Veneziano Amplitudes Consistent with PCAC Conditions. I

Progress of Theoretical Physics, Vol. 45, No. 2, February 1971

Veneziano Amplitudes Consistent with PCAC Conditions. I

Mihoko IIDA and Reijiro KUBO

Research Institute for Theoretical Physics Hiroshima University, Takehara, Hiroshima-ken

(Received September 8, 1970)

425

The nn~nA1 reaction is studied on the basis of a Veneziano model and a local chiral SU (2) X SU (2) algebra of currents. A Veneziano-type scattering amplitude for this reaction is constructed so as to satisfy Adler's selfconsistency condition and Ward-like identity. Relations among various coupling constants are derived. The electromagnetic form factor of the pion is evaluated by applying a soft pion technique to the scattering amplitude. The rms radius of the pion is calculated to be v' <r2)=0.68 fm. The A1~Pn decay width is also calculated and is compared with recent experimental data.

§ 1. Introduction

A number of remarkable results has recently been obtained from a simple scattering model proposed by V eneziano.1l Lovelace 2

l constructed a Veneziano

type amplitude for nn scattering, which satisfies the Adler selfconsistency condition, in addjtion to crossing symmetry and Regge asymptotic behavior. On the

basis of this model he obtained many successful results and found some parallelisms between current algebra and the Veneziano model. Although Veneziano's model

makes the same predictions as some of those made by current algebra, it is still

uncertain whether or not current algebra is really compatible with the Veneziano

model. A clue for exploring the compatibility of the current algebra description

with the expression obtained in terms of the Veneziano model may be found in

the problem of deriving form factors of hadrons. Because this problem is considered to be appropriate for examining whether or not the Veneziano model

satisfies partial conservation of axial-vector current (PCAC) and various low

energy theorems derived from current algebra. Some methods for obtaining Veneziano-type form factors have recently been exploited by several authors.3

l'4l

Oyanagi3l has derived a V eneziano-type electromagnetic form factor of the pion, by applying a soft pion technique of current algebra for the Veneziano amplitudes o.f nn~nn and nn~nA1. One of the results of Oyanagi's model is that the D-wave A 1pn coupling constant GD should vanish on account of conservation of vector current (CVC). Geffen5l pointed out that the matrix elements of the axial vector current may generally involve subtraction terms, so that Oyanagi's

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426 M. !ida and R. Kubo

result is a particular one because he did not take into account those terms.

Rosner and Suura6) have shown in a general treatment of 7rn"--?-7tA1 in the Ven

eziano model that the subtraction terms are necessary in order to maintain PCAC,

if one uses the simple Veneziano form given by Goebel, Blackmon and W ali/)

together with Lovelace's amplitude for nn~nn. They have also derived a general

form of the Veneziano-type pion electromagnetic form factor, which is explicitly

written as

F7< (t) = -Jnr (1- a (t)) [_!_- cl (-t-) + c2 (-t-) 2], T(5/2-a(t)) n 4m/, 2m/

(1·1)

where T (x) is the usual Gamma function, a (t) = 1/2 + (1/2m/) t the p Regge

trajectory, mp the p-meson mass, cl and c2 are subtraction constants. If we put

cl = c2 = 0, then the expression (1·1) reduces to the one originally given by

Oyanagi.3) It can readily be seen from (1·1) that, unless C 2 = 0, the last term

in the square bracket gives rise to divergence when t becomes infinite.

