Bounds on Form Factors and Propagators

Fort.sc11ritte der Physik 27, 561 -579 (1979)

Bounds on Form Factors and Propagators

VIRENDRA SINGH and A. K. RAINA~)

Taka Institute of Fundamental Research Bombay 400 0005, Indiu

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 2. Pionic Contribution to the Muon Anomaly . . . . . . . . . . . . . . . . . . . . . 561 3. Kl, Decay Form Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 4. BoundsonZ, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 5. Rigorous Phenomenology for Form Factors . . . . . . . . . . . . . . . . . . . . . 566 6. Meiman problem with Space-like Data . . . . . . . . . . . . . . . . . . . . . . . 566

. . . . . . . . . . . . . . . . . 569 . . . . . . . . . . . . . . . 571

. . . . . . . . 572

Bibliogrsphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

7. Bounds Involving Time-like Data on the Phase Alone 8. Bounds Involving Time-like Data on the Modulus Alone 9. Bounds with Space-like and Time-like Data on both Phase and Modulus

10. Bounds Derived from the Modulus Representation . . . . . . . . . . . . . . . . . . 575

1. Introduction

In the theory of strong interactions it has been realised for quite some time that general principles like unitarity, analyticity and crossing symmetry lead to rather powerful restrictions on scattering amplitudes. This is especially so in the study of the high energy behaviour of aniplitudes as well as for the pion-pion interaction. The purpose of these lectures is to point out that unitarity and analyticity also lead to useful restrictions on form factors and propagators. The subject began with the work of Geshkenbein and Ioffe [45, 461 though their work was reanalysed later because of some of their technical assumptions [40]. This work, however, led MEIMAN [79] to develop some techniques which have been useful in many other problems. For an early review see OKUBO [92]. We begin by describing in the following four sections some of the problems to which these techniques have been applied. Sections 6- 10 describe the mathematical methods of solving the problems formulated in sections 2-5.

2. Pionic Contribution to the Muon Anomaly

The recent precise measurement of ap = (g - 2)/2 = (1 166922 & 9) x by the CERN Muon Storage Ring Collaboration [a] represents an order of magnitude improvement over earlier measurements. The best estimate of the pure &ED contribution [30] is

l) Notes by A. K. Raina of lectures given by V. Singh a t the Winter School on High Energy Physics, Punchgani (Dec. 12-31, 1977).

43 *

562 VIRENDRA S I N ~ H and A. K. RAINA

ap = (1 165851.8 & 3.3) x The main source of the large discrepancy is the hadronic contribution which arises mainly from the second order vacuum polarization diagram :

There is a very small contribution from fourth order hadronic vacuum polarization and CALIVIET et al. [30] have estimated the total hadronic contribution as (66.7 & 9.4) x Uk9. In fact, after including weak interaction effects as well, they estimate up (theoret.) = (1 165920.6 f 12.9) x which is in excellent agreement with experiment. The calculation of the non &ED contribution is, of course, model dependent. Por example the calculation of the weak contribution can change from, at a very rough estimate, about 10-12 assuming a four fermion interaction, to about 3 x in a renormalizable gauge theory [30]. Various extensions of Meiman’s methods have been used to obtain lower bounds to the contribution to up coming from a x---x- intermediate state in the vacuum polarization diagram above. Phenomenological estimates [18] put it a t about 45 x leg, making it by far the largest single contribution to (hadr.). The reason for the large contribution is the existence of the g resonance. In fact BOUCHIAT and MICHEL [23] originally suggested that accurate measurements of up might provide evidence in favour of the e. However the effect was beyond the accuracy of measurements of up at that time. The total hadronic contribution is [23, 411

cn,(hadr.) = - dtK@)(t) o( t ) ; 4x3 li

4mnZ

o(t) = cross-section for e+e- --f hadrons at c.0.m. energy 1

h””(t) = J dZL( t - 21) 71yu‘) t (1 - u) ( t / t t ? , 2 ) ] - 1 . 0

whew 312 (t - 4n2,2)3/2

KI2)(t) . t5 i2 w(t) = - 127?

We note the general form of the expression for a,(zn), viz. an integrand consisting of a positive weight function w(t) and the modulus-square of a function (the pion e.m. form factor F,(t)) which is analytic in the t-plane cut form t = u(u = 4rnx2) to 00, with the integral running over the entire cut.

