- Email: [email protected]
Compositeness and vanishing of renormalization constants
~
Nuclear Physics B46 (1972) 497-504. North-Holland Publishing Company
COMPOSITENESS AND VANISHING OF RENORMALIZATION CONSTANTS J.N. PASSI Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, USSR
Received 23 February 1972
Abstract: It is demonstrated within the framework of a model field theory that the vanishing of wavefunction and vertex-function renormalization constants is not a sufficient condition to make an elementary particle equivalent to a three particle bound state. It is noticed that the Jin-MacDoweU cancellation for the three particle bound state does not hold good.
1. INTRODUCTION Many authors [1 ] have demonstrated within the framework of model field theories that under vanishing of wavefunction and vertex-function renormalization constants an elementary particle can be considered equivalent to a bound state. Similar conclusions [2] have been obtained for the equivalence between an unstable elementary particle and a resonant state. The above conclusions have been obtained for the case in which the bound state or the resonant state in question coltsists of two particles. The aim of this paper is to investigate under what conditions an elementary particle can be regarded as equivalent to a three particle bound state. To consider this problem we work within the framework of a soluble model field theory which consists of five particles U, V, N, 0 and 01 . The particles U, V and N are fixed in space but 0 and 01 can have nonrelativistic motion. We demonstrate that the irreducible part of the N001 ~N001 scattering amplitude develops a pole corresponding to an N001 bound state, this pole induces a pole in the reducible part of the scattering amplitude and these two poles do not cancel with each other i.e. the Jin-MacDowell [3] cancellation does not hold good for a three particle bound state. Conditions are found under which the U - particle can be regarded as equivalent to the N001 -bound state. It is noticed contrary to the two particle case that vanishing of wave function and vertex-function renormalization constants are not sufficient conditions for the equivalence between the U -particle and the N001-bound stare.
J.N. PassLRenormalization constants
498
2. MODEL FIELD THEORY The Hamiltonian of the model field theory under consideration is given by
H : m(~ fd3p ~ (p) flu(p) + m ( v 0 ) f d 3 p , ~ ( p ) +
+
f ~ - + / l ) ff+(k)if(k)+
X
6~
fd3k 1
v(p) + mNf d3qf~(q) fiN(q)
+//1) ~1 ( k l ) ~ 1 (kl)
l~dakd3kld3q dak'dgk: d3q' 6(3) (q + k + kl - q ' - k' _ k'l)
X ~q (q') ~ + ( k ' ) ~ (k'l) ~N(q ) ,~ (k) ~1
(kl)f(k)]i (kl)f(k')fl(k'l)
+ [ ~ g l 3 fd3p d3k I l~u(p) l/]V(p-kl)l~(kl)fl(kl)+h.c]
"(2~F
+[-Xl fdapd3 L(2n)3
d3,,
l~/~(p)~N(p_k_kl)¢(k)~l(kl)f(k)fl(kl)+h.c; (1)
The scattering amplitude for the process NO01 ~NO01 in the c.m.s, can be written as
T(E,k) =f2 (k)ff (k) r2(E,k) A U (E,k) + T 1 (E,k),
(2)
where T1 (E,k) is that part of the N001 -+ N00 scattering amplitude which contains n o contribution . . from the U -pamcle, . , Au(E,k )1 is the U .particle complete propagator and F(E,k)is the UN001 vertex function. T 1 (E,k) satisfies an integral equation which is represented diagramatically in fig. 1. The solution of this integral equation is discussed in the appendix and we find
T 1 (E,k) = D(E,k) + C(E,k), N.
O~ t-"
N
H
"n'O, O~ N
N N
O~ N
N
O~N
N
N
N
"~
-"
"" Oi
Fig. 1. Integral equation for the irreducible part of the N001 --~N001 scattering amplitude.
