- Email: [email protected]
Meson production in nucleon-nucleon collisions
Nudeaf Physics 10 (1959) 160--166;~)Nofth-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
MESON PRODUCTION IN NUCLEON-NUCLEON COLLISIONS A L L A D I R A M A K R I S H N A N , N. R. R A N G A N A T H A N t a n d S. K. S R I N I V A S A N tt
Depa~'trn~nt o/ Physics, University of Madl, as, India Received 28 N o v e m b e r 1958 A calculation of t h e a m p l i t u d e for meson production in a nucleon-nucleon collision is presented using t h e m e t h o d of Low. B y m a k i n g a suitable T a m m - D a n e o f f approximation, t h e m a t r i x element for t h e process is expressed in terms of t h e two nucleon potential, pion nucleon scattering m a t r i x element a n d nucleon vertex operator.
Abstract:
1. Introduction
It is well recognized that the description of nuclear forces depends to a very large extent on a correct theory of pion-nucleon interaction. As an initial step considerable attention has been devoted by theoreticians to the problem of meson nucleon scattering. In particular due to work of Chew i) Low 2) and Wick 8), we now have, at least what is believed to be, a correct description of the pion-nucleon scattering up to about a few hundred MeV. The agreement of this theory now known as "fixed extended source theory" with the data obtained by cyclotron experiments at Berkeley and Brookhaven is satisfactory and the resonances are predicted accurately. In view of the satisfactory agreement with experiments, the theory has been applied to the problem of nuclear forces by Gartenhaus 4), Miyazawa 6) and Klein and McCormic e) (see also ref. v, s)). The Miyazawa-Klein potential as well as the Gartenhaus potential suitably modified by Signell and Marshak 9) (who have added a spin orbit interaction using phenomenological arguments) have m a n y essential features which the previous potentials lacked. Hence it is worthwhile to a t t e m p t the problem of meson production in nucleon nucleon collisions using the present knowledge of pion phenomena. We shall follow the method of Low and express the matrix element in terms of known quantities like pion nucleon scattering matrix element, nucleon potential and vertex operator. t Atomic E n e r g y Commission Junior Research Fellow. tt National I n s t i t u t e of Sciences Senior Research Fellow. While t h e work here reported was in progress, this a u t h o r was a t t h e Adolph Basser C o m p u t i n g Labora ory a n d F.B.S. Falkiner Nuclear Research Laboratories, School of Physics, University of Sydney. 160
MESON p R O D U C T I O N IN N U C L E O N - N U C L E O N COLLISIONS
161
2. M a t r i x E l e m e n t for the P r o c e s s
Let Pa, ql be the e n e r ~ - m o m e n t u m four-vectors of the nucleons in the initial state; we require the amplitude for finding two nucleons with momenta Pn, qn and a meson of ~=th type of momentum k. The spins of the nucleons are suppressed in this notation *. The amplitude for this process is given b y (Plq2k~ISEplql)
= (~ o , b~,)b(q~)a~,(k) ~"Z=o
. . . . .
•
b*
where L (x), the interaction Lagrangian density, is given b y
here the ~ ' s are the usual isobaric spin matrices and am and ~/~ are the nucleon and meson mass renormalizations, ~0 is the bare vacuum and b and a are the annihilation operators for the nucleon and the meson respectively• The Lagrangian (2) leads to the meson and nucleon current operators
J (x) = ge61r,V(x)ff,(x)+~Smv)(x).
(4)
The S-matrix element (1) can be written as
(P, q2ko~lSlPa qx) 1
•
(5)
I;., q.,>
where the current operators j~(x), J(y) are in Heisenberg representation tt and [Pxqx) is the two-nucleon state vector consisting of "incoming" plane waves and "outgoing" scattered waves while (P2[ is a one-nucleon state vector with "outgoing" plane wave and "incoming" scattered waves. ( )+ is the time ordered product defined according to Wick ~0). The equivalence of (5) and (1) can be established b y the method of Lehmann, Symanzik and Zimmerman n) or b y the simpler approach of Low 3). j~(x) and J(y) are given b y t T h r o u g h o u t t h i s p a p e r , we s h a l l w r i t e a four v e c t o r a s x a n d a n o r d i n a r y v e c t o r b y t h e c o r r e s p o n d i n g b o l d l e t t e r x . T h e s c a l a r p r o d u c t of t w o four v e c t o r s x, y w i l l be d e n o t e d b y xy = x0Y0--x~y , = x o y o - - x , y. I n t e g r a t i o n s o v e r four d i m e n s i o n a l v o l u m e s a r e d e n o t e d b y dx, t h o s e o v e r t h r e e - d i m e n s i o n a l v o l u m e s b y d x . ?? W e s h a l l d i s t i n g u i s h H e i s e n b e r g o p e r a t o r s f r o m i n t e r a c t i o n r e p r e s e n t a t i o n o p e r a t o r s b y u s i n g b o l d t y p e for t h e former.
