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Rigorous absolute bounds for pion-pion scattering
Nuclear Physics B94 (1975) 311-336 © North-HoUand Publishing Company
RIGOROUS ABSOLUTE BOUNDS FOR PION-PION SCATTERING (III). Dispersion relations on algebraic manifolds and c o m p u t a t i o n o f bounds G. A U B E R S O N , L. E P E L E *, G. M A H O U X and F.R.A. SIMAO ** Service de Physique Thdorique, Centre d'Etudes Nucldaires de Saclay, BP n°2- 91190 Gif-sur-Yvette, France Received 1 May 1975
In this last paper of a series devoted to the derivation of absolute bounds for strong interactions, we calculate upper and lower bounds on the ~r0~r0 --' ~r01r0 amplitude _4 F(s,t,u) at the symmetry point s = t = u - 5. Use is made of dispersion relations on algebraic manifolds. We also rely on the solution of the extremum problem given in the first paper, and on the Hardy space techniques developed in the second one. The final 4 4 result is - 7 3 < F(5, ~, 4~) < 7. These bounds are rigorous consequences of axiomatic analyticity, as well as crossing and unitarity properties.
1. Introduction This paper is the third, and last one, o f a series devoted to the derivation o f rigorous absolute b o u n d s for pion-pion scattering. In the two previous papers [ 1 , 2 ] (hereafter d e n o t e d by (I) and (II)), we have prepared the way by solving various e x t r e m u m problems. Here, as a n n o u n c e d in the sect. 1 o f (I), we i n t r o d u c e dispersion relations on algebraic manifolds and c o m p l e t e our program. In sect. 2, we first switch f r o m the s,t,u Mandelstam variables to the c o m p l e t e l y symmetrical x , y Wanders variables [3] which are particularly suited to a study o f the rr07r0 --, rr07r0 amplitude; then we make choice o f linear and h y p e r b o l i c manifolds in the x , y space, and set up sufficient conditions for such manifolds to be contained in the axiomatic analyticity domain. In sect. 3, we follow the m e t h o d outlined at the end o f the i n t r o d u c t i o n o f (I): to begin with, we write a dispersion relation on the linear m a n i f o l d ; then by using the solution o f the e x t r e m u m problem considered in (I), this dispersion relation is translated into an inequality that leads us to another e x t r e m u m p r o b l e m on the h y p e r b o l i c manifold, o f the type solved in sect. 3 o f (II). Actually, this is n o t quite so because o f parasitic cuts that enter the game. The new e x t r e m u m p r o b l e m is solved in sect. 4. In sect. 5, we pro* Fellow of the Consejo Nacional des Investigaciones Cientfficas y T6cnicas. Present address: University of La Plata, La Plata, Argentina. ** Fellow of Conselho Nacional de Pesquisas. On leave of absence from IEN and UFJR, Brazil.
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G. A uberson et al. / Rigorous absolute bounds for 7r~rscattering Illl)
ceed to the numerical calculation of bounds. We concentrate on the upper and the lower bounds of the rt07r0 ~ 7r°~r0 amplitude at the symmetry point s = t = u = 4, and optimize them by varying the above mentioned manifolds. Finally, we conclude by discussing some possible extensions of this work.
2. Choice of algebraic manifolds Let F(s,t,u) be the 7T°lr0 -~ ~°Tr° scattering amplitude (s + t + u = 4). The normalization is fixed by the partial-wave expansion: F(s,t,u)=
(
leyden (2l+ 1 ) f l ( S ) P 1 1 + s
-4]
'
where the 3~(s) satisfy the unitarity condition: Im fl(s) >1 I~(s)l 2,
(s/> 4, l = 0,2 .... ).
(2.2)
F(s,t,u) is an analytic function of two complex variables in the "axiomatic" domain.
Let us denote by ~ this domain. Both F and M are invariant in all the permutations of the variables s, t and u. Therefore it is convenient to use, in place o f s and t, the symmetrical variables: x = st + tu + u s ,
y = stu.
(2.3)
Given x and y, the corresponding values o f s, t and u are the roots o f the third degree equation: Z3 - 4Z 2 + x Z - y = 0 .
(2.4)
The six-to-one mapping s,t ~ x , y has already been discussed in various places [ 3 - 5 ] . Let us however recall a few geometrical features of this mapping (see fig. 1): (1) The real s,t plane is mapped onto a part'-~ of the real x , y plane. This region c/~ is limited by a cubic curve Q, image of the lines s = t, t = u, u = s. (ii) From any point (x,y) inC?~, one can draw three real tangents to e , the slopes of which are equal to s, t, u. This remark provides us with a very simple and useful way of visualizing the inverse m~aping (x,y) -* (s, t, u). (iii) The cusp P0 (x = ~ , y = ~ ) o f e is the image of the symmetry point s = t = ~. The three physical regions map onto the shaded area 9 limited by ~ and the axis y = 0. (iv) The image ~ o f the physical cut ci = (s,t: s/> 4 or t/> 4 or s + t ~< 0 } is the set of points (x,y)where there passes a tangent to the cubic e w i t h real slope larger or equal to 4 (the equation o f such a tangent is given by eq. (2.4) with Z / > 4). The real sectionc5 r of this image is the region in the real x , y plane covered by those tangents, d r is limited partly by the line y = 4x, partly by e . Through the mapping (2.3), the amplitude F becomes a function F ( x , y ) =
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y
X
m
Fig. 1. The real x,y plane.
F(s,t,u). Due to the symmetry properties o f F , F is obviously a one-valued function wherever it is defined. In fact, the symmetry o f F(s,t,u) entails a stronger result: the function F ( x , y ) is analytic on the image ~ o f the analyticity domain a( o f F(s,t,u). That such a proposition must hold is intuitively quite clear *, and we shall omit the proof. The domain ¢t has the form ~ = c/) _ c5 where ~ is some (symmetric) domain in the s,t space. Hence, ~ = c-/) _ ~" where ~ is the image of c/). We now introduce (s,t,u) symmetrical algebraic manifolds. The simplest ones are linear in x and y : y = ~
+ t~.
(2.5)
However, they will not be sufficientfor our purpose, and we have also to consider quadratic manifolds of hyperbolic type: (y - ax - b) (y - a'x - b ' ) = c .
(2.6)
In eqs. (2.5) and (2.6), we restrict all coefficients to be real. In the following, we shall write dispersion relations on linear manifolds D of the form (2.5) and solve Szeg6-Meiman problems on hyperbolic manifolds H o f the form (2.6). To this end, it is necessary that the only singularities o f the amplitude on D and H be in the image o f the physical cuts. This means that D and H must be contained in the domain c-/) defined above. In the remaining o f this section, we * Let us stress its similarity with the Hall-Wightman theorem [6]. The group involved in this theorem is the Lorentz group whereas in the above proposition it is the symmetric group ~3. The finiteness of cJ3 makes the proof much easier in the latter case.
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give sufficient conditions to satisfy this requirement and find out the singularities on D and H. It is known from previous works [7,8] that c-/)contains domain c/)x of the form:
~ x = {s,t: I(s - X ) ( t - X ) ( u - X)I < K x } ,
(2.7)
where, for - 2 8 < )~ < 4, K x is given by: K x = ([8 + x/(4 _ )~) (16 - )0] 3 , C],
0~
(2.8)
Here C x is not given by a closed formula but has been computed by Martin (see table 1 of ref. [7]). From the relation: y - e~x - t3 = (s - o~) (t 7- °0 (u - oe) + a3 _ 4a2 _ / ~ ,
(2.9)
it immediately follows that if: la 3 - 4a 2 -/31 < K s ,
(2.10)
then D is contained in the domain ~ , image of the domain c/)~ defined in eq. (2.7). Therefore, eq. (2.10) is a sufficient condition for the manifold (2.5) to be in the axiomatic domain @. As for H, one sees that if there exist C a n d C ' such that: {x,y! [y - ax - bl < C } C ~ a ,
(2.11')
{x,y: IV - a'x - b'l < C ' } c ~ a ' ,
(2.11")
and
with C C ' > c, then H is contained in ~ . Let us show that eqs. (2.11) are fulfdled as soon as:
la3
-
4a 2 -
b[ +
c
a ,
la'3 - 4a '2 - b '[ + C' < K a, .
(2.12') (2.12")
Indeed, for any point in the domain { x , y : [y - ax - b[ < C}: ] ( s - a) ( t - a) (u - a)i ~< Iv - ax - bl + la 3 - 4a 2 - bl < C + ia 3 - 4a 2 - b[ , (2.13) and condition (2.12') obviously implies that this domain is in ~ a . The same argument holds for eq. (II.12"). Inequalities (2.12) with CC' > c (and - 2 8 < a, a' < 4) are the sufficient conditions for the manifold (2.6) to be in the axiomatic domain ~ , that will be used later on.
Our aim is to obtain bounds of F(x,y) at the symmetry point P0- Let us recall that the method proposed by Martin to get bounds at this point was, first to find
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bounds outside the small Mandelstam triangle (see sect. 1 in (I)), and then to exploit the convexity properties of the amplitude. One advantage of introducing manifolds non linear in (s,t,u) is that it enables us to reach directly the symmetry point, simply by using a straight line D and a hyperbola H that both go through P0. Fig. 2 exhibits the real points of such manifolds D and H. Notice that H also passes through the point Q(x = - 1 6 , y = - 6 4 ) , image of the point s = t = 4, and through the origin x = y = 0, where it is tangent to the axis y = 0. This particular choice corresponds to the hyperbolas that have given the best bounds in our calculations. In the following, for the sake of simplicity, we shall confine ourselves to this restricted class of hyperbolas, and shall not report on the calculations we actually did with a wider class. Assuming that conditions (2.10) and (2.12) are satisfied (this will be checked in sect. 5), it remains to find the singularities of the amplitude on the two manifolds D and H. It turns out that with the configuration of fig. 2, these singularities are located on real points (x,y). Indeed, any complex line LZ of eq. (II.4) with Z real ~ 4 intersects both manifolds D and H at real points. This is obvious for D. Concerning the hyperbola, we remark that for any Z ) 4, LZ intersects H at two real points, and these ones are the only intersection points. As a consequence, the singularities of~(x,y) in D and H lie on the intersection of these manifolds with the region @ defined previously. They correspond to cuts that are displayed in fig. 2. y
g,
Fig. 2. Configuration of the straight line D and of the hyperbola H. The cuts on these manifolds are drawn in heavy lines.
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3. Dispersion relation on manifolds and coupled extrema problem Before proceeding to the calculations announced in sect. 1, some geometrical preliminaries are necessary. Since the chosen linear manifold D goes through the point P0, its coefficients satisfy: /3 = r7-64 !6 a.
(3.1)
Similarly, since the chosen hyperbolic manifold H goes through the points P0, Q and the origin, and is tangent at this last point to the axisy = 0, its coefficients satisfy four relations which can be written as: 112 = 0 ,
27aa'-36(a+a')+
b= 2 (4-a)(a-9a)(28-9a') 27 a - a'
'
b' - 2 (4 - a') (4 - 9a') (28 - 9a) 27 a' - a '
(3.2)
c = bb'.
Thus, our manifolds depend on two (real) parameters, say a and a, which are respectively the slope of D and the slope of one of the asymptotes of H. The upper and lower bounds of the amplitude at the point P0 will be eventually optimized with respect to these parameters. It is convenient to describe the manifold H by means of a unicursal variable ~. Choosing ~ = y - ax - b (see fig. 3), we obtain the following parametric representation of H: X ~
a
y-.~'
a --a *
- a'~ - a'b + a b '
.
(3.3)
Besides the point P0, the manifolds D and H have another common point, P1. The coordinates ~0 and ~1 of P0 and P1 are readily found to be: ~0 ~1
_64
27
k~a_
(a-o
=--
---tr
Let us note also the ~O = - ( 6 4 -
b
(3.4)
" coordinate
16a + b ) .
~Q of the point Q defined above:
(3.s)
We now turn to writing d o w n a dispersion relation for F ( x , y ) on D. According to the considerations of sect. 2, F ( x , a x + [3) is analytic in the complex x-plane cut
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along [_0%/3/(4 - c0] , where the right extremity x =/3/(4 - a) of the cut is given by the intersection of D with the line y = 4x (see fig. 2). Furthermore, one easily finds that when Lx[ ~ 0% s "" x/Lx and t ~ a. Then, for values of a such that D is inside the domain c~, one deduces from well-known asymptotic bounds that I~(x, ax + 13)1 < const Ix[ 1-E. Thus, if(x, txx +/3) satisfies a once-subtracted dispersion relation. For reasons that will become clear later on, we subtract at the point P1. Writing the dispersion relation at P0, we get: Xl _ x0 t3/(4-a) 2irr f dx'
~(xl'Yl)-F(xo'Yo)-
_~
z ~ ( x ' , coc' +/3)
(3.6)
(x' - x l ) (X' - X o ) '
where (x0, Y0~ and (Xl, Yl) a ~ the coordinates of P0 andP1, respectively. The discontinuity A F ( x ' , ~ c ' +/3) = F ( x ' + iO, a ( x ' + iO) +/3) - F ( x ' - iO, a ( x ' - iO) +/3) on the cut is readily expressed in terms of the s-channel absorptive part A (s,t) of the amplitude. Here, s and t are two of the three roots of the equation: Z 3 - 4Z 2 + x ' Z - (ooc' +/3) = 0 .
(3.7)
According to the geometrical remark given in sect. 2 about the inverse mapping ( x , y ) ~ (s,t,u), when x ' ~ 4, which
we call s, and one and only one root between 0 and 4, which we call t. Moreover, s is a manifestly decreasing function o f x ' , so that: AF
', a x ' +/3) = F ( s - iO, t) - F ( s + i0, t) = -2/A(s,t) .
