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Threshold effects and unstable particles
Nuclear Physics 58 (1964) 374--384; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
T H R E S H O L D E F F E C T S AND U N S T A B L E P A R T I C L E S LUCIANO FONDA and GIAN CARLO GI-IIRARDI Istituto di Fisica dell'Universith,
Trieste, Italy
and lstituto Nazionale di Fisica Nucleate, Sottosezione di Trieste, Trieste, Italy
Received 11 February 1964 Abstract: The anomalous behaviour of cross sections at the threshold for a reaction involving
unstable particles is derived by making use of the tmitarity of the S-matrix and of analyticity arguments. Both the case of metastable particle production and that of virtual particle (antibound state) production are discussed. 1. Introduction It has been first pointed out by Wigner t) and then systematically investigated by various authors that scattering and reaction cross sections exhibit anomalous energy behaviour of the kind of a cusp or of a rounded step at the threshold for a new twoparticle channel. A detailed analysis of such effects proves to be particularly useful for the determination of relative parities and spins of the reaction products, of scattering phase shifts, o f cross sections for processes which are often outside the range of experiment and in the search for new particles t Most of the previous work has taken into account only the possibility of stable particles in the new channel. However, it is clear that for practical purposes the results so obtained still apply to the case of unstable particles in the new channel if the inverse lifetime of these particles is smaller than the experimental energy resolution. Nevertheless it is interesting on theoretical grounds to consider the possibility, which can in fact occur for certain experimentalsituations, of unstable particles in the new channel, and recently attention has been paid to this problem a-5). In this paper we try to deduce the anomalous behaviour o f the scattering S-matrix by using one of the unitarity equations a n d m a k i n g some considerations onthe analytic properties of the S-matrix. Explicit formulae are then obtained by estimating the contributions to the S-matrix which vary rapidly with the energy. I f one assumes that the reaction channel contains one unstable particle which has a two-body decay, one is really faced with a two-to-three-particle transition. For simplicity we shall therefore consider only the case of two channels, the initial one being a two-particle channel i = it + i2 and the other being a three-particle channel = ~tt + ~tz + 0ta. It is assumed that in channel ~ there appears a resonance phenomt For a review the reader is referred to the recent article by Fondas). 374
THRESHOLD EFFECTS
375
enon between particles ~x2 and %. This allows a factorization of some elements o f the S-matrix which exploits the dependence of the S-matrix elements on the scattering properties of the two resonant particles % and %. Two simple situations will be discussed in this paper: the possibility that particles ~2 and % form either a metastable state 13 or a virtual (antibound) state ft. F o r instance, the former case occurs experimentally in the reaction 4) zr + nucleon ~ p + nucleon. The p meson decays with short lifetime into two pions and plays the role of our fl particle. The discussion of the case of the virtual particle applies to the description of situations like the reaction p+d ~ p+n+p near threshold, which is influenced by the presence of the virtual deuteron. In sect. 2 the unitarity equations for the S-matrix are written down for S-waves and some considerations on the analytic properties of the S-matrix are given. The factorization is performed; all mathematical details axe confined to the appendix. In sect. 3 explicit calculations are given for the metastable particle case. The analogous calculations for the virtual particle ease are given in sect. 4.
2. Formalism
We assume non-relativistic kinematics and work in the centre-of-mass system, introducing relative coordinates and m o m e n t a in the initial two-body channel i and in the three-body channel ~. The particles ~1, u2, 0% have masses ml, m2, m3, coordinates R1, R2, R3 and m o m e n t a P1, P2, P3. For channel ~ we then choose rx = m ~ Rx,
r2 = R a - R 2 ;
Pl = h k l = t ' t ,
d¥ 2
P2 = hk2 = ( m 2 P 3 - m 3 P 2 ) / ( m 2 + m 3 ) = / 1 2 - , dt
(1)
where Pl _ m,(m2 + ma) ,
U2 - - m 2- m 3
ml+n12+m3
.
(2)
m2+nl 3
For the total energy W we get W
----- E 1 + E 2
-
h2k?
A,
2/h where E, = h2k2/2#l,
E 2 = hZk~/2~z,
(3)
376
L. F O N D A A N D G. C. G H I R A R D I
hk~ and p~ are the relative momentum and reduced mass in channel i, A/c 2 = rni + m 2 + m 3 - r n l ~ - m 2 ~ ,
ml~ and m2i being the masses of particles i 1 and i2. For simplicity all particles in channels i and ~ will be assumed to be spinless. We suppose that a resonance phenomenon appears in the S-wave of the ~2 + % system at a relative energy E z not very large, so that both for the motion of particle % relative to particle % and for the motion of particle ~lrelative to the centre-of-mass of the system ~2 + % only S-waves can be used, all other waves being negligible for the considered W energy region• It follows that we can restrict our considerations to the elements of the S-matrix relative to the S-wave of channel i and to the S-waves of channel ~. The unitarity equations for these elements, at the total energy W ~ 0, are
IS,,(W+)I2+
dE2IS,,(W+, e[+)[ 2 = 1,
f0 ~ *
.
P
S,i(W+)S,,(W+, e 2 + ) + ;
S,i(W÷
+
*
",
dE2S,,(W+, "
(4)
t .
'
*
t • e2÷, e2÷) = 0,
(5)
)
fo''
E2+, d E 2 S ~ ( W + , • E2+, E2+)S,~(W+; ' * " E2+) = 6(E2-EI').
(6)
By W+ and E 2 + we mean W + i8 and E2 + i8, respectively, with positive and vanishingly small 8. The various elements are on the energy shell E t + E 2 = E~ + E~ = E~' + E6' = W. The S-matrix elements giving rise to the i ~ g and g--¢ g transitions, owing to time reversal invariance 6), satisfy S,,(W+;E2+ ) = S~,(W+;E2+),
S~(W+;E2+,E'2+) = S,,(W+;E'2+,E2+).
