dynamics¶

import ampform.dynamics

Lineshape functions that describe the dynamics of an interaction.

See also Form factor.

(1)¶\[\begin{split}B_{L}^2\left(z\right) = \begin{cases} 1 & \text{for}\: L = 0 \\\frac{2 z}{z + 1} & \text{for}\: L = 1 \\\frac{13 z^{2}}{9 z + \left(z - 3\right)^{2}} & \text{for}\: L = 2 \\\frac{277 z^{3}}{z \left(z - 15\right)^{2} + \left(2 z - 5\right) \left(18 z - 45\right)} & \text{for}\: L = 3 \\\frac{12746 z^{4}}{25 z \left(2 z - 21\right)^{2} + \left(z^{2} - 45 z + 105\right)^{2}} & \text{for}\: L = 4 \\\frac{998881 z^{5}}{z^{5} + 15 z^{4} + 315 z^{3} + 6300 z^{2} + 99225 z + 893025} & \text{for}\: L = 5 \\\frac{118394977 z^{6}}{z^{6} + 21 z^{5} + 630 z^{4} + 18900 z^{3} + 496125 z^{2} + 9823275 z + 108056025} & \text{for}\: L = 6 \\\frac{19727003738 z^{7}}{z^{7} + 28 z^{6} + 1134 z^{5} + 47250 z^{4} + 1819125 z^{3} + 58939650 z^{2} + 1404728325 z + 18261468225} & \text{for}\: L = 7 \\\frac{4392846440677 z^{8}}{z^{8} + 36 z^{7} + 1890 z^{6} + 103950 z^{5} + 5457375 z^{4} + 255405150 z^{3} + 9833098275 z^{2} + 273922023375 z + 4108830350625} & \text{for}\: L = 8 \end{cases}\end{split}\]

is_commutative = True¶

class PhaseSpaceFactor(*args, **kwargs)[source]¶

Bases: Protocol

Protocol that is used by coupled_width().

Use this Protocol when defining other implementations of a phase space factor. Compare for instance phase_space_factor() and phase_space_factor_analytic().

__call__(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr [source]¶: Expected signature.

breakup_momentum_squared(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr [source]¶

Squared value of the two-body breakup-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 \(R\).

Parameters

s – Mandelstam variable \(s\). Commonly, this is just \(s = m_R^2\), with \(m_R\) the invariant mass of decaying particle \(R\).
m_a – Mass of decay product \(a\).
m_b – Mass of decay product \(b\).

See PDG2020, §Kinematics, p.3.

(2)¶\[q^2(s) = \frac{\left(s - \left(m_{a} - m_{b}\right)^{2}\right) \left(s - \left(m_{a} + m_{b}\right)^{2}\right)}{4 s}\]

coupled_width(s: Symbol, mass0: Symbol, gamma0: Symbol, m_a: Symbol, m_b: Symbol, angular_momentum: Symbol, meson_radius: Symbol, phsp_factor: Optional[PhaseSpaceFactor] = None) → Expr [source]¶

Mass-dependent width, coupled to the pole position of the resonance.

See PDG2020, §Resonances, p.6 and Asner et al. 2006, equation (6). Default value for phsp_factor is phase_space_factor().

Note that the BlattWeisskopfSquared of AmpForm is normalized in the sense that equal powers of \(z\) appear in the nominator and the denominator, while the definition in the PDG (as well as some other sources), always have \(1\) in the nominator of the Blatt-Weisskopf. In that case, one needs an additional factor \(\left(q/q_0\right)^{2L}\) in the definition for \(\Gamma(m)\).

With that in mind, the “mass-dependent” width in a relativistic_breit_wigner_with_ff becomes:

(3)¶\[\Gamma(s) = \frac{B_{L}^2(q) \Gamma_{0} \rho(s)}{B_{L}^2(q_{0}) \rho(m_{0})}\]

where \(B_L^2(q)\) is defined by (1), \(q(s)\) is defined by (2), and \(\rho(s)\) is (by default) defined by (4).

phase_space_factor(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr [source]¶

Standard phase-space factor, using breakup_momentum_squared().

