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On the equivalence of elementary and composite particles
ANNALS
On
OF PHYSICS:
The
B. W.
26, 466-477
Equivalence LEE,
Department
K. T.
(1964)
of Elementary
Composite
I. S. GERSTEIN,
MAHANTHAPPA,
of Physics,
and
University
of Pennsylvania,
AND
M. L. WHIPPMAN
Philadelphia,
Equivalence between composite particles, defined as the zeroes tion of the N/D method, and elementary particles with vanishing renormalization constants has been proved in the approximation ing only two particle intermediate states. A precise formulation method is presented to facilitate the comparison with field theory. ing of the wave function renormalization constant is formulated Lehmann representation. We are led to a detailed study of the the proper vertex part in order to establish the equivalence of procedures. I. INTRODUCTION
AND
Particles*
Pennsylvania
of the D-funcwave function of retainof the N/D The vanishin terms of the properties of the above two
SUMMARY
Recent investigations of some field theoretic models (1, 2) have shown the possibility of a meaningful distinction between elementarity and compositeness of a particle, and the equivalence of those two concepts in the limit in which the wave function renormalization constant Z3 goes to zero. It has been suggested by Salam (3) that this equivalence continues to hold in a full field theory. If this notion were true then the suggestion of Chew and Frautschi (4) that all particles are equally composite (or elementary) could be materialized in a field theory through the vanishing of all of their wave function renormalization constants. There have been attempts (see, for example, ref. 5) to prove this equivalence within the framework of the N/D method (6). In the N/D method, a composite particle corresponds to a zero of the D function which satisfies the usual analyticity requirements and a certain asymptotic condition for large values of energy. Some authors (7) have attempted to show that the above circumstance arises precisely when the renormalization constant Z3 of the particle is zero. In particular, Rockmore considered the usual first order determinental approximation for the denominator function and was then forced to make the simplifying but unjustified approximation of approximating the numerator function by a constant in his equation for the residue of the bound state pole. Having done this, he finds that the residue equation implies 23 = 0 to second order. However. * Work
supported
in part
by the
U. S. Atomic 466
Energy
Commission.
EQUIVALENCE
OF
ELEMENTARY
AND
COMPOSITE
467
PARTICLES
these approximations are exact in the Zachariasen model (8) and thus Rockmore’s proof reduces to the treatment of this model by Dowker (1). In addition, a claim is made that the residue equation is equivalent to one given by BGKT (9) but this is not apparent. The purpose of this paper is to prove the equivalence between bound states (as defined by the N/D method as above) and elementary particles with their wave function renormalization constants set equal to zero in a full field theory, making only the approximation of neglecting all but two particle intermediate states. We differ from previous authors in that we make no assumptions about the nature of interactions, except requiring a relatively mild condition which amounts to saying that the theory be not too singular. At this point we may recall that Fubini has questioned lately the validity of the usual association of compositeness and the condition Z 3 = 0 in the case of singular theories (10). We shall make use of various well-known results of renormalized field theory, rather than having recourse to perturbation theory. In the elastic unitarity approximation it will be shown that the equation for bound state residues derived in BGKT for potential theory is identical (except for some kinematical factors) to the one obtained by setting Z3 = 0. In the next section we shall summarize the content of the N/D method in terms of the phase integral a, with particular emphasis on the asymptotic behavior of 3 for high energies. In Section III, we study the Lehmann representation of the propagator which in the two particle approximation yields an expression for Z3 in terms of the vertex function. The relation between the vertex function and the a function on the one hand, and the connection between the asymptotic behavior of %J and the existence of a bound state on the other imply the equivalence of Z. = 0 and the existence of a zero in the D-function. In order to show the identity of the residue equation in the N/D method and the condition Z3 = 0, we are led to study analytic properties of the proper vertex part. Here our results are compared to the similar results in potential theory. Section IV contains some concluding remarks. II.
N/D
METHOD
Our basic assumption in both field theory and the N/D method (6) is that the partial wave T-matrix for two-particle (B and C) scattering is described by a dispersion relation. We specialize to the s-wave, and assume that there is a particle A of mass m having the quantum numbers of this channel. We write the dispersion relation in the form (6) A(s) = a + ‘+
O” ds’ Im A(s’) + s - so ds’w( s’) s sI (s, _ s)(s, - so) __ lr s L (s’ - s)(s’ - so) 3
-s -
s - so ___ m2 m2 -
SO
(1)
468
LEE
ET
AL.
where A(s)
= sin 6(s>ei*‘“‘/p(s)
(2)
w(s) is the discontinuity across the left hand cut L, p(s) phase space factor with the property that limp(s) s-00
is the appropriate
= const.,
6(s) is the phase shift for the s-wave, s is the square of the c.m. total energy, sl = (mB + mc)2, and a is the subtraction constant at s = so . We define the phase integral a)(s) by
and the quantities
K(s,s’)= [~(s)P(s’)l”2 - s,) 7r J,(s,,14”‘;;;‘s”“’ r](s)=g-+ a+ sy sr, (s’ Warnock
ds‘w( s’) - s)(s’ - so)
1
(4)
(11) has shown that if the conditions’ ds’(K(s,
s’))~
< cc
and OD dsh(4)2 $81
< 00
are satisfied, and if 6(s) satisfies the Holder condition lima)(s) 8-m where cr is a nonpositive integer. logarithmic factors.” Since there 0 and -1.2 Under the condition neither zeroes nor poles. In the N/D method, the most
in any finite interval,
= Q(s~)
then (5)
The symbol Q means “of order of, disregarding is only one pole in Eq. (1) a! is restricted to stated above, it can be shown that B(S) has general form of the D-function
may be written
1 These conditions imply that the N/D equation (see Eqs. (9) and (10) below) can be transformedinto a Fredholm type. These conditions follow from Warnock @I), Eq. (2.10), by putting 11= 1 (no inelasticity). 2 This statement
is equivalent
to Levinson’s
theorem generalized
to our case. See (11).