Bounds on Form Factors and Propagators 563

: Data is availabIe on F’,(t) in the space-like region t < 0. Recently good data has been obtained by the scattering of pions on atomic electrons [38], but more commonly it has been extracted from the electroproduction reaction ep -+ exn [19]: We [lo61 recently obtained the bound ucr(zn) 2 22.8 x using only space-like data on FZ(t). However, time-like data on IFn(t)l is available for t 2 4mn2 from the reaction e+e- --f ;;+x-. RASZILLIER et al. [118] used both space-like data and time-like data on

P

f , +

IFz(t)} on an interval T 2 t 2 to to get acr(z+z-) 2 42 x lo-$. This does not exhaust the data, however, since the phase of F,(t) can be fixed in the elastic region 16mn2 2 t 2 4mn2 (in practice the inelastic threshold is much higher, about 0.9 GeV2 because of Q dominance) by nnitarity:

n 1T

Im F,(t) = Fz*(t) sin 6 exp (id), where 6 is the I = J = 1 xx phase-shift. We have recently used data on FJt) in the elastic (a-dominant) region to obtain [lor] ccp(mz) 2 42 x 10P. Spacelike data was not used although it present’s only extra computation.

3. Ki, Decay Form Factors

The semi-leptonic decays of K mesons are described by two form factors /* ( t ) :

1 (n0(p)I V w K + ( 0 ) lK+(fi)) = [(k + 34p f+W + ( k - Pl, /-(t)l

where t = ( k - P ) ~ . The matrix element of the divergence of the vector current VpK‘(.r) is given by

where

The fiinction f ( t ) is assumed to be real analytic in the t-plane cut along [ (wK + m,)*, 00).

The physical region of the K13 decay is (m,( - mx)2 2 t 2 mL2. The propagator A(t) is defined by

~ ( t ) = J d4seiqq0i T [ ~ ~ V ~ K + ( . ~ ) i ~ 7 ~ ~ - ( 0 ) 1 lo), t =- Q 2 .

564 VIRENDRA SINQH and A. K. RAINA

It was shown by LI and PAGELS [70] that, assuming d( t ) satisfies an unsubstracted KALLIZN-LEHMANN represent>ation,

where

MATHUR and OKUBO [78] have derived the inequality

with A = 0.837na m,, B = 0.659 n i 2 .

Further information on f ( t ) comes from unitarity which says that the phase of f(t + i0) is related to the I = 112, s-wave KT: phase-shift in the elastic region (mr; + ma)z 5 t 5 tinel., tlnel. = (mK + 3n~J'. The problem then is to find bounds on f ( t ) / f ( O ) in the physical region m12 5 t (nzK - ~ n , ) ~ . This problem has been solved by AUBERSOS et al. [4'J The continuous Iines in the figure show the upper and lower limits obtained. The dashed lines are the bounds obtained when the further input that f ( t ) = 1 (this is a consequence

of the Ademollo-Gatto theorem) is also used. The agreement with experiment is fair. The solution of this problem is quite complicated and involves extensive numerical work. The main work is in the numerical solution of a Fredholm integral equation. This is, in fact, a success of the method of Auberson et al. since in other approaches a singular integral equations is required to be solved numerically with all the consequent numerical instabilities and uncertainties.

Bounds on Form Factors and Propagators 56.5

4. Bounds on 2,

bound^ have been obtained also on the wave-function renormalization constant 2,. This is supposed to provide an estimate of the extent to which a particle, sag a nucleon, is a bound state (Zp = 0) or an elementary particle (2, = 1). Recall that 1 2 2, 2 0. In the early days of field theory one started with the prejudice that Z2 was 1. Later, in the days of ‘nuclear democracy’, one had the belief that every particle was a bound state and that 2, = 0. The first rigorous bound on 2, for a nucleon was given by DRELL et al. [40] who obtained the valiie 2, 5 0.85. The most recent bound is by BALUXI and RROADHURST [I71 who obtain Z L 5 0.25. They use, apart from analyticity, both elastic and inelastic unitarity. A, is given by the sum rule

7-1 J2 - 1 =- (+J ’> r ) dw Im ~ ( w + io)

where J ( w ) is the nucleon propagator and m and ,u are the nucleon and pion masses, respectively. The form factor F(w) is defined in terms of the coupling of an off-shell nucleon to a physical pion (monentum k, isospin e ) and physical nucleon (momentum p , isospin oc, helicity I.) by

--oo ( m i / < )

whelc q = p + k, q2 = w2, ~ ( z ) = (iB - m) $(z) is the nucleon source, operator and -c,(e = 1,2, 3) are Pauli isospin matrices. We have F(m) = 1 so that g is the xNN coupling constant which is known from xN dispersion analysis. We indicate the unitarity relations diagrammatically:

I ,

x;) /

+ ----

Here T represents the nN -+ X X amplitude. We refer to the original paper [ 27’1 for further details about this very complicated problem.