(7)
The transition amplitude S~(W+; E 2 +, E~ + ) - c5(E2 - E~) contains the contribution from the disconnected diagram which represents the scattering % + % ~ % + % , ~t being a spectator. All other contributions to the above mentioned transition amplitude are free from 6 functions on the energy of particle gl. Denoting by San(E2+) the Swave part of the S-matrix for the scattering process % + % ~ g2 + g3 (which is clearly unitary), we write S~,( W+ ; E2+ , E; +) = 3 ( e 2 - E'~)Spp(ee +) + S ~ ( W+ ; E2+, E~+).
(8)
The elements S~, and ,~,, have the following low energy properties 6): S~,(W+ ; E2 + ) ~ o ( k t k2)* constant,
S,,(W+ ; E2 +, E~ + ) ~ o ( k l k2 k~ k~)¢ constant.
(9)
THRESHOLD EFFECTS
377
We shall assume that the S-matrix elements satisfy the Schwarz reflection principle
(lO)
S*(W+; E2+ . . . . ) = S(W_, E2_ , . . .), with W_ = W-ie and E 2_ = of the energies, the functions
We then define, for real and positive values
E 2 -ig.
S(+)(W; E2 . . . . ) = S(W+;E2+ . . . . ),
01)
S(-)(W, E 2. . . . ) = S(W_; E 2. . . . . ),
in terms of which we may rewrite all the unitarity equations. w e are interested in eq. (5). We consider it, in fact, as an equation for the unknown S[~+)(W). By dividing eq. (5) by S[~)(W; E2) we get for S~+)(W) the following expression:
S ~ )( W)
sl:)(w; =
-
(+) (Ee)S,(-) (w," spp
s[:)(w; -
dE'=
E2,
(12)
,]0
which is valid for W > 0. The various quantities on the right-hand side of eq. (12), however, can be continued analytically. The analytic continuation of eq. (12) then defines Sh+)(W) also for W < 0 and for complex energies. Just the same can be said for S~i-)(W) defined by an expression similar to eq. (12). As functions of the complex variable W, Sh+)(W) and S~-)(W) are branches of the same analytic function S,(W) which exhibits normal threshold two-body and three-body cuts with branch points W = - A and W = 0, representing the opening up of channel i and channel ~, respectively. In general S,(W) does not exhibit singularities on the physical sheet of the Riemann surface, i.e. on that sheet which is reached from the real axis by continuation o f S~[)(W) to positive Im W or, equivalently by continuation of S~-)(W) to negative Im W. The location of the singularities of S , ( t V ) on unphysical sheets, due to a production process, has been the subject of various recent attempts. In particular it is known from general considerations that Sil(W) exhibits branch points singularities if unstable particles are produced 7). Speaking qualitatively, we see in fact from eq. (12) that a pole of S~+)(W; E~) in the complex E i plane, which would correspond to an unstable state for the system ~2 + a3, yields a branch point for S~i+)(W) once the integration is performed. The case discussed in the literature is that of metastable particle production. The discussion can, however, be extended to cover the case of virtual particle production. We shall accordingly assume that for the case of production of a metastable state of the system a 2 + a3 in channel ~, S,(W) exhibits two branch points in the unphysical sheets at the position of the complex mass of the metastable particle, and of the complex conjugate mass. In what follows we refer to the cuts corresponding to the production of this kind of unstable particle as metastable particle cuts. Since we are
378
L. FONDA AND O. C. GHIRARDI
interested in evaluating Su(W) on the upper rim of the three-particle cut, the metastable particle cut which is reached by crossing the three-particle cut from above will give a contribution to S[I+)(W) varying rapidly with the energy. The contribution from the complex conjugate cut will be slowly energy-varying if we are sufficiently above the threshold since we have at least to go around the three-particle cut by crossing the two-particle cut below W --- 0, in order to reach it. The situation is most easily understood by looking at the complex k~ plane as shown in fig. 1 (a). We remind that the physical region stretches immediately above the positive real axis. (a)
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kl-Plane
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X
X
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X
0
(b)
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k,l-plane
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--t--
-X
o- -, . . . . . . . . . . . . .
~
Fig. I. Singularities in the k~ plane of the scattering matrix element Su(W) for the considered cases of metastable and virtual particle production. The k~-plane is cut from the points corresponding to the three-particle threshold to infinity. The dashed branch cuts lie on unphysical sheets of the Riemann surface; those whose branch points are indicated with a circle are reached directly from the physical region of the real axis.
For the case of production of a virtual state o f the system ~2 + ~3 in channel ~, we shall assume that S u ( W ) exhibits one cut with branch point onto the unphysical sheet for real and negative W. The branch point of this cut, referred to as virtual particle cut in the following, can be reached by crossing the three-particle cut from above and then moving towards real and negative values of IV. Only the contribution of the virtual particle cut reached by crossing the three-particle cut from above will be rapidly energy-varying. Fig. 1(b) shows the position of this branch cut in the complex k~ plane. As already pointed out, the metastable or the virtual particle cuts of Su(W) are furnished by the integral appearing on the right-hand side of eq. (12). In order to obtain Su(W) on the upper rim of the three-particle cut, we must evaluate in an approximate way the integral appearing on the right-hand side of eq. (12). We shall for this purpose follow a line of thought analogous to the method of Watson s) for the treatment of final state interactions, and we shall use the low energy behaviour
THRESHOLD EFFECTS
379
of the matrix elements S~ and Su. Unfortunately such a procedure implies a violation of the analicitity properties of Su(W) in such a way that also the cuts which are far away from the physical region give a contribution to S[~+)(W) which varies rapidly with the energy. It is therefore necessary to restrict ourselves only to the explicit evaluation of the part of the integral involving the contribution of the metastable or virtual praticle cut which is reached by crossing the three-particle cut from above. All other contributions to the imegral are slowly energy-dependent. We write therefore for Su(W) for real W, calculated on the upper rim of the three-particle cut, -
S,:)(W; E,) (+) (-) ( W ,
S , p (E2)S,~
.
(f ~
dF.2
, S~+)(W; E~)S~-)(W; E2,E'2!)
