- Email: [email protected]
Field theory of unstable particles
ANNALS
OF
PHYSICS:
9,
169-193
Field
(1960)
Theory
of Unstable
JULIAN Harvard
University,
Particles*
SCHWINGER Cambridge,
Massachusetts
Using the example of a spinless boson field, the structure of the simplest Green’s function is developed to provide a uniform theory of particles, stable and unstable. Some attention is given to the time decay law of unstable particles and it is emphasized that a full account of the relevant physical situation must be contained in its mathematical representation, leading to the conclusion that an essential failure of the exponential decay law marks the limit of applicability of the physical concept of unstable particle. There is a brief discussion of the rr and K mesons.
Some attention’ has been directed recently to the field theoretic description of unstable particles. Since this question is conceived as a basic problem for field theory, the responseshave been somespecial device or definition, which need not do justice to the physical situation. If, however, one regards the description of unstable particles to be fully contained in the framework of the general theory of Green’s functions, it is only necessaryto emphasize the relevant structure of these functions, That is the purpose of this note. What is in essencethe same question, the propagation of excitations in many-particle systems where stable or longlived “particles” can occur under exceptional circumstances, has already been discussed(2) along these lines. A relativistic field describes a localized excitation which produces a spectrum of energies or massesthat ranges down to a lower limit characteristic of the field type. Thus for any electrically charged boson or lepton field this theoretical lower limit is the electron massm, while for electrically neutral fields of these two varieties the massspectrum in principle extends down to zero. These limits are set by the massesof the absolutely stable’ particles to which one is led by decay processesthat respect essentially only the conservation of electric charge Q, in addition to the usual mechanical properties. With fermion fields carrying nucleonic charge N, however, the absolute conservation of the latter evidenced by the sta* This work was supported in part by the U.S. Air Force. It was reported at the Ninth International Conference on High Energy Physics, Kiev, U.S.S.R., July 15-25, 1959. 1 See, for example, Matthews and Salam (1) and a subsequent work of these authors, together with a recent paper by Levy (1). * At least on a sub-cosmological scale. 169
170
SCHWINGER
bility of the proton implies a mass lower limit that equals M, the proton mass, for fields with Q = N, becomes M + m for electrically neut’ral fields, and is M + 2m for Q = -N (omitting the binding energy of hydrogeo at’om or ion!). The propagation of such excitations, superimposed on the vacuum state, is described by the Green’s functions as time ordered field correlation functions. For a single spinless boson field 4(z), the simplest example is SC” - z’> = mw#w)+) =
(dk) s
where, according to the elementary
eik(S-z’)s(k)
(27r)4
relativistic
’
structure
s(k) =s
dB[t?] k2 + K2 -
of its spectral form,
ic
Here B[K~] is a real non-decreasing function that equals zero for sufficiently small K2 and approaches unity as ~~ -+ to. Another form appears on exploiting the latter property as a minimum formal characterization of a local field,
S(k’)- k2, We first remark
k2-+
that the complex variable
co.
function
can have no complex zeros, since (K2
s
(K2
-
2)
dB[?] + iy
-
X)”
+
implies y = 0. Accordingly
dB[ttl s
y2
(K”
-
2)”
+
y2
=
’
the function P(x)
= s-‘(z)
+ 2
is regular everywhere in the finite complex plane, with the exception of the positive real axis, a semi-infinite portion of which constitutes a branch line for this function corresponding to the branch line of S(z) associated with the continuous spectrum. It is also possible to have isolated poles, which refer to real zeros of S(Z) appearing at points within the spectrum where B[K’] is constant. The function z-‘P(z) vanishes at infinity in the cut plane and also has a pole at the origin, from which it follows that z-‘p(x)
ds[K2] = x” z s r2>0 K 2-z
UNSTABLE
171
PARTICLES
or
The real function
s[K’] is shown to be nondecreasing
on translating
the property
& Pax:+ +I) - $32 - Q/)1> 0 into
In writing the residue of z-‘P(z) at the origin as X2 > 0 we have anticipated the expression of a physical requirement, that S(Z) have no singularity for negative z or space-like3 k. The function s-‘(z) begins at + 00, for z = - 00, and decreases monotonically along the real z axis, according to
If S-‘(Z) is to have no zero for x I 0, it is necessary and sufficient that 6-l (0) = X2 > 0. Alternatively, we recognize from the comparison of the two expressions for s(O) that
dS[K2] -l S(k) =k2 +x2 -it+lc2 sk2 +K2 -‘k1’
The second form thus obtained for s(rC) is
the assumed existence of which requires the convergence at infinity of the integral j-( K2)-’ dS[K2]. Th is convergence condition also assures the validity of the asymptotic form (k2)-’ as k2 approaches infinity either in a space-like or time-like direction. Should more be true and $ ds[~‘] converge, which implies a finite total increase in S[K2] from an initial zero value, we could infer the more detailed asymptotic property S(k)
-
(k2 + POY
where ~02 = x2 + s[oc] > x2. 3 This includes k2 = 0. The exceptional netic field. presumably do not occur with
circumstances a spinless field.
encountered
for
the electromag-
172 An inspection
SCHWINGER
of the spectral form for s(k)
Under the conditions
shows that ~02 is also given by
dB[K2]. iJo2 =sK2
which permit the identification
of cl0 it is possible to write
k2+ K2- in1’ The constant PO can be interpreted as the mass parameter associated with the field independently of interactions, at, least when these couplings are linear in the boson field. One anticipates two possible spectral conditions from the general physical conviction that a continuous energy or mass spectrum never terminates nor has jump discontinuities. Either B[K’] is a continuous increasing function for all K > KO , or, there are one or more positive steps in B[K~] followed by a continuous increasing domain. In the first circumstance there is no discrete mass value, which is to say, no stable particle associated with the field, while the alternative corresponds to one or more such particles. If there is no isolated singularity in s(z) the function s-‘(x) can have no zero for 0 < z < ~02, where ~~ marks the threshold of the continuous spectrum. The condition for this property, which expresses the absence of a stable particle, is s-‘(Ko” - 0) > 0, or
x2>.o++J.,s2]s
Thus it is necessary, but not sufficient, for X to exceed exist. This also follows from the observation that
KO
if no stable particle is to
The sign of S-l( ~02- 0) must become negative if a discrete mass value is to occur in 8. Indeed, the condition that S-‘(z) vanish at z = p2 > 0 is given by x2 = $[l < Kofl
+ SK*]
+/-$$]r
under the assumption that s[K~]is a continuous function. The inequality is obtained by remarking that the function of p2increasesmonotonically in the interval 0 < p2 < KO’. A value of X2that satisfiesthe inequality defines a unique p2 in this segment.
UNSTABLE
173
PARTICLES
To consider in more detail the situation of a stable particle we indicate explicitly the nature of the spectrum by writing
s(z) = s2
+ /* dK”p - 2’ ‘Lo2
where 0 < Bo < 1, and the continuous function B( K”) , which is positive for K’ > co2and vanishes at the beginning of this domain, obeys 0 dK” B(/c’) = 1 - Ba . s102 If we place B(2)
= B&K?,
the latter property appears as Bo = [ 1 + Ia dK” &)I-‘. v2 We also have
from which some inequalities
can be inferred. Thus Xm2 > Bopp2 and therefore u” > BoA’,
while the property 6” >
~02
> p2 shows that X2 < F-‘, or X2 > $ > BOX’.
A more precise version of the second inequality is 1 1 - Bc, ,
As 2 varies from p2 + 0 to ~02 - 0 the function from --oo to the limiting value
~(2)
increases monotonically
174
SCHWINGER
which must still be negative if 9 is to have no zero in the interval. under the restriction
the function
s-‘(z)
has no isolated singularity.
Accordingly,
This is made explicit by writing
where the continuous positive function s( K”) , K > KO , vanishes for K = KO . As one application of the inequality that characterizes this situation we note that
BO>smd2B(K’) >smd/(-z-w2B(K2) 1&
Kc+ -
which
j.12
K2 -
4
shows the necessity
Ko2
go2
K2
x2
of the property
Bo> (D’[l - ($1. A more stringent
condition
merges from the evaluation
Bo’ = -&l(z)
lzzp2
together with
1
6’) dK2K2 P2 ’ since BF’
x”= s(K2) < P2 smd2 l-J2 K02
K
(K”
-
/.L2)2
and therefore -
1 - (P/Ko>~ b/~>2hd’d2
’
/.12
’
UNSTABLE
Related inequalities
are co dKz K24K2) sd (K’ /.L2)’
175
PARTICLES
= B,’
_ 1 < x2 - pz K02 Ko2 -
/A2 ,.L2
and m
2
dtc” s kT2 (K” If the structure
this function
B,+z(K~) ’
of s(z)
1
1 -
-
wu2
(P/Ko)~(P/~)~
*
is such that
has a zero and s-‘(z)
has a pole at x = v”, /.k <
Accordingly,
1 - Bo <
=
/A’)’
U <
Kos
we now have s-l(Z)
= ii2 - x - x -+
-
x
z I- d:& d
2
where the constant s and the continuous function s( K’), K > KO , are positive, the latter approaching zero at the threshold of the continuous spectrum. It follows, conversely, from the monotonic decrease to - 00 as z increases to V” - 0, and the positive value at z = 0, that g-l(z) has a single zero at z = J.L~< v”, determined by x2 = p2 1 + -& [
p2
+
[,
dK”
$4
*
Again we calculate B, from
B;’= -g f(z) lz=p2 =; +P2 [ (v2 _s&2 +/-4 A2(K2s!2;2)2] which implies a similar inequality
for B. , with
Bo > (i-m2 On writing
the expression
1 -
for BZ1 as 2
BT”
we learn that
=
1
+
(v2
‘“,2j2
Y replacing
1 - (P/W2 (&)2(p/“)2’
KO ,
)
176
SCHWINGER
2 ($ I p2)2 Be-s < lUnder our present assumptions S(KO - 0) is positive, which is to say that the monotonic decrease of $‘(z) in the interval between V” + 0 and Q’ - 0 does not change the sign of this function. Hence
and, if X2 is eliminated
u” (
Kc?
