== 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
[[Image(https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/TwoHiggsDoublet/equation1_twohiggs.jpeg)]]
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
$\mathcal{L}''=Y\frac{\overline{Q''L}\sqrt{2}}{v}\left[(M_d H_1 + Y_d H_2)d_R+(M_u \tilde{H}''1 + Y''u \tilde{H}''2)u''R\right] +\frac{\overline{E_L}\sqrt{2}}{v}\left[(M_e H_1 + Y_e H_2)e_R\right]\,.$
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