i-ri NBS TECHNICAL NOTE 957

%*U Of

U.S. DEPARTME
NATIONAL BUREAU OF STANDARDS

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OF STANDARDS

LIBRARY .-_. SEP 2 2 197/

Threshold Photo and Electroproduction

of Pions from Nuclei

&c

±

T ^ c ^ w \ caJt ftaW w^. ^^T^

E.T. Dressier

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

*< ">%.

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

S , NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued September 1977
National Bureau of Standards Technical Note 957

Nat. Bur. Stand. (U.S.), Tech. Note 957, 38 pages (Sept. 1977) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $1.50—Stock No. 003-003-01834-0
Contents

Page

1. Introduction 1

2. Pseudoscalar vs. Pseudovector Coupling 2

3. Nonrelativistic Amplitudes .. 5

4. Threshold Photoproduction 12

4.1. Kinematics and Gauge Conditions 12

4.2. Pseudoscalar Photoproduction ... 13

4.3. Pseudovector Photoproduction 15

4.4. Discussion 16

5. Threshold Electroproduction 16

5.1. Current Conservation 16

5.2. Pseudoscalar Electroproduction 17

5.3. Pseudovector Electroproduction 20

6. Cross Section Equations 21

6.1. Coordinate Space Representation 21

6.2. Photoproduction Cross Section 22

6.3. Electroproduction Cross Section 24

7. Discussion of Results 29

8. Appendix 30

9. References '• • 30

ill

Threshold Photo and Electroproduction

of Pions from Nuclei

E. T. Dressier

The nonrelativistic amplitudes for photo and electroproduction of pions are derived using effective pseudoscalar and pseudovector Lagrangian densities. The results for pseudoscalar and

pseudovector coupling are compared at threshold for charged

and neutral pion photo and electroproduction. Cross section

formulas and kinematical conditions are also presented.

Key words: Effective Lagrangian; electroproduction; impulse

approximation; photoproduction; pseudoscalar and pseudovector

coupling; threshold pion production.

1. Introduction

In the calculation of any nuclear cross section that involves a small perturbation to the nuclear system, the standard approach is to use

the impulse approximation,, That means that one assumes that an external

perturbation acts on the nucleons in a nucleus as if they were free

particles. Using this approximation or model, then if one knows the

amplitude for the reaction off of a single nucleon, we just sum the

contributions from each of the nucleons in the nucleus (a sum correlated

by the nuclear wave functions). The purpose of this paper is to present

the nonrelativistic amplitudes for photo and electroproduction of pions

that can be used in an impulse approximation calculation for nuclear

processes.

To find the amplitude from a single nucleon, one usually starts by

considering which relativistic Feynman diagrams are important in the

particular energy region under consideration. For the case of a purely

electromagnetic interaction as in electron scattering, photo or electrodisintegration, or Compton scattering, one can use perturbation theory

and quantum field theory to give the appropriate Feynman diagrams to a particular order in the coupling constant. However, in photo or electroproduction of pions, we now have electromagnetic and strong interactions,

and since the coupling constant in strong interactions is large, the

perturbation expansion in powers of the coupling constant will not con- verge. In spite of this fact, the lowest order term in the expansion

using pseudovector pion-nuclear coupling, gives results which under certain conditions agree very well with experiment and with the low energy

theorem derived from the theory of the PCAC (partially conserved axial

current) Ll], Therefore, we will use the lowest order perturbation term

as our model for the threshold photo pion production amplitude off of a nucleon.

Two of the simplest pion-nucleon couplings are pseudoscalar (ps)

and pseudovector (pv) [2J. A comparison of the two couplings is made in

Section 2. In order to use the amplitudes in a nuclear calculation, we
must first make the nonrelativistic approximation of the relativistic

Feynman amplitudes. This is done in Section 3. The nonrelativistic

amplitudes are then specialized to the cases of threshold photo and

electroproduction in Sections 4 and 5, respectively. In Section 6 the cross section formulas are given.

2. Pseudoscalar vs. Pseudovector Coupling

In the Lagrangian field theory approach to photo pion production,

two of the simplest interaction Lagrangian densities between the pion and

nuclear fields that one can choose are called the pseudoscalar and pseudovector interactions. In the pseudoscalar case, the interaction Lagrangian density for the pion-nucleon interaction is

X^s) = iGN(x) T • 5( X) y 5N(x) (1)

where

N(x) = nucleon field operator

cp(x) = pion field operator (a vector in isospin space)

G

3 -j— = 15.0 = pion-nucleon coupling constant

T = nucleon isospin operator

For photo pion production, the total interaction Lagrangian density

is

£

(ps) = x

(ps) + z + z (2)

I 7tN yN y%

where

-f,

. ... yN 'u £ .. = ieN(x)y A

y (x)N(x) (3)

1

YTT = iecp

+

(x)(g|- - -J-) cp(x)Ay

(x) (4)

A (x) = electromagnetic field vector

When this interaction Lagrangian density is used in the second order term

of the perturbation expansion, one finds that there are three Feynman

diagrams (shown in Figs. 1-3) that can be associated with the second

order term. (Solid lines are nucleons, wiggly lines are photons, and

broken lines are pions. Each line is labeled by the 4-momenta of the

particle.