S~:)(W'~ E 2 )
E2)
uPc ( 1 3 )
where by UPC we mean the contribution from the unstable particle cut reached by crossing the three-particle cut from above. In fig. 1 (a) and 1 (b) this cut has a branch point indicated by a circle. We come now to the introduction of the various approximations which enable us to evaluate both the right-hand side of eq. (13) and the reaction cross section. Due to the presence of the resonant state fl (metastable or virtual) in channel ,t between particles u2 and a3, we can approximate the elements S~ and S~ by following the same line of thought as Watson s) for the treatment of final state interactions. The mathematical details, obtained by making use of the Hamiltonian formalism, are given in the appendix. Moreover we use the low energy behaviour eq. (9) for the amplitudes which involve transitions to or from channel c~. We get
E2) = T~;)(e2)(k, k2)½~O~,., E2 E'2) T~;)(E2)(k, ½ ' v,~'rt+)tr,~ S[+)(W;
g(+)/'W.
(14) (15)
where i
(16)
and ~0~t,, ~ are constants. It is worthwhile noticing that our normalization is such that the modulus square of Tpp (E2) is the S-wave differential cross section for the scattering of particle ~2 on particle ~3. T0~ satisfies the Schwarz reflection principle. Substitution of eqs. (14)-(16) into eq. (13) gives
SIg)(W) ~
e ''a°
E (£dE2iTpp(E2)l k, k2)J , 1+~0~*
r
t
9 t
t
(17)
UP
where 3 o = (phase of ~lffi) + ½~. We see that singular points for S[i+) (W) are generated by the singularities of TpB(E6). For completeness we write here also the expression for the reaction cross section ) ) It is obvious that a~t(W), as a function of the complex variable IV, has no normal treshold cut; follows that in eq. (18) no UPC device has to be used.
there
380
L. FONDA AND G. C. GHIRARDI
a~a(W) integrated over angles and over the available energy in the three-particle channel 0c %,(W) -
IS,,(W; E2)I2dE2 = kff - - 1~'~12
fo
' Tpa(E2)l , 2 kx , k2. , dEzl
(18)
We now come to the explicit evaluation of eqs. (17) and (18) for the metastable particle case and virtual particle case, respectively. We rteed only to specify the kind of singularities of Tpa (E;) for these two cases. 3. Metastable Particle We write Spa (E2) in terms of the Jost function S,p(e2)-
f(k2) .
(19)
f(-k~)
The function f ( k 2 ) satisfies the generalized unitarity relation f*(-k~) = f(k2). We describe the metastable particle fl by means of a couple o f simple zeros o f f ( k 2 ) in the upper half of the complex k2 plane near the real axis at the positions - £ and £*. We shall assume that both the reactions i --* • and the reaction • --. ~ proceed via the formation of the metastable particle p. A good approximation for the Jost function in the energy interval of interest is therefore f ( k 2 ) = (k2 + £)(k2 - £*).
(20)
It follows that To# (e2)
=
2 Ira
£/(k]-I£12-2ik2 Ira £).
(21)
The reaction cross section %~(W), according to eqs. (18) and (21), is
a,,(W) =
~(~ 2h2k~(Re ~2)~(1~'~12F) £) 2 r f : dE(E_Eo)2 [E(W-E)]~ +¼r 2 = n2(~ 1 ~2)~(1~,~121.) {Re [(Eo - ½ir)(W- Eo + ½it)] ~- ½r}. h~k~(Re £)2
(22)
The determination of the square root is chosen such that its real part is positive; Eo and F are defined as follows: Eo = h 2 (Re £ ) 2 _ ( i m £)2 ," 2/zz
F = 2h21Im £ Re El /t2
In the limit as F tends to zero we must take care of the dependence of ~J~ on F obtained from the theory of resonance reactions involving three-particle channels 9): lira ~02~,F~ = constant.
(23)
F~0
In the limit F ~ 0 the metastable particle fl in channel • becomes stable and ¢r,~ becomes a two-particle to two-particle reaction cross section. In particular %~ vanishes (linearly in F ) for W < E o and it varies like (W-Eo) t for W > Eo; %~ therefore exhibits the familiar low-energy behaviour at the point W = Eo.
THRESHOLD EFFECTS
381
For non-vanishing F, a~t behaves like W 2 near W = 0, it has an inflection point at W ~ E o + (1/2x/3)F and then it behaves like (W-Eo) t for W - E o > F. We now come to the consideration o f the element S,(W) describing the elastic scattering it +i2 --* i1+i2 • We have to use the UPC device in order to isolate from eq. (17) the contribution of the relevant branch point. We split the term [ ( E - E o ) 2 +¼F2] -1 in four parts and take the one in which the denominator [X/E-
x/F.o- ½iF]- ~ appears. This term in fact gives rise to the relevant metastable particle cut. The other terms will give rise slowly to energy dependent contributions. The integration of the so isolated term is straightforward, we finally get: S~+ )(W) ~ e2U°{1 + o:(W)+ fl(W)x/~V - Eo + ½iF},
(24)
where a(W) contains parts from the integration of the Ix~E-x/Eo-½iF]- ~ term and also the slowly varying energy distributions o f the order not explicitly considered terms. The quantity fl(W) is given by
(E°-½ir)½(~tu2)*(~%r) fl(W) =
[n+i log
4Eo-½ir ].
X/W-X/W-Eo+½iFJ
2h2(Re/~)2
Taking into consideration the fact that S~i+)(W) is unitary at W=Eo in the limit as F tends to zero, there follows that lim a(Eo) = 0. Since a(W) is also slowly energy /"--* 0
dependent we can write for St~t+)(W) for small FlEe and W m, Eo:
,2PE)2o e 2"° {1 + 2h2(Re/
r)(w- Eo+½iF)*}.
(25)
In the limit as F tends to zero, generalizing the argument given for ~0~ to the a -~ ,t reaction, we get lim 9X~F = constant.
