Strong decays with the boost-corrected wave functions

Abstract

Strong decay probabilities are calculated using the Lorentz contracted wave functions of decay products, determined in the arbitrary dynamical scheme with the instantaneous interaction. It is shown that the decay width acquires an additional factor, defined by the contraction coefficient $C_m(s)$, which for the two-body equal mass decays is $C^2_m(s)= 4m^2/s$ , $s= E^2$. The resulting decay widths are compared to experimental data, where, in particular the $\rho (770),\rho (1450)$ decay data, require an additional 1/s dependence of the width to fit the data. Important consequences for the dynamics of hadron decays and scattering are shortly discussed.

Appendices

Appendix A1: The L dependence of the decay matrix element

Starting from the (8) one can write the decay matrix element $J_L(p)$, when the string extending from $x_Q$ to $x_{{\bar{Q}}}$ decays at point x into two strings with the appropriate wave functions $\psi _2,\psi _3$

$$\begin{aligned} J_L(p)= & {} \int d^3x d^3(x_Q-x_{{\bar{Q}}}) \exp {ip(x_Q- x_{{\bar{Q}}})} \nonumber \\&\Psi _1^{(L)}(x_Q- x_{{\bar{Q}}}) \psi _2(x_Q -x) \psi _3(x-x_{{\bar{Q}}}). \end{aligned}$$

(A1.1)

Going over into the momentum space one obtains

$$\begin{aligned} J_L(p)= \int d^3q \Psi _1^{(L)}(p+q) \psi _2(q) \psi _3(q) \end{aligned}$$

(A1.2)

Here $\Psi _1^L(p) ~ p^L$ and for $L=1$ one obtains the linear dependence on p. One should stress the important property of (9) and (A1.1) - both hadrons 2 and 3 have the same internal momentum q which leads finally to the effects of Lorentz contraction in the decay width, while in absence of string breaking at the point x both momenta q and $q'$ in $\psi _2(q)$ and $\psi _3(q')$ are independent, they are integrated separately and no contraction effects are visible.

Appendix A2. The pion wave function

The problem of chiral wave functions was studied in a series of papers during last 20 years (see [50] and refs therein) using the Chiral Confining Lagrangian (CCL) where CSB is directly connected with confinement and all known chiral relations like GMOR [51] are directly deduced from CCL. The main outcome of the CCL for the pion Green’s function in [50] is that it can be expanded as a series of $q{\bar{q}}$ wave functions $(\phi _n(x)$ while the pion mass is strongly shifted from the set $m_n$ and is defined by the vacuum condensate $\langle {\bar{q}} q\rangle$ as it is seen in GMOR relations. As a result in [50] the pion Green’s function was obtained in the form $G(k)= \frac{\Psi (k)}{(k^2 + m_{\pi }^2) \Phi (k)}$, where both $\Psi (k),\Phi (k)$ are expressed via $\phi _n(x)$ only and the lowest eigenfunction $\phi _0(x)$ gives the dominant contribution. In a similar way the basic chiral parameters $\langle {\bar{q}} q \rangle$ and $f_\pi$ are expressed in [50] as a sum of $\phi _n$ contributions with the dominant role of $\phi _0$. Summarizing one can conclude that the pion is well described by the standard $q {\bar{q}}$ nonchiral wave functions except for its mass which is strongly decreased by the vacuum effects.