On the other hand, if we allow some satellite terms added to the nn~nn

r(a(s) -1/2) (a(t) -1/2) T(1-a(s))T(1-a(t)) T(2- a (s) -a(t))

= - (1/ 4) r (2 Vm + Vm- 4 Vm),

( V~mn= T(l-a(s))T(nz-a(t))) T(n-a(s) -a(t))

(1·2)

which satisfies the Adler selfconsistency condition by itself, we can get a con

sistent PCAC equation satisfied by Veneziano amplitudes. This possibility was

suggested also by Geffen5) and extensively investigated by Arnowitt, Friedman

and Nath. 8)

In the present work we give a Veneziano-type nn~nA1 amplitude, which:

has the form that a contact term as well as some sa~ellite terms is added to

the one given by GBW.7) In § 2 we derive the Ward-like identity, which the

scattering amplitudes for 1t1t--?-1t7t and nn~nA1 with each one meson off the mass

shell have to satisfy. The assumptions we shall make in § 2 are (1) the local

chiral SU (2) X SU (2) algebra for vector and axial vector currents, (2) PCAC

for the axial vector current A/ (x):

(1·3)

as well as eve for the vector current, where cpa (x) the renormalized pion field,

the superscript a being the isospin index, mrr and frr stand for the mass and the

decay amplitude of the pion, respectively, (3) the field-current identity with pole

dominance approximation for the axial vector current:

(1·4)

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Veneziano Amplitudes Consistent with PCAC Conditions. 1 427

where g A is defined by

<A1°IA/(O) IO)= EpUA ,*)

with E"' being the polarization vector of the A1-meson. In § 3 we introduce the Veneziano amplitudes for rcn~rm and rcn~nA1 and

by using these amplitudes we derive some relations among various coupling constants. The Veneziano-type electromagnetic form factor of the pion is derived in § 4 on the basis of the amplitudes considered in § 3. In § 5 the A1~PTC decay rate is evaluated and compared with experimental data. Discussion of the results

is also made in the same section.

and

§ 2. Scattering amplitudes and Ward-like identity

· We shall consider the following two processes,

n- (ql) + n+ (q2) ~n- (qa) + A1 + (q4),

the S-matrix elements of which we define respectively by

Sn-•rnr = o ti + i (2rc)4o4 (ql + q2- qs- q4) M

and

(2·1)

(2·2)

(2·3)

(2·4)

Now we assume the scattering amplitudes M and M"' are able to describe the scattering processes with n+ (q4) and A1+ (q4) extrapolated off their mass shells. Then, according to the convenient description of hard pion technique exploited by Schnitzer and Weinberg, 9) the scattering amplitudes M and M"' are found to be related to the following matrix element of the axial vector current:

- i<n- (qs) lAP(-) (0) In- (ql) rc+ (q2) >=gA-l A A. (q4)pj)Ml)- (q4p/mn2) 117( (q4) M' (2·5)

A/-) (x) = (1/ .J2) (A_} (x) - iA"'2 (x)),

where ·use has he en made of the field-current identity (1· 4) and

(2·6)

and

(2·7)

*> It should be understood that the normalization volume factor 1/ V2mA V has to be multiplied on the right-hand side of this relation. Hereafter we define matrix elements without this volume factor.

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428 M. lida and R. Kubo

are the covariant spin-1 and spin-0 parts of the axial vector propagators respec

tively, and mA the mass of the A 1-meson.

The Ward-like identity can be obtained by multiplying q4"' on both sides of

(2 · 5) and by taking into account PeAe (1· 3):

(UA/m})q4"'M"'=frrM. (2·8)

The off-shell scattering amplitude for reaction (2 · 2) may be written as

M"'= (qa-q2)"'A(s, t) + (qa+q2)"'B(s, t) +q4"'C(s, t), (2·9)

where A(s,t), B(s,t) and C(s,t) are invariant functions of s=-(q1+q2y and

t = - (q1- q8y, A (s, t) and C (s, t) being symmetric and B (s, t) antisymmetric

under interchange of va.riables s~t (q2~- q8).