(3.8)
The next step consists in making use of the lower bounds on A (s,t) derived in (I). These bounds are functions of the modulus IF(s,t-)l of the amplitude at the same energy s but a smaller transfer T. Of course, we choose t h e p o i n t (s,Y) on the hyperbola H, and this leads us to look for the intersection with H of the line L s of eq. (2.4) where Z = s ~> 4. Only one of the two (real) intersection points of L s with H has a transfer F smaller than t (see fig. 3). Let ~ be its unicursal coordinate on H. The variables s, t, T and ~ are functions o f x ' . As ~ is the variable we are going to work with, we rather express s,t,T and x' in terms of it. First of all, s(~) and T(~) are solutions of the third degree equation: Z 3 - 4Z 2 + Z c/~ - ~ - b + b' a
-
a r
+ ab' - a'b - ac/~ - a'~ a - a'
= O,
(3.9)
which is obtained by inserting eqs. (3.3) into eq. (3.4). For ~ > - b , this equation has one and only one root ~> 4, which is s(~);T(~) is the largest of the two remaining roots. Next, x'(~) is given by eq. (3.7) with Z = s(~):
X'(~) - / 3 + 4S2(~) -- s3(~)
s~3-- ~
(3.10)
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As for t(~), it is the positive solution of the second degree equation obtained by replacing x, s and u in the first eq. (2.3) by x'(~), s(~) and 4 - s(~) - t, respectively:
t(~)=½(s(~)-4)[~/l+4 ~(~)(s(~)-4)-/3
-1].
(3.11)
(s(~) - 4)2(s (~) - a) Now, by switching from the variable x' to ~, eqs. (3.6) and (3.8) give:
F(xl'Yl)-TF(xo'Yo)=
XO-Xl ~ 7r
d~
(dx') A(s(~),t(~)) ~-~ (x,(~)_Xl)(X,(~)_Xo) .
(3.12) We remark that the derivative dx'/d~ is negative, so that the kernel (x 0 - Xl) (-dx'/d~)/(x'(~) - X l ) (x'(~) - x 0 ) is positive. This allows us to replace the (positive) absorptive part A (s(~), t(~)) by one of its lower bounds taken from (I), which have the form:
A(s([),t(~))>~m(~) lF(s(~),T(~))l e ,
(p ~> 2).
(3.13)
Here, with the normalization of the amplitude fixed by eqs. (2.1) and (2.2): m(~) = I_2 V s(~)
(Pp)-P ,
1
(3.14)
and Up, defined by eq. (2.17) of (I), is a known function of t(~)/(s(~) - 4), F(~)/ (s(~) - 4) and p. With the particular choice of hyperbolas we have made, T(~) is always negative, so that Up is always well defined [cf. discussion after eq. (3.6) of (I)]. Then: Y
/(~)
~,:o,'÷m~/
I
D
/
/
~"
Fig. 3. Visualizingthe variables ~, x', s, t and T.
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® ~Q
o
x
x
~o ~,-b
Fig. 4. Image of the cut manifold H in the ~ variable.
~ 1 f d~ w(t) i~(x(t),Y(t))l p , F(X l, Yl) - F(Xo, YO) >~ -~ -b where, after straightforward calculations, w(t) is found to be: x0 - x1
w (~) = m (t)
( X ' ( t ) -- Xo) ( X ' ( t ) -- X l )
I[(s-a)c/~ 2 + s - a ] [~+8~s-(4+3a)s2+2s3]l
According to the discussion at the end of sect. 2, the function F(x (~), y (t)) is analytic in the complex t-plane cut along [~O, 0] and [-b, oo] (see fig. 4). These two disconnected cuts are the image in t of the intersection of the two real branches of H with ~r' Next we map the cut t-plane onto the unit disk [zl < 1 in such a way that the cut [ - b , oo] is sent onto the circle [zJ = 1 and the point t0 onto z = 0: _ x/zb-
z - _x/_Zb~_t0 + x / Z ~
(3.17)
(the surd -x/Zb-S-t is defined in the ~-plane cut along [-b, oo] and is positive on [_0% - b ]). The function:
f(z) =-F(x (t), Y (t))
(3.18)
0 -1
-
1
Fig. 5. Image o f the cut manifold H in the z-variable.
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G. Auberson et al. /Rigorous absolute bounds for 7rTrscattering (111)
is analytic in the unit disk cut along [-/31, -/32] , where -/31 = Z(~Q) and -/32 = z(0) (see fig. 5). Furthermore:
F(xo,YO) = f(0),
F(Xl, Yl) = f ( z l ) ,
(3.19)
with z 1 = z (~1). Inequality (3.15) becomes: 7r
f(zx)-f(o)~> f --
dO p(O) If(ei°)IP ,
(3.20)
lr
where the (even and positive) weight function p(O) is given by:
• . lsin }01 o ( o ) = ( - b - ~0) cos3 ½0 w ( ~ ) ,
~ = - b + ( - b - GO) tg z ~ 0 .
(3.21)
On the other hand, the imaginary part o f f ( z +/13) is positive on the cut z E [-/31 , -/32]. Indeed, if ~ is the unicursal coordinate of a point moving on the cuts of H, and s is the largest corresponding Mandelstam variable, then applying once more the geometrical remark of sect. 2, one sees that d~/ds is positive. Moreover, a glance at figs. 4 and 5 shows that, when z is real, dz/d~ is positive, so that dz/ds > 0 when z E [-/31, -~2 ]. Consequently f(z + iO) = F(s ÷ iO, t), and Im f(z + iO) = A (s,t). Since, by construction, t is positive (and < 4) for all these points: Imf(z +/0)>/0,
z E [-/31,-/32].
(3.22)
Now we are in a position to formulate our last extremum problem: find the range of the point (f(0), f(z 1)) in the two-dimensional plane, knowing (i) the analyticity properties o f f ( z ) described above, (ii) the inequality (3.20), (iii) the positivity condition (3.22). This problem is similar to the one solved in sect. 3 of (II). It differs from it, firstly by the presence o f f ( z l ) - f ( 0 ) in place of 1 in the left-hand side of eq. (3.20) (a modification which is trivially dealt with), and secondly by the occurrence of a cut inside the uhit circle. A first method to solve our present problem would consist in reducingit to the one of sect. 3 of (II), and this is actually the reason for which we gave formulas (3.16) and (3.17) in (II). Here, we shall rather use an alternative method which we found to be easier, and which does not appeal to the algorithm described in appendix (Ae) of (II). That alternative method is developed in sect. 4. Before doing so, it is convenient to get rid of the function O(0) in eq. (3.20) by introducing, as in sect. 3a of (II), the outer function:
G(z)=exp
(
-p
1 i dO e iO __+z l o g p ( 0 ) ) . 2rr eiO z --
Tr
That log p (0) is integrable is easily checked. The function:
(3.23)
G. Auberson et al. / Rigorous absolute bounds for 7rlrscattering (III) h ( z ) = fG(z3)
321 (3.24)
is analytic in the unit disk cut along the interval I = [-/31, -/32], and its discontinuity across I has the same sign as the one o f f ( z ) . Then the inequality (3.20) and the positivity property (3.22)become:
f
dOlh(ei°)lP <~G(Zl) h(Zl) - G(0) h(0) 27r
(3.25)
--'/t
Im h (z + i0)/> 0 ,
z C I.
(3.26)
4. Solution of the coupled extrema problem We have first to define the space of functions h (z) where the extrema are to be found. Let H~be the class of functions h(z) such that: (i) h (z) is "real analytic" (h *(z) = h (z*)) in the unit disk cut along 1, (ii) h (z) has boundary values on I in the sense of distributions, denoted by h(x +-iO), and Im h(x + i0) = - I m h(x - iO) is a (real) measure lah(X),
1 (iii) d (z) -- h (z) - ~-
fdUh (x) x
-1 z
(4.1)
I belongs to the (real) Hardy space HP [2, 9]. Assumptions (i) and (ii) are clearly realized as consequences of axiomatic field theory. That the distribution Im h (x + i0) must be a measure is not a restriction since, according to eq. (3.26), we are really interested only in functions h (z) whose boundary values Im h (x + i0) (x E I ) are represented by a positive distribution, thus by a (positive) measure. As for assumption (iii), we first remark that it makes sense since the function d(z) is obviously analytic in the unit disk. The requirement that d(z) belongs to I-IP is stronger than simply assuming d(e i°) E I_P, and constitutes a technical hypothesis equivalent to the one already discussed in subsect. 2a of (II). Notice that, under these conditions, the integral appearing in eq. (3.25) always exists:
N(h) =-
E~ -~d°lh (ei°)i p
< oo
(4.2)
Actually d(e i°) C LP implies h (ei°) E LP since, as long as the interval I does not meet the unit circle, the function fI dtZh (x - z) -1 is analytic on Izl --- 1. The classH F is manifestly a (real) linear space. We equip it with a norm llhll defined by:
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Iih[l = [[dllHp + 17r
fdlUhl(X),
(4.3)
I
where [[dllHp is the usual norm in kiP:
[Id[lHp =
~-
The norm (4.3) has been chosen in such a way that the space Hp is in fact the direct sum of two Banach spaces, namely the Hardy space HP and the space of measures on the interval I, which is known to be a Banach space when endowed with the norm [[/ah [1 = fl d[/Jh I (x). It follows that Hp is itself a (real) Banach space *. Now our extremum problem can be formulated as follows. Refering to eqs. (3.25) and (3.26), we introduce the convex cone ~ of functions h(z) in HP for which the corresponding measure ~h(X) is positive, and the set cg defined by: c'ff. = (h E H P : N ( h ) p <~G(Zl) h(Zl) - G(0) h(0), h E ~ }.
(4.5)
Notice that c~ is convex. This is due to the fact that N ( h ) is a Lp norm, which implies that N(h)P is a convex function (for p >~ 1). The mappings h ~ h (0) and h -+ h(Zl) obviously define two continuous linear functionals on H/p denoted by A 0 and A t respectively: h(0) = A0(h),
h(Zl) = Al(h).
(4.6)
Let A be the continuous linear mapping H/p -->R 2 : h ( h ) = (h (0), h (z 1)).
(4.7)
The problem is to find the (convex) image A(Cg) i n n 2 of the set c-g C Hp. It will be solved by constructing explicitely all functions h (z) in cg which are mapped into the boundary 0A(ql). Such a procedure requires however that the boundary aA(C'g ) belongs to A(Cg), in order that the corresponding functions h (z) belong themselves to cg. This requirement is guaranteed by the following Proposition: The set A(Cg) is compact in 112. This important property is by no means obvious and its proof, which involves some technicalities, is reported in the appendix. We now introduce the subset cly of
cg: clP= {h @ H f : N(h)P = G(Zl) h(Zl) - G(0) h ( 0 ) , h E ~ } .
(4.8)
* Note that the quantity N(h) defined by eq. (4.2) is also a norm on Hf, which is not equivalent to Ilhql. Although N(h) is the quantity of interest in our problem, it is not a "good" norm because it does not make Hya complete space.
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323
Let us show that A(Cl~) = A(q/). It is sufficient to prove that any point in A(Cg) is the image of an element of cly. Given a point P E A(q/), there exists at least one function h E c/g such that A(h) = P. Then the quantity N(h)P - [ G ( Z l ) h (z 1) G(0) h (0)] is negative or zero. If it is zero, h E c)y. If it is strictly negative, let us consider the family of functions in H~: kx(z ) = h(z) + ~z(z - z 1) g(z) ,
(4.9)
where g(z) is any given function in HP. Then kx(z ) belongs to ~ and A(kx) = P for all real values of ~. Moreover N(kx)P - [G (z 1) kh(Zl ) - G (O)k h (0)] is obviously a continuous function of ~ which is unbounded from above. Hence this function necessarily vanishes for some value X0 of X, and kx0 E c~, which establishes our assertion. Thanks to this result, we can solve our problem by constructing all functions in cl~ which are mapped onto the boundary aA(C)Y). Actually, by using a variational method, we shall work out necessary conditions for a function in c,~ to map into aA(q~). Let h(z) be such a function. The most general variation o f h ( z ) in H~ has the form: 5h(z) = e g(z) +
. [duh(x)~(x) + dX(x)] ~
+ o(e),
(4.10)
where g E HP. According to the Lebesgue decomposition theorem, the discontinuity of 6h (z) on I has been written as the sum of a measure dX(x) singular with respect to d/Jh(x ) and a measure absolutely continuous with respect to d~h(x), which, according to the Radon-Nikodym theorem, is expressible in the form d~h(x ) 4~(x) where ~(x) E Ll(~uh). The function h(z) + 6h(z) belongs to the cone 5~ if and only if the measure X(x) is positive and 4~(x)/> - 1/e (x E l ) , with e > 0. It belongs to c,~ if furthermore:
[N(h)P] = G(Zl) ~h(Zl) - G(0)~h(0) .
(4.11)
To first order in e, this equation becomes:
p f
~d0 Ih(ei°)lp-2h*(e i°) 6h(e io) = G(Zl) 6h (Zl) - G(0) 6h (0).
(4.12)
Let us first consider a variation 6h(z) in HP, that is to say ~(x) - 0, 4~(x) - 0. We get from eqs. (4.10), (4.12) and Cauchy formula: dO ei° e -~f 2~ e i o ,- - Xi g ( e i °_) = ~ h ( x i )
i = 0 , 1 (x 0 = 0 , X 1 = Z l ) ,
(4.13)
Ir
ep f
~lh(ei°)[P -2 h* (ei°)g(ei°) = G(Zl) 6h(Zl) - G(O) 6h(O).
