IC/79/21

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

LOW-ENERGY PHOTO- AND ELECTROPRODUCTION FOR PHYSICAL PIONS - II:

PHOTOPRODUCTION PHENOMENOLOGY AND EXTRACTION OF THE QUARK MASS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

J .T . MacMullen

and

M.D. Scadron

1979 MIRAMARE-TRIESTE

IC/T9/S1 A. Introduction

Interactional Atomic Energy Agency

and

United nations Educational Scientific and Cultural Organiirtion

UTTEFHATIONAL CENTRE FOR THEORETICAL PHYSICS

PHOTO- AITO ELECTHOPBODUCTIOIf TOR PHYSICAIr PIQIS - U s

PHOTOPRODUCTIOM PHEHOMEHOLOGY AND EXTRACTION OF THE QUARK MASS •

J . T . MacMullen

Phys ic s Department, U n i v e r s i t y o f Arizona, Tucson, A2 85TS1, DBA,

and

M.D. Scadron ••

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

The Background resonance A(l23O), H«(l520), N».(lltTO) and N«{l535)plon and

rector amplitudes are first calculated in the soft pion and cm-shell configuration,

respectively. Then a comparison is made with the usual soft pion theorems and

on-atell low-energy expansions of current algebra as worked out in the previous

paper. The agreement is good and ve also deduce a nucleon dipole form factor

axial-vector mass at m » 1.23 GeV. Finally an approximate value for the non-

strange current quark mass of m • 0.6k ± l.llu is extracted from the data.

KIBAMABE - TRIESTE

February 1979

Given the theoretical analysis of the previous paper* ^ (ref. I), we

are prepared to make a comparison with the data. To do this we must make

use of the pion and radiative couplings of the baryon resonances which dom-

inate the photoproduction backgrounds: the A(1230), N*(1520), N*(1470) and N*(1635).

The required covariant resonance couplings are worked out in Appendix I.

In Sec. B we approximate the soft pjem background amplitudes by these isobar

resonance poles and find excellent agreement with the standard soft pion

theorems.^'3'

Then we proceed to compare the on-shell photoproduction amplitudes with

the data. He analyze threshold photoproduction data h e r e wtth the idea of testing

the on-shell low energy theorems of I. Our threshold electroproduction

current algebra theorems will be analyzed elsewhere(4) The background axial-

vector amplitudes B . ' + * ' ' 0 ' are obtained in Appendix I I using both dis-

pcrsion-theoretic and field-theoretic techniques with results which di f fer

l i t t l e in magnitude. In Sec. C these background amplitudes are decomposed

into pole and non-pole parts for general values of the variables v and t ,

but with q^2 = M2. Finally in Sec. D these background amplitudes are sub-

stituted into the low-energy current algebra photoproduction amplitudes

of (1) in ref. I. Good agreement with threshold data is found for A 3 ^ + ' 0 ' ,

\ ' + ' ~ ' ° ' , and A / " ' . The threshold amplitude A ^ " ' leads to a value for the

axial-vector mass of m.=1.23 SeV assuming a dipole structure for the axial-

vector-nucleon form factor, in good agreement with a recent neutrino scattering

analysis^5' and with threshold electroprodi'Ction^4'.

• Sulmltted for p u t l i c a t i o n .

• • Supported in part t y the Science Hesearch Council and the US National ScienceFoundation.(Permanent address: Department of Physics , University o f Arizona, Tucson,AZ 85T21, USA.

- 2 -

This encourages us to make a detailed investigation of the non-strange

current quark mass m occurring in the chiral breaking z terms in A i^ + > 0 ^ .

Looking at threshold multipole analyses for these two amplitudes, we find in

Sec. D that m ;v 0.64 ± 1 . l i p . While the quark mass is not well determined,

the fact that i t must be of 3(w) and positive on average is consistent with

the almost exact agreement of the soft theorems of Sec. B when saturated by

the isobar poles. In Sec. E we summarize the situation and note the need

for more accurate low-energy photoproduction data.

8. The soft amplitudes

Me begin by testing the soft theorems. We use the off-shell analog of

equation ( l .a) in ref. I for photoproduction (k2 = 0):

where A, = A,-(A,) , . We are interested in the soft pion l imi t (q •+ 0)1 * * polecorresponding to v = t = 0, so we recover the soft pion result^ ' from (1),

A, (+>o)(0)4mf 4m1

(2)

where we have used the Goldberger-Treiman relation mg, - f^gfO). Recall that

8, ' * 0 ' comes from the ax ia l -vec tor background term q̂ M v ^ > 0 ' which vanishes

as q + 0. Since K ' does not vanish as q + 0 J B 1 ' + > 0 ' must do so. Then using

the values g(0) = 1Z£ KV = 3.7, KS = -0.12 we f ind

(3.a)

0.008 u"<3.b)

Me expect that A, < + )(0) will be dominated by the A(1230) intermediate

state, while there will be contributions to A / ° ' { 0 ) from the N*(1520). N*(1470)

and N*(1535). Of course the 4(1230) will not contribute to A ^ 0 ' since this

amplitude represents 1 = 1 pion production from an isoscalar photon which

cannot couple to the I = '? A(1Z30>. In the soft pion momentum limit these

various isobar (narrow width) contributions to A^*' 0' are unique (see

Appendix II), and give

-3- -k-

— - . -t----w-.^"t-aur.*•*•• iir,

WH30)] - % g* [-

<4.b)

(4.c)

"*"S3S) -(4-d)

The relative sign of g*Gjj vs. g*v is positive as determined by SUg,

consistent with (2). We use data comparison (as given in Sec. D) to choose

the relative signs of the N*(152O) and the N*(1470) to H*(1535) contributions!

