan *β* region (where tan *β* ≡ *v*2/*v*1, the ratio of the vacuum expectation values (vevs) of the two complex doublets).

Among the models which contain a second complex Higgs doublet one of the best motivated is the Minimal Supersymmetric Standard Model (MSSM). This model requires a second Higgs doublet (and its supersymmetric (SUSY) fermionic partners) in order to preserve the cancellation of gauge anomalies [1]. The Higgs sector of the MSSM contains two Higgs supermultiplets that are distinguished by the sign of their hypercharge, establishing an unambiguous theoretical basis for the Higgs sector. In this model the structure of the Higgs sector is constrained by supersymmetry, leading to numerous relations among Higgs masses and couplings. However, due to supersymmetry-breaking effects, all such relations are modified by loop-corrections, where the effects of supersymmetry-breaking can enter [1].

derived from the MSSM, we will also introduce the relevant SUSY-QCD and higgsino-stop loop correction factors to the relevant charged Higgs boson fermion couplings. Using this formalism we shall study in section 3 the possibility of determining the charged Higgs boson properties at the LHC using *H*<sup>±</sup> → *τν* and *H*<sup>±</sup> → *tb*. In Section 4 we shall present the results of *B*-decays, namely *B* → *τν* and *B* → *Dτν*, as studied in Ref.[8]. Finally, we shall combine the *B*-decay results with our LHC simulations to determine the charged Higgs boson properties

Constraining the Couplings of a Charged Higgs to Heavy Quarks 31

In this section we shall develop the general form of the effective Lagrangian for the charged Higgs interactions with fermions. As already discussed in the introduction of this chapter, at tree-level the Higgs sector of the MSSM is of the same form as the type-II 2HDM, also in (at least in certain limits of) those of type-III. In these 2HDMs the consequence of this extended Higgs sector is the presence of additional Higgs bosons in the physics spectrum. In the MSSM

We shall begin by recalling that we require at least two Higgs doublets in SUSY theories, where in the SM the Higgs doublet gave mass to the leptons and down-type quarks, whilst the up-type quarks got their mass by using the charge conjugate (as was required to preserve all gauge symmetries in the Yukawa terms). In the SUSY case the charge conjugate cannot be used in the superpotential as it is part of a supermultiplet. As such the simplest solution is to introduce a second doublet with opposite hypercharge. So our theory will contain two chiral multiplets made up of our two doublets *H*<sup>1</sup> and *H*<sup>2</sup> and corresponding higgsinos *H*<sup>1</sup> and *H*<sup>2</sup> (fields with a tilde () denote squarks and sleptons); in which case the superpotential in the

**<sup>y</sup>***uQ* <sup>−</sup> *<sup>H</sup>*1*E<sup>c</sup>*

, *Q* =

 *U D* 

<sup>3</sup> ), (**3**, **<sup>1</sup>**, <sup>−</sup><sup>4</sup>

<sup>1</sup> or *H*<sup>∗</sup>

, *L* =

*<sup>a</sup>* (**y***d*) *j i Qa*

 *N E* 

**y***eL* + *μH*1*H*<sup>2</sup> . (1)

<sup>3</sup> ), (**1**, **1**, 2); where the gauge and

<sup>2</sup> , consistent with the fact that the

. (2)

*<sup>j</sup>α�αβ* with *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3

1 ,

(such as its mass, tan *β* and SUSY loop correction factors) and give our conclusions.

**2. Effective Lagrangian for a charged Higgs boson**

we will have 5 Higgs bosons, three neutral and two charged.

*<sup>W</sup>* <sup>=</sup> <sup>−</sup>*H*1*D<sup>c</sup>*

 *H*<sup>0</sup> 1 *H*− 1

Note that Eq.(1) does not contain terms with *H*∗

*H*<sup>1</sup> =

**y***d*, **y***<sup>u</sup>* and **y***<sup>e</sup>* are 3 × 3 unitary matrices.

*<sup>E</sup><sup>c</sup>* are (**1**, **<sup>2</sup>**, <sup>−</sup>1), (**1**, **<sup>2</sup>**, 1), (**3**, **<sup>2</sup>**, <sup>1</sup>

supersymmetry transformation.

The components of the weak doublet fields are denoted as:

, *H*<sup>2</sup> =

1, 2 being the *SU*(2)*<sup>L</sup>* isospin indices and *H*1*Dc***y***dQ* = (*H*1)*βDci*

**<sup>y</sup>***d<sup>Q</sup>* <sup>+</sup> *<sup>H</sup>*2*U<sup>c</sup>*

 *H*<sup>+</sup> 2 *H*0 2

<sup>3</sup> ), (**1**, **<sup>2</sup>**, <sup>−</sup>1), (**3**, **<sup>1</sup>**, <sup>2</sup>

The quantum numbers of the *SU*(3) <sup>×</sup> *SU*(2) <sup>×</sup> *<sup>U</sup>*(1) gauge groups for *<sup>H</sup>*1, *<sup>H</sup>*2, *<sup>Q</sup>*, *<sup>L</sup>*, *<sup>D</sup>c*, *<sup>U</sup>c*,

family indices were eliminated in Eq.(1). For example *μH*1*H*<sup>2</sup> = *μ*(*H*1)*α*(*H*2)*β�αβ* with *α*, *β* =

as the family indices and *a* = 1, 2, 3 as the colour indices of *SU*(3)*c*. As in the SM the Yukawas

superpotential is a holomorphic function of the supermultiplets. Yukawa terms like *UQH* ¯ <sup>∗</sup>

which are usually present in non-SUSY models, are excluded by the invariance under the

**2.1 The MSSM charged Higgs**

MSSM is:

Thus, one can describe the Higgs-sector of the (broken) MSSM by an effective field theory consisting of the most general 2HDM, which is how we shall develop our theory in section 2.

Note that in a realistic model, the Higgs-fermion couplings must be chosen with some care in order to avoid FCNC [2, 3], where 2HDMs are classified by how they address this: In type-I models [4] there exists a basis choice in which only one of the Higgs fields couples to the SM fermions. In type-II [5, 6], there exists a basis choice in which one Higgs field couples to the up-type quarks, and the other Higgs field couples to the down-type quarks and charged leptons. Type-III models [7] allow both Higgs fields to couple to all SM fermions, where such models are viable only if the resulting FCNC couplings are small.

Once armed with a model for a charged Higgs boson, we must determine how this particle will manifest and effect our experiments. Of the numerous channels, both direct and indirect, in which its presence could have a profound effect, one of the most constraining are those where the charged Higgs mediates tree-level flavour-changing processes, such as *B* → *τν* and *B* → *Dτν* [8]. As these processes have already been measured at *B*-factories, they will provide us with very useful indirect probes into the charged Higgs boson properties. Furthermore, with the commencement of the Large Hadron Collider (LHC) studies involving the LHC environment promise the best avenue for directly discovering a charged Higgs boson. As such we shall determine the properties of the charged Higgs boson using the following processes:


The processes mentioned above have several common characteristics with regard to the charged Higgs boson couplings to the fermions. Firstly, the parameter region of tan *β* and the charged Higgs boson mass covered by charged Higgs boson production at the LHC (*pp* <sup>→</sup> *<sup>t</sup>*(*b*)*H*+) overlaps with those explored at *<sup>B</sup>*-factories. Secondly, these processes provide four independent measurements to determine the charged Higgs boson properties. With these four independent measurements one can in principle determine the four parameters related to the charged Higgs boson couplings to *b*-quarks, namely tan *β* and the three generic couplings related to the *b* − *i* − *H*<sup>±</sup> (*i* = *u*, *c*, *t*) vertices. In our analysis we focus on the large tan *β*-region [9], where one can neglect terms proportional to cot *β*, where at tree-level the couplings to fermions will depend only on tan *β* and the mass of the down-type fermion involved. Hence, at tree-level, the *b* − *i* − *H*<sup>±</sup> (*i* = *u*, *c*, *t*) vertex is the same for all the three up-type generations. This property is broken by loop corrections to the charged Higgs boson vertex.