J.N. Passi,Renormalization constants
m ,
_
499
@
u (.~^RE) +
@ Fig. 2. Integral equation for the U - particle propagator. where D(E,k) and C(E,k) represent contributions from disconnected and connected Feynman diagrams respectively. The integral equation of the U particle complete propagator, in terms of the connected part C, is given in fig. 2. In these diagrams the thick lines represent complete.U - particle propagator and double lines represent complete V- particle propagator. Therefore the U- particle complete propagator is given by AU(Lw)= [ E - m
)-~(E)]-I,
(3)
Z(E)= g2 ~ fd3qlff(ql)Av(E-~lql)+~n)6fd3qd3qi (2,n.)~
+
~2g2 fd3ql d3/d31'f21 (ql)f2(1)f2(l ') A V (E (2,)9
f2(q)f2(ql) E-t-'Oq-COl q 1-m N
~Olq)
(E -¢Ol-(~lq I --raN) (E -~o l, -¢Olq 1--raN)
+ 2?~lgglfd3qd3ql f2(q)f2 (ql) (2706 E - ~ q - ~ l ql --mN
+
~2 '
fd3qd3qld3g'
(2,)6
3 ' d ql
f(q)f(q')fl(ql)fl(q'l) C(q:q'l;q'ql) (E-COq -¢Olq 1--raN) (E-~q ,--~lq~--mN)
+ Xlgg I fd3qd3ql d3q,d3q,1 f(q)f(q')fl(ql)f 1 (q'l) C(q', q'l;q,ql ) (2~.)6 (E-COq-~lq 1--mN) (E-COq ,-COlq ,--raN) X
1
1
,
,
d ql f(q)f(q')fl (ql)fl (q'l) C(a',ql;q,ql) (2~')6 (E-C°q-C°l q l --mN)(E--~ q'--~lq ~1--ran )(E-~lq 1-m(0)) (E-~lq; -m~))
J.N. Passi,Renormalization constants
500 where k2 60/(k) = ~
+ Lti/at•
¢
A V (E) is the complete V - particle propagator.
The renormalized mass, m U and the wave function renormalization constant Z U of the U - particle are defined by m U = m(uO) + Z ( m u ) ,
(4a)
z?j~ _ a zta [
(4b)
~E
] E=mu
Fig. 3. UN001 vertex function expressed in terms of the connected part of the irreducible part part of the N001 ~ N001 scattering amplitude. The UN001 - vertex function is represented in fig. 3. Therefore ~1
gl
P(E,k) = (27r)~ Pl(E,k) + ~ r
(5)
2 (E,k),
where P 1 (E,k)
2 f. d3qf2(q) = 1 +g-~--~A'v(E--w~ ) (27r)o
+ fd3q d 3 q l
-
...
aE-%-%~-mN
f(q)fl (q 1) f - l(k) f l 1(k t ) C(q,ql ; k) E-Wq-COlq 1- m N
%(E,k)-- g~ av(E-%k) ' + g (2rr)~
f d3 q d3 q l f(q)fl (ql ) f - l ( k )f l 1 C(q'q l ;k) (g--('dq--~'Olq I --mN )(E--6Olq I --m(vO))
J.N. Passi,Renormalization constants
501
The UN001 and UV01 vertex renormalization constants are defined by zll
= Pl (E,k)
Z21 = r 2
The renormalized
(E,k)
E = m U'
(6a)
F. = m U
(6b)
UN001 and UV01 coupling constants
are given by
Xlr = ZkU Z l l ~'1'
(7)
1
glr = Z ~ Z21 gl"
3. POLES OF THE T(E,k)
T(E,k) the
total scattering amplitude for
NO01-->NO01 is given by
T(E,k) = f2(k)fl2 (k) r 2 (E,k) A U (E,k) + D(E,k) +C(E,k).
(8)
The expression (A5) indicates that C can develop a pole for sufficiently large g and corresponding to the N001 bound state, i.e.
d(E = roB) = 0 where m B is the mass of the N001 bound state. Eqs. (3) (5) and (2) indicate that this pole induces a pole in the reducible part of the scattering amplitude at E = m B. It can easily be checked that R 1 ¢ - R 2,
(9)
where R 1 is the residue* of the pole of the irreducible part at E = m B and R 2 is the residue of the induced pole of the reducible part at E = m B. It implies that the pole of T1 the irreducible part of the scattering amplitude does contribute to the total scattering amplitude contrary to the assertion made by Jill and MacDowell [3] for 2 particle ~ 2 particle scattering amplitude. The pole structure of T(E,k) is given by 1~___..__ + glr "] 2
r(e,k)= t.(2rr) 3 (2~r)~J f 2 ( k ) f ? ( k ) E - mU
¢$
.