al l relative signs are also consistent with the multipole analysis of Adler and

Gilman^3'. The coupling strength g* = 15.39 m~ is narrow width-determined

from the A + + •+ pit+ decay. In Appendix I the remaining coupling constants in

(4) are found to be G *̂ - 3.07, G, = 2.90 GeV'1, G2 « -1.50 GeV"2;

gN* = -23.11 m'1, H^ = -1.95 GeV~\ H^ = -3.24 GeV"1, H2S = 1.52 GeV'Z,

Hz 1.331 = -5.23. ^ = 0.19, ^ = 1.58; g" = 0.68, ^ = -0.34,

K" = 2.22, where m = m̂ and M is the mass of the appropriate Isobar. These

coupling constants applied to (4) give

[NM1520)] + \[$t [N*(1470)]+ * , $ , . N*(1535)

= f-0.306 ± 0.013 y"') + (-0.006 + 0.0OB v'

l\, [4(1230)]isobar

(0.023 ± 0.004 v~

(-0.016 t 0.003w'?') = -0.305 ± 0.016p" ,

- 5 -

isobar

* (O.010± 0.O08 v'1) + fo.003 ± 0.004 V~*)+ (0.002 ± 0.00 M'2)

= 0.015 ± 0.009g" .(5 .b )

The values of K^ • ' given by equations (5) are only s l ight ly larger

than the PCAC predictions (3). This is to be expected since we do not make

narrow width corrections for the various ANTT and N*Nw coupling constants as

stressed by Adler and Gilman' ' . Indeed i f narrow width corrections are made

for the & alone in (5a)(not a consistent procedure), then A * j ^ | t becomes

-0.243(j" which agrees very well with (3a) and Adler-G11man,who obtain

A~i (o) = -0.236u~ (Vi + -2Aj).

Isobar saturation of A V + * ° * was f i r s t considered by Fubini e t _ a 1 . ' 6 ' ,

but the data are better now and the consistency with both PCAC theorems is

worth emphasizing. From our point of view for what is to follow, the soft

theorems provide a test of the relative order of magnitudes and especially

the signs of the various isobar coupling constants. The good agreement

between (3) and (5) gives us further confidence to use the same type of

isobar saturation to test the on-shell photoproduction theorems of ref I in

the low-energy region.

C. The on-shell axial-rector background amplitude

We develop the background contribution to the pion photoproduction

amplitude in terms of the quantities B. defined by the following expression:

(6)

where the K^J are the CGLN photoproduction covariants. Our plan is to saturate

these axial-vector background amplitudes B. for an on-shell kinematical con-

figuration of v and t and q2 ~ IJ2 using the samp typ*- of isobar poles which

were so successful in saturating the soft giqn Iwkgruund amplitudes A~/ p0'

in Sec. 8.

The details of the calculation of the resonant contributions to the photo-

production background amplitudes are given in Appendix II. We concentrate on the

subtle spin 3 /2 A(1230) contribution since from it the spin 3/2 N*(1520) contribution

is easily obtainable. The spin % N*(147Q) and N*(1535) isobar contributions

are found in a straightforward manner.

There are two methods which can be used to calculate the spin % A(1230)

contribution. One method is a dispersion theory approach,which allows the use

of the on-shell spin % propagator; alternatively a field theory approach necessitates

the use of the ambiguous off-shell spin 3/2 propagator. Actually, what is usually

done in the field theory approach is to redefine the couplings so that the

ambiguous part of the propagator does not contribute. This technique Intro-

duces two new parameters (Y at the photon vertex and Z at the pion vertex),

which are determined by fitting to data. ' The pole terms one obtains from

these two methods are the same, the two methods differ only in the non-pole

terms, as 1s discussed in Appendix II. The discrepancy between dispersion

theory and field theory is only important in B* ' and B3* , it is of <3"(u2)

in B, \ We now list the pole parts of the M1230) contribution to the back-

ground amplitudes as derived in Appendix II:

3M;+2mM+m2

y ! + k-q] G,

(7.a)

B J ^ - [G, + >i(M-m) G 2] ,

Dl (H+fn) - n.

(7.b)

(7-c)

( 7 d )

" 27-B j p(+ ) [A{1230)] , j f 3

B.pt-'tAdZSO)] - - ^ B j p( + ) [ i (1230)] . <7.f)

where M = Mfi - 1230 MeV, m = mN = 939 HeV, v •> (s-u)/4 and ^ = |(M2-mz-k-q).

Note that B J+*[&(1230)] evaluated at the physical CM threshold

B,p(t)[A(I230)]

th= -0.362 u" (8)

This result is of &(\i/m) different from the soft value obtained in Sec. B.

This is reasonable since the discrepancy between the resonance denominators

for the two cases is of &{\\/m).

We also note at this time the necessity of introducing the previously dis-

cussed non-pole terms. The background amplitude (i.e. q^M ) must vanish as

q •+ 0. Since the CGLN covariants Kya, Ky

3, and Kj> vanish as q + 0, this soft

-7- -8 -

Hm1t places no constraints on B2, B3, or B^. However, Kv* does not vanish

as q + 0 and i t is therefore necessary for B, to vanish In this l im i t .

B p as given by {9.a) does not satisfy this condition, underlying the

importance of the extra non-pole term of B / + ' . AS 1S derived In Appendix

I I , the non-pole term is given for dispersion theory by

B lNpt+)[6(1230)] - - V, g* [-m(M-Hn) [Mz-m2+k-q] G2 • ( 9 . a )

BjNJ[4(1230)] = 0i = {+) , j = 2, 3, 4 > or

i M + ) , J = 1, 2, 3, 4 . (9-b)

The non-pole terns for field theory are given in ref. 6 as (in our notation)

g*G, , (lO.a)

(lO.b)

W(lO.c)

(10.d)

where a = 1+4Z, B = 1 + 4Y and all other B.^ 1 are zero. Note that equations

(10) do not include the G2 coupling which is included in equations (9): both

couplings are necessary to produce the observed Ml or G * A dominance. It

= B ] p + B I N P vanishes in the soft pion limitis easy to see that B

if either (9.3) or (10.a) 1s used for B 1 N p (+ ) , we shall henceforth consider

only the dispersive approach.