Our strategy in this pedagogical study will be to determine the charged Higgs boson properties first through the LHC processes. Note that the latter have been extensively studied in many earlier works (see Ref.[10], for example) with the motivation of discovering the charged Higgs boson in the region of large tan *β*. We shall assume that the charged Higgs boson is already observed with a certain mass. Using the two LHC processes as indicated above, one can then determine tan *β* and the *b* − *t* − *H*<sup>±</sup> coupling. Having an estimate of tan *β* one can then study the *B*-decays and try to determine the *b* − (*u*/*c*) − *H*<sup>±</sup> couplings from *B*-factory measurements. This procedure will enable us to measure the charged Higgs boson couplings to the bottom quark and up-type quarks [11].

The chapter will therefore be organised in the following way: In Section 2 we shall discuss the model we have considered for our analysis. As we shall use an effective field theory derived from the MSSM, we will also introduce the relevant SUSY-QCD and higgsino-stop loop correction factors to the relevant charged Higgs boson fermion couplings. Using this formalism we shall study in section 3 the possibility of determining the charged Higgs boson properties at the LHC using *H*<sup>±</sup> → *τν* and *H*<sup>±</sup> → *tb*. In Section 4 we shall present the results of *B*-decays, namely *B* → *τν* and *B* → *Dτν*, as studied in Ref.[8]. Finally, we shall combine the *B*-decay results with our LHC simulations to determine the charged Higgs boson properties (such as its mass, tan *β* and SUSY loop correction factors) and give our conclusions.

#### **2. Effective Lagrangian for a charged Higgs boson**

In this section we shall develop the general form of the effective Lagrangian for the charged Higgs interactions with fermions. As already discussed in the introduction of this chapter, at tree-level the Higgs sector of the MSSM is of the same form as the type-II 2HDM, also in (at least in certain limits of) those of type-III. In these 2HDMs the consequence of this extended Higgs sector is the presence of additional Higgs bosons in the physics spectrum. In the MSSM we will have 5 Higgs bosons, three neutral and two charged.

#### **2.1 The MSSM charged Higgs**

2 Will-be-set-by-IN-TECH

Thus, one can describe the Higgs-sector of the (broken) MSSM by an effective field theory consisting of the most general 2HDM, which is how we shall develop our theory in section 2. Note that in a realistic model, the Higgs-fermion couplings must be chosen with some care in order to avoid FCNC [2, 3], where 2HDMs are classified by how they address this: In type-I models [4] there exists a basis choice in which only one of the Higgs fields couples to the SM fermions. In type-II [5, 6], there exists a basis choice in which one Higgs field couples to the up-type quarks, and the other Higgs field couples to the down-type quarks and charged leptons. Type-III models [7] allow both Higgs fields to couple to all SM fermions, where such

Once armed with a model for a charged Higgs boson, we must determine how this particle will manifest and effect our experiments. Of the numerous channels, both direct and indirect, in which its presence could have a profound effect, one of the most constraining are those where the charged Higgs mediates tree-level flavour-changing processes, such as *B* → *τν* and *B* → *Dτν* [8]. As these processes have already been measured at *B*-factories, they will provide us with very useful indirect probes into the charged Higgs boson properties. Furthermore, with the commencement of the Large Hadron Collider (LHC) studies involving the LHC environment promise the best avenue for directly discovering a charged Higgs boson. As such we shall determine the properties of the charged Higgs boson using the following processes:

• **LHC:** *pp* <sup>→</sup> *<sup>t</sup>*(*b*)*H*+: through the decays *<sup>H</sup>*<sup>±</sup> <sup>→</sup> *τν*, *<sup>H</sup>*<sup>±</sup> <sup>→</sup> *tb* (*<sup>b</sup>* <sup>−</sup> *<sup>t</sup>* <sup>−</sup> *<sup>H</sup>*<sup>±</sup> coupling).

The processes mentioned above have several common characteristics with regard to the charged Higgs boson couplings to the fermions. Firstly, the parameter region of tan *β* and the charged Higgs boson mass covered by charged Higgs boson production at the LHC (*pp* <sup>→</sup> *<sup>t</sup>*(*b*)*H*+) overlaps with those explored at *<sup>B</sup>*-factories. Secondly, these processes provide four independent measurements to determine the charged Higgs boson properties. With these four independent measurements one can in principle determine the four parameters related to the charged Higgs boson couplings to *b*-quarks, namely tan *β* and the three generic couplings related to the *b* − *i* − *H*<sup>±</sup> (*i* = *u*, *c*, *t*) vertices. In our analysis we focus on the large tan *β*-region [9], where one can neglect terms proportional to cot *β*, where at tree-level the couplings to fermions will depend only on tan *β* and the mass of the down-type fermion involved. Hence, at tree-level, the *b* − *i* − *H*<sup>±</sup> (*i* = *u*, *c*, *t*) vertex is the same for all the three up-type generations.

Our strategy in this pedagogical study will be to determine the charged Higgs boson properties first through the LHC processes. Note that the latter have been extensively studied in many earlier works (see Ref.[10], for example) with the motivation of discovering the charged Higgs boson in the region of large tan *β*. We shall assume that the charged Higgs boson is already observed with a certain mass. Using the two LHC processes as indicated above, one can then determine tan *β* and the *b* − *t* − *H*<sup>±</sup> coupling. Having an estimate of tan *β* one can then study the *B*-decays and try to determine the *b* − (*u*/*c*) − *H*<sup>±</sup> couplings from *B*-factory measurements. This procedure will enable us to measure the charged Higgs boson

The chapter will therefore be organised in the following way: In Section 2 we shall discuss the model we have considered for our analysis. As we shall use an effective field theory

• *B***-factories:** *B* → *τν* (*b* − *u* − *H*<sup>±</sup> coupling), *B* → *Dτν* (*b* − *c* − *H*<sup>±</sup> coupling).

This property is broken by loop corrections to the charged Higgs boson vertex.

couplings to the bottom quark and up-type quarks [11].

models are viable only if the resulting FCNC couplings are small.

We shall begin by recalling that we require at least two Higgs doublets in SUSY theories, where in the SM the Higgs doublet gave mass to the leptons and down-type quarks, whilst the up-type quarks got their mass by using the charge conjugate (as was required to preserve all gauge symmetries in the Yukawa terms). In the SUSY case the charge conjugate cannot be used in the superpotential as it is part of a supermultiplet. As such the simplest solution is to introduce a second doublet with opposite hypercharge. So our theory will contain two chiral multiplets made up of our two doublets *H*<sup>1</sup> and *H*<sup>2</sup> and corresponding higgsinos *H*<sup>1</sup> and *H*<sup>2</sup> (fields with a tilde () denote squarks and sleptons); in which case the superpotential in the MSSM is:

$$W = -H\_1 \mathbf{D}^\varepsilon \mathbf{y}\_d \mathbf{Q} + H\_2 \mathbf{U}^\varepsilon \mathbf{y}\_u \mathbf{Q} - H\_1 \mathbf{E}^\varepsilon \mathbf{y}\_\varepsilon L + \mu H\_1 H\_2 \,. \tag{1}$$

The components of the weak doublet fields are denoted as:

$$H\_1 = \begin{pmatrix} H\_1^0 \\ H\_1^- \end{pmatrix}, \ H\_2 = \begin{pmatrix} H\_2^+ \\ H\_2^0 \end{pmatrix}, \ Q = \begin{pmatrix} U \\ D \end{pmatrix}, \ L = \begin{pmatrix} N \\ E \end{pmatrix}. \tag{2}$$

The quantum numbers of the *SU*(3) <sup>×</sup> *SU*(2) <sup>×</sup> *<sup>U</sup>*(1) gauge groups for *<sup>H</sup>*1, *<sup>H</sup>*2, *<sup>Q</sup>*, *<sup>L</sup>*, *<sup>D</sup>c*, *<sup>U</sup>c*, *<sup>E</sup><sup>c</sup>* are (**1**, **<sup>2</sup>**, <sup>−</sup>1), (**1**, **<sup>2</sup>**, 1), (**3**, **<sup>2</sup>**, <sup>1</sup> <sup>3</sup> ), (**1**, **<sup>2</sup>**, <sup>−</sup>1), (**3**, **<sup>1</sup>**, <sup>2</sup> <sup>3</sup> ), (**3**, **<sup>1</sup>**, <sup>−</sup><sup>4</sup> <sup>3</sup> ), (**1**, **1**, 2); where the gauge and family indices were eliminated in Eq.(1). For example *μH*1*H*<sup>2</sup> = *μ*(*H*1)*α*(*H*2)*β�αβ* with *α*, *β* = 1, 2 being the *SU*(2)*<sup>L</sup>* isospin indices and *H*1*Dc***y***dQ* = (*H*1)*βDci <sup>a</sup>* (**y***d*) *j i Qa <sup>j</sup>α�αβ* with *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, 2, 3 as the family indices and *a* = 1, 2, 3 as the colour indices of *SU*(3)*c*. As in the SM the Yukawas **y***d*, **y***<sup>u</sup>* and **y***<sup>e</sup>* are 3 × 3 unitary matrices.