EQUIVALENCE
OF
ELEMENTARY
AND
COMPOSITE
PARTICLES
469
5tS
D(s)
=
rl[
ts
-
Si>
$
(s
_
sj)
as>
(6)
and N(s)
= A(s)
D(s)
(7)
In order to identify a zero of D(s) with a bound state, it is necessary to specialize to a particular form of (6). We require that D(s) have no poles and N(s) and D(s) have no common zeroes, and’ lim D(s) s-em
= Q(l)
(81
Since B(S) has no zeroes the polynomial in the denominator on the right hand side of Eq. (6) should be absent. The degree of the polynomial in the numerator is fixed by (8) and the requirement that N and D have no common zeroes. We may now distinguish between two cases: Case I. B(s) = Q(1). In this case D(s) = ~~o(s)/B(a) (we normalize D(s) such that D(G) = 1) and D(s) N(s)
= l - s+ = a + ‘+
O1 ds’p(s’)N(s’) s,, ($ - s)(s’ _ so) w(s’)D(s’) g2D(m2> s - so / as (s, _ s)(s, _ so> - ~ ___ s - m2 m2 - s0 L
In this case g” can be chosen arbitrarily. Case II. a)(s) = Q(s-‘), and D(s) = (s - m”)D(s)/(s, - m”>B(s0>; m ,&‘W(s’) D(s) = 1 - ‘%?? ds’ 7r $81 (s’ - s) (s’ - so) N(s)
= a + ‘9
w( s’)D( s’) j- ds’ (s, _ s> (s, _ so> L
(9)
(10)
and
N(m2)_ ” = (dD(s)/ds>,,z D(m’>
(11)
= 0
3 These conditions imply that there be no CDD poles (19) in the D-function. We may note that if Eqs. (19) and (20) below are transformed into a Fredholm type, and solved by the usual method, then D does not have any CDD poles.
470
LEE
ET
AL.
In this case N(s) no longer has a pole. Note that if si were chosen at some point other than &, then N and D would have a common zero. The cases I and II correspond, respectively, to the elementary particle and bound state poles at s = m2. III.
FIELD
THEORY
A. PROPAGATOR The Lehmann representation (12) for the propagator for particle A is, in the two-particle approximation A,(s) =
(12)
In the subsequent discussionsthroughout, we shall assume that the Lehmann representation (12) is valid. The matrix element (0 1$A(0)j BC)i, may be written4
(0 ] b(O) 1BC)c = A2
F(s) = A,(s>I’(s)
(13)
where F(s) is the form factor of the particle A, and I’(s) is the renormalized proper vertex part of Dyson. We have the dispersion relation for F(s) :
F(s) =g+s+ SW
ds’ Im F(s’) 81 (s’ - s) (s’ - my
(14)
We make the explicit assumption that F(s) has no zeroes. Since Im F(s) = sin 6(s) in our approximation, F(s) may be written by the usual F*Wi6’“) technique (13) as F(s) = gB-‘(s)
(15)
The assumptions that Eqs. (12) and (14) are valid without further subtractions imply that there are enough powers of 1log s / in the asymptotic behavior of a>(s) when D(S) = Q(s-‘), to ensure the convergence of integrals in these equations, and similar expressionsbelow. Substitution of Eq. (15) into Eq. (12), and the use of the definition of 2,’ = lim,,, sA,(s) (12) give
4 Equation (13) may be derived by the technique electromapnetic vertices.
of (16). See also (17) in the case of
EQUIVALENCE
OF ELEMENTARY
AND COMPOSITE
PARTICLES
471
It is also useful to consider the inverse of the propagator (12)
Aye = (s_ m2)1 s - mz O”ds’ds’)IUs’)I2 n- ss1(s- s’)(s’- m)” [
(171
1
s - m2 +Cs - s, sz - m2 The last term in the bracket is present if Aa seen from (17) to be
has a zero at s = sZ. Z3 may be
(18) B. PROPER VERTEX
PART
We shall consider the properties of the renormalized vertex part r(s) in some detail. From (13) and (17) we have r(s) = F(s)
[
1 + (s - m2) SW d* 7r Sl
(s, “,,),
I&)
I’]-’
(19)
From (13) and (12) we have
F(S) = r(s)
[
i - ‘+
1r(d) 1’ + A-/ md~‘p(SI) s (s, _ m2>2 s 81
e]-’ -
s1 ss
(19’)
mz
Defining J?(s) = g/C(s), we get
zo(s) = C(s)
2
d&(i) (s - s’)(s’ - m2)2
1
s - m2 +C-s - sss, - m2
(20’)
From (20 ), we seethat (a) C(s) can be continued to the entire s-plane cut along the real axis from s1to ~0,and is analytic there. (b) C(m”) = 1. From (20’) we see that if c # 0 in Eq. (17), then C(s) must have a zero at s = sZ. Consequently I’(s) has a pole at s = sE. This point has been emphasized by Goebel and Sakita (14) in another context. First we shall confine ourselves to the case c = 0. We define
jqs) = _ as + ie>- as - ie) %3(s)
(21)
452
LEE
with s on the right-hand we have
ET
AL.
cut, and by analytic
continuation
elsewhere.