As Is shown in Appendix I I the N*(152O) contribution is obtained from

that of the A(.123O) by let t ing M • -M ( I .e . \

62 + !SH2V>S where the

% g* -* *sgN*, Gr> -

^ contribute to the (+) amplitude and the H.S to

the (0) amplitude. To obtain the contribution to the (-) amplitude,the

following relations should be used:

= ^ - B,p( + )[N*(1520)] , j f 3 , (11.a]

(11-b]

where \>N+ = Js(MJ-m2-k-q).

Finally, we write down the N*(1470) contributions:

(12.a',

B2pt+>O)EN*(1470)] - 0 (12. b]

B,,p(+'o)[N*(1470)] ft*4 fHvN#^v =

{12.d]

B ipC")[N*(l47O)] = £- B {+)[N*(1470)] , j i 3

-10-

B,p

{" J - 83p(+)[M*(1470)J , (12.0

BlNpl+'° V(1470)] (12.g)

BjNpn[N*(1470)] = 0

i = (+.0), j = 2, 3, 4.or

i = (-) • j = '. 2, 3. 1 , (12.h)

where g' = -5.23is obtained from the on-mass-shell decay rate for N* -* NIT and

K£ = 0.19 and Ky = 1.50 are obtained in Appendix I. The N*(1535) contribution is

obtained from that of the N*(1470) by letting g'-+g", K'+K" and H+ -H in (12).

D. Data comparison

We are now in a position to compare theory with data. KcClaskey, Jacob,

and Hite (MJH) have obtained values for the CGLN photoproduction amplitudes

at threshold through the use of interior dispersion relations and we begin by

comparing our predictions with their results. We will at first exclude the

j = 1, i = (+), (0) cases,since these contain the chirai breaking E terms which

we wish toextract from the data,and will also exclude the j - 2 cases.since

results for A2 are not given by MJH. The first thing we do is evaluate the

background contributions at the physical threshold in the CM frame for the four

isobars that we consider. These results are given in Table I. The back-

ground contributions are to be combined with the other terms contained in

equations (1) of ref. I in order to obtain predictions of the photoproduction

invariants that can be compared to the phenamenoiogical values found by HJH.

Note that HJH differ from us in their definitions of the A.1, namely,

- 1 1 -

J- AW .

US MJH(13)

Also, the numbers quoted in MJH are in units of GeV~n (n = 2 for A ] t n = 3 for

A, t ) so we must multiply by a numerical factor iin to get numbers in units of

inverse pion masses. The tabulated results are given in Table I I . As can be

seen, the data and theory agree fa i r ly well,with the exception of A i , ' 0 ' . This

discrepancy is not serious since A i / 0 ' i t se l f is very small.

The low-energy theorem for A3* ' contains the form factor

G(tl =_ gAtt) -

t - V* (14)

To estimate G(t) we have used a dipole form for

9A< 0 )

(1 - (15a)

which gives at threshold for g«(0) £ 1.25

G(t) ~-2gA(°)

(15b)

I f we demand the low-energy theorem for A3* ' in ref. I to be exactly satisfied >,

for the reconstructed experimental value at threshold' ' , A 3 ' " ' = -0.074 ± O.OOIVJ" j,

then we can solve for (15b) and Table [ as

- " 3 x — — - t j , • ' , .

fT 8 m3 (1 + M/2m)

= -0.074 + 0.019 + 0.103 = 0.048,,"3 ,

(16a)

-12-

mA = 1.23 GeV . (16.b)

This value of m» is in good agreement with the recent determination based

upon neutrino scattering ' ' and also threshold electroproduction* '.

Since the data and the theory appear to be in good agreement, we pro-

ceed to a determination of the non-strange quark mass in which is contained

In the I {+.0) terms of equation ( l .a) of ref, I. There are two d i f f icu l t ies

which prevent an accurate determination of rft. First, the values for the

Eo+ multipoles at threshold given in the literature vary quite a bit

from author to author. Second, since m is derived from a small number which

is the difference between two large numbers, even small fractional uncer-

tainties in the data will yield large fractional uncertainties in m.

There are two ways to obtain a value of rii. It can be evaluated by using

the recent determinations of A^ ' and A ^ ' made by MJH or from threshold

values of E . and E r^given in the literature.o+ o +

To obtain in directly from A, ' , we use equation (l.a) of ref. I2

evaluated at physical threshold in the CM frame v = mu, t = -u ,

(+.0)

t h

+ B (+.0)1' t h

(17)

where

0.049 (18.a)

and

(o)I -r^^T^h* 0.019 m/u3

- 1 3 -

To use E>0 data for the determination of m, we need E * *0'

evaluated at the CM threshold:

r (+.0)o+

th

t+.o)0+

th

Hu/2m » (+,o)1+u/m Ms (19)

where we have defined E ^ '°'E . to make the numerical analysis easier, with

E o + = 47.55 E0+/u at the CM threshold. Using equations (1) of ref. I to solve

(19) for the chiral breaking terms we find (g,(pZ) » 1.30 from (15))

1Eo+ + E

0+th th

(20)

where the background threshold multipoles are from table I,

0+ -B,

th th t h th

0.002 i 0.024 o

0.001 ± 0.014 u'

-2(+)

(o)

The values of in obtained independently from the isovector and Isoscaiar

are tabulated in Table I I I . N a r r o w v i d t h corrm:rloi» to (Z1) can be expected

to be negligible. Thus we nay apply the various multipole analyses considered

in the l i terature*9 " 20K

-Ik-

As can be seen from Table I I I , there is a large variation in the

values of the extracted m with reasonably larqe uncertainties. Nevertheless,

the scale of the quark mass is small, since i t is obtained from the difference

between much larger terms. Therefore the few (unphysical) negative values of

n in Table I I I are of no great concern. I f we average over al l the predictions

of m in table I I I we obtain m = 0.64 ± l . l l u . Clearly,better low-energy data

is needed in order to zero in on the quark mass.