See PDG2020, §Resonances, p.4, Equation (49.8).

(4)¶\[\frac{\sqrt{q^{2}(s)}}{8 \pi \sqrt{s}}\]

with \(q^2(s)\) defined as (2).

phase_space_factor_abs(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr [source]¶

Phase space factor square root over the absolute value.

As opposed to phase_space_factor(), this takes the Abs value of breakup_momentum_squared(), then the sqrt().

This version of the phase space factor is often denoted as \(\hat{\rho}\) and is used in analytic continuation (phase_space_factor_analytic()).

(5)¶\[\hat{\rho} = \frac{\sqrt{\left|{q^{2}(s)}\right|}}{8 \pi \sqrt{s}}\]

with \(q^2(s)\) defined as (2).

phase_space_factor_analytic(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr [source]¶

Analytic continuation for the phase_space_factor().

See PDG2014, §Resonances, p.8 and Analytic continuation.

Warning: The PDG specifically derives this formula for a two-body decay with equal masses.

(6)¶\[\begin{split}\begin{cases} \frac{i \hat{\rho} \log{\left(\left|{\frac{\hat{\rho} + 1}{\hat{\rho} - 1}}\right| \right)}}{\pi} & \text{for}\: s < 0 \\\frac{i \hat{\rho} \log{\left(\left|{\frac{\hat{\rho} + 1}{\hat{\rho} - 1}}\right| \right)}}{\pi} + \hat{\rho} & \text{for}\: s > \left(m_{a} + m_{b}\right)^{2} \\\frac{2 i \hat{\rho} \operatorname{atan}{\left(\frac{1}{\hat{\rho}} \right)}}{\pi} & \text{otherwise} \end{cases}\end{split}\]

with \(\hat{\rho}\) defined by phase_space_factor_abs() (5).

phase_space_factor_complex(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr [source]¶

Phase-space factor with ComplexSqrt.

Same as phase_space_factor(), but using a ComplexSqrt that does have defined behavior for defined for negative input values.

(7)¶\[\hat{\rho} = \frac{\sqrt[\mathrm{c}]{q^{2}(s)}}{8 \pi \sqrt{s}}\]

with \(q^2(s)\) defined as (2).

relativistic_breit_wigner(s: Symbol, mass0: Symbol, gamma0: Symbol) → Expr [source]¶

Relativistic Breit-Wigner lineshape.

See Without form factor and Asner et al. 2006.

(8)¶\[\frac{\Gamma_{0} m_{0}}{- i \Gamma_{0} m_{0} + m_{0}^{2} - s}\]

relativistic_breit_wigner_with_ff(s: Symbol, mass0: Symbol, gamma0: Symbol, m_a: Symbol, m_b: Symbol, angular_momentum: Symbol, meson_radius: Symbol, phsp_factor: Optional[PhaseSpaceFactor] = None) → Expr [source]¶

Relativistic Breit-Wigner with BlattWeisskopfSquared factor.

Note that this lineshape is set to zero for \(s < (m_a + m_b)^2\) as a continuation of the BlattWeisskopfSquared damping factor behavior at \(s = (m_a + m_b)^2\).

See With form factor and PDG2020, §Resonances, p.6.

The general form of a relativistic Breit-Wigner with Blatt-Weisskopf form factor is:

(9)¶\[\begin{split}\frac{\Gamma_{0} m_{0} \left(\begin{cases} 0 & \text{for}\: s < \left(m_{a} + m_{b}\right)^{2} \wedge L \neq 0 \\\sqrt{B_{L}^2\left(q(s)\right)} & \text{otherwise} \end{cases}\right)}{- i \Gamma(s) m_{0} + m_{0}^{2} - s}\end{split}\]

where \(\Gamma(s)\) is defined by (3), \(B_L^2(q)\) is defined by (1), and \(q(s)\) is defined by (2).

Submodules and Subpackages