566 VIRENDRA SINQH and A. K. RAINA

5. Rigorous Phenomenology for Form Factors

A lot of work has been done, mainly by Raszillier and co-workers, on the problem of finding rigorous bounds for the pion form factor PJt) in the space-like region using time- like data and also some limited space-like data. In an early work RASZILLIER [I101 solved the problem of finding a bound on FJt) (t < 0) and F,'(O) given an upper bound in the time-like region :

iFn(t)l 5 S( t ) , t 2 4nt,2.

He showed that a lower bound provides no extra constraint. In a later paper RASZILLIER and SCHMIDT [I131 used some space-like data as input as well. They investigated the consistency requirements that, data on FJt) a t some space- like points imposes on its value a t other points and they obtained tight bounds. They incorporated experimental errors in the space-like data by dividing the error ranges into a number of parts and testing each of the data sets obtained by combining these points in all possible ways. Thus for five experimental points they generated 115 = 161,051 data sets by dividing each error range into 10 parts. Only 5- 10 data sets were found to be compatible. Instead of S(t) one could also use u,(nn) as the time-like input since, as we know,

c(,(z...) = 1 dtw(t) lFz(L)]2. 4mT2

This has been the subject of recent work by RASZILLIER [120, 1221 and coworkers [ZOO]. We refer to the bibliography for references to related work.

6. Meiman Problem with Space-Like Data

We mill consider the problem of finding a lower bound to a quantity I given by an es - pression of the form

oi)

I = J dtw(t) IF(t)j2. (6

Here w(t) is a positive weight function and F ( t ) is a function analytic in the comples t-plane cut from a to 00. We saw several examples of expressions of this forni earlier. We will consider the case when F( t ) is known at a few points below t = a(a > 0). In the case of the pion form factor FJt ) we know that Fn(0) = 1 and Fn'(0) is related to the pion charge radius. Hence we want a bonnd of the forni

I 2 f ( F ( O ) , F'(O), F(t , ) , ...). The only result of t,his kind in standard textbooks on complex analysis, e.g. RUDIX [122] , is Jensen's theorem from which we get the inequality

for a function f(z) analytic in the unit circle and such that f (0) =+ 0. We could use this formula in our problem by mapping the cut t-plane into theunit circle in such a way that if we are given F(t) , a t t = b < u then the point b is mapped to the origin. In the

Bounds on Form Fact'ors and Propagators 36i

expression 2n

we have also the positive weight function p(8). Using the inequality of arithmetic and geometric means [122], viz.

where

we get

exp (1 1% f dP) 5 .I- f dP

J d p = 1 ,

1 17

I 2 exp r 2n 1 *% d6 log p ( 6 ) ) exp (7 r.7 " d8 log p7(e4y)12).

The first term on the right is completely known whlle we can apply Jensen's theorem to the second. We refer t o DRELL et al. [40] for further details. It is difficult to generalize this procedure to the case when there can be any finite number of linear constraints, e.g. values and various derivatives of F(t ) in the analyticity region. Various generali- sations have been obtained by OKUBO [92] and XESCIV and RASZILLIER [87]. We will describe another procedure which enables one to solve any problem of this kind in a simple and economical manner (RAINA and SINGH [106]). The mapping

(2 - 1);(2 + 1) = q / ( ( t ! U ) - 1)

maps the t-plane cut from t = a to cm onto / z ] < 1 with t = 0 mapped onto u = 0 and t = a mapped onto z = 1. Then I takes the form

Defining D(z) by 0

we get a function analyt,ie in jz/ < 1 with the property that,

We then define h ( z ) = D(z) f ( z ) and now

I n

I = (I,%) de i h ( & y . 0

We iwt,rict, h(z) to those functions analytic in / z / < 1 with Taylor coefficients in 12, i.e if

lZ(2) = C(0 + n,z + n / z ~ + . * I .

then

568 VIRENDRA SINGH and A. h. RMSA

fmin J l J ,

J , 2 endn 6 dn2 J , cn2 2 c,dn

Such functions belong to the Hilbert space HZ of the unit disc (DURES [42 ] , RUDIX [I221 but for our purpose the above definition is convenient. This definition is equivalent to 1 equiring h(ei@) to be of integrable square and have vanishing negative Fourier coefficient. From Parseval’s theorem we get