-
V”)(U”
-
/AZ)
by employing
the equation
m s
s>l+
p2, we get
K2S(K”)
K02 dK”
This shows the necessity
that determines
(K”
-
J)(K”
-
Ko2)
of the condition Ko2
-
5’2
Ko2
-
/.k2
2
if both s(x) and s-‘(x) contain a single pole. Other significant implied by the lower limit for Bos, together with
inequalities
are
2 B.
+
(v2
1
II”)”
Bos
<
1.
Thus
and on comparison
with the lower bound for B.
we find that
h > /J(Ko/U). If the various
parameters x 2
in ~-I(Z) [
l-
&
were altered to satisfy +
J-y*
dK2
&2]
the inequality ’
this function would pass through zero in the interval between V” + 0 and Kt - 0, which implies a second pole for s(z) at z = p12, p < Y < $ < Kg . We have now sufficiently traced the unfolding of the Green’s function’s spectral structure to permit the visualization of a hypothetical situation where the field has n kinematically similar stable particles associated with it, of masses ,.Q , . . . , fin , represented by poles of $j( z) at z = pk2, k = 1, . . . , n. There are also zeros of S(Z)
UNSTABLE
177
PARTICLES
or poles of s-‘(z) at z = vk2, where ~~ < vl < ~2 , . . . , pnWl < vnwl < pn , and there can, but need not necessarily appear an nth zero v,, located between pn and KO , the threshold of the continuum. For the situation in which s(z) exhibits n = 1, 2, . . . poles and n - 1 zeros, we define F(z) such that
+ [, &”$f&, =
$j-‘(2)= A”- 2- 2
It will be observed that F(z) approaches unity at infinity, has no zero or poles, and possesses a branch line extended from Ko2 to to. Accordingly, there is a branch of log F that vanishes at infinity and has no singularity other than the branch point at Q~, which enables us to write
The real continuous function (P(K~) approaches zero at infinity and must vanish at K = Kc) , since log F(x) is regular at this point. The resulting expression for $(z) is dK2
‘Ph2> K2-Z
1 ’
which must still be qualified by the restriction on (o(K~) that is equivalent to the positiveness of s(K~), K > KO . That characteristic of the Sk is already contained in this representation through the property F(x) > 0, x < ~02, and the relationship of the Elk and Vk . We let z = x $ iy, x > KO~, y + +0, and compare the ratios of real and imaginary parts of $j-‘( z) for the two forms, using the evaluation
1 ,
which
gives ?fK2s(
K2)
C!Ot(o(K2)
=
K”
The function on the right-hand side has a positive value at K = ~0, according to the nature of the spectrum, which implies that P( K”) is positive in a neighborhood of the continuum threshold. We conclude from the physical hypothesis of a nonzero S( K2), K > K~ , that tan (p cannot vanish and therefore that the values of cp lie entirely in the interval
178
SCHWINGER
The lower limit is attained at the threshold of t’he continuum ically approached at infinity. Let us note here that
and is asymptot-
If it should be true that a( K”) vanishes more rapidly than (x2)-’ as K” approaches infinity, so also would P(K~), and one could make the identification
In the simplest have
example of this type, with
one pole and no zero in S(Z), we
dK” dK2> K2-2
1 ’
or
S(k) =
k2
+fi
-
ie
ex+2
+
2)
;
SWK02 dK”
(K”
_
p2)$2;
K2 _
J
where it has been recognized that
1
dK2 dK2> ra P2 Other quantities
that can be derived from this form are
P(K2)= --+ P2exp[ - (K2- p2); p [, Al2 CK’2 _
;!;;31:,
_
K2,]
x
and S(K2)
=
(1
-
~)e~p[~P~~~dKt2K~2]~Sin~(KP).
The related function CT(K”)
which obeys
=
BUS,
i
Sin
(p(K’)
UNSTABLE
179
PARTICLES
K~u(K~)/~(K~) = [k sin &xp)]3, is introduced
by eliminating
X2 to make explicit the zero of $(z)
at z = c:
K”) 1 =(p2 - 2)[B;’- (pZ - x)SW dK2K2S( Ql* (K2
-
j42)2(K2
-
2)
and thus
S(k) :=
1
Bo
k2 + jh2 - ie *
The asymptotic
1 -
(k2 + P2) srn dK2 (K2 _ &&) .o*
form of this function
m sgo2
2
dK2
shows directly
K2 _ ic>
that
2
(K;2K;2)2
1 -
=
B, < 1.
A stronger inequality has already been found. It can be rederived relation following from s(O) = XB2, ! 0
2
B.
z
1
-
p2
/-
‘J(K2)
dK2
(K”
*02
P
>
1
_
;
m
by using the
dK2
-
/.b2)2
K”‘J(K”>
Ko2 s ~~2
(K2
-
/.42)2’
to obtain m s .o* When this u construction T F$
Cot
(T(K2)
K20(
dK2 (K2
K”)
-
1
/A2)2
<
1
-
-
(P/U2
(P/‘d2(P/~)2’
of $j is employed, (p is computed (p(K2)
=
1 +
(K”
-
/.t’)P
/-
dK12
from (K,2
-K;$!;;A
_
K2),
dK2)
I’
,0*
and
2 dK2) K‘J 1 [l
+
(K2
-
P”>P
/
dKf2
(K,2
-K;2;:;;;
p2 _
K2,J
+
[r
-&P2
180
SCEIWINGER
If S(Z) possesses an equal number n = 0, 1, . . *, of poles and zeros, we write
s-‘(z) = gI ( g This F(x) coincident at infinity
>
(K02 - z)F(z).
approaches unity at infinity, and its singularities are comprised in a pole and branch point at z = ~0~.Hence the log F(x) that vanishes has a branch line extended from ~0~to to,
where the initial value of cpis chosen to reproduce
the pole of F(z)
at Q”,
(p(K02) = 7r. The equation to determine
(o(K~) has the same appearance as before,
7rK2S(K2)cot (p(K2) = K2 1 - c
(n)
but now the right-hand
.-SCK2 -
+ P $,, d/c’2 g2]
Vk2
side is negative at
K
=
Kg
-
h2,
. Accordingly,
0<~07r, with the upper limit being attained at the continuum threshold limit approached asymptotically as K” + 00. Note that
and the lower
with the exception of n = 0, where
When S(K’) and ‘p( K”) tend to zero with we can make the identification
The simplest type of Green’s function,
increasing
K”
more rapidly
than
(K”)-‘,
with no zero or pole, is represented
For completeness we record the resulting functions acterize the spectral forms of S(k) and g-‘(k),
B(K~)
and
s(K’),
which
by
char-
UNSTABLE
B(K2) = -+-
K**
-- l p P
exp
II
181
PARTICLES
00&‘2 1
A+( ‘2) -K*
XOZ
1
.-’ sin (p(fr2) ?r
>
K~s(K~)B(K~) = [i The construction the aid of
of B(K~) in terms of a(~~) follows
7rS(K2) cot Q(K2) = 1 the inverse evaluation
sin p(12)]?.
from the latter relation with
$ + P [,
dK’2 g*
;
is expressed by
7rB(K2) cot Q(K2> = -P
s., dK’2 a.
It is worth noting that these forms, appropriate to purely continuous are flexible enough to include the more general possibilities of discrete This extension is attained most simply from the q construction of S(z) mitting p( FC”)to be a discontinuous function. Thus, if we now designate imum value of K as KC,’ and choose
where
KO
Q = r,
0’
5K-Cl.h
Q = 0,
PI
< K < VI
Q=T,
VI
Vn-I
<
K <
jh
lJ.n <
K 6
KQ ,
Q =
rp
Q =
0,
marks the beginning d 2 Q(K2> KF_
of the continuous
spectrum,
spectra, spectra. by perthe min-
we get
1
K; - ZVI - 2 . . . v”,-1 - x = ~___ 2 /.L1* 2 p$ 2 l-b z
&K2
Qh2> K*-2
1 ’
and there emerges &)
=
E-1;(v21
;’
(n) cc2
exp
2
the form of S(z) appropriate to n poles and n - 1 zeros. Similarly, the Green’s
182 function with quence with
SCHWIKGER
n zeros and poles appears on terminating cp = T,
Vn-I < K < I&
P = 0,
/-h
cp =
vn
<
r,
K 5
the discontinuous
se-
Kg.
