Description of Unstable Particles in Quantum Field Theory

Description of Unstable Particles in Quantum Field Theory

M . L 6 V Y

Ecole Normale Swperieure, Universite de Paris, Paris, France

I. Introduction

Dr. Zumino has probably convinced most of you that the problem of defining unstable particles exists, and that it is a difficult one.

The present lectures will eventually tend to show that the difficulties lie even deeper than one might think at first sight; we shall see also what exactly these difficulties are.

In the first place one should keep in mind that the uncertainty principle fixes severe limits to the « definition » of the mass and life­

time of an unstable particle.

If m is the mass, and τ = 1/Γ the life-time, there will be an un­

certainty in m of the order (1)

Furthermore, since Γ can be considered as a kind of mean square deviation from the average of a mass distribution, there is also an uncertainty on Γ itself of the order

(2) Δ Γ - — . Γ*

m

It is therefore clear that m and Γ can be defined uniquely only if Γ is small (i.e., if the life-time is long). In particular, if the decay occurs through an interaction, the strength of which is characterized by a coupling constant the mass will be defined with an arbitrariness of the order of g²; the life-time can be defined uniquely to the order g², but there will be an arbitrariness of order g*.

In other words, there will be a large variety of manners to define m and Γ, all consistent with uncertainties (1) and (2), and any of them can be chosen for a given specific problem. One could then stop to worry about unstable particles, and be contented with the usual per-

Μ. USvY

turbation calculation of life-times, since any attempt to go further will only be a matter of personal choice. We think, however, that, from a fundamental point of view, a problem still exists for several reasons:

(a) It is desirable to calculate the iife-time to the lowest order in a completely consistent way. This is not true in the usual methods of atomic physics. In addition, the extension of these methods to the case of unstable «elementary» particles would imply for these par­

ticles not only the existence of a field φ, but also the definition of

«free » outgoing and ingoing states which is certainly not possible, except in a very approximate and unphysical way.

(b) Eenormalization of mass and charge is usually based on the existence of «physical states » of the particles under consideration.

Special methods must therefore be used for the renormalization of the mass of unstable particles and of the coupling constant of decay inter­

actions.

(c) It would be convenient to give a «natural» definition of the parameters m and JP, for any value of the coupling constant, which would be independent of the production process. The definition would apply to metastable states of the type of the (f, f) resonance in π-ρ scattering as well as to «true » elementary unstable particles.

(d) Finally, there is a practical problem which is probably the most important and difficult one: although an unstable particle cannot be an exact eigenstate of the total Hamiltonian, we are still able to observe it, if its life-time is not too short, and to make measurements of its properties. We should therefore be able to calculate, for example, the probability to observe the particle at time ί as a function of i, and this, apart from some normalizing factor, should not depend strongly on the production mechanism. In other words, we should have a satisfactory prescription to describe the unstable particle as a wave packet of exact states of the Hamiltonian and thereby to construct its time graph without using unphysical initial conditions, such as the assumption of a «free » state at t - > — oo, etc.

In principle, a fairly unambiguous (if not straightforward) way to define the properties of unstable particles is to investigate scattering cross-sections of the decay products, as well as various reactions in­

volving these products initially or finally. The unstable particle ap­

pears then as one of the possible intermediate states in the process, and this should manifest itself as a resonance. However, the defi­

nition of mass and life-time obtained from the resonance will vary

D E S C R I P T I O N O F U N S T A B L E P A R T I C L E S

from one process to the other. Besides, the method may not be very useful in pratical cases: the detection of the Λ⁰—for example—by means of a resonance in π-ρ scattering would imply an energy reso­

lution of the order of 10~^β eV!

The method of the propagator, therefore, seems to us the most natural way to define unstable particles, although it may not be pos­

sible in all cases. I t has the advantage that it involves only the as­

sumption of a field φ(χ), but no asymptotic «free» states into which the particle goes when t - > ± oo.

In the following, we shall first discuss this method in the case of the Lee model. Then, we shall examine the conditions for analy­

tical continuation in a more general theory. We shall finally make some remarks on the «time-plot» problem.