From (20)
(s- s')(s' - m") ds’p( i)
where we have defined 31(s) by s-L(s) = for s on the right-hand
iD(s + i,) - a(s %(s)
cut, and by analytic A(s)
- iE)
continuation
elsewhere.
Also
= x(s)+(s)
We can show that (a) M(s) is analytic everywhere in s except on the left-hand cut (the apparent right-hand cuts and poles at s = m” cancel) ; (b) on the left-hand cut Im M(s)
= w( s)9( S)
[
12 - m2)2 iiS(iT) ! II
1+ +
g2 S,, (s _ $~~~,s’)
= w(s>C(s) This shows that B(s) is a unitary relation B(s)
= !!-I!! T
amplitude,
= M(s)/C(s)
i.e., Im B(s)
(22)
= p(s) 1B( s)l” satisfying
the dispersion
1
d&( s’) - dip(d) /B(i) I2 (s’ - s)(s’ - so) + s,, (s’ - s)(s’ - so) +bSL-
The last term is present if c # 0 in (21). The constant b = Mso) =a l-g2 C(so> {
[
(so - m2)~(so)
x
r,,
( (&
s - SIJ s - s* s, - so
(23)
b is related to a by
1 + ‘YE-& li-d;y(y,
m2)
/5&‘} g
(24)
EQUIVALENCE OF ELEMENTARY AND COMPOSITE PARTICLES
473
From (20’) andthe definitionsof ‘X(S) and ill(s), we seethat A(s) = xWa>(s>
s -
l--
a
[
= B(s)
m2
m s81 (s -
ds’p( s’) s’)(s’ - nay
2-l (25)
I II &)
+ A,(s)[ir(s)l”
The meaning of the decomposition of A(s) in (25) can be seen as follows. The first part, B(s), is the unitary amplitude constructed without the A particle pole at s = m”. The second term is the contribution of the particle A (in the Feynman-Dyson sense) renormalized by the rest of the interactions. In Eq. (24) the subtraction constant b differs from a since b should not include the renormalization effect from the particle A, Since B(s) is unitary, we may write for s 2 s1 , “(‘) (‘) sin scO)(S) B(s) = e (26) P(S)
Then we see that s%(Q)(8) ZZ---I as> I2 e (C(s)>”
(27)
and the amplitude -g2 (C(s) (s
-
m”)
[-’
1 _ s - m2 m s *1 (8 n[
dip( s’) -
s’)(s’
-
2-l
I II 9
. Ge+
m2)2 C(d)
(28)
is also unitary. Now combining Eqs. (25), (26), (27), and (28), we seethat “(8) sin 6(s) = e”(‘)(s) sin *‘O’(s) 2i6(@) weicm sin (y (s) A(s) = e +e (29) PCS>
P(S)
P(S)
and 6 = So+ cr. The above decomposition of the T-matrix is well-known: it has been discussed by Drell and Zachariasen (15) and appears naturally when the interactions can be split up into two parts. Two well-known examples of Eq. (29) are the interference of the Coulomb effect and the nuclear effect (where 6’ is the Coulomb phase shift and LYthe nuclear phaseshift) and the interference between the “potential” scattering and the Breit-Wigner resonance. What we have said above may be illustrated in a version of the Zachariasen model (2). We consider the ABC coupling as well as B2C2coupling (Fig. 1(a) ). The amplitude A(s) is the sum of all iterations of these two interactions (Fig. l(b)). The proper vertex part is given by the diagrams in Fig. 2(a) and B(s) is the sum of all diagrams containing only the B2C2coupling (Fig. 2(b)).
+ >o( +--lb) FIG. 1. (a) The basic interactions scattering amplitude.
of the Zachariasen
+
model;
(b) the expansion
of the
---
(a)
8(.)=X + )o( + )oo( +--FIG. 2. (a) The expansion of the proper vertex part; (b) the expansion 474
of the B-amplitude
EQUIVALEKCE
OF
IV.
