Form Factor Parametrizations

This page provides a description of the form factor parametrizations used in the HadronicLineshapes package. The implementation is consistent with the Review on Resonances by Particle Data Group up to an overall normalization factor.

The form-factor is a function that provide an energy-behavior to a one-to-two transition amplitude. The physical interpretation of commonly-used form factors is limited. This package implements,

BlattWeisskopf{L}(d::Float64)
MomentumPower{L}()

Both form-factors can be called on a single momentum variable p, but also on squared masses of the decay products,

ff(p) # as a function of momentum
ff(m0^2, m1^2, m2^2) # computes momentum from masses and calls ff(p)

Blatt-Weisskopf Form Factors

The BlattWeisskopf form factors are used to regularize $p^l$ behavior of partial waves with orbital angular momentum $l$. The factors appear in the parametrization of enertgy-dependent width of resonances, and in the vertex functions when describing amplitude with fixed orbital angular momentum, e.g using spin-orbit coupling in hadronic decays. They are defined as:

\[F_l^2(z) = \frac{z^{2l}}{\chi_l(z^2)}\]

where:

• \[z = d \cdot p\]

  is the scaled momentum

• \[d\]

  is the scale parameter (typically the interaction radius)

• \[p\]

  is the breakup momentum

• \[\chi_l(z^2)\]

  is an order-$l$ polynomial of $z^2$

The function returns $F_l$, taking a square root of the $F_l^2$ expression.

ff = BlattWeisskopf{1}(1.5) # P-wave
# explicit way: compute momentum first
p = breakup(m0, m1, m2)
result = ff(p)
# concise way: pass squared masses directly
result = ff(m0^2, m1^2, m2^2)

Momentum Power Form Factors

The MomentumPower form factors represent simple momentum-dependent terms in scattering:

\[F_l(p) = p^l\]

where $p$ is the breakup momentum and $l$ is the orbital angular momentum.

pL0 = MomentumPower{0}()  # Constant (S-wave)
pL1 = MomentumPower{1}()  # Linear (P-wave)
pL2 = MomentumPower{2}()  # Quadratic (D-wave)