W

I = Z ’ I unl2- n = O

The problem is then to minimise I subject to various linear constraints. Such constraints (e.g. value, derivatives etc.) will always be of the form

03

X C , C ( , = J . n-0

Let us take two constraints for simplicity:

= 0 .

co 2 cna, = J , n - 0

M

where the coefficients c,, d, are, of course, known while the a, must be varied so as to niinimise I . We set up the Lagrangian

from which we find

Then,

Jn the above we were able to assume that the coefficientsa, were real because in the physical problems in which these techniques are used the functions is a real analytic function and so has real Taylor coefficients. Of course, if the coefficients could be coni- plex the problem is quite as simple. It is also obvious how to generalise this solution to any finite number of constraints. Prequently it is only known that J1 and Jz lie in 3 range, i.e. have error bars. In that case we must minimise I,,,,,, which is expressed through the above determinants1 equation as a quadratic form over J, and J2, by varying JI and J, in their allowed ranges. This is a problem of quadratic programming.


This solution was used to find a bound for op(zz) from space-like data. In our calculation nine constraints were imposed and of these eight had error bars (naturally F.7(0) = 1 exactly). We obtained a,(..t) 2 22.8 x 10-9. If we had used only the constraint FJO) = 1 we would have obtained, by the method described earlier, op(z.z) 2 1.58 x (PAL- MER [981). Of course this bound does not involve any data.

I I

-11 -b

7. Bounds Involving Time-like Data on the Phase Alone

We saw in the last section that the introduction of space-like data improved the bound on ap(izz) from 1.58 x The question naturally arises of including some time-like data. We saw in sections 2 and 3 that in both the muon problem and the K,, problem we know the phase of the form factor near threshold by unitarity. In the latter problem this is, in fact, the only time-like data available. It is therefore of some interest to consider the case when the phase alone is known in the time-like region. We will first describe briefly a method of solving this problem due to OKUBO [96].

to 22.8 x

I I 0 '0

Consider the complex t-plane with two cuts t 2 to and t 5 -tl. We assume that F(t) is analytic in the cut t-plane and that the phase 8(t) of P(t + i0) is known on the intervals to 2 t 5 a and -b 5 t 2 -tl. Define

1 G(t) = exp [; (n - t ) lI2 ( b + t ) l l z

G( t ) is, by construction, holomorphic in the sa.me cut plane as F( t ) and on the intervals to 5 t 5 a and -b 5 t 5 -tl, G(t 3: i0) is continuous with

arg G(t + i0) = *-ts(t).

IG(t_+ i0)l = 1 for t > CI or t < -b . Moreover.

If we define Fi(t) = F(t ) /G( t )

then we find that F,(t) is analytic in the cut plane with cuts t 2 ri and t 5 -b. We can write

R F

I = I I + I?

570 V~~LENDRA SINCH and A. K. RAINA

where

where kl(t) = k( t ) IG(t + i0)l2.

-4 conformal map of the cut plane to the unit disc gives us an expression of the form

where %(z) is a non-negative, integrable function on - 1 < x < 1. The relevant function space is still H2 but I is no longer the H 2 norm. OKUBO [96] has shown that this expression defines a new norm on the space H 2 if %(%) is bounded. Otherwise it merely defines a subspace of H2. The problem can then be solved if one can find the ‘reproducing kernel’ of the space which is given as the solution of the singular integral equation

1

This method is very complicated and apparently no actiinl problem has been solved so far in this way. We have already described in section 4 the work of BALUN and BROADHURST [17]. They solved a similar problem and also obtained a singular integral equation which they had to solve numerically. BOURRELY [25] simplified the problem by not demanding the optimal bound and thus was able to avoid an integral equation. MICU [8.3] and AUBERSOX et al. [4] also solved problems of this kind but obtained non-singular integral equations. We will describe here the work of MICU [83]. For the work of AUBERSON et al. [a] we refer to the Les Houches lectures of MAHOUX [75] . The problem is reduced as before to finding a bound for

--x

where F(z) is analytic in jzI < 1. We are given that the phase of P(eiD) is @(O) for 101 < a. In the simplest case considered by Micu the only other information given is F(0). Ex- panding P(z) in a Taylor series,