)
/

*

/

Figure 1. Nueleon pole diagram.

Figure 2. Crossed nucleoli pole diagram.

Figure 3. Pion current diagram.

Figures 1-2 are called the nueleon pole contributions to the Born

diagrams. Figure 3 is called the pion current diagram, because it re- sults from the interaction of the pion's electromagnetic (e-m) current

with the e-m field operator^ given by £y%.

For pseudovector pion-nucleon coupling, the interaction Lagrangian

density for the interaction of the pion and nueleon field operators is

given by

t ^ if

£<PV) = _JE n(x

tcN m

(*) yu

r 5

T d?(x)

N(x) (5)
where f

l = °- 08' (a?-)"

= 4* (s%

Now in order to insure gauge invariance of our total interaction Lagrangian density, £ , in the presence of the electromagnetic field, we have to let [3]

This leads to

, (pv) , <PV>

&** - ^

+ ieA^x)

ttN ttN

fir — -» -» u (Pv) - e — N(x) y y, T.cp( x)N(x)A^(x) = JC71N + X M (6) m 'p. 5 TtNy

By inspection, we can see that this leads to a new term which has the

interaction of the nucleon, pion, and electromagnetic fields at a point.

There is no term like this in ps coupling, and inserting this into the

perturbation expansion leads to an additional Feynman diagram (fig. 4)

in pv theory.

Figure 4. Point contact or "seagull" diagram*

One should note that although the diagrams in figs. 1-3 are also

present in pv theory, they won't necessarily give the same size contribution as in the ps case, because the form of the 71 - N vertex is different for the two couplings (fig. 5).

y_ for ps

J. yJ for pv

2m

Figure 5. n - N vertex operators for ps and pv coupling.

Even though the size of the contribution to the production amplitude

from each of diagrams 1-3 may be different for the ps and pv cases,

the sum of the contribution from diagrams 1-3 using ps coupling gives
essentially the same result as the sum of the contributions from diagrams 1-4 using pv coupling for charged pion production at threshold.

However, pv and ps couplings give quite different results for tc producrion. pv coupling gives the best agreement with experiment and is usually chosen as the "effective Lagrangian" [4J. In spite of this, we will

derive our nonrelativistic amplitudes in the next section using ps cou- pling and show how these amplitudes are modified for the pv case.

3. Nonrelativistic Amplitudes

To calculate the amplitudes corresponding to the diagrams in figs.

1-3 using ps coupling, we can use the Feynman rules in appendix B of

ref. [5]. The relativistic amplitudes for these diagrams are defined as

Aw1 T3(*

p

- k

n )\j _ t(Pi)€

u

(7)

\ 2 /Jp' 2 - m

2

r 5 a s,T v 1'

/ x C T , T J (q' + q)

M

?

3

PS) = * «s V<V V 5 -V- S V"^f V<V (9>

where a = - 1, + 1, for % , 7t~, and % production, respectively,

o 3

and

T_

L |P> = /£ |n> ,

l~T >%1 = T a 3J a\3

- V3 a >

In order to use these amplitudes in a nonrelativistic calculation,

one has to expand these matrices by going from the 4 component Dirac

spinors, U S (P), to the two component Pauli spinors, Xs» This is done

by letting
us;T

(P) = Vl(^)v-(^)v + KS) (10)

where I =<l)

and x T = X XT

> X being the spin and XT the isospin Pauli spinors,

S a I S ' S I respectively.

Keeping only terms to

amplitudes by M, where

.(j). we denote our nonrelativistic

M

(P S > = -i l«

yl k / o ' o

/-k \ I 2P -k + k

a-k

^

+ a.q \i + -^ (ID

+

o 2m 2m

P *e -> i k

a«k ( ^-

m 2m / \ 2m

+i e, ( k x q)

2m a\ 2 / k

o

o \ 2m

+

V 2m 2m/ \ 2m/ 2m

^ (

fp + K n) + (V K n) M

M

(ps) . JL. y2 k o

1 ~) + cf-q ' 2P

f

«k - k

2mk

(12)

+ ^(-K-i+^i) .. ^I m 2m / 2m

, . £'(k x q) + 1 —aT"^

1 + T 37 1 - t + —- e a*k

2 a k o o 2m

+
> - / o

q

o k-q \ - r / e.q

2m 2m / \ 2m

. e»(k x q)

K+K) ( K - (C ) T \

p n/ M P n/ 3

T 2 /a

M

<ps> = . ^<q - g

re /2 q

. k \ . ;. (2 q . k ) y3 / 2k»q - k

2 o\ o o/ M

For threshold pion production, we can let q

-» which leaves

>]|M < 13)

