| 1 | |
| 2 | |
| 3 | == The 2HDM implementation == |
| 4 | |
| 5 | The two-Higgs-doublet model (2HDM) has been extensively studied for |
| 6 | more than twenty years, even though it has often been only |
| 7 | considered as the scalar sector of larger models like the MSSM |
| 8 | \citeGunion:1989we} or Little Higgs models |
| 9 | \cite{Arkani-Hamed:2002qx}. The generic 2HDM considered here may |
| 10 | display by itself an interesting phenomenology justifying its |
| 11 | study. As a non exhaustive list, let us mention new sources of %$CP$% |
| 12 | violation in scalar-scalars interactions \cite{Branco:1999fs}, |
| 13 | tree-level flavor changing neutral currents (FCNCs) due to non |
| 14 | diagonal Yukawa interactions, dark matter candidates |
| 15 | \cite{Barbieri:2006dq} or Higgs bosons lighter than the LEP bound |
| 16 | \cite{Gerard:2007kn}. |
| 17 | |
| 18 | In the ``full'' version of the model (2hdm_full), no particular |
| 19 | restrictions are imposed on the interactions allowed by gauge |
| 20 | invariance, except electric charge conservation. Many diagrams |
| 21 | involving tree-level FCNCs and violating the %$CP$% symmetry are thus |
| 22 | present. The user who is not interested in these phenomena should use |
| 23 | the ``simplified'' version of the model (2hdm), where the number |
| 24 | of generated diagrams is in general much smaller. |
| 25 | |
| 26 | The following naming convention is used: h+ and h- stand |
| 27 | for the positively and negatively charged Higgs bosons and h1, |
| 28 | h2 and h3 stand for the neutral ones. Since the %$CP$% |
| 29 | invariance of the potential is not assumed, the neutral bosons are not |
| 30 | necessarily $CP$ eigenstates and the standard naming convention in |
| 31 | this case (ie, h1 being the lightest one and h3 the |
| 32 | heaviest one) is used. |
| 33 | |
| 34 | TwoHiggsCalc is the calculator associated with the model. It has been |
| 35 | written in C and is accessible from a web interface. It has been |
| 36 | designed to compute input values for the 2HDM extension of |
| 37 | MadGraph/MadEvent but it can also be used as an independent |
| 38 | tool. Starting from various parameters of the Lagrangian, such as the |
| 39 | vacuum expectation values (vevs) or the Yukawa couplings, the program |
| 40 | computes useful secondary physical quantities at leading order such as |
| 41 | the scalar mass spectrum, the mixing matrix, the total decay widths |
| 42 | and the branching ratios. |
| 43 | |
| 44 | TwoHiggsCalc reads input and writes out results in a specific format |
| 45 | close to the ``SUSY Les Houches Accord 1.0" convention for SUSY |
| 46 | parameters \cite{Skands:2003cj}. This format can later be read by |
| 47 | MadEvent to perform numerical calculations for 2HDM processes. A |
| 48 | README file describing this modified version of the LHA format used |
| 49 | as input convention is available. To ease the use of TwoHiggsCalc, a |
| 50 | web form has been designed to automatize the parameter card writing |
| 51 | process. Numerical values for the parameters (units being fixed when |
| 52 | needed) can be entered on this form. Some simple algebraic expressions |
| 53 | can also be used. The +,-,*,/ operators and the |
| 54 | reserved keyword PI, eg, PI/2+3*PI/2, are correctly interpreted. |
| 55 | |
| 56 | |
| 57 | In the general 2HDM, one has the freedom to choose a specific basis |
| 58 | for entering parameters. All the possible choices are physically |
| 59 | equivalent (see eg \cite{Davidson:2005cw} for a |
| 60 | discussion). TwoHiggsCalc and the 2HDM model both assume that the |
| 61 | parameters are given in a particular basis, called the ``Higgs basis'' |
| 62 | where only one Higgs doublet gets a vacuum expectation value. An |
| 63 | independent program , Gen2HB, has been written to convert parameters |
| 64 | given in an arbitrary basis (where both Higgs doublets get vevs), |
| 65 | called ``generic'', to parameters in the Higgs basis. See |
| 66 | \cite{Branco:1999fs} for more information on basis invariance and on |
| 67 | the notation used. |
| 68 | |
| 69 | The scalar potential in the Higgs basis reads |
| 70 | |
| 71 | %$ |
| 72 | 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)\\ |
| 73 | & & \lambda_1 \left(H_1^\dag H_1\right)^2+ \lambda_2 \left(H_2^\dag |
| 74 | H_2\right)^2\\ & & + \lambda_3 \left(H_1^\dag |
| 75 | H_1\right)\left(H_2^\dag H_2\right)+ \lambda_4 \left(H_1^\dag |
| 76 | H_2\right)\left(H_2^\dag H_1\right)\\ & & +\left[\left( \lambda_5 |
| 77 | H_1^\dag H_2 + \lambda_6 H_1^\dag H_1+ \lambda_7 H_2^\dag |
| 78 | H_2\right)\left(H_1^\dag H_2\right)+\mathrm{h.c.}\right]\,. |
| 79 | $% |
| 80 | |
| 81 | All parameters in front of quartic terms and the charged Higgs mass |
| 82 | are input parameters, while %$\mu_1$%, %$\mu_2$% and %$\mu_3$% are fixed by |
| 83 | minimization constraints and by the vev extracted from the observed SM |
