i-ri NBS TECHNICAL NOTE 957%*U Of

U.S. DEPARTME

NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to

strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this

end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific andtechnological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to pro-mote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute

for Applied Technology, the Institute for Computer Sciences and Technology, the Office for Information Programs, and theOffice of Experimental Technology Incentives Program.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consist-

ent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essen-

tial services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry,

and commerce. The Institute consists of the Office of Measurement Services, and the following center and divisions:

Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center for Radiation Research — Lab-oratory Astrophysics 2 — Cryogenics 2 — Electromagnetics 2 — Time and Frequency 3

.

THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized materials needed by industry, commerce, educational insti

tutions, and Government; provides advisory and research services to other Government agencies; and develops, produces, and

distributes standard reference materials. The Institute consists of the Office of Standard Reference Materials, the Office of Air

and Water Measurement, and the following divisions:

Analytical Chemistry — Polymers — Metallurgy — Inorganic Materials — Reactor Radiation — Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services developing and promoting the use of avail

able technology; cooperates with public and private organizations in developing technological standards, codes, and test metfr

ods; and provides technical advice services, and information to Government agencies and the public. The Institute consists of

the following divisions and centers:

Standards Application and Analysis — Electronic Technology — Center for Consumer Product Technology: Product

Systems Analysis; Product Engineering — Center for Building Technology: Structures, Materials, and Safety; Building

Environment; Technical Evaluation and Application — Center for Fire Research: Fire Science; Fire Safety Engineering.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services

designed to aid Government agencies in improving cost effectiveness in the conduct of their programs through the selection,

acquisition, and effective utilization of automatic data processing equipment; and serves as the principal focus wthin the exec

utive branch for the development of Federal standards for automatic data processing equipment, techniques, and computer

languages. The Institute consist of the following divisions:

Computer Services — Systems and Software — Computer Systems Engineering — Information Technology.

THE OFFICE OF EXPERIMENTAL TECHNOLOGY INCENTIVES PROGRAM seeks to affect public policy and process

to facilitate technological change in the private sector by examining and experimenting with Government policies and prac

tices in order to identify and remove Government-related barriers and to correct inherent market imperfections that impede

the innovation process.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and accessibility of scientific informa

tion generated within NBS; promotes the development of the National Standard Reference Data System and a system of in'

formation analysis centers dealing with the broader aspects of the National Measurement System; provides appropriate services

to ensure that the NBS staff has optimum accessibility to the scientific information of the world. The Office consists of the

following organizational units:

Office of Standard Reference Data — Office of Information Activities — Office of Technical Publications — Library -

Office of International Standards — Office of International Relations.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

2 Located at Boulder, Colorado 80302.

OF STANDARDSLIBRARY .-_.

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

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICEWASHINGTON: 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 Electroproductionof Pions from Nuclei

E. T. Dressier

The nonrelativistic amplitudes for photo and electroproduc-tion of pions are derived using effective pseudoscalar and pseudo-vector Lagrangian densities. The results for pseudoscalar andpseudovector coupling are compared at threshold for chargedand neutral pion photo and electroproduction. Cross sectionformulas and kinematical conditions are also presented.

Key words: Effective Lagrangian; electroproduction; impulseapproximation; photoproduction; pseudoscalar and pseudovectorcoupling; 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 usethe impulse approximation,, That means that one assumes that an externalperturbation acts on the nucleons in a nucleus as if they were freeparticles. 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 correlatedby the nuclear wave functions). The purpose of this paper is to presentthe nonrelativistic amplitudes for photo and electroproduction of pionsthat can be used in an impulse approximation calculation for nuclearprocesses.