ELEMENTARY
Z8 = 0 AND
First we consider the case I in Eq. (16) converges for this In the bound state case the Z3 = 0 in Eq. (18) determines g2=
AND
COMPOSITE
PARTICLES
475
EQUIVALENCE
of Section II in which a(s) = B( 1). The integral case and hence 23 # 0. integral in Eq. (16) diverges and hence & = 0. the coupling constant g” (with c = 0);
[I
1
m ds’p(s’) i n- .T1(s’ - my
1 -l 1C(s’) 12
(30)
To show that this is equivalent to ( 11 >, we make use of Eq. (20) ; since & = 0, we may undo the subtraction in the square bracket and write Eq. (25) as -1
1
(31)
The pole comes from the second term on the right of Eq. (31) and
1
&p(s’) 1 -’ (s’ - m2)2 1 C(s’) I2
(30’)
When c # 0, the algebra involved becomes complicated, but one can show that the equivalence still holds. Finally we note that Eq. (30’) is precisely the condition on the residue of an s-wave bound state derived by BGKT. In potential scattering the amplitude B(s)
i6(o)(s) sin P(s) = e PCS>
is the contribution from the Mandelstam double spectral function which can be shown to be unitary (9). The difference A(s) - B(s) = Ap(s)[iI’(s)]2 corresponds to the contribution from the Mandelstam single spectral function. Since5
C(s)= exp[ -‘+ as follows front Eqs. (21), to the BGKT condition” -2
9
=-
l, (s,-dzi’w s,]
(22), and (26), we see that Eq. (30’)
1 O” ds’p(s’) exp [+ a s s1 (s’ - m2)2
P 1:
(SN f’zy;i;“’
is equivalent
s,J
6 We are still assuming c = 0. The case c # 0 requires a further discussion which the reader can supply. The phase 6 (0) is assumed to satisfy the Hijlder condition in any finite interval. See (11). energy. 6 In potential scattering p(s) = 4s and rn 2 = -SB where ss is the binding
476
LEE
ET
AL.
V. CONCLUDING
REMARKS
In this section we shall point out some problems we have left unsolved in this paper. The equivalence between the two methods has been proved taking into account only two particle intermediate states. Within the framework of the same approximation, we can treat the multichannel problem with similar conclusions. At present, it is not known whether inclusion of intermediate states of more than two particles modifies t,he above conclusions. In the above treatment an assumption has been made that there are no other particles with the same quantum numbers as the particle A. If the possibility of having more than one particle with the same quantum numbers is included, the problem becomes complicated.7 In this paper we have not explored the possibility of using crossing symmetry and seeing whether the vanishing of the wave function renormalization constant of the particle A implies any condition on the wave function renormalization constants of particles B and C. This problem deserves further investigation. ACKNOWLEDGMENT The authors One of the Foundation. RECEIVED:
are grateful to Prof. Susumu (B. W. L.) wishes to
authors
February
Okubo for acknowledge
a remark. the support
of the
Alfree
Sloan
4, 1964 REFERENCES
I. J. HOUARD AND B. JOUVET, Nuovo Cimento 18, 466 (1960); M. T. VAUGHN, R. AARON, AND R. D. AMADO, Phys. Rev. 134, 1258 (1961); R. ACHARYA, Nuovo Cimento 24. 870 (1962); J. D. DOWKER, Nuovo Cimento 26, 1135 (1962). R. M. L. WHIPPMAN AND I. S. GERSTEIN, Phys. Rev. (to be published). .,!I. A. SALAM, Nuovo Cimento 26,224 (1962) ; see also, R. AMADO, Phys. Rev. 127, 261 (1962); S. WEINBERG, Phys. Rev. 130, 776 (1963). 4. G. F. CHEW AND S. C. FRAUTSCHI, Phys. Rev. Letters 7,394 (1962). 6. F. ZACHARIASEN AND C. ZEMACH, Phys. Rev. laS, 849 (1962). 6. G. F. CHEW AND S. MANDELSTAM, Phys, Rev. 119, 467 (1960). 7. R. M. ROCKMORE, Phys. Rev. 132, 878 (1963); see aIso J. S. DOWKER AND J. E. PATON, Nuovo Cimento 30. 456 (1963). 8. F. ZACHARIASEN, Phys. Rev. 121, 1851 (1961). 9. R. BLANBENBECLER, M. L. GOLDBERGER, N. N. KHURI, AND S. B. TREIMAN, Ann. Phys. (N. I’.) 10, 62 (1969); hereafter referred to as BGKT. 10. S. FUBINI, report at the Stanford Conference on Nucleon Structure, June, 1963 (unpublished); Nuovo Cimento 30, 1512 (1963).
from
7 Such a problem ours.
has been
studied
in (28) ; the
emphasis
in the above
work
is different
EQUIVALENCE
OF
ELEMENTARY
AND
COMPOSITE
PARTICLES
477
11. R. L. WARNOCK, Phys. Rev. 131, 1320 (1963) ; seealso, G. FRYEANDR. L. WARNOCK, Phys. Rev. 130, 478 (1963); H. SUGAWARA ANDA. KANAZAWA,Phys. Rev. 126, 2251 (1962).
12. H. LEHMANN,Nuovo Ciwlento 11, 342 (1954). 1% R. OMNES, Nuovo Cimento 8, 316 (1958); ibid. 21, 524 (1961); R. BLANKENBECLER, Phys. Rev.
123, 983 (1961).
14. C. J. GOEBELANDB. SAKITA,Phys. Rev. Letters 11,293 (1963). 16. S. D. DRELLANDF. ZACHARIASEN, Phys. Rev. 106, 1407 (1957). 16. H. LEHMANN,K. SYMANZX,ANDW. ZIMMERMAN, Nuovo Cimento 2,425 (1955). 17. P. FEDERBUSH, M. L. GOLDBERGER, ANDS. B. TREIMAN,Phys. Rev. 112, 642 (1958). 18. G. FELDMANANDP. T. MATTHEWS,Phys. Rev. 133, 823 (1963). 19. L. CASTILLEJO, R. H. DALITZ,ANDF. J. DYSON,Phys. Rev. 101,453 (1956).
