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A family of sum rules for weak, electromagnetic and strong processes
~ _7 7 ~]. _2 _~ ]
Nuclear Physics B47 {1972) 569-588. North-llolland Publishing Company
A FAMILY OF SUM RULES FOR WEAK, ELECTROMAGNETIC AND STRONG PROCESSES L. L U K A S Z U K Institute o f Nuclear Research, Warsaw *
Received 21 April 1972
Abstract: New sum rules are derived for the neutrino-electron and pion-pion Compton forward scattering amplitudes and for the pion's electromagnetic form factor. I t is also shown that a peak in the differential cross section or form factor has to be connected with the positive derivative of the phase at the maximum, provided that the peak is steep enough and/or dominating over a sufficiently large region.
1. I N T R O D U C T I O N The sum rules o b t a i n e d in this p a p e r relate the integrals over l o g a r i t h m s o f differential cross s e c t i o n s (or cross sections t h e m s e l v e s ) to the characteristic c o n s t a n t s o f i n t e r a c t i o n like F e r m i ' s c o u p l i n g c o n s t a n t G, e 2, the s c a t t e r i n g l e n g t h a]r__~for t h e rr -- n a m p l i t u d e s or the w i d t h F of" the r e s o n a n c e p. These relations hold u n d e r q u i t e w e a k a n a l y t i c i t y a s s u m p t i o n s w h i c h are stated in sect. 2. In this section the f o r n m l a e are derived w h i c h will be applied to the physical processes. T h e q u i c k derivation goes as follows. Let us start w i t h the m o d u l u s r e p r e s e n t a t i o n I I 1, 12] o f the anaplitude (or f o r m f a c t o r ) F = IFI e i9 (assume F ( 0 ) = 1, z = x + iy is a variable with respect to w h i c h l= has a cut along real axis). D i f f e r e n t i a t i n g it with respect to x at z = 0 we get a~
ax
z=0_
1 [
,
111 I f ( dx t ) Fx ( - x~l ) l 2 -
- x
1
i
(1.1)
0 Now tire derivative o f p h a s e m a y be related t h r o u g h the k n o w n low e n e r g y b e h a #Tr viour o f the c o n s i d e r e d physical processes to G, e -"), al= 0 or, if e.g. one takes x = 0 at the r e s o n a n c e ' s p o s i t i o n , to the 1/F. In fact one has to be m o r e careful t h a n we were above: we have here the limiting process, it m a y also h a p p e n (as it d o e s in the case o f ue ~ ue) t h a t a ~ / O x is n o t cont i n u o u s at x = 0, care s h o u l d also b e t a k e n o f zeroes and o f an a s y m p t o t i c b e h a * Postal address: Warszawa 10, Zaklad VII 1 BJ, ul. Hoza 69, Poland.
570
L. ~ u k a s z u k , S u m rules"
viour. We deal in detail with these problems in sect. 2, where the generalized form of relation (1.1) is derived from Nevannlina's theorem 111. In sect. 3 we discuss necessary assumptions and derive sum rules for the ue, ;re forward scattering amplitudes. This is made (see the inequality (3.10)) for the weak analyticity assumption (ahnost exponential behaviour admitted) and also for the case when the twice subtracted dispersion relations hold (eq. (3.17)). In the latter case the knowledge of the forward differential cross section is enough to determine the sign of Ferret's coupling constant G , e. In the general case a theorem is proved which asserts that 1t,~. l'i,et cannot fall to much below perturbative (first order) resuits. Sect. 4 deals with tire forward Compton scattering (TP, "ye, or 7rr). The low-energy behaviour coming here from Q.E.D. and the weak analyticity assumptions allow the derivation of inequalities (4.9} (or equiwdently (4.12)) for the amplitude ./1 and inequalities (4.8)(or equivalently (4.13)) for the helicity amplitudes {@ ".11~1.As a consequence lhese amplitudes cannot fall too much below the unitarized Thomson's amplitudes. If one assumes substracted dispersion relations then the equalities are obtained (see eqs. (4.16), (4.17)). The results of sects. 3, 4 hold to all orders of G, e 2, respectively, once the initial analyticity assumptions are made. In sect. 5 we derive new sum rule (5.6) relating the integral over the spacelike pion's e.m. form factor to the integral over the timelike (s ~ 4m~) region. Application to the Chou-Yang model [21] is made. In sect. 6 we arrive at the consistency relation (6.4), (6.5) for rr rr scaltering amplitudes; the assumptions made here result from tire AFT. Finally, in sect. 7 we show that the modulus representation can be used to determine the sign of the phase derivative at the peak of the differential cross section. A possibility of finding the lower bound 1o the resonances reduced width is pointed out.
Theorems by Nevannlina [1] and Muskhelishvili [2] are quoted in the appendix.
2. MATIIEMAT1CAL PRELIMINARIES ha this section we shall derive tire following theorem which will be applied to the physical processes in subsequent sections. Given a function F ( z ) such that (i) F ( z ) is regular in the upper half plane, continuous m the closed half plane of z " z = x + iy = r e i ° ;
(it) lira r l+r~In IF(re i°)] = 0; r ~
hn F( + x + i c) (iii) F ( O ) = 1,
-+x
=
x~0+
a+ + O(x~/); where e, r/, r/ are arbitrarily
small positive constants, a . are arbitrary constants and 0 ~< 0 ~< 7r.
L. Lukaszuk, Suez rules
571
Then In IF(x) F ( - x ) l
½(a++a ) ~ l_Tr o
where F(+-x) =
dx
(2.1)
x2
lim F(+-x+ie). e~ 0 +
Moreover, if F ( z ) has at most finite number of zeros in the upper half plane then eq. (2.1) can be replaced by oo
2Yn
1 f
23 2+-n~ 0J n xn
In I Y ( x ) Y ( X ) l d x
(2.2)
X2
(Throughout this paper symbols r/, r~', e, denote arbitrarily small positive constants. Symbol O(x) (for x ~ 0 ) denotes terms which are at most of order x). First let us derive formula (2.1). Function F(z) satisfies (i), (ii), and therefore, from Nevannlina's theorem it has to satisfy inequality (A.2) for any z in the upper half plane. Let us choose z = iv, then In [F(iy)l ~< 1 [ y zr j
In IF(t)[ dt t2 +y2
(2.3)
(In our case C = 0 in (A.2) due to the condition (ii).) The inequality (2.3) is fulfilled for any y > 0. What we shall prove now is the existence of limits lim In [F(iy)l y--+O Y lira ~
y-+O J
71( a + + a
In If(t)[ dt =
)
(2.4)
In IF(t) F ( t)l dt < o o
t2 + y2
--~
,
(2.5)
t2 0
In order to do this, it is enough to know the behaviour o f F ( z ) around z = 0 above and on the real axis. Making the subtraction at z = 0 (possible because Im F(x)/x has at most a finite discontinuity at x = 0) and using a finite contour e.g. a semicircle with chord covering segment [ - 1 , 1] along the real axis - one gets from Cauchy formula 1
F(z) = 1 + zrt I.
ImF(x+ie)
x(x-z)
dx + z O(z) ,
-I where cb(z)is holomorphic in the neighbourhood o f z = 0:
(2.6)
L. ¢~ukaszuk, S u m rules
572
1 ;
[F(ets°)12.1 + F(ei~)- 1 ] dso
(2.7)
0 Hence, around z = 0 q~(z) = ~ ( z ) and q~(0) is real
(2.8)
It will be convenient to rewrite eq. (2.6) in the form 1
F(z) = l + Zrr I
l- mxF(~(xx +zi )e- )- - d x + cz + O(z2) ,
(2.9)
I
where the constant
c =- ~b(0) is real It is possible, due to the assumption (iii), to write the integral from eq. (2.9) as 1
I
1
1
hn F(x + i e) ; a + dx x(x-z) dx = x z
I
0
- 1
a x + z dX+¢l(Z ) ,
(2.10)
0
where 1
~1(z)
= ~
~0(x) ax X-Z
J
-1 with real ~;(x)satisfying ttolder's condition H(r~')(compare (iii))in the neighbourhood o f z = O. Therefore, using theorem B from the appendix, we can write ~;t(z) in the form el(Z) = c ' + O ( z 7~) ,
(2.11)
where c' is real constant. Finally, eq. (2.9) can be written in the form: F(z)= l+-Z [a_lnz
a+ln(ze
i~r)+c l + O ( z n ' ) ]
(2.12)
7r
where c 1 is a real constant. Using eq. (2.12) we can easily calculate the limit lira In _v~0
[F(iy)[ 2 2y
lira 1Gin I[1
½y(a++ a _]2+o0'l+r/!)[
--½(a++a )
r~0
which agrees with eq. (2.4). In the case a+ = a , i.e. when the derivative of the phase o f F - = IFI e i~' with respect to x is continuous at x = 0 (compare (iii)), eq. (2.4) means a lnay[F[ z=0 -
34~3x z=0
L. 15ukaszuk, Sum rules
573
This is the limiting value o f the Cauchy - Riemann relation between modulus and phase. In order to show that eq. (2.5) holds, it is enough to find the behaviour of IF(-+ x + ie)l 2 around x = 0. Eq. (2.12) gives x2
IF(x + ie) F( x + ie)l x ~.0 = 1 - ~ [a+ - a _ ) In x - c I ] 2 + O (x 1+~')(2.13) Hence 1
f
1
lnlF(t)F(-t)l
dt = f
t2+y2 0
0
~(r) dr r + y2 '
(2.14)
where s0(r) satisfies the condition H(~r/). Therefore, from the theorem B dix) the integral
(see
appen-
1
~l(V2) = I
~(r) dr r+y 2
0 has a limit and is continuous i n y 2 = 0 in accordance with eq. (2.5) We have proved eq. (2.1). In the case when the zeros are given, 9ne has to calculate the additional limit lira 1 v~O y
In l-I n
tl z/zn
in eq. ( A . I ) which gives the sum in eq. (2.2). The direct consequence of formula (2.1) is the following statement: Let us assume that the derivative of phase, 3~/3x, of the function F(z) exists at z = 0. If F ( z ) satisfies (i), (ii), and if l°g+
F(x) F(-x)
F2(0) I
x2 0
oo dx <
I
F2(0) log+
I
F(x) F(-x) [ dx x2
,
(2.15)
0
then lb3
=ax z=0 > 0 3. W E A K
INTERACTIONS: ve SCATTERING
3. I. Assumptions It has
already been shown that analyticity, when applied to the neutrino
(2.16)
L. Lukaszuk, Sum rules
574
electron scattering amplitudes gives quite restrictive results [3,4, 19]. In this section we are going to write down sum rules which work under weaker conditions than those assumed up to now. Let us state assumptions under which the results of this section will be derived. (Wl) rn v = me= 0, right-handed electrons do not take part in the interaction. (W2) Ana/yticity: The forward scattering amplitude Fve(S ) satisfies condition (i) from sect. 2 (complex variable s is the total c.m.s, energy squared for the real, positive values). (W3) Crossing: The forward scattering amplitude for re-+re, Fve(s ) is obtained from Fve(S):
Fve(S )
= FFe ( s )
(W4) Behaviour at s = 0: In the limit Re s-+0 the re, ue interactions are described by the V - A type of interaction. We assume also that the total cross section oto t for, ve(~e) scattering is equal to the total elastic cross section Otot elastic (s -----0)(1 + O(s ~7)). (W5) Behaviour at infinity - we shall distinguish here two cases: (a) Fve(S ) can be almost exponential when s -+ oo lira r - l + e In
IFvc(r e/°)[
= O for O ~< 0 ~< 7r ,
r --;'- o o
or (b) Fve(s) satisfies twice subtracted dispersion relations. Hence, the total cross section has to satisfy condition Oto t
~
CS
[
C
Once we have put m e = O in condition (W 1), the rejection of the right-handed electrons is a natural consequence of the V - A theory's success. The condition (WI) is not necessary for existence of the sum rule and is put here in order to simplify kinematics: we have one chiral amplitude describing the scattering ve (compare ref. [3]). Thus ve and/~e scattering is described by the same analytic function due to the condition (W3). The conditions (W2), (W3), (W5) have been nowhere proved; the axiomatic approach is not valid for massless particles, on tile other hand the existing theory of weak interactions is non-renormalizable. However, :hese conditions seem to be the weakest ones which might be deduced from our theoretical experience with strong interactions or perturbation theory [3] once we decided to make some use of the analyticity in the weak processes. In fact one usually [3, 4, 19] starts from stronger assumptions than ours. The condition (W4) is consistent with the experimental evidence about other weak processes. In calculating the total cross sections at the threshold we assume that the contribution from the higher unitarity corrections [4, 19], creations of additional v~ pairs, etc. is contained in the term O(s l+n3. Such a condition is easily satisfied in the dynamical approach of Appelquist and Bjorken [4] and of Dolqov, Zachanov and Okun [19], where a correction O(s 2 In s) appears.