]02
ALLADI RAMAKRI SHNAN, N. R. RANGANATHAN A N D S. K. SRINIVASAN
( - []
+~,) ~
(x) =gCp (x)~,+..+-++,,,p(.) +
(iy, ~
~ ~ (x)- ~ (~)~, (x)~, (x) = j,, (x),
--m)~(x) ----gys~,¢(x)O,(x)4-~z~(x) = J(x).
(6) (7)
Next we use translational invariance: r(~) = e '~" r(o)e -'~" where F(x) is any field operator. This yields
.2E(~8)w(k) <~,1 {f~'+,,(jo(o)Je))+ d+,+im'+,,e(o)} i~q,>.
(P,f,k~lSl~g,) = --2=a@x-Fq~--p,--9,-a)~(¢,)
(8)
The second term is easily evaluated:
(Pnl¢(O)lp~qx>=
(o... (Sv(O))+b*(fl)O,,)
(9)
where S is the S-matrix and Op's are state vectors in interaction representation. Using the relations (see, for example, ref. 2,xn)) /~p(O),b,(fl) } = ~ w1 ( f l )
( ~ )m
+,
~s d~, (S,b*(q,)}= (z-~f e-"""(qO(~(~O)+~(x)
(10)
(~1)
we obtain @~[~(0)ltbxql) = w(fl)(E__~fl))i. 1 (2~)t
[<,,',D- ('... f ( &
,(0))+e-".'&+,..)]. (12)
The functional derivative of S with respect to ~(x) is related to the current
operator by J(x) =
~S i~-~)St,
~S J(z) = i~-~St
(13)
so that
f (~,., (~x) v(o)) +~,,) e-"'~ = -if(~,,,
= -i f<~.l(3(=)e(o))+l~,>e-'*"~
= --ifd.[--ft
e:"'<,.l~(o)J(x)l,.>~o
* fo o--,-<~,lJ(,)e(o)l~,>~o].
(14)
MESON
PRODUCTION
IN N U C L E O N - N U C L E O N
Expansion in terms of complete set of
1 (.:f(o.
~S
"incoming" states yields
+o.) e <#21.X(O)In>
<#~['P(o)In>
---
~ tt
P~=Px+qt
163
COLLISIONS
E(ql)+E(px)--E,,+i, + ~
Pn~Pl--qx
(15)
E(p2)--E(fl)--E,,+i,"
In a similar manner, we expand the first term of the right-hand side of (8):
E(q2)+E(p~)_E,,+i~
<#2lj~(0)ln>
= --i ~
E(q2)+E_E(p~)_E(fl)_i~ + i ~
pn~pl-l-q~--qs Defining
Pn~P|+ql
= (2~)4i~(px+qx--p2--q~--k) , (17) we obtain using (8), (12), (15) and (16)
ra ½ =w(q')(2E(q~)w(k)) [ ~
pm*Spx+qt--qs
E(q.)TE,,.--E(pl)--E(fl)--ie i
-
-
Z
E(qs)WE(p2)_E.+i e --gYsz'~,w(fl)~-~
P~Pt+qs X <#21~(O)]fJg'> <~']J(O)[#l> --
•
pm'mpvFq1
E(qx)+E~,)--E,,,,-q-ie
X ps'-~q~l--q1
(
<~/~[J(0)In'>
(,s)
/1
E(p,)--E(q,)--E,,,--ie 13
where we have dropped the contribution due to the first term of the righthand side of (12) since it corresponds to the possibility where the nucleon of momentum Px goes unscattered, the final state meson being produced b y the other nucleon.
3. Tamm-Dancoff Approximation Equation (18) as it stands is very general in character. To make the problem simpler, we shall assume that the m and n states do not contain more than two particles. The motivation to make such a drastic assumption is twofold: (1) If the m and n states were to contain more than two particles, we are forced to introduce a separate equation similar to (18) for double meson production amplitudes, a circumstance which we wish to avoid.