On
OF PHYSICS:
The
B. W.
26, 466-477
Equivalence LEE,
Department
K. T.
(1964)
of Elementary
Composite
I. S. GERSTEIN,
MAHANTHAPPA,
of Physics,
and
University
of Pennsylvania,
AND
M. L. WHIPPMAN
Philadelphia,
Equivalence between composite particles, defined as the zeroes tion of the N/D method, and elementary particles with vanishing renormalization constants has been proved in the approximation ing only two particle intermediate states. A precise formulation method is presented to facilitate the comparison with field theory. ing of the wave function renormalization constant is formulated Lehmann representation. We are led to a detailed study of the the proper vertex part in order to establish the equivalence of procedures. I. INTRODUCTION
AND
Particles*
Pennsylvania
of the D-funcwave function of retainof the N/D The vanishin terms of the properties of the above two
SUMMARY
Recent investigations of some field theoretic models (1, 2) have shown the possibility of a meaningful distinction between elementarity and compositeness of a particle, and the equivalence of those two concepts in the limit in which the wave function renormalization constant Z3 goes to zero. It has been suggested by Salam (3) that this equivalence continues to hold in a full field theory. If this notion were true then the suggestion of Chew and Frautschi (4) that all particles are equally composite (or elementary) could be materialized in a field theory through the vanishing of all of their wave function renormalization constants. There have been attempts (see, for example, ref. 5) to prove this equivalence within the framework of the N/D method (6). In the N/D method, a composite particle corresponds to a zero of the D function which satisfies the usual analyticity requirements and a certain asymptotic condition for large values of energy. Some authors (7) have attempted to show that the above circumstance arises precisely when the renormalization constant Z3 of the particle is zero. In particular, Rockmore considered the usual first order determinental approximation for the denominator function and was then forced to make the simplifying but unjustified approximation of approximating the numerator function by a constant in his equation for the residue of the bound state pole. Having done this, he finds that the residue equation implies 23 = 0 to second order. However. * Work
supported
in part
by the
U. S. Atomic 466
Energy
Commission.
EQUIVALENCE
OF
ELEMENTARY
AND
COMPOSITE
467
PARTICLES
these approximations are exact in the Zachariasen model (8) and thus Rockmore’s proof reduces to the treatment of this model by Dowker (1). In addition, a claim is made that the residue equation is equivalent to one given by BGKT (9) but this is not apparent. The purpose of this paper is to prove the equivalence between bound states (as defined by the N/D method as above) and elementary particles with their wave function renormalization constants set equal to zero in a full field theory, making only the approximation of neglecting all but two particle intermediate states. We differ from previous authors in that we make no assumptions about the nature of interactions, except requiring a relatively mild condition which amounts to saying that the theory be not too singular. At this point we may recall that Fubini has questioned lately the validity of the usual association of compositeness and the condition Z 3 = 0 in the case of singular theories (10). We shall make use of various well-known results of renormalized field theory, rather than having recourse to perturbation theory. In the elastic unitarity approximation it will be shown that the equation for bound state residues derived in BGKT for potential theory is identical (except for some kinematical factors) to the one obtained by setting Z3 = 0. In the next section we shall summarize the content of the N/D method in terms of the phase integral a, with particular emphasis on the asymptotic behavior of 3 for high energies. In Section III, we study the Lehmann representation of the propagator which in the two particle approximation yields an expression for Z3 in terms of the vertex function. The relation between the vertex function and the a function on the one hand, and the connection between the asymptotic behavior of %J and the existence of a bound state on the other imply the equivalence of Z. = 0 and the existence of a zero in the D-function. In order to show the identity of the residue equation in the N/D method and the condition Z3 = 0, we are led to study analytic properties of the proper vertex part. Here our results are compared to the similar results in potential theory. Section IV contains some concluding remarks. II.
N/D
METHOD
Our basic assumption in both field theory and the N/D method (6) is that the partial wave T-matrix for two-particle (B and C) scattering is described by a dispersion relation. We specialize to the s-wave, and assume that there is a particle A of mass m having the quantum numbers of this channel. We write the dispersion relation in the form (6) A(s) = a + ‘+
O” ds’ Im A(s’) + s - so ds’w( s’) s sI (s, _ s)(s, - so) __ lr s L (s’ - s)(s’ - so) 3
-s -
s - so ___ m2 m2 -
SO
(1)
468
LEE
ET
AL.
where A(s)
= sin 6(s>ei*‘“‘/p(s)
(2)
w(s) is the discontinuity across the left hand cut L, p(s) phase space factor with the property that limp(s) s-00
is the appropriate
= const.,
6(s) is the phase shift for the s-wave, s is the square of the c.m. total energy, sl = (mB + mc)2, and a is the subtraction constant at s = so . We define the phase integral a)(s) by
and the quantities
K(s,s’)= [~(s)P(s’)l”2 - s,) 7r J,(s,,14”‘;;;‘s”“’ r](s)=g-+ a+ sy sr, (s’ Warnock
ds‘w( s’) - s)(s’ - so)
1
(4)
(11) has shown that if the conditions’ ds’(K(s,
s’))~
< cc
and OD dsh(4)2 $81
< 00
are satisfied, and if 6(s) satisfies the Holder condition lima)(s) 8-m where cr is a nonpositive integer. logarithmic factors.” Since there 0 and -1.2 Under the condition neither zeroes nor poles. In the N/D method, the most
in any finite interval,
= Q(s~)
then (5)
The symbol Q means “of order of, disregarding is only one pole in Eq. (1) a! is restricted to stated above, it can be shown that B(S) has general form of the D-function
may be written
1 These conditions imply that the N/D equation (see Eqs. (9) and (10) below) can be transformedinto a Fredholm type. These conditions follow from Warnock @I), Eq. (2.10), by putting 11= 1 (no inelasticity). 2 This statement
is equivalent
to Levinson’s
theorem generalized
to our case. See (11).