Substituting (2 · 9) into (2 · 5) and taking the soft pion limit q2-'>0, we have

- i(tr- (qs) lA"' <-) (O) In- (ql) n+ (q2) )--'> frr- 1<n- (qa) I v"' <o) (O) In- (q1))

= { g A (A (mrr2, t) + B (mrr2, t)) (ql + qg)# 2(m}-t)

+ [gA (c(mrr\ t) _ _.!_A(mrr2, t) _ _.!_B(m/, t))- frrM Jq4"'}· (2·10)

m} 2 2 mrr2-t

eve requires that (tr- (qs) I v#(O) (0) In- (ql) > should be proportional to (ql + qg)"',

hence

(2 ·11)

On the other hand,· substituting (2 · 9) into (2 · 8) and making q2-'>0, one obtains

(2 ·12)

which agrees with (2 ·11) in the limit mrr--'>0. Therefore, we note that our theory

is considered to be selfconsistent in the region where the square of the pion mass

1s negligibly small compared with t.

§ 3. Veneziano amplitudes and relations between coupling constants

In this section we shall introduce the Veneziano amplitudes for the reactions

nn-'>rcn (2 ·1) and nn-'>nA1 (2 · 2). For the former we employ the simple Lovelace

amplitude :2)

M = 2G;u Vm (s, t), (3 ·1)

which is normalized at the p-pole with Gpn being the renormalized pnn coupling

constant. We assume that this amplitude can be analytically extrapolated off the

mass shell. For the latter reaction with A 1+ (q4) off the mass shell, we take the

following expressions for the invariant amplitudes in (2 · 9) :

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Veneziano Amplitudes Consistent with PCAC Conditions. 1 429

A (s, t) = r Vm + r' Vm- 4r v223 + r (a (s) -a (t) Yf(s, t)' (3. 2)

B(s, t) =r(a(s) -a(t)) (Vm- Vl13+ (1-a(s) -a(t))f(s, t)) (3·3)

and

C(s, t) =A(s, t), (3 ·4)

where r and r' are adjustable parameters and the terms involving the function f(s, t) symmetric in the variables s and t may be considered as the sum of all the possible satellite terms. However, we assume, for the sake of simplicity, that f(s, t) has no poles and it must behave as sa(t)-

2 for t fixed and s~ oo.

If V113 and V223 as well as f(s, t) are absent in (3 · 2) and (3 · 3), then A (s, t) and B (s, t) turn out to be the ones given by GBW.7

) The term E • q4C (s, t) ap~

pears only in the off-shell scattering amplitude, then it cannot be determined by any experiment. The presence of this term gives rise to a term proportional to q4

2 in the Ward-like identity. In order to get rid of those terms in the Ward

like identity, we have taken C (s, t) as (3 · 4) without any loss of generality.

Thus, substituting (3 · 2), (3 · 3) and (3 · 4) into the Ward-like identity (2 · 8), one

obtains

(3·5)

Now we mention the Adler selfconsistency condition, which demands that

E "'M"' should vanjsh when one of the three pion momenta tends to the soft pion limit. The scattering amplitude (2 · 9) can be rewritten in terms of (3 ·2), (3 · 3) and (3 ·4) as

M"' = qlp, (r Vm + r' vl12- 4r v223 + (a (s) -a (t) Yf(s, t))

+ (q2+q3)'"(a(s) -a(t))r(Vl12- Vl13+ (1-a(s) -a(t))f(s, t)). (3·6)

In the limit q1~0, we have a(s) =a(t) =1/2(s=t=m,.2), so that we see EM p, I"

certainly vanishes in this limit. On the other hand, the condition that E ""M'" should vanish in the limit q2~0 (s=m,.2, t=m}) yields the usual relation

a(m}) =3/2 or (3. 7)

provided that

(3·8)

The Born term of p and rJ mesons in the s-channel of nn~nA1 is given by

(3 ·9)

where we have used Fayyazuddin and Riazuddin's10) convention of the coupling

constants which are defined by the following effective Lagrangians:

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430 M. lida and R. Kubo

L (A1pn) = GsAittP/n- + 2GDAittPv08 p8vn-,

L (pnn) = iGP"" (n+8 p7r-) P/,

L (rJnn) = GuurJn+n-,

L (A1rJn) = G"AI1Attt (rJB pn-).

