phasespace#
import ampform.dynamics.phasespace
Different parametrizations of phase space factors.
Phase space factors are computed by integrating over the phase space element given by Equation (49.12) in PDG2021, §Kinematics, p.2. See also Equation (50.9) on PDG2021, §Resonances, p.6. This integral is not always easy to solve, which leads to different parametrizations.
This module provides several parametrizations. They all comply with the
PhaseSpaceFactorProtocol
, so that they can be used in parametrizations like
EnergyDependentWidth
.
- class PhaseSpaceFactorProtocol(*args, **kwargs)[source]#
Bases:
Protocol
Protocol that is used by
EnergyDependentWidth
.Use this
Protocol
when defining other implementations of a phase space factor. Some implementations:Even
BreakupMomentumSquared
andchew_mandelstam_s_wave()
comply with this protocol, but are technically speaking not phase space factors.- __call__(s, m1, m2) Expr [source]#
Expected
signature
.- Parameters:
s – Mandelstam variable \(s\). Commonly, this is just \(s = m_R^2\), with \(m_R\) the invariant mass of decaying particle \(R\).
m1 – Mass of decay product \(a\).
m2 – Mass of decay product \(b\).
- class BreakupMomentumSquared(s, m1, m2, *args, evaluate: bool = False, **kwargs)[source]#
Bases:
Expr
Squared value of the two-body break-up momentum.
For a two-body decay \(R \to ab\), the break-up momentum is the absolute value of the momentum of both \(a\) and \(b\) in the rest frame of \(R\). See Equation (49.17) on PDG2021, §Kinematics, p.3, as well as Equation (50.5) on PDG2021, §Resonances, p.5.
It’s up to the caller in which way to take the square root of this break-up momentum, because \(q^2\) can have negative values for non-zero \(m1,m2\). In this case, one may want to use
ComplexSqrt
instead of the standardsqrt()
.(1)#\[\begin{split} \begin{array}{rcl} q^2\left(s\right) &=& \frac{\left(s - \left(m_{a} - m_{b}\right)^{2}\right) \left(s - \left(m_{a} + m_{b}\right)^{2}\right)}{4 s} \\ \end{array}\end{split}\]
- class PhaseSpaceFactor(s, m1, m2, *args, evaluate: bool = False, **kwargs)[source]#
Bases:
Expr
Standard phase-space factor, using
BreakupMomentumSquared()
.See PDG2021, §Resonances, p.6, Equation (50.9). We ignore the factor \(\frac{1}{16\pi}\) as done in [6], p.5.
(2)#\[\begin{split} \begin{array}{rcl} \rho\left(s\right) &=& \frac{2 \sqrt{q^2\left(s\right)}}{\sqrt{s}} \\ \end{array}\end{split}\]with \(q^2\) defined as (1).
- class PhaseSpaceFactorAbs(s, m1, m2, *args, evaluate: bool = False, **kwargs)[source]#
Bases:
Expr
Phase space factor square root over the absolute value.
As opposed to
PhaseSpaceFactor
, this takes theAbs
value ofBreakupMomentumSquared
, then thesqrt()
.This version of the phase space factor is often denoted as \(\hat{\rho}\) and is used in
EqualMassPhaseSpaceFactor
.(3)#\[\begin{split} \begin{array}{rcl} \hat{\rho}\left(s\right) &=& \frac{2 \sqrt{\left|{q^2\left(s\right)}\right|}}{\sqrt{s}} \\ \end{array}\end{split}\]with \(q^2(s)\) defined as (1).
- class PhaseSpaceFactorComplex(s, m1, m2, *args, evaluate: bool = False, **kwargs)[source]#
Bases:
Expr
Phase-space factor with
ComplexSqrt
.Same as
PhaseSpaceFactor()
, but using aComplexSqrt
that does have defined behavior for defined for negative input values.(4)#\[\begin{split} \begin{array}{rcl} \rho^\mathrm{c}\left(s\right) &=& \frac{2 \sqrt[\mathrm{c}]{q^2\left(s\right)}}{\sqrt{s}} \\ \end{array}\end{split}\]with \(q^2(s)\) defined as (1).
- class PhaseSpaceFactorSWave(s, m1, m2, *args, evaluate: bool = False, **kwargs)[source]#
Bases:
Expr
Phase space factor using
chew_mandelstam_s_wave()
.This
PhaseSpaceFactor
provides an analytic continuation for decay products with both equal and unequal masses (compareEqualMassPhaseSpaceFactor
).(5)#\[\begin{split} \begin{array}{rcl} \rho^\mathrm{CM}\left(s\right) &=& - \frac{i \left(\frac{2 \sqrt[\mathrm{c}]{q^2\left(s\right)}}{\sqrt{s}} \log{\left(\frac{m_{a}^{2} + m_{b}^{2} + 2 \sqrt{s} \sqrt[\mathrm{c}]{q^2\left(s\right)} - s}{2 m_{a} m_{b}} \right)} - \left(m_{a}^{2} - m_{b}^{2}\right) \left(- \frac{1}{\left(m_{a} + m_{b}\right)^{2}} + \frac{1}{s}\right) \log{\left(\frac{m_{a}}{m_{b}} \right)}\right)}{\pi} \\ \end{array}\end{split}\]
- chew_mandelstam_s_wave(s, m1, m2)[source]#
Chew-Mandelstam function for \(S\)-waves (no angular momentum).
(6)#\[\frac{\frac{2 \sqrt[\mathrm{c}]{q^2\left(s\right)}}{\sqrt{s}} \log{\left(\frac{m_{a}^{2} + m_{b}^{2} + 2 \sqrt{s} \sqrt[\mathrm{c}]{q^2\left(s\right)} - s}{2 m_{a} m_{b}} \right)} - \left(m_{a}^{2} - m_{b}^{2}\right) \left(- \frac{1}{\left(m_{a} + m_{b}\right)^{2}} + \frac{1}{s}\right) \log{\left(\frac{m_{a}}{m_{b}} \right)}}{\pi}\]with \(q^2(s)\) defined as (1).
See also
- class EqualMassPhaseSpaceFactor(s, m1, m2, *args, evaluate: bool = False, **kwargs)[source]#
Bases:
Expr
Analytic continuation for the
PhaseSpaceFactor()
.See PDG2018, §Resonances, p.9 and Analytic continuation.
Warning: The PDG specifically derives this formula for a two-body decay with equal masses.
(7)#\[\begin{split} \begin{array}{rcl} \rho^\mathrm{eq}\left(s\right) &=& \begin{cases} \frac{i \log{\left(\left|{\frac{\hat{\rho}\left(s\right) + 1}{\hat{\rho}\left(s\right) - 1}}\right| \right)} \hat{\rho}\left(s\right)}{\pi} & \text{for}\: s < 0 \\\frac{i \log{\left(\left|{\frac{\hat{\rho}\left(s\right) + 1}{\hat{\rho}\left(s\right) - 1}}\right| \right)} \hat{\rho}\left(s\right)}{\pi} + \hat{\rho}\left(s\right) & \text{for}\: s > \left(m_{a} + m_{b}\right)^{2} \\\frac{2 i \operatorname{atan}{\left(\frac{1}{\hat{\rho}\left(s\right)} \right)} \hat{\rho}\left(s\right)}{\pi} & \text{otherwise} \end{cases} \\ \end{array}\end{split}\]with \(\hat{\rho}\left(s\right)\) defined by
PhaseSpaceFactorAbs
(3).