II. Properties of Scattering Amplitudes in the Lee Model

1. T H E E E N O R M A L I Z E D H A M I L T O N I A N

When a stable state of the V-particle exists, the renormalization of mass and charge of the Lee model is, to a large extent, unambiguous.

When the V-particle is unstable, the renormalization process becomes quite different and cannot be separated from the determination of the life-time. In this section the renormalization constants will be exhibited formally, no distinction being made between the two possible cases.

We write the Hamiltonian:

(3) H = H 0 + m + H^{i n t},

where, with the same notations as Kallen and Pauli, we have set:

(4) B⁰+M= (m^v- $m)N* ΣV*ip)Ψ*(ρ) +

ρ

+ w

Σ

+ Σ

*

Ρ *

(5) J T^{l B t}= 9

— Σ 4^^v(p)^(p>

VQp'p +k V2(D^k

Here, ψ^ν, ψ^Ν9 and a^k are the usual annihilation operators of the V, ΕΓ, and θ particles, respectively, and the corresponding starred quan­

tities the creation operators of the same particle. V and Κ are treated

Μ . ΙΛΥΥ

non-relativistically with no recoil; ω^]ε = (&² + μ²)* is the energy of a θ-particle of momentum k; g and m^y are the «renormalized» charge and mass of the V-particle. What the latter quantity actually means will have to be defined precisely in each physical case. The unrenor- malized mass and charge are given in terms of the two constants Sm and Ν by the relations:

(6) ^{9Θ - - y ,}

m° = m^y — Sm

The only conditions which will be required in all cases are (a) 8m is real, (b) N² is real and positive.

Ν is also related, as is well known, to the V-field renormalization:

ψν = yfyjN, the corresponding anticommutation relation being written:

(7) {ψ*(ρ),ψΛρ')} = ^δ^ΡΡ·.

The relations for the Ν and 0 fields are the usual ones. The total Hamiltonian of the Lee model has two constants of the motion (8) = * +

I Qt = Η* — VQ ,

where Η^Λ is the operator representing the number of particles of type (a).

Bigenstates of Η can therefore be constructed as linear combinations of eigenstates of H⁰ corresponding to fixed values of q^x and q^2J the eigenvalues of Q^x and Q². As usual, we shall write the eigenstates of H⁰ corresponding to given numbers of each particles in the form:

\n^y, %, nly.

In the following, we shall only be interested in the eigenstates of Η corresponding to q^x = 1, q² = 0. They are related either to the

« physical» state of the V-particle (when it exists), or to the (Ν, Θ) scattering states). We consequently consider the eigenvalue problem:

(9)

ΒI

where ω can take any value between — oo and + oo (it should not be confused with ω^Λ). We can now distinguish the solution corresponding to a possible stable physical state of the V-particle and those which are related to | JV, Θ) scattering states.

D E S C R I P T I O N O F U N S T A B L E P A R T I C L E S

2. T H E P H Y S I C A L V - P A R T I C L E

The condition for the existence of a stable V-particle is

(10) Μω⁰) = 0 ,

where we have set ω⁰ = ηι^ν— m^N and defined (11) h{co) = N² g2 f2(a>k.) 1

This function can be defined in the complex ω-plane with a cut on the real axis from +μ to + o o . Going to continuous values of ω*, and defining Γ(ζ) as:

(12)

we have in the cut plane

(13) h(z) = N2

. 1 [Γ(ω')άω' , » πΝ²] ω — ζ

If Eq. (10) has a root for ω = ω⁰<μ, we have then:

( « ) , η ^ - ' ί ΰ ^ .

πΝ² J ω — ω⁰ μ

Writing the corresponding states as |F>, we can also determine N*

by the two equivalent conditions:

(15)

< 0 h M F > : or

<7V|j|F> = <0, 1^N, 0|?⁰|l^v, 0, 0> , where j is the renormalized current of the O-field (16) j = 0 ( γ £ γ ν + β. c.)

and j^Q the corresponding unrenormalized current. The physical state

\N} of the N-particle is identical with the free state |0,1^N, 0>. Note also that both \V} and |iV> have to be normalized to unity.