M

(ps) _ „ G «k "* -*

i , o o 2m \ 2m

(

l + T

3 ) a 2

+

m

- e

71 o 2mk

a-k (14)

m

2m

t -E

a

V p n/+Vp n /

- K \ T P n y 3

M

(ps)

y2

ck -* -» e „ ~ot;

+ G#e

O Al

k

1 + ^

(

X + T

3)

CI m

7t -> -> 71 - "* - e T53— G *k + -sr G *e o 2mk 2m

(<

K + K . K - K T

p n/+\ p n/ 3

2 /a

(15)

M

(Ps) . y3

ack

2k m ^~W

O 71

e ( 2m - k

o V 71 o

\ + e«k [ IVJ (16)

By replacing e Q

-» kG and e -* k in equations 14 - 16 we see that

M , P + M « + M , -» , therefore, our nonrelativistic amplitudes

yl y2 y3 are exactly gauge invariant.

Now consider how these amplitudes are modified for the case of pv

coupling. Using eq„ 4d of refo [lJ, the pv amplitudes are given by
M

,1 - - u

s

, )T ,( Pf ) iy 5 \^-r^

1 + T.

H 2

C /(k + K

i _£v k

v K_P n

2m \

+( K - K ]t

P n / 3

U s,x(Pi) e "

M

(pv)

y2 V,T.<ef

>

1 + t ; a2m

k

'n 2

(VM^n) C

+ "

fly. T U T (p.)«l

) 2 - m

3 "'5 a s,T vr i

7

(17)

(18)

M

(pv) - , v ,., ,. fra,T3] 1 = U Q

, T

,( Pf ) (K ff) y, —J- - y3 s',T'^f 5 2 „,q* . gg (19)

(q« + q) U

s>T (p.)^

TM (Pv ) t"t t \ ^ T a,T3 ] , N

p. M

y4 = V,!'^ —2— V5 U

s,T ( Pi )e (20)

Consider now the following identities:

5s',T-<Pf>4r 5

-^^=Gs

, >T ,<P f )r5

2m (j6 + m)

p

2 - m2 + 1 (21)

p'3 - rf ^5 U

s,T<Pi) =

2m(jS' + m)

p'2 - m2 + 1 ?5 U s,T^i> (22)

By comparing eqs. 7-9 with eqs. 17 - 19 and using identities 21 -22, we can see the following:
M .^Mr^v.T. ^y T

5 a

(l + T. V 2 (23)

(J I [ K + K )+ K - K +i JiZ h

v l \ P nj_ \ P 2m a^J U ( P .) e ^M (ps)

s,t i' + M

1 yl yl

M 1

PV> = M 1

PS) + 5., T , (p

J

y2 y2 s',T' Nr f

1 + T ) a V 2 2m (24)

K + K ) + K - K \t

p n/ ip bJ-J ^) T y u _ (p.)e^M < ps) + M.' / a 5 s,t r i v2 v

For M \ * toe use

y3

5

8',T" (P f

} (K

" « ?5

U

S,T (P i

) = ^ U

S ',T' (P f

} ^5 °s,T <P

I

) (25)

Therefore M < pv) = M < ps)

y3 y3

Taking the nonrelativistic reduction of M' and M

1 gives

y2

M' = < - e yl I o

tf'k O'q "P,- +

2m 2m m

+ tf.e f T (

1 + T s)

a 2

j. ii^tM, ((

K

p

+ K n)+ (%- K n) T

3

)

j o 2m 2m \ a \ 2 /

(26)

M

y2

-* -» -» -» 0'»D 2m 2m 2m

tf-e |

i- 1+T

3) T a

+ ) e o 2m 2m j \ 2 /a

(27)
Combining M

1 and M' with M and M

PS

, respectively (eqs. 11, 12),

^ ^ ^

^ (dv) (dv) we can now get the pv nonrelativistic amplitudes to M , and M . yl y2

M < pv) = ( • yl 1 o

"2 7* a *P. cr«k *i

m m

+ °- q lr + —sp—

/

\ o o I -* (28)

;.e -2 - 3^i

2m 2mk / 2mk

o / o

-» -*/ k«e

CJ.q (^—

:

l!V ,. «•(* '*)

2mk mk

o o

'+ i 2mk

a 2 o I 2m 2mk )

T

\2m 2m 2mk ^ L °

+ ;.i;!x +

16> (£ * q) T

i e n ^ L-£ n I 3

) 2mk 2mk a \ 2 /

M

(pv)

y2

tf'p,

-»f 1 -5r +tf,

« ir +

2p -k + k

2 am?

o

+

2m

+ asH- °- k l5r +0^ asr-sr

o/ o \ o o

+i e»(k x q)

2mk

1 + t.

