---
author:
- 'CP-PACS collaboration:'
- |
    S. Aoki,$^{a,b}$ M. Fukugita,$^c$ K.-I. Ishikawa,$^d$ ,$^{a,e}$[^1] K. Kanaya,$^a$ Y. Kuramashi,$^{a,e}$ Y. Namekawa,$^{f}$ M. Okawa,$^d$ K. Sasaki,$^{e}$ A. Ukawa,$^{a,e}$ T. Yoshié$^{a,e}$\
    \
    \
    Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan\
    Riken BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA\
    Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan\
    Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan\
    Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan\
    Department of Physics, Nagoya University, Nagoya 464-8602, Japan
title: ' $\rho$ meson decay from the lattice '
---

 Introduction 
==============

Lattice study of the $\rho$ meson decay is an important step for understanding of the dynamical aspect of hadron reactions induced by strong interactions. There are already three studies [@rhpipi_GMTW; @rhpipi_LD; @rhpipi_MM]. The earlier two studies employed the quenched approximation ignores the decay into two ghost pions. The most recent work, while using the $N_f=2$ dynamical configurations, concentrated on the $\rho\to\pi\pi$ transition amplitude rather than the full matrix including the $\rho\to\rho$ and $\pi\pi\to\pi\pi$ amplitudes. All three studies were carried out at an unphysical kinematics $m_\pi/m_\rho > 1/2$.

In this work we attempt to carry out a more rigorous approach. We estimate the decay width from the scattering phase shift for the $I=1$ two-pion system. The finite size formula presented by Rummukainen and Gottlieb [@fm_RG] is employed for an estimation of the phase shift. Calculations are carried out with $N_f=2$ full QCD configuration previously generated for a study of light hadron spectrum with a renormalization group improved gauge action and a clover fermion action at $\beta=1.8$, $\kappa=0.14705$ on a $12^3\times 24$ lattice [@conf_NMC]. The parameters determined from the spectrum analysis are $1/a=0.92\ {\rm GeV}$, $m_\pi/m_\rho=0.41$, and $L=2.53\ {\rm fm}$. All calculations of this work are carried out on VPP5000/80 at the Academic Computing and Communications Center of University of Tsukuba.

 Method 
========

In order to realize a kinematics such that the energy of the two-pion state is close to the resonance energy $m_\rho$, we consider the non-zero total momentum frame (the moving frame) [@fm_RG] with the total momentum ${\bf p}=2\pi / L \cdot {\bf e}_3$. The initial $\rho$ meson is assigned a polarization vector parallel to ${\bf p}$. One of the final two pions carry the momentum ${\bf p}$, while the other pion is at rest. The energies ignoring hadron interactions are then given by $E_1^0 = \sqrt{ m_{\pi }^2 + p^2 } + m_\pi$ for the two-pion state and $E_2^0 = \sqrt{ m_{\rho}^2 + p^2 }        $ for the $\rho$ meson. We neglect higher energy states whose energies are much higher than $E_1^0$ and $E_2^0$. On our full QCD configurations, the invariant mass for the two-pion state takes $\sqrt{s} = 0.97\times m_\rho$, while $1.47\times m_\rho$ is expected for the zero total momentum. The $\rho$ meson at zero momentum cannot decay energetically, so that it can be used to extract $m_\rho$.

