Module flavio.physics.bdecays.bll
Functions for the branching ratios and effective lifetimes of the leptonic decays $B_q \to \ell^+\ell^-$, where $q=d$ or $s$ and $\ell=e$, $\mu$. or $\tau$.
Functions
def ADG_func(wc_obj, par, B, lep)-
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def ADG_func(wc_obj, par, B, lep): scale = config['renormalization scale']['bll'] label = meson_quark[B]+lep+lep wc = wctot_dict(wc_obj, label, scale, par) return ADeltaGamma(par, wc, B, lep) def ADeltaGamma(par, wc, B, lep)-
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def ADeltaGamma(par, wc, B, lep): P, S = amplitudes(par, wc, B, lep, lep) # cf. eq. (17) of arXiv:1204.1737 return ((P**2).real - (S**2).real)/(abs(P)**2 + abs(S)**2) def ADeltaGamma_func(B, lep)-
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def ADeltaGamma_func(B, lep): def ADG_func(wc_obj, par): scale = config['renormalization scale']['bll'] label = meson_quark[B]+lep+lep wc = wctot_dict(wc_obj, label, scale, par) return ADeltaGamma(par, wc, B, lep) return ADG_func def amplitudes(par, wc, B, l1, l2)-
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def amplitudes(par, wc, B, l1, l2): r"""Amplitudes P and S entering the $B_q\to\ell_1^+\ell_2^-$ observables. Parameters ---------- - `par`: parameter dictionary - `B`: should be `'Bs'` or `'B0'` - `l1` and `l2`: should be `'e'`, `'mu'`, or `'tau'` Returns ------- `(P, S)` where for the special case `l1 == l2` one has - $P = \frac{2m_\ell}{m_{B_q}} (C_{10}-C_{10}') + m_{B_q} (C_P-C_P')$ - $S = m_{B_q} (C_S-C_S')$ """ scale = config['renormalization scale']['bll'] # masses ml1 = par['m_'+l1] ml2 = par['m_'+l2] mB = par['m_'+B] mb = running.get_mb(par, scale, nf_out=5) # get the mass of the spectator quark if B=='Bs': mspec = running.get_ms(par, scale, nf_out=5) elif B=='B0': mspec = running.get_md(par, scale, nf_out=5) # Wilson coefficients qqll = meson_quark[B] + l1 + l2 # For LFV expressions see arXiv:1602.00881 eq. (5) C9m = wc['C9_'+qqll] - wc['C9p_'+qqll] # only relevant for l1 != l2! C10m = wc['C10_'+qqll] - wc['C10p_'+qqll] CPm = wc['CP_'+qqll] - wc['CPp_'+qqll] CSm = wc['CS_'+qqll] - wc['CSp_'+qqll] beta_m = sqrt(1 - (ml1 - ml2)**2/mB**2) beta_p = sqrt(1 - (ml1 + ml2)**2/mB**2) P = beta_m * ( (ml2 + ml1)/mB * C10m + mB * mb/(mb + mspec) * CPm ) S = beta_p * ( (ml2 - ml1)/mB * C9m + mB * mb/(mb + mspec) * CSm ) return P, SAmplitudes P and S entering the $B_q\to\ell_1^+\ell_2^-$ observables.
Parameters
par: parameter dictionaryB: should be'Bs'or'B0'l1andl2: should be'e','mu', or'tau'
Returns
(P, S)where for the special casel1 == l2one has- $P = \frac{2m_\ell}{m_{B_q}} (C_{10}-C_{10}') + m_{B_q} (C_P-C_P')$
- $S = m_{B_q} (C_S-C_S')$
def bqll_obs(function, wc_obj, par, B, l1, l2)-
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def bqll_obs(function, wc_obj, par, B, l1, l2): scale = config['renormalization scale']['bll'] label = meson_quark[B]+l1+l2 if l1 == l2: # include SM contributions for LF conserving decay wc = wctot_dict(wc_obj, label, scale, par) else: wc = wc_obj.get_wc(label, scale, par) return function(par, wc, B, l1, l2) def bqll_obs_function(function, B, l1, l2)-
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def bqll_obs_function(function, B, l1, l2): return lambda wc_obj, par: bqll_obs(function, wc_obj, par, B, l1, l2) def bqll_obs_function_lsum(function, B, l1, l2)-
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def bqll_obs_function_lsum(function, B, l1, l2): return lambda wc_obj, par: bqll_obs_lsum(function, wc_obj, par, B, l1, l2) def bqll_obs_lsum(function, wc_obj, par, B, l1, l2)-
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def bqll_obs_lsum(function, wc_obj, par, B, l1, l2): if l1 == l2: raise ValueError("This function is defined only for LFV decays") scale = config['renormalization scale']['bll'] wc12 = wc_obj.get_wc(meson_quark[B]+l1+l2, scale, par) wc21 = wc_obj.get_wc(meson_quark[B]+l2+l1, scale, par) return function(par, wc12, B, l1, l2) + function(par, wc21, B, l2, l1) def br_inst(par, wc, B, l1, l2)-
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def br_inst(par, wc, B, l1, l2): r"""Branching ratio of $B_q\to\ell_1^+\ell_2^-$ in the absence of mixing. Parameters ---------- - `par`: parameter dictionary - `B`: should be `'Bs'` or `'B0'` - `lep`: should be `'e'`, `'mu'`, or `'tau'` """ # paramaeters GF = par['GF'] alphaem = running.get_alpha(par, 4.8)['alpha_e'] ml1 = par['m_'+l1] ml2 = par['m_'+l2] mB = par['m_'+B] tauB = par['tau_'+B] fB = par['f_'+B] # appropriate CKM elements if B == 'Bs': xi_t = ckm.xi('t','bs')(par) elif B == 'B0': xi_t = ckm.xi('t','bd')(par) N = xi_t * 4*GF/sqrt(2) * alphaem/(4*pi) beta = sqrt(lambda_K(mB**2,ml1**2,ml2**2))/mB**2 prefactor = abs(N)**2 / 32. / pi * mB**3 * tauB * beta * fB**2 P, S = amplitudes(par, wc, B, l1, l2) return prefactor * ( abs(P)**2 + abs(S)**2 )Branching ratio of $B_q\to\ell_1^+\ell_2^-$ in the absence of mixing.
