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.sympy.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
BreakupMomentumSquared()
).
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 [9], p.59, [4], p.415, and [10]. For a good overview of where to use these Blatt-Weisskopf functions, see [11].
See also Form factor.
(1)¶\[\begin{split}B_{L}^2\left(z\right) = \begin{cases} 1 & \text{for}\: L = 0 \\\frac{2 z}{z + 1} & \text{for}\: L = 1 \\\frac{13 z^{2}}{9 z + \left(z - 3\right)^{2}} & \text{for}\: L = 2 \\\frac{277 z^{3}}{z \left(z - 15\right)^{2} + \left(2 z - 5\right) \left(18 z - 45\right)} & \text{for}\: L = 3 \\\frac{12746 z^{4}}{25 z \left(2 z - 21\right)^{2} + \left(z^{2} - 45 z + 105\right)^{2}} & \text{for}\: L = 4 \\\frac{998881 z^{5}}{z^{5} + 15 z^{4} + 315 z^{3} + 6300 z^{2} + 99225 z + 893025} & \text{for}\: L = 5 \\\frac{118394977 z^{6}}{z^{6} + 21 z^{5} + 630 z^{4} + 18900 z^{3} + 496125 z^{2} + 9823275 z + 108056025} & \text{for}\: L = 6 \\\frac{19727003738 z^{7}}{z^{7} + 28 z^{6} + 1134 z^{5} + 47250 z^{4} + 1819125 z^{3} + 58939650 z^{2} + 1404728325 z + 18261468225} & \text{for}\: L = 7 \\\frac{4392846440677 z^{8}}{z^{8} + 36 z^{7} + 1890 z^{6} + 103950 z^{5} + 5457375 z^{4} + 255405150 z^{3} + 9833098275 z^{2} + 273922023375 z + 4108830350625} & \text{for}\: L = 8 \end{cases}\end{split}\]
- class BreakupMomentumSquared(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶
Bases:
ampform.sympy.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
.(2)¶\[ \begin{eqnarray} q^2\left(s\right) & = & \frac{\left(s - \left(m_{a} - m_{b}\right)^{2}\right) \left(s - \left(m_{a} + m_{b}\right)^{2}\right)}{4 s} \end{eqnarray}\]
- 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:
ampform.sympy.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
isPhaseSpaceFactor()
.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)¶\[ \begin{eqnarray} \Gamma_{0}\left(s\right) & = & \frac{\Gamma_{0} B_{L}^2\left(q^2\left(s\right)\right) \rho\left(s\right)}{B_{L}^2\left(q^2\left(m_{0}^{2}\right)\right) \rho\left(m_{0}^{2}\right)} \end{eqnarray}\]where \(B_L^2\) is defined by (1), \(q\) is defined by (2), and \(\rho\) is (by default) defined by (4).
- phsp_factor: PhaseSpaceFactorProtocol¶
- class PhaseSpaceFactor(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶
Bases:
ampform.sympy.UnevaluatedExpression
Standard phase-space factor, using
BreakupMomentumSquared()
.See PDG2020, §Resonances, p.4, Equation (49.8).
(4)¶\[ \begin{eqnarray} \rho\left(s\right) & = & \frac{\sqrt{q^2\left(s\right)}}{8 \pi \sqrt{s}} \end{eqnarray}\]with \(q^2\) defined as (2).
- class PhaseSpaceFactorAbs(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶
Bases:
ampform.sympy.UnevaluatedExpression
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 analytic continuation (
PhaseSpaceFactorAnalytic
).(5)¶\[ \begin{eqnarray} \hat{\rho}\left(s\right) & = & \frac{\sqrt{\left|{q^2\left(s\right)}\right|}}{8 \pi \sqrt{s}} \end{eqnarray}\]with \(q^2(s)\) defined as (2).
- class PhaseSpaceFactorAnalytic(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶
Bases:
ampform.sympy.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.
(6)¶\[\begin{split} \begin{eqnarray} \rho^\mathrm{ac}\left(s\right) & = & \begin{cases} \frac{i \log{\left(\left|{\frac{\hat{\rho}\left(s\right) + 1}{\hat{\rho}\left(s\right) - 1}}\right| \right)} \hat{\rho}\left(s\right)}{\pi} & \text{for}\: s < 0 \\\frac{i \log{\left(\left|{\frac{\hat{\rho}\left(s\right) + 1}{\hat{\rho}\left(s\right) - 1}}\right| \right)} \hat{\rho}\left(s\right)}{\pi} + \hat{\rho}\left(s\right) & \text{for}\: s > \left(m_{a} + m_{b}\right)^{2} \\\frac{2 i \operatorname{atan}{\left(\frac{1}{\hat{\rho}\left(s\right)} \right)} \hat{\rho}\left(s\right)}{\pi} & \text{otherwise} \end{cases} \end{eqnarray}\end{split}\]with \(\hat{\rho}\left(s\right)\) defined by
PhaseSpaceFactorAbs
(5).
- class PhaseSpaceFactorComplex(s: Symbol, m_a: Symbol, m_b: Symbol, **hints: Any)[source]¶
Bases:
ampform.sympy.UnevaluatedExpression
Phase-space factor with
ComplexSqrt
.Same as
PhaseSpaceFactor()
, but using aComplexSqrt
that does have defined behavior for defined for negative input values.(7)¶\[ \begin{eqnarray} \rho^\mathrm{c}\left(s\right) & = & \frac{\sqrt[\mathrm{c}]{q^2\left(s\right)}}{8 \pi \sqrt{s}} \end{eqnarray}\]with \(q^2(s)\) defined as (2).
- class PhaseSpaceFactorProtocol(*args, **kwds)[source]¶
Bases:
typing_extensions.Protocol
Protocol that is used by
EnergyDependentWidth
.Use this
Protocol
when defining other implementations of a phase space factor. Compare for instancePhaseSpaceFactor
andPhaseSpaceFactorAnalytic
.
- 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 (3), \(B_L^2\) is defined by (1), and \(q^2\) is defined by (2).
Submodules and Subpackages