(4.14)
G. Auberso;: et aL / Rigorous absolute bounds for 7rwscattering [IIl)
324
The left-hand sides of these three equations define three continuous linear functionals on HP. If these functionals are linearly independent, given any 6h(0), 6h (z 1), there exists at least one function g(z) in HP satisfying eqs. (4.13) and (4.14), and the point (h (0), h (z 1)) cannot belong to the boundary OA(QY). Consequently a necessary condition for (h (0), h (z 1)) to be on 3A(C~) is that the three functionals be linearly dependent. This implies the existence of real s 0 and a 1 such that:
-~
-~ [
eio ei° _ z 1 + o~oG(O)] g(ei°) = O (4.15)
Plh(ei°)lP-2h*(ei°) - ~ l G ( Z l )
for all g E HP. In order that this condition be fulfilled, the bracket in the integrand must belong to the annihilator of HP, which is known to be the set of functions in Hq vanishing at the origin [10]. Thus,
p h (ei°)lp-2h*(e i°) - cqG(Zl)
ei0
+ a0G(0) = ei°v (ei°),
el° - Zl
o(z) E Hq . (4.16)
It turns out that this equation determines partly the structure of the function h (z). To see this, let us introduce the (unique) "real" outer function [2,9] R (z) defined by:
IR(ei°)l=lh(ei°)l ~(p-2),
R(x)>0
for
-l
(4.17)
Then eq. (4.16) can be rewritten as:
pR(eiO)h(eiO)_
1 Vffl_G(zl) R(e -io) L1 - zleio
O~oG(O) +e.iOu(e_iO)] ' -n<~O <~rt. (4.18)
From eq. (4.17), the function R (e i°) h (e i°) clearly belongs to L 2. Thus U(z) =R (z) h (z) is in H? since R (z) belongs to the Smirnov class N + [2,9]. This allows us to reconstruct U(z) from its boundary value on the unit circle and the interval I via the Cauchy formula:
pU(z)
1 = P f dUh(X) ~R ( x ) + f dO ei° nI -n 2rr eiO _ z R(e -iO)
× ,1F l Zl z + 1 f dz' 1 2in Iz'l=l (1 - z z ' ) R ( z ' )
-
z
F_ l_C(zo oC(O) ] LZ --z 1 z' +u(z')
(4.19)
For fixed z, Izi < 1, the last integrand belongs to L 2. Furthermore, when multiplied by z'(z' - Zl) , it is the boundary value of a function which is in N +, and thus
325
G. Auberson et al. / Rigorous absolute bounds for nTr scattering ([ti)
in H 2. Hence the Cauchy theorem applies again [11 ], and one gets: OtlG(Zl) pU(z) = Pn -If
diSh(X)
-
ot0G(0)
+R(Zl)(1--zzl)
(4.20)
R(O)
This equation shows that the function U ( z ) has an analytic continuation outside the unit circle, which, according to eq. (4.18), is given by P U \(zt l = R ~1
ZlI°tlG(Zl) t- z z _ aoG(O)+zv(z) ] ,
([z[ ~< 1).
(4.21)
Indeed, both sides of eq. (4.21), when multiplied by (z - z 1), belong to H 2 and coincide on [z[ = 1. Under these conditions, eqs. (4.13) and (4.14) have a solution if and only if 6h(0) and 6h(Zl) verify the relation: (1 - a l ) G(Zl) 6 h ( Z l ) - (1 - s0) G(0) 6h(0) = 0 .
(4.22)
This is nothing but the equation of a supporting line to the convex set A(°Y) (actually of a tangent to its boundary). It remains to determine the structure of the measure lah(x ). This is done by considering the full variation of h (z). Inserting eqs. (4.10) and (4.16) into eq. (4.12), and using Cauchy formula, we obtain: eg(xi) + e f
[d/ah(X ) q~(x) + d?,(x)] x -1 x i _ 6 h ( x i ) '
i=0,1
(4.23)
I e
alG(Zl)g(zl)
- aoG(O)g(O ) _
1 f[duh(x) ~(x) + d?~(x)] o(x)} 1
= G(Zl)6h(Zl)
- G(0)Sh(0).
(4.24)
Since we have already made full use of the variation o f h (z) due to the term eg(z), let us eliminate g(0) and g ( z l ) between the three eqs. (4.23) and (4.24): -
--Tr I
t- x - - z 1
X
+ o(x)
= (1 -- a l ) G(Zl) ~h(Zl) - (1 - s0) G(0) 5 h ( 0 ) .
(4.25)
With the help of eq. (4.21), this equation can be rewritten as:
e--P.f[dun(x)i~(x)+dX(x)] 1R(x)U(1) I
x
= (1 - a l ) G(Zl) 6 h ( Z l ) - (1 - s0) G(O) 6 h ( 0 ) .
(4.26)
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G. Auberson et al. / Rigorous absolute bounds for 7r~rscattering (III)
We recognize in the right-hand side the quantity which vanished in eq. (4.22). Now eq. (4.22) is the equation of a supporting line to the convex set A(qY) only if for all admissible variations 6h (that is to say h + 6h E c y ) , the points (h(0) + 6h(0), h (z i ) + 5h(Zl) ) lie on the same side of this line, or equivalently if (1 - cq) G(Zl) 6 h ( Z l ) (1 - o~0) G(0)6h(0) keeps a constant sign under the same conditions. This provides us with a second necessary condition for (h(0), h ( z l ) ) to be in 3A(Cl~): the left-hand side of eq. (4.26) must keep the same sign for all admissible functions O(x) and measures k(x) [q~(x) >t - l/e, k(x) positive measure singular with respect to/ah(X)]. Let us take first k(x) - 0. Noticing that the function ( I / z ) R (z) U ( 1 / z ) is analytic on I and that the sign of the admissible functions ~(x) is not constrained, we immediately deduce from the necessary condition just stated that the support of the measure/ah(X) is contained in the necessarily finite) set of zeros of U ( 1 / x ) on I (recall that R(x) > 0 on I). Thus, since/ah(X) is a positive measure, it must have the form: -
N d/an(x) = ~ /ai 6 (x - vi) dx , i=1
(4.27)
where/ai • 0 and v i E I, U ( 1 / v i ) = 0 (i = 1 . . . . . AT). Turning back to eq. (4.20), we now see that U ( z ) is a rational function which has (i) (N + 1) simple poles at z = v i (i = 1 ..... N ) and z = 1/z 1 ; (ii) N zeros at z = 1/v i (i = 1, ..., N), and (iii) behaves like a constant at infinity. Hence it has the form: N
o -- z 1 -- z v i U ( z ) = A 1 - z z l i=ll-I ~ ,
(4.28)
where A and o are two real parameters. Moreover, the positivity of the measure /ah(x) implies that the residues of U ( z ) at each pole z = v i are negative, and consequently that between two such successive poles, there must be an odd number of zeros. As U ( z ) has only one possible zero on l(at z = o), N is at most equal to 2. Finally, taking non-zero measures ~,(x) in eq. (4.26), we deduce from the second necessary condition that the sign of the function: N X - - Z 1 i=1 1 - x v i
(4.29)
must not change on I - - [-/31, -/32]. At this point, we have to consider separately the three casesN = 0,1 and 2. Case N ; 2: The condition on the sign of U ( 1 / x ) implies v 1 = -/31, v2 = -/32 and the posRivity of/ah(X ) implies --/31 < o < --/32. Case N - - 1: If U ( 1 / x ) has two simple zeros (1/o 4= Vl) , the condition on the sign of U ( 1 / x ) implies v 1 = -/31 or -/32, and 1/o ~ I. If U ( 1 / x ) has a double zero (1/.~ -- Vl) , then the pole of U ( z ) may stand anywhere on I, -/31 ~ v 1 ~ -/32.
G. A uberson et al. /Rigorous absolute bounds for rr~rscattering (III)
327
Case N = O: a may range from _oo to +oo. The structure of the function U(z) = R (z)h (z) is now completely fixed. In order to extract that of h (z), it is convenient first to construct R (z). From eqs. (4.28) and (4.17), noticing that the modulus o f the Blaschke product IIN 1 (z - vi)/(1 - zvi) is equal to 1 on the unit circle, we readily get in all cases: IR (ei°)i
IA
o -- e iO
1
1 - ei°zl
1 - 2/p (4.30)
This is sufficient to determine unambiguously the outer function R (z) (positive for - 1 < z < 1):
[I+ R(z) =
Finally:
h(z) =
0-2 nl-2/p JAf _--2~Zl 1 j ,
IFLAI 1-°z] 1-2/p /L 1 -zztJ
+_o~>l,
'
I ( 0 -- Z ~21p fi B + 1 -ZZl] i=l
tel< 1.
1 -
(4.31)
zvi ,+o>11,
z - vi N B 1 oz \21p o - z r[ 1-zvi ( ~ ) 1-oz i=l z _ v i
,
lel ~<1.
(4.32)
where B is a real normalization factor, the modulus of which is fixed by the condition N ( h ) P = G (z 1 )h (z 1 ) - G(O)h (0), and the sign o f which is controlled by the positivity requirement. The determination o f the quantities ( )21p is chosen to be the real positive one for - 1 < z < I. Once B has been calculated in terms o f the parameter e, eqs. (4.32) provide us with several one-parameter families of functions h ( z ) in c)y, and any point in 3A(OY) is the image (h(0), h ( z l ) ) of one o f these functions *. The explicit calculation o r b depends on the particular case (N = 0, 1,2) which is considered. As this final step is straightforward, we treat here only the case N = 1 and v 1 = -/32 (this case turns out to be eventually the relevant one for the determination o f the lower bound on h(0)]:
a--z 2/1) 1 +z[32 B(+I-ZZl)
z + ~2 "'
+o>~l,
h(z) =
B(1-oz)2/P \l-~z 1
o-z i---~z
1 + z/32 z+~2'
Iol~<1 .
(4.33)
* It is worth mentioning that in the simplest case N = 0, the functions h(z) given by eqs. (4.32) are holomorphic in the unit disk and belong in fact to HP. In this case, eqs. (4.32) are nothing but the known expressions for the extremal function in HP corresponding to rational kernels [9] [compare these equations e.g. with eqs. (H.2) and (H.7) of (II)]. The method developed in this section thus appears as an alternative way for deriving extremal functions which allows us to cope with more complicated situations.
328
G. A uberson et al. /Rigorous absolute bounds for nTr scattering (111)
The positivity condition imposes B ~< 0 when + o/> 1, and B(e +/32) ~< 0 when iol ~< 1. An elementary calculation gives:
N(h)P = [B[P
1 - 2 Z l o + 02 1
(4.34)
-z 2
Equating this quantity with G (z 1) h (z 1) - G (0)h (0), we get the expression of B: o~<-
(i) for
1
or-
~2'
1
~1 ~< o ~< - 1 ,
o r o / > --/32,
and K (o) <<.0, 2 B =- I -
1-Zl 1 -- 2 Z l O +
e2
K(cr) 11/(p-l)
(4.35)
(ii) for - 1 ~< o ~< --/3 2 and K(o) >~0,
l-z1
K(o)II/(p-1) .
1 - 2 Z l o + 02
(iii) the other values of a must be ignored. Here K(o) is the quantity: G(Zl)
K(o) =
[Io-z11~2/p 1+Zl/32
~1--~12 !
a ( 0 ) - [o[2/P ----,
zl + th
(1--Oz l~2[p O - Z 1 1 +Zl/32 G(Zl) k l _ z ~ / l _ o z I z1+/32
Iol~> 1
th o
G(O)~2,
Iol~
The formulas (4.33), (4.35) and (4.36) provide us with the parametric equations of a piece of curve: h(0) = X(e) and h(Zl) = Y(o). Repeating the same calculation in all other cases discussed above, we obtain a set of curves which must contain all boundary points of the convex A(qY), and possibly interior points. Finally, the set A(O/) = A(q~) we were looking for is simply the convex hull of all these points. This completes the solution of our extremum problem. 5. Numerical calculations and concluding remarks The computation of the upper and lower bounds on f(0) = F(~-, 4, 4) now proceeds as follows: (i) Choice of the power p (~> 2), of the hyperbola H and of the straight line D. With the restricted class of hyperbolas considered here (a one-parameter family), we have three parameters at hand: p, a and a.
G. Auberson et al. / Rigorous absolute bounds for lrzr scattering (III)
329
(ii) Calculation of G(0) and G(Zl) , by using eqs. of sect. 3. There, we need the function Vp def'med by eq. (2.17) of (I). The computer program set up in (I) has provided us with it. (iii) Calculation of the two-dimensional region A(ql) from eqs. (4.33), (4.35) and (4.36) in the case N = 1, v 1 = -/32, and from similar equations in the other cases. Then, upper and lower bounds on F ( 4, 4 , 4 ) are given by: G(0) min h(0)~
(5.1)
(iv) Optimization of each of these bounds with respect to the parameters p, a and {~. (v) Check that the optimal curves D and H are inside the axiomatic domain of analyticity, by means of inequalities (2.10) and (2.12). The calculations have been made on a computer IBM 360. The best bounds obtained are: 4 4 - 7 3 ~
(5.2)
The corresponding values o f the parameters p, a and a are, for the maximum: p = 2,
a = 2.39,
a = 0.548,
(5.3)
and for the minimum: p = 3.60,
a = 2.38,
~ = 0.448.
