The spin factors for \texttt{Zemach\_P} are those given in Equations~(\ref{eq:ZTFactors})--(\ref{eq:ZTFactors-end}), which differ from the expressions of Equations~(\ref{eq:Legendre-TFactors})--(\ref{eq:Legendre-TFactors-end}) by extra factors of $-2\,p\,q$.
Similarly, those for \texttt{Zemach\_Pstar} are the same as those for
\texttt{Zemach\_P} but with the bachelor momentum evaluated in the rest frame
of the parent particle~($p^{\ast}$), rather than that of the resonance~($p$).
The angular distributions have been implemented in \laura\ up to $L=5$, which is two units larger than the maximum spin of any resonance observed to be produced in any Dalitz plot to date~\cite{Aaij:2014xza,Aaij:2014baa,Aaij:2015sqa}.
The angular distributions discussed above are based on a non-relativistic assumption.
For certain channels, this may not be sufficiently precise, and therefore the \texttt{Covariant} formalism is also made available.
The first three of these expressions are derived in
Ref.~\cite{Filippini:1995yc} and, based on that work, the last two were
derived in Ref.~\cite{Aaij:2015sqa}.
-{\bf ideally we should add the expression for $L=5$.}
+{\bf Ideally we should add the expression for $L=5$. Wenbin is looking into it.}
As can be seen from the expressions, the differences between formalisms are more significant for higher spin resonances, and particularly affect tails of the distributions.
To give an idea of the effect, the lineshapes for the $f_2(1270)$ and $\rho_3(1690)^0$ resonances decaying to $\pip\pim$ are shown in Fig.~\ref{fig:angular-formulae}.
where in this example the momentum of the bachelor in the rest frame of the parent ($p^{\ast}$) is to be used.
Again, this operation must be performed before constructing any resonances.
In addition, it is possible to change the form of the Blatt--Weisskopf factors,
with the different types being defined by the \texttt{LauBlattWeisskopfFactor::BarrierType} enumeration.
The default setting, corresponding to Equations~(\ref{eq:BWFormFactors})--(\ref{eq:BWFormFactors-end}),
is given by \texttt{LauBlattWeisskopfFactor::BWPrimeBarrier} and is recommended when the angular terms contain momentum factors.
One possible alternative is to use the \texttt{LauAbsResonance::Legendre} angular terms and the \texttt{LauBlattWeisskopfFactor::BWBarrier} form for the Blatt--Weisskopf factors:
where in this example the forms in Equations~(\ref{eq:BWFormFactors-NonPrimed})--(\ref{eq:BWFormFactors-NonPrimed-end})
are to be used.
Again, this operation should be performed before constructing any resonances.
+
As for the $T(\vec{p},\vec{q})$ terms, the differences between Blatt--Weisskopf form factor formalisms are more significant for higher spin resonances, and far from the peak of the resonance.
An illustrative comparison of the shapes is given in Fig.~\ref{fig:BWFF-formulae}.
+ Lineshapes for the (left) $f_2(1270)$ and (right) $\rho_3(1690)^0$
+ resonances decaying to $\pip\pim$ (in the $\Bp\to\Kp\pip\pim$ Dalitz
+ plot) with (blue) no Blatt--Weisskopf factors, and with the (red)
+ \texttt{ResonanceFrame}, (green) \texttt{ParentFrame} and (magenta)
+ \texttt{Covariant} settings for evaluating the momentum that enters the
+ Blatt--Weisskopf factor associated with the decay of the parent particle.
+ In all cases the relativistic Breit--Wigner description is used, with
+ mass and width parameters as given in App.~\ref{sec:resNames} and the
+ \texttt{Zemach\_P} formalism for the spin factors.
}
\label{fig:BWFF-formulae}
\end{figure}
-It is possible to make all of the changes discussed in this Appendix at the level of individual resonances, using the functions \texttt{LauAbsResonance::setSpinType}, \texttt{LauAbsResonance::setBarrierRadii},
+It is possible to make all of the changes discussed in this Appendix at the
+level of individual resonances, using the functions
+\texttt{LauAbsResonance::setSpinType},
+\texttt{LauAbsResonance::setBarrierRadii},
but this requires much care to be taken and is not generally recommended.
This section provides the complete set of available resonances, indicating the
name, mass $m_0$, width $\Gamma_0$, spin, charge and Blatt--Weisskopf barrier radius $r_{\rm BW}$.
Table~\ref{tab:resNames1} contains information for light meson resonances,
Table~\ref{tab:resNames2} for charm, charmonium, strange-charm, beauty and strange--beauty resonances,
Table~\ref{tab:resNames3} for $K^*$ resonances and
Table~\ref{tab:resNames4} for non-resonant terms.
Most data are taken from Ref.~\cite{PDG2016}.
The tables list the information contained in the information records for both neutral and positively-charged resonances.
Negatively-charged resonance records are implemented as charge-conjugates of
the positively charged ones; the plus sign in the name is replaced with a minus sign.
In case a user wishes to modify the values of the parameters from those given in the tables, the {\tt LauAbsResonance::changeResonance} function, which takes the mass, width and spin as arguments, can be used.
The Blatt--Weisskopf barrier radius can be changed with the {\tt LauAbsResonance::changeBWBarrierRadii} function, and other parameters specific to particular lineshapes can be changed with the {\tt LauAbsResonance::setResonanceParameter} function.
The same approach can be used to include a resonance that is not available in these tables, by using any of the existing states of appropriate charge and redefining its properties.
-%
\begin{table}[!htb]
\caption{Standard light meson resonances defined in \laura.}