EQUIVALENCE
OF
ELEMENTARY
AND
COMPOSITE
PARTICLES
469
5tS
D(s)
=
rl[
ts
-
Si>
$
(s
_
sj)
as>
(6)
and N(s)
= A(s)
D(s)
(7)
In order to identify a zero of D(s) with a bound state, it is necessary to specialize to a particular form of (6). We require that D(s) have no poles and N(s) and D(s) have no common zeroes, and’ lim D(s) s-em
= Q(l)
(81
Since B(S) has no zeroes the polynomial in the denominator on the right hand side of Eq. (6) should be absent. The degree of the polynomial in the numerator is fixed by (8) and the requirement that N and D have no common zeroes. We may now distinguish between two cases: Case I. B(s) = Q(1). In this case D(s) = ~~o(s)/B(a) (we normalize D(s) such that D(G) = 1) and D(s) N(s)
= l - s+ = a + ‘+
O1 ds’p(s’)N(s’) s,, ($ - s)(s’ _ so) w(s’)D(s’) g2D(m2> s - so / as (s, _ s)(s, _ so> - ~ ___ s - m2 m2 - s0 L
In this case g” can be chosen arbitrarily. Case II. a)(s) = Q(s-‘), and D(s) = (s - m”)D(s)/(s, - m”>B(s0>; m ,&‘W(s’) D(s) = 1 - ‘%?? ds’ 7r $81 (s’ - s) (s’ - so) N(s)
= a + ‘9
w( s’)D( s’) j- ds’ (s, _ s> (s, _ so> L
(9)
(10)
and
N(m2)_ ” = (dD(s)/ds>,,z D(m’>
(11)
= 0
3 These conditions imply that there be no CDD poles (19) in the D-function. We may note that if Eqs. (19) and (20) below are transformed into a Fredholm type, and solved by the usual method, then D does not have any CDD poles.
470
LEE
ET
AL.
In this case N(s) no longer has a pole. Note that if si were chosen at some point other than &, then N and D would have a common zero. The cases I and II correspond, respectively, to the elementary particle and bound state poles at s = m2. III.
FIELD
THEORY
A. PROPAGATOR The Lehmann representation (12) for the propagator for particle A is, in the two-particle approximation A,(s) =
(12)
In the subsequent discussionsthroughout, we shall assume that the Lehmann representation (12) is valid. The matrix element (0 1$A(0)j BC)i, may be written4
(0 ] b(O) 1BC)c = A2
F(s) = A,(s>I’(s)
(13)
where F(s) is the form factor of the particle A, and I’(s) is the renormalized proper vertex part of Dyson. We have the dispersion relation for F(s) :
F(s) =g+s+ SW
ds’ Im F(s’) 81 (s’ - s) (s’ - my
(14)
We make the explicit assumption that F(s) has no zeroes. Since Im F(s) = sin 6(s) in our approximation, F(s) may be written by the usual F*Wi6’“) technique (13) as F(s) = gB-‘(s)
(15)
The assumptions that Eqs. (12) and (14) are valid without further subtractions imply that there are enough powers of 1log s / in the asymptotic behavior of a>(s) when D(S) = Q(s-‘), to ensure the convergence of integrals in these equations, and similar expressionsbelow. Substitution of Eq. (15) into Eq. (12), and the use of the definition of 2,’ = lim,,, sA,(s) (12) give
4 Equation (13) may be derived by the technique electromapnetic vertices.
of (16). See also (17) in the case of
EQUIVALENCE
OF ELEMENTARY
AND COMPOSITE
PARTICLES
471
It is also useful to consider the inverse of the propagator (12)
Aye = (s_ m2)1 s - mz O”ds’ds’)IUs’)I2 n- ss1(s- s’)(s’- m)” [
(171
1
s - m2 +Cs - s, sz - m2 The last term in the bracket is present if Aa seen from (17) to be
has a zero at s = sZ. Z3 may be
(18) B. PROPER VERTEX
PART
We shall consider the properties of the renormalized vertex part r(s) in some detail. From (13) and (17) we have r(s) = F(s)
[
1 + (s - m2) SW d* 7r Sl
(s, “,,),
I&)
I’]-’
(19)
From (13) and (12) we have
F(S) = r(s)
[
i - ‘+
1r(d) 1’ + A-/ md~‘p(SI) s (s, _ m2>2 s 81
e]-’ -
s1 ss
(19’)
mz
Defining J?(s) = g/C(s), we get
zo(s) = C(s)
2
d&(i) (s - s’)(s’ - m2)2
1
s - m2 +C-s - sss, - m2
(20’)
From (20 ), we seethat (a) C(s) can be continued to the entire s-plane cut along the real axis from s1to ~0,and is analytic there. (b) C(m”) = 1. From (20’) we see that if c # 0 in Eq. (17), then C(s) must have a zero at s = sZ. Consequently I’(s) has a pole at s = sE. This point has been emphasized by Goebel and Sakita (14) in another context. First we shall confine ourselves to the case c = 0. We define
jqs) = _ as + ie>- as - ie) %3(s)
(21)
452
LEE
with s on the right-hand we have
ET
AL.
cut, and by analytic
continuation
elsewhere.