(5.4)
Fig. 6 exhibits, as an example, the convex region A(C//) that corresponds to the minimum, conveniently rescaled so as to give directly (f(0), f ( z 1)). In that case, the boundary of A(Q/) is made of two pieces; on the first one the function h ( z ) has no singularity on I (N = 0), on the other one it has one pole (N = 1) at z = -/32; notice also that there are no points corresponding t o N = 2, and to N = 1 with a double zero of U ( l [ x ) . The bounds (5.2) have to be compared with those derived by previous authors. Using the method of Lukaszuk and Martin [12], we have found - 1 5 9 ~< F ( 4, 4], ~3) ~< 16. With the improvement suggested by Healy [13], one gets - 1 4 9 ~
330
G. Auberson et al. / Rigourous absolute bounds for rr~rscattering (III) (zw)
- 7 ~
1o
~ tf(o)
Fig. 6. The r e g i o n A ( q t ) f o r the values of the parameters 7, a and c~giving the best lower bound at the symmetry point. The curve A corresponds to extremal functions without singularity on the interval I, the curve B to functions with a pole at z = -/31, and the curve C to functions with a pole at z = -~32.
would be the application of our techniques to the derivation of semi-phenomenological bounds *, already closer to the experimental data, in which case an improvement by a factor say, of the order of two, would be quite significant. Concerning our initial aim of including as much axiomatic information as possible, an overall look at the present derivation still reveals a noticeable shortcoming. Actually, by imposing the positivity of the discontinuity o f f ( z ) along the cut I, only _apartial use of the unitarity condition has been made. As a result, the extremal function corresponding to the minimum displays an unpleasant pole at z = -~32. The occurrence of this pole is responsible for the large value of the lower bound as compared to the upper one, as it can be observed on fig. 6: the region limited by the curve A, which is precisely the range of 0r(0), f(z)) when f(z) has no singularity on 1, is only a small fraction of A(QL). Therefore, one finds here still a possibility of improving the situation. One way to do it would lead us, after changing the manifolds D and H, to consider Hardy spaces of analytic functions in an annulus (15). Another way, which would allow us to keep the same manifolds, leads to the construction of a new convex set smaller than ~2~. Whether such further refinements would significantly strengthen our bounds or not, is an open question. In that respect, it would be quite interesting to approach the bounds from the other side, that is to say to construct symmetrical amplitudes as large as possible, satisfying analyticity and inelastic unitarity. Even without demanding elastic unitarity for 4 ~< s ~< 16, this is already a difficult task. Let us make a last remark. Although in our method a rather extensive use of analyticity has been made as compared to previous works, still one variable dispersion relations do not turn the analyticity properties of the amplitude to the best account. Therefore, it would be desirable to have a two-variable integral representa* Like those investigated by Chung and Vinh Mau (private communication), where the phenomenological input is the D-wave scattering length.
G. A uberson et al. /Rigorous absolute bounds for nTr scattering (III)
3 31
tion at our disposal, and to state the extremum problems in a suitable space of twovariable analytic functions. This would be indeed quite a different approach for deriving bounds, which would possibly lead to further progress.
Two of us (L.E. and F.S.) are grateful to C. de Dominicis for the hospitality extended to them-at Saclay.
Appendix P r o o f o f the proposition o f sect. 4
In order to establish the compactness of A(QZ) in R 2, where the set 9Z E H/p and the continuous linear mapping A of H/p into R 2 are defined by eqs. (4.5) - (4.7), we cannot rely on the usual weak compactness argument [16], because the Banach space H/p is not reflexive. Actually, according to eqs. (4.1) and (4.3), H~ is isomorphic to the direct sum: Hp= Hp mc/~ ,
(A.1)
where c-fig, the Banach space of (real) measures over the closed interval 1, is known to be non-reflexive. Therefore, in place of the weak topology of H/p , we shall use the weak-* topology, which turns out to be sufficient for our purpose, due to the particular form of our functionals A0 and A t . To this end, we have to consider the Banach space: X = (HP) * e e ,
(A.2)
where Q is the (non-reflexive!) Banach space of (real) continuous functions on I, and (HP)* is the dual of liP. Then, since ~ * = ~ and (liP)** = HP (lip is reflexive), one has:
H/~=X*,
(A.3)
and the weak-, topology on H/p is the topology o(H/p, X) [17]. It is shown below that: (i) the functionals A 0 and A 1 belong to X, (ii) the set 9/ is weak-, compact. Then from (i) the mapping A is obviously continuous on H/p in the weak-, topology. This, together with ii), entails the compactness of A(QZ) i n n 2 and achieves the proof of the proposition. (i) Ai E X, (i = 0,1). Indeed, for any h E H/p, eqs. (4.1) and (4.6) give: lah(X) x - X~---i"'
Ai(h) = d ( x i ) + 1
(X0 = 0, X l = Z l ) ,
(A.4)
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G. Auberson et al. /Rigorous absolute bounds for 7rrrscattering (III]
where the functionals d -~ d ( x i ) belong to (HP)* (since d E liP) and the functionals
,h --l f d,h(X)(X- Xi)-1 I belong to C since the Cauchy kernels (x - x i ) - I are continuous o n / . (ii) -'-// is weak-, compact. Recall that the convex set 9 / i s defined by: cg = {h E H/P: N ( h ) P <~ A ( h ) , h ~
),
(A.5)
where A E X is the continuous linear functional:
IX(h) = G(z l ) Al (h ) -- G (O) £x0(h ) .
(A.6)
Let us show first that q/ is bounded and weak-, closed. (a) q/ is bounded. We need the following lemma: L e m m a 1: For any function h(z) C H~ , N ( h ) = lim rt 1
Ff
dO h(reiO)l p I1/P LJ ~ -it
Proof.' Given h (z) E H/p , let us define the fnnctions hr(z ) = h (rz), where 1 -/31 < r ~< 1. Then hr(z ) ~ H~r with 1r = [-/~l/r, -/~2/r], and we can write:
hr(z) = dr(z) + ar(z) ,
(A.7)
where dr(z ) E HP and
ar(z ) = 1 f d P h ( X ) x I
(A.8)
Now dr(e i°) converges to d l ( e i°) in the I_P norm when r t 1 [this is a well-known property of HP, see e.g. theorem A.1 of (II)]. Similarly, ar(eiO)~ al(e io) in the LP norm since:
tar(eiO)_al(eiO)l
x
1
- re io
_
1
x - e io
< const (1 - r ) ,
(A.9)
uniformly in 0. Thus hr(e i°) ~ h(e i° ) in the I.P norm, which implies:
lira N ( h r ) = N ( h ) , rtl
q.e.d.
(A. 10)
Now, from the Cauchy theorem, one can write for any h E Hp and I -/31 < r <1:
G. Auberson et al. /Rigorous absolute bounds for lrlr scattering (111)
1
2Br f dzh(z)=-~ Izl=r
1
fdP.h(X ).
333
(A.11)
I
When moreover h E 5~, the measure
lah(X) is positive,
and one deduces:
II/.thll = f dlUhl(X) = ~ I f ~-~nr dO ei°h(rei°) < ' l r l / 1
~ ih(reiO)lP 1 1/p <,TrN(h),
-~r
(A.12)
where we have used successively the H61der inequality and lemma 1. On the other hand, we get from eq. (4.1):
Id(ei°)l <~Ih (e/°)[
+ 7r(1 1- 131) II/ahII
a.e.,
(A.13)
and by applying Minkowski inequality in Lp : IldllHp
<<,N(h) + zr(1
1 -/31) IIt~hll.
(m.14)
Inserting eqs. (A.14) and (A.12) into eq. (4.3), one obtains: 3 - 2/31 IlhlI -~< 1 _/31 N(h),
(hE~).
(A.15)
Finally, eq. (A.5) gives, for any h E QZ :
N(h)P <~II/Xllxllhll,
(A.16)
and, together with eq. (A.15):
{ 3 - 2/31 )p/(p -1) Ilhl[ ~< \ 1 - ~
1
l/(p-1) IIAIIx
,
(A.17)
which establishes the boundedness of c2g.* (/3) q/ is weak-, closed. Given any sequence {h n } in 9Z converging to h E Hp in the weak-, topology, one has to show that h E 9/. First of all, the weak-, convergence h n ~ h implies the weak-, convergence t~hn -+ t~h . Thus:
lirn fdUhn(X) 4fix) = f dUh(X) ¢(x), n-'~ °° I
,¢c~E e
(hAS)
I
Since h n E ~ for any n, fldt~hn(X) ¢(x) is non-negative for all positive functions in C, and eq. (A.18) shows that the same is true for fl d/ah(X) ¢(x). This means that the measure lah(X) is positive and that h E ~ .
¢(x)
* Incidentally, let us notice that eq. (A.17) may be used to derive at once rough (and in fact very bad) upper bounds for Ih(0)l and Ih(z1 )1.
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G. Auberson et al. / Rigorous absolute bounds for 7rlrscattering {III)
It remains to prove that N ( h ) P <~ A(h). We shall use the following lemma: L e m m a 2: Let (gn ~ -+ g be a weakly convergent sequence in a reflexive Banach space. Then I[g[[ ~< lira inf I~nlln --~. o a
The proof stems from the weak compactness of the unit ball. It is a simple exercise which is left to the reader. Let us rewrite the difference N ( h ) P - A ( h ) in the form: N ( h ) P - A(h) = N ( h ) p - N ( h n ) P + [N(hn)P - A(hn) ] + [A(hn) - A(h)] . (A.19)
Then, since h n E Q.l. : N ( h n ) P - A(hn) ~< 0,
~¢n,
(A.20)
and since A @ X: lim A(hn) = A ( h ) .
(A.21)
n--~
It follows that: N ( h ) P - A(h) <<.N(h)P - lira sup N ( h n ) P .
(A.22)
n ..--~ o o
Next, decomposing hn(z ) according to eq. (4.1): (A.23)
hn(z ) = dn(z ) + an(Z),
we see from eq. (A.18) that: an (eiO) ==1 f dl~hn (x) (x - e i° ) - 1. 1
(A.24)
converges pointwise to a(ei°) =--~rl fdUh(X ) (x - ei°) -1 , I
(-zr ~< 0 ~< zr).
Moreover, ( a n ( z ) ) is a uniformly bounded sequence of holomorphic functions in any compact Q c C which does not intersect I. Indeed, from eq. (A.24) and the boundedness of q£ : fan(z)l <<.Z IlUhnll ~
(z E Q ) ,
(A.25)
where the constants A and B are independant of n and z in Q.* Then, the Vitali theorem [19] tells us that:
* The fact that the sequence {ll~hnll ) is bounded is also a dkect consequence of the principle of uniform boundedness [ 18 ].
G. Auberson et al. / Rigorous absolute bounds ]or irlr scattering {III) lim an(e io ) = a(e i° ) uniformly on [-Tr, 7r] .
335
(A.26)
On the other hand, the weak-, convergence of h n to h obviously implies the weak convergence of dn(e i°) to d(e i°) in LP:
lira dn(e io ) = d(e i° ) weakly in L p .
(A.27)
n.-~
From eqs. (A.23), (A.26) and (A.27), we deduce that:
lira hn(e io) = h(e iO) weakly in L p .
(A.28)
n--~
Noticing that N ( h ) is nothing but the norm o f h in LP and applying the Lemma 2 to the (reflexive) Banach space L p, one gets:
N ( h ) <<,lira i n f N ( h n ) .
(A.29)
n.-..>, e,o
Returning to eq. (A.22), we are led to the desired result:
N ( h ) P <~ A ( h ) .
(A.30)
Finally, to derive the weak-, compactness of 9 / f r o m (a) and (/3), it is sufficient to remark that q / is contained in some ball [Jhlt ~
References [1] G. Auberson, L. Epele, G. Mahoux and F.R.A. Sima'o, Nucl. Phys. B73 (1974) 314. [2] G. Auberson, L. Epele, G. Mahoux and F.R.A. Sim$o, Saclay preprint DPh-T/74-76, to appear in Ann. Inst. Henri Poincar& [3] G. Wanders, Helv. Phys. Acta 39 (1966) 228. [4] G. Mahoux, S.M. Roy and G. Wanders, Nucl. Phys. B70 (1974) 297. [5] G. Mahoux, Proc. of the 2rid Int. Winter Meeting on fundamental physics, Formigal 1974 (Instituto de Estudios Nucleares, Madrid). [6] D. Hall and A.S. Wightman, Mat. Fys. Medd. Dan. Vid. Selsk, 31 no. 5 (1957). [7] A. Martin, Nuovo Cimento 44 (1966) 1219. [8] G. Auberson and N.N. Khuri, Phys. Rev. D6 (1972) 2953. [9] P.L. Duren, Theory of HP spaces (Academic Press, New-York and London, 1970). [10] See ref. [9], sect. 7.2. [11] See ref. [2], theorem A.5, [12] L. Lukaszuk and A. Martin, Nuovo Cimento 52A (1967) 122. [13] J.B. Healy, Phys. Rev. D8 (1973) 1904. [14] B. Bonnier and R. Vinh Mau, Phys. Rev. 165 (1968) 1923. [15] D. Sarason, Mem. Amer. Math. Soc., No 56 (1965). [16] See ref. [1], sect. 3, and ref. [2], appendix C.
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G. Auberson et al. / Rigorous absolute bounds for lrrr scattering (111)
[17] M. Reed and B. Simon, Method of modern mathematical physics, I: Functional analysis (Academic Press, New-York and London, 1972), sect. IV.5. [18] See ref. [17], sect. 1II.5. [ 19 ] A.I. Markushevich, Theory of functions of a complex variable, vol. I (Prentice-Hall, Englewood Cliffs, N.J., 1965), sect. 86.