164
ALLADI RAMAKRISHIqAN, N. R. RANGAIqATHAN AND S. K. SRINIVASAN
(2) Effects due to virtual pair creation are eliminated and this simplifies the evaluation of the vertex operator and other related quantities. Of course, here we are guided by the fact that pion nucleon scattering properties are known only for energies less than a GeV. Next we shall examine the various possibilities of m and n states: m state: The non-vanishing contribution comes from one-nucleon and one-nucleon + one-meson states. n state: The only state that contributes is the two-nucleon state. m' state: Here also only the two-nucleon state gives a contribution. n' state: The state that gives a contribution is the one-meson state. Thus (18) can be written as (P2qsk~lTl~xql) = ~ (q') [ (
(
J J E(
(q~w (k)) ½ [(#,lJ, (0)IPl+qx--q,)(lblq-ql--q,I J (0)[Pxqx)
(t,2lj=(O)lIYk'#)
q
~
~
. . q IJ,()l~bzql) 0 g I"(P
--,J| a
/ .
~e
~.
~e
---ied~'dk'6 ( p ' + k ' - - p , - - q 1 + q , )
...... aP aq o ( p , + q , - - p ' - - q ' ) 0
. . . .
0
(1
MESON P R O D U C T I O N I N N U C L E O N - N U C L E O N COLLISIONS
165
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
G. F. Chew, Phys. Rev. 94 (1954) 1748, 1755; 95 (1954) 1669 F. E. Low, Phys. Rev. 97 (1955) 1392 G. C. Wick, Revs. Mod. Phys. 27 (1955) 339 S. Gartenhaus, Phys. Rev. 100 (1955) 900 H. Miyazawa, Phys. Rev. 104 (1956) 1741 A. Klein and B. H. McCormick, Phys. Rev. 104 (1956) 1747 A. Klein, Prog. Theor. Phys. 20 (1958) 257 Yu. V. Novozhilov, Journ. Exp. Theor. Phys. (Trans.) 6 (1958) 692 P. S. Signell and R. E. Marshak, Phys. Rev. 109 (1958) 339 G. C. Wick, Phys. Rev. 80 (1950) 268 H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cimento I (1955) 205 A. A. Loganov and A. N. Tavkhalidze, Journ. Exp. Theor. Phys. (Trans) 5 (1957) 1134
MESON PRODUCTION IN NUCLEON-NUCLEON COLLISIONS A L L A D I R A M A K R I S H N A N , N. R. R A N G A N A T H A N t a n d S. K. S R I N I V A S A N tt
Depa~'trn~nt o/ Physics, University of Madl, as, India Received 28 N o v e m b e r 1958 A calculation of t h e a m p l i t u d e for meson production in a nucleon-nucleon collision is presented using t h e m e t h o d of Low. B y m a k i n g a suitable T a m m - D a n e o f f approximation, t h e m a t r i x element for t h e process is expressed in terms of t h e two nucleon potential, pion nucleon scattering m a t r i x element a n d nucleon vertex operator.
Abstract:
1. Introduction
It is well recognized that the description of nuclear forces depends to a very large extent on a correct theory of pion-nucleon interaction. As an initial step considerable attention has been devoted by theoreticians to the problem of meson nucleon scattering. In particular due to work of Chew i) Low 2) and Wick 8), we now have, at least what is believed to be, a correct description of the pion-nucleon scattering up to about a few hundred MeV. The agreement of this theory now known as "fixed extended source theory" with the data obtained by cyclotron experiments at Berkeley and Brookhaven is satisfactory and the resonances are predicted accurately. In view of the satisfactory agreement with experiments, the theory has been applied to the problem of nuclear forces by Gartenhaus 4), Miyazawa 6) and Klein and McCormic e) (see also ref. v, s)). The Miyazawa-Klein potential as well as the Gartenhaus potential suitably modified by Signell and Marshak 9) (who have added a spin orbit interaction using phenomenological arguments) have m a n y essential features which the previous potentials lacked. Hence it is worthwhile to a t t e m p t the problem of meson production in nucleon nucleon collisions using the present knowledge of pion phenomena. We shall follow the method of Low and express the matrix element in terms of known quantities like pion nucleon scattering matrix element, nucleon potential and vertex operator. t Atomic E n e r g y Commission Junior Research Fellow. tt National I n s t i t u t e of Sciences Senior Research Fellow. While t h e work here reported was in progress, this a u t h o r was a t t h e Adolph Basser C o m p u t i n g Labora ory a n d F.B.S. Falkiner Nuclear Research Laboratories, School of Physics, University of Sydney. 160
MESON p R O D U C T I O N IN N U C L E O N - N U C L E O N COLLISIONS
161
2. M a t r i x E l e m e n t for the P r o c e s s
Let Pa, ql be the e n e r ~ - m o m e n t u m four-vectors of the nucleons in the initial state; we require the amplitude for finding two nucleons with momenta Pn, qn and a meson of ~=th type of momentum k. The spins of the nucleons are suppressed in this notation *. The amplitude for this process is given b y (Plq2k~ISEplql)
= (~ o , b~,)b(q~)a~,(k) ~"Z=o
. . . . .