L. t;ukaszuk, Sum rules
575
3.2. S u m rule f o r Fue(S ) - the general case The amplitude Fue(S ) satisfying (W2), (W5) (in this section we shall deal with the more general case (a)) fulfills conditions (i), (ii) o f sect. 2. The low-energy behaviour (W4) guarantees that (iii) is fulfilled too. Let us express a+, a_ from eq. (2. I) in terms of the ve coupling constant G. From (W4) the phenomenological interaction Hamiltonian of the form G (~e 7c~(1 + 75) ~v ) ( f v 7a(1 + 3'5) ~e ) (3.1) is responsible for the re, ~e scattering in the limit s-->O. From (3.1) one gets R e F v e ( S , t ) = 4 v/2 G s = F °
,
Re F~e(S , t) = - 4 x/2 G(s + t)
(3.2) .
(3.3)
The imaginary amplitudes for forward scattering can be determined through the optical theorem: lm F = SOtot(S )
.
(3.4)
Let us fix units introducing the dimensionless variable z -~ 4 x/21GIs
(3.5)
then F o = z sgn(G) (compare (3.2)). Further on we shall use z instead o f s and x = Rez instead if Res. If, e.g, one assumes universal value for G, IGI = 10 - 5 m ~ 2, then x = 1 corresponds to x/~ = 130 GeV. Now, from eqs. (3.2) (3.4) and condition (W4) one gets x2 lm F e ( X + ie) - 167r ' (3.6) Im F e ( - X + ie) = - Im F-e(X + ie)
x2 31 16rr
_
(3.7)
Let us normalize Fve(Z) to unity in z = 0 : Fe(Z) F(z) =
(3.8)
z sgn (G)
Then, applying the definition of a_+ from (iii) to the function F we obtain -~
l(a++ a )_
a
1
3 167r s g n G
(3.9)
Therefore, inequality (2.1) looks as follows in the ve case: Fve F~e
F e F~e oo
48
48
x2 0
-
x2 0
-
L. Lukaszuk, Sum rules
576
The relation (3.10) is true in the general case (a) of (W5); e.g. the behaviour
F(z) ~
exp (z l - e )
is still leading to the inequality (3.10). The inequality (3.10) is sensitive with respect to the behaviour of amplitudes at low values o f x and might be used in checking eventual models. If'we knew the amplitudes from threshold up to, say, x 1, then the integral over differential cross sections - taken from x l to oo _ could be bounded from below. Before we demonstrate it let us define R 2 = [F~e Fve(X)[
(3.11)
x-
X/~ve/dt dOFe/dt~
t=0
where doo/dt = 2 G2/Trdenotes the differential cross section with Fve replaced by Fo([F o] - x, comp. eq. (3.5)) i.e. the first order perturbative result. Let us also denote the integral over " k n o w n " region as XI
I_
|r J 0
lnR~2 dx x2
(3.12)
Then we have the following theorem. I f R 2 ~< x l - e for x "+ ~ , then the following inequality holds:
=l/d°"e f
?
~
e] It=O
d-d;a ° t=0
1 exp(
Xl/48-XlI )
(3.13)
XXl
-
2G 2 gX
exp ( - x ]/48 - x 1 I)
If, moreover, R 12 ~< x - 1 - e for x ~ 0% then
f
/ d ° v e dOTe t=0
2 G2
x 1 exp ( - X l / 4 8 - x 1 I)
.
(3.14)
XI
Eq. (3.13) is satisfied if e.g. the number L of partial waves giving finite contributions 3 at infinity satisfies condition L ~< c x~ - e . Let us also notice that conditions leading to eq. (3.14) are fulfilled if e.g. only a finite number of partial waves contribute to the cross section at high energies.
L. £ukaszuk, Sum rules
577
In order to prove (3.13), (3.14) let us notice that from the conditions of the theorem and from (3.10) 1-4~
< ~
lnR~2 dx x2
Xl
Using now a theorem about geometrical and arithmetical means, we get I - I f ~<
° l n R 2 cLv ~< i ln(x 1 ~ R2 dx) x-
x5
Xl
'
Xl
which gives (3.13) and (3.14) 3.3. Twice subtracted case In this case Fve(Z ) has no zeros in the upper half plane. In order to show this, let us write the dispersion formula (we work with variable z = 4 x/2 [G[ s):
z2
ve
Otot --dx+
Fve(Z ) = sgn (G) z +
x(x z)
2 z-
o
j jetot - -
~
x(x+z)
dx
(3.15)
0
Let us assume that Fve(Z ) = O at z o = x o + iYo, y o ~ O. Then, from eq. (3.15) we have
yo I = O ,
xo I1 = O
,
(3.16)
where I - sgn (G)
lZot2
1 I n 0
t o t dx + 1 j xlx Zo 12 n 0
ore ii = 1 [ 7r
ff
to___.~t dx + 1
- - -°tet -dx xlX+Z o 12
,
~e ] °t°~t dx > 0 iX+Zol2
iX_Zo[2 n o o We are interested in the case Yo 4: 0; eqs. (3.16) would then lead to the condition I 1 = 0 which is impossible unless there is no interaction at all. Now we use formulas (2.2), (3.9) and we get the equality
i Fve Fve 1 sgn(G) = [ In 48 j x2 dx (3.17) 0 Our sum rule makes it possible to determine the sign of G once the information about forward differential cross sections is given.
578
L. £ukaszuk, Sum rules
4. FORWARD COMPTON SCATTERING 4.1. A s s u m p t i o n s
Let us consider forward Compton scattering on the proton or electron. Then the process is described by two amplitudes, e.g. fl(p), f2(u) or fp(U), fa(U) (for definition see e.g. [5]). Let us formulate properties of these amplitudes which will be demanded in this section. (C1) Analyticity and crossing: The forward scattering amplitudes fl(v), f2(u) satisfy condition (i) from sect. 2, fl(u) being symmetric, f2(u) antisymmetric with respect to operation of crossing (complex variable u is lab, energy of the photon on the real axis). (C2) Behaviour at v = 0: given rigorously by Q.E.D. (C3) Behaviour at infinity: We shall distinguish here three cases: ( a ) f l , f 2 can be almost exponential lim r
l+e
log [f(rei°)l = 0
;
(b) f l satisfies twice subtracted dispersion relations; (c)f2 satisfies unsubtracted dispersion relations, f l satisfying (b). Analyticity assumed here can be checked in perturbation theory for the 3' e case; e.g. from the w o n of Cheng and Wu [6] one knows that (C3, c) is satisfied up to the sixth order of perturbation tl~ory in Q.E.D. Of course the cases (a), (b) of (C3) are then fulfilled, too. 4.2. S u m rule - the general case
Now we assume the most general asymptotic behaviour from (C3), namely "almost exponential" case (a), Therefore our sum rule will be the inequality (2.1); we are going to determine constants a+, a_ in terms of r = ~ / m . Consider amplitudes fp(u), fa(U) corresponding to spins parallel and antiparallel, respectively. The crossing properties of fl, f2 (see (C 1)) are equivalent to the condition (compare e.g. [5]) ¢p ( - ~ ) = fa(U)
(4.1)
The behaviour around u = 0 results from the low-energy theorem [7, 8]. We are interested in the value Offp and a T = ½(Op + %) at u = 0: fl(0) = fp(0) = fa(0) = OT
~/m ~ -r
= l ( o +Oa ) = 8~ r2(1 +O(v))
,
,
(4.2) (4.3)
.5
where, from the optical theorem P
Im fp,a(V) = ~
lop, a(0) + O(v)]
(4.4)
579
L. t;ukaszuk, Sum rules
Eq. (2.1) will be applied to = fp(P) F(u) fl (0) (4.5) From assumption (C 1) and (C3, a) it is evident that conditions (i), (ii) of sect. 2 are satisfied. The low-energy behaviour given by eqs. (4.2), (4.4) yields F(u) satisfying condition (iii): F ( u + ie)
=
v~0+
F(v+ie)
P
l +i
=
47r fl(0) P
l-i----
v~0+
[Op(0)+O(v)]
4n'fl(0)
[Oa(0 ) + O@)1
(4.6)
Tile constant ~(a+ + a ) from eq. (2.1) is therefore equal to 1 OT(0 ) = }(a++a ) - 41rfl(0)
~- r
(4.7)
It is also evident from eqs. (4.1), (4.5) that + ic)
IF( v+ie)[ =
¢1(0 )
Finally, eq. (2.1) can be written as
r ~<
i
) J:r
In
| dp p_) J 0 Similar result can be obtained for the amplitude
(4.8)
7T
In this case
fl(P).