These remarks indicate that a Green’s function containing poles and zeros can be regarded as the limit of one with purely continuous spectra in which regions of rapid variation of the phase p(K2) have become points of discontinuity. Evidently there are circumstances in which a suffciently rapid variation is indistinguishable from a discontinuous one, which serves to remind us that a particle description of physical phenomena is an idealization, the validity of which depends upon the nature and accuracy of the description. To study these situations of concentrated phase variation in more detail we first observe that if (o is close to 0 or ?r, both B(K~) and s(K~) are correspondingly small, whereas one or the other of these functions acquires large values as (p(K2) deviates from the extreme limits. If cp diminishes from essentially the value ?r to 0, with increasing K2, it is B(K~) that changes rapidly, while S(K2) is the strongly varying function when (o increases from 0 to ?r. In either situation the center of the region of rapid variation can be placed at the point where (P(K~) = n/2. We shall designate such a center as P if 9 is decreasing, and as v for increasing (c. When K - p the equation to determine cot (p from S( K”) reads, approximately, ?rjb2s(/.b2)
Cot
where B is a positive constant,
(O(K2)
=
B-‘(K2
-
/.d2),
and therefore
w2Bsb2) K -
‘:
B(K2)
=
;
B
(K2
_
p2)2
+
(Tp2BS(p2))2’
If we now consider the limit s(p2) --+ 0, we get K -
B(K~)
p:
=
B~(K’
-
p2)
and the contribution from this neighborhood to s(z) is the term B/(p’ - z), describing a pole at ~1~.With K - v, we can write an approximate equation to determine cot cpfrom B( K”) , rB(v2)
where the constant
cot
s is positive.
(o(K2)
=
-s-‘(K2
-
v”),
Then mB( v”)
K -
‘:
dK2)
=
;’
(K2
_
y”)2
+
(,&(4)2’
UNSTABLE
183
PARTICLES
and in the limit B( v’) + 0, K N
V:
S(K’)
=
S&(K”
-
V”).
The resulting contribution to s-*(x) is - ZS/ ( Y” - z). The physical situation we are primarily concerned with is one where .s(K~) is very small below a crit’ical value of K, above which substantially larger values of s( K’) occur. This is usually described by a decomposition of physical interactions into t’wo categories, strong (S) and weak (W) , which is roughly the content of the separation S(K’) = %b2)
+ Sw(K2),
where Ss(K2) = 0,
K 5 Kos K > Kos ,
Sw(K2) << %(K2), and the positive We also have
continuous
function
So
approaches
zero at K =
KoW
<
Kos .
x2 = Aa + xw2, xw << As . In the approximation that sets sw = Xw = 0, t.he Green’s function has (for example) no zero, and one pole at the mass value pLg, which is to say that
1 defines a value of pa < KOB , and the description of some physical phenomena can be given in terms of a stable particle of mass ps . With a more accurate treatment, however, no stable particle exists, which is expressed by the inequality
or K”
dK2 $s(K’) (2
-
/Js2)($
-
Kiw)
1
> KiwJ -
UJW2
Evidently the smaller the magnitude dition satisfied by ~~~ < pcls, which
dK2
sw(K2) K2 -
_
xw2.
K&r
of sw and Xw , the more closely is this conis the elementary criterion for instability.
184
SCHWINGER
It may be noted here that, under the conditions the interaction independent mass ~0 , we have
PO2 =
interaction
Cot
(p(K’)
of
scheme and to the strong
side of the above inequality is the positive quantity m K2 dK2 Sw(‘&, s QW2 K2 K&r
which shows that PS must exceed Kow by a corresponding stability to appear. In the region K - PS < KO~ the equation to determine ?rK2Sw(K2)
the introduction
X2 + /-- dK2 a(~~), go2
which relation applies to the complete interactions separately. Hence
and the right-hand
permitting
=
K2
-
A:
+
K2 SW
kvP
finite amount
for in-
is
+o(K~)
di2 $$
- A,” + K2p ,“* dk2 $d!? J QW2
-
K2,
or, approximately, ~&~(p~)
cot
(o(K2)
=
BY&c2 - ps2) - hw2 + /.t2P * dK12p s QW2 - P2 + (K2 - M~>P j-i2
The value of
K
at which
$2
Q = ?r/2 is not quite PS , but is given by
p2 = /.Ls~+ Bos and the constant
di2 K$--+
Xw2 - /J~P
t2 SW(K’~)
- dK p--s QW2 -
iJ2
1
B in ~~~s,(p~)
cot
(o(K~)
=
B-‘(K~
-
~1~)
is obtained from B-’
= B&l + P /=
dtcr2-++
$2
( K’~SW(
K’~)
) .
WDW2
If the equation that determines
,A is further
approximated
by inserting
ps for ~1
UNSTABLE
on the right-hand Bos with .B,
185
PARTICLES
side, a slight increase in accuracy
/.t2 = ps2 + B
A,” - /LIMP
K -
p:
by replacing
00 s - 1 dK
12
in the neighborhood
hv(K’2)
p
QlW2
We see that the continuum by
is produced
PfJ2
*
of mass value P is characterized
Y/J B(K~) = B i 7r (2 - P2j2 + (r/J)“’
where
Y = wBs&~), and this .is equivalent to specifying a discrete mass only when the physical circumstances do not permit the constant y to be distinguished from zero. We have now reached the point in our examination of the structure of a particular type of Green’s function where an explicit consideration of time dependence is needed to complete the physical interpretation. It suffices for this purpose to use a mixed description in which the spatial momentum is specified, the time dependent Green’s function s(k,t) being derived from SW”) as the frequency
= /
K2 -
km
-
ie
integral s(k,t)
On performing
dB[K2] k2 $
=
the k” integration
m$ s--(o
e-ikots(k,ko).
we get
s(k,t)
=
s
dB[K2] &
I
e--iB~lt’,
with E, = (k2 + K~)I’~. We first consider the situation
of a single discrete mass, for which + 1,
dK2 ,@K2)
;&
e--iEK!
tI
1 .
Our principal concern here is with the asymptotic form of this function for large t (which is henceforth taken to be positive). The predominant contributions to the asymptotic value will come from the single term of definite frequency asso-
186
SCHWINGER
ciated with the discrete mass, and from the immediate neighborhoods of the lowest order thresholds. At the mass value which is the threshold of a new physical process, some derivative of the function /3( K”) is singular. A first-order threshold is one where the first derivat#ive is singular, as in K 2
Kj:
We shall assume here that
f
Kg
/C(K’)
-
f&GJ2.
, the lowest
of all thresholds,
is of first order. Then
s
., dK2@(K’) -$ e--iEKt = ; SW e-iEfit dfl(K2) x 4 -F
;
e-i~,jt
SW d(K
_
Kj) ;
0
[ 1 j3
h
Ki K -
I’”
e-WErj)t(W
Kj
provided Kjt
which
is satisfied for all thresholds
Erj/Kj 7
>>
whenever Kot >>
-&,/Ko
,
and
The absolute magnitude of the threshold contributions, discrete mass term, is less than /?(EKo/K~t)3’2, where
If ,B does not exceed unity the asymptotic evaluation
compared with the single
(it is certainly finite), this ratio is already small when applies, and for large /3 it becomes so when Kgt >>
@‘3(
EJKO).
We have now suitably qualified the statement that, eventually, the only significant contribution to the Green’s function comes from the term referring to the stable particle of mass IL, 0
$j(k,t)wBo--e
i
2
1 E,
--iE#t
=
B”
f:
$
k2
+
p2e-l*
;02
-
ie’
Without the constant B. this is the simple propagation function that displays the complete set of states accessible to a single particle of maw /.L.The constant
UNSTABLE
187
PARTICLES
B. appears here as the reminder of what is fundamental in the theory-the field, not the particle. Accordingly this constant is removed on transferring the description to the more immediate physical basis provided by the stable particles and their interactions. That, in a few words, is the general physical meaning of renormalization, as distinguished from the more prevalent notion in which it is equated to a formal subtraction device for the elimination of the divergences that appear in perturbation calculations. Finally, we come to the time dependence of the Green’s function when there is no stable particle. From what has already been said it is clear that the true asymptotic form contains only the contributions of the first-order thresholds.4 There is, however, another significant contribution under the circumstances outlined in the discussion of weak and strong interactions. In the vicinity of mass value p, the function B(tc2) varies strongly and attains large values. The contribution of this neighborhood to S(k,t) is
izs
rNCl dK2B(K2) $ eeiBKt II
m E;i Bk eeiEpt s d(K
=
;
B $
-
-or)
P
e-iE,d
P) L 2?r (K
-i(r/E,J -
Pj2’+
(4’)‘)”
t (K-/d
e
e-l12(riE,ht,
P
provided
that
which evidently describes the exponential ~1and proper mean life time 7, given by
decay of an unstable particle
1 - = y = - k Im B $‘(k) 7 Thus, a more complete asymptotic
evaluation Kot >>
of mass
1k~+,~=o. of S(k,t)
for
&/Ko
is given by
The threshold summation “strong thresholds”, pi 2 4 More
accurately,
includes both “weak thresholds”, ~j 2 KOW , and The lowest threshold KO may refer either to ~0~
~08 .
the first-order
thresholds
for the production
of stable
particles.