To find the amplitude from a single nucleon, one usually starts byconsidering which relativistic Feynman diagrams are important in the

particular energy region under consideration. For the case of a purelyelectromagnetic interaction as in electron scattering, photo or electro-disintegration, or Compton scattering, one can use perturbation theoryand quantum field theory to give the appropriate Feynman diagrams to a

particular order in the coupling constant. However, in photo or electro-production 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 expansionusing pseudovector pion-nuclear coupling, gives results which under cer-

tain 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 relativisticFeynman amplitudes. This is done in Section 3. The nonrelativisticamplitudes are then specialized to the cases of threshold photo andelectroproduction in Sections 4 and 5, respectively. In Section 6 thecross 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 andnuclear fields that one can choose are called the pseudoscalar and pseud-ovector interactions. In the pseudoscalar case, the interaction La-grangian density for the pion-nucleon interaction is

X^ s)= iGN(x) T • 5(X ) y 5

N(x) (1)

whereN(x) = nucleon field operator

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

G3

-j— = 15.0 = pion-nucleon coupling constant

T = nucleon isospin operator

For photo pion production, the total interaction Lagrangian densityis

£(ps) = x(ps) + z + z (2)I 7tN yN y%

where

-f,.

...

yN 'u£ .. = ieN(x)y Ay (x)N(x) (3)

1YTT

= 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 termof the perturbation expansion, one finds that there are three Feynmandiagrams (shown in Figs. 1-3) that can be associated with the secondorder term. (Solid lines are nucleons, wiggly lines are photons, andbroken 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 Borndiagrams. Figure 3 is called the pion current diagram, because it re-

sults from the interaction of the pion's electromagnetic (e-m) currentwith the e-m field operator^ given by £y%.

For pseudovector pion-nucleon coupling, the interaction Lagrangiandensity for the interaction of the pion and nueleon field operators is

given by

t ^if

£<PV) = _JE n(xtcN m (*) yur 5

Td?(x)

N(x) (5)

where fl = °- 08

' (a?-)"= 4* (s%

Now in order to insure gauge invariance of our total interaction La-

grangian 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 contri-

bution as in the ps case, because the form of the 71 - N vertex is dif-

ferent 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 dia-grams 1-4 using pv coupling for charged pion production at threshold.However, pv and ps couplings give quite different results for tc produc-rion. pv coupling gives the best agreement with experiment and is usual-

ly chosen as the "effective Lagrangian" [4J. In spite of this, we willderive 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 ofref. [5]. The relativistic amplitudes for these diagrams are defined as

Aw 1 T3(*p

- kn )\j _ t(Pi)€

u(7)

\ 2 /Jp' 2 - m2 r 5 a s,T v 1'

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

M?3

PS)= * «sV<V V 5 -V- SV"^f V<V (9>

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

o 3

and

T_L|P> = /£ |n> ,

l~T >%1 = T \ - Va

3Ja 3 3 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(PS

> = -i l«yl k / o' o

/-k \ I 2P -k + ka-k ^ + a.q \i + -^ (ID

+o 2m 2m

P *e ->

i ka«k ( ^-

m 2m / \ 2m

+ie,

( k x q)2m a\ 2 / k

oo \ 2m

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

^ (fp

+ Kn) + (V K

n) M

M(ps)

. JL.

y2 k o1 ~) + cf-q

'

2Pf«k - k

2mk(12)

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

, . £'(k x q)+ 1 —aT"^1 + T

37 1- t + —- e a*k2 a k o

o2m

+

> - / oqo 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 - k2 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 2mka-k (14)

m

2mt -Ea

V p n/+Vp n /

- K \ TP n y 3

M(ps)

y2

ck -* -»

e„ ~ot; + G#eO Al

k

1 + ^ (X + T

3)

CI m7t -> ->

71 - "*

- e T53— G *k + -sr G *eo 2mk 2m

(<

K + K . K - K Tp n/+\ p n/ 3

2 /a

(15)

M (Ps) .

y3

ack2k m ^~W

O 71

e ( 2m - ko V 71 o

\ + e«k [

IVJ(16)

By replacing eQ

-» 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 pvcoupling. 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

kv K_P n

2m \

+( K - K ]tP n / 3

Us,x(Pi)e "

M(pv)

y2 V,T.<ef>

1 + t;

a

2mk

'n 2

(VM^n) C+ "

fly.T U T

(p.)«l)

2 - m3 "'5 a s,T vri

7

(17)

(18)

M (pv) - , v ,., ,. fra,T3] 1= UQ , T ,( Pf ) (K ff) y, —J- - „, . ggy3 s',T'^f 5 2 q* (19)

(q« + q) Us>T

(p.)^

™ (Pv ) t"t t \ ^ Ta,T3 ] , N p.My4

= V,!'^ —2— V5 Us,T

( Pi)e (20)

Consider now the following identities:

5s',T-<Pf>4r 5-^^=G

s, >T ,<P f)r 5

2m (j6 + m)

p2 - m2

+ 1 (21)

p'3 - rf ^5 Us,T<Pi) =2m(jS' + m)p'2 - m2 + 1 ?5 Us,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 T5 a

(l + T.