.
.
R1 R2 + - - + ~ + E - mB E - m B
......
(10)
The exphclt expressxon for R 1 can be obtained from eq. (A1) and for R2 from eqs. (3) (5) and (2).
J.N. Passi, Renormalizalion constants
502
4. EQUIVALENCE OF U- PARTICLE AND N001 - BOUND STATE The conditions under which the U particle pole disappears and the N001 bound state pole replaces the U- particle pole are a.
mB
~ mu,
b.
R1
[-Xlr glr ~ ~ + L(27r) 3 +(27r)~
2
f2(k)f?(kl)' 2
c.
_
R2
[-~lr glr ] [_(27r)3 + (27r)~
f 2 ( k ) fl 2 (k 1).
Eqs. (6a and (6b) indicate that the condition (a) implies Z 1 = 0 and Z 2 = 0. A little algebra indicates that the condition (c) is equivalent to lim. Z U = 0. Z 1 "+0
z2~o Condition (b) is completely independent o f Z 1 = 0, Z 2 = 0 and Z U =. 0 and is not satisfied under Z 1 = 0, Z 2 = 0 and Z U = 0. We, therefore, conclude that Z 1 = 0, Z 2 = 0 and Z U = 0 are not sufficient conditions for the equivalence of the U - particle and the N001 bound state.
The author would like to thank Prof. T. Pradhan for his interest in this work. He would also like to thank the International Atomic Energy Agency for financial assistance and the Joint Institute for Nuclear Research for hospitality.
APPENDIX The integral equation for T 1 (E,k) represented diagramatically in fig. 1 is
,, T 1 (k,k 1 ;k,k 1) =
, g2 ~---~-Tf(k')f,( k l ) f ( k ) f 1 (k 1) + (2n)o
+ (270 6X
f d3q d3ql
+ g2 fd3q (2")3
~
(21r)3
8(3)(kl-k'l)f(k)f(k' ) E-Wlk 1- m v ( 0 )
f(q)fl (ql) f ( k ) f l (kl)Tl (k ;k ~ ;q, ql ) E-Wq-Wlq 1-m N
f(k)f(q)oT1 (k"k'l'q'kl) ' (E--C°lk 1-mV( )) (E-¢Oq-COlk 1- m s )
(A.1)
J.N. Passi, R enormalization constants
503
where k,k I and k', k'1 represent momenta of incoming and outgoing 0,01" Here momentum dependence is written explicitly for the sake of keeping track of different momenta while solving the integral equation. To solve this integral equation we make use of Weinberg's method of decomposing T1 the irreducible part of N001~ N001 scattering amplitude, into disconnected part and connected part, and then solving the integral equation for the connected part; (A.2)
T 1 (k',k' 1 ;k,kl) = D(k',k' 1 ;k,kl) + C(k; k' 1 ;k,kl) where the disconnected part t
(A.3)
D(k;k' 1 ;k,k 1) = 6(3)(k 1-k'l) Av(E)
and a little manipulation shows that the connected part satisfies an integral equation as represented digramatically in fig. 4. Therefore we have )t Jd£-3qd3ql f ( k ) f l ( k l ) f ( q ) f l ( q l ) C(k"k'l;q'ql) C(k,k . 1 ;k, . kl) . =. A (k,k 1 ;k,kl) + ,-..,~6 E-co,~-COlq I ~m N
+
-
X -
fd3l t (l,k)
fd3q d3qlf(l)f(q)f 1 (kl)f1(ql) C(k',k'l ;q,ql )
(2n) 6 E - - w l - W l k x - m N
E-COq --~lq 1 --m N (A.4)
where
A (k',k'1 ;k,k 1) = ~ f(k')fl (k'l)f(k)fl (kl) -Ifd3 q d3 qlf(k)fl (kl )f(q)fl(k' 1) (2rr) 6 (2~)" X
t(q,k) E-~ ~-~I~ ' -m
+
N
X fd3q f(k)fl(kl)f(q)fl(k'l) t(k;q) E-(°q-~°lk' 1- m N
(2rr) 6
+ X (2rr)6
fd3q
f(q)fl ( k l ) f ( q ' ) f l (k'l) t(q,k) t(k',q') d3ql ,(E-~ q_COlk I - m N) (E-~q'--COl k ,1 --m N '
Fig. 4. Integral equation for the connected part of the irreducible part of the N001 "* N001 scattering amplitude.