188
SCHWINGER
KOS , depending upon the desired accuracy of the discussion. (A weak threshold at zero massrequires a separate treatment.) When the threshold asymptotic evaluation has become valid, or at the possibly greater time produced by the factor /32’3,the argument of the exponential decay factor is somelarge numerical multiple of
Or
P -So Y -----NW* E p KO KO
Y KO
Hence, if y is sufficiently small,
there is an extended time interval during which the contribution of the thresholds has become negligible and the Green’s function is completely dominated by the term referring to the unstable particle, --i!& O”dk” e 6(W) - B f’ & e-~E,te-l/2b/E,ht = jj l m_2a k2 + (/,J - +ir)” - kE P The exponential time decay of this term, in comparison with the algebraic decrease associated with the stable particle threshold contributions, indicates however that the latter are eventually the sole survivors of the asymptotic evaluation, as we have remarked. Does this mean that the intrinsic decay law of unstable particles ceasesto be of exponential form after a sufficiently long time? The answer to this question (which has occasionally been given by a categorical affirmative) illustrates the necessity of including suitable idealizations of all relevant aspects of the physical situation in its mathematical representation. We first remark that any physical process can be conceived as initiating in and terminating with stable particles. An example is the reaction p + p ---) p + p + e+ + em+ 2y + 21, + 2i;. The multipoint Green’s function that gives an account of this process will be constructed, in part, from the simplest Green’s functions, describing the propagation of the various stable particles into or out of the interaction region, and from a function that gives the details of the interactions among these particles. The nature of this interaction is specified more closely when we add the information that the above reaction has proceeded through the following stages: p+p+K++C++n, K+ + fi + (p+ + e+ + 2v), c+-, ?a--,
Q + ho-+&), p+e-+i;.
UNSTABLE
189
PARTICLES
Correspondingly, the function that characterizes the stable particle interactions will contain the Green’s functions that refer to the propagation of the intermediate unstable particles, together with details of their localized interactions. But this is not all. The statement that the reaction has involved the particle K+, for example, and not some kinematically equivalent combination of particles, must have a physical counterpart in an aspect of the measurement apparatus that serves to display the presence of the particle, and a mathematical counterpart, in an operation on the relevant Green’s function, that conveys the effect of the selective measurement. The only valid basis for asserting that a particular unstable particle has been involved is through an energy-momentum or mass determination of sufficient accuracy to rule out all other possibilities. This need not be a disturbing measurement in any other respect, however. The mathematical representation of this mass filter is supplied by the modified Green’s function
where
M(K)
is roughly
characterized
by the properties
M(K)
= 1,
1% - ~11 5 Au
M(K)
= 0,
\K - ~11 2 A/J
and Ap, measuring the precision of the mass determination, is restricted by y << A~.L<< /.L. Thus the threshold contributions, referring to recognizably distinct physical processes,are removed from the Green’s function and
where
g(S) =,[lC&i - P> &cK _$+(lr>2 e-‘(‘-p)aM(K) 2 _s00 =e -w2)Ya
d(K
-
EL)
$
(K
_
p)2y+
--m
(Lr)2 2
(l
-
M(K))
e--i(r-r)nv
Since 1 - M(K) does not differ from zero until ) K - p / >> y, the denominator of the lat,ter integral can be approximated by (K - P)~. On performing a partial integration that becomesasymptotically valid when A/.Ls>> 1,
190
SCHWINGER
we obtain g(s) N e-(l’Z)rs -
yli _-AlAps
co d(K - p) e-i(n-p)s g M(p), 21r s--m
which contains a further approximation that is justified by the localized nature of dM/dK. The second term of g(s) is evidently quite negligible, at least for moderate values of ys. The question before us, however, concerns the behavior of g(s) after very many lifetimes have elapsed. We must now introduce more detailed assumptions about M(K) in order to provide analytical equivalents of realizable experimental procedure. The major hypothesis is the boundedness of all derivatives of M(K) . If this property failed at some value of K, that mass value would be specifically distinguished, in contrast with the limited selectivity of any finite act of measurement. We shall also assume, for simplicity, that the shape of M(K) in the neighborhood of p + Ap = PZ is the mirror image of that near ~1- Ap = p1 . Then, confining the integration to the latter neighborhood, we have
g(s)N e-~~~~)rs + 2 sin - Aw1 Ap
Aficcs ?r s
d(K
-
pl)
e--i(r-sl)s
d”, M(K).
According to the boundedness hypothesis for all derivatives of M(K), the latter integral decreases asymptotically more rapidly than any power of (l/s). The asympt,otic behaviour is essentially exponential, of the form exp [-f( I’s)], where r >> y measures t.he boundary width of M(K), or physically, the latitude in the precise specification of the mass interval accepted by the filter. Thus during the extended interval r-l
<< s 5 y-l,
the second term of g(s) is not only small in comparison with the exponential, but is also decreasing much more rapidly. Whether or not this remains true for all time depends upon the precise character of the function that gives the exponential dependence on I’s. If f( r s ) is ultimately some finite numerical multiple of rs, the first term of g(s) will dominate for all s, whereas should the growth of f( Ik) with increasing s be less than linear, the roles of the two terms in g ( s ) will eventually reverse. The situation can be studied with the aid of the following class of functions, which satisfy the requirements that have been imposed on M(K) :
UNSTABLE
191
PARTICLES
Here X and a are positive numbers and c
-1
s 1
=
dx
e-(h/2a)
(l-22)-0
-1
The required asymptotic form can be obtained gration path into the complex plane, and
s 1
dx
e-o;2.)(1-z~)-Q
e--ire
-a(,
1 ;
by suitably
deforming
the inte-
yya>ll(l’*)~I
-1
exp[-(cos~)&(~F]sin[~+,
- (sin~)-&(~~],
in which ,=&
(r/71a >> 1,
the decaying exponential term, e-(1’2)‘us,ceasesto dominate g(s). But, precisely when this will occur, and the form of g(s) that ensues,is unavoidably dependent upon the details of the measurement process and is not intrinsic to the unstable particle. We conclude that with the failure of the simple exponential decay law we have reached, not merely the point at which some approximation ceasesto be valid, but rather the limit of physical meaningfulness of the very concept of unstable particle. As it stands, this discussion of a simple boson Green’s function applies only to the Hermitian field associated with the unstable particle s’,(where the “weak” decay mechanism is electromagnetic in nature). An extension to a two-component Hermitian field is needed to discussthe particles T*, K*, and ICI,2 . The Green’s function s(x
- x’)d
= i <
(&(x)&,(x’))+
>
= / g4
ezk(z-s’)S(k),b
is a symmetrical function of the indices x,a and x’, b. The space-time coordinates can appear only in the translational, rotational invariant combination (x - x’)* (an otherwise conceivable dependence upon the algebraic sign of x0 - x0’, in the interior of the light cone, is excluded by the requirement of a lowest energy
192
SCHWINGER
state) and therefore S(J: - ~‘),b is necessarily symmetrical in the indices a and b. When these indices refer to the two axes of an electric charge space, the structure of the matrix $,b is completely restricted by the conservation of electric charge
[%Sl =0,
q=
for there is only one type of two-dimensional with the antisymmetrical charge matrix: g(k)ab
=
O-i i 0’
(
)
symmetrical
matrix that commutes
&b$j(k).
Thus the problem is reduced to that of a single component field, and this situation is not changed by introducing the non-Hermitian fields that diagonalize q. The value of the electric charge does not affect the properties contained in the Green’s function and masses and lifetimes are identical for K+ and 6, K+ and
K-. It is a different matter when the two-fold multiplicity refers to a hypercharge space for this property is not conserved by the weak interactions, and the designations K”( Y = + 1) and K”( Y = - 1) are meaningful only for sufficiently short times. We accept the conservation law implied by invariance under space and charge reflection and conclude that S is invariant under charge reflection, since it is explicitly invariant to the reflection of the space coordinates. The two Hermitian field components are distinguished by the response to charge reflection, as it operates in the hypercharge space, and Qeb must be a diagonal matrix. However, there is no necessary connection between the diagonal elements 61 and ~2 , except in the approximation that ignores weak interactions where they are equal. Thus from $jl and $ we infer two distinct sets of masses and lifetimes, which characterize the unstable particles K, and Kt . The unstable particle contribution to S(k$), is
or, in view of the small difference anticipated
s(W),
i -zige
B
--iE,,t
between
B1 and Bz , p1 and pZ ,
{~.,,p[-~(611--iyl)s]+6anexp[-~(--6p--iyl)s]}.
Here
while p and s = t(p/Ep) refer to the average mass. If 6~ << Ap, the mass interval that is accepted in selecting electrically neutral K-particles, the asymptotic
UNSTABLE
193
PARTICLES
structure of $,b is left intact and we encounter a kind of mass interferometer. We need not repeat the discussion, based previously on more elementary considerations, of the experimental possibilities thus implied. If 6~ is so large that a mass separation of K1 and Kz becomes feasible, no significant interference effects can occur. The analogous treatment for fermion fields is sufficiently different in detail that it will be deferred to another publication. Note added in proof: It should have been mentioned in the text that these considerations are incomplete in one important respect. The full dynamical effect of the electromagnetic field is not represented since a more elaborate spectral form is required for the Green’s functions of charge-bearing fields in the physical radiation gauge. This refinement will not alter our physical conclusions, however. RECEIVED:
August
21, 1959 REFERENCES
1. P. T. MATTHEWS AND A. SALAM, 13, 115 (1959). 8. P. C. MARTIN AND J. SCHWINGER, 1342 (1959).
Phys. Bull.
Rev. 112,283 Am.
Phys.
(1958); Sot.
[II],
M.
Llivy,
3,202
Nuovo (1958);
cimento
Phys.