E. Conclusion

We conclude that the low-energy photoproduction data is reasonably con-

sistent with the soft pion predictions of current algebra. Furthermore the

on-Shell Ward identities derived in ref. I appear to be in agreement with a

recent extraction of the CM threshold invariant amplitudes.' ' Finally,

various determinations of the chiral breaking quark mass roughly average to

a value of m = 0.64 +_ Lily.

In order that future data analyses zero in on the quark mass, we suggest

the data be analyzed in terms of the invariant amplitudes below threshold, as

is now done for nN scattering (see e.g. Nielsen and Oades v" ;) rather

than in terms of multipoles at threshold or slightly above threshold. In

this connection it should prove useful to work with the amplitudes F, A2, A,,

A, with

(22)

replacing A . This amplitude F is analogous to the nN amplitude F = A + uB

and both are useful because the non-pole parts (in dispersion theory) vanish

-15-

identically in the single soft pion l imit. That i s , the large contact terms

of g2/m in nN scattering and gic/4m2 for photoproduction are automatically

subtracted out of the background amplitudes V. For photoprodur.tion,it is the

difference A, - gic/4m' which is the quantity of interest in determining the

chiral breaking E-terms. I f we use equations (1) of ref. I to write an off-

(the pion) mass-shell expression for F we find

(+.0)

' WV)*T res ( + '0 ) ' (23

( + ' o ) * B , ( + ' o ) + \ B , < + ' c ) + j ^ B , ( + ' o ) .where F r e s( + ' o ) * B , ( + ' o ) + \ B , < + ' c ) + j ^ B, From this expression

it is clear that f •* 0 as q •* 0,since the B amplitudes vanish in the soft pion

limit. Furthermore,we see that PCAC requires that F be small on the pion mass

shell. Thus it is the natural amplitude to use to investigate chiral breaking

terms.

Finally, if our low-energy analysis is to be sharpened, the narrow width connect!

to all four isobar coupling constants should be found by folding in the relevant

phase shift data near resonance. This has be^n considered for the 5(1230)

,(3) (22)resonance in photoproduction^' and in *N sca t te r ing 1 " ' , but also must be done

for the 1520, 1470 and 1535 photoproduction resonances.

Onr df u4iM-»4*) «"•!* i±*e to t&anX R. Crmrford; If.F Osalt, Q. Hit9, tU JacoT; and O. Mooffecnue for helpful tdvics. H* iaalso grateful to Prbfee«or £bdu* S«l«m, toe International Atomle QxwgyAgency and UfgSCO f « hospitality at the International Cantr* for Tb«or«tl4slfhys io , Trieste. Hospitality at the University af Durham, 9BA Eopmrlalollege, London, vher« this work was coctpl^ted, Is aleo gratefully

ocJmowl«dged.

APPENDIX I

Resonance coupiIngs

There are a total of eight couplings required to calculate the resonance

background contributions to photoproduction. They are: yNA, TN6 and yHH*

and vNN* for the N*(15Z0), N*(1470) and N*(1535). We take these couplings to

be given by the following:= -V T, = K

(1.1.a)

<AJ(K)|VvY|N(p)> - 6 j '

<N*{K)|j iT1!N(p)>1520 = -

<W*(K)|Vv¥|H(p)>|520 = N*a

y2 ] N(p) ,

N(p) ,

(N*(K)|Vv

(I.1.C)

(I.l-d)

(I.l.e)

(I.l.f)

( i . i . g )

(I.i.h)

r.' = i <<s' + «„' T3)

K* = J

The values of g*. g^*, g ' and g" are obtained from the decay rates

(as given in ref. (9) for resonance •* nN and are given by g* « 15.39 m" ,. 1

gN» = -23.11 m , gu = -5.22 and g* = 0.68. The radiative coupling

constants are obtained from ref. 23 (BCP).

The manner in which the photon coupling constants are obtained

deserves conment. The coupling constant data of BCP are given in terms of

he l i d t y amplitudes whereas our coupling constants do not correspond to

definite hel ic i ty states. I t is easy to relate our coupling constants to

hel ic i ty coupling constants by expanding the relevant spinors into heTicity

states. When this 1s done tlie following results are obtained

(also see ref.(24):

^ ^ - [2mG, - M{M-m)

(M 2 -m 2 ) [G ,

(1.2. a)

(1.2.0)

with Q » J(K + p) and where

-1T- - 1 8 -

M(M+m) H J

(Mtin) H j ,

(I.Z.c)

(I.Z.d)

These coupling constants are still not those of the BCP tables.

In order to determine the relationship between our constants and the A3.

of the BCP tables,we compare rate expressions. We obtain the following:

(I.3.a)

]

rN*(1470) ' e 16̂ W

while the PDG tabl«'25'give

*-m*)* 2(1.4)

where j = spin of the decaying resonance.

-19-

Coinparing (t.4) and (1.3) to obtain the G's, H's and F in terms of the

A's and then combining the results with the inverted form of (1.2), we find

H - m ) ri N*

A'/2 (I.5.e)

where a = e2/4Tr.

These expressions imply from the BCP helidty couplings' '

G, = Z.90 ± 0.09 6eV"

-1.5 t 0.55 GeV

H,p = -2.60 ±0.22 GeV"1 , (1.6-c)

H,n = 0.64 ± 0.40 Ge

-20-

= 1.43 ± 0.17 GeV

H2" = 0.09 i 0.30

K' = 0.88 ± 0.18 ,

K^ = -0.69 ± 0.19 ,

K* = 0.94 ± 0.22 ,

-1.28 ± 0.39 .

( 1 . 6 . 1 )

( I . 6 . J )

We need the isovector and isoscalar coupling constants HV = Hp - Hn, HS = Hp + Hn ,

etc.

HjV = -3.24+ 0.46 GeV"1 , (1.7.a)

H2V = 1.33 ± 0.34 GeV"2 ,

H ^ = -1.95 ± 0.46 GeV-' ( I .7 .C)

= 1.52 ± 0.34

= 1.58 ± 0.26 ,

j = 0.19 ± 0.26 .

