dynamics¶

import ampform.dynamics

Lineshape functions that describe the dynamics of an interaction.

See also Form factor.

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

max_angular_momentum: Optional[int] = None¶

Limit the maximum allowed angular momentum \(L\).

This improves performance when \(L\) is a Symbol and you are note interested in higher angular momenta.

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. Compare for instance PhaseSpaceFactor and PhaseSpaceFactorAnalytic.

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

class PhaseSpaceFactor(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶

Bases: UnevaluatedExpression

Standard phase-space factor, using BreakupMomentumSquared().

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

(2)¶\[ \begin{eqnarray} \rho\left(s\right) & = & \frac{\sqrt{q^2\left(s\right)}}{8 \pi \sqrt{s}} \end{eqnarray}\]

with \(q^2\) defined as (7).

class PhaseSpaceFactorAbs(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶

Bases: UnevaluatedExpression

Phase space factor square root over the absolute value.

As opposed to PhaseSpaceFactor, this takes the Abs value of BreakupMomentumSquared, then the sqrt().

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

(3)¶\[ \begin{eqnarray} \hat{\rho}\left(s\right) & = & \frac{\sqrt{\left|{q^2\left(s\right)}\right|}}{8 \pi \sqrt{s}} \end{eqnarray}\]

with \(q^2(s)\) defined as (7).

class PhaseSpaceFactorAnalytic(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶

Bases: UnevaluatedExpression

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.

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

with \(\hat{\rho}\left(s\right)\) defined by PhaseSpaceFactorAbs (3).

class PhaseSpaceFactorComplex(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶

Bases: UnevaluatedExpression

Phase-space factor with ComplexSqrt.

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

(5)¶\[ \begin{eqnarray} \rho^\mathrm{c}\left(s\right) & = & \frac{\sqrt[\mathrm{c}]{q^2\left(s\right)}}{8 \pi \sqrt{s}} \end{eqnarray}\]

with \(q^2(s)\) defined as (7).

class EnergyDependentWidth(s: Symbol, mass0: Symbol, gamma0: Symbol, m_a: Symbol, m_b: Symbol, angular_momentum: Symbol, meson_radius: Symbol, phsp_factor: Optional[PhaseSpaceFactorProtocol] = None, name: Optional[str] = None, evaluate: bool = False)[source]¶

Bases: UnevaluatedExpression

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

See PDG2020, §Resonances, p.6 and [11], equation (6). Default value for phsp_factor is PhaseSpaceFactor().

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:

(6)¶\[ \begin{eqnarray} \Gamma_{0}\left(s\right) & = & \frac{\Gamma_{0} B_{L}^2\left(q^2\left(s\right)\right) \rho\left(s\right)}{B_{L}^2\left(q^2\left(m_{0}^{2}\right)\right) \rho\left(m_{0}^{2}\right)} \end{eqnarray}\]

where \(B_L^2\) is defined by (1), \(q\) is defined by (7), and \(\rho\) is (by default) defined by (2).

phsp_factor: PhaseSpaceFactorProtocol¶

class BreakupMomentumSquared(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶

Bases: UnevaluatedExpression

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 \(R\).

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 \(b\).

It’s up to the caller in which way to take the square root of this break-up momentum.See Analytic continuation and ComplexSqrt.

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

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

Relativistic Breit-Wigner lineshape.

See Without form factor and [11].

(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[PhaseSpaceFactorProtocol] = None) → Expr[source]¶

Relativistic Breit-Wigner with BlattWeisskopfSquared factor.

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

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

(9)¶\[\frac{\Gamma_{0} m_{0} \sqrt{B_{L}^2\left(d^{2} q^2\left(s\right)\right)}}{m_{0}^{2} - i m_{0} \Gamma_{0}\left(s\right) - s}\]

where \(\Gamma(s)\) is defined by (6), \(B_L^2\) is defined by (1), and \(q^2\) is defined by (7).

Submodules and Subpackages