Rev.
[lo], 116,
OF
PHYSICS:
9,
169-193
Field
(1960)
Theory
of Unstable
JULIAN Harvard
University,
Particles*
SCHWINGER Cambridge,
Massachusetts
Using the example of a spinless boson field, the structure of the simplest Green’s function is developed to provide a uniform theory of particles, stable and unstable. Some attention is given to the time decay law of unstable particles and it is emphasized that a full account of the relevant physical situation must be contained in its mathematical representation, leading to the conclusion that an essential failure of the exponential decay law marks the limit of applicability of the physical concept of unstable particle. There is a brief discussion of the rr and K mesons.
Some attention’ has been directed recently to the field theoretic description of unstable particles. Since this question is conceived as a basic problem for field theory, the responseshave been somespecial device or definition, which need not do justice to the physical situation. If, however, one regards the description of unstable particles to be fully contained in the framework of the general theory of Green’s functions, it is only necessaryto emphasize the relevant structure of these functions, That is the purpose of this note. What is in essencethe same question, the propagation of excitations in many-particle systems where stable or longlived “particles” can occur under exceptional circumstances, has already been discussed(2) along these lines. A relativistic field describes a localized excitation which produces a spectrum of energies or massesthat ranges down to a lower limit characteristic of the field type. Thus for any electrically charged boson or lepton field this theoretical lower limit is the electron massm, while for electrically neutral fields of these two varieties the massspectrum in principle extends down to zero. These limits are set by the massesof the absolutely stable’ particles to which one is led by decay processesthat respect essentially only the conservation of electric charge Q, in addition to the usual mechanical properties. With fermion fields carrying nucleonic charge N, however, the absolute conservation of the latter evidenced by the sta* This work was supported in part by the U.S. Air Force. It was reported at the Ninth International Conference on High Energy Physics, Kiev, U.S.S.R., July 15-25, 1959. 1 See, for example, Matthews and Salam (1) and a subsequent work of these authors, together with a recent paper by Levy (1). * At least on a sub-cosmological scale. 169
170
SCHWINGER
bility of the proton implies a mass lower limit that equals M, the proton mass, for fields with Q = N, becomes M + m for electrically neut’ral fields, and is M + 2m for Q = -N (omitting the binding energy of hydrogeo at’om or ion!). The propagation of such excitations, superimposed on the vacuum state, is described by the Green’s functions as time ordered field correlation functions. For a single spinless boson field 4(z), the simplest example is SC” - z’> = mw#w)+) =
(dk) s
where, according to the elementary
eik(S-z’)s(k)
(27r)4
relativistic
’
structure
s(k) =s
dB[t?] k2 + K2 -
of its spectral form,
ic
Here B[K~] is a real non-decreasing function that equals zero for sufficiently small K2 and approaches unity as ~~ -+ to. Another form appears on exploiting the latter property as a minimum formal characterization of a local field,
S(k’)- k2, We first remark
k2-+
that the complex variable
co.
function
can have no complex zeros, since (K2
s
(K2
-
2)
dB[?] + iy
-
X)”
+
implies y = 0. Accordingly
dB[ttl s
y2
(K”
-
2)”
+
y2
=
’
the function P(x)
= s-‘(z)
+ 2
is regular everywhere in the finite complex plane, with the exception of the positive real axis, a semi-infinite portion of which constitutes a branch line for this function corresponding to the branch line of S(z) associated with the continuous spectrum. It is also possible to have isolated poles, which refer to real zeros of S(Z) appearing at points within the spectrum where B[K’] is constant. The function z-‘P(z) vanishes at infinity in the cut plane and also has a pole at the origin, from which it follows that z-‘p(x)
ds[K2] = x” z s r2>0 K 2-z
UNSTABLE
171
PARTICLES
or
The real function
s[K’] is shown to be nondecreasing
on translating
the property
& Pax:+ +I) - $32 - Q/)1> 0 into
In writing the residue of z-‘P(z) at the origin as X2 > 0 we have anticipated the expression of a physical requirement, that S(Z) have no singularity for negative z or space-like3 k. The function s-‘(z) begins at + 00, for z = - 00, and decreases monotonically along the real z axis, according to
If S-‘(Z) is to have no zero for x I 0, it is necessary and sufficient that 6-l (0) = X2 > 0. Alternatively, we recognize from the comparison of the two expressions for s(O) that
dS[K2] -l S(k) =k2 +x2 -it+lc2 sk2 +K2 -‘k1’
The second form thus obtained for s(rC) is
the assumed existence of which requires the convergence at infinity of the integral j-( K2)-’ dS[K2]. Th is convergence condition also assures the validity of the asymptotic form (k2)-’ as k2 approaches infinity either in a space-like or time-like direction. Should more be true and $ ds[~‘] converge, which implies a finite total increase in S[K2] from an initial zero value, we could infer the more detailed asymptotic property S(k)
-
(k2 + POY
where ~02 = x2 + s[oc] > x2. 3 This includes k2 = 0. The exceptional netic field. presumably do not occur with
circumstances a spinless field.
encountered
for
the electromag-
172 An inspection
SCHWINGER
of the spectral form for s(k)
Under the conditions
shows that ~02 is also given by
dB[K2]. iJo2 =sK2
which permit the identification
of cl0 it is possible to write
k2+ K2- in1’ The constant PO can be interpreted as the mass parameter associated with the field independently of interactions, at, least when these couplings are linear in the boson field. One anticipates two possible spectral conditions from the general physical conviction that a continuous energy or mass spectrum never terminates nor has jump discontinuities. Either B[K’] is a continuous increasing function for all K > KO , or, there are one or more positive steps in B[K~] followed by a continuous increasing domain. In the first circumstance there is no discrete mass value, which is to say, no stable particle associated with the field, while the alternative corresponds to one or more such particles. If there is no isolated singularity in s(z) the function s-‘(x) can have no zero for 0 < z < ~02, where ~~ marks the threshold of the continuous spectrum. The condition for this property, which expresses the absence of a stable particle, is s-‘(Ko” - 0) > 0, or
x2>.o++J.,s2]s
Thus it is necessary, but not sufficient, for X to exceed exist. This also follows from the observation that
KO
if no stable particle is to
The sign of S-l( ~02- 0) must become negative if a discrete mass value is to occur in 8. Indeed, the condition that S-‘(z) vanish at z = p2 > 0 is given by x2 = $[l < Kofl
+ SK*]
+/-$$]r
under the assumption that s[K~]is a continuous function. The inequality is obtained by remarking that the function of p2increasesmonotonically in the interval 0 < p2 < KO’. A value of X2that satisfiesthe inequality defines a unique p2 in this segment.
UNSTABLE
173
PARTICLES
To consider in more detail the situation of a stable particle we indicate explicitly the nature of the spectrum by writing
s(z) = s2
+ /* dK”p - 2’ ‘Lo2
where 0 < Bo < 1, and the continuous function B( K”) , which is positive for K’ > co2and vanishes at the beginning of this domain, obeys 0 dK” B(/c’) = 1 - Ba . s102 If we place B(2)
= B&K?,
the latter property appears as Bo = [ 1 + Ia dK” &)I-‘. v2 We also have
from which some inequalities
can be inferred. Thus Xm2 > Bopp2 and therefore u” > BoA’,
while the property 6” >
~02
> p2 shows that X2 < F-‘, or X2 > $ > BOX’.
A more precise version of the second inequality is 1 1 - Bc, ,
As 2 varies from p2 + 0 to ~02 - 0 the function from --oo to the limiting value
~(2)
increases monotonically
174
SCHWINGER
which must still be negative if 9 is to have no zero in the interval. under the restriction
the function
s-‘(z)
has no isolated singularity.
Accordingly,
This is made explicit by writing
where the continuous positive function s( K”) , K > KO , vanishes for K = KO . As one application of the inequality that characterizes this situation we note that
BO>smd2B(K’) >smd/(-z-w2B(K2) 1&
Kc+ -
which
j.12
K2 -
4
shows the necessity
Ko2
go2
K2
x2
of the property
Bo> (D’[l - ($1. A more stringent
condition
merges from the evaluation
Bo’ = -&l(z)
lzzp2
together with
1
6’) dK2K2 P2 ’ since BF’
x”= s(K2) < P2 smd2 l-J2 K02
K
(K”
-
/.L2)2
and therefore -
1 - (P/Ko>~ b/~>2hd’d2
’
/.12
’
UNSTABLE
Related inequalities
are co dKz K24K2) sd (K’ /.L2)’
175
PARTICLES
= B,’
_ 1 < x2 - pz K02 Ko2 -
/A2 ,.L2
and m
2
dtc” s kT2 (K” If the structure
this function
B,+z(K~) ’
of s(z)
1
1 -
-
wu2
(P/Ko)~(P/~)~
*
is such that
has a zero and s-‘(z)
has a pole at x = v”, /.k <
Accordingly,
1 - Bo <
=
/A’)’
U <
Kos
we now have s-l(Z)
= ii2 - x - x -+
-
x
z I- d:& d
2
where the constant s and the continuous function s( K’), K > KO , are positive, the latter approaching zero at the threshold of the continuous spectrum. It follows, conversely, from the monotonic decrease to - 00 as z increases to V” - 0, and the positive value at z = 0, that g-l(z) has a single zero at z = J.L~< v”, determined by x2 = p2 1 + -& [
p2
+
[,
dK”
$4
*
Again we calculate B, from
B;’= -g f(z) lz=p2 =; +P2 [ (v2 _s&2 +/-4 A2(K2s!2;2)2] which implies a similar inequality
for B. , with
Bo > (i-m2 On writing
the expression
1 -
for BZ1 as 2
BT”
we learn that
=
1
+
(v2
‘“,2j2
Y replacing
1 - (P/W2 (&)2(p/“)2’
KO ,
)
176
SCHWINGER
2 ($ I p2)2 Be-s < lUnder our present assumptions S(KO - 0) is positive, which is to say that the monotonic decrease of $‘(z) in the interval between V” + 0 and Q’ - 0 does not change the sign of this function. Hence
and, if X2 is eliminated
u” (
Kc?