RIGOROUS ABSOLUTE BOUNDS FOR PION-PION SCATTERING (III). Dispersion relations on algebraic manifolds and c o m p u t a t i o n o f bounds G. A U B E R S O N , L. E P E L E *, G. M A H O U X and F.R.A. SIMAO ** Service de Physique Thdorique, Centre d'Etudes Nucldaires de Saclay, BP n°2- 91190 Gif-sur-Yvette, France Received 1 May 1975
In this last paper of a series devoted to the derivation of absolute bounds for strong interactions, we calculate upper and lower bounds on the ~r0~r0 --' ~r01r0 amplitude _4 F(s,t,u) at the symmetry point s = t = u - 5. Use is made of dispersion relations on algebraic manifolds. We also rely on the solution of the extremum problem given in the first paper, and on the Hardy space techniques developed in the second one. The final 4 4 result is - 7 3 < F(5, ~, 4~) < 7. These bounds are rigorous consequences of axiomatic analyticity, as well as crossing and unitarity properties.
1. Introduction This paper is the third, and last one, o f a series devoted to the derivation o f rigorous absolute b o u n d s for pion-pion scattering. In the two previous papers [ 1 , 2 ] (hereafter d e n o t e d by (I) and (II)), we have prepared the way by solving various e x t r e m u m problems. Here, as a n n o u n c e d in the sect. 1 o f (I), we i n t r o d u c e dispersion relations on algebraic manifolds and c o m p l e t e our program. In sect. 2, we first switch f r o m the s,t,u Mandelstam variables to the c o m p l e t e l y symmetrical x , y Wanders variables [3] which are particularly suited to a study o f the rr07r0 --, rr07r0 amplitude; then we make choice o f linear and h y p e r b o l i c manifolds in the x , y space, and set up sufficient conditions for such manifolds to be contained in the axiomatic analyticity domain. In sect. 3, we follow the m e t h o d outlined at the end o f the i n t r o d u c t i o n o f (I): to begin with, we write a dispersion relation on the linear m a n i f o l d ; then by using the solution o f the e x t r e m u m problem considered in (I), this dispersion relation is translated into an inequality that leads us to another e x t r e m u m p r o b l e m on the h y p e r b o l i c manifold, o f the type solved in sect. 3 o f (II). Actually, this is n o t quite so because o f parasitic cuts that enter the game. The new e x t r e m u m p r o b l e m is solved in sect. 4. In sect. 5, we pro* Fellow of the Consejo Nacional des Investigaciones Cientfficas y T6cnicas. Present address: University of La Plata, La Plata, Argentina. ** Fellow of Conselho Nacional de Pesquisas. On leave of absence from IEN and UFJR, Brazil.
312
G. A uberson et al. / Rigorous absolute bounds for 7r~rscattering Illl)
ceed to the numerical calculation of bounds. We concentrate on the upper and the lower bounds of the rt07r0 ~ 7r°~r0 amplitude at the symmetry point s = t = u = 4, and optimize them by varying the above mentioned manifolds. Finally, we conclude by discussing some possible extensions of this work.
2. Choice of algebraic manifolds Let F(s,t,u) be the 7T°lr0 -~ ~°Tr° scattering amplitude (s + t + u = 4). The normalization is fixed by the partial-wave expansion: F(s,t,u)=
(
leyden (2l+ 1 ) f l ( S ) P 1 1 + s
-4]
'
where the 3~(s) satisfy the unitarity condition: Im fl(s) >1 I~(s)l 2,
(s/> 4, l = 0,2 .... ).
(2.2)
F(s,t,u) is an analytic function of two complex variables in the "axiomatic" domain.
Let us denote by ~ this domain. Both F and M are invariant in all the permutations of the variables s, t and u. Therefore it is convenient to use, in place o f s and t, the symmetrical variables: x = st + tu + u s ,
y = stu.
(2.3)
Given x and y, the corresponding values o f s, t and u are the roots o f the third degree equation: Z3 - 4Z 2 + x Z - y = 0 .
(2.4)
The six-to-one mapping s,t ~ x , y has already been discussed in various places [ 3 - 5 ] . Let us however recall a few geometrical features of this mapping (see fig. 1): (1) The real s,t plane is mapped onto a part'-~ of the real x , y plane. This region c/~ is limited by a cubic curve Q, image of the lines s = t, t = u, u = s. (ii) From any point (x,y) inC?~, one can draw three real tangents to e , the slopes of which are equal to s, t, u. This remark provides us with a very simple and useful way of visualizing the inverse m~aping (x,y) -* (s, t, u). (iii) The cusp P0 (x = ~ , y = ~ ) o f e is the image of the symmetry point s = t = ~. The three physical regions map onto the shaded area 9 limited by ~ and the axis y = 0. (iv) The image ~ o f the physical cut ci = (s,t: s/> 4 or t/> 4 or s + t ~< 0 } is the set of points (x,y)where there passes a tangent to the cubic e w i t h real slope larger or equal to 4 (the equation o f such a tangent is given by eq. (2.4) with Z / > 4). The real sectionc5 r of this image is the region in the real x , y plane covered by those tangents, d r is limited partly by the line y = 4x, partly by e . Through the mapping (2.3), the amplitude F becomes a function F ( x , y ) =
G. A uberson et al. / Rigorous absolute bounds for ~Ir scattering (III]
313
y
X
m
Fig. 1. The real x,y plane.
F(s,t,u). Due to the symmetry properties o f F , F is obviously a one-valued function wherever it is defined. In fact, the symmetry o f F(s,t,u) entails a stronger result: the function F ( x , y ) is analytic on the image ~ o f the analyticity domain a( o f F(s,t,u). That such a proposition must hold is intuitively quite clear *, and we shall omit the proof. The domain ¢t has the form ~ = c/) _ c5 where ~ is some (symmetric) domain in the s,t space. Hence, ~ = c-/) _ ~" where ~ is the image of c/). We now introduce (s,t,u) symmetrical algebraic manifolds. The simplest ones are linear in x and y : y = ~
+ t~.
(2.5)
However, they will not be sufficientfor our purpose, and we have also to consider quadratic manifolds of hyperbolic type: (y - ax - b) (y - a'x - b ' ) = c .
(2.6)
In eqs. (2.5) and (2.6), we restrict all coefficients to be real. In the following, we shall write dispersion relations on linear manifolds D of the form (2.5) and solve Szeg6-Meiman problems on hyperbolic manifolds H o f the form (2.6). To this end, it is necessary that the only singularities o f the amplitude on D and H be in the image o f the physical cuts. This means that D and H must be contained in the domain c-/) defined above. In the remaining o f this section, we * Let us stress its similarity with the Hall-Wightman theorem [6]. The group involved in this theorem is the Lorentz group whereas in the above proposition it is the symmetric group ~3. The finiteness of cJ3 makes the proof much easier in the latter case.
G. Auberson et al. / Rigorous absolute bounds for mt scattering (III)
314
give sufficient conditions to satisfy this requirement and find out the singularities on D and H. It is known from previous works [7,8] that c-/)contains domain c/)x of the form:
~ x = {s,t: I(s - X ) ( t - X ) ( u - X)I < K x } ,
(2.7)
where, for - 2 8 < )~ < 4, K x is given by: K x = ([8 + x/(4 _ )~) (16 - )0] 3 , C],
0~
(2.8)
Here C x is not given by a closed formula but has been computed by Martin (see table 1 of ref. [7]). From the relation: y - e~x - t3 = (s - o~) (t 7- °0 (u - oe) + a3 _ 4a2 _ / ~ ,
(2.9)
it immediately follows that if: la 3 - 4a 2 -/31 < K s ,
(2.10)
then D is contained in the domain ~ , image of the domain c/)~ defined in eq. (2.7). Therefore, eq. (2.10) is a sufficient condition for the manifold (2.5) to be in the axiomatic domain @. As for H, one sees that if there exist C a n d C ' such that: {x,y! [y - ax - bl < C } C ~ a ,
(2.11')
{x,y: IV - a'x - b'l < C ' } c ~ a ' ,
(2.11")
and
with C C ' > c, then H is contained in ~ . Let us show that eqs. (2.11) are fulfdled as soon as:
la3
-
4a 2 -
b[ +
c
a ,
la'3 - 4a '2 - b '[ + C' < K a, .
(2.12') (2.12")
Indeed, for any point in the domain { x , y : [y - ax - b[ < C}: ] ( s - a) ( t - a) (u - a)i ~< Iv - ax - bl + la 3 - 4a 2 - bl < C + ia 3 - 4a 2 - b[ , (2.13) and condition (2.12') obviously implies that this domain is in ~ a . The same argument holds for eq. (II.12"). Inequalities (2.12) with CC' > c (and - 2 8 < a, a' < 4) are the sufficient conditions for the manifold (2.6) to be in the axiomatic domain ~ , that will be used later on.
Our aim is to obtain bounds of F(x,y) at the symmetry point P0- Let us recall that the method proposed by Martin to get bounds at this point was, first to find
G. Auberson et al. /Rigorous absolute bounds for ~rlr scattering (Ill)
315
bounds outside the small Mandelstam triangle (see sect. 1 in (I)), and then to exploit the convexity properties of the amplitude. One advantage of introducing manifolds non linear in (s,t,u) is that it enables us to reach directly the symmetry point, simply by using a straight line D and a hyperbola H that both go through P0. Fig. 2 exhibits the real points of such manifolds D and H. Notice that H also passes through the point Q(x = - 1 6 , y = - 6 4 ) , image of the point s = t = 4, and through the origin x = y = 0, where it is tangent to the axis y = 0. This particular choice corresponds to the hyperbolas that have given the best bounds in our calculations. In the following, for the sake of simplicity, we shall confine ourselves to this restricted class of hyperbolas, and shall not report on the calculations we actually did with a wider class. Assuming that conditions (2.10) and (2.12) are satisfied (this will be checked in sect. 5), it remains to find the singularities of the amplitude on the two manifolds D and H. It turns out that with the configuration of fig. 2, these singularities are located on real points (x,y). Indeed, any complex line LZ of eq. (II.4) with Z real ~ 4 intersects both manifolds D and H at real points. This is obvious for D. Concerning the hyperbola, we remark that for any Z ) 4, LZ intersects H at two real points, and these ones are the only intersection points. As a consequence, the singularities of~(x,y) in D and H lie on the intersection of these manifolds with the region @ defined previously. They correspond to cuts that are displayed in fig. 2. y
g,
Fig. 2. Configuration of the straight line D and of the hyperbola H. The cuts on these manifolds are drawn in heavy lines.
G. Auberson et al. /Rigorous absolute bounds for lrTrscattering (III)
316
3. Dispersion relation on manifolds and coupled extrema problem Before proceeding to the calculations announced in sect. 1, some geometrical preliminaries are necessary. Since the chosen linear manifold D goes through the point P0, its coefficients satisfy: /3 = r7-64 !6 a.
(3.1)
Similarly, since the chosen hyperbolic manifold H goes through the points P0, Q and the origin, and is tangent at this last point to the axisy = 0, its coefficients satisfy four relations which can be written as: 112 = 0 ,
27aa'-36(a+a')+
b= 2 (4-a)(a-9a)(28-9a') 27 a - a'
'
b' - 2 (4 - a') (4 - 9a') (28 - 9a) 27 a' - a '
(3.2)
c = bb'.
Thus, our manifolds depend on two (real) parameters, say a and a, which are respectively the slope of D and the slope of one of the asymptotes of H. The upper and lower bounds of the amplitude at the point P0 will be eventually optimized with respect to these parameters. It is convenient to describe the manifold H by means of a unicursal variable ~. Choosing ~ = y - ax - b (see fig. 3), we obtain the following parametric representation of H: X ~
a
y-.~'
a --a *
- a'~ - a'b + a b '
.
(3.3)
Besides the point P0, the manifolds D and H have another common point, P1. The coordinates ~0 and ~1 of P0 and P1 are readily found to be: ~0 ~1
_64
27
k~a_
(a-o
=--
---tr
Let us note also the ~O = - ( 6 4 -
b
(3.4)
" coordinate
16a + b ) .
~Q of the point Q defined above:
(3.s)
We now turn to writing d o w n a dispersion relation for F ( x , y ) on D. According to the considerations of sect. 2, F ( x , a x + [3) is analytic in the complex x-plane cut
G. Auberson et aL / Rigorous absolute bounds for 7rn scattering (III)
317
along [_0%/3/(4 - c0] , where the right extremity x =/3/(4 - a) of the cut is given by the intersection of D with the line y = 4x (see fig. 2). Furthermore, one easily finds that when Lx[ ~ 0% s "" x/Lx and t ~ a. Then, for values of a such that D is inside the domain c~, one deduces from well-known asymptotic bounds that I~(x, ax + 13)1 < const Ix[ 1-E. Thus, if(x, txx +/3) satisfies a once-subtracted dispersion relation. For reasons that will become clear later on, we subtract at the point P1. Writing the dispersion relation at P0, we get: Xl _ x0 t3/(4-a) 2irr f dx'
~(xl'Yl)-F(xo'Yo)-
_~
z ~ ( x ' , coc' +/3)
(3.6)
(x' - x l ) (X' - X o ) '
where (x0, Y0~ and (Xl, Yl) a ~ the coordinates of P0 andP1, respectively. The discontinuity A F ( x ' , ~ c ' +/3) = F ( x ' + iO, a ( x ' + iO) +/3) - F ( x ' - iO, a ( x ' - iO) +/3) on the cut is readily expressed in terms of the s-channel absorptive part A (s,t) of the amplitude. Here, s and t are two of the three roots of the equation: Z 3 - 4Z 2 + x ' Z - (ooc' +/3) = 0 .
(3.7)
According to the geometrical remark given in sect. 2 about the inverse mapping ( x , y ) ~ (s,t,u), when x ' ~ 4, which
we call s, and one and only one root between 0 and 4, which we call t. Moreover, s is a manifestly decreasing function o f x ' , so that: AF
', a x ' +/3) = F ( s - iO, t) - F ( s + i0, t) = -2/A(s,t) .
