The 2HDM implementation

The two-Higgs-doublet model (2HDM) has been extensively studied for more than twenty years, even though it has often been only considered as the scalar sector of larger models like the MSSM \cite{Gunion:1989we} or Little Higgs models \cite{Arkani-Hamed:2002qx}. The generic 2HDM considered here may display by itself an interesting phenomenology justifying its study. As a non exhaustive list, let us mention new sources of $CP$ violation in scalar-scalars interactions \cite{Branco:1999fs}, tree-level flavor changing neutral currents (FCNCs) due to non diagonal Yukawa interactions, dark matter candidates \cite{Barbieri:2006dq} or Higgs bosons lighter than the LEP bound \cite{Gerard:2007kn}.

In the full version of the model (2hdm_full), no particular restrictions are imposed on the interactions allowed by gauge invariance, except electric charge conservation. Many diagrams involving tree-level FCNCs and violating the $CP$ symmetry are thus present. The user who is not interested in these phenomena should use the simplified version of the model (2hdm), where the number of generated diagrams is in general much smaller.

The following naming convention is used: h+ and h- stand for the positively and negatively charged Higgs bosons and h1, h2 and h3 stand for the neutral ones. Since the $CP$ invariance of the potential is not assumed, the neutral bosons are not necessarily $CP$ eigenstates and the standard naming convention in this case (ie, h1 being the lightest one and h3 the heaviest one) is used.

TwoHiggsCalc is the calculator associated with the model. It has been written in C and is accessible from a web interface. It has been designed to compute input values for the 2HDM extension of MadGraph/MadEvent but it can also be used as an independent tool. Starting from various parameters of the Lagrangian, such as the vacuum expectation values (vevs) or the Yukawa couplings, the program computes useful secondary physical quantities at leading order such as the scalar mass spectrum, the mixing matrix, the total decay widths and the branching ratios.

TwoHiggsCalc reads input and writes out results in a specific format close to the SUSY Les Houches Accord 1.0" convention for SUSY parameters \cite{Skands:2003cj}. This format can later be read by MadEvent to perform numerical calculations for 2HDM processes. A README file describing this modified version of the LHA format used as input convention is available. To ease the use of TwoHiggsCalc, a web form has been designed to automatize the parameter card writing process. Numerical values for the parameters (units being fixed when needed) can be entered on this form. Some simple algebraic expressions can also be used. The +,-,*,/ operators and the reserved keyword PI, eg, PI/2+3*PI/2, are correctly interpreted.

In the general 2HDM, one has the freedom to choose a specific basis for entering parameters. All the possible choices are physically equivalent (see eg \cite{Davidson:2005cw} for a discussion). TwoHiggsCalc and the 2HDM model both assume that the parameters are given in a particular basis, called the Higgs basis where only one Higgs doublet gets a vacuum expectation value. An independent program , Gen2HB, has been written to convert parameters given in an arbitrary basis (where both Higgs doublets get vevs), called generic, to parameters in the Higgs basis. See \cite{Branco:1999fs} for more information on basis invariance and on the notation used.

The scalar potential in the Higgs basis reads

$V = \mu_1 H_1^{\dag H_1 +\mu_2 H_2}\dag H_2-\left(\mu_3 H_1^{\dag H_2+\mathrm{h.c.}\right) \lambda_1 \left(H_1}\dag H_1\right)^{2+ \lambda_2 \left(H_2}\dagH_2\right)^{2 + \lambda_3 \left(H_1}\dag H_1\right)\left(H_2^{\dag H_2\right)+ \lambda_4 \left(H_1}\dag H_2\right)\left(H_2^{\dag H_1\right) +\left[\left( \lambda_5 H_1}\dag H_2 + \lambda_6 H_1^{\dag H_1+ \lambda_7 H_2}\dag H_2\right)\left(H_1^{\dag H_2\right)+\mathrm{h.c.}\right]\,. $}

All parameters in front of quartic terms and the charged Higgs mass are input parameters, while $\mu_1$, $\mu_2$ and $\mu_3$ are fixed by minimization constraints and by the vev extracted from the observed SM parameters. $\lambda_1$ to $\lambda_4$ are real while $\lambda_5$ in general is complex. However, since only the phase differences between %$\lambda_5$, $\lambda_6$, $\lambda_7$ and $\mu_3$ matter, the phase of %$\lambda_5$ can always be rotated out. It is thus considered as a real parameter while $\lambda_6$ and $\lambda_7$ are a priori complex.