(26)
F~O
In this limit S . (W) behaves like
S"(W)e2'6°+r-o "' (li)A(IW-E°I)~"
(27)
with A constant. The symbol (~) indicates that for W > E0 the upper quantity has to be used, for W < Eo the lower one. The element Sii(W), and therefore the scattering cross section a . (W), exhibits, in the limit as F tends to zero, the familiar "cusp" behaviour at the point W = E o. For non-vanishing '~ we look for stationary points of k2a.(W). It is seen that there is either one stationary point which corresponds to a maximum, or no stationary point. Moreover, the occurrenc~ o f the former or o f the
382
L. FONDA AND G. C. OHIRARDI
latter possibility is approximately governed by the sign of tg 60 in such a way that a positive sign corresponds to a maximum. This m a x i m u m becomes a cusp for vanishing F. 4. Virtual Particle Let us now take into account the case in which particles % and % form a virtual particle (antibound state)//. This situation is described by means of a purely imaginary simple zero of the Jost function, near the real axis at the position kz = il~, with positive ~. Also in this case the transitions to or from channel u are supposed to go via the formation of the virtual particle/L A good approximation for f ( k 2 ) for low energies is therefore f ( k 2 ) = i(k2-iJi).
(28)
We get i Taa(E2) - k2 + il~ -
ih 1 (2#2)t x/E + ix/7 ,
(29)
where x/7 = h£/(2#2) ½ > 0 and ~/E is positive for real and positive E. F o r the reaction cross section %~(W) we get
For non-vanishing 7, % i ( B 0 behaves like W 2 at W = 0. The main difference between the reaction cross section for our virtual case and the reaction cross section for the metastable case with small E o lies in the fact that for the virtual case %~(W) does not exhibit an inflection point near zero energy. We come now to the consideration o f S , ( W ) . We split the term [ E + 7 ] - 1 in two parts and take the one in which the denominator [%/E+~/7]-1 appears. This term in fact gives rise to the relevant virtual particle cut. After integration we get:
i7' ]/ x log W½_(W+7)½] j .
(31)
Since 7~ > 0, S~t~+)(W) does not exhibit an infinite derivative at W = - 7 (as it would for 7~ < 0, i.e. when our particle/~ ia a bound state of the % + % system). As far as the energy dependence o f St#+)(W) for W ~ 0 is concerned, we have:
sI:)(w)%
{1
'
"
•
THRESHOLD EFFECTS
383
The quantity R~ a ~ ( W ) exhibits at W = 0 a stationary point. From considerations of conservation of total flux we infer that this point is a maximum for R~2a,(W). Note that if there is no resonance phenomenon in the three particle channel S~i+)(W) behaves like the expression given in eq. (32) without the appearence of the constant ~, in the denominator of the W 2 term. We acknowledge discussions with Professors P. Budini, A. Rimini and L. Taffara.
Appendix We shall here derive eqs. (14) and (15) by making use of the Hamiltonian formalism. We separate from the interaction Hamiltionian the part ~g'2a which commutes with operations on particle cq: Hi = ~(/~23 .-t- V,
(A.1)
i.e., Y/'23 is the interaction Hamiltonian for the ~2 +u3 system and cannot give i - , transitions. For the T-operator we have the following expressions: T ~ HI--FHIGH I
= T~-+ (I + ~e'23G~,)V(I + Gilt) = T ~ + (1 +
3(/'23G.t-)(V-I
-
(A.2)
VGV)(G~-'](/'23-t- 1),
(A.3)
where T~ is the T-operator for the interaction Hamiltonian ~(/'23; G and G~ are the Green functions at the energy W, relative to the interaction Hamiltonians H l and 3r23, respectively, defined with an outgoing wave boundary condition. The elements of T giving rise to the transitions i - , ~ and ~ --, ~ are given by ( ~ ( k l , k2) , Tdp~(k~)) = (q~-)(kx, k2) , Vd/(+)(k,)),
(A.4)
(~,(kl, k2), T~p~(k'l k'2) ) = ( ~ ( k l , k2), T.r~(k'l , k'2)) + (~bt~-)(kl, k2), [ V +
VGV]q~c.+)(k;, k'2)),
(A.5)
where we have used eq. (A.2) and the fact that "//'23 cannot give i --+ ct transitions in order to get eq. (A.4), and eq. (A.3) in order to get eq. (A.5). The state vectors ~b~ and ~ are free states for channels i and ct, respectively; ~+)(~bt~-)) is the outgoing (incoming) wave solution for channel c~ when only the interaction ~¢'23 is taken into account. The first term on the right-hand side of eq. (A.5) is easily seen to be (~b,(kx, k2) , T.r~,(kl, k'2) ) --- (~23(k2), T~rfb23(k'2))t~a(kx-kl) ,
(A.6)
42a being the free wave of particle ~3 relative to particle a2. We now follow the same line of thought as Watson s). Owing to the assumed low-
384
L. FONDA AND G. C. GHIRARDI
energy r e s o n a n t b e h a v i o u r o f the u2 + u3 scattering cross section we can limit our= selves to S-waves a n d factorize ~b~+) as follows: ~+)=
rp(~')(E2)x(k,),
(A.7)
the wave function x ( k l ) v a r y i n g very slowly with kz. T h e c o m p l e x conjugate o f eq. (A.7) gives the e q u a t i o n for ~ - ) . F r o m eqs. ( A . 4 ) - ( A . 7 ) follow eqs. (14) a n d (15).