<* = 2.Z2 + 0.45

* = -0.34 + 0.45 .s

- 2 1 -

APPENDIX II

Calculation of the axtaKvector background

As mentioned in the text, we choose to evaluate the background con-

tribution of spin % objects through the use of dispersion theory. Since

dispersion theory calculations keep the intermediate states on-mass-shell,

It Is possible to use the unambiguous on-sheli spin Vz propagator. Ambiguities

would be present if subtraction constants were needed, but the high-energy

behavior of the CGLN invariant amplitudes guarantees that no subtractions are_ i

necessary because each amplitude falls off at least as fast as w

To calculate the background we must actually calculate a part of the

a contribution to the axial-vector photoproduction amplitude (R,v j . He

do not need the full amplitude,since many of the double-Index covariants

required for the description of axial-vector photoproduction have zero

contraction with q*\ There are 16 double Index covariants needed:

K.,,/ = (k'<» 9,.,, - KflJi* , (II.1.a)

' ( ^V (H.l.b)

K U U} = <k<c> V *uK

l >u u v( I I . l . c )

( I l . l . d )

K " = p K ' - v(k Y - iliv p v * u'v

( I l . l . e )

-22-

(Il.l.f)

(Il.l.g)

Here the single index K's are the standard CGLN covariants. Note that all

of these covariants are electromagnetic current conserv1ng(wh1le only the

first 12 conserve the axial-vector current. Along with these 16 covariants

there are two "equivalence theorems" (see, e.g. refs.(Z6).

V

(II.1.1)

(Il.l.j)

(ii.i.k)

(ii. l.i

(Il.l.m)

While i t is true that these 16 covariants form a complete set, using them as

such necessitates the introduction of kinematic s ingular i t ies fnto the invar-

iant amplitudes mult iplying them in processes involving an axial-vector current1

While th is practice presents no problem in any dynamical calculat ion, i t v / i l l

lead to erroneous results 1n dispersion theory. We therefore need to introduce

two more covariants, the use of which w i l l eliminate the need for the introductior

of kinematic s ingular i t ies :

K l" = {k P - vg W (Il.i.n)

(II.1.0)

(ll.l.p)

K(IV

uv q KK >

( I I . 3 . a )

( I I . 3 . b )

- 2 3 - - 2 U -

We need only consider contributions to our background amplitude from the

covariants K lv13 through K V

1B since

= 0 , j = 1 - 12 ,

( I I . 4 . b )

( I I .4 .C)

( I I . 4 . d )

<•'V - ( I I . 4 . e )

( H . 4 . f )

qwK l e = q 2 K ' ( I I . 4 . g )

Next we discuss the coupling at the ANA (A -> axial - vector) vertex,

defined as in the *N scattering case {see, e.g. ref. 27),

<Ak|A[i1|N) = - f i 1 k f 1 16 aCK)[g*(q 2 ) (g a i J -^7) + (three other covariants)] N(P) .

( I I . 5 . a )

The three other covariants are divergenceless and would only contribute to the

- 2 5 -

K l through K , l s covariants, contributions in which we are not Interested since

they do not correspond to pion amplitudes. The term proportional to q ^ has a

pole for physical pions and cannot contribute to the on-shell pion amplitude.

I t must therefore give the A contribution to the f i r s t term in R J of equation (4;

of rcf. I . To avoid double counting we ignore i t here.

We are now ready to evaluate the the background amplitudes B(A ). We use

equation ( I . i . b ) for the photon vertex coupling and

ga)JN(p) (II.6)

at the axial-vector vertex. Using these couplings we find that

"V

for the s-channel .where (? (K) is the on-shell spin % projection operator

(the 6j, - VsT.jT. isospin part of (II.8) yields for the isospin couplings

the structure I+ - >sl_ in (11.7)):

(II-8)

We now work out (11.71 ignoring contributions from K ' through K,v12 which

will not contribute to our final result. We obtain

-26-

w

i [V 6H6HT

3M7

()6M ' 6M I)} ( I I . 9 )

- fi"

o f r |(H+m)(H-2m)

+ K

A 1 ," W I +

r [m(H-in) vTR1 VA ' 3FTI ' G?l 6H VA " 6H

The u-channei result follows from this expression by let t ing s •+ u, v •+ -v,

KUU

1 * K J 1 ' 13' 14- 16> 18> KHv 1 * " K p J ' '- 15' 17' and I-1 + -1-1" After

adding the s and u channel contributions together we find the non-contribution

to M which w i l l not vanish in

w

- 2 7 - - 2 8 -

(11.10)

To obtain the dispersion theory expression from (11.10) we let v2 •+ v 2 in

the coefficients of the crossing even quantities, that is, we let v2 + v 2

inside the square brackets. After making this substitution, vie contract with

qv to obtain the contribution to the pion photoproduction amplitude. Note

that when qu is contracted with the K j y1 7 term of the I+' part of the amplitude,

a tern proportional to \>J is introduced. To obtain the pole part of B / + ' ,

(9.a),this v2 should be replaced by v 2, while the non-pole tern is obtained

by replacing this \>z by v3 - v.2. That is, if one wishes to write Bj 'in

the natural form + B]Np

^+' one should replace v2 with

As was mentioned in the body of the paper, the contribution from the

N*(15Z0) is easily obtainable from that of the i. This can be seen from the

expression for (%')f cnar]nel and using T| . -1, ̂ y s = v

channel

H--HJ

- 2 9 -

M+-M

where we have used in analogy with ( I I .6)

<N*|Aui|N> = - g ^ M K V l J T ^ Y s N t p ) • (11.12)

Comparing ( I I .7) and (11.11) we can see ft •+ N* implies that we should let

%g* * VzgJJ. G,+ -l/zH^'S and G2 + ViH2V>S for (+,o), and M + -W. The {-)

amplitude is then obtained from the following expressions:

(-)«* . ^ L B C+)N*i . , (11.13 a)

(11.13 b)

For completeness the alternate field theory background as obtained in

n>f. 6 is given by equations (7) and (10).