The hadron interactions shift the energy from $E_n^0$ to $E_n$ ($n=1,2$). These energies $E_n$ are related to the two-pion scattering phase shift $\delta(\sqrt{s})$ through the Rummukainen-Gottlieb formula [@fm_RG], which is an extension of the Lüscher formula [@fm_LU] to the moving frame. The formula for the total momentum ${\bf p}=p {\bf e}_3$ and the ${\bf A}_2^-$ representation of the rotation group on the lattice reads $$\frac{1}{\tan\delta(\sqrt{s})}
= \frac{ 1 }{ 2\pi^2 q \gamma }\
  \sum_{ {\bf r} \in \Gamma }  \
     \frac{ 1 + ( 3 r_3^2 - r^2 )/q^2 }{ r^2 - q^2 }
\ ,
\label{eq:RG_F}$$ where $\sqrt{s} = \sqrt{ E^2 - p^2 }$ is the invariant mass, $k$ is the scattering momentum ($\sqrt{s} = 2 \sqrt{ m_\pi^2 + k^2 }$), $\gamma$ is the Lorentz boost factor ($\gamma=E/\sqrt{s}$), and $q=k L / (2\pi)$. The summation for ${\bf r}$ in (\[eq:RG\_F\]) runs over the set $$\Gamma
   = \{ {\bf r} |
        \ r_1 = n_1 \ ,
        \ r_2 = n_2 \ ,
        \ r_3 = ( n_3 + \frac{p}{2} \frac{L}{2\pi} )/\gamma \ ,
        \ {\bf n}\in {\bf Z}^3 \}
\ .$$ The right hand side of (\[eq:RG\_F\]) can be evaluated by the method described in Ref. [@phsh_YAMA].

In order to calculate $E_1$ and $E_2$ we construct a $2\times 2$ matrix time correlation function, $$G(t) =
\left(
\begin{array}{llll}
      \displaystyle \langle 0 | \ (\pi\pi)^\dagger(t) \ (\pi\pi) (t_s) \ | 0 \rangle
& & & \displaystyle \langle 0 | \ (\pi\pi)^\dagger(t) \   \rho_3 (t_s) \ | 0 \rangle  \\
      \displaystyle \langle 0 | \   \rho_3^\dagger(t) \ (\pi\pi) (t_s) \ | 0 \rangle
& & & \displaystyle \langle 0 | \   \rho_3^\dagger(t) \   \rho_3 (t_s) \ | 0 \rangle  \\
\end{array}
\right)
\ .
\label{eq:G}$$ Here, $\rho_3(t)$ is an interpolating operator for the neutral $\rho$ meson with the momentum ${\bf p}=2\pi/L\cdot {\bf e}_3$ and the polarization vector parallel to ${\bf p}$; $(\pi\pi)(t)$ is an interpolating operator for the two pions given by $$(\pi\pi)(t) = \frac{1}{\sqrt{2}}
    \Bigl(
         \pi^{-}({\bf p},t) \pi^{+}({\bf 0},t)
       - \pi^{+}({\bf p},t) \pi^{-}({\bf 0},t)
    \Bigl)
\ ,
\label{eq:pp_op}$$ which belongs to the ${\bf A}_2^-$ and iso-spin representation with $I=1$, $I_z=0$.

We can extract the two energy eigenvalues by a single exponential fitting of the two eigenvalues $\lambda_1 (t,t_R)$ and $\lambda_2 (t,t_R)$ of the normalized matrix $M(t,t_R) = G(t) G^{-1}(t_R)$ with some reference time $t_R$ [@method_diag] assuming that the lower two states dominate the correlation function.

In order to construct the meson state with non-zero momentum we introduce a $U(1)$ noise $\xi_j({\bf x})$ in three-dimensional space whose property is $$\frac{1}{N_R}
\sum_{j=1}^{N_R}
\xi_j^\dagger ({\bf x}) \xi_j ({\bf y})
= \delta^3( {\bf x} - {\bf y} )
             \qquad \mbox{ for \ $N_R \to \infty$ }
\ .
\label{eq:xi_prop}$$ We calculate the quark propagator $$Q( {\bf x}, t | {\bf q}, t_s, \xi_j )
  = \sum_{\bf y} ( D^{-1} )({\bf x}, t ; {\bf y}, t_s )
    \cdot \Bigl[ \ {\rm e}^{ i{\bf q}\cdot{\bf y}} \xi_j({\bf y}) \ \Bigr]
\ ,
\label{eq:QP_Q}$$ regarding the term in the square bracket as the source. The two point function of the meson with the spin content $\Gamma$ and the momentum ${\bf p}$ can be constructed from $Q$ as $$\frac{1}{N_R} \sum_{j=1}^{N_R} \
\sum_{\bf x} {\rm e}^{ - i {\bf p}\cdot {\bf x} }
\cdot
\Bigl\langle
  \ \gamma_5
  \ Q^\dagger ( {\bf x}, t | {\bf 0}, t_s, \xi_j ) \ \gamma_5 \Gamma^\dagger
  \ Q         ( {\bf x}, t | {\bf p}, t_s, \xi_j ) \          \Gamma
  \
\Bigr\rangle
\ ,$$ where the bracket refers to the trace for color and spin indeces.