(3.8)
The next step consists in making use of the lower bounds on A (s,t) derived in (I). These bounds are functions of the modulus IF(s,t-)l of the amplitude at the same energy s but a smaller transfer T. Of course, we choose t h e p o i n t (s,Y) on the hyperbola H, and this leads us to look for the intersection with H of the line L s of eq. (2.4) where Z = s ~> 4. Only one of the two (real) intersection points of L s with H has a transfer F smaller than t (see fig. 3). Let ~ be its unicursal coordinate on H. The variables s, t, T and ~ are functions o f x ' . As ~ is the variable we are going to work with, we rather express s,t,T and x' in terms of it. First of all, s(~) and T(~) are solutions of the third degree equation: Z 3 - 4Z 2 + Z c/~ - ~ - b + b' a
-
a r
+ ab' - a'b - ac/~ - a'~ a - a'
= O,
(3.9)
which is obtained by inserting eqs. (3.3) into eq. (3.4). For ~ > - b , this equation has one and only one root ~> 4, which is s(~);T(~) is the largest of the two remaining roots. Next, x'(~) is given by eq. (3.7) with Z = s(~):
X'(~) - / 3 + 4S2(~) -- s3(~)
s~3-- ~
(3.10)
318
(7,.Auberson et aL / Rigorous absolute bounds for n~rscattering (IlI)
As for t(~), it is the positive solution of the second degree equation obtained by replacing x, s and u in the first eq. (2.3) by x'(~), s(~) and 4 - s(~) - t, respectively:
t(~)=½(s(~)-4)[~/l+4 ~(~)(s(~)-4)-/3
-1].
(3.11)
(s(~) - 4)2(s (~) - a) Now, by switching from the variable x' to ~, eqs. (3.6) and (3.8) give:
F(xl'Yl)-TF(xo'Yo)=
XO-Xl ~ 7r
d~
(dx') A(s(~),t(~)) ~-~ (x,(~)_Xl)(X,(~)_Xo) .
(3.12) We remark that the derivative dx'/d~ is negative, so that the kernel (x 0 - Xl) (-dx'/d~)/(x'(~) - X l ) (x'(~) - x 0 ) is positive. This allows us to replace the (positive) absorptive part A (s(~), t(~)) by one of its lower bounds taken from (I), which have the form:
A(s([),t(~))>~m(~) lF(s(~),T(~))l e ,
(p ~> 2).
(3.13)
Here, with the normalization of the amplitude fixed by eqs. (2.1) and (2.2): m(~) = I_2 V s(~)
(Pp)-P ,
1
(3.14)
and Up, defined by eq. (2.17) of (I), is a known function of t(~)/(s(~) - 4), F(~)/ (s(~) - 4) and p. With the particular choice of hyperbolas we have made, T(~) is always negative, so that Up is always well defined [cf. discussion after eq. (3.6) of (I)]. Then: Y
/(~)
~,:o,'÷m~/
I
D
/
/
~"
Fig. 3. Visualizingthe variables ~, x', s, t and T.
G. Auberson et al. /Rigorous absolute bounds lbr 7rlr scattering (III)
319
® ~Q
o
x
x
~o ~,-b
Fig. 4. Image of the cut manifold H in the ~ variable.
~ 1 f d~ w(t) i~(x(t),Y(t))l p , F(X l, Yl) - F(Xo, YO) >~ -~ -b where, after straightforward calculations, w(t) is found to be: x0 - x1
w (~) = m (t)
( X ' ( t ) -- Xo) ( X ' ( t ) -- X l )
I[(s-a)c/~ 2 + s - a ] [~+8~s-(4+3a)s2+2s3]l
According to the discussion at the end of sect. 2, the function F(x (~), y (t)) is analytic in the complex t-plane cut along [~O, 0] and [-b, oo] (see fig. 4). These two disconnected cuts are the image in t of the intersection of the two real branches of H with ~r' Next we map the cut t-plane onto the unit disk [zl < 1 in such a way that the cut [ - b , oo] is sent onto the circle [zJ = 1 and the point t0 onto z = 0: _ x/zb-
z - _x/_Zb~_t0 + x / Z ~
(3.17)
(the surd -x/Zb-S-t is defined in the ~-plane cut along [-b, oo] and is positive on [_0% - b ]). The function:
f(z) =-F(x (t), Y (t))
(3.18)
0 -1
-
1
Fig. 5. Image o f the cut manifold H in the z-variable.
320
G. Auberson et al. /Rigorous absolute bounds for 7rTrscattering (111)
is analytic in the unit disk cut along [-/31, -/32] , where -/31 = Z(~Q) and -/32 = z(0) (see fig. 5). Furthermore:
F(xo,YO) = f(0),
F(Xl, Yl) = f ( z l ) ,
(3.19)
with z 1 = z (~1). Inequality (3.15) becomes: 7r
f(zx)-f(o)~> f --
dO p(O) If(ei°)IP ,
(3.20)
lr
where the (even and positive) weight function p(O) is given by:
• . lsin }01 o ( o ) = ( - b - ~0) cos3 ½0 w ( ~ ) ,
~ = - b + ( - b - GO) tg z ~ 0 .
(3.21)
On the other hand, the imaginary part o f f ( z +/13) is positive on the cut z E [-/31 , -/32]. Indeed, if ~ is the unicursal coordinate of a point moving on the cuts of H, and s is the largest corresponding Mandelstam variable, then applying once more the geometrical remark of sect. 2, one sees that d~/ds is positive. Moreover, a glance at figs. 4 and 5 shows that, when z is real, dz/d~ is positive, so that dz/ds > 0 when z E [-/31, -~2 ]. Consequently f(z + iO) = F(s ÷ iO, t), and Im f(z + iO) = A (s,t). Since, by construction, t is positive (and < 4) for all these points: Imf(z +/0)>/0,
z E [-/31,-/32].
(3.22)
Now we are in a position to formulate our last extremum problem: find the range of the point (f(0), f(z 1)) in the two-dimensional plane, knowing (i) the analyticity properties o f f ( z ) described above, (ii) the inequality (3.20), (iii) the positivity condition (3.22). This problem is similar to the one solved in sect. 3 of (II). It differs from it, firstly by the presence o f f ( z l ) - f ( 0 ) in place of 1 in the left-hand side of eq. (3.20) (a modification which is trivially dealt with), and secondly by the occurrence of a cut inside the uhit circle. A first method to solve our present problem would consist in reducingit to the one of sect. 3 of (II), and this is actually the reason for which we gave formulas (3.16) and (3.17) in (II). Here, we shall rather use an alternative method which we found to be easier, and which does not appeal to the algorithm described in appendix (Ae) of (II). That alternative method is developed in sect. 4. Before doing so, it is convenient to get rid of the function O(0) in eq. (3.20) by introducing, as in sect. 3a of (II), the outer function:
G(z)=exp
(
-p
1 i dO e iO __+z l o g p ( 0 ) ) . 2rr eiO z --
Tr
That log p (0) is integrable is easily checked. The function:
(3.23)
G. Auberson et al. / Rigorous absolute bounds for 7rlrscattering (III) h ( z ) = fG(z3)
321 (3.24)
is analytic in the unit disk cut along the interval I = [-/31, -/32], and its discontinuity across I has the same sign as the one o f f ( z ) . Then the inequality (3.20) and the positivity property (3.22)become:
f
dOlh(ei°)lP <~G(Zl) h(Zl) - G(0) h(0) 27r
(3.25)
--'/t
Im h (z + i0)/> 0 ,
z C I.
(3.26)
4. Solution of the coupled extrema problem We have first to define the space of functions h (z) where the extrema are to be found. Let H~be the class of functions h(z) such that: (i) h (z) is "real analytic" (h *(z) = h (z*)) in the unit disk cut along 1, (ii) h (z) has boundary values on I in the sense of distributions, denoted by h(x +-iO), and Im h(x + i0) = - I m h(x - iO) is a (real) measure lah(X),
1 (iii) d (z) -- h (z) - ~-
fdUh (x) x
-1 z
(4.1)
I belongs to the (real) Hardy space HP [2, 9]. Assumptions (i) and (ii) are clearly realized as consequences of axiomatic field theory. That the distribution Im h (x + i0) must be a measure is not a restriction since, according to eq. (3.26), we are really interested only in functions h (z) whose boundary values Im h (x + i0) (x E I ) are represented by a positive distribution, thus by a (positive) measure. As for assumption (iii), we first remark that it makes sense since the function d(z) is obviously analytic in the unit disk. The requirement that d(z) belongs to I-IP is stronger than simply assuming d(e i°) E I_P, and constitutes a technical hypothesis equivalent to the one already discussed in subsect. 2a of (II). Notice that, under these conditions, the integral appearing in eq. (3.25) always exists:
N(h) =-
E~ -~d°lh (ei°)i p
< oo
(4.2)
Actually d(e i°) C LP implies h (ei°) E LP since, as long as the interval I does not meet the unit circle, the function fI dtZh (x - z) -1 is analytic on Izl --- 1. The classH F is manifestly a (real) linear space. We equip it with a norm llhll defined by:
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G. Auberson et al. / Rigorous absolute bounds for 7r~rscattering (III)
Iih[l = [[dllHp + 17r
fdlUhl(X),
(4.3)
I
where [[dllHp is the usual norm in kiP:
[Id[lHp =
~-
The norm (4.3) has been chosen in such a way that the space Hp is in fact the direct sum of two Banach spaces, namely the Hardy space HP and the space of measures on the interval I, which is known to be a Banach space when endowed with the norm [[/ah [1 = fl d[/Jh I (x). It follows that Hp is itself a (real) Banach space *. Now our extremum problem can be formulated as follows. Refering to eqs. (3.25) and (3.26), we introduce the convex cone ~ of functions h(z) in HP for which the corresponding measure ~h(X) is positive, and the set cg defined by: c'ff. = (h E H P : N ( h ) p <~G(Zl) h(Zl) - G(0) h(0), h E ~ }.
(4.5)
Notice that c~ is convex. This is due to the fact that N ( h ) is a Lp norm, which implies that N(h)P is a convex function (for p >~ 1). The mappings h ~ h (0) and h -+ h(Zl) obviously define two continuous linear functionals on H/p denoted by A 0 and A t respectively: h(0) = A0(h),
h(Zl) = Al(h).
(4.6)
Let A be the continuous linear mapping H/p -->R 2 : h ( h ) = (h (0), h (z 1)).
(4.7)
The problem is to find the (convex) image A(Cg) i n n 2 of the set c-g C Hp. It will be solved by constructing explicitely all functions h (z) in cg which are mapped into the boundary 0A(ql). Such a procedure requires however that the boundary aA(C'g ) belongs to A(Cg), in order that the corresponding functions h (z) belong themselves to cg. This requirement is guaranteed by the following Proposition: The set A(Cg) is compact in 112. This important property is by no means obvious and its proof, which involves some technicalities, is reported in the appendix. We now introduce the subset cly of
cg: clP= {h @ H f : N(h)P = G(Zl) h(Zl) - G(0) h ( 0 ) , h E ~ } .
(4.8)
* Note that the quantity N(h) defined by eq. (4.2) is also a norm on Hf, which is not equivalent to Ilhql. Although N(h) is the quantity of interest in our problem, it is not a "good" norm because it does not make Hya complete space.
G. Auberson et al. / Rigorous absolute bounds for lrTrscattering (III)
323
Let us show that A(Cl~) = A(q/). It is sufficient to prove that any point in A(Cg) is the image of an element of cly. Given a point P E A(q/), there exists at least one function h E c/g such that A(h) = P. Then the quantity N(h)P - [ G ( Z l ) h (z 1) G(0) h (0)] is negative or zero. If it is zero, h E c)y. If it is strictly negative, let us consider the family of functions in H~: kx(z ) = h(z) + ~z(z - z 1) g(z) ,
(4.9)
where g(z) is any given function in HP. Then kx(z ) belongs to ~ and A(kx) = P for all real values of ~. Moreover N(kx)P - [G (z 1) kh(Zl ) - G (O)k h (0)] is obviously a continuous function of ~ which is unbounded from above. Hence this function necessarily vanishes for some value X0 of X, and kx0 E c~, which establishes our assertion. Thanks to this result, we can solve our problem by constructing all functions in cl~ which are mapped onto the boundary aA(C)Y). Actually, by using a variational method, we shall work out necessary conditions for a function in c,~ to map into aA(q~). Let h(z) be such a function. The most general variation o f h ( z ) in H~ has the form: 5h(z) = e g(z) +
. [duh(x)~(x) + dX(x)] ~
+ o(e),
(4.10)
where g E HP. According to the Lebesgue decomposition theorem, the discontinuity of 6h (z) on I has been written as the sum of a measure dX(x) singular with respect to d/Jh(x ) and a measure absolutely continuous with respect to d~h(x), which, according to the Radon-Nikodym theorem, is expressible in the form d~h(x ) 4~(x) where ~(x) E Ll(~uh). The function h(z) + 6h(z) belongs to the cone 5~ if and only if the measure X(x) is positive and 4~(x)/> - 1/e (x E l ) , with e > 0. It belongs to c,~ if furthermore:
[N(h)P] = G(Zl) ~h(Zl) - G(0)~h(0) .
(4.11)
To first order in e, this equation becomes:
p f
~d0 Ih(ei°)lp-2h*(e i°) 6h(e io) = G(Zl) 6h (Zl) - G(0) 6h (0).