J.N. Passi,Renormalization constants
504 and 2
t (l,k) = ~ f ( l ) f ( k )
A v.
(2~) ~
The solution of the integral equation can be obtained by method of iteration, the final result obtained is
C(k', k I ; k,k 1) = A (k,k 1 ;k,k 1) + ?t
fd3l t (l,k)
(2rr)6
E-co/
COlk1
rnN
3q d3ql
f ( k ) f l ( k l ) f ( q ) f l ( q l ) A (k,,k,1
E--COq-Wlq 1--my
f f d3q d3ql f(l)fl(kl)f(q)fl(ql)A(k:k'l E-COq COlql-m N
X d - 1 (E),
;q'ql ) ;q'ql )] (A.5)
with a(E)=
~_ _
X
(2~) 6
fd3qd3ql
-(22)6fd3qd3ql
f2(q)f?(q 1) E-COq-COlq x -m N
f(q)fl2(ql ) fdal Y(Z,qly(q q l~--COq ~iq 1 E-C°l-C°lql-mN j
REFERENCES [1] P.E. Kaus and F. Zachariasen, Phys. Rev. 138 (1965) 1304; T. Saito, Phys. Rev. 152 (1966) 1339; T. Pradhan and J.N. Passi, Phys. Rev. 160 (1967) 1336; J.M. Cornwall and D. Levy, Phys. Rev. 178 (1968) 2356. [2] J.N. Passi, Phys. Rev. 2 (1970) 310. [3] Y.S. Jin and S.W. MacDowell, Phys. Rev. 137 (1965) 688. [4] S. Weinberg, Phys. Rev. 133 (1964) B 232.
Nuclear Physics B46 (1972) 497-504. North-Holland Publishing Company
COMPOSITENESS AND VANISHING OF RENORMALIZATION CONSTANTS J.N. PASSI Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, USSR
Received 23 February 1972
Abstract: It is demonstrated within the framework of a model field theory that the vanishing of wavefunction and vertex-function renormalization constants is not a sufficient condition to make an elementary particle equivalent to a three particle bound state. It is noticed that the Jin-MacDoweU cancellation for the three particle bound state does not hold good.
1. INTRODUCTION Many authors [1 ] have demonstrated within the framework of model field theories that under vanishing of wavefunction and vertex-function renormalization constants an elementary particle can be considered equivalent to a bound state. Similar conclusions [2] have been obtained for the equivalence between an unstable elementary particle and a resonant state. The above conclusions have been obtained for the case in which the bound state or the resonant state in question coltsists of two particles. The aim of this paper is to investigate under what conditions an elementary particle can be regarded as equivalent to a three particle bound state. To consider this problem we work within the framework of a soluble model field theory which consists of five particles U, V, N, 0 and 01 . The particles U, V and N are fixed in space but 0 and 01 can have nonrelativistic motion. We demonstrate that the irreducible part of the N001 ~N001 scattering amplitude develops a pole corresponding to an N001 bound state, this pole induces a pole in the reducible part of the scattering amplitude and these two poles do not cancel with each other i.e. the Jin-MacDowell [3] cancellation does not hold good for a three particle bound state. Conditions are found under which the U - particle can be regarded as equivalent to the N001 -bound state. It is noticed contrary to the two particle case that vanishing of wave function and vertex-function renormalization constants are not sufficient conditions for the equivalence between the U -particle and the N001-bound stare.