The quark contraction for the $\pi\pi\to\pi\pi$ and the $\pi\pi\to\rho$ components of $G(t)$ are given by $$\includegraphics[width=14.5cm]{Fig/quark_c.eps}
%
\label{eq:quark_c}$$ where the four verteces for the $\pi\pi\to\pi\pi$ and three verteces for the $\pi\pi\to\rho$ components refer to the pion or the $\rho$ meson with definite momentum. The time direction is upward in the diagrams, and the $\rho\to\pi\pi$ component is given by changing the time direction.

The first term of the $\pi\pi\to\pi\pi$ component in (\[eq:quark\_c\]) can be calculated by introducing another $U(1)$ noise $\eta_j({\bf x})$ having the same property as $\xi_j({\bf x})$ in (\[eq:xi\_prop\]); $$\frac{1}{N_R}
\sum_{j=1}^{N_R}
\sum_{{\bf x}, {\bf y}} {\rm e}^{ - i {\bf p}\cdot {\bf x} }
    \cdot
    \Bigl\langle
         Q^\dagger ( {\bf x}, t | {\bf 0}, t_s, \xi_j  ) \
         Q         ( {\bf x}, t | {\bf p}, t_s, \xi_j  ) \Bigr\rangle
    \Bigl\langle
         Q^\dagger ( {\bf y}, t | {\bf 0}, t_s, \eta_j ) \
         Q         ( {\bf y}, t | {\bf 0}, t_s, \eta_j ) \Bigr\rangle
\ .
\label{eq:D1_quark_c}$$ The second term of (\[eq:quark\_c\]) is obtained by exchanging the momentum of the sink in (\[eq:D1\_quark\_c\]).