Note that Eq.(1) does not contain terms with *H*∗ <sup>1</sup> or *H*<sup>∗</sup> <sup>2</sup> , consistent with the fact that the superpotential is a holomorphic function of the supermultiplets. Yukawa terms like *UQH* ¯ <sup>∗</sup> 1 , which are usually present in non-SUSY models, are excluded by the invariance under the supersymmetry transformation.

D<sup>L</sup> D<sup>R</sup>

D<sup>L</sup> D<sup>R</sup>

h˜− <sup>1</sup> <sup>h</sup>˜<sup>−</sup> 2

1,2.

� *<sup>M</sup>dV*†

�

tan *βH*−*D*<sup>R</sup>

cot *βH*+*U*<sup>R</sup>

<sup>2</sup> *<sup>I</sup>*[*<sup>M</sup><sup>h</sup>*, *MU*<sup>L</sup>

*<sup>b</sup>*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*2*c*<sup>2</sup> ln *<sup>b</sup>*<sup>2</sup>

**<sup>E</sup>***<sup>g</sup>* and **<sup>E</sup>***<sup>h</sup>* are gluino and charged higgsino contributions shown in Fig.1(a) and (b) respectively. Note that these corrections for Yukawa couplings are calculated in the unbroken

Up to now we have assumed all squark mass matrices are proportional to a unit matrix at the EW scale, as shown in Eqs.(4). However, models with Minimal Flavour Violation (MFV) correspond to more general cases. For instance, the assumption of Eqs.(4) is not satisfied in minimal supergravity, where all squarks have a universal mass at the Planck scale, not at the EW scale. In Ref.[8] they derive the charged Higgs coupling in a more general case of MFV.

*<sup>u</sup>***y***<sup>u</sup>* + *<sup>b</sup>*2**y**†

*<sup>u</sup>*]*<sup>M</sup>*<sup>2</sup> ,

The corresponding corrections to the up-type couplings can be calculated from Eq.(6). Since we are interested in the large tan *β* case, these corrections are very small. In the following we neglect such corrections, and the Lagrangian for the up-type quarks is given as follows:

For the case of the charged-lepton, we can derive the relevant parts of the Lagrangian in a similar way to the case of the down-type quark by choosing an appropriate basis choice.

In the present case with Eqs.(4) Δ*md* receives contributions from gluino and down-type squark,

and higgsino and up-type squark diagrams. The explicit form is given as follows:

<sup>16</sup>*π*<sup>2</sup> *<sup>A</sup>*u|**<sup>y</sup>**<sup>u</sup><sup>|</sup>

*a*2*b*<sup>2</sup> ln *<sup>a</sup>*<sup>2</sup>

= [*a*1**1** + *b*1**y**†

= [*a*2**1** + *b*5**y***u***y**†

= [*a*3**<sup>1</sup>** + *<sup>b</sup>*6**y***d***y**†

√2 *v*

U˜<sup>R</sup> U˜<sup>L</sup>

H<sup>0</sup><sup>∗</sup> 2 (b)

CKM*<sup>R</sup>*<sup>−</sup><sup>1</sup> *<sup>d</sup> U*�

*M uV*CKM*D*�

<sup>Δ</sup>*md* <sup>=</sup> **<sup>E</sup>***<sup>g</sup>* <sup>+</sup> **<sup>E</sup>***<sup>h</sup>* , (12)

**1***μ*∗*Mg*˜ *I*[*Mg*˜, *MD*˜ <sup>L</sup> , *MD*˜ <sup>R</sup> ] , (13)

*a*2 (*a*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*2)(*b*<sup>2</sup> <sup>−</sup> *<sup>c</sup>*2)(*a*<sup>2</sup> <sup>−</sup> *<sup>c</sup>*2) . (15)

, *MU*<sup>R</sup>

*<sup>c</sup>*<sup>2</sup> <sup>+</sup> *<sup>c</sup>*2*a*<sup>2</sup> ln *<sup>c</sup>*<sup>2</sup>

*<sup>d</sup>***y***d*]*<sup>M</sup>*<sup>2</sup> ,

*<sup>d</sup>*]*<sup>M</sup>*<sup>2</sup> . (16)

<sup>L</sup> + h.c. (10)

<sup>L</sup> + h.c. (11)

] , (14)

g˜<sup>R</sup> g˜<sup>L</sup>

D˜<sup>L</sup> D˜<sup>R</sup>

H<sup>0</sup><sup>∗</sup> 2

(a)

induced by (a) gluino *<sup>g</sup>*L,R and (b) charged higgsino *<sup>h</sup>*<sup>−</sup>

LD−quark = −*D*<sup>R</sup>

LU−quark = −*U*<sup>R</sup>

where

phase of SU(2) × U(1).

we obtain the following Lagrangian for down-type quarks.

� *MdD*� <sup>L</sup> +

> � *M uU*� <sup>L</sup> +

**<sup>E</sup>***<sup>g</sup>* <sup>≡</sup> <sup>2</sup>*α<sup>s</sup>* 3*π*

**<sup>E</sup>***<sup>h</sup>* ≡ − *<sup>μ</sup>*

*I*[*a*, *b*, *c*] =

*M*<sup>2</sup> *Q*L

*M*<sup>2</sup> *U*R

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

Namely the squark mass matrix is taken to be

Fig. 1. Non-holomorphic radiative corrections to the down-type quark Yukawa couplings

Constraining the Couplings of a Charged Higgs to Heavy Quarks 33

√2 *v*

The soft SUSY breaking masses and trilinear SUSY breaking terms (*A*-term) are given by:

$$\mathcal{L}\_{\text{soft}} = -\tilde{\mathsf{Q}}\_{\text{L}}^{\dagger}M\_{\text{\tilde{Q}}}^{2}\tilde{\mathsf{Q}}\_{\text{L}} - \tilde{\mathsf{U}}\_{\text{R}}^{\dagger}M\_{\text{\tilde{U}}\_{\text{R}}}^{2}\tilde{\mathsf{U}}\_{\text{R}} - \tilde{\mathsf{D}}\_{\text{R}}^{\dagger}M\_{\text{\tilde{D}}}^{2}\tilde{\mathsf{D}}\_{\text{R}} - \tilde{\mathsf{L}}\_{\text{L}}^{\dagger}M\_{\text{\tilde{L}}\_{\text{L}}}^{2}\tilde{\mathsf{L}}\_{\text{L}} - \tilde{\mathsf{E}}\_{\text{R}}^{\dagger}M\_{\text{\tilde{E}}\_{\text{R}}}^{2}\tilde{\mathsf{E}}\_{\text{R}} $$

$$+ H\_{1}\tilde{D}\_{\text{R}}^{\dagger}\mathbf{A}\_{d}\tilde{\mathsf{Q}}\_{\text{L}} - H\_{2}\tilde{\mathsf{U}}\_{\text{R}}^{\dagger}\mathbf{A}\_{\text{u}}\tilde{\mathsf{Q}}\_{\text{L}} + H\_{1}\tilde{\mathsf{E}}\_{\text{R}}^{\dagger}\mathbf{A}\_{\text{c}}\tilde{\mathsf{L}}\_{\text{L}} + \text{h.c.} \tag{3}$$

