Application of Dispersion Relations to the Determination of Coupling Constants

Application of Dispersion Relations to the Determination of Coupling Constants

M. C I N I

Istituto di Fisica delVUniversita di Roma, Roma, Italy

I

i .

The use of pion nucleon scattering dispersion relations for the pur pose of determining the coupling constant f² and selecting between the Fermi and Yang sets of phaseshifts is well known ( i ) . The basis for such determinations is the possibility of writing relations of the type *

,Τ ν - ^{m /} , j. CI (ω) , ΓΐπιΤ(ω'ϊ , flm Τ(ω') , , ( L I ) Be Τ ω = /² - + —-—-— άω' + ——^κ— άω',

μ - οο

for scattering amplitudes in a given charge state in the forward di­

rection (or the corresponding subtracted relations at two different energies) in which, by means of crossing symmetry the integration of Ι ι η Τ ( ω ) in the unphysical region of negative values of ω' can be transformed in an integration on positive values of ω of the imaginary part of the scattering amplitude in a different charge state. The re­

lations therefore involve only combinations of phaseshifts at physical values of the energy and can be used without any further manipu­

lation for the above mentioned purposes.

2.

The dispersion relations for nucleon-nucleon scattering have been proven to be valid only in perturbation theory. If one assumes how-

* f2 = coupling constant squared, μ = meson mass.

Μ . C I N I

ever, their validity, as an heuristic way of studying nucleon-nucleon phenomena, it turns out that a similar relation as ( L I ) can be written.

However in this case while the first integral can still be expressed (in the forward case) only in terms of nucleon-nucleon scattering phaseshifts it happens that the second integral can be expressed in terms of nucleon-antinucleon scattering phaseshifts only for a part of the range of integration namely for — oo < ω^Γ< Μ. In the range from — Μ to the upper limit of integration Μ — (2μ²/Μ) the integrand is the analytic continuation of the nucleon-antinucleon scattering am­

plitude for unphysical values of the energy, obviously a very difficult thing to know precisely. This fact made people believe for a long time that nucleon-nucleon dispersion relations were useless from the point of view of practical applications. We will see that this is probably not completely true.

3.

In this situation Chew (2) made an interesting proposal which we will try to summarize now. Chew conjectured that, if one considers the real part of the n-n scattering amplitude as a function of A² (square of momentum transfer) at fixed W² (square of total energy in cm. system) there is a pole of residue g² [</²(μ²/4 i f²) = /"] located at Α²= — μ².

In the c m . system

(1.2) A² = 2k²(l — co&0)

and this pole corresponds therefore to the unphysical value of cos θ (1.3) eosfl = 1 + | J ·

This means that if this pole is actually present, and no other singu­

larities are in the immediate neighbourhood, then the residue of this pole determines the asymptotic behaviour of the real part of the forward scattering amplitude completely. I t seems possible therefore to extrapolate from the physical values of cos θ our knowledge of the scattering amplitude and to determine the residue at the value (1.3).

The investigation of the singularities of Τ (A², W²) as a function of A² for fixed W² has not been made with exact methods but only on the basis of plausible conjectures. Confining the discussion to

D E T E R M I N A T I O N OF C O U P L I N G C O N S T A N T S

perturbation theory it is possible to show that in fact only the con­

tribution of the lowest order Feynman graph (Fig. 1) becomes infinite as Zl²-> — μ², all others being finite in this limit. I t is the meson propagator which gives of course the term

(1.4)

(Ρι-Ρ*)² + μ² Δ²+μ²

F I G . 1.

The whole contribution to the transition matrix is ΰ(ρ²)γ!>τ^Λ η{ρ^λ)ΰ(α²)γ^^Λ u(q¹) (1.5)

Δ²+μ²

with standard notations for nucleon spinors.

F I G . 2.

The exchange graph (Fig. 2) on the other hand, gives a con­

tribution

^ ( a i) y e T * t t( P i ) ΰ(ρ²)τ^αγ^δ ufa)

(1.6) g²

fa-ViY + μ2

which has a singularity at

(1.7) cos θ = — 1 — (μ²/2&²)

in the cm., since q² = (E^k, — k') if p² == (E^k, k') and q^x == (E^k, —k), p^x= (E^k, k) since co&6 = kk^r. This pole is located at A² = W²—

— ±Μ² + μ².