In order to construct the other terms of (\[eq:quark\_c\]) we calculate a quark propagator of another type by the source method, $$W( {\bf x}, t | {\bf k}, t_1 | {\bf q}, t_s, \xi_j )
= \sum_{\bf z} ( D^{-1} )({\bf x}, t ; {\bf z}, t_1 )
    \cdot \Bigl[ {\rm e}^{ i{\bf k}\cdot{\bf z}}
                  \gamma_5 \ Q( {\bf z}, t_1 | {\bf q}, t_s, \xi_j )
          \Bigr]
\ ,
\label{eq:QP_W}$$ where the term in the square bracket is regarded as the source in solving the propagator. Using $W$ we can construct the third to sixth terms in the $\pi\pi\to\pi\pi$ component of (\[eq:quark\_c\]) by $$\begin{aligned}
\mbox{ 3rd } &=&
\frac{1}{N_R}
\sum_{j=1}^{N_R}
\sum_{\bf x}  {\rm e}^{ - i {\bf p}\cdot {\bf x} } \cdot
    \Bigl\langle
      \  W^\dagger ( {\bf x}, t | {\bf 0}, t_s | - {\bf p}, t_s, \xi_j ) \
         W         ( {\bf x}, t | {\bf 0}, t   |   {\bf 0}, t_s, \xi_j ) \
    \Bigr\rangle
\ ,
\cr
\mbox{ 4th } &=&
\frac{1}{N_R}
\sum_{j=1}^{N_R}
\sum_{\bf x}  {\rm e}^{ - i {\bf p}\cdot {\bf x} } \cdot
    \Bigl\langle
      \  W         ( {\bf x}, t | {\bf 0}, t_s |   {\bf p}, t_s, \xi_j ) \
         W^\dagger ( {\bf x}, t | {\bf 0}, t   |   {\bf 0}, t_s, \xi_j ) \
    \Bigr\rangle
\ ,
\cr
\mbox{ 5th } &=&
\frac{1}{N_R}
\sum_{j=1}^{N_R}
\sum_{\bf x}  {\rm e}^{ - i {\bf p}\cdot {\bf x} } \cdot
    \Bigl\langle \
      \  W         ( {\bf x}, t | {\bf p}, t_s | {\bf 0}, t_s, \xi_j ) \
         W^\dagger ( {\bf x}, t | {\bf 0}, t   | {\bf 0}, t_s, \xi_j ) \
    \Bigr\rangle
\ ,
\cr
\mbox{ 6th } &=&
\frac{1}{N_R}
\sum_{j=1}^{N_R}
\sum_{\bf x}  {\rm e}^{ - i {\bf p}\cdot {\bf x} } \cdot
    \Bigl\langle \
      \  W^\dagger ( {\bf x}, t | - {\bf p}, t_s | {\bf 0}, t_s, \xi_j ) \
         W         ( {\bf x}, t |   {\bf 0}, t   | {\bf 0}, t_s, \xi_j ) \
    \Bigr\rangle
\ .
\label{eq:QC_B}\end{aligned}$$ The two terms of $\pi\pi\to\rho$ of (\[eq:quark\_c\]) can be similarly constructed by $$\begin{aligned}
\mbox{ 1st } &=&
\frac{1}{N_R}
\sum_{j=1}^{N_R}
\sum_{\bf x}  {\rm e}^{ - i {\bf p}\cdot {\bf x} } \cdot
  \Bigl\langle
    \  W^\dagger ( {\bf x}, t | - {\bf p}, t_s | {\bf 0}, t_s, \xi_j ) \ (\gamma_5 \gamma_3 ) \
       Q         ( {\bf x}, t                  | {\bf 0}, t_s, \xi_j ) \
  \Bigr\rangle
\ ,
\cr
\mbox{ 2nd } &=&
\frac{1}{N_R}
\sum_{j=1}^{N_R}
\sum_{\bf x}  {\rm e}^{ - i {\bf p}\cdot {\bf x} } \cdot
  \Bigl\langle
    \  Q^\dagger ( {\bf x}, t                | {\bf 0}, t_s, \xi_j ) \ (\gamma_5 \gamma_3 ) \
       W         ( {\bf x}, t | {\bf p}, t_s | {\bf 0}, t_s, \xi_j ) \
  \Bigr\rangle
\ .
\label{eq:QC_pprh}\end{aligned}$$ In this work we set the source at $t_s=4$ and impose the Dirichlet boundary condition in the time direction. We calculate the $Q$-type propagators for four sets of ${\bf q}$ and the $U(1)$ noise in (\[eq:QP\_Q\]) : $
( {\bf q}, {\rm noise})=\{
( {\bf 0}, \xi  ) ,
( {\bf 0}, \eta ) ,
( {\bf p}, \xi  ) ,
(-{\bf p}, \xi  )       \}$. The $W$-type propagators are calculated for 22 sets of ${\bf k}$, $t_1$ and ${\bf q}$ in (\[eq:QP\_W\]) : $
(  {\bf k}, t_1      |  {\bf q} )=\{
(  {\bf p}, t_s      |  {\bf 0} ) ,
( -{\bf p}, t_s      |  {\bf 0} ) ,
(  {\bf 0}, t_s      |  {\bf p} ) ,
(  {\bf 0}, t_s      | -{\bf p} ) ,
(  {\bf 0}, t_1=4-21 |  {\bf 0} ) \}$, with the same $U(1)$ noise $\xi$. All diagrams for the time correlation function can be calculated with combinations of these propagators. We choose $N_R=10$ for the number of $U(1)$ noise. We carry out additional measurements to reduce statistical errors using the source operator is located at $t_s+T/2$ and the Dirichlet boundary condition is imposed at $T/2$. We average over the two measurements for the analysis. Thus we calculate $520$ quark propagators for each configuration. The total number of configurations analyzed are 800 separated by 5 trajectories [@conf_NMC].