Μ . I&VY

(17)

If then follows that

Γ(ω) άω

j r ,

=

-/-»

μ

If JV² > 0, it can be proved that the system does not contain any other root of 7ι(ω) = 0 for ω<μ. If ^ T²< 0, there appears the well known ghost state of the V-particle. We exclude this possibility in the following.

3. T H E S C A T T E R I N G S T A T E S (ω > ρ)

The quantity of interest here is the forward scattering amplitude:

(18) * ™

h(o)

+

(19) h^± == h{(o ± ie) = h(w) ± ιΓ(ω),

where %(ω) is the same function as h(w) given by Eq. (11) but with the integral replaced by its principal part.)

We also consider the function:

(20) 9+{<o) = U ( ( 0 ) 1

Γ (ω) Μω + ιε) '

This function satisfies quite generally a dispersion relation of the type

0 0

(21) g+((o) = - — τ - αω -] . π J ω — ω — ιε ω — ω⁰

μ

where the inhomogeneous term on the right-hand side appears only when ω⁰< μ (the function θ(χ) is defined as θ(χ) = 0 for x< 0 and θ(χ) = 1 for x> 0).

The forward scattering amplitude itself does not obey in general a dispersion relation. This is not surprising, since we are dealing with a fixed source theory, for which causality requirements are not auto­

matically fulfilled. If we want /+ to obey a dispersion relation, we have to require that Γ(ζ) be an analytical function in the cut plane except for

D E S C R I P T I O N OF U N S T A B L E P A R T I C L E S

4. P R O P E R T I E S O F h(z)

We state now two rather well-known properties of h(z):

(a) If N² > 0, h(z) cannot have any complex zeros (regardless of the value of ω⁰ real). This can be seen directly from the expression (13) by calculating

?Γ(ω') dco'l (23) Im h(z) = Im (z) 1 +

nN

J

which cannot vanish if N² > 0.

Zf N² < 0, which corresponds to the possibility of ghost states, it is well known that the two real zeros ω„ and λ can be replaced by two complex conjugate zeros which are obtained by equating to zero the right-hand side of (23) (dm being determined by the condition Ee h(z) = 0).

(b) If the cut off function /(ω) does not have zeros for ω > μ, then h^± = h(co ± is) cannot vanish for ω > μ. This property seems, at first sight, to be a special case of the preceding one. However, this is not quite true when Im ζ -> 0. We have in this case

(24) Im *^± = ± ^ .

The usual method of perturbation theory for treating an unstable particle consists in assuming that there exists an approximate complex zero of h(z) of the type ζ ^ ω⁰ — iy where γ is supposed to be a po­

sitive quantity infinitely small compared to co⁰. This method is easily possible poles. We would then have a relation of the type

(22) ,+( „ ) = i f I m ^ , do/ + ^ > ^ - ω . ) + Σ

where the ω^ί are the poles of Γ which are not situated on the cut of h(z).

The condition of analyticity of Γ(ζ) does not follow «naturally»

from the Lee model. However, if we want to consider this model as a kind of simplified image of a more accurate and complicated theory, then we may be led to require this condition. This question will be discussed more in detail later.

Μ . L J i I V Y

seen to be inconsistent, since we have, in this case

(25) Im Μω⁰ - ιγ) - - + ,

which cannot be made to vanish with a positive γ (assuming always N² > 0).

III. Definition of the Unstable V-Particles by Means of the Propagator

1. G E N E R A L P R O P E R T I E S O F T H E P R O P A G A T O R

As was explained in the introduction, we are looking for a « natural»

definition of the mass and life-time of the unstable particle which would be, as much as possible, independent of the production process.

For this purpose we consider the propagator, as defined by Lehmann:

+ CO

(26) ^ S'^v(t) = i- J8'^y{<o) exp [ - <(m^K + ω)*] da>,

— CO

with

(27) ^

= - i f 4 ^ t .

J ω — ω — ιε

— CO

In this case

(28) ρ(ω) = Σ Κ 0| ν ν Ι ί, ω > | · ,

i

where the summation is extended over all the possible states of Ή of energy co+m^N = JE. It is easily seen that the only states for which the expectation value of the right-hand side does not vanish are the states corresponding to q^x = 1, q^% = 0 considered in the last section.