Let us first discuss the simplest case where soft breaking masses are proportional to a unit matrix in the flavour space, and **A***u*, **A***<sup>d</sup>* and **A***<sup>e</sup>* are proportional to Yukawa couplings. Their explicit forms being:

$$\begin{aligned} \mathcal{M}^2\_{\tilde{Q}\_{\text{Lij}}} &= a\_1 \tilde{\mathcal{M}}^2 \delta\_{\text{ij}} \ \mathcal{M}^2\_{\tilde{\mathcal{U}}\_{\text{Bij}}} = a\_2 \tilde{\mathcal{M}}^2 \delta\_{\text{ij}} \ \mathcal{M}^2\_{\tilde{\mathcal{D}}\_{\text{Bi}}} = a\_3 \tilde{\mathcal{M}}^2 \delta\_{\text{ij}} \ \mathcal{M}^2\_{\tilde{\mathcal{L}}\_{\text{Li}}} &= a\_4 \tilde{\mathcal{M}}^2 \delta\_{\text{ij}} \ \mathcal{M}^2\_{\tilde{\mathcal{U}}\_{\text{Li}}} \\ \mathcal{M}^2\_{\tilde{E}\_{\text{Bi}j}} &= a\_5 \tilde{\mathcal{M}}^2 \delta\_{\text{ij}} \ \mathcal{A}\_{\text{uij}} = A\_u \mathbf{y}\_{\text{uij}} \ \mathcal{A}\_{\text{dij}} = A\_d \mathbf{y}\_{\text{dij}} \ \mathcal{A}\_{\text{eij}} = A\_e \mathbf{y}\_{\text{eij}} \end{aligned} \tag{4}$$

where *ai*(*i* = 1 − 5) are real parameters.

At tree-level the Yukawa couplings have the same structure as the above superpotential, namely, *H*<sup>1</sup> couples to *D<sup>c</sup>* and *Ec*, and *H*<sup>2</sup> to *Uc*. On the other hand, different types of couplings are induced when we take into account SUSY breaking effects through one-loop diagrams. The Lagrangian of the Yukawa sector can be written as:

$$\begin{split} \mathcal{L}\_{\text{Yukawa}} &= -H\_{\text{1}} \overline{D}\_{\text{R}} \mathbf{y}\_{d} Q\_{\text{L}} + H\_{\text{2}} \overline{U}\_{\text{R}} \mathbf{y}\_{u} Q\_{\text{L}} - H\_{\text{1}} \overline{E}\_{\text{R}} \mathbf{y}\_{\text{c}} L\_{\text{L}} - i \sigma\_{2} H\_{\text{2}}^{\*} \overline{D}\_{\text{R}} \Delta \mathbf{y}\_{d} Q\_{\text{L}} \\ &+ i \sigma\_{2} H\_{\text{1}}^{\*} \overline{U}\_{\text{R}} \Delta \mathbf{y}\_{u} Q\_{\text{L}} - i \sigma\_{2} H\_{\text{2}}^{\*} \overline{E}\_{\text{R}} \Delta \mathbf{y}\_{\text{c}} L\_{\text{L}} + \text{h.c.} \end{split} \tag{5}$$

where Δ**y***d*, Δ**y***u*, and Δ**y***<sup>e</sup>* are one-loop induced coupling constants, and we recall that gauge indices have been suppressed; for example *σ*2*H*∗ <sup>2</sup> *D*RΔ**y***dQ*<sup>L</sup> = (*σ*2)*αβ*(*H*<sup>∗</sup> <sup>2</sup> )*β*(*D*R)*<sup>i</sup> <sup>a</sup>*(Δ**y***d*) *j i* (*Q*L)*<sup>a</sup> <sup>j</sup>α*. From the above Yukawa couplings, we can derive the quark and lepton mass matrices and their charged Higgs couplings. For the quark sector, we get

$$\mathcal{L}\_{\text{quark}} = -\frac{\upsilon}{\sqrt{2}} \cos \beta \overline{D}\_{\text{R}} \mathbf{y}\_d [1 + \tan \beta \Lambda\_{\text{m}\_d}] D\_{\text{L}} + \sin \beta H^- \overline{D}\_{\text{R}} \mathbf{y}\_d [1 - \cot \beta \Delta\_{\text{m}\_d}] \mathbf{L}\_{\text{L}} \tag{6}$$

$$ -\frac{\upsilon}{\sqrt{2}} \sin \beta \overline{\mathcal{U}}\_{\text{R}} \mathbf{y}\_u [1 - \cot \beta \Lambda\_{\text{m}\_u}] \mathbf{L}\_{\text{L}} + \cos \beta H^+ \overline{\mathcal{U}}\_{\text{R}} \mathbf{y}\_u [1 + \tan \beta \Delta\_{\text{m}\_u}] D\_{\text{L}} + \text{h.c.} $$

where we define <sup>Δ</sup>*md* (Δ*mu* ) as <sup>Δ</sup>*md* <sup>≡</sup> **<sup>y</sup>**−<sup>1</sup> *<sup>d</sup>* <sup>Δ</sup>**y***<sup>d</sup>* (Δ*mu* <sup>≡</sup> **<sup>y</sup>**−<sup>1</sup> *<sup>u</sup>* <sup>Δ</sup>**y***u*), and *<sup>v</sup>* � 246GeV. Notice that Δ**y***<sup>d</sup>* is proportional to **y***<sup>d</sup>* or **y***d***y**† *<sup>u</sup>***y***<sup>u</sup>* in this case. We then rotate the quark bases as follows:

$$\mathrm{d}\mathrm{L}\_{\mathrm{L}} = \mathrm{V\_{\mathrm{L}}(\mathrm{Q})} \mathrm{L}\_{\mathrm{L}}^{\prime} \text{ , } \mathrm{D\_{\mathrm{L}}} = \mathrm{V\_{\mathrm{L}}(\mathrm{Q})} V\_{\mathrm{CKM}} \mathrm{D\_{\mathrm{L}}^{\prime}} \text{ , } \mathrm{U\_{\mathrm{R}}} = \mathrm{V\_{\mathrm{R}}(\mathrm{U})} \mathrm{U\_{\mathrm{R}}^{\prime}} \text{ , } \mathrm{D\_{\mathrm{R}}} = \mathrm{V\_{\mathrm{R}}(\mathrm{D})} \mathrm{D\_{\mathrm{R}}^{\prime}} \text{ , } \tag{7}$$

where the fields with a prime (� ) are mass eigenstates. In this basis, the down-type quark Lagrangian is given by

$$\mathcal{L}\_{\text{D-quark}} = -\frac{v}{\sqrt{2}} \cos \beta \overline{D}\_{\text{R}} \,^{\prime} V\_{\text{R}}^{\dagger}(D) \mathbf{y}\_{d} V\_{\text{L}}(Q) \widehat{R}\_{d} V\_{\text{CKM}} D\_{\text{L}}^{\prime}$$

$$+ \sin \beta H^{-} \overline{D}\_{\text{R}} \,^{\prime} V\_{\text{R}}^{\dagger}(D) \mathbf{y}\_{d} V\_{\text{L}}(Q) \mathcal{U}\_{\text{L}}^{\prime} + \text{h.c.}, \tag{8}$$

where *<sup>R</sup><sup>d</sup>* <sup>≡</sup> <sup>1</sup> <sup>+</sup> tan *<sup>β</sup>*<sup>Δ</sup>*md* and <sup>Δ</sup>*md* <sup>≡</sup> *<sup>V</sup>*† <sup>L</sup> (*Q*)Δ*mdV*L(*Q*). Hereafter, a matrix with a hat () represents a diagonal matrix. Since the down-type diagonal mass term is given by

$$
\hat{M}\_d \equiv \frac{v}{\sqrt{2}} \cos \beta V\_\mathbf{R}^\dagger(D) \mathbf{y}\_d V\_\mathbf{L}(Q) \hat{R}\_d V\_{\mathbf{CKM}} \,\prime \tag{9}
$$

Fig. 1. Non-holomorphic radiative corrections to the down-type quark Yukawa couplings induced by (a) gluino *<sup>g</sup>*L,R and (b) charged higgsino *<sup>h</sup>*<sup>−</sup> 1,2.

