master
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a Code for the Combination of Indirect and Direct Constraints on High Energy Physics Models |
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master
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a Code for the Combination of Indirect and Direct Constraints on High Energy Physics Models |
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A class for the \(M \to V \gamma\) decay. More...
#include <MVgamma.h>
A class for the \(M \to V \gamma\) decay.
This class is used to compute all the functions needed in order to compute the observables relative to the \(M \to V \gamma\) decays, where \(M\) is a generic meson and \(V\) is a vector meson.
The mandatory parameters of MVll are summarized below:
| Label | LaTeX symbol | Description | |||
|---|---|---|---|---|---|
| a_0T1 | \(a_0^{T_1}\) | The fit parameters for the form factor \(T_1\) of the \(B\to K^*\) at \(q^2=0\). | |||
| a_0T1phi | \(a_0^{T_1}\) | The fit parameters for the form factor \(T_1\) of the \(B\to\phi\) at \(q^2=0\). | absh_p | \(\mathrm{Abs}h_+^{(0)}\) | The constant term of the absolute value of the hadronic parameter \(h_+\) of the \(B\to K^*\) at \(q^2=0\). |
| argh_p | \(\mathrm{Arg}h_+^{(0)}\) | The constant term of the argument of the hadronic parameter \(h_+\) of the \(B\to K^*\) at \(q^2=0\). | |||
| absh_m | \(\mathrm{Abs}h_-^{(0)}\) | The constant term of the absolute value of the hadronic parameter \(h_-\) of the \(B\to K^*\) at \(q^2=0\). | |||
| argh_m | \(\mathrm{Arg}h_-^{(0)}\) | The constant term of the argument of the hadronic parameter \(h_-\) of the \(B\to K^*\) at \(q^2=0\). |
This kind of decays can be described by means of the \(\Delta B = 1 \) weak effective Hamiltonian
\[ \mathcal{H}_\mathrm{eff}^{\Delta B = 1} = \mathcal{H}_\mathrm{eff}^\mathrm{had} + \mathcal{H}_\mathrm{eff}^\mathrm{\gamma}, \]
where the first term is the hadronic contribution
\[ \mathcal{H}_\mathrm{eff}^\mathrm{had} = \frac{4G_F}{\sqrt{2}}\Bigg[\sum_{p=u,c}\lambda_p\bigg(C_1 Q^{p}_1 + C_2 Q^{p}_2\bigg) -\lambda_t \bigg(\sum_{i=3}^{6} C_i P_i + C_{8}Q_{8g} \bigg)\Bigg] \,, \]
involving current-current, chromodynamic penguin and chromomagnetic dipole operators, while the second one, given by
\[ \mathcal{H}_\mathrm{eff}^\mathrm{\gamma} = - \frac{4G_F}{\sqrt{2}}\lambda_t C_7Q_{7\gamma} \,, \]
includes the electromagnetic penguin operator.
Considering the matrix element of \(\mathcal{H}_\mathrm{eff}^{\Delta B = 1}\) between the initial state \(M\) and the final state \(V \gamma\), only the contribution of \(\mathcal{H}_\mathrm{eff}^\mathrm{\gamma}\) clearly factorizes into the product of hadronic form factors and leptonic tensors at all orders in strong interactions. Following [Jager:2012uw], we implemented the amplitude in the helicity basis; hence we made use of the helicity form factor \( T_-(0)\), which is related to the ones in the transverse basis through the following relation:
\[ T_{-}\left( q^{2}\right) = \frac{m_M^2 - m_V^2}{m_M^2}T_1\left( q^{2}\right)\,. \]
The effect of the operators of \(\mathcal{H}_\mathrm{eff}^\mathrm{had}\) due to exchange of soft gluon can be reabsorbed in the following parameterization,
\[ h_\lambda(q^2) = \frac{\epsilon^*_\mu(\lambda)}{m_M^2} \int d^4x e^{iqx} \langle \bar V \vert T\{j^{\mu}_\mathrm{em} (x) \mathcal{H}_\mathrm{eff}^\mathrm{had} (0)\} \vert \bar M \rangle = h_\lambda^{(0)}\,, \]
while the effect due to exchange of hard gluons can be parametrized following the prescription of [Bosch:2001gv] as a shift to the Wilson coefficient \(C_7\) :
\[ \Delta C_{7} = \frac{\alpha_s(\mu) C_F}{4\pi} \left( C_1(\mu) G_1(s_p)+ C_8(\mu) G_8\right) + \frac{\alpha_s(\mu_h) C_F}{4\pi} \left( C_1(\mu_h) H_1(s_p)+ C_8(\mu_h) H_8\right)\,, \]
where the terms proportional to \(G_i\) are the ones describing the corrections where the spectator quark is connected to the hard process only through soft interactions, while the ones proportional to \(H_i\) (involving leading twist light-cone distributions) are the ones describing the corrections where the spectactor quark is involved in the hard process, and \(s=\frac{m_c^2}{m_b^2}\).