Parameters
par: parameter dictionaryB: should be'Bs'or'B0'lep: should be'e','mu', or'tau'
def br_lifetime_corr(y, ADeltaGamma)-
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def br_lifetime_corr(y, ADeltaGamma): r"""Correction factor relating the experimentally measured branching ratio (time-integrated) to the theoretical one (instantaneous), see e.g. eq. (8) of arXiv:1204.1735. Parameters ---------- - `y`: relative decay rate difference, $y_q = \tau_{B_q} \Delta\Gamma_q /2$ - `ADeltaGamma`: $A_{\Delta\Gamma_q}$ as defined, e.g., in arXiv:1204.1735 Returns ------- $\frac{1-y_q^2}{1+A_{\Delta\Gamma_q} y_q}$ """ return (1 - y**2)/(1 + ADeltaGamma*y)Correction factor relating the experimentally measured branching ratio (time-integrated) to the theoretical one (instantaneous), see e.g. eq. (8) of arXiv:1204.1735.
Parameters
y: relative decay rate difference, $y_q = \tau_{B_q} \Delta\Gamma_q /2$ADeltaGamma(): $A_{\Delta\Gamma_q}$ as defined, e.g., in arXiv:1204.1735
Returns
$\frac{1-y_q^2}{1+A_{\Delta\Gamma_q} y_q}$
def br_timeint(par, wc, B, l1, l2)-
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def br_timeint(par, wc, B, l1, l2): r"""Time-integrated branching ratio of $B_q\to\ell^+\ell^-$.""" if l1 != l2: raise ValueError("Time-integrated branching ratio only defined for equal lepton flavours") lep = l1 br0 = br_inst(par, wc, B, lep, lep) y = par['DeltaGamma/Gamma_'+B]/2. ADG = ADeltaGamma(par, wc, B, lep) corr = br_lifetime_corr(y, ADG) return br0 / corrTime-integrated branching ratio of $B_q\to\ell^+\ell^-$.
def tau_ll(wc, par, B, lep)-
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def tau_ll(wc, par, B, lep): r"""Effective B->l+l- lifetime as defined in eq. (26) of arXiv:1204.1737 . This formula one either gets by integrating eq. (21) or by inverting eq. (27) of arXiv:1204.1737. Parameters ---------- - `wc` : dict of Wilson coefficients - `par` : parameter dictionary - `B` : should be `'Bs'` or `'B0'` - `lep` : lepton: 'e', 'mu' or 'tau' Returns ------- $-\frac{\tau_{B_s} \left(y_s^2+2 A_{\Delta\Gamma_q} ys+1\right)}{\left(ys^2-1\right) (A_{\Delta\Gamma_q} ys+1)}$ """ ADG = ADeltaGamma(par, wc, B, lep) y = .5*par['DeltaGamma/Gamma_'+B] tauB = par['tau_'+B] return -(((1 + y**2 + 2*y*ADG)*tauB)/((-1 + y**2)*(1 + y*ADG)))Effective B->l+l- lifetime as defined in eq. (26) of arXiv:1204.1737 . This formula one either gets by integrating eq. (21) or by inverting eq. (27) of arXiv:1204.1737.
Parameters
wc: dict of Wilson coefficientspar: parameter dictionaryB: should be'Bs'or'B0'lep: lepton: 'e', 'mu' or 'tau'
Returns
$-\frac{\tau_{B_s} \left(y_s^2+2 A_{\Delta\Gamma_q} ys+1\right)}{\left(ys^2-1\right) (A_{\Delta\Gamma_q} ys+1)}$
def tau_ll_func(wc_obj, par, B, lep)-
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def tau_ll_func(wc_obj, par, B, lep): scale = config['renormalization scale']['bll'] label = meson_quark[B]+lep+lep wc = wctot_dict(wc_obj, label, scale, par) return tau_ll(wc, par, B, lep)