1 o I 2m 2mk 1 V 7

+ 6 e (k q r

-* \ _2 . _2 k»q I -* -j* q«e

"2m 2m " 2mk / 2mk o/ o

• e*(k x q) -1 " 2mk

o (

(

K

p

+KJ

+ (V K

n)

T

3

) T \ 2 /a

(29)

10
Taking the nonrelativistic reduction of M , gives

M

(pv) =

y4

*i g»k

m 2m 2m

- a •e

[

T a'

T 3]

(30)

Making the substitutions e

-» k and e -» k, the nonrelativistic

o o amplitude M

<Pv) + M

y2

(PV) + M

y3

Cpv) + K^ (pv) is also gauge

invariant.

For threshold production, letting q

-» 0,

k

o \ 2m 2mk

(31)

e y2m 2m/ a \ 2 /

M

(pv) _

y2

S

tf ' P

i -- k

o> (

1+T

3)

, m

o

/m k

. -1 ~*

( It O

2m 2mk / e

\ 2m " 2m

^P + K n)

+

( K - (C _J2

(32)

n^Va

M

M

(pv)

y3

(pv) =

r? fe f2m - k \+ e .k

m - k5 oVtc o / O 7X l J [VT 3]

2m m

<3* e

[

T a'

T 3]

(33)

(34)

We should note at this point that in deriving these expressions, we

have not used any particular gauge or mentioned any specific reference

frame in order to eliminate terms. However, by stating that threshold means "q = o, we are by implication in the center of momentum frame where

k = - p.. In the laboratory frame, q F0 even at threshold.

11
4. Threshold Photoproduction

4.1. Kinematics and Gauge Conditions

For real photons we are free to choose the gauge where e*k = and

e =0. This is the transverse or Coulomb gauge. By inspection of M „, the contribution from the pion current diagram, we can see that this

term vanishes at threshold.

Choosing to work in the center of momentum frame of the incoming

photon and the target, gives

k + p. = q + p. (35)

1 i

and at threshold

q = = p

f

(36)

where p. an^ P^ are the momenta of the initial and final nucleus assum- r

i f

ing the final state is a bound nucleus. Then at threshold, using

and

k + E. = m + m£ (37)

o 1 71 f

gives

E. = Vm2

. + k

2 (38)

1 1 o

m + mJ - m. k = \ I

*L * (39)

o 2(% + *

f )

"""For transitions to continuum states, the kinematics are a bit more

complicated.

12
4.2. Pseudoscalar Photoproduction

+

Case 1. y + p

-^ tc + n

(ps) The largest contribution comes from M .. as can be seen by consid-

(ps) ering diagram 2 which corresponds to M . In this diagram the proton

has already emitted the % and become a neutron. Since the neutron has no charge, the incoming photon can only interact with the anomalous

magnetic moment, K . ° n

+

<M(7t )> = - /2 <(*•£>

- v2

I k \ m

f i

. _£) + _E (K + K \

\ 2m/ 2m I p n)

R<cr«e> m

1 - -- (K + K ) 2m p n

(40)

In the cross section we have

+ s |<M(w )H

2 I <c. e:m

1 + -5

m

m

^

(K

p

+ V (41)

where we have used

k \-

2m/ m

1 +

m

(42)

in order to put the expression in a form which can be compared with the

completely relativistic calculation in ref. [7J.

Since (K + K ) = -.12, the anomalous moment gives only a small

P n +

correction and can be neglected in 7t production.

Case 2. y + n -> -n: +p

13
In this case, the largest contribution comes from M

(ps)

y2

, since the

isospin operators allow only contributions from the neutron's anomalous

moment part of M

(ps) Again this can be seen intuitively by considering

diagrams 1 and 2 as was done in Case 1.

<M(tc )> = /2 <d

<* */2

JL<<5'e>

~*>> L

mm +

7i +

n

. ~ni \ m 2m

1 + TC \ L

+ K ) P n

2m

(K + K ) P n

(43)

and

|<M(if)>P« 2

<tf-e>r

1-+-5

m

_j

m

m m

m *£n p n

(44)

where again we have arranged terms to put this in the form of the

relativistic results.

Case 3. y+p^K +p

Diagrams 1 and 2 contribute on an equal footing in this case, however, the leading terms of the C*e operator cancel each other, and we are

left with only higher order terms.

<M(p,7t )> = <(?•£>

k. m

m

2 + -2k (45)

|<M(p,7T°)>| 2 « \<^>\\^) (1 + K p

> 2 (46)

Case 4. y + n -* 71 +n

14
Only the anomalous magnetic moment parts of M , and M „ can

yl y2

contribute, because (1 + T ) |n> = 0, which reflects the fact that the

neutron has no charge.

o -* -. /m

<M(n,ir )> = - <(S 'H^) K n

(47)

|<M(n,7i°)>| 2 = l<tf'1>| a f^) ( K n

) 2 (48)

By inspection, we can see that charged pion production is much

o o larger than n production since the n cross sections have an overall

(m\2 factor of(

) % .02 not present in the charged pion case. Note that the

/

m

m V1 o factor (l + — ) cs .87 is not present in our cross sections for n

\ m /

production although it is present in the fully relativistic cross sections

in ref. [6J. This factor would come from corrections of 0( ——)to our

amplitudes which have been neglected in the spirit of our nonrelativistic

approximation. As we can see, these factors are not insignificant, and

obviously a nonrelativistic calculation of ti production is not very

reliable unless terms of 01 —*- have been carefully retained in the

amplitudes from the beginning.