Μ . C I N I

Besides these poles, the scattering matrix is expected to have cuts on the real axis of the complex A² plane from

(1.8) — oo > A²> — 4μ² W²— ±M² + 4 μ²< μ²< oo .

This might be justified by considering the process of nucleon-anti- nucleon scattering which can be obtained with the substitution (1.9) PI=PI, P*=P2, ϊΊ= — Ϊ2, 2 2 = - f c .

In this case the total energy squared of the process in the c m . is (1.10) W '² = — {p[ + q[)² = - (PI — & )² = W²—±M²—A². The other nucleon antinucleon scattering process obtained with the substitution

(1.11) pi = ρ^γ, pi = g², ql= — p2, 9.1 = — 9.I has total energy in the c m .

(1.12) TP" = - (pi + ql)² = - (^Pl- p²)² = - A ².

Therefore if we consider the analyticity properties of T(W^r2, A'²) and T(W"², A"²) as function of W* and W"², respectively, at fixed momentum transfer we can obtain the analyticity properties of T(W% A²) as a function of A², at fixed W². Now the total energy of a nucleon-anti­

nucleon system can be either

(LIS) W '² = μ², Ψ"² = μ or

4 μ²< W²< o o ,

2

( I'^{1 4 ) 1} 4 μ²< W"²< o o ,

the values (1.13) give the poles in A² already found, and the ranges (1.14) give the cuts in A² given by (1.8). In order to determine g² it is now possible in principle to consider the quantities

(1.15) F(A², W²) = (A² + μ²)(Ψ²— ±M² — A² + a²) T(W², A²) as a function of cos 0 at a fixed energy and state that, because of the analyticity properties conjectured, F(A², W²) has no singularities in the range

2u² 2u²

(1.16) i + _ £ > c o e e > - i - J i -

D E T E R M I N A T I O N O F C O U P L I N G C O N S T A N T S

and can therefore be represented in the physical region (1.17) 1 > cos 0 > — 1

by a polynomial in cos 0. If one determines from experiment the coefficients of the polynomial

(1.18) F(A², W²) = 2 Cⁿ(fc^a)(eos0)^w

η

in the region (1.17) the extrapolation of (1.18) at the poles

J? ( - , i « , W*)

F(W² — O P + μ % W²)

will be directly proportional to g² and can give a value of this quan­

tity. The actual residua of the two poles will be given in detail later.

The actual procedure used by Chew is even more directly related to the experimental data than outlined above. Since the Τ matrix has first-order poles in A² the differential cross section

(άσ/άΩ) =\T\²

will have second-order poles from the squares of the first-order poles, and first-order poles from the interference terms of the singular and non-singular terms. Therefore the quantity

(A² + μ²)²(άσΙάΩ)

can be approximated by a polynomial in cos 0 in the physical region and its extrapolated value at A²= — μ² is proportional to /*. The application of the procedure outlined above led to the value

f² ^ 0.08

in good agreement with the determination from pion-nucleon scat­

tering.

4.

The physical idea underlying Chew's proposal can be briefly sum­

marized as follows. The terms with the poles give the whole con­

tribution of the one pion exchange processes to the Τ matrix, and correspond therefore to the longest range interaction. All other con­

tributions have a range at least half (~ 1/2μ) the simple pion exchange ones. Therefore the scattering amplitude for waves of sufficiently high angular momentum will be almost completely determined by the

Μ . C I N I

simple pion exchange amplitude. These waves of high angular mo­

mentum appear in the forward and backward direction and form the part of the scattering amplitude rapidly varying with the angle, which is the only part which survives after the multiplication by the deno­

minators of the poles. The idea is therefore that one has to approach as much as possible the region where the scattering amplitude is given exactly by the lowest order perturbation expression (with renormalized coupling constant of course). The possibility of such a procedure rests on the existence of a large gap between the beginning of the cuts and the poles.

