dynamics¶
import ampform.dynamics
Lineshape functions that describe the dynamics of an interaction.
See also
-
class
BlattWeisskopfSquared
(angular_momentum: Symbol, z: Symbol, **hints: Any)[source]¶ Bases:
ampform.dynamics.decorator.UnevaluatedExpression
Blatt-Weisskopf function \(B_L^2(z)\), up to \(L \leq 8\).
- Parameters
angular_momentum – Angular momentum \(L\) of the decaying particle.
z – Argument of the Blatt-Weisskopf function \(B_L^2(z)\). A usual choice is \(z = (d q)^2\) with \(d\) the impact parameter and \(q\) the breakup-momentum (see
breakup_momentum_squared()
).
Note that equal powers of \(z\) appear in the nominator and the denominator, while some sources have nominator \(1\), instead of \(z^L\). Compare for instance PDG2020, §Resonances, p.6, just before Equation (49.20).
Each of these cases for \(L\) has been taken from Pychy 2016, p.59, Chung et al. 1995, p.415, and Chung 2015. For a good overview of where to use these Blatt-Weisskopf functions, see Asner et al. 2006.
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, **kwds)[source]¶ Bases:
typing_extensions.Protocol
Protocol that is used by
coupled_width()
.Use this
Protocol
when defining other implementations of a phase space factor. Compare for instancephase_space_factor()
andphase_space_factor_analytic()
.
-
breakup_momentum
(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr[source]¶ 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)¶\[q(s) = \frac{\sqrt{\frac{\left(s - \left(m_{a} - m_{b}\right)^{2}\right) \left(s - \left(m_{a} + m_{b}\right)^{2}\right)}{s}}}{2}\]
-
breakup_momentum_squared
(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr[source]¶ Squared value of the two-body
breakup_momentum
.This version of the break-up momentum is useful if you do not want to take a simple square root. See
breakup_momentum()
and Analytic continuation.(3)¶\[q^2(s) = \frac{\left(s - \left(m_{a} - m_{b}\right)^{2}\right) \left(s - \left(m_{a} + m_{b}\right)^{2}\right)}{4 s}\]
-
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
isphase_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:(4)¶\[\Gamma(s) = \frac{B_{L}^2(q) \Gamma_{0} \rho(s)}{B_{L}^2(q_{0}) \rho(m_{0})}\]where \(B_L^2(q)\) is defined by (1), \(q(s)\) is defined by (3), and \(\rho(s)\) is (by default) defined by (5).
-
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).
(5)¶\[\frac{\sqrt{q^{2}(s)}}{8 \pi \sqrt{s}}\]with \(q^2(s)\) defined as (3).
-
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 theAbs
value ofbreakup_momentum_squared()
, then thesqrt()
.This version of the phase space factor is often denoted as \(\hat{\rho}\) and is used in analytic continuation (
phase_space_factor_analytic()
).(6)¶\[\hat{\rho} = \frac{\sqrt{\left|{q^{2}(s)}\right|}}{8 \pi \sqrt{s}}\]with \(q^2(s)\) defined as (3).
-
phase_space_factor_analytic
(s: Symbol, m_a: Symbol, m_b: Symbol) → Expr[source]¶ Analytic continuation for the
phase_space_factor()
.See PDG2018, §Resonances, p.9 and Analytic continuation.
Warning: The PDG specifically derives this formula for a two-body decay with equal masses.
(7)¶\[\begin{split}\begin{cases} \frac{i \hat{\rho} \log{\left(\left|{\frac{\hat{\rho} + 1}{\hat{\rho} - 1}}\right| \right)}}{\pi} & \text{for}\: s < 0 \\\frac{i \hat{\rho} \log{\left(\left|{\frac{\hat{\rho} + 1}{\hat{\rho} - 1}}\right| \right)}}{\pi} + \hat{\rho} & \text{for}\: s > \left(m_{a} + m_{b}\right)^{2} \\\frac{2 i \hat{\rho} \operatorname{atan}{\left(\frac{1}{\hat{\rho}} \right)}}{\pi} & \text{otherwise} \end{cases}\end{split}\]with \(\hat{\rho}\) defined by
phase_space_factor_abs()
(6).
-
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 aComplexSqrt
that does have defined behavior for defined for negative input values.(8)¶\[\hat{\rho} = \frac{\sqrt[\mathrm{c}]{q^{2}(s)}}{8 \pi \sqrt{s}}\]with \(q^2(s)\) defined as (3).
-
relativistic_breit_wigner
(s: Symbol, mass0: Symbol, gamma0: Symbol) → Expr[source]¶ Relativistic Breit-Wigner lineshape.
See Without form factor and Asner et al. 2006.
(9)¶\[\frac{\Gamma_{0} m_{0}}{- i \Gamma_{0} m_{0} + m_{0}^{2} - s}\]
-
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:
(10)¶\[\frac{\sqrt{B_{L}^2\left(q(s)\right)} \Gamma_{0} m_{0}}{- i \Gamma(s) m_{0} + m_{0}^{2} - s}\]where \(\Gamma(s)\) is defined by (4), \(B_L^2(q)\) is defined by (1), and \(q(s)\) is defined by (3).
Submodules and Subpackages