(4.12)
Let us first consider a variation 6h(z) in HP, that is to say ~(x) - 0, 4~(x) - 0. We get from eqs. (4.10), (4.12) and Cauchy formula: dO ei° e -~f 2~ e i o ,- - Xi g ( e i °_) = ~ h ( x i )
i = 0 , 1 (x 0 = 0 , X 1 = Z l ) ,
(4.13)
Ir
ep f
~lh(ei°)[P -2 h* (ei°)g(ei°) = G(Zl) 6h(Zl) - G(O) 6h(O).
(4.14)
G. Auberso;: et aL / Rigorous absolute bounds for 7rwscattering [IIl)
324
The left-hand sides of these three equations define three continuous linear functionals on HP. If these functionals are linearly independent, given any 6h(0), 6h (z 1), there exists at least one function g(z) in HP satisfying eqs. (4.13) and (4.14), and the point (h (0), h (z 1)) cannot belong to the boundary OA(QY). Consequently a necessary condition for (h (0), h (z 1)) to be on 3A(C~) is that the three functionals be linearly dependent. This implies the existence of real s 0 and a 1 such that:
-~
-~ [
eio ei° _ z 1 + o~oG(O)] g(ei°) = O (4.15)
Plh(ei°)lP-2h*(ei°) - ~ l G ( Z l )
for all g E HP. In order that this condition be fulfilled, the bracket in the integrand must belong to the annihilator of HP, which is known to be the set of functions in Hq vanishing at the origin [10]. Thus,
p h (ei°)lp-2h*(e i°) - cqG(Zl)
ei0
+ a0G(0) = ei°v (ei°),
el° - Zl
o(z) E Hq . (4.16)
It turns out that this equation determines partly the structure of the function h (z). To see this, let us introduce the (unique) "real" outer function [2,9] R (z) defined by:
IR(ei°)l=lh(ei°)l ~(p-2),
R(x)>0
for
-l
(4.17)
Then eq. (4.16) can be rewritten as:
pR(eiO)h(eiO)_
1 Vffl_G(zl) R(e -io) L1 - zleio
O~oG(O) +e.iOu(e_iO)] ' -n<~O <~rt. (4.18)
From eq. (4.17), the function R (e i°) h (e i°) clearly belongs to L 2. Thus U(z) =R (z) h (z) is in H? since R (z) belongs to the Smirnov class N + [2,9]. This allows us to reconstruct U(z) from its boundary value on the unit circle and the interval I via the Cauchy formula:
pU(z)
1 = P f dUh(X) ~R ( x ) + f dO ei° nI -n 2rr eiO _ z R(e -iO)
× ,1F l Zl z + 1 f dz' 1 2in Iz'l=l (1 - z z ' ) R ( z ' )
-
z
F_ l_C(zo oC(O) ] LZ --z 1 z' +u(z')
(4.19)
For fixed z, Izi < 1, the last integrand belongs to L 2. Furthermore, when multiplied by z'(z' - Zl) , it is the boundary value of a function which is in N +, and thus
325
G. Auberson et al. / Rigorous absolute bounds for nTr scattering ([ti)
in H 2. Hence the Cauchy theorem applies again [11 ], and one gets: OtlG(Zl) pU(z) = Pn -If
diSh(X)
-
ot0G(0)
+R(Zl)(1--zzl)
(4.20)
R(O)
This equation shows that the function U ( z ) has an analytic continuation outside the unit circle, which, according to eq. (4.18), is given by P U \(zt l = R ~1
ZlI°tlG(Zl) t- z z _ aoG(O)+zv(z) ] ,
([z[ ~< 1).
(4.21)
Indeed, both sides of eq. (4.21), when multiplied by (z - z 1), belong to H 2 and coincide on [z[ = 1. Under these conditions, eqs. (4.13) and (4.14) have a solution if and only if 6h(0) and 6h(Zl) verify the relation: (1 - a l ) G(Zl) 6 h ( Z l ) - (1 - s0) G(0) 6h(0) = 0 .
(4.22)
This is nothing but the equation of a supporting line to the convex set A(°Y) (actually of a tangent to its boundary). It remains to determine the structure of the measure lah(x ). This is done by considering the full variation of h (z). Inserting eqs. (4.10) and (4.16) into eq. (4.12), and using Cauchy formula, we obtain: eg(xi) + e f
[d/ah(X ) q~(x) + d?,(x)] x -1 x i _ 6 h ( x i ) '
i=0,1
(4.23)
I e
alG(Zl)g(zl)
- aoG(O)g(O ) _
1 f[duh(x) ~(x) + d?~(x)] o(x)} 1
= G(Zl)6h(Zl)
- G(0)Sh(0).
(4.24)
Since we have already made full use of the variation o f h (z) due to the term eg(z), let us eliminate g(0) and g ( z l ) between the three eqs. (4.23) and (4.24): -
--Tr I
t- x - - z 1
X
+ o(x)
= (1 -- a l ) G(Zl) ~h(Zl) - (1 - s0) G(0) 5 h ( 0 ) .
(4.25)
With the help of eq. (4.21), this equation can be rewritten as:
e--P.f[dun(x)i~(x)+dX(x)] 1R(x)U(1) I
x
= (1 - a l ) G(Zl) 6 h ( Z l ) - (1 - s0) G(O) 6 h ( 0 ) .
(4.26)
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G. Auberson et al. / Rigorous absolute bounds for 7r~rscattering (III)
We recognize in the right-hand side the quantity which vanished in eq. (4.22). Now eq. (4.22) is the equation of a supporting line to the convex set A(qY) only if for all admissible variations 6h (that is to say h + 6h E c y ) , the points (h(0) + 6h(0), h (z i ) + 5h(Zl) ) lie on the same side of this line, or equivalently if (1 - cq) G(Zl) 6 h ( Z l ) (1 - o~0) G(0)6h(0) keeps a constant sign under the same conditions. This provides us with a second necessary condition for (h(0), h ( z l ) ) to be in 3A(Cl~): the left-hand side of eq. (4.26) must keep the same sign for all admissible functions O(x) and measures k(x) [q~(x) >t - l/e, k(x) positive measure singular with respect to/ah(X)]. Let us take first k(x) - 0. Noticing that the function ( I / z ) R (z) U ( 1 / z ) is analytic on I and that the sign of the admissible functions ~(x) is not constrained, we immediately deduce from the necessary condition just stated that the support of the measure/ah(X) is contained in the necessarily finite) set of zeros of U ( 1 / x ) on I (recall that R(x) > 0 on I). Thus, since/ah(X) is a positive measure, it must have the form: -
N d/an(x) = ~ /ai 6 (x - vi) dx , i=1
(4.27)
where/ai • 0 and v i E I, U ( 1 / v i ) = 0 (i = 1 . . . . . AT). Turning back to eq. (4.20), we now see that U ( z ) is a rational function which has (i) (N + 1) simple poles at z = v i (i = 1 ..... N ) and z = 1/z 1 ; (ii) N zeros at z = 1/v i (i = 1, ..., N), and (iii) behaves like a constant at infinity. Hence it has the form: N
o -- z 1 -- z v i U ( z ) = A 1 - z z l i=ll-I ~ ,
(4.28)
where A and o are two real parameters. Moreover, the positivity of the measure /ah(x) implies that the residues of U ( z ) at each pole z = v i are negative, and consequently that between two such successive poles, there must be an odd number of zeros. As U ( z ) has only one possible zero on l(at z = o), N is at most equal to 2. Finally, taking non-zero measures ~,(x) in eq. (4.26), we deduce from the second necessary condition that the sign of the function: N X - - Z 1 i=1 1 - x v i
(4.29)
must not change on I - - [-/31, -/32]. At this point, we have to consider separately the three casesN = 0,1 and 2. Case N ; 2: The condition on the sign of U ( 1 / x ) implies v 1 = -/31, v2 = -/32 and the posRivity of/ah(X ) implies --/31 < o < --/32. Case N - - 1: If U ( 1 / x ) has two simple zeros (1/o 4= Vl) , the condition on the sign of U ( 1 / x ) implies v 1 = -/31 or -/32, and 1/o ~ I. If U ( 1 / x ) has a double zero (1/.~ -- Vl) , then the pole of U ( z ) may stand anywhere on I, -/31 ~ v 1 ~ -/32.
G. A uberson et al. /Rigorous absolute bounds for rr~rscattering (III)
327
Case N = O: a may range from _oo to +oo. The structure of the function U(z) = R (z)h (z) is now completely fixed. In order to extract that of h (z), it is convenient first to construct R (z). From eqs. (4.28) and (4.17), noticing that the modulus o f the Blaschke product IIN 1 (z - vi)/(1 - zvi) is equal to 1 on the unit circle, we readily get in all cases: IR (ei°)i
IA
o -- e iO
1
1 - ei°zl
1 - 2/p (4.30)
This is sufficient to determine unambiguously the outer function R (z) (positive for - 1 < z < 1):
[I+ R(z) =
Finally:
h(z) =
0-2 nl-2/p JAf _--2~Zl 1 j ,
IFLAI 1-°z] 1-2/p /L 1 -zztJ
+_o~>l,
'
I ( 0 -- Z ~21p fi B + 1 -ZZl] i=l
tel< 1.
1 -
(4.31)
zvi ,+o>11,
z - vi N B 1 oz \21p o - z r[ 1-zvi ( ~ ) 1-oz i=l z _ v i
,
lel ~<1.
(4.32)
where B is a real normalization factor, the modulus of which is fixed by the condition N ( h ) P = G (z 1 )h (z 1 ) - G(O)h (0), and the sign o f which is controlled by the positivity requirement. The determination o f the quantities ( )21p is chosen to be the real positive one for - 1 < z < I. Once B has been calculated in terms o f the parameter e, eqs. (4.32) provide us with several one-parameter families of functions h ( z ) in c)y, and any point in 3A(OY) is the image (h(0), h ( z l ) ) of one o f these functions *. The explicit calculation o r b depends on the particular case (N = 0, 1,2) which is considered. As this final step is straightforward, we treat here only the case N = 1 and v 1 = -/32 (this case turns out to be eventually the relevant one for the determination o f the lower bound on h(0)]:
a--z 2/1) 1 +z[32 B(+I-ZZl)
z + ~2 "'
+o>~l,
h(z) =
B(1-oz)2/P \l-~z 1
o-z i---~z
1 + z/32 z+~2'
Iol~<1 .
(4.33)
* It is worth mentioning that in the simplest case N = 0, the functions h(z) given by eqs. (4.32) are holomorphic in the unit disk and belong in fact to HP. In this case, eqs. (4.32) are nothing but the known expressions for the extremal function in HP corresponding to rational kernels [9] [compare these equations e.g. with eqs. (H.2) and (H.7) of (II)]. The method developed in this section thus appears as an alternative way for deriving extremal functions which allows us to cope with more complicated situations.
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G. A uberson et al. /Rigorous absolute bounds for nTr scattering (111)
The positivity condition imposes B ~< 0 when + o/> 1, and B(e +/32) ~< 0 when iol ~< 1. An elementary calculation gives:
N(h)P = [B[P
1 - 2 Z l o + 02 1
(4.34)
-z 2
Equating this quantity with G (z 1) h (z 1) - G (0)h (0), we get the expression of B: o~<-
(i) for
1
or-
~2'
1
~1 ~< o ~< - 1 ,
o r o / > --/32,
and K (o) <<.0, 2 B =- I -
1-Zl 1 -- 2 Z l O +
e2
K(cr) 11/(p-l)
(4.35)
(ii) for - 1 ~< o ~< --/3 2 and K(o) >~0,
l-z1
K(o)II/(p-1) .
1 - 2 Z l o + 02
(iii) the other values of a must be ignored. Here K(o) is the quantity: G(Zl)
K(o) =
[Io-z11~2/p 1+Zl/32
~1--~12 !
a ( 0 ) - [o[2/P ----,
zl + th
(1--Oz l~2[p O - Z 1 1 +Zl/32 G(Zl) k l _ z ~ / l _ o z I z1+/32
Iol~> 1
th o
G(O)~2,
Iol~
The formulas (4.33), (4.35) and (4.36) provide us with the parametric equations of a piece of curve: h(0) = X(e) and h(Zl) = Y(o). Repeating the same calculation in all other cases discussed above, we obtain a set of curves which must contain all boundary points of the convex A(qY), and possibly interior points. Finally, the set A(O/) = A(q~) we were looking for is simply the convex hull of all these points. This completes the solution of our extremum problem. 5. Numerical calculations and concluding remarks The computation of the upper and lower bounds on f(0) = F(~-, 4, 4) now proceeds as follows: (i) Choice of the power p (~> 2), of the hyperbola H and of the straight line D. With the restricted class of hyperbolas considered here (a one-parameter family), we have three parameters at hand: p, a and a.
G. Auberson et al. / Rigorous absolute bounds for lrzr scattering (III)
329
(ii) Calculation of G(0) and G(Zl) , by using eqs. of sect. 3. There, we need the function Vp def'med by eq. (2.17) of (I). The computer program set up in (I) has provided us with it. (iii) Calculation of the two-dimensional region A(ql) from eqs. (4.33), (4.35) and (4.36) in the case N = 1, v 1 = -/32, and from similar equations in the other cases. Then, upper and lower bounds on F ( 4, 4 , 4 ) are given by: G(0) min h(0)~
(5.1)
(iv) Optimization of each of these bounds with respect to the parameters p, a and {~. (v) Check that the optimal curves D and H are inside the axiomatic domain of analyticity, by means of inequalities (2.10) and (2.12). The calculations have been made on a computer IBM 360. The best bounds obtained are: 4 4 - 7 3 ~
(5.2)
The corresponding values o f the parameters p, a and a are, for the maximum: p = 2,
a = 2.39,
a = 0.548,
(5.3)
and for the minimum: p = 3.60,
a = 2.38,
~ = 0.448.