-
V”)(U”
-
/AZ)
by employing
the equation
m s
s>l+
p2, we get
K2S(K”)
K02 dK”
This shows the necessity
that determines
(K”
-
J)(K”
-
Ko2)
of the condition Ko2
-
5’2
Ko2
-
/.k2
2
if both s(x) and s-‘(x) contain a single pole. Other significant implied by the lower limit for Bos, together with
inequalities
are
2 B.
+
(v2
1
II”)”
Bos
<
1.
Thus
and on comparison
with the lower bound for B.
we find that
h > /J(Ko/U). If the various
parameters x 2
in ~-I(Z) [
l-
&
were altered to satisfy +
J-y*
dK2
&2]
the inequality ’
this function would pass through zero in the interval between V” + 0 and Kt - 0, which implies a second pole for s(z) at z = p12, p < Y < $ < Kg . We have now sufficiently traced the unfolding of the Green’s function’s spectral structure to permit the visualization of a hypothetical situation where the field has n kinematically similar stable particles associated with it, of masses ,.Q , . . . , fin , represented by poles of $j( z) at z = pk2, k = 1, . . . , n. There are also zeros of S(Z)
UNSTABLE
177
PARTICLES
or poles of s-‘(z) at z = vk2, where ~~ < vl < ~2 , . . . , pnWl < vnwl < pn , and there can, but need not necessarily appear an nth zero v,, located between pn and KO , the threshold of the continuum. For the situation in which s(z) exhibits n = 1, 2, . . . poles and n - 1 zeros, we define F(z) such that
+ [, &”$f&, =
$j-‘(2)= A”- 2- 2
It will be observed that F(z) approaches unity at infinity, has no zero or poles, and possesses a branch line extended from Ko2 to to. Accordingly, there is a branch of log F that vanishes at infinity and has no singularity other than the branch point at Q~, which enables us to write
The real continuous function (P(K~) approaches zero at infinity and must vanish at K = Kc) , since log F(x) is regular at this point. The resulting expression for $(z) is dK2
‘Ph2> K2-Z
1 ’
which must still be qualified by the restriction on (o(K~) that is equivalent to the positiveness of s(K~), K > KO . That characteristic of the Sk is already contained in this representation through the property F(x) > 0, x < ~02, and the relationship of the Elk and Vk . We let z = x $ iy, x > KO~, y + +0, and compare the ratios of real and imaginary parts of $j-‘( z) for the two forms, using the evaluation
1 ,
which
gives ?fK2s(
K2)
C!Ot(o(K2)
=
K”
The function on the right-hand side has a positive value at K = ~0, according to the nature of the spectrum, which implies that P( K”) is positive in a neighborhood of the continuum threshold. We conclude from the physical hypothesis of a nonzero S( K2), K > K~ , that tan (p cannot vanish and therefore that the values of cp lie entirely in the interval
178
SCHWINGER
The lower limit is attained at the threshold of t’he continuum ically approached at infinity. Let us note here that
and is asymptot-
If it should be true that a( K”) vanishes more rapidly than (x2)-’ as K” approaches infinity, so also would P(K~), and one could make the identification
In the simplest have
example of this type, with
one pole and no zero in S(Z), we
dK” dK2> K2-2
1 ’
or
S(k) =
k2
+fi
-
ie
ex+2
+
2)
;
SWK02 dK”
(K”
_
p2)$2;
K2 _
J
where it has been recognized that
1
dK2 dK2> ra P2 Other quantities
that can be derived from this form are
P(K2)= --+ P2exp[ - (K2- p2); p [, Al2 CK’2 _
;!;;31:,
_
K2,]
x
and S(K2)
=
(1
-
~)e~p[~P~~~dKt2K~2]~Sin~(KP).
The related function CT(K”)
which obeys
=
BUS,
i
Sin
(p(K’)
UNSTABLE
179
PARTICLES
K~u(K~)/~(K~) = [k sin &xp)]3, is introduced
by eliminating
X2 to make explicit the zero of $(z)
at z = c:
K”) 1 =(p2 - 2)[B;’- (pZ - x)SW dK2K2S( Ql* (K2
-
j42)2(K2
-
2)
and thus
S(k) :=
1
Bo
k2 + jh2 - ie *
The asymptotic
1 -
(k2 + P2) srn dK2 (K2 _ &&) .o*
form of this function
m sgo2
2
dK2
shows directly
K2 _ ic>
that
2
(K;2K;2)2
1 -
=
B, < 1.
A stronger inequality has already been found. It can be rederived relation following from s(O) = XB2, ! 0
2
B.
z
1
-
p2
/-
‘J(K2)
dK2
(K”
*02
P
>
1
_
;
m
by using the
dK2
-
/.b2)2
K”‘J(K”>
Ko2 s ~~2
(K2
-
/.42)2’
to obtain m s .o* When this u construction T F$
Cot
(T(K2)
K20(
dK2 (K2
K”)
-
1
/A2)2
<
1
-
-
(P/U2
(P/‘d2(P/~)2’
of $j is employed, (p is computed (p(K2)
=
1 +
(K”
-
/.t’)P
/-
dK12
from (K,2
-K;$!;;A
_
K2),
dK2)
I’
,0*
and
2 dK2) K‘J 1 [l
+
(K2
-
P”>P
/
dKf2
(K,2
-K;2;:;;;
p2 _
K2,J
+
[r
-&P2
180
SCEIWINGER
If S(Z) possesses an equal number n = 0, 1, . . *, of poles and zeros, we write
s-‘(z) = gI ( g This F(x) coincident at infinity
>
(K02 - z)F(z).
approaches unity at infinity, and its singularities are comprised in a pole and branch point at z = ~0~.Hence the log F(x) that vanishes has a branch line extended from ~0~to to,
where the initial value of cpis chosen to reproduce
the pole of F(z)
at Q”,
(p(K02) = 7r. The equation to determine
(o(K~) has the same appearance as before,
7rK2S(K2)cot (p(K2) = K2 1 - c
(n)
but now the right-hand
.-SCK2 -
+ P $,, d/c’2 g2]
Vk2
side is negative at
K
=
Kg
-
h2,
. Accordingly,
0<~07r, with the upper limit being attained at the continuum threshold limit approached asymptotically as K” + 00. Note that
and the lower
with the exception of n = 0, where
When S(K’) and ‘p( K”) tend to zero with we can make the identification
The simplest type of Green’s function,
increasing
K”
more rapidly
than
(K”)-‘,
with no zero or pole, is represented
For completeness we record the resulting functions acterize the spectral forms of S(k) and g-‘(k),
B(K~)
and
s(K’),
which
by
char-
UNSTABLE
B(K2) = -+-
K**
-- l p P
exp
II
181
PARTICLES
00&‘2 1
A+( ‘2) -K*
XOZ
1
.-’ sin (p(fr2) ?r
>
K~s(K~)B(K~) = [i The construction the aid of
of B(K~) in terms of a(~~) follows
7rS(K2) cot Q(K2) = 1 the inverse evaluation
sin p(12)]?.
from the latter relation with
$ + P [,
dK’2 g*
;
is expressed by
7rB(K2) cot Q(K2> = -P
s., dK’2 a.
It is worth noting that these forms, appropriate to purely continuous are flexible enough to include the more general possibilities of discrete This extension is attained most simply from the q construction of S(z) mitting p( FC”)to be a discontinuous function. Thus, if we now designate imum value of K as KC,’ and choose
where
KO
Q = r,
0’
5K-Cl.h
Q = 0,
PI
< K < VI
Q=T,
VI
Vn-I
<
K <
jh
lJ.n <
K 6
KQ ,
Q =
rp
Q =
0,
marks the beginning d 2 Q(K2> KF_
of the continuous
spectrum,
spectra, spectra. by perthe min-
we get
1
K; - ZVI - 2 . . . v”,-1 - x = ~___ 2 /.L1* 2 p$ 2 l-b z
&K2
Qh2> K*-2
1 ’
and there emerges &)
=
E-1;(v21
;’
(n) cc2
exp
2
the form of S(z) appropriate to n poles and n - 1 zeros. Similarly, the Green’s
182 function with quence with
SCHWIKGER
n zeros and poles appears on terminating cp = T,
Vn-I < K < I&
P = 0,
/-h
cp =
vn
<
r,
K 5
the discontinuous
se-
Kg.