(3 ·10)

(3 ·11)

(3 ·12)

(3 ·13)

Comparing the residues of (3 · 6) at p and rJ poles with those of (3 · 9) we have

and

r=GpnnGD,

r'= _l_cpuGs m2 p

(3 ·14)

(3 ·15)

(3 ·16)

The relation (3 ·16) is different from the one derived by Fayyazuddin and Riazuddin/0

) and Rosner and Suura.6) It follows from (3 · 5), (3 ·14) and (3 ·15)

that

(3 ·17)

According to the hard-pion method of Schnitzer and Weinberg, 9) G8 and GD are respectively proportional to 2 + (J and (], where (J is the standard SchnitzerWeinberg parameter. Then we have from (3 ·17)

Gs= 2f;~pnm/(2+(J) (3 ·18)

and

• 2fnGpn GD= UA (J. (3 ·19)

If we further define nAf(JP = 2+) coupling by the following effective Lagrangian:

then we obtain

and

where G1u IS defined by

L (fnn) = Gfn (n+8 p8vn-)ftt0v •

(3. 20)

(3. 21)

(3 ·22)

(3 ·23)

(3 ·24)

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Veneziano Amplitudes Consistent with PCAC Conditions. I 431

§ 4. The electromagnetic form factor of the pion

The isovector electromagnetic form factor of the pion F" (t) is defined by the following relation,

F" (t) (p + q)p_ = <n- (p) IV" <o) (O) In- (q) > . (4·1)

Then it follows from (2 ·10) that

F (t) = gAf" .{A(m 2 t) +B(m 2 t)}. 7r 2( 2 ) "' "' mA-t

(4·2)

Substituting (3 · 2) and (3 · 3) into this expression, we obtain

F (t) = gAf" [r' -lnT(l-a(t)) + 1 T( 4m/ T(5/2-a(t)) 3/2-a(t)

x {-r(1+2a(t)) ../7iT(2 -a(t)) +2r(1/2-a(t)Yf(m/, t)}], (4·3) T(5/2-a(t)) ·

and in terms of (3 ·14) and (3 ·15) this is rewritten as

F (t) = gAfT( ..JnG [Gs T(l-a(t)) T( 4m/ pn m/ T(5/2-a(t))

+ . Gn {- (1+2a(t)) T(2 -a(t)) + 2 (1/2-a(t)Yf(m"2' t)}J·

3/2-a(t) T(5/2-a(t)) ../n (4·4)

It should be remarked that the expression ( 4 · 4) does not have a pole at a (t) = 3/2 by virtue of the condition (3 · 8). From the requirement FT( (0) = 1 follows the modified KSRF relation :12

)

(4·5)

where we have used the relation (3 ·17). The rms radius can readily be calculated from ( 4 · 4) irrespective of the

unknown function f(m/, t) to be

.j <r2) = [6F/ (0) ]112 = 0.68 fm, (4·6)

where we have used (3 ·17) and ( 4 · 5). This is in fairly good agreement with recent experimental value, 0.86 ± 0.14 fm. 13

)

§ 5. Discussion of results

We constructed in a previous section a V eneziano-type scattering amplitude for mr~nA1. which, together with the nn~nn amplitude, satisfies the Ward-like identity and Adler's selfconsistency condition. We have invoked advantage of the Lovelace-Veneziano amplitude for TCn~nn without satellite terms or subtraction terms, while the nn~nA1 amplitude is different from the one given by GBW/)

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432 M. Iida and R. Kubo

in such a way that satellite terms and an unknown function f(s, t) are added to GBW's amplitude.

