UDC 539.12.01+539.128.417

PHYSICS

NGUEN VAN HIEU

SOME EXPERIMENTAL CONSEQUENCES OF THE ANALYTICITY OF THE FORM FACTOR

(Presented by Academician N. N. Bogolyubov, 15 III 1968)

Consider a certain form factor \(F(t)\) and suppose that it is an analytic function in the complex \(t\)-plane with a cut on the real axis from \(t=4m_\pi^2\) to \(\infty\). In local field theory \(F(t)\), as \(t\to\infty\), can grow only more slowly than any linear exponential of \(\sqrt{t}\):

\[ |F(t)| \leqslant \exp[\varepsilon |t|^{1/2}], \qquad t\to\infty \tag{1} \]

for any \(\varepsilon>0\) \((^{1,2})\). In a number of works \((^{3-8})\) it was shown that from the analytic properties of the form factor one can obtain a number of experimentally verifiable consequences. In the present work we consider some other consequences.

1. Let us first note that as \(t\to+\infty\) (in the physical region of the annihilation channel) \(F(t)\) cannot decrease faster than \(\exp[-\operatorname{const}\sqrt{t}]\). More precisely, there exists some sequence of points \(t_n\to+\infty\) such that on it the inequality

\[ |F(t_n)| \geqslant \operatorname{const}\cdot \exp[-a\sqrt{t_n}], \qquad a>0, \qquad t_n\to+\infty \tag{2} \]

holds.

In order to prove this assertion, it is sufficient to make the change of variables \(t=z^2\) and then apply to the function \(f(z)\equiv F(t)\), analytic in the upper half-plane \(z\), the following theorem.

Theorem 1. Let the function \(f(z)\) be analytic in the upper half-plane \(\operatorname{Im} z>0\) and bounded at every finite point of the real axis. If \(f(z)\) grows no faster than some linear exponential in the upper half-plane,

\[ |f(z)| \leqslant \operatorname{const}\cdot \exp[b|z|], \qquad b>0, \qquad z\to\infty,\quad \operatorname{Im} z>0, \]

and decreases exponentially on the real axis:

\[ |f(z)| \leqslant \operatorname{const}\cdot \exp[-c|z|], \qquad c>0, \qquad z\to\pm\infty, \]

then \(f(z)\equiv 0\).

A similar theorem was proved in \((^9)\) (see Theorem 5.8) for functions analytic also on the real axis. The theorem formulated here can be proved analogously if, instead of the maximum principle (see \((^9)\), Theorem 5.1), one applies the generalized maximum principle (see \((^{10})\), Chapter VI, § 5).

Inequality (2) can also be obtained in a more general case, when the function \(F(t)\) is analytic only outside some finite region of the \(t\)-plane with a cut. For this purpose it is sufficient to apply a suitable conformal mapping and use the theorem formulated above.

1. If we further suppose that \(F(t)\) as \(t\to-\infty\) and \(|F(t)|\) as \(t\to+\infty\) do not oscillate, but have some regular behavior (which can be checked experimentally), then we can obtain stronger results. Applying the Phragmén—Lindelöf theorem in the general form-

level, given, for example, in works \((^{5,11})\), one can show that if

\[ F(t)\to a/|t|^n,\qquad t\to -\infty, \]

then

\[ |F(t)|\gtrsim |a|/t^n,\qquad t\to +\infty; \]

if

\[ F(t)\to a\exp[-b|t|^\alpha],\qquad t\to -\infty,\qquad b>0,\qquad 0<\alpha\leq \tfrac12, \]

then

\[ |F(t)|\gtrsim |a|\exp[-b\sin\pi\alpha\, t^\alpha],\qquad t\to +\infty. \]

In particular, if the interaction is minimal in the sense of Martin \((^3)\) (see also \((^{12})\)), i.e.

\[ F(t)\to a\exp[-b\sqrt{|t|}],\qquad t\to -\infty,\qquad b>0, \]

then

\[ |F(t)|\gtrsim |a|,\qquad t\to +\infty. \]

In the case when \(|F(t)|\) oscillates as \(t\to +\infty\), there must exist a sequence of points \(t_n\to +\infty\) on which one of the above inequalities holds, provided the corresponding condition on \(F(t)\) is satisfied as \(t\to -\infty\).

1. Let us further assume that \(F(t)\) is bounded on the cut:

\[ |F(t)|\leq M,\qquad t\geq 4m_\pi^2, \tag{3} \]

and show that for the values of \(|F(t)|\) in the region \(t<0\) there exists a certain lower bound. For this purpose we first make the change of variables
\(w=[t/4m_\pi^2+\alpha]^{1/2}\), where \(\alpha\) is a positive sufficiently large number, and set \(F(t)\equiv g(w)\). The \(t\)-plane with a cut is transformed into the upper half-plane \(w\). Since \(g(w)\) takes real values on the interval
\(-\sqrt{1+\alpha}<w<\sqrt{1+\alpha}\), by the Riemann—Schwarz symmetry principle it can be analytically continued into the lower half-plane. Thus, \(g(w)\) is an analytic function in the \(w\)-plane with cuts \((-\infty,-\sqrt{1+\alpha})\) and \((\sqrt{1+\alpha},\infty)\).