We have therefore

(29) ρ(ω) = ^ \ ^{] 2} θ (ω — μ) + θ(μ — ω⁰) δ (ω — ω⁰) ,

where the second term on the right-hand side appears only when a stable V particle exists ( ω⁰< μ ) . If ω⁰> μ, Glaser and Kallon have proved that the scattering states form a complete set of states.

D E S C R I P T I O N O F U N S T A B L E P A R T I C L E S

Putting the expression (29) into (27), we find, by making use of the dispersion relation for g+ (21) that we have, in all cases

(30) s ;^{( c o) =} _ ? _

Λ⁺(ω) so that, in all cases, we can write

(31) £ ⁼ _ e x p [ - ^ ]

7

W l U S

2 2n% J h(z)

2. A N A L Y T I C A L C O N T I N U A T I O N O F T H E P R O P A G A T O R

The function h(w) has a jump of 2ιΓ(ω) when we cross the cut on the real axis. The analytical continuation of h(z) in the second sheet should be therefore

(32) H(z) = h(z) + 2ιΓ(ζ) .

However, such a continuation would be permissible only if Γ(ζ) is an analytic function, at least in a certain region in the neighbourhood of a finite segment of the cut. We have already mentioned in the last section that an analyticity condition may be imposed on Γ(ζ) if we want to have a dispersion relation for the forward scattering amplitude, in spite of the fact that we are dealing with a fixed source theory. On the other hand, it is clear also that special forms of the cutoff function can be chosen which will insure this analytical property. For example, since we cannot take an infinite cutoff, because of the ghost difficulty, we could decide that f(z) = 1 inside a circle of large radius Β and f(z) = 0 outside. We would then have the possibility to continue h(z) inside the circle of radius Β and that would be sufficient provided that Β is large enough.

However, we are not interested in such special choices of the cutoff function, since our purpose in using the Lee model is not to solve a mathematical exercise, but to illustrate in a simplified manner the conceptual features of a more general and realistic theory. We shall therefore perform the analytical continuation in a different manner, where the cutoff function never appears, making use in a rather crucial way of the unitarity of the S-matrix. The Lee model is then only con­

venient in the sense that the unitarity of the $-matrix can be expressed

Μ . L^JVY

very simply by the relation

(33) Ι π ι /⁺( ω ) = | /⁺( ω ) | « , because only one intermediate state contributes to the ΪΓ-Θ scattering process.

Going back to the definition of the propagator, we write in the first sheet:

(34) ^ » ( , ) = - i f ^ ^ , J ω —z

μ

and in the second sheet:

S'«>(2) = 8^(z) + 2πρ(ζ) .

ρ(#), defined by (28), is the absolute square of a vertex function, and even if we can prove a dispersion relation for the vertex function itself, we have no guarantee that its absolute square will be analytical.

We can write, however, making use of (29) (in the case ω⁰ > μ) and (18)

(35, IW-'I'

where /_ is obtained from /+ by changing h+ into Ji-. Now, /_(ω) and Λ_(ω), which are already defined in the lower half-plane can be continued immediately; on the other hand, |/+(ω)|² can be written (36) |/⁺|« = /.(ω) [/-(ω) + 2i Im/+(co)],

and in general, we will have no information on the analytical pro­

perties of Im/+, from causality alone. However, the simple unitary relation (33) allows us to write

and consequently ρ(ω) can be written

(38) ^{ρ (}α » = — ¹ >-<»>

π M< » ) [ 1 -2 » / _ ( ω ) ] '

ρ(ω) can then, in this form, be continued in the lower half plane.

We find, in this way

(39) ^{W a M =} _ i _ _ ,

D E S C R I P T I O N O F U N S T A B L E P A R T I C L E S

which is, of course, the same analytical continuation as (32). I t is clear that this method of analytical continuation will be valid in a more general theory where the ^-matrix contains only one term. This will not be the case in general, except for certain energy regions of the cut where some intermediate processes are forbidden by energetical conside­

rations. However, another difficulty will arise in a more general theory, from the fact that a consistent definition of the ^-matrix will never make use of the unstable V-particle itself, but only of its stable decay products. This difficulty will be discussed in the next Section.