 Results 
=========

In Fig. \[fig:G\_t\] we plot the real part of the diagonal components ($\pi\pi\to\pi\pi$ and $\rho\to\rho$) and the imaginary part of the off-diagonal components ($\pi\pi\to\rho$, $\rho\to\pi\pi$) of $G(t)$. Our construction of $G(t)$ is such that the sink and source operators are identical for a sufficiently large number of the $U(1)$ noise. In this case we can prove that $G(t)$ is an Hermitian matrix and the off-diagonal parts are pure imaginary from $P$ and $CP$ symmetry. We find that this is valid within statistics, but the statistical errors of the $\rho\to\pi\pi$ component is larger than those of $\pi\pi\to\rho$ in Fig. \[fig:G\_t\]. In the following analysis we substitute $\rho\to\pi\pi$ by $\pi\pi\to\rho$ to reduce the statistical error.

![ $G(t)$ \[fig:G\_t\] ](Fig/Fig.H.pp.pp2rI.rI.eps "fig:"){width="7.5cm"} ![ $G(t)$ \[fig:G\_t\] ](Fig/Fig.H.pp2rI.rI2pp.eps "fig:"){width="7.5cm"}

The two eigenvalues $\lambda_1(t,t_R)$ and $\lambda_2(t,t_R)$ for the matrix $M(t,t_R) = G(t) G^{-1}(t_R)$ are shown in Fig. \[fig:Lambda\_t\]. We set the reference time $t_R=9$ and normalize the eigenvalues by the correlation function for the free two-pion system, $\langle 0| \pi(-{\bf p},t) \pi({\bf p},t_s) |0\rangle
 \langle 0| \pi( {\bf 0},t) \pi({\bf 0},t_s) |0\rangle
$. Thus the slope of the figure corresponds to the energy difference $\Delta E_n=E_n-E_1^0$. We observe that the energy difference for $\lambda_1$ is negative and that for $\lambda_2$ is positive. This means that the two-pion scatting phase shift is positive for the lowest state and negative for the next higher state.

![ Normalized eigenvalues $\lambda_1(t,t_R)$ and $\lambda_2(t,t_R)$. \[fig:Lambda\_t\] ](Fig/Fig.H.Ev1_ppN.eps "fig:"){width="7.5cm"} ![ Normalized eigenvalues $\lambda_1(t,t_R)$ and $\lambda_2(t,t_R)$. \[fig:Lambda\_t\] ](Fig/Fig.H.Ev2_ppN.eps "fig:"){width="7.5cm"}

We extract the energy difference $\Delta E_n$ for both states by a single exponential fitting of the normalized eigenvalues $\lambda_1$ and $\lambda_2$ for the time range $t=10-16$. Then we reconstruct the energy $E_n$ in the moving frame by adding the energy of the two free pions, [*i.e.,*]{} $E_n=\Delta E_n + E_1^0$, and convert it to the invariant mass $\sqrt{s}$. Substituting $\sqrt{s}$ into the Rummukainen-Gottlieb formula (\[eq:RG\_F\]) we obtain the scattering phase shift : $$\begin{array}{clccccc}
\hline
\hline
& \displaystyle a \sqrt{s}        &&&& \displaystyle  \tan\delta(\sqrt{s}) & \\
\hline
& \displaystyle 0.7880 \pm 0.0082 &&&& \displaystyle   0.0773 \pm 0.0033   & \\
& \displaystyle 0.962  \pm 0.024  &&&& \displaystyle  -0.43   \pm 0.12     & \\
\hline
\hline
\end{array}
\label{eq:tanD}$$ The $\rho$ meson mass obtained at zero momentum is $a m_\rho =0.858 \pm 0.012$. Hence the sign of the scattering phase shifts at $\sqrt{s}< m_\rho$ is positive (attractive interaction) and that at $\sqrt{s}> m_\rho$ is negative (repulsive interaction) as expected. The corresponding results for $\sin^2\delta(\sqrt{s})$, which is proportional to the scattering cross section of the two-pion system, are plotted in Fig. \[fig:sin2Del\] together with the position of $m_\rho$.