There is no exact proof at the moment that this happens in field theory, but this property seems to be satisfied in potential scattering for an arbitrary superposition of Yukawa potentials. Beside it has been proved that this happens up to fourth-order perturbation theory.

5.

Other applications of the idea of Chew have been made by Taylor, Moravcsik, and Uretsky (3) (see Fig. 3) in order to determine the

F I G . 3.

coupling constant f² from photoproduction experiments at high ener­

gies; by Taylor (4) with the purpose of obtaining information on the

D E T E R M I N A T I O N OF C O U P L I N G C O N S T A N T S

F K ; . 4.

where k and co^Q are the photon and meson energies and vⁿ is the pion velocity. We report here the extrapolation curve taken from (3) of the quantity

(άσ/άΩ)(1 — vⁿ cos 0)²

vs cos θ in the c m . system for 260 MeV photon energy in the lab system. The extrapolated part of the curve in the unphysical region leads to a value of the residue at cos θ = 1.31 corresponding to f² =

= 0.108 ± 0.064.

The error is seen to be fairly large, and the average of all the available data give

f² = 0.064 ± 0.041.

It is suggested, nevertheless, that the possibility of obtaining reason­

ably precise values of f² is not outside of the present experimental relative parity of K⁺K ° ; and by Moravcsik (5) to obtain coupling constant and parity of K⁺ relative to Λ in pbotoproduction. All these works use the presence of poles in the scattering matrix as functions of the momentum transfer.

In the charged pion photoproduction the so called «direct inter­

action term» in which the photon is absorbed by the photon arises from the diagram given as Fig. 4. Here again the meson propagator gives a term

1

which in the c m . becomes

1

2k(D^Q(l — vⁿ cosfl) '

Μ . CIISI

possibilities. The same technique is used by Moravcsik to analyse the K+ photoproduction data. He concludes that if the K+ meson is pseudoscalar it should have a coupling constant of the order of unity, while if it is scalar the coupling constant should be of the order of 0.1.

Since, on the other hand, we know from other data that the coupling constant ought to be in either case of the order of unity, this result can be considered as an evidence that K+ is pseudoscalar.

II 1.

The idea of using a polynomial approximation to represent an analytic function obtained by suitably eliminating the singularities of the scattering amplitude, and of extrapolating this polynomial from the physical region to the often more interesting unphysical region can be extended further.

One can consider at first the forward dispersion relation for proton- neutron scattering which we write as (6)

, l m g y , , π] ω -+ ω + Μ^{2 1} ω + ω^υ Here Τ is the forward proton-neutron scattering amplitude, a the total cross section, and Τ the forward neutron-antiproton scattering amplitude. The last term is a pole due to the deuteron, being ω⁰ Mb with b the deuteron binding energy, and Λ the residue which is known.

The variable ω is ft², the square of the c m . momentum. One sees by inspection that Ee Τ(ω) as a function of the complex variable ω has a pole at ω = —

(μ

²

Μ),

a pole at ω = — co^D and cuts at

0 < ω < oo (Π.2)

— oo < ω < —μ¹ Therefore the function

(II.3) J?(a>) - (4ω + μ*)(ω + ft>^D) Ee Τ (ω) •

4π² Μ J ω— ω

D E T E R M I N A T I O N O F C O U P L I N G C O N S T A N T S

is free from singularities in the range

(II.4) ~μ²<ω<ω^ΒΪΛΧ.

F(co) can therefore be approximated with a polynomial

(IL5) F(co) =

2 ,

whose coefficients can be determined from the knowledge of the expe­

rimental phaseshifts and cross sections in the physical region 0 < ω < co^max .

Once the coefficients 0, have been determined one should be able to obtain with sufficient accuracy

'(-?)-£* M).

and hence determine f². The procedure we have outlined is clearly the same used by Chew, except that one represents here the analytic function -^(ω) as a polynomial in ω and extrapolates it to the value ω = — (μ²/⁴)> while in the former case one extrapolated by means of a polynomial in cos 0 an analytic function of cos 0 to the value cos 0 = 1 + (μ²121c²). With this procedure f² turns out to be of the correct order of magnitude.