3. E X I S T E N C E O F A P O L E A N D E E N O R M A L I Z A T I O N O F M A S S A N D C O U P L I N G C O N S T A N T

Assuming then that the propagator can be continued, we evaluate iS'^y(t) on the contour of Fig. 1. This gives (for ω⁰> μ)

(40) ±S'^v(t) =

•• jJr-, exp [— i ( w^N + z⁰)t] - -*^{P [}

(*o)

- im^Nt] Γ Γ(ω) exp [— ιωί]άω π J Μω)Η(ω) '

\

- \

\^

\

\

\

\

\ /

\ /

F I G . 1.

where the function Η is defined by (32); z⁰ is the root of the equation

(41) 3(z⁰) = 0 .

That such a root exists can be seen directly in the weak coupling limit where it can be calculated without specializing the form of Γ(ω).

Writing

(42) z⁰ = x^Q — iy^Q,

Μ . L^JVY

assuming that y⁰ cz. 0((</²/4π)<#⁰ and that Γ(ζ) is continuous around z⁰, we can write

CO

(43) *(*,) ~ N*(x⁰ - iy0) + N*(8m - ω„) + i Ρ (Γ(ω)άω _ {Γ(χα).

71 J CO — X⁰ Μ

Equation (41) can then be split into two parts:

h(x0) 0, (44)

N*y⁰-r(x0)=0

The second equation gives y⁰ = r(x^Q)jN²-, the first can always be solved by an appropriate choice of 8m, which we only require to be real. The value of y⁰ (the inverse half-life) which is so obtained is in agreement with the value deduced from perturbation theory by a special limiting process.

There remains to determine the value of the renormalization con­

stants 8m and N². There is no difficulty for 8m, since it is natural to ask that the real part x⁰ of the pole coincides with ω⁰ (the splitting of the unrenormalized mass into m^v — 8m was prepared just for that):

(45) First condition: x⁰ = ω⁰.

The determination of N² is more ambiguous, and various pre­

scriptions can be given which are all consistent and differ only by terms of the order g*. By analogy with the stable case, one could think of requiring the condition

B'M = 1,

but this would imply a choice of a complex N² which leads clearly to difficulties with the hermitic properties of the field ψ^ν, since N² is also a field renormalizing factor (ψ^γ and \p% would not be hermi- tically conjugate). The

(46) second condition: | 2 T( 2⁰) | = 1

has its advantages, because the coefficient of the exponential in (40) reduces then to an arbitrary phase factor. The same would be true for the amplitude b(t) = <0 |exp [— iBt] |0> which is related, in the case of the Lee model, to #v(i) the relation $S'^y(i) = N~²b(t). Another way to define the coupling constant renormalizing factor ^² would be to require that the N-scattering cross-section reduces to its Born approximation when Ε = m^s + mv. When ω⁰< μ (stable case), this

D E S C R I P T I O N O F U N S T A B L E P A R T I C L E S

prescription is identical with conditions (15), since these are equivalent to h'(a)⁰) = 1 . When ω⁰> μ the prescription based on the scattering cross section would correspond to

(47) |*'(ω.)| = 1.

With the choice (44) of x^0J it can be seen that conditions (46) and (47) agree with each other to the order g* as could be expected on the basis of uncertainty relation (2). However, we feel, in the spirit of our general discussion of Section 1, that prescriptions derived from scattering cross sections should be avoided. In any case, it should be pointed out that this discussion of the coupling constant renor­

malization is rather academic. Indeed, if the life-time is long (as in the decays due to weak interactions), then the corrections to the renormalization constant due to the decay interaction can easily be neglected (this corresponds, in the present model, to taking ^² = 1);

and if it is short (as in the 7r°-decay, for example) the coupling con­

stant renormalization is already known from other phenomena.

IV. Analytical Continuation of the Propagator in a More General Field Theory

We now discuss the analytical properties of the propagator of an unstable particle in a more general field theory. For simplicity, we shall assume that it is a scalar « V » particle represented by a « field » φ(χ). We do not discuss here the manner in which this field can be defined or constructed. This V particle decays into two stable scalar particles Ν and θ of masses Μ and μ. We suppose also that V is not coupled to particles of lower masses, but we do not restrict the number of θ particles which can be emitted.