4.3. Pseudovector Photoproduction

For pv pion-nuclear coupling, there are three terms which contribute

to the cross section at threshold,

M

(PV) = M

(pv) + M

( Pv) +

(pv)

(49)

y yl y2 y4

Within the approximations we have been using, we can see that the

anomalous moment parts vanish. This does not have any significant effect

for charged pion production; however, it makes the cross section for

o

y + n -* n + n equal to 0.

Case 1. y+p-»7i +n

15
The largest contribution now comes from diagram 4 as opposed to the

ps case where diagram 1 gave the largest term

<M(7t+)> = .. V 2 <5.> (l - 75;) (50)

Case 2. y + n -* 71 +p

Again diagram 4 is the largest contribution with a small correction

coming from diagram 2.

<M(ti")> /2 <tf->(i + 2^) (51)

o Case 3. y + p

-» 71 +p

gives

Diagram 4 can't contribute in this case, and M + I (pv) (pv)

yl y2

m

<M(p,7C°)> = — <tf-e> (52)

m

4.4. Discussion

In the ps case, if we neglect the very small anomalous moment cor- rections to the charged pion cross sections, we get exactly the same result as with pv coupling. However, for 7t° production, the ps cross sections give results much closer to the experimental values (See ref.

[6] for numerical values) even using our approximate amplitudes. In con- clusion, we can say that for threshold photoproduction of charged pions,

either eqs. 14-16 or eqs. 31-34 can be used. For 71° production, these

equations can be in error by as much as 207» due to neglect of terms of

0(p

2 /m2 ).

5. Threshold Electroproduction

5.1. Current Conservation

For electroproduction of pions where the incident photon is now

—> —> virtual (k2 7^0), we. can no longer use k«e = to eliminate terms. This

means that we now have contributions from the pion current diagram (fig.

3) even at threshold. However, we can use current conservation of the

electron and nucleon currents to eliminate the e terms. For electroproduction, e^ now represents the electrons current and is not the polarization vector of a real photon.

16
as We cati define a nuclear current J by writing our matrix elements

M = e^J = e «J - e'J , H o o (53)

then current conservation gives

k^s = k e - k«e =

[I o o

k^j = k J - k.J =

u o o

(54)

(55)

Using eqs. 54-55 to solve for e and J , we can write

o o

M- — — - e.J= e.|k rr

o o \ o (56)

Therefore- we need not consider the terms in M that have e o

5.2. Pseudoscalar Electroproduction

For electroproduction of pions the incident photon in figs. 1-3 is

virtual (k2 ^0), therefore, we have to introduce form factors at each of

the vertices that a photon is attached. Eqs. 7-9 now become.

M

(ps) _ ~ rT / \ A, T (f$ + m) M

el - ^sV^V *5 a P

2

- m^

F

S + T FV

r

l 3

f l

cj ( F

S + T Fv +ij£ k

U

\ 2

Q

32

2m 2

U S,T^i)

£t

(57)

M

(ps)

e2

2mU

s (p)

F

S + T fv

) a (f

s + T F

V

1 Vy + i

_b£hi k

Uil2__3_2

ru 2 + X 2m

k

2

(58)

17
{£ + m)

r«T u (p,)^

M

e3

PS> = «..,T -<'f> V5

^le^^> '2 - F„U. „<p.) m3

t,; s J *1

7C (59)

where

F^k2

) = F

P + F^ (60)

F^) = FP.

F«

(61)

F

S2

(k2 )=K pFP + K n

F^ (62)

F>

2 >=VP2-V2 (63)

Fj(o) = F

P (o) = Fg(o) = 1 (64)

n, F (o) = (65)

Rearranging terms, we can let [lJ

(PS) = M 5 PS) + M < ps) + M

< ps) = U , ,( Pf ) J- y F^

el e2 e3 s ,t' f ' 5) V 1

<2q - k).. r^rj <2h

+ k)

u (f; + T 3

F[)

2

" * p

3 - m

3 a 2

M

v K^l

q

2 - nr ti

"

fl 2 > - nf3 T o\ 2 2 /

+

(66)
p' 2 - m2 2 2 p' 2 - m

2 \ 2

J

r«pSL|rt T< p, 2 / a[ s,T i

If we use the test for gauge invariance on this amplitude, we find v

that we must let F, = F . Since all of the form factors are assumed to

1 TC be independent, an alternate way to make M

(ps)

[auge invariant is to

make each term in eq. 66 independently gauge invariant. This can be

done by letting

y

F

v 7i

v.kk

: t k

2 (67)

(

2q

- k

)u

n ' 2 _ m2 (2q-k)

(2q-k)»kk

k2 Jq 7^ m

c (68)

(

2P

l + k

) , p

2 - m2 (2Pl + k) .