J.N. PassLRenormalization constants
498
2. MODEL FIELD THEORY The Hamiltonian of the model field theory under consideration is given by
H : m(~ fd3p ~ (p) flu(p) + m ( v 0 ) f d 3 p , ~ ( p ) +
+
f ~ - + / l ) ff+(k)if(k)+
X
6~
fd3k 1
v(p) + mNf d3qf~(q) fiN(q)
+//1) ~1 ( k l ) ~ 1 (kl)
l~dakd3kld3q dak'dgk: d3q' 6(3) (q + k + kl - q ' - k' _ k'l)
X ~q (q') ~ + ( k ' ) ~ (k'l) ~N(q ) ,~ (k) ~1
(kl)f(k)]i (kl)f(k')fl(k'l)
+ [ ~ g l 3 fd3p d3k I l~u(p) l/]V(p-kl)l~(kl)fl(kl)+h.c]
"(2~F
+[-Xl fdapd3 L(2n)3
d3,,
l~/~(p)~N(p_k_kl)¢(k)~l(kl)f(k)fl(kl)+h.c; (1)
The scattering amplitude for the process NO01 ~NO01 in the c.m.s, can be written as
T(E,k) =f2 (k)ff (k) r2(E,k) A U (E,k) + T 1 (E,k),
(2)
where T1 (E,k) is that part of the N001 -+ N00 scattering amplitude which contains n o contribution . . from the U -pamcle, . , Au(E,k )1 is the U .particle complete propagator and F(E,k)is the UN001 vertex function. T 1 (E,k) satisfies an integral equation which is represented diagramatically in fig. 1. The solution of this integral equation is discussed in the appendix and we find
T 1 (E,k) = D(E,k) + C(E,k), N.
O~ t-"
N
H
"n'O, O~ N
N N
O~ N
N
O~N
N
N
N
"~
-"
"" Oi
Fig. 1. Integral equation for the irreducible part of the N001 --~N001 scattering amplitude.
J.N. Passi,Renormalization constants
m ,
_
499
@
u (.~^RE) +
@ Fig. 2. Integral equation for the U - particle propagator. where D(E,k) and C(E,k) represent contributions from disconnected and connected Feynman diagrams respectively. The integral equation of the U particle complete propagator, in terms of the connected part C, is given in fig. 2. In these diagrams the thick lines represent complete.U - particle propagator and double lines represent complete V- particle propagator. Therefore the U- particle complete propagator is given by AU(Lw)= [ E - m
)-~(E)]-I,
(3)
Z(E)= g2 ~ fd3qlff(ql)Av(E-~lql)+~n)6fd3qd3qi (2,n.)~
+
~2g2 fd3ql d3/d31'f21 (ql)f2(1)f2(l ') A V (E (2,)9
f2(q)f2(ql) E-t-'Oq-COl q 1-m N
~Olq)
(E -¢Ol-(~lq I --raN) (E -~o l, -¢Olq 1--raN)
+ 2?~lgglfd3qd3ql f2(q)f2 (ql) (2706 E - ~ q - ~ l ql --mN
+
~2 '
fd3qd3qld3g'
(2,)6
3 ' d ql
f(q)f(q')fl(ql)fl(q'l) C(q:q'l;q'ql) (E-COq -¢Olq 1--raN) (E-~q ,--~lq~--mN)
+ Xlgg I fd3qd3ql d3q,d3q,1 f(q)f(q')fl(ql)f 1 (q'l) C(q', q'l;q,ql ) (2~.)6 (E-COq-~lq 1--mN) (E-COq ,-COlq ,--raN) X
1
1
,
,
d ql f(q)f(q')fl (ql)fl (q'l) C(a',ql;q,ql) (2~')6 (E-C°q-C°l q l --mN)(E--~ q'--~lq ~1--ran )(E-~lq 1-m(0)) (E-~lq; -m~))
J.N. Passi,Renormalization constants
500 where k2 60/(k) = ~
+ Lti/at•
¢
A V (E) is the complete V - particle propagator.
