We present the dispersive approach of the decays $\omega/\phi\to 3\pi$ published in [Dan14a]. The double and single differential decay width are calculated below. Here we repeat the main ideas of the approach, leaving all the details to [Dan14a]. The code is provided at the end.
Formalism
The matrix element for the three pion decay of a vector particle is given in terms of a helicity amplitude $H^{abc}_\lambda$, \begin{gather} \langle \pi^a(p_1) \pi^b(p_2) \pi^c(p_3)\,|\,T\,|\,V(p_V,\lambda)\rangle=(2\pi)^4\,\delta(p_V-p_1-p_2-p_3)H^{abc}_{\lambda}\nonumber\\ H^{abc}_{\lambda }= i \, \epsilon _{\mu \nu \alpha \beta }\,\epsilon ^{\mu}(p_V,\lambda )\, p_1^{\nu }\,p_2^{\alpha }\,p_3^{\beta }\,\frac{P^1_{abc}}{\sqrt{2}}\,F(s,t,u)\,, \label{Eq1} \end{gather} Here $p_V$ and $\lambda$ are the momentum and helicity of the vector particle, $V=\omega/\phi$ in our case, $p_1,p_2,p_3$ are the momenta of outgoing pions with $a,b,c$ denoting their Cartesian isospin indices. The Lorentz-invariant Mandelstam variables are defined by $s=(p_V-p_3)^2$, $t=(p_V-p_1)^2$, $u=(p_V-p_2)^2$ and satisfy the relation $s+t+u=M^2+3\,m_\pi^2$. In (\ref{Eq1}) the helicity amplitude $H^{abc}_\lambda$ is expressed in terms of a single scalar function of the Mandelstam variables $F(s,t,u)$, since Lorentz and parity invariance imply that. We made sure that the invariant amplitude $F(s,t,u)$ is free from kinematical singularities as it should satisfies Mandelstam analyticity which postulates that it is an analytic function everywhere except for cuts required by unitarity. $P^1_{abc} = -i\,\epsilon_{abc}/\sqrt{2}\,$ is the isospin factor corresponding to the coupling of three isospin-1 pions to a state with total isospin-0
From the invariant amplitude, it is straightforward to compute the Dalitz plot distribution, the partial decay and the total, $3\pi$ decay widths, [PDG] \begin{gather} \frac{d^2\Gamma}{ds\,dt}=\frac{1}{(2\pi)^3}\frac{1}{32M^3}\frac{1}{3}\sum_{\lambda}|H^{\omega\rightarrow \pi^+\pi^-\pi^0}_{\lambda}|^2 \nonumber =\frac{1}{(2\pi)^3}\frac{1}{32M^3}\frac{1}{3}P(s,t)\,|F(s,t,u)|^2\,,\nonumber\\ P(s,t)=\frac{1}{4}\left(s\,t\,u-m_\pi^2\left(M^2-m_\pi^2\right)^2\right)\,, \label{Eq2} \end{gather} where $P(s,t)$ is the kinematic factor.
Crossing symmetry implies that the function $F(s,t,u)$ describes the decay $V \to 3\pi$ and also the three $V \pi \to 2\pi$ scattering channels. We adopt scattering kinematics where $s$ corresponds to the square of the center of mass energy and $t$ is related to the cosine of the $s$-channel scattering angle by \begin{gather} z_s = \cos\theta_s = \frac{t -u}{4\,p(s)\,q(s)} \equiv \frac{t -u}{k(s)}\nonumber\\ q(s)=\frac{\lambda^{1/2}(m_\pi^2,m_\pi^2,s)}{2\sqrt{s}}\,,\quad p(s)=\frac{\lambda^{1/2}(M^2,m_\pi^2,s)}{2\sqrt{s}} \label{zs} \end{gather} where $q(s)$ and $p(s)$ are the magnitude of the relative momentum between the outing pions in the $s$-channel center of mass frame and the magnitude of the incoming pion's momentum in the same frame, respectively. $\lambda(x,y,z)=x^2+y^2+z^2-2\,(xy+yz+xz)$ is the Kallen triangle function. The $s$-channel partial wave decomposition is given by \begin{eqnarray}\label{Eq:p.w._expansion} H^{abc}_{\lambda}=\frac{P^1_{abc}}{\sqrt{2}}\, \sum_{J=1,3,\dots}^{\infty}\, (2J+1)\,d_{\lambda 0}^J(\theta_s)\,f^J_\lambda(s) \end{eqnarray} with $d_{\lambda 0}^J(\theta_s)$ being the Wigner d-functions and we choose the $x-z$ plane as the reaction plane. Due to Bose symmetry of pions the sum over partial waves is restricted to odd values of $J$.