References
1) E. P. Wigner, Phys. Rev. 73 (1948) 1002 2) L. Fonda, Nuovo Cim. Suppl. 20 (1961) 116 3) A. I. Baz, JETP (USSR) 40 (1961) 1511, translation: JETP (Soviet Physics) 13 (1961) 1058 4) J. S. Ball and W. R. Frazer, Phys. Rev. Lett. 7 (1961) 204 5) M. Nauenberg and A. Pais, Phys. Rev. 126 (1962) 360 6) L. Fonda and R. G. Newton, Phys. Rev. 119 (1960) 1394 7) R. Blankenbecler, M. L. Goldberger, S. W. MacDowell and S. B. Treiman, Phys. Rev. 123 (1961) 692; D. Zwanziger, Phys. Rev. 131 (1963) 888 8) K. M. Watson, Phys. Rev. 88 (1952) 1163 9) L. Fonda, Ann. of Phys. 12 (1961) 476
T H R E S H O L D E F F E C T S AND U N S T A B L E P A R T I C L E S LUCIANO FONDA and GIAN CARLO GI-IIRARDI Istituto di Fisica dell'Universith,
Trieste, Italy
and lstituto Nazionale di Fisica Nucleate, Sottosezione di Trieste, Trieste, Italy
Received 11 February 1964 Abstract: The anomalous behaviour of cross sections at the threshold for a reaction involving
unstable particles is derived by making use of the tmitarity of the S-matrix and of analyticity arguments. Both the case of metastable particle production and that of virtual particle (antibound state) production are discussed. 1. Introduction It has been first pointed out by Wigner t) and then systematically investigated by various authors that scattering and reaction cross sections exhibit anomalous energy behaviour of the kind of a cusp or of a rounded step at the threshold for a new twoparticle channel. A detailed analysis of such effects proves to be particularly useful for the determination of relative parities and spins of the reaction products, of scattering phase shifts, o f cross sections for processes which are often outside the range of experiment and in the search for new particles t Most of the previous work has taken into account only the possibility of stable particles in the new channel. However, it is clear that for practical purposes the results so obtained still apply to the case of unstable particles in the new channel if the inverse lifetime of these particles is smaller than the experimental energy resolution. Nevertheless it is interesting on theoretical grounds to consider the possibility, which can in fact occur for certain experimentalsituations, of unstable particles in the new channel, and recently attention has been paid to this problem a-5). In this paper we try to deduce the anomalous behaviour o f the scattering S-matrix by using one of the unitarity equations a n d m a k i n g some considerations onthe analytic properties of the S-matrix. Explicit formulae are then obtained by estimating the contributions to the S-matrix which vary rapidly with the energy. I f one assumes that the reaction channel contains one unstable particle which has a two-body decay, one is really faced with a two-to-three-particle transition. For simplicity we shall therefore consider only the case of two channels, the initial one being a two-particle channel i = it + i2 and the other being a three-particle channel = ~tt + ~tz + 0ta. It is assumed that in channel ~ there appears a resonance phenomt For a review the reader is referred to the recent article by Fondas). 374
THRESHOLD EFFECTS
375
enon between particles ~x2 and %. This allows a factorization of some elements o f the S-matrix which exploits the dependence of the S-matrix elements on the scattering properties of the two resonant particles % and %. Two simple situations will be discussed in this paper: the possibility that particles ~2 and % form either a metastable state 13 or a virtual (antibound) state ft. F o r instance, the former case occurs experimentally in the reaction 4) zr + nucleon ~ p + nucleon. The p meson decays with short lifetime into two pions and plays the role of our fl particle. The discussion of the case of the virtual particle applies to the description of situations like the reaction p+d ~ p+n+p near threshold, which is influenced by the presence of the virtual deuteron. In sect. 2 the unitarity equations for the S-matrix are written down for S-waves and some considerations on the analytic properties of the S-matrix are given. The factorization is performed; all mathematical details axe confined to the appendix. In sect. 3 explicit calculations are given for the metastable particle case. The analogous calculations for the virtual particle ease are given in sect. 4.
2. Formalism
We assume non-relativistic kinematics and work in the centre-of-mass system, introducing relative coordinates and m o m e n t a in the initial two-body channel i and in the three-body channel ~. The particles ~1, u2, 0% have masses ml, m2, m3, coordinates R1, R2, R3 and m o m e n t a P1, P2, P3. For channel ~ we then choose rx = m ~ Rx,
r2 = R a - R 2 ;
Pl = h k l = t ' t ,
d¥ 2
P2 = hk2 = ( m 2 P 3 - m 3 P 2 ) / ( m 2 + m 3 ) = / 1 2 - , dt
(1)
where Pl _ m,(m2 + ma) ,
U2 - - m 2- m 3
ml+n12+m3
.
(2)
m2+nl 3
For the total energy W we get W
----- E 1 + E 2
-
h2k?
A,
2/h where E, = h2k2/2#l,
E 2 = hZk~/2~z,
(3)
376
L. F O N D A A N D G. C. G H I R A R D I
hk~ and p~ are the relative momentum and reduced mass in channel i, A/c 2 = rni + m 2 + m 3 - r n l ~ - m 2 ~ ,
ml~ and m2i being the masses of particles i 1 and i2. For simplicity all particles in channels i and ~ will be assumed to be spinless. We suppose that a resonance phenomenon appears in the S-wave of the ~2 + % system at a relative energy E z not very large, so that both for the motion of particle % relative to particle % and for the motion of particle ~lrelative to the centre-of-mass of the system ~2 + % only S-waves can be used, all other waves being negligible for the considered W energy region• It follows that we can restrict our considerations to the elements of the S-matrix relative to the S-wave of channel i and to the S-waves of channel ~. The unitarity equations for these elements, at the total energy W ~ 0, are
IS,,(W+)I2+
dE2IS,,(W+, e[+)[ 2 = 1,
f0 ~ *
.
P
S,i(W+)S,,(W+, e 2 + ) + ;
S,i(W÷
+
*
",
dE2S,,(W+, "
(4)
t .
'
*
t • e2÷, e2÷) = 0,
(5)
)
fo''
E2+, d E 2 S ~ ( W + , • E2+, E2+)S,~(W+; ' * " E2+) = 6(E2-EI').
(6)
By W+ and E 2 + we mean W + i8 and E2 + i8, respectively, with positive and vanishingly small 8. The various elements are on the energy shell E t + E 2 = E~ + E~ = E~' + E6' = W. The S-matrix elements giving rise to the i ~ g and g--¢ g transitions, owing to time reversal invariance 6), satisfy S,,(W+;E2+ ) = S~,(W+;E2+),
S~(W+;E2+,E'2+) = S,,(W+;E'2+,E2+).