The renormalized mass, m U and the wave function renormalization constant Z U of the U - particle are defined by m U = m(uO) + Z ( m u ) ,
(4a)
z?j~ _ a zta [
(4b)
~E
] E=mu
Fig. 3. UN001 vertex function expressed in terms of the connected part of the irreducible part part of the N001 ~ N001 scattering amplitude. The UN001 - vertex function is represented in fig. 3. Therefore ~1
gl
P(E,k) = (27r)~ Pl(E,k) + ~ r
(5)
2 (E,k),
where P 1 (E,k)
2 f. d3qf2(q) = 1 +g-~--~A'v(E--w~ ) (27r)o
+ fd3q d 3 q l
-
...
aE-%-%~-mN
f(q)fl (q 1) f - l(k) f l 1(k t ) C(q,ql ; k) E-Wq-COlq 1- m N
%(E,k)-- g~ av(E-%k) ' + g (2rr)~
f d3 q d3 q l f(q)fl (ql ) f - l ( k )f l 1 C(q'q l ;k) (g--('dq--~'Olq I --mN )(E--6Olq I --m(vO))
J.N. Passi,Renormalization constants
501
The UN001 and UV01 vertex renormalization constants are defined by zll
= Pl (E,k)
Z21 = r 2
The renormalized
(E,k)
E = m U'
(6a)
F. = m U
(6b)
UN001 and UV01 coupling constants
are given by
Xlr = ZkU Z l l ~'1'
(7)
1
glr = Z ~ Z21 gl"
3. POLES OF THE T(E,k)
T(E,k) the
total scattering amplitude for
NO01-->NO01 is given by
T(E,k) = f2(k)fl2 (k) r 2 (E,k) A U (E,k) + D(E,k) +C(E,k).
(8)
The expression (A5) indicates that C can develop a pole for sufficiently large g and corresponding to the N001 bound state, i.e.
d(E = roB) = 0 where m B is the mass of the N001 bound state. Eqs. (3) (5) and (2) indicate that this pole induces a pole in the reducible part of the scattering amplitude at E = m B. It can easily be checked that R 1 ¢ - R 2,
(9)
where R 1 is the residue* of the pole of the irreducible part at E = m B and R 2 is the residue of the induced pole of the reducible part at E = m B. It implies that the pole of T1 the irreducible part of the scattering amplitude does contribute to the total scattering amplitude contrary to the assertion made by Jill and MacDowell [3] for 2 particle ~ 2 particle scattering amplitude. The pole structure of T(E,k) is given by 1~___..__ + glr "] 2
r(e,k)= t.(2rr) 3 (2~r)~J f 2 ( k ) f ? ( k ) E - mU
¢$
.
.
.
R1 R2 + - - + ~ + E - mB E - m B
......
(10)
The exphclt expressxon for R 1 can be obtained from eq. (A1) and for R2 from eqs. (3) (5) and (2).
J.N. Passi, Renormalizalion constants
502
4. EQUIVALENCE OF U- PARTICLE AND N001 - BOUND STATE The conditions under which the U particle pole disappears and the N001 bound state pole replaces the U- particle pole are a.
mB
~ mu,
b.
R1
[-Xlr glr ~ ~ + L(27r) 3 +(27r)~
2
f2(k)f?(kl)' 2
c.
_
R2
[-~lr glr ] [_(27r)3 + (27r)~
f 2 ( k ) fl 2 (k 1).
Eqs. (6a and (6b) indicate that the condition (a) implies Z 1 = 0 and Z 2 = 0. A little algebra indicates that the condition (c) is equivalent to lim. Z U = 0. Z 1 "+0
z2~o Condition (b) is completely independent o f Z 1 = 0, Z 2 = 0 and Z U =. 0 and is not satisfied under Z 1 = 0, Z 2 = 0 and Z U = 0. We, therefore, conclude that Z 1 = 0, Z 2 = 0 and Z U = 0 are not sufficient conditions for the equivalence of the U - particle and the N001 bound state.
The author would like to thank Prof. T. Pradhan for his interest in this work. He would also like to thank the International Atomic Energy Agency for financial assistance and the Joint Institute for Nuclear Research for hospitality.