Finally we outline the derivation of the soft isobar expressions for the

pion amplitude, equations (4.a) and (4.b). We first note that contracting

{II.6) with q11 and dividing by f^ yields the coupling for fiNTt. This means,

that if we first contract (II.10) with qv and then let v2 •* vft' (rather than

performing these operations In the opposite order as was done above) we will

obtain the on-shell expression for the pion amplitude. Letting q + 0 in the

coefficient of I+1 <v

l will yield equation (4.a) and the use of the s + n*

prescription given after equation (11.11) will then yield equation (4.b).

-30-

REFERENCES

1) J.T. MacMulien and M.D. Scadron, to be published.

2 ) S. Fubini , G. Furian and C. Rossett i , Nuovo Ciinento 40 T̂?7T (1965).

3) S.L. Adler and F.J. Gilman, Phys. Rev. 252,1460 (1966).

4) F.D. Gault and M.D. Scadron, Durham prepr int , 1979-

5) J . Bell et aL.Phys. Rev. Le t t . 41 , 1012 (1978).

6) S. Fubini, G. Furlan and C. Rossett i , Nuovo Ciinento 43161 (1966),

7) H.G. Olsson and E.T. Osypowski, Nucl. Phys. 687,399 (1975).

8) R. McClaskey, R.J. Jacob, and G.E. Hi te, Arizona State University prepr in t ,

and Phys. Rev. ^18, 660 (1978).

9) R.G. Moorhouse, H. Oberiack, and A.H. Rosenfeid, Phys. Rev. WJ (1974).

10) F.A. Berends and A. Donnachie, Nucl. Phys. B84,342 (1975).

11) W. Pfe i i and D. Schwela, Nucl. Phys. 845,379 (1972).

12) G. von Gehien, Bonn University preprint PI 2-80 (1970),

13) M.I . Adamovich et a l , , Sov. J . Nucl. Phys. _H,369 (1970).

14) P. Nol le, Diploma Thesis, Bonn, 1972 (unpublished),

15) A. Donnachie and G. Shaw, Ann. Phys. (N.Y.) 31, 333 (1966),

16) P. de Boenst, Nucl. Phys. J324, 633 (1970),

17) M.G. Olsson and E.T. Osypowski, Phys. Rev. DU, 174 (1978).

18) F.A. Berends, A. Donnachie, and D.L. Weaver, Nucl. Phys. JJ4, 54 (1974).

19) G. Shaw and A. Donnachie, Nucl. Phys. 87, 556 (1967).

20) The values for the A.1 and E A were obtained using hyperbolic boundary

conditions and were given to us by MJH in a private communication.

H. Nielsen and G.C. Oades, NucT. Phys. B72, 310 (1974).2 l )

22) G. Hflhler, H.P. Jacob, and R. Strauss, Nucl. Phys. B39, 237 (1972).

23) I.M. Barbour, R.L. Crawford, and N.H. Parsons, Nucl. Phys. B141, 253 (1978),

24) H.F. Jones and M.D. Scadron, Ann. Phys. (N.Y.) 81 , 1 (1973).

25) Part ic le Data Group, Rev. Mod. Phys. 48, SI (1978).

- 3 1 -

26) H.F. Jones and M.D. Scadron, Phys. Rev. .173. 1734 (1968); Nucl. Phys.

BIO, 17 (1969);and M.D. Scadron, Univ. of Arizona prepr int , 1970 (unpublishe<

27) M.D. Scadron and L.R. Thebaud, Phys. Rev. D9, 1544 (1974).

- 3 2 -

TABLE I

Isobar background contributions

A(1230) N*(1520) N*{1470) N*(1535) Total

>

I

- 0 .

+0.

- 0 .

+0.

+0,

- 0 .

+0,

0.(

-0

+0

,362

,305

,057

.045

,096

• 432

.078

1098

.110

.104

4-

±

±

t

t

±

0

0

0

0

0

0

±

±

t

+

0.

.0.

0.

0.

0.

0.

0.

0.

0.

0.

012

013

018

002

007

017

003

0009

,008

,004

-0.010

+0.006

-0.004

-0.006

+0.003

-0.009

-0.010

-0.010

0.000

0.000

-0.001

-0.004

-0.002

-0.0010

+0.0143

-0.0017

±

±

±

±

•4-

±

±

±

±

±

t

±

±

t

0.009

0.008

0.012

0.002

0,001

0.002

0.009

0.008

0.012

0.002

0.001

0.002

0.002

0.0004

0.008

0.0003

T0.024

-0.023

+0.001

-0.0013

-0.006

+0.003

-0.003

0.000

-0.0002

-0.001

0.005

-0.006

-0.0013

±

±

t

0

±

±

±

±

±

0

•4-

±

±

0

±

±

0.004

0.004

0.006

0.001

0.001

0.004

0.004

0.006

0.0002

0.001

0.001

0.001

0.0002

-0.017 ± 0.003

0.016 ± 0.003

-0.001 i 0.005

0

-0.00017 i 0.00004

-0.0010 ± 0.0002

0.003 ± 0.003

-0.003 ± 0.003

0.000 ± 0.005

0

0.00003 ± 0.00004

0.0O01 ± 0.0002

-0.003 ± 0.001

0

-0.0010 ± 0.0002

-0.00017 ± 0.00004

-0.365

+0.304

-0.061

+0.039

+0.098

-0.498

+0.016

-0.016

0.000

0.000

-0.001

-0.005

0.078

-0.011

-0.103

0.101

±

t

*

*

i

i

±

*

t

±

t

±

t

±

±

0.

0.

0.

0,

0,

0.

0,

0.