(7)
The transition amplitude S~(W+; E 2 +, E~ + ) - c5(E2 - E~) contains the contribution from the disconnected diagram which represents the scattering % + % ~ % + % , ~t being a spectator. All other contributions to the above mentioned transition amplitude are free from 6 functions on the energy of particle gl. Denoting by San(E2+) the Swave part of the S-matrix for the scattering process % + % ~ g2 + g3 (which is clearly unitary), we write S~,( W+ ; E2+ , E; +) = 3 ( e 2 - E'~)Spp(ee +) + S ~ ( W+ ; E2+, E~+).
(8)
The elements S~, and ,~,, have the following low energy properties 6): S~,(W+ ; E2 + ) ~ o ( k t k2)* constant,
S,,(W+ ; E2 +, E~ + ) ~ o ( k l k2 k~ k~)¢ constant.
(9)
THRESHOLD EFFECTS
377
We shall assume that the S-matrix elements satisfy the Schwarz reflection principle
(lO)
S*(W+; E2+ . . . . ) = S(W_, E2_ , . . .), with W_ = W-ie and E 2_ = of the energies, the functions
We then define, for real and positive values
E 2 -ig.
S(+)(W; E2 . . . . ) = S(W+;E2+ . . . . ),
01)
S(-)(W, E 2. . . . ) = S(W_; E 2. . . . . ),
in terms of which we may rewrite all the unitarity equations. w e are interested in eq. (5). We consider it, in fact, as an equation for the unknown S[~+)(W). By dividing eq. (5) by S[~)(W; E2) we get for S~+)(W) the following expression:
S ~ )( W)
sl:)(w; =
-
(+) (Ee)S,(-) (w," spp
s[:)(w; -
dE'=
E2,
(12)
,]0
which is valid for W > 0. The various quantities on the right-hand side of eq. (12), however, can be continued analytically. The analytic continuation of eq. (12) then defines Sh+)(W) also for W < 0 and for complex energies. Just the same can be said for S~i-)(W) defined by an expression similar to eq. (12). As functions of the complex variable W, Sh+)(W) and S~-)(W) are branches of the same analytic function S,(W) which exhibits normal threshold two-body and three-body cuts with branch points W = - A and W = 0, representing the opening up of channel i and channel ~, respectively. In general S,(W) does not exhibit singularities on the physical sheet of the Riemann surface, i.e. on that sheet which is reached from the real axis by continuation o f S~[)(W) to positive Im W or, equivalently by continuation of S~-)(W) to negative Im W. The location of the singularities of S , ( t V ) on unphysical sheets, due to a production process, has been the subject of various recent attempts. In particular it is known from general considerations that Sil(W) exhibits branch points singularities if unstable particles are produced 7). Speaking qualitatively, we see in fact from eq. (12) that a pole of S~+)(W; E~) in the complex E i plane, which would correspond to an unstable state for the system ~2 + a3, yields a branch point for S~i+)(W) once the integration is performed. The case discussed in the literature is that of metastable particle production. The discussion can, however, be extended to cover the case of virtual particle production. We shall accordingly assume that for the case of production of a metastable state of the system a 2 + a3 in channel ~, S,(W) exhibits two branch points in the unphysical sheets at the position of the complex mass of the metastable particle, and of the complex conjugate mass. In what follows we refer to the cuts corresponding to the production of this kind of unstable particle as metastable particle cuts. Since we are
378
L. FONDA AND O. C. GHIRARDI
interested in evaluating Su(W) on the upper rim of the three-particle cut, the metastable particle cut which is reached by crossing the three-particle cut from above will give a contribution to S[I+)(W) varying rapidly with the energy. The contribution from the complex conjugate cut will be slowly energy-varying if we are sufficiently above the threshold since we have at least to go around the three-particle cut by crossing the two-particle cut below W --- 0, in order to reach it. The situation is most easily understood by looking at the complex k~ plane as shown in fig. 1 (a). We remind that the physical region stretches immediately above the positive real axis. (a)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
kl-Plane
.
.
.
.
X
X
.
X
0
(b)
.
.
.
.
.
.
.
.
.
.
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.
.
.
.
k,l-plane
.
.
.
.
.
--t--
-X
o- -, . . . . . . . . . . . . .
~
Fig. I. Singularities in the k~ plane of the scattering matrix element Su(W) for the considered cases of metastable and virtual particle production. The k~-plane is cut from the points corresponding to the three-particle threshold to infinity. The dashed branch cuts lie on unphysical sheets of the Riemann surface; those whose branch points are indicated with a circle are reached directly from the physical region of the real axis.
For the case of production of a virtual state o f the system ~2 + ~3 in channel ~, we shall assume that S u ( W ) exhibits one cut with branch point onto the unphysical sheet for real and negative W. The branch point of this cut, referred to as virtual particle cut in the following, can be reached by crossing the three-particle cut from above and then moving towards real and negative values of IV. Only the contribution of the virtual particle cut reached by crossing the three-particle cut from above will be rapidly energy-varying. Fig. 1(b) shows the position of this branch cut in the complex k~ plane. As already pointed out, the metastable or the virtual particle cuts of Su(W) are furnished by the integral appearing on the right-hand side of eq. (12). In order to obtain Su(W) on the upper rim of the three-particle cut, we must evaluate in an approximate way the integral appearing on the right-hand side of eq. (12). We shall for this purpose follow a line of thought analogous to the method of Watson s) for the treatment of final state interactions, and we shall use the low energy behaviour
THRESHOLD EFFECTS
379
of the matrix elements S~ and Su. Unfortunately such a procedure implies a violation of the analicitity properties of Su(W) in such a way that also the cuts which are far away from the physical region give a contribution to S[~+)(W) which varies rapidly with the energy. It is therefore necessary to restrict ourselves only to the explicit evaluation of the part of the integral involving the contribution of the metastable or virtual praticle cut which is reached by crossing the three-particle cut from above. All other contributions to the imegral are slowly energy-dependent. We write therefore for Su(W) for real W, calculated on the upper rim of the three-particle cut, -
S,:)(W; E,) (+) (-) ( W ,
S , p (E2)S,~
.
(f ~
dF.2
, S~+)(W; E~)S~-)(W; E2,E'2!)