From (20)
(s- s')(s' - m") ds’p( i)
where we have defined 31(s) by s-L(s) = for s on the right-hand
iD(s + i,) - a(s %(s)
cut, and by analytic A(s)
- iE)
continuation
elsewhere.
Also
= x(s)+(s)
We can show that (a) M(s) is analytic everywhere in s except on the left-hand cut (the apparent right-hand cuts and poles at s = m” cancel) ; (b) on the left-hand cut Im M(s)
= w( s)9( S)
[
12 - m2)2 iiS(iT) ! II
1+ +
g2 S,, (s _ $~~~,s’)
= w(s>C(s) This shows that B(s) is a unitary relation B(s)
= !!-I!! T
amplitude,
= M(s)/C(s)
i.e., Im B(s)
(22)
= p(s) 1B( s)l” satisfying
the dispersion
1
d&( s’) - dip(d) /B(i) I2 (s’ - s)(s’ - so) + s,, (s’ - s)(s’ - so) +bSL-
The last term is present if c # 0 in (21). The constant b = Mso) =a l-g2 C(so> {
[
(so - m2)~(so)
x
r,,
( (&
s - SIJ s - s* s, - so
(23)
b is related to a by
1 + ‘YE-& li-d;y(y,
m2)
/5&‘} g
(24)
EQUIVALENCE OF ELEMENTARY AND COMPOSITE PARTICLES
473
From (20’) andthe definitionsof ‘X(S) and ill(s), we seethat A(s) = xWa>(s>
s -
l--
a
[
= B(s)
m2
m s81 (s -
ds’p( s’) s’)(s’ - nay
2-l (25)
I II &)
+ A,(s)[ir(s)l”
The meaning of the decomposition of A(s) in (25) can be seen as follows. The first part, B(s), is the unitary amplitude constructed without the A particle pole at s = m”. The second term is the contribution of the particle A (in the Feynman-Dyson sense) renormalized by the rest of the interactions. In Eq. (24) the subtraction constant b differs from a since b should not include the renormalization effect from the particle A, Since B(s) is unitary, we may write for s 2 s1 , “(‘) (‘) sin scO)(S) B(s) = e (26) P(S)
Then we see that s%(Q)(8) ZZ---I as> I2 e (C(s)>”
(27)
and the amplitude -g2 (C(s) (s
-
m”)
[-’
1 _ s - m2 m s *1 (8 n[
dip( s’) -
s’)(s’
-
2-l
I II 9
. Ge+
m2)2 C(d)
(28)
is also unitary. Now combining Eqs. (25), (26), (27), and (28), we seethat “(8) sin 6(s) = e”(‘)(s) sin *‘O’(s) 2i6(@) weicm sin (y (s) A(s) = e +e (29) PCS>
P(S)
P(S)
and 6 = So+ cr. The above decomposition of the T-matrix is well-known: it has been discussed by Drell and Zachariasen (15) and appears naturally when the interactions can be split up into two parts. Two well-known examples of Eq. (29) are the interference of the Coulomb effect and the nuclear effect (where 6’ is the Coulomb phase shift and LYthe nuclear phaseshift) and the interference between the “potential” scattering and the Breit-Wigner resonance. What we have said above may be illustrated in a version of the Zachariasen model (2). We consider the ABC coupling as well as B2C2coupling (Fig. 1(a) ). The amplitude A(s) is the sum of all iterations of these two interactions (Fig. l(b)). The proper vertex part is given by the diagrams in Fig. 2(a) and B(s) is the sum of all diagrams containing only the B2C2coupling (Fig. 2(b)).
+ >o( +--lb) FIG. 1. (a) The basic interactions scattering amplitude.
of the Zachariasen
+
model;
(b) the expansion
of the
---
(a)
8(.)=X + )o( + )oo( +--FIG. 2. (a) The expansion of the proper vertex part; (b) the expansion 474
of the B-amplitude
EQUIVALEKCE
OF
IV.