V 2(23)

(J I [ K + K )+ K - K

+i JiZ hv

l \ P nj_ \ P

2m a^J U ( P .) e^M (ps)

s,t i'+ M 1

yl yl

M 1PV> = M 1

PS) + 5., T , (p

J

y2 y2 s',T' Nrf

1 + T ) a

V 2 2m(24)

K + K ) + K - K \tp

n/ ip bJ-J^) T y u _ (p.)e^M <

ps)+ M.'

/ a 5 s,t ri v2 v

For M \ * toe usey3

58',T" (P

f} (K

" « ?5US,T (P i

)= ^ U

S ',T' (Pf

} ^5 °s,T <PI ) (25)

Therefore M <pv)

= M <ps)

y3 y3

Taking the nonrelativistic reduction of M' and M 1 givesy2

M' = < - eyl I o

tf'k O'q "P,-+

2m 2m m+ tf.e f T

(1 + T

s)a 2

j. ii^tM, ((Kp+ K

n)+ (%- Kn)

T3)

j o 2m 2m\ a \ 2 /

(26)

My2

-* -» -» -» 0'»D

2m 2m 2mtf-e |

i-

1+T3) T

a

+ ) eo 2m 2m j \ 2 /a

(27)

Combining M 1 and M' with M and MPS

, 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^i2m 2mk / 2mk

o / o

-» -*/ k«eCJ.q (^—

:

l!V ,.

«•(* '*)2mk mk

o o

'+ i2mk

a 2 o I 2m 2mk )T

\2m 2m 2mk ^L °

+ ;.i;!x +16>(£ * q) T i e n ^ L-£ n

I3

)2mk 2mk a \ 2 /

M(pv)

y2

tf'p,-»f 1

-5r+tf,

« ir+

2p -k + k2

am?o

+2m

+ asH- °- k l5r +0^ asr-sro/ o \ o o

+ie»(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 / 2mko/ o

• e*(k x q)-1" 2mk

o

((Kp+KJ +(V K

n)T3) T

\ 2 /a

(29)

10

Taking the nonrelativistic reduction of M , gives

M(pv) =

y4

*i g»k

m 2m 2m- a •e

[Ta'

T3]

(30)

Making the substitutions e -» k and e -» k, the nonrelativistico o

amplitude M <Pv) + My2

(PV) + My3

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) _

y2S

tf ' Pi -- ko> (

1+T3)

, m

o

/m k.

-1 ~*(

It O

2m 2mk /

e\ 2m " 2m

^P + Kn)

+(K - (C

_J2

(32)

n^Va

M

M

(pv)

y3

(pv) =

r? fe f2m - k \+ e .km - k5 oVtc o /O 7X l J

[V T3]

2m m<3* e

[Ta'

T3]

(33)

(34)

We should note at this point that in deriving these expressions, wehave not used any particular gauge or mentioned any specific referenceframe in order to eliminate terms. However, by stating that thresholdmeans "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 thisterm vanishes at threshold.

Choosing to work in the center of momentum frame of the incomingphoton and the target, gives

k + p. = q + p. (35)1 i

and at threshold

q = = pf

(36)

where p. an^ P^ are the momenta of the initial and final nucleus assum-ri 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. + k2 (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 anomalousmagnetic moment, K .° n

+<M(7t )> = - /2 <(*•£>

- v2

I k \ m

f i . _£) + _E (K + K \

\2m/ 2m I p n)

R<cr«e>

m1 - -- (K + K )

2m p n

(40)

In the cross section we have

+ s

|<M(w )H2 I <c. e:

m1 + -5

m

m

^ (Kp+ V (41)

where we have used

k \-

2m/ m1 +

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 smallP 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~*>> mm

<<5'e> L+ 7i + n

. ~ni \ m 2m1 + TC \ L

+ K )

P n

2m

(K + K )P n

(43)

and

|<M(if)>P« 2<tf-e>r

1-+-5m_jm

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, how-ever, 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

m2 + -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 „ canyl y2

contribute, because (1 + T ) |n> = 0, which reflects the fact that theneutron has no charge.

o -* -. /m

<M(n,ir )> = - <(S'H^) Kn (47)

|<M(n,7i°)>|2 = l<tf'1>|

af^) ( K

n)2

(48)

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

/

mm V1

ofactor (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 nonrelativisticapproximation. 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 contributeto 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 effectfor charged pion production; however, it makes the cross section for

oy + 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 correctioncoming from diagram 2.