APPENDIX The integral equation for T 1 (E,k) represented diagramatically in fig. 1 is
,, T 1 (k,k 1 ;k,k 1) =
, g2 ~---~-Tf(k')f,( k l ) f ( k ) f 1 (k 1) + (2n)o
+ (270 6X
f d3q d3ql
+ g2 fd3q (2")3
~
(21r)3
8(3)(kl-k'l)f(k)f(k' ) E-Wlk 1- m v ( 0 )
f(q)fl (ql) f ( k ) f l (kl)Tl (k ;k ~ ;q, ql ) E-Wq-Wlq 1-m N
f(k)f(q)oT1 (k"k'l'q'kl) ' (E--C°lk 1-mV( )) (E-¢Oq-COlk 1- m s )
(A.1)
J.N. Passi, R enormalization constants
503
where k,k I and k', k'1 represent momenta of incoming and outgoing 0,01" Here momentum dependence is written explicitly for the sake of keeping track of different momenta while solving the integral equation. To solve this integral equation we make use of Weinberg's method of decomposing T1 the irreducible part of N001~ N001 scattering amplitude, into disconnected part and connected part, and then solving the integral equation for the connected part; (A.2)
T 1 (k',k' 1 ;k,kl) = D(k',k' 1 ;k,kl) + C(k; k' 1 ;k,kl) where the disconnected part t
(A.3)
D(k;k' 1 ;k,k 1) = 6(3)(k 1-k'l) Av(E)
and a little manipulation shows that the connected part satisfies an integral equation as represented digramatically in fig. 4. Therefore we have )t Jd£-3qd3ql f ( k ) f l ( k l ) f ( q ) f l ( q l ) C(k"k'l;q'ql) C(k,k . 1 ;k, . kl) . =. A (k,k 1 ;k,kl) + ,-..,~6 E-co,~-COlq I ~m N
+
-
X -
fd3l t (l,k)
fd3q d3qlf(l)f(q)f 1 (kl)f1(ql) C(k',k'l ;q,ql )
(2n) 6 E - - w l - W l k x - m N
E-COq --~lq 1 --m N (A.4)
where
A (k',k'1 ;k,k 1) = ~ f(k')fl (k'l)f(k)fl (kl) -Ifd3 q d3 qlf(k)fl (kl )f(q)fl(k' 1) (2rr) 6 (2~)" X
t(q,k) E-~ ~-~I~ ' -m
+
N
X fd3q f(k)fl(kl)f(q)fl(k'l) t(k;q) E-(°q-~°lk' 1- m N
(2rr) 6
+ X (2rr)6
fd3q
f(q)fl ( k l ) f ( q ' ) f l (k'l) t(q,k) t(k',q') d3ql ,(E-~ q_COlk I - m N) (E-~q'--COl k ,1 --m N '
Fig. 4. Integral equation for the connected part of the irreducible part of the N001 "* N001 scattering amplitude.
J.N. Passi,Renormalization constants
504 and 2
t (l,k) = ~ f ( l ) f ( k )
A v.
(2~) ~
The solution of the integral equation can be obtained by method of iteration, the final result obtained is
C(k', k I ; k,k 1) = A (k,k 1 ;k,k 1) + ?t
fd3l t (l,k)
(2rr)6
E-co/
COlk1
rnN
3q d3ql
f ( k ) f l ( k l ) f ( q ) f l ( q l ) A (k,,k,1
E--COq-Wlq 1--my
f f d3q d3ql f(l)fl(kl)f(q)fl(ql)A(k:k'l E-COq COlql-m N
X d - 1 (E),
;q'ql ) ;q'ql )] (A.5)
with a(E)=
~_ _
X
(2~) 6
fd3qd3ql
-(22)6fd3qd3ql
f2(q)f?(q 1) E-COq-COlq x -m N
f(q)fl2(ql ) fdal Y(Z,qly(q q l~--COq ~iq 1 E-C°l-C°lql-mN j
REFERENCES [1] P.E. Kaus and F. Zachariasen, Phys. Rev. 138 (1965) 1304; T. Saito, Phys. Rev. 152 (1966) 1339; T. Pradhan and J.N. Passi, Phys. Rev. 160 (1967) 1336; J.M. Cornwall and D. Levy, Phys. Rev. 178 (1968) 2356. [2] J.N. Passi, Phys. Rev. 2 (1970) 310. [3] Y.S. Jin and S.W. MacDowell, Phys. Rev. 137 (1965) 688. [4] S. Weinberg, Phys. Rev. 133 (1964) B 232.