P
Ifl(--u)l = Ifl0')[
'
l m f l = 4-rr °T
'
and instead of (4.8) one has V 2
~ r ~< 1 [ 71" J
In ~ r l - du
(4.9)
p_
0
Eqs. (4.8), (4.9) are valid in all orders of strong and electromagnetic interactions once properties (CI), (C3, a) are assumed. We are going to demonstrate that eq. (4.9) prevents the situation in which e.g. iflL would be everywhere smaller than the unitarised Thomson's amplitude, fTh.U.: fTh.U. -= - - r + [
P 4~
OT(0)
(4.10)
Indeed, fTh.U./(--r) satisfies the conditions of the theorem from sect. 2 with no zeroes
in the upper half plane. Hence from eq. (2.2)
580
L. £ukaszuk, Sum rules
i fTh.U. 2 1 [
~ r -=
In
-r--
dv
") p_
17" J
(4.11)
0 Subtracting eq. (4.1 !) from (4.9) one gets
0 ~< |J~ In
dv
v2
(4.12)
0 Similar result will be obtained from eq. (4.8) fp fa in ]faTh U
f
v 2 " ' dv . (4.13) 0 The inequalities (4.12), (4.13) are true for electron, proton or pion (where sum rule f o r f 1 applies) in all orders of electromagnetic and strong interactions, provided that our rather weak analyticity assumptions are nrade. 0 <~
4.3. N o z e r o e s case We shall assume now that fl satisfies condition (C3, b). Then such an amplitude has no zeroes in the upper half-plane [9]. If, moreover, f l satisfies an unsubtracted dispersion relation (i.e. condition (C3, c) is fulfilled) then also fp(V) and, equivalently, fa(V) has no zeroes in the upper half-plane: fp(V) = - - r + ~
(x-v)--~
dx
-
(x+v)x
dx
(4.14)
0 0 Let us assume that fp(V0) = 0 for v 0 = x 0 + iYo, YO =/: O. Then, taking the imaginary part OfJp(Vo)/V 0 one gets Qo
oo
-m iv012
dr+7r
x l x - Vo12
0
dr > 0 7r
0
(4.15)
X[X + Vo 12
i.e. inconsistency. A similar proof works for ft" Now we can apply eq. (2.2) and, in the case (C3. b), -
,f
In I f l/r[ 2 -
-
-) p-
dv
,
0 or
In [ f l / f T h . U ) v2
0 = o
dv .
(4.16)
L. t~.ukaszuk, Sum rules
58 l
Eq. (4.16) applies also to the Compton scattering on spinless particles e.g. pions. If we assume that also (C3, c) holds, then we get additionally
r
~-r = _1 [ 7r
j
In
du
(4.17)
p-
0
i Jpi; [
or
( in I2'.U. 0 =
"~ p-
0 It seems remarkable that, once the mass of the particle is fixed, the values of the integrals in eqs. (4,16), (4.17)are insensitive to the influence of strong interactions. Let us also notice that e.g. 3'P amplitudes [f112, ]fpfa ]have to fall below ]J~rh.U.I2 in some region above threshold. This does not depend on the values of the strong interactions coupling constants and is a non-trivial statement because at very low energies [8] the amplitudes have to be larger than [fTh.U.]2. Eqs. (4.16) and (4.17) seem to be worthy of checking in the 7P case. On the other hand, if we assume them to be true, eqs. (4.16), (4,17) might be of use in determination of proton's mesic polarizabilities from the low-energy Compton scattering [10l.
5. PION ELECTROMAGNETIC FORM FACTOR The pion form factor has been already extensively examined within the modulus representation [1 1, 12]. We shall add here a new relation between integrals over spacelike m o m e n t u m transfers and timelike (t >~ 4 rn~) transfers of form factors. This will be done under somewhat weaker assumptions than used up to now in similiar considerations. We demand that the form factor F(t) satisfies (F 1) F(t) is analytic in the cut t plane with a cut from 4 m 2 to oo. (F2) Behaviour at infinity: lira r l+e log IF(rei°)l = 0 for 0 ~<0 ~< rr. r~
The condition (F1) results from the local field theory [13], while the "almost exponential" behaviour admitted by (F2) seems to be a quite weak condition: the lower bound for the rate of decrease of form factor [20] in local theories makes it plausible that lim Ir- 1 log IFII <~ const is fulfilled. Instead o f F ( t ) we shall take f ( t ) defined by (we put m 2 = 1)
582
L. ~ukaszuk, Sum rules
f(t) = F(t)F(4-t)
(5.1)
.
This is a symmetric function with respect to the change t ~, 4 - t. The domain of analyticity for f ( t ) is the cut t plane with cuts from 4 to o~ and from ,~ to 0. After introducing the symmetric variable w
w = ~(t-2)
2
,
(5.2)
our w - c u t extends from 1 to ~o Next, the transformation z = ,,/w- 1
(5.3)
maps this cut onto the real axis. Therefore f ( z ) (we replace f ( t ( z ) ) by f ( z ) ) satisfies the conditions of the theorem from sect. 2; a+ = a_ = 0 because the phase at the threshold ~ ~ (t - 4)~-. Of course f ( z = O) = F ( t = O)F(t = 4) = F(4),
(5.4)
and the relation (2.1) takes the form F(t(x)) F(4 - t(x))
In
0 ~<
x2
dx
,
(5.5)
0 or oo
f
0
In I FF(t_~) (4) x2
i dXdt dt ~> -
J
In IF(x2 t ) [
dXdt dt
,
(5.6)
0
with x = 1 ~ / t (t --4-). The timelike region of F ( t ) has not been measured below m~rTr= 600 MeV [14]. The spacelike data [15] at the m o m e n t indicate the magnitude of pion's radius only. Before more experimental information about the low energy and low transfer region is available, the relation (5.6) could be used if some model considerations were added to the existing experimental results. Let us notice, that "unphysical" region 0 ~< t ~< 0.08 GeV 2 does not contribute to the integral from inequality (5.6). However, as mentioned above, the physical low energy region (0.08, 0.34) GeV 2 of reaction e + e ~ n+n - has not yet been experimentally explored. Therefore, the form factors in this interval have to be described using a smooth extrapolation of data from higher energies down to the point t = 0. (Notice that F ( 0 ) = 1 and that F ' ( 0 ) = -~(r2) is approximately known). Such extrapolation will cover unphysical piece (0,0.08) GeV 2, too. Therefore, instead of inequality (5.6) we can use more handy formulae (compare eq. (2.1 .) with Re z = t and ~(a+ + a ) = M / / ~ t t t 0 now):
L. Lukaszuk, Sum rules
1
fj~
IF(t) ] In F(to) dt >1
_~ ( t - %)2
~[ t=to
583
(5.7)
To give an example, we shall discuss inequality (5.7) in connection with the ChouYang model [21 ]. Choosing t o = 0 one has O~/Ot = 0 and F ( 0 ) = 1. Next, if we make the hypothesis [21 ], successful in the case of proton, that the "hadronic stuff" form factor of the pion is proportional to the electromagnetic one, then the contribution from ( - 6 . 0 , 0.36) GeV 2 can be calculated from Chou-Yang's result and is equal 5.2 GeV -2. The form factor is experimentally determined in the interval (0.34, 1) GeV 2 and the contribution from this region does not exceed 2.9 GeV -2. Finally, the negative contribution from ( - oo, _ 10) GeV 2 will be majorized by zero. Inserting these numbers into inequality (5.7) one gets 0.34 I
foo In[F[ dt + t2
0.34
l n l F I dt /> 2.3 GeV - 2 t2
(5.8)
1
Now we want to show that (a) smooth extrapolation of IFI to the region (0,0.34) and (b) assumption that IFI ~< 1 for t > 1 GeV 2 are inconsistent with inequality (5.8). In the interval ( 0 . 3 4 , 0 ) we use the Chou-Yang model and In IFI = 2.36t. If we extrapole this behaviour up to point t = 0.34, we get ]FI 2 = 5 while experimentally IF[ 2 = 4.5 -+ 2. However, the first integral is zero in the linear approximation of In If]. If we put IFL2 = 7 at t = 0.34 then the quadratic correctiol is needed: In [FI "~ 2.36t+ + at 2, a = 1.5 (t ~> 0). In such a case contribution of first integral is (0.51) GeV - 2 i.e. inequality (5.8) is still far from being satisfied. In fact, if one assumes a to be slowly changing with t, the value a--~ 7 is needed in order to fulfill inequality (5.8). Then IF(0.34)12 = 26 instead of experimental 7. Therefore, as long as we keep condition (b), a considerable structure in a(t) is necessary in the (0,0.34) interval. If we tried to release condition (b) having (a) satisfied, then situation would be even more drastic then before. The mean value of In [FL in (1, ~ ) should be between 1.8 and 2.3, i.e. iFI 2 > 4 0 over quite a large region. The alternative would be LFI -+ oo for t -~ ~ , the growth however should be quite quick; Assuming e.g. IF[ = e~',fi-for t > 1 GeV 2 one gets c~ -----1 G e V - 1.