Model
The relation between $H^{abc}_\lambda$ and $F(s,t,u)$ in Eq. \eqref{Eq1} enables the determination of the kinematical singularities of the partial wave amplitudes $f_J(s)$. We remove this kinematical constraints by introducing reduced partial wave amplitudes $F_{J}(s)$ which are proportional to $f_J(s)$. Finally, when the sum in (\ref{Eq:p.w._expansion}) is truncated, in order to retain dynamical singularities of $F(s,t,u)$ in all three variables, the scalar amplitude is approximated by a linear combination of truncated partial wave series in the three channels simultaneously \begin{equation}\label{Eq:covariant_form_2} F(s,t,u) = \sum_{J=1,3,\dots}^{J_{max}} (p(s) q(s))^{J-1} P'_J(z_s)\,F_{J}(s) + (s \to t) + (s\to u) \end{equation} The ansatz \eqref{Eq:covariant_form_2} satisfies crossing symmetry and single-variable dispersion relation. We note that a similar decomposition for $\pi\pi\rightarrow\pi\pi$ and $\eta\rightarrow3\pi$ is sometimes called as reconstruction theorem and it was aslo shown to be true up to NNLO in $\chi$PT.
In the following we consider only the P-waves, i.e. take $J_{max} = 1$, since in the kinematical region in $s,t,u$ corresponding to the $\omega/\phi$ decays the spin-3 and higher partial waves are expected to be insignificant. By imposing elastic unitarity we obtain the following discontinuity relation \begin{gather} \text{Disc}\,F(s)=\rho(s)\,t^*(s)\left(F(s)+\hat{F}(s)\right)\,,\quad \hat{F}(s)=3\int _{-1}^{+1}\frac{dz_s}{2}\left(1-z_s^2\right)F(t(s,z_s))\label{Eq:Disc2} \end{gather} and, for real $s$, \begin{equation} \label{di} F(s) = \frac{1}{\pi} \int_{4\,m_{\pi}^2}^{\infty} ds' \frac{\mbox{Disc} F(s')}{s' - s - i\,\epsilon} \end{equation} where $F(s)\equiv F_{J=1}(s)$, $t(s)\equiv t_{J=1}(s)$ is the isospin-1 p-wave pion-pion scattering amplitude, $\rho(s) = \sqrt{1 - 4m_\pi^2/s}$ is the phase space factor, and dependence on $t$ under the integral should be expressed in terms of $s$ and $z_s$ using Eq. \eqref{zs}. Also, Eq. \eqref{Eq:Disc2} was derived in the physical region of the $s$-channel, which corresponds to $s\geq (M + m_\pi)^2$ and $|z_s|\leq 1$. To obtain $F(s)$ in the decay region $4\,m_\pi^2 \leq s \leq (M-m_\pi)^2$ the right hand side of Eq. \eqref{Eq:Disc2} has to be analytically continued in $s$. Analytical continuation of the direct-channel contribution is well known. However, analytical continuation of the exchange contribution is more difficult and has been extensively studied in [Kambor:1995yc] and references therein.
The solution of Eq. \eqref{di} takes into account the final state interactions due to all possible two-pion rescattering (see figure). We solve Eq. \eqref{di} by factoring out the Omnes function and spliting the dispersive integral into two parts: The first part is determined entirely by elastic scattering while the second part takes into account inelastic effects. The inelastic contribution is parametrized through an expansion in a conformal variable which maps the right-hand cut in the complex $s$-plane onto the unit disk. The integral equation for the KT amplitude takes the form of \begin{gather} F(s)=\Omega(s)\left(\frac{1}{\pi} \int_{s_\pi}^{s_i}ds' \frac{\rho(s')\,t^*(s')}{\Omega^{*}(s')}\frac{\hat{F}(s')}{s'-s}+\Sigma(s)\right)\,\nonumber\\ \Sigma (s)=\sum_{k=0}^{\infty}a_k\,\omega^k(s)\,,\quad \omega(s)=\frac{\sqrt{s_i-s_E}-\sqrt{s_i-s}}{\sqrt{s_i-s_E}+\sqrt{s_i-s}} \label{Eq:Integral_eq} \end{gather} where $s_E=0$, $s_\pi=4\,m_\pi^2$ and $s_i=1$ GeV is identified with the point where inelastic contributions are expected to become relevant. The unknown coefficients $a_k$ in (\ref{Eq:Integral_eq}) are to be determined by comparing with experimental data.