00

S(2) = F(0) + 2 P@. ,i=1

F(0) is given while the other coefficients are constrained by the condition

cxp (-i@(~)) F(ei8) = exp ( i ~ ~ ( 6 ) ) [F(eio)l*

for 101 < a. We rewrite this as

3ounds on Form Factors and Propagators 571

The limiting procedure is introduced for later convenience. We now have to find the ininiinum of

1 = IV)I2 + i IPA2 n= 1

subject to the above condition, which we take to hold for any r < 1. We will taken the limit as r --f 1 - later. We then find, after setting up a Lagrangian, that

where A(0) is a real Lagrange multiplier function and satisfies A(@ = -A(--O) because F(z) is real analytic. Substituting the F, into the phase condition and summing the series we get

Taking the limit as r -+ 1 - one obtains the integral equation

This is a Fredholm equation. The extremal value Imi, is then given by

We note that if we had also put in P'(0) as a constraint we would have had a second integral equation to solve. Putting in more space-like data is consequently very difficult. Again there has apparently been no (published) numerical computation based on this work. The equations of AUBERSON et al. [a] are very similar and as we have already remarked they obtained excellent bounds for the Kl, case in this way.

8. Bounds Involving Time-like Data on the Modulus Alone

In the case of the a,(nn) bound we have data on IF&)/ in the time-like region. This has been handled in a simple way by NENCIU and RASZILLIER [87]. Our starting point is as usual the expression

2.2 . , .

We now suppose that Ih(eis)[ = m(O), a known function, on an are r which subtends an angle 2ny. Then

572 VIRENDRA SINCH and A. K. RAINA

and by the inequality of arithmetic and geometric means (RUDIN [122])

This does not seem to lead to much improvement but we now recall Jensen’s inequality which we used in section 6:

so that’

We thus get

We see that use of analyticity allows us to obtain a lower bound in terms of known quantities. We refer to RASZILLIER et al. [I181 for later developments.

9. Bounds with Space-like and Time-like Data on both Phase and Modulus

As usual F(t) is analytic in the t-plane cut from a to 00. We now suppose that we know both IB’(t)j and argP(t), i.e. P(t) itself, on an interval a 5 t 5 T and we want a bound on

00

I = J dtwft) IF(t)/2. U

This problem has been solved by us recently [lor]. From our previous experience with the problem when the phase was specified we should expect an integral equation. This is, in fact, the case, but the integral equation, which is singular, can be solved exactly. Space-like data changes only the inhomogeneous term in the integral equation and so causes no difficulty in principle. It turns out to be convenient to map the cut t-plane onto the upper half-plane using the mapping w f u + iv = f M / l / ( T - a). The real analytic function F ( t ) becomes the hermitian analytic function f (w) = f * ( -w*) and now

m

I = J duP(u) If@)l2. - W

The integral has been extended over the whole real line. This can be done using the hermitian symmetry of f(u) and the properties of p(u) in the physical problems. Because of the conformal transformation the function f(u) is now assumed to be known on the interval [-1, 11. A logical problem now arises since an analytic function with continuous boundary values is determined uniquely by its values on an interval of the boundary. In fact the requirement of continuity can be greatly weakened but this is enough to


indicate the problem. We are therefore forced to assume that f ( x ) is only approximately known on [ - 1, 11. We remove the weight function by defining

I q(w) = exp [ h J d u t - W (1 + u'2) (u' - w )

W

(1 + d w ) In p(u')

so t,hat W

I = J du lq(u)12. - W

We now assume that we are given gexp(u) on the interval [ - 1, 11 and an error E with

1

J = J du Ig(u) - qexp(~)I' 5 E .

Any finite number of linear constraints can be added and so space-like data can be taken into account with no important changes in the problem. They affect only the inhonio- geneous term of the int.egra1 equation we are going to obtain. At this point we must specify the space of functions over which we perform the variation. We take the space H 2 (DEREW [a,?]) of functions analyt,ic in the upper half-plane and such that

-1

W

sup J du If(u + iwj' < 00. 0<ll<W -W

This is the class of functions for which the Titchmarsh theorem, familiar from the early days of dispersion theory, is valid. Such functions are also characterised by the property of being Fourier transforms of causal functions. This is knom-n as the Paley-Wiener theorem 1421 according to which g(w) is in H 2 if and only if

where G(z) is square-integrable on [0, 00).

The introduction of the space H 2 into a physical problem does, however, require some justification. This is provided by the following theorem (RAINA [208]) : Theorem : A function f ( z ) analytic in the upper half-plane is in H 2 if and only if there is an 16 s.t. in every half-plane Im z 2 6 > 0

If(z)l 5 Ca(1 + 14fn

is satisfied and f ( z ) converges in the sense of distributions to f ( x ) in L2(--w, 00).

