Double Phase Representation of Analytic Functions

Abstract

The double phase representation is discussed for the elastic scattering amplitude $A(s, t, u)$ as a function of the covariant Mandelstam variables $s$, $t$, and $u$. This representation is written as $A(s, t, u)=\left[\frac{P_1(s, t, u)}{P_2(s, t, u)}\right]Q(s, t, u)$, where $P_1(s, t, u)$ and $P_2(s, t, u)$ are both finite polynomials in $s$, $t$, and $u$, and $Q(s, t, u)$ has no zeros or poles except at infinity and is expressed in terms of the phase of $A(s, t, u)$ along the cuts. Thus, $P_1(s, t, u)$ and $P_2(s, t, u)$ account for all the zeros and poles of $A(s, t, u)$, respectively, except for a zero or a pole at infinity. The conditions for the above double phase representation to exist are, besides the usual Mandelstam assumption, that a finite polynomial $P_1(s, t, u)$ accounts for all the zeros of $A(s, t, u)$ except for the one at infinity and no others, and that $A(s, t, u)$ has even or odd crossing symmetry with respect to the interchange of some pair of $s$, $t$, and $u$. These conditions imply that the phase of $A(s, t, u)$ has no extra branch points in the momentum-transfer plane other than those which belong to $A(s, t, u)$ and remains finite in the physical regions even in the limit of infinite energy. The asymptotic forms of this double phase representation when some of $s$, $t$, and $u$ become infinite are derived in the case when the phase approaches the limit at infinity not too slowly. This is the case when the elastic scattering amplitude exhibits asymptotically a power behavior in energy (usually called the Regge behavior). In particular, the case when the forward peak of high-energy elastic scattering does not shrink is examined closely. The case of no shrinkage is found to be the case when the phase in the crossed channel does not diverge logarithmically at infinity in its momentum-transfer plane. If the forward peak shrinks, the above phase diverges logarithmically at infinity. In the case of no shrinkage, the asymptotic shape of the forward peak is determined solely by the phase in the crossed channel. Furthermore, the above shape assumes a pure exponential function of the covariant momentum-transfer squared when momentum transfer is small, and approaches a power-law behavior in the same variable for large momentum transfer. In the case of the ${\pi }^0+{\pi }^0\to {\pi }^0+{\pi }^0$ amplitude, high symmetry available in this amplitude enables one to determine almost uniquely the polynomials in the double phase representation. In particular, the only possibility in the case of no shrinkage is $\frac{P_1(s, t, u)}{P_2(s, t, u)}=c_0+c_2(s^2+t^2+u^2)$, where $c_0$ and $c_2$ are real constants. No shrinkage also implies that the $S$-wave scattering length must not be negative for the ${\pi }^0+{\pi }^0\to {\pi }^0+{\pi }^0$ amplitude. Some of the specific predictions of the phase representation approach to highenergy elastic scattering are listed at the end of the last section.