6. CONSISTENCY RELATION F O R THE PION-PION SCATTERING
-
The forward pion-pion scattering amplitudes fulfil within the AFT assumptions twice subtracted dispersion relations [ 16]. This together with the positivity of the
L. tukaszuk, Sum rules
584
absorptive part means that the crossing symmetric amplitude (e.g. f ( n ° n ° ~ n°Tr°) , f ( n ° n ÷-~ n°n+)) has at most one pair of zeroes in its analyticity domain (if, moreover, the amplitude at the threshold F(s = 4) < 0 then there are no zeroes). Therefore, there will be at most one zero on the real negative axis in the w-plane, where w is the symmetric variable (m 2 -= l): w
=
~(s(s
-
4))
(6.1)
The scattering amplitude is analytic in a cut w plane with the cut from 0 to ~ . The transformation z = x/w
(6.2)
maps this cut into the real axis. Again, there can be at most one zero in the upper half plane (on the imaginary axis). The conditions of the theorem from sect. 2 are satisfied and we can apply formula (2.2); if F(z) = ]FI ei~,z = x + iy and eventual zero is at z 0 = iYo, then in the case of e.g. non+ scattering Ox x=O = ~rt'~7r°rr+ k=O =½ aTr/=0~r = F o
~(s=4)-=F(z=O)
,
(6.3)
and the sum rule (2.2) now reads
F(x) 2
-~a •°'÷=-F(z:O)=--2 l=0
Y0
+
In F((O)) x2
1
¢r
dx
,
(6.4)
0 or in the case F(z = O) < 0 +
~
½ a~r
l=lF(z=O)l= 1
l=0
~
F(x) 2 In ~ x2
dx '
(6.5)
0 with x = ~ w/s(s 4). Let us remark that nlr forward differential cross sections can be extracted from experiments through C h e w - L o w ' s extrapolating procedure [22]. At the m o m e n t statistics are rather poor, however eqs. (6.4) or (6.5) can be useful as additional consistency relation in model considerations.
7. DERIVATIVE OF THE PHASE AT THE PEAK OF THE AMPLITUDE First let us recall the well known result of Wigner [17] that if the wave packet is accelerated in the region of interaction then, due to causality, its time advance cannot be too large. This works for the scattering on the short range potential. The
L. Lukaszuk,Sum rules
585
measure of delay (advance) is d61/dk and this derivative cannot be too large negative. In other words if there is a sufficiently narrow bump in the cross section then it has to be connected with the positive time delay and we can speak about a quasi bound state. Results of similar nature were obtained-within the AFT assumption for rrrr scattering [18]: e.g. it was found, that the 7rrr scattering length has a lower bound depending on the pion's mass alone. In this section we formulate a theorem working for the AFT forward scattering amplitude (e.g. ~rlr, #N) or for the pion's electromagnetic form factor which makes it possible to check whether the bump in differential cross section is connected with the positive time delay. Let z be variable for which F(z) (amplitude or form factor) is analytic in the upper half plane with eventual cut along the real axis (e.g. z = ~- X/~s 4) for 7r~ scattering, z = t or z = ~ x/t 4 for the pion's form factor). I f F ( x ) - [FIe i¢ has a bump dominating over the region (x 0 Xl, x0 + Xl) with maximum at x = x 0 and if
F(x O) / °+x' In ~ (X
F(x) dx>(f=
+
/° x' ) h'lF(xO~ dx (X
Xo)2
X0 - X 1
XO)2
(7.1) '
Xo+X 1
then
3~axX=Xo >
0
(7.2)
This result comes immediately from the second theorem of sect. 2 (compare eqs. (2.15), (2.16)). Imagine now that the peak is sufficiently steep (i.e. left-hand side of (7.1) is large) and/or it is dominating over a large region (i.e. x I is large). Then the right-hand side of (7.1) will be negligible in comparison with the left-hand side and the phase derivative at the peak will be positive. A good example of such situation would be the pion's the form factor F~(t) (we need ~ ~> t/> 4). Unless F~ is wildly increasing in hitherto unmeasured regions, then d6 1/dklk=kp > 0 from the knowledge of the process e ÷ e ~ 7r+Tr alone. If the detailed knowledge of form ['actor [F(t)l is given, then it is possible to establish an upper bound for the reduced width of rho meson, Fp (compare eq. (2.2) with d~5/=_1 -
we put z =
t
~(a++a
dx for definiteness)
);
= ~ ] ['p mp
F(rn~)
l I ln F 5> ~ - ~ (t
rn2) P- 2
dt
(7.3)
586
L, £ukaszuk, Sum rules
In particular, if the form factor IF[ is majorized by simple Breit-Wigner shape with [FBw(m2)I = IF(m2)l and width I'B, then P o < P B independently of the distribution of zeros':
I If IF(t)l ~< IFB.w,(t)I --
m ° FB
F(m~)
t-~m2p-~i r B m p l
(7.4)
then d6/=_1 ] - 1
z
P
~ FB
8. CONCLUSIONS Weak analyticity assumptions led to several consistency relations for scattering processes. These relations apply to quite unlike reactions involving leptons or hadrons, or both of them. They seem to be strongly restrictive: a good example may be the discussion of pion's form factor in sect. 5. The applications made and main results are sketched in sect. 1 and here we would like to point out possible future developements: similar sum rules should work for up -+ e-n scattering, weak form factors, forward virtual Compton scattering. It would be also interesting to generalize our relations for the case of 3' - nucleus Compton scattering. The author is indebted to Professor J.D. Bjorken for communicating the results of ref. [4] before publication.
APPENDIX The following theorems due to Nevannlina [1] and Muskhelishvili [2] are used in deriving the results of sect. 2: Theorem A (Nevannlina)
|f f(z) is regular in the upper half plane, continuous in the closed half plane, its zeros in the upper half plane have no finite limit point, and it satisfies a = lira inf r -1 log M ( r ) < p - - > oo
(M (r) = max if(z)[ for Iz[ = r) and
J log* V(x)l ~ < 1 +x 2 --oo
~,
L. Lukaszuk, Sum rules
587
or
then
f
x 2 log [f(x)l dx and J
1
1
log If(z)t = log
1
z/~ n
x 21og lf( x)l dx exist ,
7r
(t x)2+V 2
"
where z n are the zeros o f f ( z ) in the upper half plane and n c = lim -2 r 1 [
J
7"f
r~
~
log b"(r ei°)l sin 0 dO
0
(z n is c o m p l e x conjugate of Zn, z - x + iy). Eq. (A. 1 ) still can be replaced by inequality if no assumption about distribution of zeroes is made. Then [1]: f
log if(z)l
log If(t)]
dt + cy
(A.2)
Theorem B (Muskhelishvili) Let L be c o n t o u r and let ~p(t) satisfies the H O ) condition on L; let S ÷, S - be the regions of plane b o u n d e d by L, then in each of the regions S÷+ L, S - + L, the function
1
O(z) = U~i satisfies condition
[ ,,~(t) J
t~
dt
L
IO(z2)-O(Zl)l~clz 2
ZllU for/J < 1
or
[O(z2)-O(Zl)l~clz 2
2'1 I1 e f o r / . t = l
,
where e is arbitrary small positive constant, c is a constant and by 0(z) with z E L must bw u n d e r s t o o d the corresponding limiting value 0÷ or 0 - .
REFERENCES I l] 12] [3] [4] [5]
R.P. Boas, Entire functions (Academic Press, New York, 1954). N.I. Muskhclishvili, Singular integral equations (Groningen, 1953). l.Ya. Pomeranchuk, Yad. Fiz. 11 (1970) 852. T. Appelquist, J.l). Bjorkjen, Phys. Rev. 4D (1971) 3726. M. Damashck, 17.J. Gilman, Phys. Rev. ID (1970) 1319.
588
L. Lukaszuk, S u m rules
[6] tt. Cheng, T.T. Wu, Phys. Rev. 182 (1969) 1852. 171 F.E. Low, Phys. Rev. 96 (1954) 1428; M. Gell-Mann, M.L. Goldberger, Phys. Rev. 96 (1954) 1433; K.Y. Lin, Nuovo Cimento 2A (1971) 695; in this paper further references can be found. 181 S.B. Gerasimov, L.D. Soloviev, Nucl. Phys. 74 (1965) 589. 19] T.N. Truong, Phys. Letters 31B (1970) 461. [101 P.S. Baranov, L.V. Filkov, G.A. Sokol, Fortschr. Phys. 16 (1968) 595. [ 11 ] T.N. Truong, R. Vinh-Mau, Phan Xuan Yem, Phys. Rev. 172 (1968) 1645 ; T.N. Truong, R. Vinh-Mau, Phys. Rev. 177 (1969) 2494. [121 J.E. Bowcock, Th. Kannelopoulos, Nucl. Phys. B3 (1968)417. 113] R. Oehme, Phys. Rev. 111 (1958) 1430. [14] V.L. Auslander et al, Phys. Letters 25B (1967) 433; J.E. Augustin et al. Phys. Rev. Letters 20 (1968) 126; Phys. Letters 28B (1968) 508. 115] C.W. Akerlof et al. Phys. Rev. Letters 16 (1966) 147. [ 16] A. Martin, Nuovo Cimento 42A (1966) 930; Y.S. Jim A. Martin, Phys. Rev. 135B (1964) 1369. [17] E.P. Wigner, Phys. Rev. 98 (1955) 145. [ 18] k. Lukaszuk, A. Martin, Nuovo Cimento 52 (1967) 122; B. Bonnier, R. Vinh-Mau Phys. Rev. 165 (1968) 1923; A.K. Common, Nuovo Cimento 63A (1969) 451. [19] Dolgov, Zacharov, Okun, Yad. Fiz. 14 (1971) 1247. [20] A. Martin, Nuovo Cimento 37 (1965)671; A.M. Jaffe, Phys. Rev. Letters 17 (1966)661. [211 T.T. Chou, C.N. Yang, ttigh-energy physics and nuclear structure, ed. G. Alexander (North ttolland, 1967) p. 348. [22] G.F. Chew, F.E. Low, Phys. Rev. 113 (1959) 1640; J.P. Baton et al. Phys. Letters 25B (1967) 419.
NOTE ADDEDINPROOF: In sect. 5, referring to tire c o n c l u s i o n s o f C h o u and Y a n g we have o v e r l o o k e d their p a p e r p u b l i s h e d in Phys. Rev. ( 1 9 6 9 ) 2469. T h e f o r m f a c t o r given there is c o n s i s t e n t w i t h o u r i n e q u a l i t y (5.7).