We refer to the paper [lo81 for a specification of the precise class of distributions. One can also raise the question as to whether il solution to this problem necessarily exists with arbirtrary space-like and time-like data. This can in fact be shown [ lor ] . One can also prove that the lower bound is attained uniquely and also, as one would expect,

574 VIRENDRA SINGE and A. K. RAINA

that, there is no upper bound. Moreover, the iiiiniinising function is hermitian symmetric. It then follows from the Paley-Wiener theorem that

where G(x) is a real function. We now express I as well as the constraint term J in terms of a. We find that

a: I = J d x ( G ( ~ ) ) ~

0

00 m a: 011

J = s dxG(z) dyG(y) L(z - y) - 2 1 dxG(z) H ( s ) + s dz (H(z))2 0 0 0 0

where

and

sin (z - y) L(z - y) = ?c(x - y)

1 1 P

We form the Lagrangian

and from SLjSG = 0 we obtain the integral equation

I, = I + ( l i p ) (J - E )

a:

pG(z) + J dyL(x - y) G(?/i = H ( r ) (X 2 0). 0

If we are given space-like data then we will have an inhomogeneous term of the form

I f 1 n X ( x ) = H ( x ) + 2 ai exp (-crix) + 2 pi z exp (--biz)

j = 1 j - 1

where the ai, b, > 0 and the ctj , pi are Lagrange multipliers. This form of R(x) corresponds to the case when the values and first derivatives of F(t ) are specified. This integral equation has appeared in the work of KREIN and NUDEL’W [63] who have, however, only made a brief announcement of their results. We have solved it by a Fourier transform technique which reduces i t to a Cauchy singular equation which is much easier to solve. This analysis proves that

where I(?.) = ( L - tanh (nA))/2

Bounds on Form Factors and Propagst'ors 575

+(x, I . ) = 1/(1 + tanh (XI . ) ) $(z, I.)

1 4(x, I.) = cosh (zj.)J du (1 + u)iieiuz

z V(l -d) 1 -u

The &.r, 1) are eigenfunctions of the operator

satisfying the boundary condition 4(O, i) = 1. We can verify that L and A commute. The solution of the integral equation is now

whcte ,LA > 0. M7e can insert this solution into the expressions for I and J . If there ale space-like (i.e. linear) constraints they can be eliminated and we are left with just one nonlinear equation to be solved niinierioally. We refer to the paper for complete details.

10. Bounds Derived from the Modulus Representation

G . KOZSEAU and F. MARTIN [22] have recently clarified the conditions under which the niodiilus representation holds for the pion form factor and have used it to obtain bounds. Starting from the results of local field theory they give a careful discussion of the extra assumptions required to .prove that the pion form factor (after being niapped to the iiiiit circle) belongs to the Nevanlinna class N . This is the class of functions analytic in t h e unit disc such that

2n

J do i f ( re t6) / 0

is l~oi~ndcd as r 4 1 -. where

log+ .c = log x if z > 1 1 - - 0 if 2 2 1 .

This class of functions has the Fatou property that radial boundary values exist alniost cvery where and functions in N are uniquely determined on a set of positive measure on the boundary. This class appears to be the largest class for which these two properties hold. This has been shown by explicit counter-example and this indicates the importance of N .

44 %c~itsc.liriit ,,'Fortscliribte der Pliysik", Heft 1 1 - 12/79

576 V ~ E N D R A SINGH and A. B. RAINA

By a theorem of SMIRNOV [42] every f ( z ) E+ 0 of the class N has a canonical factorization of the form

f(2) = w W ) [f4(z)/&(z)l where

i.e. B(z) is a Blaschke product, and

where ,ul(t) is a bounded, non-decreasing singular real function (singular' means that d,u(t)/dt = 0 a.e.-e.g. a step function). Further technical assumptions eliminate S&) and permit only a specific form for S,(z) which corresponds to the possibility of an essen- t i a l singularity a t infinity. Finally they obtain

where

The t, are real zeros in t < 4??rZ2. The complex zeros s, are given by

The norma.lization condition PJ0) = 1 immediately gives us the condition Znz + iy,, = i ( sm/(4m2) - 1).

log B(0) - c + 21' = 0, where

N - -! log I F * ( " ) l ds!. sf l/s'- 4mn2

4nrnZ

Since c 2 0 we can write this as

log B(0) + N 2 0.