we obtain the following Lagrangian for down-type quarks.

$$\mathcal{L}\_{\text{D-quark}} = -\overline{D}\_{\text{R}}{}^{\prime}\hat{M}\_{d}D\_{\text{L}}^{\prime} + \frac{\sqrt{2}}{v}\tan\beta H^{-}\overline{D}\_{\text{R}}{}^{\prime}\hat{M}\_{d}V\_{\text{CKM}}^{\dagger}\hat{R}\_{d}^{-1}\mathcal{U}\_{\text{L}}^{\prime} + \text{h.c.}\tag{10}$$

The corresponding corrections to the up-type couplings can be calculated from Eq.(6). Since we are interested in the large tan *β* case, these corrections are very small. In the following we neglect such corrections, and the Lagrangian for the up-type quarks is given as follows:

$$\mathcal{L}\_{\text{U-quark}} = -\overline{\mathcal{U}}\_{\text{R}}{}^{\prime}\hat{M}\_{\text{H}}\mathcal{U}\_{\text{L}}^{\prime} + \frac{\sqrt{2}}{v}\cot\beta H^{+} \overline{\mathcal{U}}\_{\text{R}}{}^{\prime}\hat{M}\_{\text{U}}\mathcal{V}\_{\text{CKM}}\mathcal{D}\_{\text{L}}^{\prime} + \text{h.c.}\tag{11}$$

For the case of the charged-lepton, we can derive the relevant parts of the Lagrangian in a similar way to the case of the down-type quark by choosing an appropriate basis choice.

In the present case with Eqs.(4) Δ*md* receives contributions from gluino and down-type squark, and higgsino and up-type squark diagrams. The explicit form is given as follows:

$$
\dot{\Delta}\_{m\_d} = \overleftarrow{\mathbf{E}}\_{\widetilde{\mathcal{S}}} + \overleftarrow{\mathbf{E}}\_{\widetilde{h}} \, \, \, \, \tag{12}
$$

where

4 Will-be-set-by-IN-TECH

The soft SUSY breaking masses and trilinear SUSY breaking terms (*A*-term) are given by:

*<sup>U</sup>*<sup>R</sup> <sup>−</sup> *<sup>D</sup>*†

Let us first discuss the simplest case where soft breaking masses are proportional to a unit matrix in the flavour space, and **A***u*, **A***<sup>d</sup>* and **A***<sup>e</sup>* are proportional to Yukawa couplings. Their

At tree-level the Yukawa couplings have the same structure as the above superpotential, namely, *H*<sup>1</sup> couples to *D<sup>c</sup>* and *Ec*, and *H*<sup>2</sup> to *Uc*. On the other hand, different types of couplings are induced when we take into account SUSY breaking effects through one-loop diagrams.

where Δ**y***d*, Δ**y***u*, and Δ**y***<sup>e</sup>* are one-loop induced coupling constants, and we

quark and lepton mass matrices and their charged Higgs couplings. For the quark sector, we

<sup>=</sup> *<sup>a</sup>*2*<sup>M</sup>*<sup>2</sup>*δij* , *<sup>M</sup>*<sup>2</sup>

LYukawa = −*H*1*D*R**y***dQ*<sup>L</sup> + *H*2*U*R**y***uQ*<sup>L</sup> − *H*1*E*R**y***eL*<sup>L</sup> − *iσ*2*H*<sup>∗</sup>

<sup>1</sup>*U*RΔ**y***uQ*<sup>L</sup> − *iσ*2*H*<sup>∗</sup>

recall that gauge indices have been suppressed; for example *σ*2*H*∗

<sup>L</sup> , *D*<sup>L</sup> = *V*L(*Q*)*V*CKM*D*�

cos *βD*<sup>R</sup> � *V*†

+ sin *βH*−*D*<sup>R</sup>

represents a diagonal matrix. Since the down-type diagonal mass term is given by

cos *βV*†

√2

*<sup>M</sup><sup>d</sup>* <sup>≡</sup> *<sup>v</sup>* √2 R*M*<sup>2</sup> *D*R

<sup>R</sup>**A***uQ*<sup>L</sup> <sup>+</sup> *<sup>H</sup>*1*<sup>E</sup>*†

*D*<sup>R</sup>*ij*

*D*<sup>R</sup> − *L*† L*M*<sup>2</sup> *L*L *<sup>L</sup>*<sup>L</sup> <sup>−</sup> *<sup>E</sup>*†

<sup>R</sup>**A***<sup>e</sup>*

<sup>=</sup> *<sup>a</sup>*3*<sup>M</sup>*<sup>2</sup>*δij* , *<sup>M</sup>*<sup>2</sup>

*<sup>j</sup>α*. From the above Yukawa couplings, we can derive the

*<sup>d</sup>* <sup>Δ</sup>**y***<sup>d</sup>* (Δ*mu* <sup>≡</sup> **<sup>y</sup>**−<sup>1</sup> *<sup>u</sup>* <sup>Δ</sup>**y***u*), and *<sup>v</sup>* � 246GeV. Notice that

L

<sup>L</sup> (*Q*)Δ*mdV*L(*Q*). Hereafter, a matrix with a hat ()

<sup>R</sup>(*D*)**y***dV*L(*Q*)*RdV*CKM , (9)

<sup>R</sup> , *D*<sup>R</sup> = *V*R(*D*)*D*�

*<sup>u</sup>***y***<sup>u</sup>* in this case. We then rotate the quark bases as follows:

) are mass eigenstates. In this basis, the down-type quark

R(*D*)**y***dV*L(*Q*)*U*�

cos *βD*R**y***d*[1 + tan *β*Δ*md* ]*D*<sup>L</sup> + sin *βH*−*D*R**y***d*[1 − cot *β*Δ*md* ]*U*<sup>L</sup> (6)

sin *<sup>β</sup>U*R**y***u*[<sup>1</sup> <sup>−</sup> cot *<sup>β</sup>*Δ*mu* ]*U*<sup>L</sup> <sup>+</sup> cos *<sup>β</sup>H*+*U*R**y***u*[<sup>1</sup> <sup>+</sup> tan *<sup>β</sup>*Δ*mu* ]*D*<sup>L</sup> <sup>+</sup> h.c. ,

<sup>L</sup> , *U*<sup>R</sup> = *V*R(*U*)*U*�

R(*D*)**y***dV*L(*Q*)*RdV*CKM*D*�

� *V*†

<sup>=</sup> *<sup>a</sup>*5*<sup>M</sup>*<sup>2</sup>*δij* , **<sup>A</sup>***uij* <sup>=</sup> *Au***y***uij* , **<sup>A</sup>***dij* <sup>=</sup> *Ad***y***dij* , **<sup>A</sup>***eij* <sup>=</sup> *Ae***y***eij* , (4)

 *L*L*ij* R*M*<sup>2</sup> *E*R *E*R

*L*<sup>L</sup> + h.c. (3)

<sup>=</sup> *<sup>a</sup>*4*<sup>M</sup>*<sup>2</sup>*δij* ,

<sup>2</sup> *D*RΔ**y***dQ*<sup>L</sup>

<sup>2</sup> *D*RΔ**y***dQ*<sup>L</sup> =

<sup>R</sup> , (7)

<sup>L</sup> + h.c. , (8)

<sup>2</sup> *E*RΔ**y***eL*<sup>L</sup> + h.c. , (5)

<sup>L</sup>soft <sup>=</sup> <sup>−</sup>*<sup>Q</sup>*†

*M*<sup>2</sup> *E*<sup>R</sup>*ij*

where *ai*(*i* = 1 − 5) are real parameters.

explicit forms being:

(*σ*2)*αβ*(*H*<sup>∗</sup>

get

<sup>2</sup> )*β*(*D*R)*<sup>i</sup>*

<sup>L</sup>quark <sup>=</sup> <sup>−</sup> *<sup>v</sup>*

*<sup>a</sup>*(Δ**y***d*) *j i* (*Q*L)*<sup>a</sup>*

√2

where we define <sup>Δ</sup>*md* (Δ*mu* ) as <sup>Δ</sup>*md* <sup>≡</sup> **<sup>y</sup>**−<sup>1</sup>