•
b*
where L (x), the interaction Lagrangian density, is given b y
here the ~ ' s are the usual isobaric spin matrices and am and ~/~ are the nucleon and meson mass renormalizations, ~0 is the bare vacuum and b and a are the annihilation operators for the nucleon and the meson respectively• The Lagrangian (2) leads to the meson and nucleon current operators
J (x) = ge61r,V(x)ff,(x)+~Smv)(x).
(4)
The S-matrix element (1) can be written as
(P, q2ko~lSlPa qx) 1
•
(5)
I;., q.,>
where the current operators j~(x), J(y) are in Heisenberg representation tt and [Pxqx) is the two-nucleon state vector consisting of "incoming" plane waves and "outgoing" scattered waves while (P2[ is a one-nucleon state vector with "outgoing" plane wave and "incoming" scattered waves. ( )+ is the time ordered product defined according to Wick ~0). The equivalence of (5) and (1) can be established b y the method of Lehmann, Symanzik and Zimmerman n) or b y the simpler approach of Low 3). j~(x) and J(y) are given b y t T h r o u g h o u t t h i s p a p e r , we s h a l l w r i t e a four v e c t o r a s x a n d a n o r d i n a r y v e c t o r b y t h e c o r r e s p o n d i n g b o l d l e t t e r x . T h e s c a l a r p r o d u c t of t w o four v e c t o r s x, y w i l l be d e n o t e d b y xy = x0Y0--x~y , = x o y o - - x , y. I n t e g r a t i o n s o v e r four d i m e n s i o n a l v o l u m e s a r e d e n o t e d b y dx, t h o s e o v e r t h r e e - d i m e n s i o n a l v o l u m e s b y d x . ?? W e s h a l l d i s t i n g u i s h H e i s e n b e r g o p e r a t o r s f r o m i n t e r a c t i o n r e p r e s e n t a t i o n o p e r a t o r s b y u s i n g b o l d t y p e for t h e former.
]02
ALLADI RAMAKRI SHNAN, N. R. RANGANATHAN A N D S. K. SRINIVASAN
( - []
+~,) ~
(x) =gCp (x)~,+..+-++,,,p(.) +
(iy, ~
~ ~ (x)- ~ (~)~, (x)~, (x) = j,, (x),
--m)~(x) ----gys~,¢(x)O,(x)4-~z~(x) = J(x).
(6) (7)
Next we use translational invariance: r(~) = e '~" r(o)e -'~" where F(x) is any field operator. This yields
.2E(~8)w(k) <~,1 {f~'+,,(jo(o)Je))+ d+,+im'+,,e(o)} i~q,>.
(P,f,k~lSl~g,) = --2=a@x-Fq~--p,--9,-a)~(¢,)
(8)
The second term is easily evaluated:
(Pnl¢(O)lp~qx>=
(o... (Sv(O))+b*(fl)O,,)
(9)
where S is the S-matrix and Op's are state vectors in interaction representation. Using the relations (see, for example, ref. 2,xn)) /~p(O),b,(fl) } = ~ w1 ( f l )
( ~ )m
+,
~s d~, (S,b*(q,)}= (z-~f e-"""(qO(~(~O)+~(x)
(10)
(~1)
we obtain @~[~(0)ltbxql) = w(fl)(E__~fl))i. 1 (2~)t
[<,,',D- ('... f ( &
,(0))+e-".'&+,..)]. (12)
The functional derivative of S with respect to ~(x) is related to the current
operator by J(x) =
~S i~-~)St,
~S J(z) = i~-~St
(13)
so that
f (~,., (~x) v(o)) +~,,) e-"'~ = -if(~,,,
= -i f<~.l(3(=)e(o))+l~,>e-'*"~
= --ifd.[--ft
e:"'<,.l~(o)J(x)l,.>~o
* fo o--,-<~,lJ(,)e(o)l~,>~o].