0

0

0

0

0

0

0

0

016

,016

,023

,003

.007

.017

.010

.009

.014

.002

.001

.002

.004

.001

.011

.004

TABLE I I

Data comparison with MJH for amplitudes containing no current algebra t e rm

j

I

A3< + > < / )

A, ( +V3)A3

(°V3)

Al )(°) (u-3)A r '()i~ )

A^'V3)

NUCLEON POLES

+0.275 ± 0.014

-0.019 ± 0.001

-0.0089 ± 0.0004

+0.00062 ± 0.00003

-0.998 ± 0.050

+0.275 ± 0.014

BACKGROUND

+0.098 ± 0.007

-0 .49e ± 0.017

-0.001 ± 0.001

-0.005 ± 0.002

+0.078 ± 0.004

+0.101 ± 0.004

+0

-0

-0

-0

-0

+0

TOTAL

.373

.517

.010

.004

.920

.376

± 0.016

t 0.017

i 0.001

± 0.002

± 0.050

± 0.015

MJH

+0.363 ±

-0.431 +

-0.0097 ±

-0.0135 ±

-1.000 i

+0.357 ±

0.003

0.007

0.OOO1

0.0001

0.010

0.003

8

1-

o

+o

o . • o ^-* oN og N eo n

r- —. m N

ao, uj

Q Or- -c

i— a ;o _i e:•-• 3: o

ooo

CO —9 °5 o 8 8 5o o+1 +1

S> 00co cxv•"> O

o

H

<"

O CI

tl II

+ o

<* <

oO O O

-f I +1 +1

I D m CJ* v£» I O f— ^ «Ol r - CM O CM r— oO P — O • O O •

. • . O • • OO O O I O O I

so

II It

* * CM Q CM CM

O O O O O

O O O O O

+1 +1 +1 +1 +1

§ g § 8 § 1 2 8 3 S S 8O O O

I0 0 0 0

. + + + + + + + + + 0 0 0 0 0 0 0 0 O

i ^ O O O O O O O O O O O O O O O O O<!UllUJlLJlUlLUIlUlkllliJIUlUllLJ|Ul|UJ|UI,LJ,LJ tUJ I L

O+1

LO

r - CM

<— o

o o+1 +1

Olo

o •*o oo o+i +i

sO O . r-

O O f l L O C M C J l f - -- l O i — IO CM O • " O Lft >™

enLDo

• * o o in CM

so+1

oo

( C i n ^ r F - rt cv i t j m c M_ . _ _ _ _ _ _ _ . f — O r — ( M r — i ~ r — r — r —

e M O ' O O * . ^ — . O . . . . . * . . . .• O • ' O - O ' O O O O O O O O O O

O O I O C 3 I O 1 O I I I I I I 1 I T i

II II II

4- O O*—*

•X

n n ii n

O O O O O O O Q

„ „ rt rt o o t S S o O O O O O O O O o i• t I . I t j i L L J L L J L U i t ! L L J L J J U J L t J » . i

LD

O

+1

.58

o+1

.74

DO

o+1

.07

CO

o+1

.96

—

+1

.66

o—+1

.82

00o*«o+1

.04

COCJ1

o+1

.76

00CTi

o+1

.45

98

CJ

+1

en<M

COC31

O

+1

.37

<M

r -

+1

.64

coeno+1

.80

n

—+i

.98

Ol00o+1

.93

oCOo•H

.07

o00o

+1

.03

oCOo+1CM

so

+1

.13

oCOo

+1

%

o

—+1

.07

go+1

.34

CM

r -

+1

.39

— O r — r— O C M

O O O O O OUJ UJ LU UJ UJ UJ

i — O O C M O ' — O O O O o

o o

- 3 5 -

IC/7»/lO8 >.F.IItS u d J.* of ab»art«i ultculca.

cmssvr ICTP rasraim AKS ISTEHJIAL RSPOOTS

of surfaea maffiatio fiald on diaaociatio

/A ».B. KiBCH «mi M.P. TOSIi Hjrdrktion of div»l«it lona M tea.I K .HIP.*

IC/78/122 C. FOTLJJT «md S . OAVlt JUrcm solutions in * tuii-ifaolianW J S * ier««ri«rt Hi di

for

HW.H1J?.*

1XSJ0SB.- acilutiottm eat *S» ia liquid •%• «t low t«»pw«tuT««.

XC/?tVl25 0 . StBUXOU, « . P1HHI3IBIX0 u d M.P. TOSIt Optical ttworptlon ofdl lnt* •olatldBS of metals in oolt«a •* l t» .

IC/78/126 tTJ-JOBJlDar, LX. POGHEBEIT aad JC.C. POLITASDVi On tha tolutioaa witlsoia#tla>vltiaw of th« Liouvlll* »ciu»tion O ^ ± ( » 2 / 2 ) « P « - 0 .

IC/78/12T T- 3UaUKI> Oparatar ordarinjj and ftinctional lntagrala for Farnlt

Ic/78/130

IC/78/128 A. 03i*AS« Thr««-bod7 -odal for •trippinfj nuel««r reactions.

IC/T&/129 I-A. ELTAIEB and M.I.A. HASSlH. OB the non-linaar •TOIUUOU of a and

d»e«y of th« 0.S17 1UV 7+ »t«t« in4 2So.

IC/T8/IJI B.L, AKTO. and 9 ,7 . STOTlSOYi A «up«rsymmrtrio fora of ttaa Tblrrlax•edal .

IC/7B/132 <J- (HTJ1I4II nri E. TOSJWTXt Lon^rttudinal phonon BP«OMTI« of lnoo»-" nansorat* oca—diiwnaional charga-d»n»ity-w«.Taa.

Wajrl and conforaal eovariunt

IC/7?A3* 3 . PBaSftB and. C JP. SISCEi Conatralai* dua to partial conaarvmtion ofaaial. -notar taixranl and ebaraad baryon ooupllnga.k j l . IH*>t Iao«T»la wynawtrjr in A - 9 mirror oats d«c«r>.

IC/ i8 / l3 i P. BDDZH, P. FOHLAI and fi.