<sup>L</sup>D−quark <sup>=</sup> <sup>−</sup> *<sup>v</sup>*

where *<sup>R</sup><sup>d</sup>* <sup>≡</sup> <sup>1</sup> <sup>+</sup> tan *<sup>β</sup>*<sup>Δ</sup>*md* and <sup>Δ</sup>*md* <sup>≡</sup> *<sup>V</sup>*†

− *v* √2

Δ**y***<sup>d</sup>* is proportional to **y***<sup>d</sup>* or **y***d***y**†

where the fields with a prime (�

*U*<sup>L</sup> = *V*L(*Q*)*U*�

Lagrangian is given by

*M*<sup>2</sup> *Q*<sup>L</sup>*ij* L*M*<sup>2</sup> *Q*L

<sup>=</sup> *<sup>a</sup>*1*<sup>M</sup>*<sup>2</sup>*δij* , *<sup>M</sup>*<sup>2</sup>

The Lagrangian of the Yukawa sector can be written as:

+*iσ*2*H*∗

<sup>+</sup>*H*1*<sup>D</sup>*†

*<sup>Q</sup>*<sup>L</sup> <sup>−</sup> *<sup>U</sup>*†

*U*<sup>R</sup>*ij*

R*M*<sup>2</sup> *U*R

<sup>R</sup>**A***dQ*<sup>L</sup> <sup>−</sup> *<sup>H</sup>*2*<sup>U</sup>*†

$$\hat{\mathbf{E}}\_{\widetilde{\mathcal{S}}} \equiv \frac{2a\_{\mathbf{s}}}{3\pi} \mathbf{1}\mu^\* M\_{\mathfrak{S}} I[M\_{\mathfrak{S}'} M\_{\mathcal{D}\_{\mathbf{L}'}} M\_{\mathcal{D}\_{\mathbf{R}}}] \, \, \, \, \tag{13}$$

$$\hat{\mathbf{E}}\_{\tilde{\mathbf{h}}} \equiv -\frac{\mu}{16\pi^2} A\_{\mathbf{u}} |\hat{\mathbf{y}}\_{\mathbf{u}}|^2 I[M\_{\tilde{\mathbf{h}}'} M\_{\tilde{\mathbf{U}}\_{\mathbf{L}}'} M\_{\tilde{\mathbf{U}}\_{\mathbf{R}}}] \, \, \, \, \tag{14}$$

$$I[a,b,c] = \frac{a^2 b^2 \ln\frac{a^2}{b^2} + b^2 c^2 \ln\frac{b^2}{c^2} + c^2 a^2 \ln\frac{c^2}{a^2}}{(a^2 - b^2)(b^2 - c^2)(a^2 - c^2)}\,. \tag{15}$$

**<sup>E</sup>***<sup>g</sup>* and **<sup>E</sup>***<sup>h</sup>* are gluino and charged higgsino contributions shown in Fig.1(a) and (b) respectively. Note that these corrections for Yukawa couplings are calculated in the unbroken phase of SU(2) × U(1).

Up to now we have assumed all squark mass matrices are proportional to a unit matrix at the EW scale, as shown in Eqs.(4). However, models with Minimal Flavour Violation (MFV) correspond to more general cases. For instance, the assumption of Eqs.(4) is not satisfied in minimal supergravity, where all squarks have a universal mass at the Planck scale, not at the EW scale. In Ref.[8] they derive the charged Higgs coupling in a more general case of MFV. Namely the squark mass matrix is taken to be

$$\begin{aligned} \boldsymbol{M}\_{\tilde{\mathbf{Q}}\_{\mathbb{L}}}^{2} &= [a\_{1}\mathbf{1} + b\_{1}\mathbf{y}\_{\boldsymbol{\mu}}^{\dagger}\mathbf{y}\_{\boldsymbol{\mu}} + b\_{2}\mathbf{y}\_{d}^{\dagger}\mathbf{y}\_{d}] \tilde{\boldsymbol{M}}^{2}, \\ \boldsymbol{M}\_{\tilde{\mathbf{U}}\_{\mathbb{R}}}^{2} &= [a\_{2}\mathbf{1} + b\_{5}\mathbf{y}\_{\boldsymbol{\mu}}\mathbf{y}\_{\boldsymbol{\mu}}^{\dagger}] \tilde{\boldsymbol{M}}^{2}, \\ \boldsymbol{M}\_{\tilde{\mathbf{D}}\_{\mathbb{R}}}^{2} &= [a\_{3}\mathbf{1} + b\_{6}\mathbf{y}\_{d}\mathbf{y}\_{d}^{\dagger}] \tilde{\boldsymbol{M}}^{2}. \end{aligned} \tag{16}$$

 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

The left-hand plots are for *R*−<sup>1</sup>

This effectively implies that *R*−<sup>1</sup>

treat the diagonal entries of *<sup>R</sup>*<sup>−</sup><sup>1</sup>

and numerics, but we will assume that *R*−<sup>1</sup>

*Mg*˜ = 800 GeV and *M*˜

*M*˜ *bL* = *M*˜

GeV.

R-111

R-111

10 15 20 25 30 35 40 45 50

 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

<sup>22</sup> , while those on the right are for *<sup>R</sup>*−<sup>1</sup>

<sup>33</sup> can differ from *<sup>R</sup>*−<sup>1</sup>

*<sup>d</sup>* as model-independent free parameters in our simulations

<sup>22</sup> . Note that the corresponding corrections

*<sup>b</sup>*<sup>1</sup> = *M*˜*t*<sup>1</sup> = 500 GeV. We have also assumed *M*˜*tL* = *M*˜*tR* and

*bR* . The legends in the right top and right bottom panels correspond to (*μ*, *A*) in

<sup>33</sup> can differ substantially from *<sup>R</sup>*−<sup>1</sup>

R-133

for various values of the higgsino mass parameter *μ* and the up-type trilinear coupling *A*.

case of negative *μ* in the top panels and for positive *μ* below. The other SUSY parameters are

*B* → *μμ* and *b* → *sμμ*) restricts the admissible parameter space [14]. In addition, it can be observed that the higgsino corrections are proportional to the up-type Yukawa couplings and hence can be substantial for diagrams involving the top quark as an external fermion line.

the effective couplings are invariant under a rescaling of all SUSY masses and may indeed be the first observable SUSY effect, as long as the heavy Higgs bosons are light enough. The situation is similar in other models predicting a charged Higgs boson, such as those with a Peccei-Quinn symmetry, spontaneous *CP* violation, dynamical symmetry breaking, or those based on *E*<sup>6</sup> superstring theories, but these have usually been studied much less with respect to the constraints imposed by low-energy data. In the remainder of this work, we shall thus

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

to the up-type couplings are suppressed by cot *β* and hence can be neglected in our analysis.

R-133

Constraining the Couplings of a Charged Higgs to Heavy Quarks 35

10 15 20 25 30 35 40 45 50

 (-300,-500) (-300,0) (-300,500) (-500,-500) (-500,0) (-500,500)

tan

10 15 20 25 30 35 40 45 50

(300,-500) (300,0) (300,500) (500,-500) (500,0) (500,500)

tan

*ii* on tan *β* in the exemplary case of the MSSM

<sup>33</sup> . We present the

<sup>11</sup> , where for certain SUSY

<sup>11</sup> by more than 30%. This

<sup>11</sup> . We remind the reader that

<sup>33</sup> when compared

 = -300 GeV = -500 GeV

tan

 = 300 GeV = 500 GeV

10 15 20 25 30 35 40 45 50

tan

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

difference could be observed at the LHC for processes that depend on *R*−<sup>1</sup>

with the results of *B*-factories for processes that depend on *R*−<sup>1</sup>

Fig. 2. Dependence of the general couplings *R*−<sup>1</sup>

scenarios, as shown in Fig.2, we observe that *R*−<sup>1</sup>

The final results of the charged Higgs coupling being given by

$$\mathcal{L}\_{H^{\pm}} \approx \frac{\sqrt{2}}{v} \tan \beta H^{-} \overline{D}\_{\text{R}} \frac{\hat{M}\_{\text{di}}}{1 + [E\_{\bar{\mathcal{S}}}^{(i)}] \tan \beta} V\_{\text{CKMij}}^{\dagger} \mathcal{U}\_{\text{L}j}^{\prime} + \text{h.c.}$$