(14)
MESON
PRODUCTION
IN N U C L E O N - N U C L E O N
Expansion in terms of complete set of
1 (.:f(o.
~S
"incoming" states yields
+o.) e <#21.X(O)In>
<#~['P(o)In>
---
~ tt
P~=Px+qt
163
COLLISIONS
E(ql)+E(px)--E,,+i, + ~
Pn~Pl--qx
(15)
E(p2)--E(fl)--E,,+i,"
In a similar manner, we expand the first term of the right-hand side of (8):
<#2lj~(0)ln>
= --i ~
E(q2)+E_E(p~)_E(fl)_i~ + i ~
pn~pl-l-q~--qs Defining
Pn~P|+ql
pm*Spx+qt--qs
-
-
Z
E(qs)WE(p2)_E.+i e --gYsz'~,w(fl)~-~
P~Pt+qs X <#21~(O)]fJg'> <~']J(O)[#l> --
•
pm'mpvFq1
E(qx)+E~,)--E,,,,-q-ie
X ps'-~q~l--q1
(
<~/~[J(0)In'>
(,s)
E(p,)--E(q,)--E,,,--ie 13
where we have dropped the contribution due to the first term of the righthand side of (12) since it corresponds to the possibility where the nucleon of momentum Px goes unscattered, the final state meson being produced b y the other nucleon.
3. Tamm-Dancoff Approximation Equation (18) as it stands is very general in character. To make the problem simpler, we shall assume that the m and n states do not contain more than two particles. The motivation to make such a drastic assumption is twofold: (1) If the m and n states were to contain more than two particles, we are forced to introduce a separate equation similar to (18) for double meson production amplitudes, a circumstance which we wish to avoid.
164
ALLADI RAMAKRISHIqAN, N. R. RANGAIqATHAN AND S. K. SRINIVASAN
(2) Effects due to virtual pair creation are eliminated and this simplifies the evaluation of the vertex operator and other related quantities. Of course, here we are guided by the fact that pion nucleon scattering properties are known only for energies less than a GeV. Next we shall examine the various possibilities of m and n states: m state: The non-vanishing contribution comes from one-nucleon and one-nucleon + one-meson states. n state: The only state that contributes is the two-nucleon state. m' state: Here also only the two-nucleon state gives a contribution. n' state: The state that gives a contribution is the one-meson state. Thus (18) can be written as (P2qsk~lTl~xql) = ~ (q') [ (
(
J J E(
(q~w (k)) ½ [(#,lJ, (0)IPl+qx--q,)(lblq-ql--q,I J (0)[Pxqx)
(t,2lj=(O)lIYk'#)
q
~
~
. . q IJ,()l~bzql) 0 g I"
--,J| a
/ .
~e
~.
~e
---ied~'dk'6 ( p ' + k ' - - p , - - q 1 + q , )
...... aP aq o ( p , + q , - - p ' - - q ' ) 0
. . . .
0
(1
MESON P R O D U C T I O N I N N U C L E O N - N U C L E O N COLLISIONS
165
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
G. F. Chew, Phys. Rev. 94 (1954) 1748, 1755; 95 (1954) 1669 F. E. Low, Phys. Rev. 97 (1955) 1392 G. C. Wick, Revs. Mod. Phys. 27 (1955) 339 S. Gartenhaus, Phys. Rev. 100 (1955) 900 H. Miyazawa, Phys. Rev. 104 (1956) 1741 A. Klein and B. H. McCormick, Phys. Rev. 104 (1956) 1747 A. Klein, Prog. Theor. Phys. 20 (1958) 257 Yu. V. Novozhilov, Journ. Exp. Theor. Phys. (Trans.) 6 (1958) 692 P. S. Signell and R. E. Marshak, Phys. Rev. 109 (1958) 339 G. C. Wick, Phys. Rev. 80 (1950) 268 H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cimento I (1955) 205 A. A. Loganov and A. N. Tavkhalidze, Journ. Exp. Theor. Phys. (Trans) 5 (1957) 1134