By means of the conformal mapping

\[ \xi=\frac{\sqrt{1+\alpha}}{w}\left[\sqrt{1+\alpha}-\sqrt{1+\alpha-w^2}\right] \]

we transform the \(w\)-plane with cuts into the circle \(C\) of radius \(\sqrt{1+\alpha}\) and with center at zero. The point \(w=\sqrt{\alpha}\) is transformed into the point \(\xi=a\),

\[ a=\frac{\sqrt{1+\alpha}}{\sqrt{\alpha}}\left(\sqrt{1+\alpha}-1\right), \tag{4} \]

and the points \(w=\pm\sqrt{\alpha-\gamma}\), where \(\gamma<\alpha\) is a certain fixed positive number, are transformed into the points \(\xi=\pm b\),

\[ b=\frac{\sqrt{1+\alpha}}{\sqrt{\alpha-\gamma}}\left(\sqrt{1+\alpha}-\sqrt{1+\gamma}\right). \tag{5} \]

The circle \(C\) completely contains the ellipse \(E\) with foci at the points \(\xi=\pm b\) and with major semiaxis \(\sqrt{1+\alpha}\). By means of the conformal mapping

\[ \eta=\frac{1}{b}\left[\xi+\sqrt{\xi^2-b^2}\right] \]

we transform this ellipse \(E\) into an annulus with inner radius \(1\) and outer radius \(R\):

\[ R=\frac{1}{b}\left[\sqrt{1+\alpha}-\sqrt{1+\alpha-b^2}\right], \tag{6} \]

following Cerulus and Martin \({}^{13}\). The point \(\xi=a\) (i.e., \(w=\sqrt{a}\), \(t=0\)) is transformed into the point \(\eta=r\)

\[ r=\frac{1}{b}\left[a+\sqrt{a^{2}-b^{2}}\right]. \tag{7} \]

Let \(h(\eta)\equiv g(w)\equiv F(t)\). According to the assumption

\[ \max_{|\eta|=R}|h(\eta)|\leq M \]

(see formula (3)), while \(h(r)=F(0)=1\). From Hadamard’s theorem on three circles (see \({}^{9}\), Theorem 5.3) it follows that

\[ \max_{|\eta|=1}|h(\eta)|= \max_{-\alpha\leq t/4m_\pi^{2}\leq-\gamma}|F(t)| \geq \left(\frac{1}{M}\right)^{\frac{\ln r/\ln R}{1-\ln r/\ln R}} . \]

Letting \(\alpha\) tend to infinity and using expressions (4)—(7), we obtain \({}^{*}\)

\[ \max_{t\leq -4m_\pi^{2}\gamma}|F(t)| \geq \left(\frac{1}{M}\right)^{\Phi(\gamma)}, \tag{8} \]

where

\[ \Phi(\gamma)= \frac{\left[1-(1+\gamma)^{-1/2}\right]^{1/2}} {1-\left[1-(1+\gamma)^{-1/2}\right]^{1/2}} . \tag{9} \]

If \(F(t)\) decreases monotonically with increasing \(|t|\) in the region \(t<0\), then we have

\[ F(t)\geq \left(\frac{1}{M}\right)^{\Phi(|t|/4m_\pi^{2})}. \tag{10} \]

It follows from this inequality that the form factor can decrease by a factor of \(e\) in the interval \((-t_e,0)\) only if \(t_e\) satisfies the condition

\[ t_e\geq \frac{1}{(1+\ln M)^2-1}. \tag{11} \]

In conclusion, the author expresses gratitude to N. N. Bogoliubov, D. I. Blokhintsev, and A. N. Tavkhelidze for their interest in the work.

Joint Institute
for Nuclear Research

Received
5 II 1968

CITED LITERATURE

\({}^{1}\) N. N. Meiman, ZhETF, 46, 1502 (1964).
\({}^{2}\) Nguyen van Hieu, Ann. Phys., 33, 428 (1965).
\({}^{3}\) A. Martin, Nuovo Cim., 37, 671 (1965).
\({}^{4}\) A. M. Jaffe, Phys. Rev. Lett., 17, 661 (1966).
\({}^{5}\) A. A. Logunov, N. V. Hieu, I. T. Todorov, Ann. Phys., 31, 203 (1965).
\({}^{6}\) B. V. Geshkenbein, B. L. Ioffe, ZhETF, 46, 902 (1964).
\({}^{7}\) Nguyen Van Hieu, Preprint of the Joint Institute for Nuclear Research, E2-3509, 1967.
\({}^{8}\) T. N. Tran, R. Vinh Mau, P. X. Yem, Preprint IHES, Paris, 1968.
\({}^{9}\) E. Titchmarsh, Theory of Functions, Moscow, 1951.
\({}^{10}\) C. Stoilov, Theory of Functions of a Complex Variable, IL, 1962.
\({}^{11}\) N. N. Meiman, ZhETF, 43, 2277 (1962).
\({}^{12}\) T. S. Wu, C. N. Yang, Phys. Rev., 137, B 708 (1965).
\({}^{13}\) F. Cerulus, A. Martin, Phys. Lett., 8, 70 (1964).

\({}^{*}\) For pions this relation contains only experimentally measurable quantities.