ELEMENTARY
Z8 = 0 AND
First we consider the case I in Eq. (16) converges for this In the bound state case the Z3 = 0 in Eq. (18) determines g2=
AND
COMPOSITE
PARTICLES
475
EQUIVALENCE
of Section II in which a(s) = B( 1). The integral case and hence 23 # 0. integral in Eq. (16) diverges and hence & = 0. the coupling constant g” (with c = 0);
[I
1
m ds’p(s’) i n- .T1(s’ - my
1 -l 1C(s’) 12
(30)
To show that this is equivalent to ( 11 >, we make use of Eq. (20) ; since & = 0, we may undo the subtraction in the square bracket and write Eq. (25) as -1
1
(31)
The pole comes from the second term on the right of Eq. (31) and
1
&p(s’) 1 -’ (s’ - m2)2 1 C(s’) I2
(30’)
When c # 0, the algebra involved becomes complicated, but one can show that the equivalence still holds. Finally we note that Eq. (30’) is precisely the condition on the residue of an s-wave bound state derived by BGKT. In potential scattering the amplitude B(s)
i6(o)(s) sin P(s) = e PCS>
is the contribution from the Mandelstam double spectral function which can be shown to be unitary (9). The difference A(s) - B(s) = Ap(s)[iI’(s)]2 corresponds to the contribution from the Mandelstam single spectral function. Since5
C(s)= exp[ -‘+ as follows front Eqs. (21), to the BGKT condition” -2
9
=-
l, (s,-dzi’w s,]
(22), and (26), we see that Eq. (30’)
1 O” ds’p(s’) exp [+ a s s1 (s’ - m2)2
P 1:
(SN f’zy;i;“’
is equivalent
s,J
6 We are still assuming c = 0. The case c # 0 requires a further discussion which the reader can supply. The phase 6 (0) is assumed to satisfy the Hijlder condition in any finite interval. See (11). energy. 6 In potential scattering p(s) = 4s and rn 2 = -SB where ss is the binding
476
LEE
ET
AL.
V. CONCLUDING
REMARKS
In this section we shall point out some problems we have left unsolved in this paper. The equivalence between the two methods has been proved taking into account only two particle intermediate states. Within the framework of the same approximation, we can treat the multichannel problem with similar conclusions. At present, it is not known whether inclusion of intermediate states of more than two particles modifies t,he above conclusions. In the above treatment an assumption has been made that there are no other particles with the same quantum numbers as the particle A. If the possibility of having more than one particle with the same quantum numbers is included, the problem becomes complicated.7 In this paper we have not explored the possibility of using crossing symmetry and seeing whether the vanishing of the wave function renormalization constant of the particle A implies any condition on the wave function renormalization constants of particles B and C. This problem deserves further investigation. ACKNOWLEDGMENT The authors One of the Foundation. RECEIVED:
are grateful to Prof. Susumu (B. W. L.) wishes to
authors
February
Okubo for acknowledge
a remark. the support
of the
Alfree
Sloan
4, 1964 REFERENCES
I. J. HOUARD AND B. JOUVET, Nuovo Cimento 18, 466 (1960); M. T. VAUGHN, R. AARON, AND R. D. AMADO, Phys. Rev. 134, 1258 (1961); R. ACHARYA, Nuovo Cimento 24. 870 (1962); J. D. DOWKER, Nuovo Cimento 26, 1135 (1962). R. M. L. WHIPPMAN AND I. S. GERSTEIN, Phys. Rev. (to be published). .,!I. A. SALAM, Nuovo Cimento 26,224 (1962) ; see also, R. AMADO, Phys. Rev. 127, 261 (1962); S. WEINBERG, Phys. Rev. 130, 776 (1963). 4. G. F. CHEW AND S. C. FRAUTSCHI, Phys. Rev. Letters 7,394 (1962). 6. F. ZACHARIASEN AND C. ZEMACH, Phys. Rev. laS, 849 (1962). 6. G. F. CHEW AND S. MANDELSTAM, Phys, Rev. 119, 467 (1960). 7. R. M. ROCKMORE, Phys. Rev. 132, 878 (1963); see aIso J. S. DOWKER AND J. E. PATON, Nuovo Cimento 30. 456 (1963). 8. F. ZACHARIASEN, Phys. Rev. 121, 1851 (1961). 9. R. BLANBENBECLER, M. L. GOLDBERGER, N. N. KHURI, AND S. B. TREIMAN, Ann. Phys. (N. I’.) 10, 62 (1969); hereafter referred to as BGKT. 10. S. FUBINI, report at the Stanford Conference on Nucleon Structure, June, 1963 (unpublished); Nuovo Cimento 30, 1512 (1963).
from
7 Such a problem ours.
has been
studied
in (28) ; the
emphasis
in the above
work
is different
EQUIVALENCE
OF
ELEMENTARY
AND
COMPOSITE
PARTICLES
477
11. R. L. WARNOCK, Phys. Rev. 131, 1320 (1963) ; seealso, G. FRYEANDR. L. WARNOCK, Phys. Rev. 130, 478 (1963); H. SUGAWARA ANDA. KANAZAWA,Phys. Rev. 126, 2251 (1962).
12. H. LEHMANN,Nuovo Ciwlento 11, 342 (1954). 1% R. OMNES, Nuovo Cimento 8, 316 (1958); ibid. 21, 524 (1961); R. BLANKENBECLER, Phys. Rev.
123, 983 (1961).
14. C. J. GOEBELANDB. SAKITA,Phys. Rev. Letters 11,293 (1963). 16. S. D. DRELLANDF. ZACHARIASEN, Phys. Rev. 106, 1407 (1957). 16. H. LEHMANN,K. SYMANZX,ANDW. ZIMMERMAN, Nuovo Cimento 2,425 (1955). 17. P. FEDERBUSH, M. L. GOLDBERGER, ANDS. B. TREIMAN,Phys. Rev. 112, 642 (1958). 18. G. FELDMANANDP. T. MATTHEWS,Phys. Rev. 133, 823 (1963). 19. L. CASTILLEJO, R. H. DALITZ,ANDF. J. DYSON,Phys. Rev. 101,453 (1956).