1. E E L A T I O N B E T W E E N T H E P R O P A G A T O R

A N D T H E S C A T T E R I N G A M P L I T U D E O F T H E D E C A Y P R O D U C T S

According to Lehmann, the propagator A^y(x) (48) Δ'4^ι>(χ) = — ^ (A'^Ym exp [iJcx] d⁴fc ,

can be constructed by means of a mass distribution ρ(Κ²) · Δ'^ν(Κ²) is analytic in the complex (— K²) plane with a cut starting at 2TJ =

Μ . L ^ V Y

=(Μ+μ)². I t will also have other cuts, starting at (Μ + ημ)², (η = 2, 3, etc.). Our problem is to decide whether we can continue it in the second (« unphysical») sheet of the Riemann surface, through the cut starting at K²,. For this, we shall have to define a continuation of the analytical function in the first sheet

)άΚ 1 K2

into the second sheet by the formula

(50) ^Δ?\ζ)=±Δ?\ζ)+2πρ{*),

and, therefore, we have to know if ρ(ζ) is analytic in a region of the lower half plane sufficiently large to enable us to reach a pole if there is one.

Now, ρ(Κ²) can be defined by means of a sum over a complete set of states

(51) ρ(Κ²) = 2π

21

6{K + ,

π

where Tel is the square of the energy momentum of the state \n).

In practice we shall limit ourselves to the states into which the V-par­

ticle can decay. The reason for this will be discussed later. With this approximation, ρ(Κ²) will then be expressed in terms of the ab­

solute square of the vertex function

(52) ^F ^ ( p + 1c)2) = < 0 M O ) | p , * > , through the formula

(53) ^Q(K²) =\F⁺(-K²)\².

In his second lecture, Dr. Symanzik has proved a dispersion relation for F⁺ (— Jc²) = F+ (K²):

κ2

where the lower bound of the integral will again be Kl = (Μ+μ)² (this is, in any case, easy to verify in perturbation theory). This means that F+(z) is analytic in the complex cut-plane, and, in particular,

D E S C R I P T I O N OF U N S T A B L E P A R T I C L E S

in the lower half of the K² plane. However, we are not especially interested in the continuation of F+ but of its absolute square:

\F+\² = F⁺[F⁺-2ilmF⁺].

We therefore have to study the analytical properties of Im-F + , on which causality does not give us any special information. However, we can restrict ourselves to values of K² which are limited by (55) Kl <K²<Kl = (M+ 2μ)².

If the energy of the final state is limited in this way, we can then calculate I m ^⁺ by the standard techniques which have been used already in connection with other problems. We first write

(56) F⁺ = - L = [exp [ikx] d x( 0 j (Φ(0), ,'(*))+ |p> ,

\2a)kJ

where J(x) is the current of the θ-particles. Then, in the usual way, we replace by

(57) F⁺ =

-L= ^(exp [ikx] ^άχ<0

j(x)]

χ),

and then take the imaginary part of F⁺ by inserting a set of inter­

mediate states:

(58) Im F., =

2 <0 1

I

where the spatial part of the ό-function must be understood as a Kronecker symbol. If we remain in the limited domain of K² the only states which will contribute to the sum (48) will again be the

I p, ft) states. We therefore have

(59) ImF+((p+h)²) =

= -β= I <01 Φ(0) Ip', *'> </>', V I j(0) \ρ}δ(ρ + Ίο- ρ'- V) . Kow, on the right-hand side we recognize, as the first factor, the vertex function F+(p^r+ W)². The second factor is directly related to the Ν-θ scattering amplitude Τ(ρ,ΊΘ)ρ^¥):

(60) <p^r, Jc^r\J(0)\p} = (8ω., ρ0ρΌ)-*Τ{ρ, »; p[, Jo') .

Μ. L^VY

Τ can be considered as a function of the square of the total energy W² and of the square of the momentum transfer Δ². Since F+ de­

pends only on (p+k)², one sees immediately that the summation on the right-hand side of (59) involves an average over A². For conve­

nience, we can use the center-of-mass system, and write (61) Im F⁺(- W²) = —- ⁺ ) " ^] T(W, Α²) άΑ²,

32π (Wy/ωΙ—μ*]

so that (imFjF⁺) is then simply proportional to the S-seattering ampli­

tude.