![ $\sin^2\delta(\sqrt{s})$, position of $m_\rho$ and resonance mass $M_R$. \[fig:sin2Del\] ](Fig/sin2Del.eps){width="9cm"}

In order to estimate the $\rho$ meson decay width at the physical quark mass we parameterize the scattering phase shift by the effective $\rho\to\pi\pi$ coupling constant $g_{\rho\pi\pi}$, $$\tan\delta(\sqrt{s})
 = \frac{ g_{\rho\pi\pi}^2 }{ 6\pi } \cdot
   \frac{ k^3 }{ \sqrt{s} ( M_R^2 - s ) }
\ ,
\label{eq:tanD_g}$$ with $g_{\rho\pi\pi}$ defined by the effective Lagrangian, $$L_{\rm eff.} = g_{\rho\pi\pi}
  \cdot \epsilon_{abc}
  ( k_1 - k_2 )_\mu   \rho_\mu^a(p) \pi^b(k_1) \pi^c(k_2)
\ ,$$ where $k$ is the scattering momentum and $M_R$ is the resonance mass. The coupling $g_{\rho\pi\pi}$ generally depends on the quark mass and the energy, but our present data at a single quark mass do not provide this information. Here we assume that these dependences are small and try to estimate $g_{\rho\pi\pi}$ and $M_R$ from our results in (\[eq:tanD\]). We also estimate the $\rho$ meson decay width at the physical quark mass from $$\Gamma_\rho
  = \frac{ g_{\rho\pi\pi}^2 }{ 6\pi } \cdot
    \frac{ \bar{k}_\rho^3 }{ \bar{m}_\rho^2 }
  = g_{\rho\pi\pi}^2 \times 4.128 \ \ {\rm MeV}
\ ,$$ where $\bar{m}_\rho$ is the $\rho$ meson mass at the physical quark mass and $\bar{k}_\rho$ is the scattering momentum at $\sqrt{s}=\bar{m}_\rho$.

Our final results are as follows. $$\begin{aligned}
a M_R          &=& 0.877 \pm 0.025           \cr
g_{\rho\pi\pi} &=& 6.01  \pm 0.63            \cr
\Gamma_\rho    &=& 149   \pm 31   \ \ {\rm MeV}
\ .
\label{eq:FinalR}\end{aligned}$$ The resonance mass $M_R$ obtained from the scattering phase shift is consistent with $a m_\rho=0.858\pm 0.012$ obtained from the $\rho$ meson with zero momentum. The $\rho$ meson decay width $\Gamma_\rho$ at the physical quark mass is consistent with experiment ($150\ {\rm MeV}$). In Fig. \[fig:sin2Del\] we indicate the position of $M_R$ and draw the line given by (\[eq:tanD\_g\]) with $g_{\rho\pi\pi}$ and $M_R$ in (\[eq:FinalR\]).

 Summary 
=========

We have shown that a direct calculation of the $\rho$ meson decay width from the scattering phase shift for the $I=1$ two-pion system is possible with present computing resources. However, several issues remain which should be investigated in future work. The most important issue is a proper evaluation of the quark mass and energy dependence of the effective $\rho\to\pi\pi$ coupling constant $g_{\rho\pi\pi}$. This constant is used to obtain the physical decay width at $m_\pi/m_\rho=0.18$ from our results at $m_\pi/m_\rho=0.41$ by a long chiral extrapolation. In principle we can estimate the decay width from the scattering phase shift without such a parameterization, if we have data for several energy values at or near the physical quark mass. This will be our goal toward the lattice determination of the $\rho$ meson decay.

This work is supported in part by Grants-in-Aid of the Ministry of Education (Nos. 17340066, 16540228, 18104005, 17540259, 13135216, 18540250, 15540251, 13135204, 16740147, 18740139 ). The numerical calculations have been carried out on VPP5000/80 at Academic Computing and Communications Center of University of Tsukuba.

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[^1]: E-mail : ishizuka@ccs.tsukuba.ac.jp