(2p + k)-kk

1 u k

2

3s - m2 (69)

(2p - k),

,•2 mc (2Po - k),

(2p

2

- k) - klv

k

2 Jp 1 m2 (70)

The fourth and sixth terms in eq . 66 are already gauge invariant. Note

that these "gauge corrections" only enter into the longitudinal part of

the cross section since their 3 - momentum part is always in the longitudinal direction, i.e., along k.

Keeping terms of 0(p/m) and letting q

-»

, the current operator to

be used in eq. 56 is

19
fCps) _

/_ z -. ^Tt ' t

t 1 2k m

I ~1 , k "2 7*1 a 3 _v ojt

\ ks

/ 2 1 /2km - k (?»k

k2 \ k

2

r

' T a 3l| F

2 7C (71)

2m / 1 + m

I an

m

(

F

1 + T3

FI) , (

F

S T^V

+ T F

2 3 2

2m

~

1 - ZEE

2mk

(i+_vl) , F

2

+ Vl, +

2 I "a

it

5.3. Pseudovector Electroproduction

From eq. 5.2 of ref. [l], the amplitude for electroproduction using

pseudovector coupling is

M

i

Pv):=5

s',T-<Pf>r 5 j. rFA Lp] +an (f^ i 'H A 2 q

2 - m

T ,T a 3 JL F

q' 2 - m2 % 2 (72)

/

2P

1 + k

)u T (

F

1 +T3

F

" D w ps - m2 a 2

idl - k) T i(livL) + (I1V2

2 p2 - m

2 a\ 2 2

+

(

2p 2

' k

) u /

F

i

+ T

3

F

I. + T p' 2 - m

2 * 2 a

2 p' 2 - m2 -^^.i^k*

20
Eq. 72 is made gauge invariant by making the substitutions in eqs.

67 - 70.

In ref. [l], Dombey and Read identify F^(k2

) with the nucleon axial

vector form factor. Expressions for this form factor and the other form

factors in eq. 72 are given in the appendix.

;(pv)

In the nonrelativistic limit at threshold, the current operator is

= [a + r-5- cr-k L a 3

F"

-1 F. 2k m

7t K Ck

2k m - k2 \ k2 O 71 L« 3J

2 \

(73)

k

o2m

1 + ^)

• S V \ / s V

F

l +T3

Fl) , ( F

2

+T

3

F

1

\ 2 2

2m/

m

71 2m

+

2m

k?

2mk

livD+ (livI

s V

F T + F

2 a 2

IV^}

6. Cross Section Equations

6.1. Coordinate Space Representation

The amplitudes given in the previous sections are in a momentum

space representation. It is convenient in order to use known nuclear

wave functions to express these matrix elements in a coordinate space

representation. This is done by using the fact that each of the amplitudes has a 3-momentum conserving ^-function which multiplies it. For

production off of a single nucleon, we can let

!3(VVvVx»',t'" x

S,T

(74)

J

-IP »X .7* -» _

3 e F

f -ik »x T d x

V8t?" x

s',tMxs,t

e

r

.r* -» ip.«X ik «x e i V87?"

21
~ip *x *

' 's ip *x — ' -

'

where e

" f /• 87c3 and e i /V8TC3 are now the final and initial nucleon

wave functions in coordinate space. To generalize this to a system of A

nucleons, one has to replace

T ip^x ip.*x

e f e 1 X„i T

i "' and X .. by the appropriate initial and final

' /87U3 /87T3 S,T

state nuclear wave functions and let /d3 x ->^T]/d3 x ... d'\

When we separate out the center of mass part of the initial and final

state nuclear wave functions, then the integration over the center of

mass coordinate gets back a 3-momentum conserving 6-function,

6 (P + k - P. - k.), where P. and P^ are the momenta of the center

f 7C 1 1 1 r of mass of initial and final nucleus. All that is left in the coordinate

space matrix element is the integration over the internal coordinates of

the nucleus.

If we include interactions of the^outgoing pion with the final state

- i q • r nucleus, then we can no longer use e i as the pion wave function

—

— 1 Q * TT

* where r. represents a relative coordinate. We let e i -» cp (r.) where

1 TCI

(p (r.) is now distorted by Coulomb and strong final state interactions.

71 1 This is called the distorted wave impulse approximation (DWIA). There is no general agreement as to how to generate these wave functions, and the

investigation of this part of the problem is beyond the scope of this

paper. However, an understanding of these final state interactions could

be the most important and interesting aspect of photo pion production

from nuclei.