The above proposal used the forward scattering dispersion relation (A² = 0). One would think that a generalization to all values of Δ² might be possible, but it is well known that a non-physical region between the physical threshold and the pole arises as soon as Δ²Φ 0.

The generalization of the forward scattering extrapolation procedure with variable energy has been found instead by investigating the analyticity properties of Τ(Δ², W²) as a function of ω = k² at fixed cos 0 rather than at fixed Δ² (7). Let us consider the poles in Δ² already discussed in connection with Chew's method. Their contribution to Τ(ω, cos0) can be written

μ² + 2ω(1 — cos 0) ¹ μ² + 2ω(1 + cos 0) '

Clearly these terms can be considered also as poles of the Τ matrix

Μ. CITSI

for given values of cos θ as a function of ω at μ²

(Π.8) ω - ^r 2(1 ± c o s 0 )

We remember also that we had cuts on the real A² axis ranging from

— oo to — 4μ² and from 4&² + 4μ² to oo. These can be interpreted as cuts in the complex ω plane at fixed cos 0 (for physical values of

|cos0|<l) from

(IL9)

— oo < ω <

— oo < ω <

2μ ( l + cos0) '

2μ² (1 — cos 0) *

The two cuts overlap up to the smallest of the two values

— (2μ²Ι(1 ± cos 0)) and are equivalent to a unique cut from — oo to -(2/*²/(l+|cos0|)).

Besides these singularities arising from the singularities in A², the scattering matrix has a further cut in the complex ω plane on the real axis in the range

0 < ω < oo .

This cut arises from the fact that the minimum energy of a system with nucleon number 2 is 2M. If one assumes that no other singu­

larities exist, between the cuts, besides the poles, one can immediately write the following integral representation for Τ(ω, cos 0)

(11.10) Τ(ω, cos 0) = ^A + _ _ _ ? +

* o) + (o^D

μ* +

2ω (1 — cos 0)

co 1 + lcoeOI

G 1. Γ ΐ ί( ω ' , ο ο β β ) ^Λ , L F β ( ω ' , ο ο Β β ) , ,

_ ι — — I — I — — α ω Η — Ι - - d c o ·

μ² + 2ω(1 + cos0) π) ω'—ω — ιε π] ω ' — ω 0 - οο

In order to be able to deduce some useful consequences from Eq. (11.10) we have to express the function Ω in terms of known quan­

tities. If we could prove that Ω and Θ are real, then one immediately would obtain

(11.11) Im Τ(ω, cos 0) = Ω(ω, cos 0) .

D E T E R M I N A T I O N O F C O U P L I N G C O N S T A N T S

The relation (11) follows directly from the Mandelstam representation, and can be shown to be true for potential scattering in the case of an arbitrary superposition of Yukawa potentials, and for field theory up to fourth order perturbation approximation. I t seems plausible therefore to assume Eq. (11.11) at least as a working hypothesis and investigate what physical consequences can be drawn from it. We rewrite there­

fore Eq. (11.10) as

(11.10α) Ke Τ(ω, cos 0) = + A Β

ω + ω⁰ μ2 + 2ω(1 — cos 0)

2

^μ*

1 + l c O g 01

C , 1 r i m T K, c o s 0 ) , , , 1 Γ β( ω ' , ο ο β β ) , ,

Η —ο . ^—τζ—; τττ Η— I ; αω Η— / ; αω .

μ² + 2ω(1 + cos 0) π] ω'—ω π] ω'—ω

0 - οο

We need not worry about the unknown function Θ(ω, cos Θ) be­

cause the very fact that the lower limit of integration on ω is at least four times larger than the ω corresponding to at least one of the poles will enable us to use Eq. (11.10a) as a tool in order to correlate nucleon- nucleon experimental data.

I l l

1.