On the other hand, if we use GBW's amplitude, we have to modify the 7C7C~7C7C amplitude by adding satellite terms to itself, so as to maintain PCAC

conditions. This possibility has been investigated by Arnowitt, Friedman and Nath.8

) We shall briefly explain their method, paying a particular attention to

the Ward-like identity. Substituting GBW's amplitude

M'" = (q3- q2)'" (r Vm + r' Vm) + (q3 + q2)'" (a (s) -a (t)) r Vm

for reaction (2 · 2) into the Ward-like identity (2 · 8), one gets

g A q4p,Mp, = _g A~ { (r'- 2r) Vm + 4r v222- r Vm}. m} 2

Therefore, if one takes the 7C7C->7C7C amplitude as

M = 2G ~n Vm + g A f7tr (2 v222- Vm)'

then one has

According to Fayyazuddin and Riazuddin/0)

r'-2r (Gs/m/-G»)Gpu·

(5 ·1)

(5 ·2)

(5 ·3)

(5·4)

(5 ·5)

Thus one obtains from (5 · 4) and (5 · 5) just the same relation as the one that we have derived in (3 ·17). However, it should be remarked that the satellite

term as in (5 · 3) is considered to be only a particular one, since it may contain V111 in itself (see (1· 2)), while in our case the relation (3 ·17) can be uniquely

derived. The relation (3 ·17) is combined with hard pion expressions for npA1 coupling

constants9) to yield (3 ·18) and (3 ·19). Now if we further assume the Weinberg

relation11)

(5 ·6)

with (5. 7)

which implies

gA= -Jnf"mP, (5 ·8)

by virtue of the modified KSRF relation ( 4 · 5), then we have

(5. 9)

and

(5 ·10)

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Veneziano Amplitudes Consistent with 'PCAC Conditions. I 433

Although the Weinberg relation (5 · 6) is originally derived from a p-pole dominance approximation, we here regard it rather as a universal relation.

In terms of these expressions for G8 and Gn we can calculate the Ac-'>PI! decay width by using the interaction Lagrangian (3 ·10):

.J2G;ump T(Ac-'>Pn) = 96n2 (5o2 +22o+25), (5 ·11)

while the P---'>7!7! decay width is approximately given by

(5 ·12)

Hence

(5 ·13)

Figure 1 shows the value of T (Ac·-'>pn) as a function of the parameter o, where we have taken the P---'>7!7! width as

T (P---'>7!7!) = 125 ± 20 MeV . (5 ·14)

Recent experimental data14) show the value of A 1---'>PI! width ranges from 50 MeV

to 160 MeV. If we take the presently accepted experimental width :14)

T(A1---'>pn) =95±35 MeV,

then we can deduce from (5 ·13)

(5 ·15)

which indicates a substantial D-wave contribution to the A1pn interaction. Deery et al.15

) have analyzed the data on the reaction n+ d---'>7!+7!+7!-d at 5.4 Ge V / c by taking a simple Veneziano form for 7!7!---'>7!A1 and obtained G8 /Gn

= - 0.15 ~ g:~~' from which

o is evaluated to be o = -1.7 ± 0/2, in agreement with our result. BaHam et al.16

) have· studied the orbital angular structure of the A 1-meson produced from n-p--7 n-n-n+p at 16GeV/c and

-Ls<o< -1.2,

(MeV

200

-1.2 -1.3 -1.4 -1.5 -1.6 -1.7 -1.8 0

(5 ·16)

Fig. 1. The A1~Pn' decay withs are plotted as functions of the SW-parameter o. The upper and lower solid .lines correspond to the ones taking F(p~n-n-) =145MeV and F(p~n-n-) =105 MeV respectively. Recent experimental data14> on A1~Pn' decay are almost within the region bounded by the two broken lines.

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434 M. !ida and R. Kubo

obtained the ratio of the helicity coupling constants IU1/Uol = 0.48 ± 0.12, which implies o = -1.5 ± 0.2, also consistent with our result. On the other hand, by analyzing the same reaction as that of Ballam et al. at 6 Ge V / c, Crennell et al.17

)

have obtained o= -0.32±0.31 Clul/Uol =0.89±0.07), which seems to support Schnitzer and Weinberg's prediction -1::So::SO and inconsistent with ours. After all, we cannot yet state anything definite from recent experimental data, which seem to contain a considerable amount of ambiguity. However, it is not plausible to say that there is no D-wave contribution to A 1pn jnteraction, since there are some evidences that the absolute value of o is rather large.