$$\text{for } (i, j) = (1, 1), (1, 2), (2, 1), (2, 2), \tag{17}$$

$$\mathcal{L}\_{H^{\pm}} \approx \frac{\sqrt{2}}{v} \tan \beta H^{-} \overline{D}\_{\text{Ri}}^{\prime} \frac{\hat{\mathcal{M}}\_{\text{di}}}{1 + [E\_{\hat{\mathcal{g}}}^{\prime(i)} - E\_{\hat{\mathcal{g}}}^{\prime(i\bar{j})}] \tan \beta} V\_{\text{CKM}ij}^{\dagger} \mathcal{U}\_{\text{L}j}^{\prime} + \text{h.c.}$$
 
$$\text{for } (i, j) = (3, 1), (3, 2), \tag{18}$$

$$\mathcal{L}\_{H^{\pm}} \approx \frac{\sqrt{2}}{v} \tan \beta H^{-} \overline{D}\_{\text{Rj}}^{\prime} \frac{\hat{M}\_{\text{di}}}{1 + E\_{\overline{\mathcal{g}}}^{\left( \bar{i} \right)} \tan \beta} \frac{1 + \left[ E\_{\overline{\mathcal{g}}}^{\left( \bar{3} \right)} + E\_{\overline{\mathcal{h}}}^{\left( \bar{3} \right)} \right] \tan \beta}{1 + \left[ E\_{\overline{\mathcal{g}}}^{\left( \bar{i} \right)} + E\_{\overline{\mathcal{h}}}^{\left( \bar{3} \right)} + E\_{\overline{\mathcal{g}}}^{\left( \bar{i} \right)} + E\_{\overline{\mathcal{h}}}^{\left( \bar{i} \right)} + E\_{\overline{\mathcal{h}}}^{\left( \bar{i} \right)} \right] \tan \beta}} $$
 
$$\times V\_{\text{CKMij}}^{\dagger} \mathcal{U}\_{\text{Lj}}^{\prime} + \text{h.c.} \text{ for } (i, j) = (1, 3), (2, 3), \tag{19}$$

$$\mathcal{L}\_{H^{\pm}} \approx \frac{\sqrt{2}}{v} \tan \beta H^{-} \overline{D}\_{\text{Rj}} \frac{\hat{M}\_{\text{di}}}{1 + [E\_{\hat{\mathcal{S}}}(^{\hat{\imath}}) + E\_{\hat{\mathcal{h}}}(^{\text{(3)}})] \tan \beta} V\_{\text{CKM}ij}^{\dagger} \mathcal{U}\_{\text{Lj}}^{\prime} + \text{h.c.} $$
 
$$\text{for } (i, j) = (3, 3). \tag{20}$$

The functions *Eg* (*i*) , etc. are listed in Ref.[8]. In deriving these results only the *yt* in the up-type Yukawa coupling in loop diagrams was kept and use made of the hierarchy of the CKM matrix elements. See Ref.[8] for details. Notice that the above results do not depend on the relationship between the *A*-terms and the Yukawa couplings, since only the *yt* in loop diagrams was kept, even though Eqs.(4) are assumed.

#### **2.2 Couplings to the bottom quark**

From Eq.(10) and now under the assumption of MFV, we know that trilinear couplings are in general proportional to the original Yukawa couplings. We shall therefore label the components of the diagonal matrix *<sup>R</sup>*<sup>−</sup><sup>1</sup> *<sup>d</sup>* <sup>=</sup> diag *R*−<sup>1</sup> <sup>11</sup> , *<sup>R</sup>*−<sup>1</sup> <sup>22</sup> *<sup>R</sup>*−<sup>1</sup> 33 , where the three diagonal values of *<sup>R</sup>*<sup>−</sup><sup>1</sup> *<sup>d</sup>* represent the couplings of a charged Higgs boson to the bottom quark and the three up-type quarks. At tree-level, these three couplings are equal, *R*−<sup>1</sup> <sup>11</sup> <sup>=</sup> *<sup>R</sup>*−<sup>1</sup> <sup>22</sup> <sup>=</sup> *<sup>R</sup>*−<sup>1</sup> <sup>33</sup> = 1, where this equality is broken to some extent by loop corrections to the charged Higgs vertex, and *R<sup>d</sup>* can then be written as:

$$
\widehat{R}\_d = 1 + \tan \beta \widehat{\Delta}\_{m\_d} \,. \tag{21}
$$

In the forth-coming analysis we have kept the O(*αs*) SUSY-QCD corrections and SUSY loop corrections associated with the Higgs-top Yukawa couplings (as discussed in the previous subsection) and have neglected the subleading EW corrections of the order <sup>O</sup>(*g*2) as given in Ref.[12].1 Therefore, they then depend upon the higgsino-mass parameter *μ*, the up-type trilinear couplings *A*, and the bino, bottom and top squark masses. As argued in Ref.[8] the higgsino-diagram contributions can be neglected in *R*−<sup>1</sup> <sup>11</sup> and *<sup>R</sup>*−<sup>1</sup> <sup>22</sup> , so that to a very good approximation *R*−<sup>1</sup> <sup>11</sup> <sup>≈</sup> *<sup>R</sup>*−<sup>1</sup> <sup>22</sup> . As an illustration, we show in Fig.2 the dependence of the SUSY corrections on tan *β* for some illustrative SUSY parameters. These corrections can alter the tree-level values significantly, although low-energy data (e.g. from *<sup>b</sup>* <sup>→</sup> *<sup>s</sup>γ*, *<sup>B</sup>* <sup>−</sup> *<sup>B</sup>*¯ mixing,

<sup>1</sup> For an alternative definition, in which SUSY loop effects are assigned to the CKM matrix, see Ref.[13]

6 Will-be-set-by-IN-TECH

<sup>L</sup>*<sup>j</sup>* + h.c.

*V*† CKM*ijU*�

(*i*) <sup>+</sup> *<sup>E</sup><sup>h</sup>*

*V*† CKM*ijU*�

for (*i*, *j*)=(1, 1),(1, 2),(2, 1),(2, 2), (17)

for (*i*, *j*)=(3, 1),(3, 2), (18)

(33) + *E*� *g* (*ij*) <sup>+</sup> *<sup>E</sup><sup>h</sup>*

for (*i*, *j*)=(3, 3). (20)

, etc. are listed in Ref.[8]. In deriving these results only the *yt* in the

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

<sup>L</sup>*<sup>j</sup>* + h.c.

(3) <sup>+</sup> *<sup>E</sup><sup>h</sup>*

<sup>L</sup>*<sup>j</sup>* + h.c. for (*i*, *j*)=(1, 3),(2, 3), (19)

<sup>L</sup>*<sup>j</sup>* + h.c.