Nuclear Physics B47 {1972) 569-588. North-llolland Publishing Company
A FAMILY OF SUM RULES FOR WEAK, ELECTROMAGNETIC AND STRONG PROCESSES L. L U K A S Z U K Institute o f Nuclear Research, Warsaw *
Received 21 April 1972
Abstract: New sum rules are derived for the neutrino-electron and pion-pion Compton forward scattering amplitudes and for the pion's electromagnetic form factor. I t is also shown that a peak in the differential cross section or form factor has to be connected with the positive derivative of the phase at the maximum, provided that the peak is steep enough and/or dominating over a sufficiently large region.
1. I N T R O D U C T I O N The sum rules o b t a i n e d in this p a p e r relate the integrals over l o g a r i t h m s o f differential cross s e c t i o n s (or cross sections t h e m s e l v e s ) to the characteristic c o n s t a n t s o f i n t e r a c t i o n like F e r m i ' s c o u p l i n g c o n s t a n t G, e 2, the s c a t t e r i n g l e n g t h a]r__~for t h e rr -- n a m p l i t u d e s or the w i d t h F of" the r e s o n a n c e p. These relations hold u n d e r q u i t e w e a k a n a l y t i c i t y a s s u m p t i o n s w h i c h are stated in sect. 2. In this section the f o r n m l a e are derived w h i c h will be applied to the physical processes. T h e q u i c k derivation goes as follows. Let us start w i t h the m o d u l u s r e p r e s e n t a t i o n I I 1, 12] o f the anaplitude (or f o r m f a c t o r ) F = IFI e i9 (assume F ( 0 ) = 1, z = x + iy is a variable with respect to w h i c h l= has a cut along real axis). D i f f e r e n t i a t i n g it with respect to x at z = 0 we get a~
ax
z=0_
1 [
,
111 I f ( dx t ) Fx ( - x~l ) l 2 -
- x
1
i
(1.1)
0 Now tire derivative o f p h a s e m a y be related t h r o u g h the k n o w n low e n e r g y b e h a #Tr viour o f the c o n s i d e r e d physical processes to G, e -"), al= 0 or, if e.g. one takes x = 0 at the r e s o n a n c e ' s p o s i t i o n , to the 1/F. In fact one has to be m o r e careful t h a n we were above: we have here the limiting process, it m a y also h a p p e n (as it d o e s in the case o f ue ~ ue) t h a t a ~ / O x is n o t cont i n u o u s at x = 0, care s h o u l d also b e t a k e n o f zeroes and o f an a s y m p t o t i c b e h a * Postal address: Warszawa 10, Zaklad VII 1 BJ, ul. Hoza 69, Poland.
570
L. ~ u k a s z u k , S u m rules"
viour. We deal in detail with these problems in sect. 2, where the generalized form of relation (1.1) is derived from Nevannlina's theorem 111. In sect. 3 we discuss necessary assumptions and derive sum rules for the ue, ;re forward scattering amplitudes. This is made (see the inequality (3.10)) for the weak analyticity assumption (ahnost exponential behaviour admitted) and also for the case when the twice subtracted dispersion relations hold (eq. (3.17)). In the latter case the knowledge of the forward differential cross section is enough to determine the sign of Ferret's coupling constant G , e. In the general case a theorem is proved which asserts that 1t,~. l'i,et cannot fall to much below perturbative (first order) resuits. Sect. 4 deals with tire forward Compton scattering (TP, "ye, or 7rr). The low-energy behaviour coming here from Q.E.D. and the weak analyticity assumptions allow the derivation of inequalities (4.9} (or equiwdently (4.12)) for the amplitude ./1 and inequalities (4.8)(or equivalently (4.13)) for the helicity amplitudes {@ ".11~1.As a consequence lhese amplitudes cannot fall too much below the unitarized Thomson's amplitudes. If one assumes substracted dispersion relations then the equalities are obtained (see eqs. (4.16), (4.17)). The results of sects. 3, 4 hold to all orders of G, e 2, respectively, once the initial analyticity assumptions are made. In sect. 5 we derive new sum rule (5.6) relating the integral over the spacelike pion's e.m. form factor to the integral over the timelike (s ~ 4m~) region. Application to the Chou-Yang model [21] is made. In sect. 6 we arrive at the consistency relation (6.4), (6.5) for rr rr scaltering amplitudes; the assumptions made here result from tire AFT. Finally, in sect. 7 we show that the modulus representation can be used to determine the sign of the phase derivative at the peak of the differential cross section. A possibility of finding the lower bound 1o the resonances reduced width is pointed out.
Theorems by Nevannlina [1] and Muskhelishvili [2] are quoted in the appendix.
2. MATIIEMAT1CAL PRELIMINARIES ha this section we shall derive tire following theorem which will be applied to the physical processes in subsequent sections. Given a function F ( z ) such that (i) F ( z ) is regular in the upper half plane, continuous m the closed half plane of z " z = x + iy = r e i ° ;
(it) lira r l+r~In IF(re i°)] = 0; r ~
hn F( + x + i c) (iii) F ( O ) = 1,
-+x
=
x~0+
a+ + O(x~/); where e, r/, r/ are arbitrarily
small positive constants, a . are arbitrary constants and 0 ~< 0 ~< 7r.
L. Lukaszuk, Suez rules
571
Then In IF(x) F ( - x ) l
½(a++a ) ~ l_Tr o
where F(+-x) =
dx
(2.1)
x2
lim F(+-x+ie). e~ 0 +
Moreover, if F ( z ) has at most finite number of zeros in the upper half plane then eq. (2.1) can be replaced by oo
2Yn
1 f
23 2+-n~ 0J n xn
In I Y ( x ) Y ( X ) l d x
(2.2)
X2
(Throughout this paper symbols r/, r~', e, denote arbitrarily small positive constants. Symbol O(x) (for x ~ 0 ) denotes terms which are at most of order x). First let us derive formula (2.1). Function F(z) satisfies (i), (ii), and therefore, from Nevannlina's theorem it has to satisfy inequality (A.2) for any z in the upper half plane. Let us choose z = iv, then In [F(iy)l ~< 1 [ y zr j
In IF(t)[ dt t2 +y2
(2.3)
(In our case C = 0 in (A.2) due to the condition (ii).) The inequality (2.3) is fulfilled for any y > 0. What we shall prove now is the existence of limits lim In [F(iy)l y--+O Y lira ~
y-+O J
71( a + + a
In If(t)[ dt =
)
(2.4)
In IF(t) F ( t)l dt < o o
t2 + y2
--~
,
(2.5)
t2 0
In order to do this, it is enough to know the behaviour o f F ( z ) around z = 0 above and on the real axis. Making the subtraction at z = 0 (possible because Im F(x)/x has at most a finite discontinuity at x = 0) and using a finite contour e.g. a semicircle with chord covering segment [ - 1 , 1] along the real axis - one gets from Cauchy formula 1
F(z) = 1 + zrt I.
ImF(x+ie)
x(x-z)
dx + z O(z) ,
-I where cb(z)is holomorphic in the neighbourhood o f z = 0:
(2.6)
L. ¢~ukaszuk, S u m rules
572
1 ;
[F(ets°)12.1 + F(ei~)- 1 ] dso
(2.7)
0 Hence, around z = 0 q~(z) = ~ ( z ) and q~(0) is real
(2.8)
It will be convenient to rewrite eq. (2.6) in the form 1
F(z) = l + Zrr I
l- mxF(~(xx +zi )e- )- - d x + cz + O(z2) ,
(2.9)
I
where the constant
c =- ~b(0) is real It is possible, due to the assumption (iii), to write the integral from eq. (2.9) as 1
I
1
1
hn F(x + i e) ; a + dx x(x-z) dx = x z
I
0
- 1
a x + z dX+¢l(Z ) ,
(2.10)
0
where 1
~1(z)
= ~
~0(x) ax X-Z
J
-1 with real ~;(x)satisfying ttolder's condition H(r~')(compare (iii))in the neighbourhood o f z = O. Therefore, using theorem B from the appendix, we can write ~;t(z) in the form el(Z) = c ' + O ( z 7~) ,
(2.11)
where c' is real constant. Finally, eq. (2.9) can be written in the form: F(z)= l+-Z [a_lnz
a+ln(ze
i~r)+c l + O ( z n ' ) ]
(2.12)
7r
where c 1 is a real constant. Using eq. (2.12) we can easily calculate the limit lira In _v~0
[F(iy)[ 2 2y
lira 1Gin I[1
½y(a++ a _]2+o0'l+r/!)[
--½(a++a )
r~0
which agrees with eq. (2.4). In the case a+ = a , i.e. when the derivative of the phase o f F - = IFI e i~' with respect to x is continuous at x = 0 (compare (iii)), eq. (2.4) means a lnay[F[ z=0 -
34~3x z=0
L. 15ukaszuk, Sum rules
573
This is the limiting value o f the Cauchy - Riemann relation between modulus and phase. In order to show that eq. (2.5) holds, it is enough to find the behaviour of IF(-+ x + ie)l 2 around x = 0. Eq. (2.12) gives x2
IF(x + ie) F( x + ie)l x ~.0 = 1 - ~ [a+ - a _ ) In x - c I ] 2 + O (x 1+~')(2.13) Hence 1
f
1
lnlF(t)F(-t)l
dt = f
t2+y2 0
0
~(r) dr r + y2 '
(2.14)
where s0(r) satisfies the condition H(~r/). Therefore, from the theorem B dix) the integral
(see
appen-
1
~l(V2) = I
~(r) dr r+y 2
0 has a limit and is continuous i n y 2 = 0 in accordance with eq. (2.5) We have proved eq. (2.1). In the case when the zeros are given, 9ne has to calculate the additional limit lira 1 v~O y
In l-I n
tl z/zn
in eq. ( A . I ) which gives the sum in eq. (2.2). The direct consequence of formula (2.1) is the following statement: Let us assume that the derivative of phase, 3~/3x, of the function F(z) exists at z = 0. If F ( z ) satisfies (i), (ii), and if l°g+
F(x) F(-x)
F2(0) I
x2 0
oo dx <
I
F2(0) log+
I
F(x) F(-x) [ dx x2
,
(2.15)
0
then lb3
=ax z=0 > 0 3. W E A K
INTERACTIONS: ve SCATTERING
3. I. Assumptions It has
already been shown that analyticity, when applied to the neutrino
(2.16)
L. Lukaszuk, Sum rules
574
electron scattering amplitudes gives quite restrictive results [3,4, 19]. In this section we are going to write down sum rules which work under weaker conditions than those assumed up to now. Let us state assumptions under which the results of this section will be derived. (Wl) rn v = me= 0, right-handed electrons do not take part in the interaction. (W2) Ana/yticity: The forward scattering amplitude Fve(S ) satisfies condition (i) from sect. 2 (complex variable s is the total c.m.s, energy squared for the real, positive values). (W3) Crossing: The forward scattering amplitude for re-+re, Fve(s ) is obtained from Fve(S):
Fve(S )
= FFe ( s )
(W4) Behaviour at s = 0: In the limit Re s-+0 the re, ue interactions are described by the V - A type of interaction. We assume also that the total cross section oto t for, ve(~e) scattering is equal to the total elastic cross section Otot elastic (s -----0)(1 + O(s ~7)). (W5) Behaviour at infinity - we shall distinguish here two cases: (a) Fve(S ) can be almost exponential when s -+ oo lira r - l + e In
IFvc(r e/°)[
= O for O ~< 0 ~< 7r ,
r --;'- o o
or (b) Fve(s) satisfies twice subtracted dispersion relations. Hence, the total cross section has to satisfy condition Oto t
~
CS
[
C
Once we have put m e = O in condition (W 1), the rejection of the right-handed electrons is a natural consequence of the V - A theory's success. The condition (WI) is not necessary for existence of the sum rule and is put here in order to simplify kinematics: we have one chiral amplitude describing the scattering ve (compare ref. [3]). Thus ve and/~e scattering is described by the same analytic function due to the condition (W3). The conditions (W2), (W3), (W5) have been nowhere proved; the axiomatic approach is not valid for massless particles, on tile other hand the existing theory of weak interactions is non-renormalizable. However, :hese conditions seem to be the weakest ones which might be deduced from our theoretical experience with strong interactions or perturbation theory [3] once we decided to make some use of the analyticity in the weak processes. In fact one usually [3, 4, 19] starts from stronger assumptions than ours. The condition (W4) is consistent with the experimental evidence about other weak processes. In calculating the total cross sections at the threshold we assume that the contribution from the higher unitarity corrections [4, 19], creations of additional v~ pairs, etc. is contained in the term O(s l+n3. Such a condition is easily satisfied in the dynamical approach of Appelquist and Bjorken [4] and of Dolqov, Zachanov and Okun [19], where a correction O(s 2 In s) appears.