Finally, we should mention the residual unknown function f(s, t). One method possible to obtain f(s, t) may be to determine the scattering amplitude

for nAr-"n'Ah which we denote by E/EvM,m where E/ and Ev stand for the polarization vectors of the outgoing and incoming A 1-mesons respectively. The

relation between M"'v and M"' is given by the following Ward-like identity:

(5 ·17)

In addition, M"'v satisfies

The form of f(s, t) can be determined from (5 ·17). Details will be given in a forthcoming paper.

Acknowledgements

We would like to express their thanks to Professor H. Takeno and Professor K. Yokoyama for continual encouragement. We are also grateful to Dr. M. Aoki

for discussions at an early stage of this work.

References

1) G. Veneziano, N uovo Cim. 57 A (1968), 190. 2) C. Lovelace, Phys. Letters 28B (1968), 265. 3) Y. Oyanagi, Prog. Theor. Phys. 42 (1969), 898; Nucl. Phys. B14 (1969), 375. 4) H. Suura, Phys. Rev. Letters 23 (1969), 551.

M. Namiki and I. Ohba, Prog. Theor. Phys. 42 (1969), 1166. P. H. Frampton, Lett. Nuovo Cim. 3 (1970), 229. F. Hussain and M. S. M. Razmi, Lett. Nuovo Cim. 3 (1970), 339. R. Acharya, L. Brink and R. P. Saxena, Lett. Nuovo Cim. 4 (1970), 153. T. F. Walsh, Nuovo Cim. 68 (1970), 469.

5) D. A. Geffen, Phys. Rev. Letters 23 (1969), 897. 6) J, L. Rosner and H. Suura, Phys. Rev. 187 (1969), 1905. 7) C. ]. Goebel, M. L. Blackmon and K. C. Wali, Phys. Rev. 182 (1969), 1487. Hereafter we

refe;r to this paper as GBW,

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Veneziano Amplitudes Consistent with PCAC Conditions. I 435

8) R. Arnowitt, M. H. Friedman and P. Nath, Phys. Rev. D1 (1970), 1813.

9) H. ]. Schnitzer and S. Weinberg, Phys. Rev. 164 (1967), 1828.

See also I. S. Gerstein and H. ]. Schnitzer, Phys. Rev. 170 (1968), 1638.

10) Fayyazuddin and Riazuddin, Ann. of Phys. 55 (1969), 131.

11) S. Weinberg, Phys. Rev. Letters 18 (1967), 507.

12) K. Kawarabayashi, .S. Kitakado and H. Yabuki, Phys. Letters 28B (1969), 432.

13) C. Mistretta, D. Imrie, ]. A. Appel, R. Budnitz, L. Carroll, M. Goitein, K. Hanson and R.

Wilson, Phys. Rev. Letters 20 (1968), 1523.

14) Particle Data Group, Phys. Letters 33B (1970).

15) B. ]. Deery, ]. E. Mansfield, N. N. Biswas, N. M. Cason, V. P. Kenney, ]. A. Poirier and

W. D. Shephard, Phys. Letters 31B (1970), 82.

16) ]. Ballam, A. D. Brody, G. B. Chadwick, Z. G. T. Guiragossian, W. B. Johnson, R. R. Larsen,

D. W. G. S. Leith and K. Moriyasu, Bull. Am. Phys. Soc. 14 (1969), 573.

17) D. ]. Crennell, U. Karshon, K. W. Lai, ]. M. Scarr and W. H. Sims, Phys. Rev. Letters

24 (1970)' 781.

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Veneziano Amplitudes Consistent with PCAC Conditions. I

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