| 84 | parameters. %$\lambda_1$% to %$\lambda_4$% are real while %$\lambda_5$% in |
| 85 | general is complex. However, since only the phase differences between |
| 86 | %$\lambda_5$%, %$\lambda_6$%, %$\lambda_7$% and %$\mu_3$% matter, the phase of |
| 87 | %$\lambda_5$% can always be rotated out. It is thus considered as a real |
| 88 | parameter while %$\lambda_6$% and %$\lambda_7$% are a priori complex. |
| 89 | |
| 90 | In the same basis, the Yukawa interactions read |
| 91 | |
| 92 | %$ |
| 93 | \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]\\ |
| 94 | & &+\frac{\overline{E_L}\sqrt{2}}{v}\left[(M_e H_1 + Y_e H_2)e_R\right]\,. |
| 95 | $% |
| 96 | |
| 97 | Yukawa couplings are expected to be given in the physical basis for |
| 98 | fermions, ie, in the basis where the mass matrix is diagonal. Since |
| 99 | in the Higgs basis only the first Higgs doublet gets a non zero vev, |
| 100 | the %$M$% matrices are completely fixed by the physical fermion masses |
| 101 | and CKM mixing matrix (restricted to Cabibbo angle) while the %$Y$% |
| 102 | matrices (giving the couplings of the second Higgs doublet) are a |
| 103 | priori free. For these matrices, the first index refers to doublet |
| 104 | generation while the second refers to the singlet generation. For |
| 105 | example, Y2B stands for the complex Yukawa couplings of the |
| 106 | second Higgs doublet to the second generation quark left doublet and |
| 107 | to the bottom singlet. |
| 108 | |
| 109 | In the generic basis, similar expressions are assumed. For the scalar |
| 110 | potential all parameters in front of quartic terms are inputs as well |
| 111 | as %$\tan(\beta)$%, the norm of %$\mu_3$% and the phase of %$v_2$%. The |
| 112 | overall vev is again extracted from SM parameters while mass terms |
| 113 | parameters, like %$\mu_1$%, %$\mu_2$% and the phase of %$\mu_3$%, are fixed |
| 114 | by the minimization constraints. %$\lambda_1$% to %$\lambda_4$% are real |
| 115 | parameters, %$\lambda_5$%, %$\lambda_6$% and %$\lambda_7$% are a priori |
| 116 | complex. Like in the Higgs basis, the Yukawa couplings must be given |
| 117 | in the physical basis for fermions. Since the mass matrices are fixed, |
| 118 | only the Yukawa coupling matrices of the second Higgs doublet |
| 119 | (%$\Gamma$%), is required. The other one is going to be automatically |
| 120 | evaluated to match observed fermion masses and CKM mixing matrix |
| 121 | (restricted to Cabibbo angle). For the %$\Gamma$% matrix, the first |
| 122 | index refers to doublet generation while the second one refer to the |
| 123 | singlet generation. For example, G2B stands for the complex |
| 124 | Yukawa couplings of the second Higgs doublet to the second generation |
| 125 | quark left doublet and to the bottom singlet. |
| 126 | |
| 127 | Given the above parameters and some SM parameters, TwoHiggsCalc |
| 128 | computes the following quantities |
| 129 | |
| 130 | * Scalar particles mass spectrum |
| 131 | * Normalized mixing matrix of neutral scalars (called $T$ in \cite{Branco:1999fs}) |
| 132 | * Decay widths for all scalars as well as for $W$ and $Z$ bosons |
| 133 | and the top quark. All widths are evaluated at tree-level using |
| 134 | the same couplings as in MadEvent. Below threshold formulas |
| 135 | are included for the scalar decays into two vector bosons and |
| 136 | the one loop driven scalar decay into two gluons is also |
| 137 | computed. |
| 138 | |
| 139 | The LHA blocks and parameters used by MadEvent are given in |
| 140 | Table below. All blocks in the table are provided by |
| 141 | TwoHiggsCalc. Note that if parton density functions (PDFs) are used in |
| 142 | the MadEvent run, the value for %$\alpha_s$% at %$M_Z$% and the order of |
| 143 | its running is given by the PDF. Otherwise %$\alpha_s(M_Z)$% is given by |
| 144 | block SMINPUTS, parameter 3, and the order of running is taken |
| 145 | to be 2-loop. The scale where %$\alpha_s$% is evaluated can be fixed or |
| 146 | evaluated on an event-by-event basis like in the SM. |
| 147 | |
| 148 | ||Block||Comment|| |
| 149 | ||SMINPUTS||From 1 to 4, SM parameters, see the SM section for more details|| |
| 150 | ||MGSMPARAM||Extra block with %$\sin\theta_W$% and %$M_W$%, see the SM section for more details|| |
| 151 | ||MGYUKAWA}||``Yukawa'' masses used in the Yukawa couplings evaluation|| |
| 152 | ||MGCKM ||The full CKM matrix|| |
| 153 | ||BASIS||Basis choice, must be 1 (Higgs basis) for MadEvent !|| |
| 154 | ||MINPAR||Scalar potential parameters in the Higgs basis|| |
| 155 | ||YUKAWA2||Yukawa couplings of the second Higgs doublet|| |
| 156 | ||MASS|| All SM particles masses, plus the five new Higgs boson masses|| |
| 157 | ||TMIX|| The scalar mixing matrix|| |
| 158 | ||DECAY|| For all the Higgs bosons, top, %$W^\pm$% and %$Z$%|| |
| 159 | |
| 160 | -- Main.MichelHerquet - 09 Apr 2007 |
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