L. t;ukaszuk, Sum rules
575
3.2. S u m rule f o r Fue(S ) - the general case The amplitude Fue(S ) satisfying (W2), (W5) (in this section we shall deal with the more general case (a)) fulfills conditions (i), (ii) o f sect. 2. The low-energy behaviour (W4) guarantees that (iii) is fulfilled too. Let us express a+, a_ from eq. (2. I) in terms of the ve coupling constant G. From (W4) the phenomenological interaction Hamiltonian of the form G (~e 7c~(1 + 75) ~v ) ( f v 7a(1 + 3'5) ~e ) (3.1) is responsible for the re, ~e scattering in the limit s-->O. From (3.1) one gets R e F v e ( S , t ) = 4 v/2 G s = F °
,
Re F~e(S , t) = - 4 x/2 G(s + t)
(3.2) .
(3.3)
The imaginary amplitudes for forward scattering can be determined through the optical theorem: lm F = SOtot(S )
.
(3.4)
Let us fix units introducing the dimensionless variable z -~ 4 x/21GIs
(3.5)
then F o = z sgn(G) (compare (3.2)). Further on we shall use z instead o f s and x = Rez instead if Res. If, e.g, one assumes universal value for G, IGI = 10 - 5 m ~ 2, then x = 1 corresponds to x/~ = 130 GeV. Now, from eqs. (3.2) (3.4) and condition (W4) one gets x2 lm F e ( X + ie) - 167r ' (3.6) Im F e ( - X + ie) = - Im F-e(X + ie)
x2 31 16rr
_
(3.7)
Let us normalize Fve(Z) to unity in z = 0 : Fe(Z) F(z) =
(3.8)
z sgn (G)
Then, applying the definition of a_+ from (iii) to the function F we obtain -~
l(a++ a )_
a
1
3 167r s g n G
(3.9)
Therefore, inequality (2.1) looks as follows in the ve case: Fve F~e
F e F~e oo
48
48
x2 0
-
x2 0
-
L. Lukaszuk, Sum rules
576
The relation (3.10) is true in the general case (a) of (W5); e.g. the behaviour
F(z) ~
exp (z l - e )
is still leading to the inequality (3.10). The inequality (3.10) is sensitive with respect to the behaviour of amplitudes at low values o f x and might be used in checking eventual models. If'we knew the amplitudes from threshold up to, say, x 1, then the integral over differential cross sections - taken from x l to oo _ could be bounded from below. Before we demonstrate it let us define R 2 = [F~e Fve(X)[
(3.11)
x-
X/~ve/dt dOFe/dt~
t=0
where doo/dt = 2 G2/Trdenotes the differential cross section with Fve replaced by Fo([F o] - x, comp. eq. (3.5)) i.e. the first order perturbative result. Let us also denote the integral over " k n o w n " region as XI
I_
|r J 0
lnR~2 dx x2
(3.12)
Then we have the following theorem. I f R 2 ~< x l - e for x "+ ~ , then the following inequality holds:
=l/d°"e f
?
~
e] It=O
d-d;a ° t=0
1 exp(
Xl/48-XlI )
(3.13)
XXl
-
2G 2 gX
exp ( - x ]/48 - x 1 I)
If, moreover, R 12 ~< x - 1 - e for x ~ 0% then
f
/ d ° v e dOTe t=0
2 G2
x 1 exp ( - X l / 4 8 - x 1 I)
.
(3.14)
XI
Eq. (3.13) is satisfied if e.g. the number L of partial waves giving finite contributions 3 at infinity satisfies condition L ~< c x~ - e . Let us also notice that conditions leading to eq. (3.14) are fulfilled if e.g. only a finite number of partial waves contribute to the cross section at high energies.
L. £ukaszuk, Sum rules
577
In order to prove (3.13), (3.14) let us notice that from the conditions of the theorem and from (3.10) 1-4~
< ~
lnR~2 dx x2
Xl
Using now a theorem about geometrical and arithmetical means, we get I - I f ~<
° l n R 2 cLv ~< i ln(x 1 ~ R2 dx) x-
x5
Xl
'
Xl
which gives (3.13) and (3.14) 3.3. Twice subtracted case In this case Fve(Z ) has no zeros in the upper half plane. In order to show this, let us write the dispersion formula (we work with variable z = 4 x/2 [G[ s):
z2
ve
Otot --dx+
Fve(Z ) = sgn (G) z +
x(x z)
2 z-
o
j jetot - -
~
x(x+z)
dx
(3.15)
0
Let us assume that Fve(Z ) = O at z o = x o + iYo, y o ~ O. Then, from eq. (3.15) we have
yo I = O ,
xo I1 = O
,
(3.16)
where I - sgn (G)
lZot2
1 I n 0
t o t dx + 1 j xlx Zo 12 n 0
ore ii = 1 [ 7r
ff
to___.~t dx + 1
- - -°tet -dx xlX+Z o 12
,
~e ] °t°~t dx > 0 iX+Zol2
iX_Zo[2 n o o We are interested in the case Yo 4: 0; eqs. (3.16) would then lead to the condition I 1 = 0 which is impossible unless there is no interaction at all. Now we use formulas (2.2), (3.9) and we get the equality
i Fve Fve 1 sgn(G) = [ In 48 j x2 dx (3.17) 0 Our sum rule makes it possible to determine the sign of G once the information about forward differential cross sections is given.
578
L. £ukaszuk, Sum rules
4. FORWARD COMPTON SCATTERING 4.1. A s s u m p t i o n s
Let us consider forward Compton scattering on the proton or electron. Then the process is described by two amplitudes, e.g. fl(p), f2(u) or fp(U), fa(U) (for definition see e.g. [5]). Let us formulate properties of these amplitudes which will be demanded in this section. (C1) Analyticity and crossing: The forward scattering amplitudes fl(v), f2(u) satisfy condition (i) from sect. 2, fl(u) being symmetric, f2(u) antisymmetric with respect to operation of crossing (complex variable u is lab, energy of the photon on the real axis). (C2) Behaviour at v = 0: given rigorously by Q.E.D. (C3) Behaviour at infinity: We shall distinguish here three cases: ( a ) f l , f 2 can be almost exponential lim r
l+e
log [f(rei°)l = 0
;
(b) f l satisfies twice subtracted dispersion relations; (c)f2 satisfies unsubtracted dispersion relations, f l satisfying (b). Analyticity assumed here can be checked in perturbation theory for the 3' e case; e.g. from the w o n of Cheng and Wu [6] one knows that (C3, c) is satisfied up to the sixth order of perturbation tl~ory in Q.E.D. Of course the cases (a), (b) of (C3) are then fulfilled, too. 4.2. S u m rule - the general case
Now we assume the most general asymptotic behaviour from (C3), namely "almost exponential" case (a), Therefore our sum rule will be the inequality (2.1); we are going to determine constants a+, a_ in terms of r = ~ / m . Consider amplitudes fp(u), fa(U) corresponding to spins parallel and antiparallel, respectively. The crossing properties of fl, f2 (see (C 1)) are equivalent to the condition (compare e.g. [5]) ¢p ( - ~ ) = fa(U)
(4.1)
The behaviour around u = 0 results from the low-energy theorem [7, 8]. We are interested in the value Offp and a T = ½(Op + %) at u = 0: fl(0) = fp(0) = fa(0) = OT
~/m ~ -r
= l ( o +Oa ) = 8~ r2(1 +O(v))
,
,
(4.2) (4.3)
.5
where, from the optical theorem P
Im fp,a(V) = ~
lop, a(0) + O(v)]
(4.4)
579
L. t;ukaszuk, Sum rules
Eq. (2.1) will be applied to = fp(P) F(u) fl (0) (4.5) From assumption (C 1) and (C3, a) it is evident that conditions (i), (ii) of sect. 2 are satisfied. The low-energy behaviour given by eqs. (4.2), (4.4) yields F(u) satisfying condition (iii): F ( u + ie)
=
v~0+
F(v+ie)
P
l +i
=
47r fl(0) P
l-i----
v~0+
[Op(0)+O(v)]
4n'fl(0)
[Oa(0 ) + O@)1
(4.6)
Tile constant ~(a+ + a ) from eq. (2.1) is therefore equal to 1 OT(0 ) = }(a++a ) - 41rfl(0)
~- r
(4.7)
It is also evident from eqs. (4.1), (4.5) that + ic)
IF( v+ie)[ =
¢1(0 )
Finally, eq. (2.1) can be written as
r ~<
i
) J:r
In
| dp p_) J 0 Similar result can be obtained for the amplitude
(4.8)
7T
In this case
fl(P).