This result can be expressed using Eq. (55), in the form in which Dr. Symanzik wrote it without proof in his second lecture:

(62) \F⁺\² = F\exj>[2id⁸], where δ⁸ is the 8- scattering phase-shift.

The analyticity of the #-scattering amplitude has not been de­

monstrated from dispersion relation corresponding to a unite mo­

mentum transfer A². We believe, however, that it is plausible that there is such an analyticity in the neighborhood of the real axis. This is the case for example, for potential scattering.

2. T R E A T M E N T O F T H E « S P U R I O U S » P O L E S O F T H E P R O P A G A T O R

We have already mentioned, in the above discussion, the possible existence of additional poles in the analytical continuation of the propagator through the higher cuts starting at Μ+2μ, Μ+3μ, etc....

We would like now to discuss the meaning of these poles more in detail with the help of a generalization of the Lee model, in which the V-particle is involved in two interactions instead of one:

(63a) V?±N + 9^l, fa),

(635) V*nN + e^t, (g^%).

We suppose that the second interaction is much stronger than the first:

(64)

but that the mass μ² of the θ-particle is such that the rapid decay V is energetically forbidden

(65) m^N + μ^χ< m^y< m^s + μ².

D E S C R I P T I O N O F U N S T A B L E P A R T I C L E S

This situation is very close to the physical case of the Λ⁰ decay, for example, where the reaction Λ ° ^ ± ρ + Κ occurs much more strongly than the decay Λ ° ^ ± ρ + π but is energetically forbidden unless the Λ⁰ has a high kinetic energy. The Lehmann weight function becomes the sum of two terms:

( 6 6 ) ρ ( ω ) = *\Μω + ίε)\·'

where we have put

(67) rt(a>) = |/,(o>) - μ ? 0 ( ω -μ^{), and

(68) h(z) = N² 1 f W ) + W ) ,

^ ^ πΝ² J ω'— ζ ο

The propagator retains the form (31). If we continue h(z) in the energy region between ms + μ! and m^^ + μ^ we have the continuation (69) h^{2)(z) = h(z) + 2il\(z)

and it can be seen immediately that there is a pole having the cor­

rect physical meaning; the effect of the second interaction (63b) is simply to change 8m and ^V² by what is usually called «renormal­

ization due to strong interactions». In particular, this pole will tend to the real axis and transform V into a stable particle if g^L-^ 0. How­

ever, we can also continue h(z) in the energy region higher than

#&^Ν + μ²· Then, the continuation becomes

(70) h^li)(z) = h(z) + 2ίΓ^χ(ζ) + 2ιΓ²(ζ),

which has, in the complex plane, a zero with a completely different behaviour. If we make the (physically meaningful) approximation (71) Γχ(ωο) < Γ(ω⁰) < m^v — mN ,

and write for this pole z^x = ω⁰ — iy^x, we find Γ¹( ω⁰) + 2 Γ²( ω⁰)

(72) 2/i Ν*

which does not go to zero when g^x -> 0. I t is therefore clear that this pole does not have the correct physical meaning and should not be considered for the definition of the unstable V-particle.

Μ. L^JVY

Finally, we arrive at the conclusion that the method of conti­

nuation which we used in the previous paragraph (namely, consi­

dering the energy region where the only states which are energetically allowed are the decay states) was not only convenient but necessary.

In other words, if the mass and life-time of an unstable particle are to be defined by a pole in the propagator, one has to make a definite prescription, based only on physical arguments, to choose the sheet of the Riemann surface into which the propagator has to be analytically continued.

In the above discussion we have implicitly assumed that the decay states are the lowest possible mass states of the spectrum. This may not be always the case; for example, the cascade particle Ξ " decays into ( Λ⁰,7 t_) states which have a threshold higher than the (η, π~) states which seem to be forbidden, at least partially. Another example is the 7t-decay, which we may consider more in detail. There are two possible reactions:

On the other hand, we have mⁿ> m^> m^e. If the propagator is continued into the sheet corresponding to the cut between m⁰ and mⁿ, the corresponding ^-function will be approximately equal to

where we have written the possible pole as m^ — iy; Γ⁶ and Γ^μ are some functions proportional to g\ and g², respectively. Since Γ⁶ <

we see that ¥^{2 )} has no zero in the lower half-plane. On the other hand, if the continuation is made between m^ and the next threshold (which will be, in this case, Smⁿ), a zero appears, the imaginary part of which has the expected behaviour

Consequently, no ambiguity will arise in such cases.