Another sum rule can be derived from the condition that the phase of F,(t) should have the threshold behaviour of a p-wave resonance since for 4mn2 5 t 5 16?nn2 the phase is, as we remarked earlier, equal to the I = J = Inn phase shift. Many inequalities can be derived and the representation t,heorern can be used to show


that they are optimal. An interesting bound on the e.m. radius of the pion that they obtain is

--sin11 h7 - N 1 sinh X - N 8run2 0 8m,A + R s y - ~ ~ ' s + R,

where

7z

Numerically they find 0.24 fm 5 r, 5 0.78 fm. They also obtain a lower bound for the phase

&(s) 2 8,O(s) + 2 tan-* K tanh - ( 3 where

nc

4m,Z

If one has information on the zeros of F,(s), then more refined bounds exist'.

Bibliography

The following bibliography contains the references referred to above as well as papers of related interest. We do not, however, claim that this a complete bibliography of papers on form factor bounds and prohlem using techniques of the Meiman type.

[I] AGRANOVSKII, B. R., Soviet. J. Nuclear Phys. 20 (1975) 549. [ Z ] AUBERSON, G., Li, L.-F., Phys. Rev. D 6 (1972) 2269. [3] AUBERSON, G. et al., I, Nuclear Phys. B 73 (1974) 314; 11, Ann. Inst. Henri Poinc. XXII (1975)

[4] AUBERSON, G. et sl., Nuclear Phys. B 98 (1975) 204. [5] AUBERSON, G. et al., Nuclear Phys. B 113 (1976) 285. [6] AUBRECET, G. J., 11, et al., Phys. Rev. D 4 (1971) 1423. [7] BAIER, V. N., FADIN, V. S., Nuovo Cimento A 66 (1970) 693. [8] BAILEY, J. et al., Phys. Letters B 67 (1977) 225; B 68 (1977) 191. [9] BALACHANDRAN, A. P., Nuovo Cimento 42 A (1966) 804.

[IO] BALACHANDRAN, A. P., LOEPFEL, J. J., Phys. Rev. 167 (1968) 1322. [U] BALUNI, V. Z. et al., Soviet J. Nuclear Phys. 9 (1969) 366. [I21 BALUNI, V. Z., Theor. Math. Phys. 6 (1971) 236. [I31 BALUNI, V. Z., Theor. Math. Phys. 10 (1972) 13. [I41 BALUNI, V., Phys. Letters B 38 (1972) 535. [I51 BALUNI, V., BROADHURST, D. J., Phys. Rev. D 7 (1973) 3738. [ IS ] BALUNI, V., BROADEURST, D. J., Nuclear Phys. B 84 (1975) 178. [I71 BALUNI, V., BROADEURST, D. J., Phys. Rev. D 16 (1977) 230. [I81 BARGER, V. et al., Phys. Letters B 60 (1975) 89. [I91 BEBEK, C. J. et al., Phys. Rev. D 13 (1977) 25. [ZO] BHAMATHI, G., RAQEAVAN, S., Lettere Nuovo Cimento 18 (1975) 183. [ZI] BONNEAU, G. et al., Phys. Rev. D 10 (1974) 3136. [ZZ] BONNEAU, G., MARTIN, F., Nudear Phys. B 97 (1975) 269. [23] BOUCHIAT, C., MICHEL, L., J. Phys. et Radium 22 (1961) 121. [24] BOURRELY, C., Kuclear Phys. B 43 (1972) 434.

317; 111. Nuclear Phys. B 94 (1975) 311.

44"