In the figure below we show the Dalitz plot distribution in terms of dimensioless parameters $x$ and $y$ \begin{equation}\label{Eq:x_y} x=\frac{\sqrt{3}(t-u)}{2M(M-3\,m_\pi)},\quad y=\frac{3(s_c-s)}{2M(M-3m_\pi)}\,. \end{equation} where $s_{c}=\frac{1}{3}(M^2+3\,m_\pi^2)$ represents the location of the center of the Mandelstam triangle.
Fig: The Dalitz plot distributions for $\omega\to3\pi$ and $\phi\to3\pi$ divided by the p-wave phase space $P(s,t)$ (determined in (\ref{Eq2})) and normalized to 1 at $x=y=0$. This is a parameter free result, because we kept only one term in the conformal expansion (\ref{Eq:Integral_eq}) which is responsible for the overall normalization.
Fig: Single differential decay rates. Zeroth term in the confromal expnsion (\ref{Eq:Integral_eq}) is fixed to reproduce the measured $3\pi$ decay width for $\omega$ and $\phi$, which are $\Gamma_{\omega\rightarrow 3\pi}^{exp}= 7.57\mbox{ MeV}$ and $\Gamma_{\phi\rightarrow 3\pi}^{exp} = 0.65\mbox{ MeV}$, respectively [PDG].
References
[Dan14a]
I.V. Danilkin, C. Fernandez-Ramirez, P. Guo, V. Mathieu, D. Schott, M. Shi and A.P. Szczepaniak,
``Dispersive Analysis of Ï‰/Ï• â†’ 3Ï€, Ï€Î³âˆ—,''
e-Print: arXiv:1409.7708 [hep-ph]
[PDG]
J. Beringer et al.,
``Review of particle physics,''
Phys. Rev. D, 86, 010001 (2012).
[Kambor:1995yc]
J.Kambor, C.Wiesendanger and D.Wyler,,
``Final state interactions and Khuri-Treiman equations in $\eta \to 3\pi$ decays,''
Nucl. Phys. B 465, 215 (1996).
Resources
- Publication: [Dan14a]
- Fortran: Fortran file, Input file, Output file
- Mathematica: Main notebook , Style , Masses , Pi-Pi input , Omnes
- Contact person: Igor Danilkin
- Last update: April 2015
Description of the Fortran code:
The Fortran code calculates $F(s,t,u)$ from (\ref{Eq2}) and (\ref{Eq:Integral_eq}) for the case when $\Sigma(s)$ is approximated by two terms: $\Sigma(s)\approx a_0+a_1\,\omega(s)$. Since the integral equation (\ref{Eq:Integral_eq}) is linear in $F(s)$ the solution can written as: $F(s)=F_0(s)+ \frac{a_1}{a_0}\,F_1(s)$ up to the overall constant wich can be fixed from the comparison with partial decay width. In the code $\frac{a_1}{a_0}:=param$. The functions $F_{0}(s)$ and $F_{1}(s)$ satisfy \begin{gather} F_0(s)=\Omega(s)\left(\frac{1}{\pi} \int_{s_\pi}^{s_i}ds' \frac{\rho(s')\,t^*(s')}{\Omega^{*}(s')}\frac{\hat{F_0}(s')}{s'-s}+1\right)\,\nonumber\\ F_1(s)=\Omega(s)\left(\frac{1}{\pi} \int_{s_\pi}^{s_i}ds' \frac{\rho(s')\,t^*(s')}{\Omega^{*}(s')}\frac{\hat{F_1}(s')}{s'-s}+\omega(s)\right)\,\nonumber\\ \end{gather} which were calulated in Mathematica for different s=[0, 1.21] GeV$^2$ and $M$=[0.6, 1.11] GeV. The Fortran code just upload and interpolate files.
execute the command 'gfortran AmplitudeDanilkin.F' in a terminal and call the program with './a.out'.
'param.txt' contains: param, s, t, u
List of subroutines
- upload_omega(threebody)
Upload amplitudes for different $M$=[0.6, 1.11] GeV and energy and then interpolate
- amp(ret,s,t,u,param)
return the amplitude $F(s,t,u)$ from Eq.(\ref{Eq2})
for a particular $s,t,u$ - subroutine f0f1(s,mv,f0,f1)
for the given $s$ and calculated mv=dsqrt(s+t+u-3.d0*mpi**2) returns f0 and f1 - subroutine spline(n,x,y,b,c,d)
Spline interpolation
List of variables
- param: unknown parameter (responsible for the shape of Dalitz plot)
- s: Mandelstam s in GeV$^2$
- t: Mandelstam t in GeV$^2$
- u: Mandelstam u in GeV$^2$ (for the physicsl $M_{\omega/\phi}$ put $u=M_{\omega/\phi}^2+3m_\pi^2-s-t$)