In the same basis, the Yukawa interactions read

$\mathcal{L}=Y\frac{\overline{QL}\sqrt{2}}{v}\left[(M_d H_1 + Y_d H_2)d_R+(M_u \tilde{H}1 + Yu \tilde{H}2)uR\right] +\frac{\overline{E_L}\sqrt{2}}{v}\left[(M_e H_1 + Y_e H_2)e_R\right]\,.$

Yukawa couplings are expected to be given in the physical basis for fermions, ie, in the basis where the mass matrix is diagonal. Since in the Higgs basis only the first Higgs doublet gets a non zero vev, the $M$ matrices are completely fixed by the physical fermion masses and CKM mixing matrix (restricted to Cabibbo angle) while the $Y$ matrices (giving the couplings of the second Higgs doublet) are a priori free. For these matrices, the first index refers to doublet generation while the second refers to the singlet generation. For example, Y2B stands for the complex Yukawa couplings of the second Higgs doublet to the second generation quark left doublet and to the bottom singlet.

In the generic basis, similar expressions are assumed. For the scalar potential all parameters in front of quartic terms are inputs as well as $\tan(\beta)$, the norm of $\mu_3$ and the phase of $v_2$. The overall vev is again extracted from SM parameters while mass terms parameters, like $\mu_1$, $\mu_2$ and the phase of $\mu_3$, are fixed by the minimization constraints. $\lambda_1$ to $\lambda_4$ are real parameters, $\lambda_5$, $\lambda_6$ and $\lambda_7$ are a priori complex. Like in the Higgs basis, the Yukawa couplings must be given in the physical basis for fermions. Since the mass matrices are fixed, only the Yukawa coupling matrices of the second Higgs doublet ($\Gamma$), is required. The other one is going to be automatically evaluated to match observed fermion masses and CKM mixing matrix (restricted to Cabibbo angle). For the $\Gamma$ matrix, the first index refers to doublet generation while the second one refer to the singlet generation. For example, G2B stands for the complex Yukawa couplings of the second Higgs doublet to the second generation quark left doublet and to the bottom singlet.

Given the above parameters and some SM parameters, TwoHiggsCalc computes the following quantities

• Scalar particles mass spectrum
• Normalized mixing matrix of neutral scalars (called $T$ in \cite{Branco:1999fs})
• Decay widths for all scalars as well as for $W$ and $Z$ bosons

and the top quark. All widths are evaluated at tree-level using the same couplings as in MadEvent. Below threshold formulas are included for the scalar decays into two vector bosons and the one loop driven scalar decay into two gluons is also computed.

The LHA blocks and parameters used by MadEvent are given in Table below. All blocks in the table are provided by TwoHiggsCalc. Note that if parton density functions (PDFs) are used in the MadEvent run, the value for $\alpha_s$ at $M_Z$ and the order of its running is given by the PDF. Otherwise $\alpha_s(M_Z)$ is given by block SMINPUTS, parameter 3, and the order of running is taken to be 2-loop. The scale where $\alpha_s$ is evaluated can be fixed or evaluated on an event-by-event basis like in the SM.

Block	Comment
SMINPUTS	From 1 to 4, SM parameters, see the SM section for more details
MGSMPARAM	Extra block with $\sin\theta_W$ and $M_W$, see the SM section for more details
MGYUKAWA}	Yukawa masses used in the Yukawa couplings evaluation
MGCKM	The full CKM matrix
BASIS	Basis choice, must be 1 (Higgs basis) for MadEvent !
MINPAR	Scalar potential parameters in the Higgs basis
YUKAWA2	Yukawa couplings of the second Higgs doublet
MASS	All SM particles masses, plus the five new Higgs boson masses
TMIX	The scalar mixing matrix
DECAY	For all the Higgs bosons, top, $W\pm$ and $Z$

-- Main.MichelHerquet - 09 Apr 2007