These remarks indicate that a Green’s function containing poles and zeros can be regarded as the limit of one with purely continuous spectra in which regions of rapid variation of the phase p(K2) have become points of discontinuity. Evidently there are circumstances in which a suffciently rapid variation is indistinguishable from a discontinuous one, which serves to remind us that a particle description of physical phenomena is an idealization, the validity of which depends upon the nature and accuracy of the description. To study these situations of concentrated phase variation in more detail we first observe that if (o is close to 0 or ?r, both B(K~) and s(K~) are correspondingly small, whereas one or the other of these functions acquires large values as (p(K2) deviates from the extreme limits. If cp diminishes from essentially the value ?r to 0, with increasing K2, it is B(K~) that changes rapidly, while S(K2) is the strongly varying function when (o increases from 0 to ?r. In either situation the center of the region of rapid variation can be placed at the point where (P(K~) = n/2. We shall designate such a center as P if 9 is decreasing, and as v for increasing (c. When K - p the equation to determine cot (p from S( K”) reads, approximately, ?rjb2s(/.b2)
Cot
where B is a positive constant,
(O(K2)
=
B-‘(K2
-
/.d2),
and therefore
w2Bsb2) K -
‘:
B(K2)
=
;
B
(K2
_
p2)2
+
(Tp2BS(p2))2’
If we now consider the limit s(p2) --+ 0, we get K -
B(K~)
p:
=
B~(K’
-
p2)
and the contribution from this neighborhood to s(z) is the term B/(p’ - z), describing a pole at ~1~.With K - v, we can write an approximate equation to determine cot cpfrom B( K”) , rB(v2)
where the constant
cot
s is positive.
(o(K2)
=
-s-‘(K2
-
v”),
Then mB( v”)
K -
‘:
dK2)
=
;’
(K2
_
y”)2
+
(,&(4)2’
UNSTABLE
183
PARTICLES
and in the limit B( v’) + 0, K N
V:
S(K’)
=
S&(K”
-
V”).
The resulting contribution to s-*(x) is - ZS/ ( Y” - z). The physical situation we are primarily concerned with is one where .s(K~) is very small below a crit’ical value of K, above which substantially larger values of s( K’) occur. This is usually described by a decomposition of physical interactions into t’wo categories, strong (S) and weak (W) , which is roughly the content of the separation S(K’) = %b2)
+ Sw(K2),
where Ss(K2) = 0,
K 5 Kos K > Kos ,
Sw(K2) << %(K2), and the positive We also have
continuous
function
So
approaches
zero at K =
KoW
<
Kos .
x2 = Aa + xw2, xw << As . In the approximation that sets sw = Xw = 0, t.he Green’s function has (for example) no zero, and one pole at the mass value pLg, which is to say that
1 defines a value of pa < KOB , and the description of some physical phenomena can be given in terms of a stable particle of mass ps . With a more accurate treatment, however, no stable particle exists, which is expressed by the inequality
or K”
dK2 $s(K’) (2
-
/Js2)($
-
Kiw)
1
> KiwJ -
UJW2
Evidently the smaller the magnitude dition satisfied by ~~~ < pcls, which
dK2
sw(K2) K2 -
_
xw2.
K&r
of sw and Xw , the more closely is this conis the elementary criterion for instability.
184
SCHWINGER
It may be noted here that, under the conditions the interaction independent mass ~0 , we have
PO2 =
interaction
Cot
(p(K’)
of
scheme and to the strong
side of the above inequality is the positive quantity m K2 dK2 Sw(‘&, s QW2 K2 K&r
which shows that PS must exceed Kow by a corresponding stability to appear. In the region K - PS < KO~ the equation to determine ?rK2Sw(K2)
the introduction
X2 + /-- dK2 a(~~), go2
which relation applies to the complete interactions separately. Hence
and the right-hand
permitting
=
K2
-
A:
+
K2 SW
kvP
finite amount
for in-
is
+o(K~)
di2 $$
- A,” + K2p ,“* dk2 $d!? J QW2
-
K2,
or, approximately, ~&~(p~)
cot
(o(K2)
=
BY&c2 - ps2) - hw2 + /.t2P * dK12p s QW2 - P2 + (K2 - M~>P j-i2
The value of
K
at which
$2
Q = ?r/2 is not quite PS , but is given by
p2 = /.Ls~+ Bos and the constant
di2 K$--+
Xw2 - /J~P
t2 SW(K’~)
- dK p--s QW2 -
iJ2
1
B in ~~~s,(p~)
cot
(o(K~)
=
B-‘(K~
-
~1~)
is obtained from B-’
= B&l + P /=
dtcr2-++
$2
( K’~SW(
K’~)
) .
WDW2
If the equation that determines
,A is further
approximated
by inserting
ps for ~1
UNSTABLE
on the right-hand Bos with .B,
185
PARTICLES
side, a slight increase in accuracy
/.t2 = ps2 + B
A,” - /LIMP
K -
p:
by replacing
00 s - 1 dK
12
in the neighborhood
hv(K’2)
p
QlW2
We see that the continuum by
is produced
PfJ2
*
of mass value P is characterized
Y/J B(K~) = B i 7r (2 - P2j2 + (r/J)“’
where
Y = wBs&~), and this .is equivalent to specifying a discrete mass only when the physical circumstances do not permit the constant y to be distinguished from zero. We have now reached the point in our examination of the structure of a particular type of Green’s function where an explicit consideration of time dependence is needed to complete the physical interpretation. It suffices for this purpose to use a mixed description in which the spatial momentum is specified, the time dependent Green’s function s(k,t) being derived from SW”) as the frequency
= /
K2 -
km
-
ie
integral s(k,t)
On performing
dB[K2] k2 $
=
the k” integration
m$ s--(o
e-ikots(k,ko).
we get
s(k,t)
=
s
dB[K2] &
I
e--iB~lt’,
with E, = (k2 + K~)I’~. We first consider the situation
of a single discrete mass, for which + 1,
dK2 ,@K2)
;&
e--iEK!
tI
1 .
Our principal concern here is with the asymptotic form of this function for large t (which is henceforth taken to be positive). The predominant contributions to the asymptotic value will come from the single term of definite frequency asso-
186
SCHWINGER
ciated with the discrete mass, and from the immediate neighborhoods of the lowest order thresholds. At the mass value which is the threshold of a new physical process, some derivative of the function /3( K”) is singular. A first-order threshold is one where the first derivat#ive is singular, as in K 2
Kj:
We shall assume here that
f
Kg
/C(K’)
-
f&GJ2.
, the lowest
of all thresholds,
is of first order. Then
s
., dK2@(K’) -$ e--iEKt = ; SW e-iEfit dfl(K2) x 4 -F
;
e-i~,jt
SW d(K
_
Kj) ;
0
[ 1 j3
h
Ki K -
I’”
e-WErj)t(W
Kj
provided Kjt
which
is satisfied for all thresholds
Erj/Kj 7
>>
whenever Kot >>
-&,/Ko
,
and
The absolute magnitude of the threshold contributions, discrete mass term, is less than /?(EKo/K~t)3’2, where
If ,B does not exceed unity the asymptotic evaluation
compared with the single
(it is certainly finite), this ratio is already small when applies, and for large /3 it becomes so when Kgt >>
@‘3(
EJKO).
We have now suitably qualified the statement that, eventually, the only significant contribution to the Green’s function comes from the term referring to the stable particle of mass IL, 0
$j(k,t)wBo--e
i
2
1 E,
--iE#t
=
B”
f:
$
k2
+
p2e-l*
;02
-
ie’
Without the constant B. this is the simple propagation function that displays the complete set of states accessible to a single particle of maw /.L.The constant
UNSTABLE
187
PARTICLES
B. appears here as the reminder of what is fundamental in the theory-the field, not the particle. Accordingly this constant is removed on transferring the description to the more immediate physical basis provided by the stable particles and their interactions. That, in a few words, is the general physical meaning of renormalization, as distinguished from the more prevalent notion in which it is equated to a formal subtraction device for the elimination of the divergences that appear in perturbation calculations. Finally, we come to the time dependence of the Green’s function when there is no stable particle. From what has already been said it is clear that the true asymptotic form contains only the contributions of the first-order thresholds.4 There is, however, another significant contribution under the circumstances outlined in the discussion of weak and strong interactions. In the vicinity of mass value p, the function B(tc2) varies strongly and attains large values. The contribution of this neighborhood to S(k,t) is
izs
rNCl dK2B(K2) $ eeiBKt II
m E;i Bk eeiEpt s d(K
=
;
B $
-
-or)
P
e-iE,d
P) L 2?r (K
-i(r/E,J -
Pj2’+
(4’)‘)”
t (K-/d
e
e-l12(riE,ht,
P
provided
that
which evidently describes the exponential ~1and proper mean life time 7, given by
decay of an unstable particle
1 - = y = - k Im B $‘(k) 7 Thus, a more complete asymptotic
evaluation Kot >>
of mass
1k~+,~=o. of S(k,t)
for
&/Ko
is given by
The threshold summation “strong thresholds”, pi 2 4 More
accurately,
includes both “weak thresholds”, ~j 2 KOW , and The lowest threshold KO may refer either to ~0~
~08 .
the first-order
thresholds
for the production
of stable
particles.