(33)] tan *β*

(*i*3) + *E*� *h* (*i*33)

, where the three diagonal

<sup>22</sup> , so that to a very good

<sup>22</sup> <sup>=</sup> *<sup>R</sup>*−<sup>1</sup>

<sup>33</sup> = 1,

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

*<sup>R</sup><sup>d</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> tan *<sup>β</sup>*Δ<sup>ˆ</sup> *md* . (21)

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

<sup>22</sup> . As an illustration, we show in Fig.2 the dependence of the SUSY

] tan *β*

*V*† CKM*ijU*�

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

(*i*<sup>3</sup>)] tan *β*

up-type Yukawa coupling in loop diagrams was kept and use made of the hierarchy of the CKM matrix elements. See Ref.[8] for details. Notice that the above results do not depend on the relationship between the *A*-terms and the Yukawa couplings, since only the *yt* in loop

From Eq.(10) and now under the assumption of MFV, we know that trilinear couplings are in general proportional to the original Yukawa couplings. We shall therefore label the

where this equality is broken to some extent by loop corrections to the charged Higgs vertex,

In the forth-coming analysis we have kept the O(*αs*) SUSY-QCD corrections and SUSY loop corrections associated with the Higgs-top Yukawa couplings (as discussed in the previous subsection) and have neglected the subleading EW corrections of the order <sup>O</sup>(*g*2) as given in Ref.[12].1 Therefore, they then depend upon the higgsino-mass parameter *μ*, the up-type trilinear couplings *A*, and the bino, bottom and top squark masses. As argued in Ref.[8]

corrections on tan *β* for some illustrative SUSY parameters. These corrections can alter the tree-level values significantly, although low-energy data (e.g. from *<sup>b</sup>* <sup>→</sup> *<sup>s</sup>γ*, *<sup>B</sup>* <sup>−</sup> *<sup>B</sup>*¯ mixing,

<sup>1</sup> For an alternative definition, in which SUSY loop effects are assigned to the CKM matrix, see Ref.[13]

 *R*−<sup>1</sup> <sup>11</sup> , *<sup>R</sup>*−<sup>1</sup> <sup>22</sup> *<sup>R</sup>*−<sup>1</sup> 33 

*<sup>d</sup>* represent the couplings of a charged Higgs boson to the bottom quark and the

*<sup>d</sup>* = diag

three up-type quarks. At tree-level, these three couplings are equal, *R*−<sup>1</sup>

the higgsino-diagram contributions can be neglected in *R*−<sup>1</sup>

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

The final results of the charged Higgs coupling being given by

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

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

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

<sup>×</sup>*V*†

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

*Mdi*

*Mdi*

(*i*)] tan *β*

*Mdi*

(*i*) <sup>−</sup> *<sup>E</sup>*� *g* (*ij*) ] tan *β*

(*i*) tan *β*

*Mdi*

(*i*) <sup>+</sup> *<sup>E</sup><sup>h</sup>*

CKM*ijU*�

� *i*

� *i*

� *i*

� *i*

diagrams was kept, even though Eqs.(4) are assumed.

L*H*<sup>±</sup> ≈

L*H*<sup>±</sup> ≈

L*H*<sup>±</sup> ≈

L*H*<sup>±</sup> ≈

The functions *Eg*

values of *<sup>R</sup>*<sup>−</sup><sup>1</sup>

approximation *R*−<sup>1</sup>

√2 *v*

√2 *v*

√2 *v*

√2 *v*

tan *βH*−*D*<sup>R</sup>

tan *βH*−*D*<sup>R</sup>

tan *βH*−*D*<sup>R</sup>

tan *βH*−*D*<sup>R</sup>

(*i*)

**2.2 Couplings to the bottom quark**

and *R<sup>d</sup>* can then be written as:

components of the diagonal matrix *<sup>R</sup>*<sup>−</sup><sup>1</sup>

Fig. 2. Dependence of the general couplings *R*−<sup>1</sup> *ii* on tan *β* in the exemplary case of the MSSM for various values of the higgsino mass parameter *μ* and the up-type trilinear coupling *A*. The left-hand plots are for *R*−<sup>1</sup> <sup>11</sup> <sup>=</sup> *<sup>R</sup>*−<sup>1</sup> <sup>22</sup> , while those on the right are for *<sup>R</sup>*−<sup>1</sup> <sup>33</sup> . We present the case of negative *μ* in the top panels and for positive *μ* below. The other SUSY parameters are *Mg*˜ = 800 GeV and *M*˜ *<sup>b</sup>*<sup>1</sup> = *M*˜*t*<sup>1</sup> = 500 GeV. We have also assumed *M*˜*tL* = *M*˜*tR* and *M*˜ *bL* = *M*˜ *bR* . The legends in the right top and right bottom panels correspond to (*μ*, *A*) in GeV.

*B* → *μμ* and *b* → *sμμ*) restricts the admissible parameter space [14]. In addition, it can be observed that the higgsino corrections are proportional to the up-type Yukawa couplings and hence can be substantial for diagrams involving the top quark as an external fermion line. This effectively implies that *R*−<sup>1</sup> <sup>33</sup> can differ substantially from *<sup>R</sup>*−<sup>1</sup> <sup>11</sup> , where for certain SUSY scenarios, as shown in Fig.2, we observe that *R*−<sup>1</sup> <sup>33</sup> can differ from *<sup>R</sup>*−<sup>1</sup> <sup>11</sup> by more than 30%. This difference could be observed at the LHC for processes that depend on *R*−<sup>1</sup> <sup>33</sup> when compared with the results of *B*-factories for processes that depend on *R*−<sup>1</sup> <sup>11</sup> . We remind the reader that the effective couplings are invariant under a rescaling of all SUSY masses and may indeed be the first observable SUSY effect, as long as the heavy Higgs bosons are light enough. The situation is similar in other models predicting a charged Higgs boson, such as those with a Peccei-Quinn symmetry, spontaneous *CP* violation, dynamical symmetry breaking, or those based on *E*<sup>6</sup> superstring theories, but these have usually been studied much less with respect to the constraints imposed by low-energy data. In the remainder of this work, we shall thus treat the diagonal entries of *<sup>R</sup>*<sup>−</sup><sup>1</sup> *<sup>d</sup>* as model-independent free parameters in our simulations and numerics, but we will assume that *R*−<sup>1</sup> <sup>11</sup> <sup>≈</sup> *<sup>R</sup>*−<sup>1</sup> <sup>22</sup> . Note that the corresponding corrections to the up-type couplings are suppressed by cot *β* and hence can be neglected in our analysis.

0.0001

100 150 200 250 300 350 400 450 500

μ ν

tan *β* = 5 (left panel), and tan *β* = 40 (right panel)[10].

bc

˜l ˜ν

Fig. 4. The branching ratios of charged decays into SM particles as a function of *mH*<sup>±</sup> , for

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

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* →

*<sup>b</sup>* <sup>→</sup> ¯*tH*+) process is generated, there will be an accompanying outgoing ¯

the accompanying *<sup>b</sup>*-quark one instead uses the exact matrix element of the *gg* <sup>→</sup> *<sup>t</sup>*¯

*<sup>b</sup>* <sup>→</sup> *tH*<sup>±</sup> and the *gg* <sup>→</sup> *<sup>t</sup>*¯

Ref.[24], for low transverse momenta ( 100GeV) the *gg* <sup>→</sup> *<sup>t</sup>*¯

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

¯*tbH*+) process. Whilst for low transverse momenta, this process can be described in terms of

the differential cross-section. Therefore, when the accompanying *b*-quark is observed, it is

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>*¯

the sum of the cross-sections of the two processes. The double-counting term is given by the

(*x*2, *<sup>μ</sup>F*) *<sup>d</sup><sup>σ</sup>*2→<sup>2</sup>

*dx*1*dx*<sup>2</sup>

*b* times the matrix element of the *gb* → *tH*<sup>±</sup> process. As was shown in

0.0001

*b* pairs, can be summed to all orders in perturbation theory by introducing bottom

100 150 200 250 300 350 400 450 500

hW

cs

tanβ = 40

τ ν

˜q˜q′

*b* (*b*) quark.

*bH*<sup>−</sup> (*gg* →

*bH*±

, (22)

*bH*± approach underestimates

*bH*± processes together, appropriately

*b* → *tH*<sup>±</sup> process. This term is then subtracted from

(*x*1, *x*2) + *x*<sup>1</sup> ↔ *x*<sup>2</sup>

tb

χ− χ0

mH- (GeV)

0.001

0.01

bc

μ ν

su

Branching Ratio

Constraining the Couplings of a Charged Higgs to Heavy Quarks 37

0.1

1

˜q˜q′

cs

τ ν

hW

tanβ = 5

tb

χ− χ0

mH- (GeV)

0.001

su

AW

0.01

Branching Ratio

collinear *b*¯

*tH*<sup>−</sup> (*g*¯

the gluon splitting to *b*¯

necessary to use both the *g*¯

matched to remove the double-counting.

process which is already included in the *g*¯

*σ*DC =

leading contribution of the *b* quark density as:

*dx*1*dx*<sup>2</sup>

*g*(*x*1, *μF*)*b*�

0.1

1

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