A similar problem occurs when we discuss not the propagator of the unstable particle but its decay probability. This probability can be related to the Lehmann weight function ρ(Κ2), but a great care must be exercised in its definition, in order to prevent unphysical situations to occur. For example, if g\ is large the, uncertainty on the mass of the V-particle is such that there will always be a finite

(73)

(74) fc<²> ~ N²(mⁿ - iy) + ir^e (mⁿ) - ιΓ^μ (mⁿ),

(75) γ = r^e(mⁿ) + r^mj

N²

D E S C R I P T I O N O F U N S T A B L E P A R T I C L E S

probability for the V-particle to decay through interaction (636) rather than (63a). This is physically acceptable provided that the corres*

ponding partial probability goes to zero when g\ 0, even if g\ re­

mains large.

V. Remarks on the Time-Graph of an Unstable Particle In the usual perturbation treatment, one considers the bare state

|V⁰> of the unstable particle and one writes as (76) b(t -1⁰) = <V⁰1 exp [ - iB(t -1⁰) | r^o>

the probability amplitude to observe it at time t if it was produced at <⁰. This definition is clearly objectionable on the basis of our pre­

vious discussions since use of the « bare » state of the unstable particle is certainly not very satisfactory. Another definition, used by Ma­

thews and Salam, and which has the advantage of being relativis- tically invariant is

(77) bit -t⁰)= Q^y{k*) exp [ - ik*(t —1⁰)] dk*

where r is the proper time of the particle and ρ^ν contains only the state into which the particle decays. All these definitions are ambi­

guous. They also all lead to the conclusion that the exponential term is not the only term which appears in b. This results from the fact that b(t) can always be put in the form

0 0

(78) b(t) = I φ(Ε)!² exp [— iEt] dE

simply on the basis of the requirement of hermiticity of the Hamil­

tonian and of positive energies. Then it follows from a well known theorem on the Fourier transform that b(t) behaves like a certain power of if the function \φ(Ε)\^ζ does not have all its derivatives, vanishing at Ε = 0. We would like to point out, however, that some of the deviations from the exponential are not observable in general.

Indeed, b(i) can usually be put into the form (79) ^W ~ ^{6 o} e x p [ - ^ ] + ^ ^{p C i (} ^ + ^{a ) ]}^

(Qt)^p

Μ . L ^ V Y

where Q is the Q-value of the decay, α a certain phase factor and γ a quantity proportional to the coupling constant. We then have (80) i b(t -1⁰) ]² ~ bl exp [ - 2Γ(ί - 1⁰) ] +

+

M ^

^

Now, l/Q corresponds to a time which is much shorther than 1/JH.

Therefore, the second term can be only observed if one can measure t —1⁰ with an accuracy higher than which in most of the cases, will be of the order of 10~^{2 1} to 10~²³ s. Therefore, in general, this term will disappear and one will be left with only the first and the last term, which is even much smaller than one might have thought by considering b(t) itself.

The deviations to the exponential are the only terms of the time- graph which depend on the history of the particle. They also will depend strongly on the particular choice which is made of the initial state of the unstable particle and therefore on the production me­

chanism.

K E F E R E N C E S

1. V. Glaser and G. Kallen, Nuclear Phys., 2, 706 (1956-1957).

2. P. T. Mathews and A. Salam, Phys. Rev., 112, 283 (1958); also: « Kela- tivistic theory of unstable particles - I I » (preprint).

3. H. Araki, Y . Munakata, M. Kawaguchi and T. Goto, Progr. Theoret. Phys., 17, 419 (1957).

4. Gr. Hohler, Z. Physik, 152, 546 (1958).

5. Κ. E. Peierls, Proceedings of the 1954 Glasgow Conference, p. 296.

Pergamon, New York, 1955.