IC/76/136 Sacond I«tln-itBariean workahop on aalf-conslstant thaorlaa of eon<iISTJISP,* Bjjvtta* - 9-20 OetolMr 1978 (Contrlbutloaa).

IC/78/llT ABIH3 SjIUf*« Th» alactraHMk fore*, grind unlflowtloit and eup*r-mu. float i oa .

10/78/138 P. flUDIBl, •?. FTOLAir and H. HJJCZEAt A uo«»ibla f-eo-etrlcal oriffln ofi»otcrpio I ) « M try group in 3tron« intar act lona.

on locallSJAblllty and on quantized f»ug« f lalda.J .

aportjxi*atioa in th« casa of a • n l t i -IC/7a/l«O CjXO-XUA»-CHU*»t OB th»IIT .̂HB?.*" ccmporunt potantial.

S J , FAIBBT* On cocforaal in-rarianc* of «ravitaticn*l «*w aquation*.

Poa-iol . w i n n of iartopic spin

IC/78/14.3 J . SIEIBBLEi Tha mnxaual alpebra* and taeir anpli cat ions in

/ 7 V 4 P. BUDISI, P. TORUS and H. aIHT.HEP.' from ait«nded conforaal

• • lutamal -taparta* Ltatttadrasas pasPRrirre ASS ATAZLASLZ FHOK THE PTJBLICATTOBS OFFICE, I C ? ? , P . O . BOX586, I-34-1OO TBIESTS, TTALT. IT IS SOT 5ECESSARY TO WHITE TO THE AUTH03S.

-i-

IC/78/1441ST.SEP.*

10/78/1451ST .SEP.*

Ic/78/146

IC/78/147

ic/78/148INT.REP."

10/78/149

IC/73/150I5T.R£P."Ic/78/151INT.SEP.*rc/78/152

IC/76/153IBT.HSP."Ic/78/154

L. BERTOCCHI sad D. THELEANIi K° regeneration in nuclei and the affectof tba inelastic intermediate states,J. LUKIEHSKIs Graded orthosym^Iactic (i8O"etry and OSu(4|1}—invariantfar«ionio ff-«odel3,J. LUKIEHSKIs Field operator for unstable particle and complex "seedaacriotion in local QFT.

MUBARAK AHMAD: Symmetry Tweaking mechanism, scaling transformation*and Joaephson affects in auperfluid belium-4>

A. QADIRt Field aquations in twistara.

I . I . DSLCHEV, E.K. «tARCHS» and I .J . FSTKOTs A vie» on reactions ofco«plota fusion.T. SUZUKI» An anomaly-free regularization for axial-vector currant.

* . HOVERE, R. UlttSRO, S. PARHIBSLLO and * .P . TOSIi CM-li<piidtransition in charged fluids.A.H. ASTONOV, V.A. fTIKOLAET and t .Z. PETKOVi Caharant fluctuationsof nuclear density.

X.A. NAMAZXEt On an O5p(l,4) renor-alizabla theory of suparftrarltywith higher derivatives,A. ^DIR and W.\. SATSDi Hawking radiation and strong grari t j black holt

IC/79/? MUBAJUUE AOKkSx Few-fcody aystea and particle reaona.ic««.INT.BBP.*

IC/79/5 V. ELIAS and 3 . RAJTOOT: Confinement, perturbation theory and Steraea-

IC/79/S 3KJBS3W ARXK3 and 9 . SHAFT JALUTt Heavy quark and na^netio moment.

17/79/7 X. KAWARAB1IA3HI and T. SUZUKIi Dynamical quai-k naaa and theIST.2SP.* prt>l>lea.

IC/73/> Q. HiX)3alCHt t priori laound* for solutionB of two-point 'bounc'.iry valu«TSrjSP.* problecs u»in« differential inequalit ies .IC/79AO 0 . TTDOSSICH: On tbe existence of positive solutions for non-linearIHT.RSP.* e l l i p t i c aquations.IC/79/ll 0. TISOSSICEi Comparison, existence, uniqueness and successiveTSCJtSB.* «ppxo«±a*-tiona for two-point boundary value probleoa.Ic/79/12 C. nmSSICHi Two r»m«ria OR tbe s t a o i l i t y of ordinary differentialXBT.HSP." equstloos.

Ic/79/19 1. WB3LB m l SIJ1. TOST: Inhomowneity affects in tt»«ehan«» holeTJRJtBP,*- foe * yHJ-OM 'dimensional chaia.

Ic/73/157 H.5. 3AAKLIHI and B.B. BOUZOOITAi «iantismtlon of tie- non-iineav «aper-Folncare formion In a bag-like fonsuletion,

IC/78/158 V. ELIAS and 3 . RAJPOOTt Enhanced flavour charge, in the integer onsrkIBT.HSP. oharge model.10/78/159 V. SLUL3 and 3 . RAJPOOT: Low unifying Baas saalca withowe int«ro»di»t»

ohlral colour aymmatry.I0/78/16O V. HE ALFARO, 3 . FUBIVI and G. FCHEJUTi Gas«» theory and •1WPJ0P. tbe use of ifeyl inTariaae*.

IC/7?/25

IC/79/2&

IC/W/3O

IHT.REP.*

ie/79/3i

Cbara fragmentation funotion frea neutrino data.

and U, WAEAUft Dielectrtniic reeosbinetion in eodiua la

3JC. PAL*

2.P. HTAI.eleetnmloJ.C. 3SfHt an* a.P. PACHBCOt Retioriialiiation group approach to thephas* Aieoraa of t^o-diaetieional Eeioenberer spin syrteaa,

5 . nS&TE-BBXafiBt Froduotion of l ight lejrtons in arbitrary beauvia ueuj »*g*eg boeon exohange(

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Ilia leading order behaviour of the two-photo*In QCD.

J . PSZWiKe. and A J . CHiCnHX» * oometit on th» oh«taeriterien.

3. USlTt

J , LOBSie,sobductian

-iii-