(5.4)
Fig. 6 exhibits, as an example, the convex region A(C//) that corresponds to the minimum, conveniently rescaled so as to give directly (f(0), f ( z 1)). In that case, the boundary of A(Q/) is made of two pieces; on the first one the function h ( z ) has no singularity on I (N = 0), on the other one it has one pole (N = 1) at z = -/32; notice also that there are no points corresponding t o N = 2, and to N = 1 with a double zero of U ( l [ x ) . The bounds (5.2) have to be compared with those derived by previous authors. Using the method of Lukaszuk and Martin [12], we have found - 1 5 9 ~< F ( 4, 4], ~3) ~< 16. With the improvement suggested by Healy [13], one gets - 1 4 9 ~
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G. Auberson et al. / Rigourous absolute bounds for rr~rscattering (III) (zw)
- 7 ~
1o
~ tf(o)
Fig. 6. The r e g i o n A ( q t ) f o r the values of the parameters 7, a and c~giving the best lower bound at the symmetry point. The curve A corresponds to extremal functions without singularity on the interval I, the curve B to functions with a pole at z = -/31, and the curve C to functions with a pole at z = -~32.
would be the application of our techniques to the derivation of semi-phenomenological bounds *, already closer to the experimental data, in which case an improvement by a factor say, of the order of two, would be quite significant. Concerning our initial aim of including as much axiomatic information as possible, an overall look at the present derivation still reveals a noticeable shortcoming. Actually, by imposing the positivity of the discontinuity o f f ( z ) along the cut I, only _apartial use of the unitarity condition has been made. As a result, the extremal function corresponding to the minimum displays an unpleasant pole at z = -~32. The occurrence of this pole is responsible for the large value of the lower bound as compared to the upper one, as it can be observed on fig. 6: the region limited by the curve A, which is precisely the range of 0r(0), f(z)) when f(z) has no singularity on 1, is only a small fraction of A(QL). Therefore, one finds here still a possibility of improving the situation. One way to do it would lead us, after changing the manifolds D and H, to consider Hardy spaces of analytic functions in an annulus (15). Another way, which would allow us to keep the same manifolds, leads to the construction of a new convex set smaller than ~2~. Whether such further refinements would significantly strengthen our bounds or not, is an open question. In that respect, it would be quite interesting to approach the bounds from the other side, that is to say to construct symmetrical amplitudes as large as possible, satisfying analyticity and inelastic unitarity. Even without demanding elastic unitarity for 4 ~< s ~< 16, this is already a difficult task. Let us make a last remark. Although in our method a rather extensive use of analyticity has been made as compared to previous works, still one variable dispersion relations do not turn the analyticity properties of the amplitude to the best account. Therefore, it would be desirable to have a two-variable integral representa* Like those investigated by Chung and Vinh Mau (private communication), where the phenomenological input is the D-wave scattering length.
G. A uberson et al. /Rigorous absolute bounds for nTr scattering (III)
3 31
tion at our disposal, and to state the extremum problems in a suitable space of twovariable analytic functions. This would be indeed quite a different approach for deriving bounds, which would possibly lead to further progress.
Two of us (L.E. and F.S.) are grateful to C. de Dominicis for the hospitality extended to them-at Saclay.
Appendix P r o o f o f the proposition o f sect. 4
In order to establish the compactness of A(QZ) in R 2, where the set 9Z E H/p and the continuous linear mapping A of H/p into R 2 are defined by eqs. (4.5) - (4.7), we cannot rely on the usual weak compactness argument [16], because the Banach space H/p is not reflexive. Actually, according to eqs. (4.1) and (4.3), H~ is isomorphic to the direct sum: Hp= Hp mc/~ ,
(A.1)
where c-fig, the Banach space of (real) measures over the closed interval 1, is known to be non-reflexive. Therefore, in place of the weak topology of H/p , we shall use the weak-* topology, which turns out to be sufficient for our purpose, due to the particular form of our functionals A0 and A t . To this end, we have to consider the Banach space: X = (HP) * e e ,
(A.2)
where Q is the (non-reflexive!) Banach space of (real) continuous functions on I, and (HP)* is the dual of liP. Then, since ~ * = ~ and (liP)** = HP (lip is reflexive), one has:
H/~=X*,
(A.3)
and the weak-, topology on H/p is the topology o(H/p, X) [17]. It is shown below that: (i) the functionals A 0 and A 1 belong to X, (ii) the set 9/ is weak-, compact. Then from (i) the mapping A is obviously continuous on H/p in the weak-, topology. This, together with ii), entails the compactness of A(QZ) i n n 2 and achieves the proof of the proposition. (i) Ai E X, (i = 0,1). Indeed, for any h E H/p, eqs. (4.1) and (4.6) give: lah(X) x - X~---i"'
Ai(h) = d ( x i ) + 1
(X0 = 0, X l = Z l ) ,
(A.4)
332
G. Auberson et al. /Rigorous absolute bounds for 7rrrscattering (III]
where the functionals d -~ d ( x i ) belong to (HP)* (since d E liP) and the functionals
,h --l f d,h(X)(X- Xi)-1 I belong to C since the Cauchy kernels (x - x i ) - I are continuous o n / . (ii) -'-// is weak-, compact. Recall that the convex set 9 / i s defined by: cg = {h E H/P: N ( h ) P <~ A ( h ) , h ~
),
(A.5)
where A E X is the continuous linear functional:
IX(h) = G(z l ) Al (h ) -- G (O) £x0(h ) .
(A.6)
Let us show first that q/ is bounded and weak-, closed. (a) q/ is bounded. We need the following lemma: L e m m a 1: For any function h(z) C H~ , N ( h ) = lim rt 1
Ff
dO h(reiO)l p I1/P LJ ~ -it
Proof.' Given h (z) E H/p , let us define the fnnctions hr(z ) = h (rz), where 1 -/31 < r ~< 1. Then hr(z ) ~ H~r with 1r = [-/~l/r, -/~2/r], and we can write:
hr(z) = dr(z) + ar(z) ,
(A.7)
where dr(z ) E HP and
ar(z ) = 1 f d P h ( X ) x I
(A.8)
Now dr(e i°) converges to d l ( e i°) in the I_P norm when r t 1 [this is a well-known property of HP, see e.g. theorem A.1 of (II)]. Similarly, ar(eiO)~ al(e io) in the LP norm since:
tar(eiO)_al(eiO)l
x
1
- re io
_
1
x - e io
< const (1 - r ) ,
(A.9)
uniformly in 0. Thus hr(e i°) ~ h(e i° ) in the I.P norm, which implies:
lira N ( h r ) = N ( h ) , rtl
q.e.d.
(A. 10)
Now, from the Cauchy theorem, one can write for any h E Hp and I -/31 < r <1:
G. Auberson et al. /Rigorous absolute bounds for lrlr scattering (111)
1
2Br f dzh(z)=-~ Izl=r
1
fdP.h(X ).
333
(A.11)
I
When moreover h E 5~, the measure
lah(X) is positive,
and one deduces:
II/.thll = f dlUhl(X) = ~ I f ~-~nr dO ei°h(rei°) < ' l r l / 1
~ ih(reiO)lP 1 1/p <,TrN(h),
-~r
(A.12)
where we have used successively the H61der inequality and lemma 1. On the other hand, we get from eq. (4.1):
Id(ei°)l <~Ih (e/°)[
+ 7r(1 1- 131) II/ahII
a.e.,
(A.13)
and by applying Minkowski inequality in Lp : IldllHp
<<,N(h) + zr(1
1 -/31) IIt~hll.
(m.14)
Inserting eqs. (A.14) and (A.12) into eq. (4.3), one obtains: 3 - 2/31 IlhlI -~< 1 _/31 N(h),
(hE~).
(A.15)
Finally, eq. (A.5) gives, for any h E QZ :
N(h)P <~II/Xllxllhll,
(A.16)
and, together with eq. (A.15):
{ 3 - 2/31 )p/(p -1) Ilhl[ ~< \ 1 - ~
1
l/(p-1) IIAIIx
,
(A.17)
which establishes the boundedness of c2g.* (/3) q/ is weak-, closed. Given any sequence {h n } in 9Z converging to h E Hp in the weak-, topology, one has to show that h E 9/. First of all, the weak-, convergence h n ~ h implies the weak-, convergence t~hn -+ t~h . Thus:
lirn fdUhn(X) 4fix) = f dUh(X) ¢(x), n-'~ °° I
,¢c~E e
(hAS)
I
Since h n E ~ for any n, fldt~hn(X) ¢(x) is non-negative for all positive functions in C, and eq. (A.18) shows that the same is true for fl d/ah(X) ¢(x). This means that the measure lah(X) is positive and that h E ~ .
¢(x)
* Incidentally, let us notice that eq. (A.17) may be used to derive at once rough (and in fact very bad) upper bounds for Ih(0)l and Ih(z1 )1.
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G. Auberson et al. / Rigorous absolute bounds for 7rlrscattering {III)
It remains to prove that N ( h ) P <~ A(h). We shall use the following lemma: L e m m a 2: Let (gn ~ -+ g be a weakly convergent sequence in a reflexive Banach space. Then I[g[[ ~< lira inf I~nlln --~. o a
The proof stems from the weak compactness of the unit ball. It is a simple exercise which is left to the reader. Let us rewrite the difference N ( h ) P - A ( h ) in the form: N ( h ) P - A(h) = N ( h ) p - N ( h n ) P + [N(hn)P - A(hn) ] + [A(hn) - A(h)] . (A.19)
Then, since h n E Q.l. : N ( h n ) P - A(hn) ~< 0,
~¢n,
(A.20)
and since A @ X: lim A(hn) = A ( h ) .
(A.21)
n--~
It follows that: N ( h ) P - A(h) <<.N(h)P - lira sup N ( h n ) P .
(A.22)
n ..--~ o o
Next, decomposing hn(z ) according to eq. (4.1): (A.23)
hn(z ) = dn(z ) + an(Z),
we see from eq. (A.18) that: an (eiO) ==1 f dl~hn (x) (x - e i° ) - 1. 1
(A.24)
converges pointwise to a(ei°) =--~rl fdUh(X ) (x - ei°) -1 , I
(-zr ~< 0 ~< zr).
Moreover, ( a n ( z ) ) is a uniformly bounded sequence of holomorphic functions in any compact Q c C which does not intersect I. Indeed, from eq. (A.24) and the boundedness of q£ : fan(z)l <<.Z IlUhnll ~
(z E Q ) ,
(A.25)
where the constants A and B are independant of n and z in Q.* Then, the Vitali theorem [19] tells us that:
* The fact that the sequence {ll~hnll ) is bounded is also a dkect consequence of the principle of uniform boundedness [ 18 ].
G. Auberson et al. / Rigorous absolute bounds ]or irlr scattering {III) lim an(e io ) = a(e i° ) uniformly on [-Tr, 7r] .
335
(A.26)
On the other hand, the weak-, convergence of h n to h obviously implies the weak convergence of dn(e i°) to d(e i°) in LP:
lira dn(e io ) = d(e i° ) weakly in L p .
(A.27)
n.-~
From eqs. (A.23), (A.26) and (A.27), we deduce that:
lira hn(e io) = h(e iO) weakly in L p .
(A.28)
n--~
Noticing that N ( h ) is nothing but the norm o f h in LP and applying the Lemma 2 to the (reflexive) Banach space L p, one gets:
N ( h ) <<,lira i n f N ( h n ) .
(A.29)
n.-..>, e,o
Returning to eq. (A.22), we are led to the desired result:
N ( h ) P <~ A ( h ) .
(A.30)
Finally, to derive the weak-, compactness of 9 / f r o m (a) and (/3), it is sufficient to remark that q / is contained in some ball [Jhlt ~
References [1] G. Auberson, L. Epele, G. Mahoux and F.R.A. Sima'o, Nucl. Phys. B73 (1974) 314. [2] G. Auberson, L. Epele, G. Mahoux and F.R.A. Sim$o, Saclay preprint DPh-T/74-76, to appear in Ann. Inst. Henri Poincar& [3] G. Wanders, Helv. Phys. Acta 39 (1966) 228. [4] G. Mahoux, S.M. Roy and G. Wanders, Nucl. Phys. B70 (1974) 297. [5] G. Mahoux, Proc. of the 2rid Int. Winter Meeting on fundamental physics, Formigal 1974 (Instituto de Estudios Nucleares, Madrid). [6] D. Hall and A.S. Wightman, Mat. Fys. Medd. Dan. Vid. Selsk, 31 no. 5 (1957). [7] A. Martin, Nuovo Cimento 44 (1966) 1219. [8] G. Auberson and N.N. Khuri, Phys. Rev. D6 (1972) 2953. [9] P.L. Duren, Theory of HP spaces (Academic Press, New-York and London, 1970). [10] See ref. [9], sect. 7.2. [11] See ref. [2], theorem A.5, [12] L. Lukaszuk and A. Martin, Nuovo Cimento 52A (1967) 122. [13] J.B. Healy, Phys. Rev. D8 (1973) 1904. [14] B. Bonnier and R. Vinh Mau, Phys. Rev. 165 (1968) 1923. [15] D. Sarason, Mem. Amer. Math. Soc., No 56 (1965). [16] See ref. [1], sect. 3, and ref. [2], appendix C.
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G. Auberson et al. / Rigorous absolute bounds for lrrr scattering (111)
[17] M. Reed and B. Simon, Method of modern mathematical physics, I: Functional analysis (Academic Press, New-York and London, 1972), sect. IV.5. [18] See ref. [17], sect. 1II.5. [ 19 ] A.I. Markushevich, Theory of functions of a complex variable, vol. I (Prentice-Hall, Englewood Cliffs, N.J., 1965), sect. 86.