Module flavio.physics.bdecays.formfactors.b_p.bcl

Functions

def ff(process, q2, par, n=3, t0=None)

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def ff(process, q2, par, n=3, t0=None):
    r"""Central value of $B\to P$ form factors in the standard convention
    and BCL parametrization (arXiv:0807.2722).

    The standard convention defines the form factors $f_+$, $f_0$, and $f_T$.
    """
    flavio.citations.register("Bourrely:2008za")
    pd = process_dict[process]
    mpl = par[process + ' BCL m+']
    m0 = par[process + ' BCL m0']
    mB = par['m_'+pd['B']]
    mP = par['m_'+pd['P']]
    chL = par.get(process + ' BCL chL', 1) # chiral logarithm factor (arXiv:2207.12468)
    ff = {}
    a={}
    for i in ['f+', 'fT']:
        a[i] = [ par[process + ' BCL' + ' a' + str(j) + '_' + i] for j in range(n) ]
    if t0 == 0: # z(q^2=0) = 0
        # if z(q^2=0) = 0, then the kinematic constraint f+(0)=f0(0) means
        # a['f0'][0] = a['f+'][0]
        a['f0'] = a['f+'][:1] + [ par[process + ' BCL' + ' a' + str(j) + '_f0'] for j in range(1,n) ]
    else:
        # only the first n-1 parameters for f0 are taken from par
        # the nth one is chosen to fulfill the kinematic constraint f+(0)=f0(0)
        a['f0'] = [ par[process + ' BCL' + ' a' + str(j) + '_f0'] for j in range(n-1) ]
        fplus_q20 = pole('f+', mpl, 0) * param_fplusT(mB, mP, a['f+'], 0, t0)
        f0_q20 = pole('f0', m0, 0) * param_f0(mB, mP, a['f0'], 0, t0)
        an_f0 = (f0_q20-fplus_q20)/z(mB, mP, 0, t0)**(n-1)
        a['f0'].append(an_f0)
    # evaluate FFs
    ff['f+'] = chL * pole('f+', mpl, q2) * param_fplusT(mB, mP, a['f+'], q2, t0)
    ff['fT'] = chL * pole('fT', mpl, q2) * param_fplusT(mB, mP, a['fT'], q2, t0)
    ff['f0'] = chL * pole('f0', m0, q2) * param_f0(mB, mP, a['f0'], q2, t0)
    return ff

Central value of $B\to P$ form factors in the standard convention and BCL parametrization (arXiv:0807.2722).

The standard convention defines the form factors $f_+$, $f_0$, and $f_T$.

def ff_isgurwise(process, q2, par, scale, n=3, t0=None)

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def ff_isgurwise(process, q2, par, scale, n=3, t0=None):
    r"""Central value of $B\to P$ form factors in the standard convention
    and BCL parametrization (arXiv:0807.2722) for $f_0$ and $f_+$, but using
    an improved Isgur-Wise relation in the heavy quark limit for $f_T$.
    """
    flavio.citations.register("Bourrely:2008za")
    pd = process_dict[process]
    mpl = par[process + ' BCL m+']
    m0 = par[process + ' BCL m0']
    mB = par['m_'+pd['B']]
    mP = par['m_'+pd['P']]
    ff = {}
    a={}
    a['f+'] = [ par[process + ' BCL' + ' a' + str(j) + '_f+'] for j in range(n) ]
    # only the first n-1 parameters for f0 are taken from par
    # the nth one is chosen to fulfill the kinematic constraint f+(0)=f0(0)
    a['f0'] = [ par[process + ' BCL' + ' a' + str(j) + '_f0'] for j in range(n-1) ]
    fplus_q20 = pole('f+', mpl, 0) * param_fplusT(mB, mP, a['f+'], 0, t0)
    f0_q20 = pole('f0', m0, 0) * param_f0(mB, mP, a['f0'], 0, t0)
    an_f0 = (fplus_q20-f0_q20)/z(mB, mP, 0, t0)**(n-1)
    a['f0'].append(an_f0)
    # evaluate FFs
    ff['f+'] = pole('f+', mpl, q2) * param_fplusT(mB, mP, a['f+'], q2, t0)
    ff['f0'] = pole('f0', m0, q2) * param_f0(mB, mP, a['f0'], q2, t0)
    ff = isgur_wise(process, q2, ff, par, scale=scale)
    return ff

Central value of $B\to P$ form factors in the standard convention and BCL parametrization (arXiv:0807.2722) for $f_0$ and $f_+$, but using an improved Isgur-Wise relation in the heavy quark limit for $f_T$.

def param_f0(mB, mP, a_i, q2, t0=None)

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def param_f0(mB, mP, a_i, q2, t0=None):
    Z = z(mB, mP, q2, t0)
    k = np.arange(len(a_i))
    return ( a_i * Z**k ).sum()

def param_fplusT(mB, mP, a_i, q2, t0=None)

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def param_fplusT(mB, mP, a_i, q2, t0=None):
    Z = z(mB, mP, q2, t0)
    n = len(a_i)
    k = np.arange(n)
    return ( a_i * (Z**k - (-1.)**(k - n) * k/n * Z**n) ).sum()

def pole(ff, mres, q2)

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def pole(ff, mres, q2):
    if mres == 0 or mres is None:
        return 1
    return 1/(1-q2/mres**2)