Equation (II. 10a) reduces clearly to Eq. ( I I . l ) for cos θ = 1 in the forward direction. Only in this case the optical theorem holds and one can express Im Τ in terms of the cross section σ. In other cases it is necessary to express both real and imaginary parts in terms of phaseshifts. Actually the scattering matrix Τ can be expressed by means of five independent amplitudes as follows:

Μ

( I I I . l ) Τ = U + β(σ^{1) · η)(σ<²> · η) +

+ 7(σ^{( 1 )} + σ^{( 2 )})·η + ό(σ^{( 1 )}τη)(σ^{( 2 )}·#η) + ε(σ<1}·1)(&*>-1)}9

with α, β, γ, δ, ε scalar functions of ω, cos 0, and I, m, η unit vectors in the directions g i + g²>

9ι~^2,

^{9 ι Λ 92 )} respectively. For each of the amplitudes α, · · · ε a relation of the type (11.10) holds. Only the residua A, B, C at the poles are different for each amplitude.

Μ . C I N I

They are given in the original paper (7), and we are not interested in them now. Similar relations can be obtained, in non relativistic limit, for the amplitudes T^i} which are functions of the scattering phaseshifts (Eef. (I) Eqs. B6 to B l l ) . For proton-proton scattering one has similar relations, except that the deuteron pole is not there and combinations of the amplitudes at cos θ and — cos θ have to be taken because of the identity of the particles.

One has therefore five relations of the type (11.10) which can be used for two purposes:

(a) to obtain another determination of /²;

(b) to select between different available sets of phaseshifts.

The first use may even seem not too interesting if one considers that the value of f² from nucleon-nucelon scattering data has already been obtained. Nevertheless it is useful to point out that for proton- proton scattering the real part of the forward scattering amplitude is not very precisely determinable because of the Coulomb effects, while the 90° scattering amplitude is probably better known. It is also worth pointing out that the information exploited in the present method of determining f² is completely different from the information exploited in the Chew method. This is because in the latter the part of the scattering amplitude rapidly varying with angle due to the one pion exchange was extracted, while in ours the part of the scattering amplitude rapidly varying with energy due to the one pion exchange is used. I t can also be shown easily that the high angular momentum waves are responsible for the first effect while the second one depends on the lowest angular momentum waves.

The results of the calculations performed in Bologna by Tomasini and Alles along these lines give f² = 0.082. Furthermore a theoret­

ical determination of the «shape» parameter Ρ in the effective range expansion of the ^Z8^X phaseshift turns out to be possible, as a function of /^a. For the value previously determined they find Ρ = — 0.04

2.

From Eq. (11.10) an easy way of obtaining f² can be worked out along the lines of the effective range approximation introduced by Chew and Low in the study of pion-nucleon scattering.

Let us consider the singlet scattering amplitude at 90°:

(ΙΪΙ.12) Τ ^β - ^ (k ι ^{2 ί} + 1) ^{e x}Pί^{ί δ}ι 1 ^{s i n} *^p*(°) ·

D E T E R M I N A T I O N Ο IT C O U P L I N G C O N S T A N T S

The dispersion relation is

h/tJq t r τ m ι..ι\ t ' ~ ( Ι Π . 1 3 ) T,(a>) =

rjo + μ² π J ω — ω — ιε π] ω—ω 2ω +

ο

Let us consider the function

( Ι Π . 1 4 , ^ ω > - ώ ·

Equation ( 1 1 1 . 1 3 ) shows that g(cu) has the following analytic pro­

perties: cuts between 0 and + 0 0 and between —0 0 and —2μ² and possibly some poles corresponding to the zeros of T⁸(co).

Finally we have

( 1 1 1 . 1 5 )

One can therefore write for g(co) the following representation:

( Π Ι . 1 6 ) , < « , ) = Β^{ω) + » [ Μ + Ω ( * < ¥ ϊ - ^λ ΒΩ',

π ] GJ (ω' - ω) J ω(ω^Γ—ω) 0 — αο

with Β(ω) a rational function of ω. The advantage of using Eq. ( 1 1 1 . 1 6 )

is that for small ω, Ιηΐ0(ω) can be immediately evaluated. When the contribution from the s-wave dominates in Eq. ( 1 1 1 . 1 2 ) one has

Ee g(co) = k cotg δ⁸,

\ Tm #(ω) = — κ .