188
SCHWINGER
KOS , depending upon the desired accuracy of the discussion. (A weak threshold at zero massrequires a separate treatment.) When the threshold asymptotic evaluation has become valid, or at the possibly greater time produced by the factor /32’3,the argument of the exponential decay factor is somelarge numerical multiple of
Or
P -So Y -----NW* E p KO KO
Y KO
Hence, if y is sufficiently small,
there is an extended time interval during which the contribution of the thresholds has become negligible and the Green’s function is completely dominated by the term referring to the unstable particle, --i!& O”dk” e 6(W) - B f’ & e-~E,te-l/2b/E,ht = jj l m_2a k2 + (/,J - +ir)” - kE P The exponential time decay of this term, in comparison with the algebraic decrease associated with the stable particle threshold contributions, indicates however that the latter are eventually the sole survivors of the asymptotic evaluation, as we have remarked. Does this mean that the intrinsic decay law of unstable particles ceasesto be of exponential form after a sufficiently long time? The answer to this question (which has occasionally been given by a categorical affirmative) illustrates the necessity of including suitable idealizations of all relevant aspects of the physical situation in its mathematical representation. We first remark that any physical process can be conceived as initiating in and terminating with stable particles. An example is the reaction p + p ---) p + p + e+ + em+ 2y + 21, + 2i;. The multipoint Green’s function that gives an account of this process will be constructed, in part, from the simplest Green’s functions, describing the propagation of the various stable particles into or out of the interaction region, and from a function that gives the details of the interactions among these particles. The nature of this interaction is specified more closely when we add the information that the above reaction has proceeded through the following stages: p+p+K++C++n, K+ + fi + (p+ + e+ + 2v), c+-, ?a--,
Q + ho-+&), p+e-+i;.
UNSTABLE
189
PARTICLES
Correspondingly, the function that characterizes the stable particle interactions will contain the Green’s functions that refer to the propagation of the intermediate unstable particles, together with details of their localized interactions. But this is not all. The statement that the reaction has involved the particle K+, for example, and not some kinematically equivalent combination of particles, must have a physical counterpart in an aspect of the measurement apparatus that serves to display the presence of the particle, and a mathematical counterpart, in an operation on the relevant Green’s function, that conveys the effect of the selective measurement. The only valid basis for asserting that a particular unstable particle has been involved is through an energy-momentum or mass determination of sufficient accuracy to rule out all other possibilities. This need not be a disturbing measurement in any other respect, however. The mathematical representation of this mass filter is supplied by the modified Green’s function
where
M(K)
is roughly
characterized
by the properties
M(K)
= 1,
1% - ~11 5 Au
M(K)
= 0,
\K - ~11 2 A/J
and Ap, measuring the precision of the mass determination, is restricted by y << A~.L<< /.L. Thus the threshold contributions, referring to recognizably distinct physical processes,are removed from the Green’s function and
where
g(S) =,[lC&i - P> &cK _$+(lr>2 e-‘(‘-p)aM(K) 2 _s00 =e -w2)Ya
d(K
-
EL)
$
(K
_
p)2y+
--m
(Lr)2 2
(l
-
M(K))
e--i(r-r)nv
Since 1 - M(K) does not differ from zero until ) K - p / >> y, the denominator of the lat,ter integral can be approximated by (K - P)~. On performing a partial integration that becomesasymptotically valid when A/.Ls>> 1,
190
SCHWINGER
we obtain g(s) N e-(l’Z)rs -
yli _-AlAps
co d(K - p) e-i(n-p)s g M(p), 21r s--m
which contains a further approximation that is justified by the localized nature of dM/dK. The second term of g(s) is evidently quite negligible, at least for moderate values of ys. The question before us, however, concerns the behavior of g(s) after very many lifetimes have elapsed. We must now introduce more detailed assumptions about M(K) in order to provide analytical equivalents of realizable experimental procedure. The major hypothesis is the boundedness of all derivatives of M(K) . If this property failed at some value of K, that mass value would be specifically distinguished, in contrast with the limited selectivity of any finite act of measurement. We shall also assume, for simplicity, that the shape of M(K) in the neighborhood of p + Ap = PZ is the mirror image of that near ~1- Ap = p1 . Then, confining the integration to the latter neighborhood, we have
g(s)N e-~~~~)rs + 2 sin - Aw1 Ap
Aficcs ?r s
d(K
-
pl)
e--i(r-sl)s
d”, M(K).
According to the boundedness hypothesis for all derivatives of M(K), the latter integral decreases asymptotically more rapidly than any power of (l/s). The asympt,otic behaviour is essentially exponential, of the form exp [-f( I’s)], where r >> y measures t.he boundary width of M(K), or physically, the latitude in the precise specification of the mass interval accepted by the filter. Thus during the extended interval r-l
<< s 5 y-l,
the second term of g(s) is not only small in comparison with the exponential, but is also decreasing much more rapidly. Whether or not this remains true for all time depends upon the precise character of the function that gives the exponential dependence on I’s. If f( r s ) is ultimately some finite numerical multiple of rs, the first term of g(s) will dominate for all s, whereas should the growth of f( Ik) with increasing s be less than linear, the roles of the two terms in g ( s ) will eventually reverse. The situation can be studied with the aid of the following class of functions, which satisfy the requirements that have been imposed on M(K) :
UNSTABLE
191
PARTICLES
Here X and a are positive numbers and c
-1
s 1
=
dx
e-(h/2a)
(l-22)-0
-1
The required asymptotic form can be obtained gration path into the complex plane, and
s 1
dx
e-o;2.)(1-z~)-Q
e--ire
-a(,
1 ;
by suitably
deforming
the inte-
yya>ll(l’*)~I
-1
exp[-(cos~)&(~F]sin[~+,
- (sin~)-&(~~],
in which ,=&
(r/71a >> 1,
the decaying exponential term, e-(1’2)‘us,ceasesto dominate g(s). But, precisely when this will occur, and the form of g(s) that ensues,is unavoidably dependent upon the details of the measurement process and is not intrinsic to the unstable particle. We conclude that with the failure of the simple exponential decay law we have reached, not merely the point at which some approximation ceasesto be valid, but rather the limit of physical meaningfulness of the very concept of unstable particle. As it stands, this discussion of a simple boson Green’s function applies only to the Hermitian field associated with the unstable particle s’,(where the “weak” decay mechanism is electromagnetic in nature). An extension to a two-component Hermitian field is needed to discussthe particles T*, K*, and ICI,2 . The Green’s function s(x
- x’)d
= i <
(&(x)&,(x’))+
>
= / g4
ezk(z-s’)S(k),b
is a symmetrical function of the indices x,a and x’, b. The space-time coordinates can appear only in the translational, rotational invariant combination (x - x’)* (an otherwise conceivable dependence upon the algebraic sign of x0 - x0’, in the interior of the light cone, is excluded by the requirement of a lowest energy
192
SCHWINGER
state) and therefore S(J: - ~‘),b is necessarily symmetrical in the indices a and b. When these indices refer to the two axes of an electric charge space, the structure of the matrix $,b is completely restricted by the conservation of electric charge
[%Sl =0,
q=
for there is only one type of two-dimensional with the antisymmetrical charge matrix: g(k)ab
=
O-i i 0’
(
)
symmetrical
matrix that commutes
&b$j(k).
Thus the problem is reduced to that of a single component field, and this situation is not changed by introducing the non-Hermitian fields that diagonalize q. The value of the electric charge does not affect the properties contained in the Green’s function and masses and lifetimes are identical for K+ and 6, K+ and
K-. It is a different matter when the two-fold multiplicity refers to a hypercharge space for this property is not conserved by the weak interactions, and the designations K”( Y = + 1) and K”( Y = - 1) are meaningful only for sufficiently short times. We accept the conservation law implied by invariance under space and charge reflection and conclude that S is invariant under charge reflection, since it is explicitly invariant to the reflection of the space coordinates. The two Hermitian field components are distinguished by the response to charge reflection, as it operates in the hypercharge space, and Qeb must be a diagonal matrix. However, there is no necessary connection between the diagonal elements 61 and ~2 , except in the approximation that ignores weak interactions where they are equal. Thus from $jl and $ we infer two distinct sets of masses and lifetimes, which characterize the unstable particles K, and Kt . The unstable particle contribution to S(k$), is
or, in view of the small difference anticipated
s(W),
i -zige
B
--iE,,t
between
B1 and Bz , p1 and pZ ,
{~.,,p[-~(611--iyl)s]+6anexp[-~(--6p--iyl)s]}.
Here
while p and s = t(p/Ep) refer to the average mass. If 6~ << Ap, the mass interval that is accepted in selecting electrically neutral K-particles, the asymptotic
UNSTABLE
193
PARTICLES
structure of $,b is left intact and we encounter a kind of mass interferometer. We need not repeat the discussion, based previously on more elementary considerations, of the experimental possibilities thus implied. If 6~ is so large that a mass separation of K1 and Kz becomes feasible, no significant interference effects can occur. The analogous treatment for fermion fields is sufficiently different in detail that it will be deferred to another publication. Note added in proof: It should have been mentioned in the text that these considerations are incomplete in one important respect. The full dynamical effect of the electromagnetic field is not represented since a more elaborate spectral form is required for the Green’s functions of charge-bearing fields in the physical radiation gauge. This refinement will not alter our physical conclusions, however. RECEIVED:
August
21, 1959 REFERENCES
1. P. T. MATTHEWS AND A. SALAM, 13, 115 (1959). 8. P. C. MARTIN AND J. SCHWINGER, 1342 (1959).
Phys. Bull.
Rev. 112,283 Am.
Phys.
(1958); Sot.
[II],
M.
Llivy,
3,202
Nuovo (1958);
cimento
Phys.
Rev.
[lo], 116,