P
Ifl(--u)l = Ifl0')[
'
l m f l = 4-rr °T
'
and instead of (4.8) one has V 2
~ r ~< 1 [ 71" J
In ~ r l - du
(4.9)
p_
0
Eqs. (4.8), (4.9) are valid in all orders of strong and electromagnetic interactions once properties (CI), (C3, a) are assumed. We are going to demonstrate that eq. (4.9) prevents the situation in which e.g. iflL would be everywhere smaller than the unitarised Thomson's amplitude, fTh.U.: fTh.U. -= - - r + [
P 4~
OT(0)
(4.10)
Indeed, fTh.U./(--r) satisfies the conditions of the theorem from sect. 2 with no zeroes
in the upper half plane. Hence from eq. (2.2)
580
L. £ukaszuk, Sum rules
i fTh.U. 2 1 [
~ r -=
In
-r--
dv
") p_
17" J
(4.11)
0 Subtracting eq. (4.1 !) from (4.9) one gets
0 ~< |J~ In
dv
v2
(4.12)
0 Similar result will be obtained from eq. (4.8) fp fa in ]faTh U
f
v 2 " ' dv . (4.13) 0 The inequalities (4.12), (4.13) are true for electron, proton or pion (where sum rule f o r f 1 applies) in all orders of electromagnetic and strong interactions, provided that our rather weak analyticity assumptions are nrade. 0 <~
4.3. N o z e r o e s case We shall assume now that fl satisfies condition (C3, b). Then such an amplitude has no zeroes in the upper half-plane [9]. If, moreover, f l satisfies an unsubtracted dispersion relation (i.e. condition (C3, c) is fulfilled) then also fp(V) and, equivalently, fa(V) has no zeroes in the upper half-plane: fp(V) = - - r + ~
(x-v)--~
dx
-
(x+v)x
dx
(4.14)
0 0 Let us assume that fp(V0) = 0 for v 0 = x 0 + iYo, YO =/: O. Then, taking the imaginary part OfJp(Vo)/V 0 one gets Qo
oo
-m iv012
dr+7r
x l x - Vo12
0
dr > 0 7r
0
(4.15)
X[X + Vo 12
i.e. inconsistency. A similar proof works for ft" Now we can apply eq. (2.2) and, in the case (C3. b), -
,f
In I f l/r[ 2 -
-
-) p-
dv
,
0 or
In [ f l / f T h . U ) v2
0 = o
dv .
(4.16)
L. t~.ukaszuk, Sum rules
58 l
Eq. (4.16) applies also to the Compton scattering on spinless particles e.g. pions. If we assume that also (C3, c) holds, then we get additionally
r
~-r = _1 [ 7r
j
In
du
(4.17)
p-
0
i Jpi; [
or
( in I2'.U. 0 =
"~ p-
0 It seems remarkable that, once the mass of the particle is fixed, the values of the integrals in eqs. (4,16), (4.17)are insensitive to the influence of strong interactions. Let us also notice that e.g. 3'P amplitudes [f112, ]fpfa ]have to fall below ]J~rh.U.I2 in some region above threshold. This does not depend on the values of the strong interactions coupling constants and is a non-trivial statement because at very low energies [8] the amplitudes have to be larger than [fTh.U.]2. Eqs. (4.16) and (4.17) seem to be worthy of checking in the 7P case. On the other hand, if we assume them to be true, eqs. (4.16), (4,17) might be of use in determination of proton's mesic polarizabilities from the low-energy Compton scattering [10l.
5. PION ELECTROMAGNETIC FORM FACTOR The pion form factor has been already extensively examined within the modulus representation [1 1, 12]. We shall add here a new relation between integrals over spacelike m o m e n t u m transfers and timelike (t >~ 4 rn~) transfers of form factors. This will be done under somewhat weaker assumptions than used up to now in similiar considerations. We demand that the form factor F(t) satisfies (F 1) F(t) is analytic in the cut t plane with a cut from 4 m 2 to oo. (F2) Behaviour at infinity: lira r l+e log IF(rei°)l = 0 for 0 ~<0 ~< rr. r~
The condition (F1) results from the local field theory [13], while the "almost exponential" behaviour admitted by (F2) seems to be a quite weak condition: the lower bound for the rate of decrease of form factor [20] in local theories makes it plausible that lim Ir- 1 log IFII <~ const is fulfilled. Instead o f F ( t ) we shall take f ( t ) defined by (we put m 2 = 1)
582
L. ~ukaszuk, Sum rules
f(t) = F(t)F(4-t)
(5.1)
.
This is a symmetric function with respect to the change t ~, 4 - t. The domain of analyticity for f ( t ) is the cut t plane with cuts from 4 to o~ and from ,~ to 0. After introducing the symmetric variable w
w = ~(t-2)
2
,
(5.2)
our w - c u t extends from 1 to ~o Next, the transformation z = ,,/w- 1
(5.3)
maps this cut onto the real axis. Therefore f ( z ) (we replace f ( t ( z ) ) by f ( z ) ) satisfies the conditions of the theorem from sect. 2; a+ = a_ = 0 because the phase at the threshold ~ ~ (t - 4)~-. Of course f ( z = O) = F ( t = O)F(t = 4) = F(4),
(5.4)
and the relation (2.1) takes the form F(t(x)) F(4 - t(x))
In
0 ~<
x2
dx
,
(5.5)
0 or oo
f
0
In I FF(t_~) (4) x2
i dXdt dt ~> -
J
In IF(x2 t ) [
dXdt dt
,
(5.6)
0
with x = 1 ~ / t (t --4-). The timelike region of F ( t ) has not been measured below m~rTr= 600 MeV [14]. The spacelike data [15] at the m o m e n t indicate the magnitude of pion's radius only. Before more experimental information about the low energy and low transfer region is available, the relation (5.6) could be used if some model considerations were added to the existing experimental results. Let us notice, that "unphysical" region 0 ~< t ~< 0.08 GeV 2 does not contribute to the integral from inequality (5.6). However, as mentioned above, the physical low energy region (0.08, 0.34) GeV 2 of reaction e + e ~ n+n - has not yet been experimentally explored. Therefore, the form factors in this interval have to be described using a smooth extrapolation of data from higher energies down to the point t = 0. (Notice that F ( 0 ) = 1 and that F ' ( 0 ) = -~(r2) is approximately known). Such extrapolation will cover unphysical piece (0,0.08) GeV 2, too. Therefore, instead of inequality (5.6) we can use more handy formulae (compare eq. (2.1 .) with Re z = t and ~(a+ + a ) = M / / ~ t t t 0 now):
L. Lukaszuk, Sum rules
1
fj~
IF(t) ] In F(to) dt >1
_~ ( t - %)2
~[ t=to
583
(5.7)
To give an example, we shall discuss inequality (5.7) in connection with the ChouYang model [21 ]. Choosing t o = 0 one has O~/Ot = 0 and F ( 0 ) = 1. Next, if we make the hypothesis [21 ], successful in the case of proton, that the "hadronic stuff" form factor of the pion is proportional to the electromagnetic one, then the contribution from ( - 6 . 0 , 0.36) GeV 2 can be calculated from Chou-Yang's result and is equal 5.2 GeV -2. The form factor is experimentally determined in the interval (0.34, 1) GeV 2 and the contribution from this region does not exceed 2.9 GeV -2. Finally, the negative contribution from ( - oo, _ 10) GeV 2 will be majorized by zero. Inserting these numbers into inequality (5.7) one gets 0.34 I
foo In[F[ dt + t2
0.34
l n l F I dt /> 2.3 GeV - 2 t2
(5.8)
1
Now we want to show that (a) smooth extrapolation of IFI to the region (0,0.34) and (b) assumption that IFI ~< 1 for t > 1 GeV 2 are inconsistent with inequality (5.8). In the interval ( 0 . 3 4 , 0 ) we use the Chou-Yang model and In IFI = 2.36t. If we extrapole this behaviour up to point t = 0.34, we get ]FI 2 = 5 while experimentally IF[ 2 = 4.5 -+ 2. However, the first integral is zero in the linear approximation of In If]. If we put IFL2 = 7 at t = 0.34 then the quadratic correctiol is needed: In [FI "~ 2.36t+ + at 2, a = 1.5 (t ~> 0). In such a case contribution of first integral is (0.51) GeV - 2 i.e. inequality (5.8) is still far from being satisfied. In fact, if one assumes a to be slowly changing with t, the value a--~ 7 is needed in order to fulfill inequality (5.8). Then IF(0.34)12 = 26 instead of experimental 7. Therefore, as long as we keep condition (b), a considerable structure in a(t) is necessary in the (0,0.34) interval. If we tried to release condition (b) having (a) satisfied, then situation would be even more drastic then before. The mean value of In [FL in (1, ~ ) should be between 1.8 and 2.3, i.e. iFI 2 > 4 0 over quite a large region. The alternative would be LFI -+ oo for t -~ ~ , the growth however should be quite quick; Assuming e.g. IF[ = e~',fi-for t > 1 GeV 2 one gets c~ -----1 G e V - 1.