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

oCase 3. y + p -» 71 +p

givesDiagram 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 sameresult as with pv coupling. However, for 7t° production, the ps crosssections 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(p2 /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 electropro-

duction, e^ now represents the electrons current and is not the polariza-

tion 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 writeo o

M-—— - e.J= e.|k rro o \ o

(56)

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

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)Mel - ^sV^V *5 a P

2 - m^

FS+ T F

Vrl 3

fl

cj ( FS + T F

v

+ij£ kU

\ 2Q

322m 2

US,T^i)£t

(57)

M(ps)

e22mU

s(p)

FS+ T f

v)

a (fs+ T F

V

1 Vy+ i _b£hi k

Uil2__3_2ru 2

+ X2m

k2

(58)

17

{£ + m)r«T u (p,)^

Me3

PS>= «.., T -<'f> V5

^le^^> F„U. „<p.)'2 - m3 t,; s J *17C

(59)

where

F^k2 ) = FP + F^ (60)

F^) = FP.

F«

(61)

FS

2(k

2 )=K pFP + KnF^ (62)

F>2

>=VP

2-V2 (63)

Fj(o) = FP(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; + T3F[)

2" * p3 - m3 a 2

Mv K^l

q 2 - nr ti

"fl

2 > - nf3To\ 2 2 /

+

(66)

p' 2 - m2 2 2 p' 2 - m2 \ 2

Jr«pSL| rtT<p,

2 / a[ s,T i

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

that we must let F, = F . Since all of the form factors are assumed to1 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 bedone by letting

y F7v i

v.kk: t

k2 (67)

(2q

-k)u

n ' 2 _ m2(2q-k)

(2q-k)»kk

k2 Jq 7^ mc (68)

(2P

l+ k

) ,

p2 - m2 (2Pl + k)

.

(2p + k)-kk1 u

k2 3s - m2 (69)

(2p - k),

,•2 mc(2Po - k),

(2p2

- k) - klvk2 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 longi-tudinal 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

'tt 1 2k m

I~1

,k "2 7*1 a 3 _v ojt

\ ks / 2 1 /2km -

k (?»k

k2 \ k2

r' T

3la 3|

F2 7C

(71)

2m / 1 + m

I an

m

(F1+ T

3FI)

,(FS T^V+ T F2 3 2

2m~

1 - ZEE2mk

(i+_vl) ,

F2+ Vl,+2 I "a

it

5.3. Pseudovector Electroproduction

From eq. 5.2 of ref. [l], the amplitude for electroproduction usingpseudovector coupling is

Mi

Pv):=5s',T-<Pf>r 5

j. rFA Lp] +an (f^i 'H A 2 q 2 - m

T ,Ta 3JL F

q' 2 - m2 % 2(72)

/2P

1+ k

)u T (F1+T

3FD

"w ps - m2 a 2

idl - k) T i(livL) + (I1V22 p2 - m2 a\ 2 2

+ (2p 2

'

k) u /

Fi+ T

3FI.

T+p' 2 - m2 * 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 axialvector form factor. Expressions for this form factor and the other formfactors 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 m7t K Ck

2k m - k2 \ k2O 71

L« 3J2 \ (73)

ko

2m

1 + ^)

• S V \ / s VFl+T

3Fl)

,(F2+T

3F1

\ 2 2

2m/

m71

2m

+2m

k?