From the preceding discussion it follows that the function ( i r r . i 8 ) ^{H a )} = « „ ^{) +} 2 ( * ™

π J co (co— ω)

ο

apart from the possible appearance of poles, is analytic between — 2μ² and being the value of ω for which Eqs. ( 1 1 1 . 1 7 ) start being invalid.

From the existing phaseshift analysis we know that the #-wave dominates in Eq. ( 1 1 1 . 1 2 ) at least up to 5 0 Mev. Let us now use these analytic properties in order to determine f². Equation ( 1 1 1 . 1 8 )

Μ . C I N I

tells us that on the real axis /(ω) is given by (IH.19) /(ω) = g(a>) + λ(ω), where

| λ(^ω) = - ( - ^ω) * ⁰j< 0 ,

Furthermore we have

/ (ω) = 1c cotg δ^$, ω > 0

7 \ 2/ -^Z²

(111.21)

Equations (111.21) show clearly that an extrapolation of /(ω) from ω > 0 to ω = — (μ²12) leads to a determination of f*. This program has been carried out using the data coming from low-energy proton proton scattering. In Pig. 5, we have plotted the experimental values

0.8 <

0.6

0.4 0 . 2 /

k f(o)) = q cot g<5^s ^ for ω > 0 /

<<> Ύ

j7 ^ = 0.11 ± 1 5 %

ι ι ι ι 1 1

-0-4 - 0 . 2 / 0 0.2 0.4 0.6 0.8

*/ */

// ω >•

/ /

/ /

/ /

/ /

/ /

-0-2

/ / / /

/ /

• / ' /

/ /

/ / • /

-0-4

/ /

/ /

-0-6

F I G . 5.

D E T E R M I N A T I O N OF C O U P L I N G C O N S T A N T S

K E F E R E N C E S

7. See for all preceding references, G. Puppi and A. Stanghellini, Nuovo Cimento, 5, 1305 (1957).

2. G. F. Chew, Phys. Rev., 112, 1380 (1958).

3. J. G. Taylor, M. J. Moravcsik, and J. L. Uretsky, Phys. Rev. 113, 689 (1958).

4. J. G. Taylor, Nuclear Phys. 9, 357 (1958-59).

5. M. J. Moravcsik, Phys. Rev. Letters 2, 352 (1959).

6. M. L. Goldberger, E. Oehme, and Y . Nambu, Ann. Phys. (Ν. Y.), 2, 226 (1957).

7. M, Cini, S. Fubini and A. StangheUini, Phys. Rev. 114, 1633 (1959).

8. B. Cork, Phys. Rev., 80, 323 (1950).

9. Μ. H. McGregor, Phys. Rev. Letters, 2, 106 (1959).

of cotg δ⁸ (8) shifted by -a constant amount in order to account for the Coulomb corrections to the scattering length. The solid line repre­

sents our best fit of /(ω) giving /² = 0.11 ± 1 5 % . The interpolating curve has been taken of the form

(ΙΪΙ.22) /(ω) = a + bu + οω²/(1 + dd)

The parameters a = 0.06 and b = 0.97 are fixed by the low energy points, while c = — 0.33 and d = 1.40 are determined by the theore­

tical relation /(—0.5)= —0.7 and by the high-energy experimental points. The reason for introducing a pole is that it is not possible to fit with a fourth-order polynomial the slope at low energy (Blatt- Jackson straight line) together with the high-energy points and the theoretical point at ω= — 0.5.

It might be possible to fit the points with a higher order polynomial in ω, but the curve with the least number of parameters which meets all the requirements is definitely the one given by (111.22). The zero of T^b(oS) implied by the pole in f(ay) has been found also in a slightly different position, with the direct relations (III.13).

This determination of /², which is already of the good order of magnitude, can be probably improved if one uses more recent results which give a slightly lower value of the s-phaseshift (9), which gives a variation in the right direction. These results confirm, on the whole, the view that it is the exchange of one pion which gives the longest range nuclear force, and give the hope that a selection between the different available sets of phaseshifts will be possible.