6.2. Photoproduction Cross Section

Assuming the initial and final nuclear states are bound states, the

photoproduction cross section can be written |_8J

y =

~5. a\^J (2E%

k

o

> T _ v \ ° vt

£ T \ ' *i

_ n

r *° = k {^)

^^^^^

+ E- " E

= - E"' < 75 >

6 3 (P

f

+ q - Pi - k)d3

P

f

d q

22
where |<M(r)>| c includes the sum and average over the projections of the

angular momentum of the final and initial states and the coordinate

space integrations.

v The flux factor |— - 1 | can be evaluated in the center of momentum

frame where

v =

m, m. c

c

and

v

1 + ^r >k 1 >

3 3 The integration over d P is done using 6 ( ) . Letting

A 3

d q = J% EdEdfi » (he)2 71 % 7t (76)

the integration & over dE can be performed using 6(E r + E E.- E ). But

7t & v f 7t i y

E,- is now a function of E so that we cannot use the 6-function directly

f 71

E

f ~

m

f + *r

f =

m

f +

E

3 - m

2

71 71 2m.

(77)

Then

6(E, + E - E. - E )dE =

f 71 1 V 71

E

2

- m

2 mr + —~-—— + E - E. - E

f 2m, 71 i y

dE = (78)

'( E*

- E

°)

1 +

dE

E \ 71 _7C m,

23
where E is the result one gets by setting [ J = and solving for E . O 71 Now the dE integration can be done simply.

dgV- 1 /G \

s iq, (he)* |<M(r)>|»

dQ 4ti 2mc2 V 1 / eTT E ^ wy;

/ i + _lJ/l + * m.c3 JI m c2

Note that the kinematical factors in the denominator of eq. 79 were

derived assuming a bound final state in a particular frame and are not

necessarily present or of this form for all photoproduction processes.

6.3. Electroproduction Cross Section

From eq. 56 we saw that the electroproduction amplitudes could be

written as

M = e-lk^r - J

Defining a new current operator J

1 by

3 ' iN^- - 3 > (80)

the electroproduction cross section can be written [8]

a

2 LGf fel)&T 1 Jfe .<j.>\ fr .<j.>"\ (81)

dE f dfi , leicHanc (2m2)2 / L\k / E k4 i\ e / \ e

e e

N ' \ e / n

+ ^e -<J , >*Yk;»<J , >\- |-<J»>'<J'>*| 6(E

f + E^ + E

e

, - E. - E

e

)

24
6 (p-c + q + k' - vr P . - k ) f ^ e

r

i e

where E and E , are the initial and final electron energies, k and k'

e e & ' e e are the initial and final electron momenta, and

k

2 = (k - k')2 = (k - k' ) 2

- (k - k')2 (82)

e e eo eo e e

The longitudinal part of the current J* is given by

-» k*j' -» k

2 -* J

L

= ^k =-i? J

L

< 83 > k o

and the transverse current by

J^ = J

T (84)

From eq. 73, we can see that the contribution to J from the pion current

diagram is in the k direction. Therefore, this diagram only contributes

to the longitudinal current and need not be considered in J .

Assuming the final state to be a state with one definite angular

turn, the longitudinal- 1: interference terms disappear.momentum, the longitudinal- transverse and transverse- transverse2 This is a safe assumption since we are considering only threshold

production.

25
The cross section can now be written as

dc? dE ,dQ

e e :

4 m(^j ¥^ ^ "\ «<*

f

- h> «& - ^

) (85)

E.L.

-2 k* k o

2 (k

e .k) (k;.k) - ^k2 k

2 |k-<I > +

"2k

2 k

,2 sin2

6 e e e kf_

2

[' e+ .<J

T>r + U_-<jt

>

where e+ =

e +ie

/2

and 8 is the angle between k and k' . Neglecting the electron mass,

e e e we can write for the coefficient of the longitudinal current

k o L o (1 - e) (86)

and for the coefficient of the transverse current

2k2 k'

2 sin2 e e k

2

i?

k

2

i 2 (1 - e)

(87)

Dividing through by k

2

, the cross section can now be written

dtf dE ,dO , e e

2 , r

87T3 ( G \a (K\ (hcf <fq 3p L

e)

(88)

26
[iv<yi* + I'.-^i'l k

= nlk«< T > 2

64 (P, - P ) jl'-+ " T ' " '"-

~ T ' J - e ^ - -ki.

f i

;

I 2

s k2 -2 k

_i

. - 2p- tan2 7p )

It is useful for the purpose of comparing with photoproduction data

to express parts of the cross section in center of mass variables (variables in the photon- target center of mass system will be denoted by a *)

[9]. The rest of the cross section will be evaluated in the lab system.

It is known that

.a cfp, ,3 * d

3

P

6*<p

f

_P.) = £a__l 6*(p*-P*) (90) \

E

f E

71 f

If we transform to a system where k is in the z- direction, then we know

the transverse terms won't change and

v^i2 + r^.- <j

T

>|2 = Iv<j

t

|2 + \*--<3

t

i 2 (91)

The longitudinal current transforms by

^V'2 A\2 |k-<j>,v

| 2 L 1 ° \ h (92)

k

2 \k / k

2

If we assume the electron and pion are detected, the cross section is

_of./G \ 3

( k

e\ (hc|q"|) 1

87T

3 [Z^?] UJ WT (1 - e)

d£ _JL( G \

2 e Uhcjq 1) 1 1 (93)

dE , dQ , dQ \ / \ e/ ' / E

e e' 7C N '

i +

m

f

c2

27
[l V^/l5 + Ie.-<J

T

>*| 8 1

o |k.<J> p

k

2 k

3 (93)

(cont.