2mk

livD + (livI

s VF T + F2 a 2

IV^}

6. Cross Section Equations

6.1. Coordinate Space Representation

The amplitudes given in the previous sections are in a momentumspace representation. It is convenient in order to use known nuclearwave functions to express these matrix elements in a coordinate spacerepresentation. This is done by using the fact that each of the ampli-

tudes has a 3-momentum conserving ^-function which multiplies it. For

production off of a single nucleon, we can let

!3(VVvV x»',t'" xS,T(74)

J

-IP »X .7* -» _3 e Ff -ik »x T

d xV8t?"

xs',t-Mx

s,te

r

.r* -» ip.«Xik «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 Anucleons, one has to replace

T ip^x ip.*xe 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 /d3x ->^T]/d

3x ... d'\When we separate out the center of mass part of the initial and finalstate nuclear wave functions, then the integration over the center ofmass coordinate gets back a 3-momentum conserving 6-function,

6 (P + k - P. - k.), where P. and P^ are the momenta of the centerf 7C 1 1 1 r

of mass of initial and final nucleus. All that is left in the coordinatespace matrix element is the integration over the internal coordinates ofthe 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.) where1 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%ko

> T _ v \° vt

£T \ ' *i

_ nr

*° = k {^) ^^^^^ + E- " E= -

E"' < 75 >

63(P

f+ q - Pi - k)d

3Pfd q

22

where |<M(r)>| c includes the sum and average over the projections of theangular momentum of the final and initial states and the coordinatespace integrations.

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

frame where

v =m, m. cc

andv

1 + ^r>k 1 >

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

A3d q = J% EdEdfi»(he)2 71 % 7t

(76)

the integration over dE can be performed using 6(E r + E E.- E ). But&7t

& vf 7t i y

E,- is now a function of E so that we cannot use the 6-function directlyf 71

Ef ~ m

f+ *r

f= m

f+

E3- m2

71 71

2m.(77)

Then

6(E, + E - E. - E )dE =f 71 1 V 71

E2

- m2

m r + —~-—— + E - E. - Ef 2m, 71 i y

dE = (78)

'( E* - E°)

1 +

dEE \ 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 \

siq, (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 werederived assuming a bound final state in a particular frame and are notnecessarily 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]

a2 LGf fel)&T 1 Jfe .<j.>\ fr .<j.>"\ (81)

(2mc2) LdEfdfi , leicHanc2 / \k / E k4 i\ e / \ e

e e N ' \ e / n

+ ^e-<J ,

>*Yk;»<J,>\- |-<J»>'<J'>*| 6(E

f+ E^ + E

e , - E. - Ee

)

24

6 (p-c + q + k' - P .- k )vrf ^ 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

k2 = (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' -» k2 -*

JL =^ k =-i? J

L < 83 >

k o

and the transverse current by

J^ = JT

(84)

From eq. 73, we can see that the contribution to J from the pion currentdiagram is in the k direction. Therefore, this diagram only contributesto the longitudinal current and need not be considered in J .

Assuming the final state to be a state with one definite angularturn, the longitudinal- 1:

interference terms disappear.momentum, the longitudinal- transverse and transverse- transverse-

2 This is a safe assumption since we are considering only threshold

production.

25

The cross section can now be written as

dc?

dE ,dQe e

:

=

4 m(^j ¥^^ "\ «<*f

- h> «& - ^) (85)

E.L.-2 k*k o

2 (ke.k) (k;.k) - ^k2 k2

|k-<I >

+"2k

2 k ,2 sin2 6e 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

2k2k'

2 sin2e e k2

i?

k2 i

2 (1 - e)(87)

Dividing through by k2 , 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 2sk2 -

2k

_i

.- 2p- tan2 7p )

It is useful for the purpose of comparing with photoproduction datato express parts of the cross section in center of mass variables (vari-ables 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 * d3P

6*<pf_P.) = £a__l 6*(p*-P*) (90)

\ Ef E

71 f

If we transform to a system where k is in the z- direction, then we knowthe transverse terms won't change and

v^i 2 + r^.-<jT>|2 = Iv<j

t|2 + \*--<3

t i

2(91)

The longitudinal current transforms by

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

|

2

L1 ° \ h (92)

k2 \k / k2

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

_of./G \3

(

ke\ (hc|q"|) 1

87T3 [Z^?] UJ WT (1 - e)

d£ _JL( G \2

eUhcjq 1) 1 1(93)

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

e e' 7CN '

i +mfc2

27

[lV^/l 5+ Ie.-<J

T>*| 8

1o |k.<J> p

k 2k3

(93)

(cont.