The amplitude can be therefore parametrized in terms of the following helicity amplitudes:
\[ H_V^+ = \lambda_t \Big[- C_{7}' {T}_{-} - \frac{m_M}{m_b} 8 \pi^2 h_\lambda \Big] \,, \\ H_V^- = \lambda_t \Big[ C_{7} T_{-} - \frac{m_M}{m_b} 8 \pi^2 h_\lambda \Big] \,. \]
Squaring the amplitude and summing over the spins it is possible to obtain the Branching Ratio, which is
\[ BR = \frac {\alpha_e G_F^2 M_b^2 M_M \lambda}{(4\pi)^2 4 w_M} ( |H_V^+|^2 + |H_A^+|^2 +|\overline{H}_V^-|^2 + |\overline{H}_A^-|^2) \,. \]
The class is build as follows: after the parameters are updated in updateParameters() and the form factor \( T_1 \) is computed in T_1() following [Straub:2015ica], the QCDF corrections to the Wilson coefficient \( C_7 \) is computed in the functions G1(), G8(), H1() and H8(). The helicity amplitudes \(H_V^{(+,-)},\overline{H}_V^{(+,-)}\) are build in H_V_p(), H_V_m(), H_V_p_bar() and H_V_m_bar(), in order to be further used to compute the observables.
Public Member Functions | |
| std::vector< std::string > | initializeMVgammaParameters () |
| A method for initializing the parameters necessary for MVgamma. More... | |
| MVgamma (const StandardModel &SM_i, QCD::meson meson_i, QCD::meson vector_i) | |
| Constructor. More... | |
| void | updateParameters () |
| The update parameter method for MVgamma. More... | |
| virtual | ~MVgamma () |
| Destructor. More... | |
Public Attributes | |
| double | ale |
| double | Chi1minus |
| double | Chi1plus |
| double | ChiTT |
| double | deltaC9_1 |
| double | deltaC9_2 |
| gslpp::complex | exp_Phase_1 |
| gslpp::complex | exp_Phase_2 |
| double | fB |
| double | fpara |
| double | fperp |
| double | GF |
| gslpp::complex | h [2] |
| double | lambda |
| gslpp::complex | lambda_t |
| gslpp::complex | lambda_u |
| double | Mb |
| double | mb_pole |
| double | mc_pole |
| double | MM |
| double | MM2 |
| double | Ms |
| double | mu_b |
| double | mu_h |
| double | MV |
| double | MW |
| std::vector< std::string > | parametersForMVgamma |
| double | r1_1 |
| double | r1_2 |
| double | r2_1 |
| double | r2_2 |
| double | rV |
| double | spectator_charge |
| gslpp::complex | SU3_breaking |
| double | t_m |
| double | t_p |
| double | width |
| double | z_DM |
| MVgamma::MVgamma | ( | const StandardModel & | SM_i, |
| QCD::meson | meson_i, | ||
| QCD::meson | vector_i | ||
| ) |
Constructor.
| [in] | SM_i | a reference to an object of type StandardModel |
| [in] | meson_i | initial meson of the decay |
| [in] | vector_i | final vector meson of the decay |
Definition at line 22 of file MVgamma.cpp.
|
virtual |
| std::vector< std::string > MVgamma::initializeMVgammaParameters | ( | ) |
A method for initializing the parameters necessary for MVgamma.
Definition at line 45 of file MVgamma.cpp.
| void MVgamma::updateParameters | ( | ) |
The update parameter method for MVgamma.
NOTE: ComputeCoeff with different argumetns cannot be mixed. They have to be called sequentially.
NOTE: ComputeCoeff with different argumetns cannot be mixed. They have to be called sequentially.
Definition at line 219 of file MVgamma.cpp.
| double MVgamma::fperp |
| gslpp::complex MVgamma::h[2] |
| double MVgamma::rV |
| double MVgamma::spectator_charge |
1.9.2