S~:)(W'~ E 2 )
E2)
uPc ( 1 3 )
where by UPC we mean the contribution from the unstable particle cut reached by crossing the three-particle cut from above. In fig. 1 (a) and 1 (b) this cut has a branch point indicated by a circle. We come now to the introduction of the various approximations which enable us to evaluate both the right-hand side of eq. (13) and the reaction cross section. Due to the presence of the resonant state fl (metastable or virtual) in channel ,t between particles u2 and a3, we can approximate the elements S~ and S~ by following the same line of thought as Watson s) for the treatment of final state interactions. The mathematical details, obtained by making use of the Hamiltonian formalism, are given in the appendix. Moreover we use the low energy behaviour eq. (9) for the amplitudes which involve transitions to or from channel c~. We get
E2) = T~;)(e2)(k, k2)½~O~,., E2 E'2) T~;)(E2)(k, ½ ' v,~'rt+)tr,~ S[+)(W;
g(+)/'W.
(14) (15)
where i
(16)
and ~0~t,, ~ are constants. It is worthwhile noticing that our normalization is such that the modulus square of Tpp (E2) is the S-wave differential cross section for the scattering of particle ~2 on particle ~3. T0~ satisfies the Schwarz reflection principle. Substitution of eqs. (14)-(16) into eq. (13) gives
SIg)(W) ~
e ''a°
E (£dE2iTpp(E2)l k, k2)J , 1+~0~*
r
t
9 t
t
(17)
UP
where 3 o = (phase of ~lffi) + ½~. We see that singular points for S[i+) (W) are generated by the singularities of TpB(E6). For completeness we write here also the expression for the reaction cross section ) ) It is obvious that a~t(W), as a function of the complex variable IV, has no normal treshold cut; follows that in eq. (18) no UPC device has to be used.
there
380
L. FONDA AND G. C. GHIRARDI
a~a(W) integrated over angles and over the available energy in the three-particle channel 0c %,(W) -
IS,,(W; E2)I2dE2 = kff - - 1~'~12
fo
' Tpa(E2)l , 2 kx , k2. , dEzl
(18)
We now come to the explicit evaluation of eqs. (17) and (18) for the metastable particle case and virtual particle case, respectively. We rteed only to specify the kind of singularities of Tpa (E;) for these two cases. 3. Metastable Particle We write Spa (E2) in terms of the Jost function S,p(e2)-
f(k2) .
(19)
f(-k~)
The function f ( k 2 ) satisfies the generalized unitarity relation f*(-k~) = f(k2). We describe the metastable particle fl by means of a couple o f simple zeros o f f ( k 2 ) in the upper half of the complex k2 plane near the real axis at the positions - £ and £*. We shall assume that both the reactions i --* • and the reaction • --. ~ proceed via the formation of the metastable particle p. A good approximation for the Jost function in the energy interval of interest is therefore f ( k 2 ) = (k2 + £)(k2 - £*).
(20)
It follows that To# (e2)
=
2 Ira
£/(k]-I£12-2ik2 Ira £).
(21)
The reaction cross section %~(W), according to eqs. (18) and (21), is
a,,(W) =
~(~ 2h2k~(Re ~2)~(1~'~12F) £) 2 r f : dE(E_Eo)2 [E(W-E)]~ +¼r 2 = n2(~ 1 ~2)~(1~,~121.) {Re [(Eo - ½ir)(W- Eo + ½it)] ~- ½r}. h~k~(Re £)2
(22)
The determination of the square root is chosen such that its real part is positive; Eo and F are defined as follows: Eo = h 2 (Re £ ) 2 _ ( i m £)2 ," 2/zz
F = 2h21Im £ Re El /t2
In the limit as F tends to zero we must take care of the dependence of ~J~ on F obtained from the theory of resonance reactions involving three-particle channels 9): lira ~02~,F~ = constant.
(23)
F~0
In the limit F ~ 0 the metastable particle fl in channel • becomes stable and ¢r,~ becomes a two-particle to two-particle reaction cross section. In particular %~ vanishes (linearly in F ) for W < E o and it varies like (W-Eo) t for W > Eo; %~ therefore exhibits the familiar low-energy behaviour at the point W = Eo.
THRESHOLD EFFECTS
381
For non-vanishing F, a~t behaves like W 2 near W = 0, it has an inflection point at W ~ E o + (1/2x/3)F and then it behaves like (W-Eo) t for W - E o > F. We now come to the consideration o f the element S,(W) describing the elastic scattering it +i2 --* i1+i2 • We have to use the UPC device in order to isolate from eq. (17) the contribution of the relevant branch point. We split the term [ ( E - E o ) 2 +¼F2] -1 in four parts and take the one in which the denominator [X/E-
x/F.o- ½iF]- ~ appears. This term in fact gives rise to the relevant metastable particle cut. The other terms will give rise slowly to energy dependent contributions. The integration of the so isolated term is straightforward, we finally get: S~+ )(W) ~ e2U°{1 + o:(W)+ fl(W)x/~V - Eo + ½iF},
(24)
where a(W) contains parts from the integration of the Ix~E-x/Eo-½iF]- ~ term and also the slowly varying energy distributions o f the order not explicitly considered terms. The quantity fl(W) is given by
(E°-½ir)½(~tu2)*(~%r) fl(W) =
[n+i log
4Eo-½ir ].
X/W-X/W-Eo+½iFJ
2h2(Re/~)2
Taking into consideration the fact that S~i+)(W) is unitary at W=Eo in the limit as F tends to zero, there follows that lim a(Eo) = 0. Since a(W) is also slowly energy /"--* 0
dependent we can write for St~t+)(W) for small FlEe and W m, Eo:
,2PE)2o e 2"° {1 + 2h2(Re/
r)(w- Eo+½iF)*}.
(25)
In the limit as F tends to zero, generalizing the argument given for ~0~ to the a -~ ,t reaction, we get lim 9X~F = constant.