)

where

k =o

: An. + k ^

m. + k1 o

k2 2(94)

_>+ km.k* = -p-

m + k \2

- k2(95)

Let

k'

r = S-ls.t 2ti2 \ k

g

h _1 1_Tk2! (1 - e) (he)

Then

dtf

dE ,dfi ,dQe

1 e' 7i

r _s / G f (hc )

2lq"l

t 4ti \2mc2y Kl v-

1 +m c*

(96)

V <JT>

I

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

- e

k*3

k2o

>

= ^<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, = a2mc2

(he)' 1 +mfcc

|k«<J >"| :

1j

(98)

7. Discussion of Results

In order to see where the nonrelativistic current operators are nolonger valid, the electroproduction cross section off of a proton pro-ducing 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 beyondk2 = - .l(GeV/c) 2

. In figs. 7 and 8 are plotted the transverse andlongitudinal cross sections.

In order to get better agreement between the two results, one wouldhave to include terms which depend on the initial and final 3 - momentumof 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 ai _^ f

nucleon in a nucleus P.^*k and Pf^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

FA= 1

, mA=l.

1k2

<£=

(1 + K )

Pk2

1 ".71

°£= 1

k21 ".71

GM

"Kn

1k2

".71

nGE=

(1

T Gn

M+ 4T) J 4m2

The pion form factor is

_1_

.-

F = r-g— , m = 0.765 GeVtc , 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

|K2

| (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 sectionsusing nonrelativistic amplitudes.

33

300

250

200

=1150

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^957

2. Gov't AccessionNo.

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 STANDARDSDEPARTMENT OF COMMERCEWASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

24u4?T111. Contract/Grant Nc

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as item 9.

13. Type of Report & PeriodCovered

Final

14. Sponsoring Agency Code

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 arederived using effective pseudoscalar and pseudovector Lagrangian densities. Theresults for pseudscalar and pseudovector coupling are compared at threshold forcharged and neutral pion photo and electroproduction. Cross section formulas andkinematical 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.

18. AVAILABILITY [X Unlimited

IFor Official Distribution. Do Not Release to NTIS

I

-x- ! Order From Sup. of Doc, U.S. Government Printing OfficeWashington, D.C. 20402, SD Cat. No. C13.y

|

AiQ^7

I

I Order From National Technical Information Service (NTIS)Springfield, Virginia 22151

19. SECURITY CLASS(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

38

22. Price

$1.50

USCOMM.DC 29042-P74

* U. S. GOVERNMENT PRINTING OFFICE : 1 977--21*0-848/299

NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH reports National Bureauof Standards research and development in physics,

mathematics, and chemistry. It is published in twosections, available separately:

., orking in

• Physics and Chemistry (Section A)3apers of interest primarily to scier'

these fields. This section covers a br

ral and chemical research, wit v

standards of physical measu*" . y**i

ge of physi-

emphasis onfundamental con-

stants, and properties of rrr ^<v* .ssued six times a year.

Lnnual subscription: D-v<x \

0>>-, $17.00; Foreign, $21.25.

Mathematical Sci'

Studies and com r»*

\8>

a<", Section B).is designed mainly for the math-

ematician and . ^wdtical physicist. Topics in mathemat-

cal statis f ' .^.leory of experiment design, numericalinalysi" ^ _<retical physics and chemistry, logical de-

ign ^> programming of computers and computer sys-

A°

,

v^nort numerical tables. Issued quarterly. Annual.'"ocription: Domestic, $9.00; Foreign, $11.25.

DIMENSIONS/NBS (formerly Technical News Bulle-

tin)—This monthly magazine is published to informscientists, engineers, businessmen, industry, teachers,

students, and consumers of the latest advances in

science and technology, with primary emphasis on thework at NBS. The magazine highlights and reviewssuch issues as energy research, fire protection, buildingtechnology, metric conversion, pollution abatement,health and safety, and consumer product performance.In addition, it reports the results of Bureau programsin measurement standards and techniques, properties of

matter and materials, engineering standards and serv-

ices, instrumentation, and automatic data processing.