6. CONSISTENCY RELATION F O R THE PION-PION SCATTERING
-
The forward pion-pion scattering amplitudes fulfil within the AFT assumptions twice subtracted dispersion relations [ 16]. This together with the positivity of the
L. tukaszuk, Sum rules
584
absorptive part means that the crossing symmetric amplitude (e.g. f ( n ° n ° ~ n°Tr°) , f ( n ° n ÷-~ n°n+)) has at most one pair of zeroes in its analyticity domain (if, moreover, the amplitude at the threshold F(s = 4) < 0 then there are no zeroes). Therefore, there will be at most one zero on the real negative axis in the w-plane, where w is the symmetric variable (m 2 -= l): w
=
~(s(s
-
4))
(6.1)
The scattering amplitude is analytic in a cut w plane with the cut from 0 to ~ . The transformation z = x/w
(6.2)
maps this cut into the real axis. Again, there can be at most one zero in the upper half plane (on the imaginary axis). The conditions of the theorem from sect. 2 are satisfied and we can apply formula (2.2); if F(z) = ]FI ei~,z = x + iy and eventual zero is at z 0 = iYo, then in the case of e.g. non+ scattering Ox x=O = ~rt'~7r°rr+ k=O =½ aTr/=0~r = F o
~(s=4)-=F(z=O)
,
(6.3)
and the sum rule (2.2) now reads
F(x) 2
-~a •°'÷=-F(z:O)=--2 l=0
Y0
+
In F((O)) x2
1
¢r
dx
,
(6.4)
0 or in the case F(z = O) < 0 +
~
½ a~r
l=lF(z=O)l= 1
l=0
~
F(x) 2 In ~ x2
dx '
(6.5)
0 with x = ~ w/s(s 4). Let us remark that nlr forward differential cross sections can be extracted from experiments through C h e w - L o w ' s extrapolating procedure [22]. At the m o m e n t statistics are rather poor, however eqs. (6.4) or (6.5) can be useful as additional consistency relation in model considerations.
7. DERIVATIVE OF THE PHASE AT THE PEAK OF THE AMPLITUDE First let us recall the well known result of Wigner [17] that if the wave packet is accelerated in the region of interaction then, due to causality, its time advance cannot be too large. This works for the scattering on the short range potential. The
L. Lukaszuk,Sum rules
585
measure of delay (advance) is d61/dk and this derivative cannot be too large negative. In other words if there is a sufficiently narrow bump in the cross section then it has to be connected with the positive time delay and we can speak about a quasi bound state. Results of similar nature were obtained-within the AFT assumption for rrrr scattering [18]: e.g. it was found, that the 7rrr scattering length has a lower bound depending on the pion's mass alone. In this section we formulate a theorem working for the AFT forward scattering amplitude (e.g. ~rlr, #N) or for the pion's electromagnetic form factor which makes it possible to check whether the bump in differential cross section is connected with the positive time delay. Let z be variable for which F(z) (amplitude or form factor) is analytic in the upper half plane with eventual cut along the real axis (e.g. z = ~- X/~s 4) for 7r~ scattering, z = t or z = ~ x/t 4 for the pion's form factor). I f F ( x ) - [FIe i¢ has a bump dominating over the region (x 0 Xl, x0 + Xl) with maximum at x = x 0 and if
F(x O) / °+x' In ~ (X
F(x) dx>(f=
+
/° x' ) h'lF(xO~ dx (X
Xo)2
X0 - X 1
XO)2
(7.1) '
Xo+X 1
then
3~axX=Xo >
0
(7.2)
This result comes immediately from the second theorem of sect. 2 (compare eqs. (2.15), (2.16)). Imagine now that the peak is sufficiently steep (i.e. left-hand side of (7.1) is large) and/or it is dominating over a large region (i.e. x I is large). Then the right-hand side of (7.1) will be negligible in comparison with the left-hand side and the phase derivative at the peak will be positive. A good example of such situation would be the pion's the form factor F~(t) (we need ~ ~> t/> 4). Unless F~ is wildly increasing in hitherto unmeasured regions, then d6 1/dklk=kp > 0 from the knowledge of the process e ÷ e ~ 7r+Tr alone. If the detailed knowledge of form ['actor [F(t)l is given, then it is possible to establish an upper bound for the reduced width of rho meson, Fp (compare eq. (2.2) with d~5/=_1 -
we put z =
t
~(a++a
dx for definiteness)
);
= ~ ] ['p mp
F(rn~)
l I ln F 5> ~ - ~ (t
rn2) P- 2
dt
(7.3)
586
L, £ukaszuk, Sum rules
In particular, if the form factor IF[ is majorized by simple Breit-Wigner shape with [FBw(m2)I = IF(m2)l and width I'B, then P o < P B independently of the distribution of zeros':
I If IF(t)l ~< IFB.w,(t)I --
m ° FB
F(m~)
t-~m2p-~i r B m p l
(7.4)
then d6/=_1 ] - 1
z
P
~ FB
8. CONCLUSIONS Weak analyticity assumptions led to several consistency relations for scattering processes. These relations apply to quite unlike reactions involving leptons or hadrons, or both of them. They seem to be strongly restrictive: a good example may be the discussion of pion's form factor in sect. 5. The applications made and main results are sketched in sect. 1 and here we would like to point out possible future developements: similar sum rules should work for up -+ e-n scattering, weak form factors, forward virtual Compton scattering. It would be also interesting to generalize our relations for the case of 3' - nucleus Compton scattering. The author is indebted to Professor J.D. Bjorken for communicating the results of ref. [4] before publication.
APPENDIX The following theorems due to Nevannlina [1] and Muskhelishvili [2] are used in deriving the results of sect. 2: Theorem A (Nevannlina)
|f f(z) is regular in the upper half plane, continuous in the closed half plane, its zeros in the upper half plane have no finite limit point, and it satisfies a = lira inf r -1 log M ( r ) < p - - > oo
(M (r) = max if(z)[ for Iz[ = r) and
J log* V(x)l ~ < 1 +x 2 --oo
~,
L. Lukaszuk, Sum rules
587
or
then
f
x 2 log [f(x)l dx and J
1
1
log If(z)t = log
1
z/~ n
x 21og lf( x)l dx exist ,
7r
(t x)2+V 2
"
where z n are the zeros o f f ( z ) in the upper half plane and n c = lim -2 r 1 [
J
7"f
r~
~
log b"(r ei°)l sin 0 dO
0
(z n is c o m p l e x conjugate of Zn, z - x + iy). Eq. (A. 1 ) still can be replaced by inequality if no assumption about distribution of zeroes is made. Then [1]: f
log if(z)l
log If(t)]
dt + cy
(A.2)
Theorem B (Muskhelishvili) Let L be c o n t o u r and let ~p(t) satisfies the H O ) condition on L; let S ÷, S - be the regions of plane b o u n d e d by L, then in each of the regions S÷+ L, S - + L, the function
1
O(z) = U~i satisfies condition
[ ,,~(t) J
t~
dt
L
IO(z2)-O(Zl)l~clz 2
ZllU for/J < 1
or
[O(z2)-O(Zl)l~clz 2
2'1 I1 e f o r / . t = l
,
where e is arbitrary small positive constant, c is a constant and by 0(z) with z E L must bw u n d e r s t o o d the corresponding limiting value 0÷ or 0 - .
REFERENCES I l] 12] [3] [4] [5]
R.P. Boas, Entire functions (Academic Press, New York, 1954). N.I. Muskhclishvili, Singular integral equations (Groningen, 1953). l.Ya. Pomeranchuk, Yad. Fiz. 11 (1970) 852. T. Appelquist, J.l). Bjorkjen, Phys. Rev. 4D (1971) 3726. M. Damashck, 17.J. Gilman, Phys. Rev. ID (1970) 1319.
588
L. Lukaszuk, S u m rules
[6] tt. Cheng, T.T. Wu, Phys. Rev. 182 (1969) 1852. 171 F.E. Low, Phys. Rev. 96 (1954) 1428; M. Gell-Mann, M.L. Goldberger, Phys. Rev. 96 (1954) 1433; K.Y. Lin, Nuovo Cimento 2A (1971) 695; in this paper further references can be found. 181 S.B. Gerasimov, L.D. Soloviev, Nucl. Phys. 74 (1965) 589. 19] T.N. Truong, Phys. Letters 31B (1970) 461. [101 P.S. Baranov, L.V. Filkov, G.A. Sokol, Fortschr. Phys. 16 (1968) 595. [ 11 ] T.N. Truong, R. Vinh-Mau, Phan Xuan Yem, Phys. Rev. 172 (1968) 1645 ; T.N. Truong, R. Vinh-Mau, Phys. Rev. 177 (1969) 2494. [121 J.E. Bowcock, Th. Kannelopoulos, Nucl. Phys. B3 (1968)417. 113] R. Oehme, Phys. Rev. 111 (1958) 1430. [14] V.L. Auslander et al, Phys. Letters 25B (1967) 433; J.E. Augustin et al. Phys. Rev. Letters 20 (1968) 126; Phys. Letters 28B (1968) 508. 115] C.W. Akerlof et al. Phys. Rev. Letters 16 (1966) 147. [ 16] A. Martin, Nuovo Cimento 42A (1966) 930; Y.S. Jim A. Martin, Phys. Rev. 135B (1964) 1369. [17] E.P. Wigner, Phys. Rev. 98 (1955) 145. [ 18] k. Lukaszuk, A. Martin, Nuovo Cimento 52 (1967) 122; B. Bonnier, R. Vinh-Mau Phys. Rev. 165 (1968) 1923; A.K. Common, Nuovo Cimento 63A (1969) 451. [19] Dolgov, Zacharov, Okun, Yad. Fiz. 14 (1971) 1247. [20] A. Martin, Nuovo Cimento 37 (1965)671; A.M. Jaffe, Phys. Rev. Letters 17 (1966)661. [211 T.T. Chou, C.N. Yang, ttigh-energy physics and nuclear structure, ed. G. Alexander (North ttolland, 1967) p. 348. [22] G.F. Chew, F.E. Low, Phys. Rev. 113 (1959) 1640; J.P. Baton et al. Phys. Letters 25B (1967) 419.
NOTE ADDEDINPROOF: In sect. 5, referring to tire c o n c l u s i o n s o f C h o u and Y a n g we have o v e r l o o k e d their p a p e r p u b l i s h e d in Phys. Rev. ( 1 9 6 9 ) 2469. T h e f o r m f a c t o r given there is c o n s i s t e n t w i t h o u r i n e q u a l i t y (5.7).