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.