Annual subscription: Domestic, $12.50; Foreign, $15.65.

NONPERIODICALS

Monographs—Major contributions to the technical liter-

ature on various subjects related to the Bureau's scien-

tific and technical activities.

Handbooks—Recommended codes of engineering andindustrial practice (including safety codes) developedin cooperation with interested industries, professionalorganizations, and regulatory bodies.

Special Publications—Include proceedings of conferencessponsored by NBS, NBS annual reports, and otherspecial publications appropriate to this grouping suchas wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, man-uals, and studies of special interest to physicists, engi-neers, chemists, biologists, mathematicians, com-puter programmers, and others engaged in scientific

and technical work.

National Standard Reference Data Series—Providesquantitative data on the physical and chemical proper-ties of materials, compiled from the world's literature

and critically evaluated. Developed under a world-wideprogram coordinated by NBS. Program under authorityof National Standard Data Act (Public Law 90-396).

NOTE: At present the principal publication outlet for

these data is the Journal of Physical and ChemicalReference Data (JPCRD) published quarterly for NBSby the American Chemical Society (ACS) and the Amer-ican Institute of Physics (AIP). Subscriptions, reprints,

and supplements available from ACS, 1155 Sixteenth

St. N.W., Wash. D. C. 20056.

Building Science Series—Disseminates technical infor-

mation developed at the Bureau on building materials,

components, systems, and whole structures. The series

presents research results, test methods, and perform-ance criteria related to the structural and environmentalfunctions and the durability and safety characteristics

of building elements and systems.

Technical Notes—Studies or reports which are completein themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so compre-hensive in scope or definitive in treatment of the sub-

ject area. Often serve as a vehicle for final reports of

work performed at NBS under the sponsorship of other

government agencies.

Voluntary Product Standards—Developed under proce-

dures published by the Department of Commerce in Part10, Title 15, of the Code of Federal Regulations. Thepurpose of the standards is to establish nationally rec-

ognized requirements for products, and to provide all

concerned interests with a basis for common under-standing of the characteristics of the products. NBSadministers this program as a supplement to the activi-

ties of the private sector standardizing organizations.

Consumer Information Series—Practical information,based on NBS research and experience, covering areas

of interest to the consumer. Easily understandable lang-

uage and illustrations provide useful background knowl-edge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent

of Documents, Government Printing Office, Washington,D.C. 20402.

Order following NBS publications—NBSlR's and FIPSfrom the National Technical Information Services,

Springfield, Va. 22161.

Federal Information Processing Standards Publications(FIPS PUBS)—Publications in tjiis series collectively

constitute the Federal Information Processing Stand-ards Register. Register serves as the official source of

information in the Federal Government regarding stand-

ards issued by NBS pursuant to the Federal Propertyand Administrative Services Act of 1949 as amended,Public Law 89-306 (79 Stat. 1127), and as implementedby Executive Order 11717 (38 FR 12315, dated May 11,

1973) and Part 6 of Title 15 CFR (Code of FederalRegulations).

NBS Interagency Reports (NBSIR)—A special series of

interim or final reports on work performed by NBS for

outside sponsors (both government and non-govern-ment). In general, initial distribution is handled by thesponsor; public distribution is by the National Techni-cal Information Services (Springfield, Va. 22161) in

paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICEShe following current-awareness and literature-survey

bibliographies are issued periodically by the Bureau:Cryogenic Data Center Current Awareness Service. A

literature survey issued biweekly. Annual subscrip-tion: Domestic, $25.00 ; Foreign, $30.00'.

Liquified Natural Gas. A literature survey issued quar-terly. Annual subscription: $20.00.

Superconducting Devices and Materials. A literature

survey issued quarterly. Annual subscription: $30.00 .

Send subscription orders and remittances for the pre-

ceding bibliographic services to National Bureau of

Standards, Cryogenic Data Center (275.02) Boulder,

Colorado 80302.

U.S. DEPARTMENT OF COMMERCENational Bureau of StandardsWashington. D.C. 20234

OFFICIAL BUSINESS

Penalty for Private Use. $300

POSTAGE AND FEES PAIDU.S. DEPARTMENT OF COMMERCE

COM-215

SPECIAL FOURTH-CLASS KATEBOOK