578 VIRENDRA SINQE and A. I<. RAINA

[25] BOERRELY, C., Nuclear Phys. B 53 (1973) 289. [26] BOWCOCK, J . E., KANNEPOULOS, Th., Nuclear Phys. B 4 (1968) 417. [27] BROA4nHURST, D. J., Phys. Rev. D 5 (1972) 2615. [28] BROADHURST, D. J., Nuc1ea.r Phys. B S.5 (1975) 189. [29] BUDNY, R., Phys. Letters B 59 (1975) 166. [30] CALMET, J. eta,]., Rev. Mod. Phys. 49 (1977) 21. 1311 CELIK, T., PALMER, D. R., J. Ma’th. Phys. 11 (1973) 770. [32] CO~XMON, A. K., WIT, R., Nuovo Cimento 3 A (1971) 179. [33] COOPER, F., PAGELS, H., Phys. Rev. D 2 (1970) 226. [34] CORNILLE, H., MARTIN, A., Kuclear Phys. B 93 (1975) 61. [35] CORNWALL, D. T., EIMERL, D., ?iuclear Phys. B 49 (1972) 61. [36] CREUTZ, M., Phys. Rev. D 6 (1972) 3537. [37] CRONSTROM, C., Phys. Letters f) 49 (1974) 263. [38] DALLY, E. B. et al., Phys. Rev. Letters 39 (1977) 1176. [39] DITA, P., Nnovo Cimento 10 A (1972) 753. “4 DRELL, S. D., et al., Phys. Rev. B 136 (1964) 1439. [41] DURAND, L., III., Phys. Rev. 128 (1962) 441; errata ibid 129 (1963) 3835. [42] DUREN, P. L., Theory of H p Spaces (Academic Press, 1970). [13] DYC, D. V., HIEU, N. V., Theor. Math. Phys. 3 (1970) 432. [-I4 ECKER, G., SCHWELA, D., Lettere Nuovo Cimento 9 (1974) 587. L.451 GESHIIENBEIN, B. V., ICFFE, B. L., Phys. Rev. Letters 11 (1963) 55. [46] GESHKENBEIN, B. V., IOFFE, B. L., Soviet Physics JETP 17 (1963) 830. [47] GESRKENBEIN, B. V., IOFFE, B. L., Soviet Physics-JETP 18 (1964) 382. [J8] GESHKENBEIN, B. V., IOFFE, B. L., Soviet Physics-JETP 19 (1964) 615. [49] GESHKENBEIN, B. V., Soviet J. Xnclear Phys. 9 (1969) 720. [50] GESHKENBEIF, B. V. et al., Soviet’ J. Nuclear Phys. 20 (1975) 302. [ 5 l ] GESHKENBEIN, B. V., Soviet J. Kuclear Phys. 22 (1976) 311. [52] GHIKA, G., VISINESCU, &I., Nuovo Cimento 13 A (1973) 365. [&?I GOURDIN, N., in Lectures in Theoretical Physics, vol. XII-B, ed. K. T. MAHAXTHAPPA and

W. E. BRITTW (Gordon and Breach, 1971). [54] GRUNBERG, G., Phys. Rev. D 12 (1975) 580. [55] HEALY, J. B., Phys. Rev. D 11 (1975) 580. [56] HIEU, N. V., Soviet J. Nuclear Phys. 7 (1968) 667. [57] HIEU, N. V., Soviet J. Nucleitr Phys. 13 (1969) 1056. [58] HONG, N. T., Soviet J. Nucelar Phys. 13 (1971) 241. [.591 IDA, M., Phys. Rev. B 136 (1964) 1767. [GO] JAFFE, A. PI., Phys. Rev. Let.ters 17 (1966) 661. [Sl] KIEIILMANN, H. D., SCHMIDT, W., Nuclear Phys. B 94 (1975) 67. I621 KOBAYASXI, M., Progr. Theor. Phgs. 47 (1972) 2044. [63] KREIE-, M. G., NUDEL’MAX, P. Ja., Sov. Mat’h. Dokl. 14 (1973) 435. [64] Laxa, C. B., STENANESCU, I. S., Phys. Let’ters B 58 (1975) 450. L65] hX(*.4CKER, P., SUZUKI, &I., Phys. Rev. D 4 (1971) 2160. [W] LEVIN, D. N. et al., Pliys. Rev. D 4 (1971) 1647. [67] LEVIN, D. N. et al., Phys. Re\-. U 5 (1972) 912; errata, p. 1555. [68] LEVIN, D. N., OKUBO, S., Phys. Rev. D 6 (1972) 3149. [69] LEVIX, D. N., PALMER, I>. R.. Phys. Rev. D 4 (1971) 3716. [70] LI, L.-F., PAGELS, H., Phys. Rev. D 3 (1971) 2191. [74 Li, L.-F., PAGELS, H., Phys. Rev. D 4 (1971) 255. [72] LUKASZUK, L., Nuovo Cimento 41 A (1966) 67. [73] LUKASZUK, L., MARTIN, A., Nuovo Cimento 52 A (1967) 122. [74] LUKASZUX, L., Nuclear Phys. B 1 7 (1972), 569. 1751 MAHOUX, G., ‘Bounds on Scattering Amplitudes and Porm Factors’, Structural Analysis of

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Bounds on Form Factors and Propagators

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Bounds on Form Factors and Propagators

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