)

where

k =

o

: An. + k ^

m. + k

1 o k

2 2 (94)

_>+ km.

k* = -p- m + k \ 2

- k

2 (95)

Let

k' r = S-ls.

t 2ti2 \ k

g h _1 1_

Tk2! (1 - e) (he)

Then

dtf dE ,dfi ,dQ

e

1 e' 7i

r _s / G f (hc ) 2 lq"l

t 4ti \2mc2y Kl v1 +

m c*

(96)

V<J

T

> I 2 + K^V- |k.<X>*| 2

r.

- e k*

3 k

2 o

> = ^<V e<,L>

whe re JL is the equivalent real photon energy in the lab system,

28
where

(

G

V&nc2

Dv<v>

(97)

(5, = a

2mc2 (he)' 1 +

mf

c

c

|k«<J >"| :

1j (98)

7. Discussion of Results

In order to see where the nonrelativistic current operators are no longer valid, the electroproduction cross section off of a proton producing a n+ is calculated using the fully relativistic operator in eq

.

72 and compared with the nonrelativistic results from eq. 73. In fig. 6, we see that the nonrelativistic results are not very good beyond

k2 = - .l(GeV/c) 2 . In figs. 7 and 8 are plotted the transverse and

longitudinal cross sections.

In order to get better agreement between the two results, one would

have to include terms which depend on the initial and final 3 - momentum

of the nucleon. This is no problem for pion production off of a single

—> —* —

>

nucleon, because P. = - k and P_ = 0. But for pion production off a

i _^ f

nucleon in a nucleus P.^*k and P

f^0 and these terms greatly complicate

the calculation.

The author thanks Dts. James O'Connell and Leonard Maximon for

proofreading the manuscript and for helpful suggestions and discussions

29
8. Appendix

From ref. [l] and [7], the form factors to be used in eq . (71) and

(73) are

F

A

=

1

, m

A =l.

1 k

2

<£ =

(1 + K ) P k

2

1 ".71

°£ =

1 k

2

1 ".71

G

M

"

K n

1 k2

".71

n

G

E =

(1

T GnM + 4T) J 4m2

The pion form factor is

_1_

. - F = r-g— , m = 0.765 GeV

tc , k3

p

m2P

9. References

[l] Dombey, N., and Read, B. J., Nucl. Phys. B60, 65 (1973).

[2] Mandl, F. , Introduction to Quantum Field Theory, pp. 75-78

(Interscience Publishers, Inc., New York, 1960).

[3] Sakurai, J. J., Invariance Principles and Elementary Particles,

pp. 178-184 (Princeton University Press, Princeton, New Jersey,

1964).

[4] 01s son, M. G., and Osypawski , Nucl. Phys. B87 , 399 (197 5).

[5] Bjorken, J. D. , and Drell, S. D. , Relativistic Quantum Mechanics

,

pp. 285-290 (McGraw-Hill, Inc., New York, 1964).

30
[6] DeBaenst, P., Nucl. Phys . B24, 633 (1970).

[7] Read, B. J., Nucl. Phys. B74, 482 (1974).

[8] Derived with Dr. E. L. Tomusiak.

[9] Dalitz, R. H., and Yennie, D. R., Phys. Rev. 105, 1598 (1957)

31
200

-J

b

+

6

it: cr

150

100

50

NONRELATIVISTIC

RELATIVISTIC

02 .04 .06

|K 2

| (GeV/c)

2

.08 .10

+ , Figure 6. Comparison of cross section slope for e(p,e'7C )n using fully

relativistic and nonrelativistic amplitudes.

32
300

I K 1 (GeV/c)'

Figure 7. Slope of the transverse and longitudinal cross sections

using nonrelativistic amplitudes.

33
300

250

200

=1

150

100

K | (GeV/c)'

Figure 8. Slope of the transverse and Longitudinal cross sections

using fully relativistic amplitudes.

34
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS T^9572. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Threshold Photo and Electroproduction of Pions from Nuclei

5. Publication Date September 1977

6. Performing Organization Code

7. AUTHOR(S)

E. T. Dressier

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 24u4?T1

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Same as item 9.

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15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less (actual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

The nonrelativistic amplitudes for photo and electroproduction of pions are derived using effective pseudoscalar and pseudovector Lagrangian densities. The

results for pseudscalar and pseudovector coupling are compared at threshold for

charged and neutral pion photo and electroproduction. Cross section formulas and

kinematical conditions are also presented.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Effective Lagrangian; electroproduction; impulse approximation; photoproduction;

pseudoscalar and pseudovector coupling; threshold pion production.

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