(26)
F~O
In this limit S . (W) behaves like
S"(W)e2'6°+r-o "' (li)A(IW-E°I)~"
(27)
with A constant. The symbol (~) indicates that for W > E0 the upper quantity has to be used, for W < Eo the lower one. The element Sii(W), and therefore the scattering cross section a . (W), exhibits, in the limit as F tends to zero, the familiar "cusp" behaviour at the point W = E o. For non-vanishing '~ we look for stationary points of k2a.(W). It is seen that there is either one stationary point which corresponds to a maximum, or no stationary point. Moreover, the occurrenc~ o f the former or o f the
382
L. FONDA AND G. C. OHIRARDI
latter possibility is approximately governed by the sign of tg 60 in such a way that a positive sign corresponds to a maximum. This m a x i m u m becomes a cusp for vanishing F. 4. Virtual Particle Let us now take into account the case in which particles % and % form a virtual particle (antibound state)//. This situation is described by means of a purely imaginary simple zero of the Jost function, near the real axis at the position kz = il~, with positive ~. Also in this case the transitions to or from channel u are supposed to go via the formation of the virtual particle/L A good approximation for f ( k 2 ) for low energies is therefore f ( k 2 ) = i(k2-iJi).
(28)
We get i Taa(E2) - k2 + il~ -
ih 1 (2#2)t x/E + ix/7 ,
(29)
where x/7 = h£/(2#2) ½ > 0 and ~/E is positive for real and positive E. F o r the reaction cross section %~(W) we get
For non-vanishing 7, % i ( B 0 behaves like W 2 at W = 0. The main difference between the reaction cross section for our virtual case and the reaction cross section for the metastable case with small E o lies in the fact that for the virtual case %~(W) does not exhibit an inflection point near zero energy. We come now to the consideration o f S , ( W ) . We split the term [ E + 7 ] - 1 in two parts and take the one in which the denominator [%/E+~/7]-1 appears. This term in fact gives rise to the relevant virtual particle cut. After integration we get:
i7' ]/ x log W½_(W+7)½] j .
(31)
Since 7~ > 0, S~t~+)(W) does not exhibit an infinite derivative at W = - 7 (as it would for 7~ < 0, i.e. when our particle/~ ia a bound state of the % + % system). As far as the energy dependence o f St#+)(W) for W ~ 0 is concerned, we have:
sI:)(w)%
{1
'
"
•
THRESHOLD EFFECTS
383
The quantity R~ a ~ ( W ) exhibits at W = 0 a stationary point. From considerations of conservation of total flux we infer that this point is a maximum for R~2a,(W). Note that if there is no resonance phenomenon in the three particle channel S~i+)(W) behaves like the expression given in eq. (32) without the appearence of the constant ~, in the denominator of the W 2 term. We acknowledge discussions with Professors P. Budini, A. Rimini and L. Taffara.
Appendix We shall here derive eqs. (14) and (15) by making use of the Hamiltonian formalism. We separate from the interaction Hamiltionian the part ~g'2a which commutes with operations on particle cq: Hi = ~(/~23 .-t- V,
(A.1)
i.e., Y/'23 is the interaction Hamiltonian for the ~2 +u3 system and cannot give i - , transitions. For the T-operator we have the following expressions: T ~ HI--FHIGH I
= T~-+ (I + ~e'23G~,)V(I + Gilt) = T ~ + (1 +
3(/'23G.t-)(V-I
-
(A.2)
VGV)(G~-'](/'23-t- 1),
(A.3)
where T~ is the T-operator for the interaction Hamiltonian ~(/'23; G and G~ are the Green functions at the energy W, relative to the interaction Hamiltonians H l and 3r23, respectively, defined with an outgoing wave boundary condition. The elements of T giving rise to the transitions i - , ~ and ~ --, ~ are given by ( ~ ( k l , k2) , Tdp~(k~)) = (q~-)(kx, k2) , Vd/(+)(k,)),
(A.4)
(~,(kl, k2), T~p~(k'l k'2) ) = ( ~ ( k l , k2), T.r~(k'l , k'2)) + (~bt~-)(kl, k2), [ V +
VGV]q~c.+)(k;, k'2)),
(A.5)
where we have used eq. (A.2) and the fact that "//'23 cannot give i --+ ct transitions in order to get eq. (A.4), and eq. (A.3) in order to get eq. (A.5). The state vectors ~b~ and ~ are free states for channels i and ct, respectively; ~+)(~bt~-)) is the outgoing (incoming) wave solution for channel c~ when only the interaction ~¢'23 is taken into account. The first term on the right-hand side of eq. (A.5) is easily seen to be (~b,(kx, k2) , T.r~,(kl, k'2) ) --- (~23(k2), T~rfb23(k'2))t~a(kx-kl) ,
(A.6)
42a being the free wave of particle ~3 relative to particle a2. We now follow the same line of thought as Watson s). Owing to the assumed low-
384
L. FONDA AND G. C. GHIRARDI
energy r e s o n a n t b e h a v i o u r o f the u2 + u3 scattering cross section we can limit our= selves to S-waves a n d factorize ~b~+) as follows: ~+)=
rp(~')(E2)x(k,),
(A.7)
the wave function x ( k l ) v a r y i n g very slowly with kz. T h e c o m p l e x conjugate o f eq. (A.7) gives the e q u a t i o n for ~ - ) . F r o m eqs. ( A . 4 ) - ( A . 7 ) follow eqs. (14) a n d (15).
References
1) E. P. Wigner, Phys. Rev. 73 (1948) 1002 2) L. Fonda, Nuovo Cim. Suppl. 20 (1961) 116 3) A. I. Baz, JETP (USSR) 40 (1961) 1511, translation: JETP (Soviet Physics) 13 (1961) 1058 4) J. S. Ball and W. R. Frazer, Phys. Rev. Lett. 7 (1961) 204 5) M. Nauenberg and A. Pais, Phys. Rev. 126 (1962) 360 6) L. Fonda and R. G. Newton, Phys. Rev. 119 (1960) 1394 7) R. Blankenbecler, M. L. Goldberger, S. W. MacDowell and S. B. Treiman, Phys. Rev. 123 (1961) 692; D. Zwanziger, Phys. Rev. 131 (1963) 888 8) K. M. Watson, Phys. Rev. 88 (1952) 1163 9) L. Fonda, Ann. of Phys. 12 (1961) 476