On the theory of reactions with production of three low-energy particles

F

7 .A

I

Publishing Nuclear PhWks 38 (062) 13"145 ; @ North-Holland

CO .,

Amste;,da nj

without written permission from the publij,Iler Not to be reproduced by photoprint or microfilm

V. V. ANISOVICH, A. A. ANSELM and V. N. GRIBOV

Lenin,,-rad Physical- Technical Institute, Academy of Sciences, Lcaingrad, USS Received I January 1962 Abs act : The stiv ,;ture of the amplitude of production of three low-cnergy particles is discussed (the momenta of tl-~ particles produced being small compared with 11ro, where ro is the interaction radius) . Results obtained earlier by the authors are given for the terms linear and quadratic with respect to the momenta . Detailed consideration : 1.3 given to the terms cubic with respect to the mornent!, of the particles, produced, in the case of neutral particles and -r-decay. These terms are a simple polynomial of the third degree with respect to the moments, with coefficients depending onl!, on the scattering amplitude of two particles . Comparison of the respective formulae with experimental data makes it possible in principle. to determine the amplitudes for scattering of unstable particles on each other (79-mesOn on .7 . meson, for example).

e eral Strue In reactions involving scattering or production of two low-energy particles the unitarity and analyticity conditions impose simple and well-known restriction's on the dependence of the amplitude of the partial wavef on th(. energy near the t1ireshold : a(k 2) 1 - ika(k 2)

The function a(k 2 ) is real and permits of an expansion in powers of the square of the momentum k 2 for small k 2 . There are similar restrictions on the reactions involving production of a larger number of particles . Because of thecomplexity of the analytical properties and unitarity sonditions in this case, the coroParies from these restrictions cannot, however, be formulated so simply and are not sufficiently known . Different reactions involving the production of three lomenergy particles, were considered in refs . ' - ). The amplitude of a reaction involving the production of three, particle', is a function , of five invariant quantities : the three squares of the energies of the particles produced S 112 = ( I +P2 )2, ßz3 = ( n! + n )2 and s = ( n 4- n 12 In the four%,i being 1 .1 1 .3 imensional momenta of the particles formed) and the two mo ants transferred . Just as in the case of two-particle scattering, the amplitude is an analytic Nnction of the momenta transferred in the physical region . The expansion in terms of' the niomenta tran~Ierred k equivalent to the expansion of the amplitude in partial % ,"Ves with differc!ntt values of the total rnomenturn . If one canalders the state of vanishing 132

oi~ïJC'I°I®~~ ~?~

® t®tal car t

Im®'-v~-

ée

cntu , the rorres onr'' resh®1 o ~ c eactio

~,

1LG`C

lit

is a~

IHAfl~~1

c de ~ s ® ly litu as oot-

~l

si

~ t e asses of t e articlcs ro uce ~ )~ ( °~° o e ta 1 ~, ~ 3 atura.l to c nsi er it as ~. functi®~~ of t e ro~ci atio otion of the a ticles ; i non-relativistic a 111~ 112 t

2

(111~ ~-111 1 1

ularabPs, t e a.

plittt

°

cortes ® i of t c ener 13

flz'

is

ete { 1111

an ine

t® t ®f Z39

e

°11ß2~°1113) ;

ein~ the sc~ arc ctio of t ree articlcs ass s ste ). n the vari t~~les e t ee articlcs in t e ce tre-o `arity i no - elativistic a ro i atio , t e Josition o t is sit t res ol

y t e c~~

°°}~ 111 ~ ~° 1°113 )

wers e~ an e in t e a o~~e- entioned

as a sin ularity at

SD3-F~S~3°®111ήï%ß~®111~ = (1311

Sfl~

ularities

SbB °~111 t °~ 1111)

~

t a~ sinlilarly to e . ( ) t e a litu e can t i ht see o~rever, ol . a art ro 13 and o a3 l~ear t e t res gz, Sin

°~~ ~ 3 e thcrcforc it is f t c relati~~e

s °~13

o t e

o

ition

{131 D -~131~) `

~

~131D~1313) `

~11ß~°~°1313)'

~

11ß D ~"rß3

iY1 D 111,

111, liß3

~s the kinetic~ enemy of the three particl~°s. is sin larit is lo a ith ic an tl~ere a pear

wh~re

li u c° te s rthcrr°nore it pill e e~ ®~n ed in po~~°ers r~f the ;a . cannot cont~~inin ln ~vllic e clear fro t c fo lc~w n t at scvJe al t~ rl s i~a~e c~th a root-type sin ularity z~phe ereforc ~~Pe ca not `~rrite o~~° lo Grit ~~~ic~ sin t~l~ rit ~~t°ltcr~ ° ~ C~. a 1~- 1 Z ~ aht account alI sir~ ula~ -itïes ~.into eneral o e ~ essio the type (l ) correctly takin as a li resent tlle a t can, ho`~°e~ler, ofc t e ae lir e r t e ~ res ol . there ore

series

Jl.

D2

D3

23) °`°

.in~ éa~0

z~ suc thatsac f ctio ~{ Dz f D3 ;3) Mould be o t ure os t e separa.tc ter of the expansion. he ~` anen on ~l an is t e a for exa le) t e f nctior~ ~D

litude for zero ener fD is °°`"" ~(9(I~D2`ßDr`~°1

63)

D~~`13

.

in

t

e a,

A"

and trace the strucrb ~~~~ er° ld~fl ~ ~~y ~`~ o`s ot, of course, ~~~as s o`~° earlier (see refs . ~0 )

Ds~fl3°~`l

~

where a~~ are the scatterin~ ac~-~plitudes for t`wo pa~°tides .

y y AmsoccH

134

et ai.

the amplitudes hese terms reflect the presence of the simplest singularities in (Mi + MJ)2 and are determined by the graphs of the type represented in whe Sa 1 . The quantities a il and fo correspond to the vertices b and a of this graph . f i . s°6 : .) he quantityf2 was calculated in. refs. f2 = f0[a ,2 a 3(j' 1(k 2E)+J,(k,,P)+a 2a23(j2(kj2E)+ Jf 2(k23E)) 2 2 +a,3a23(J3(k13E)+JAk23E))+ocj k 2 3+ 12 k 13+ 13 k 12], ,f, j(k il E) = I(xil E),

I#Q = - 2E

/

Xil -

î

IV,+M2+in3

1

I 7r

Pi -

Pil

-9

~ 2p a E

j2x arc cos x

171 1 1712 tîä3

il

#i+X2

-

i

1

Mj+Mj

9

1-4#i

(In E - i7r)

Jn1(M1+W4 2+M3)

(in I

=

3

1

1

)

(I + 2#j) +X2

4fli]

3

(6)

+ in 2)(M I + in 3)

In contrast to ref. 6 ) eq . (6) explicitly incorporates imaginary quadratic terms arising from the interaction in the final state and easily calculated using the unitarily conditions . Apart from these terms, in the expression for the amplitude there may appear imaginary terms of the same type, due to the interaction in the initial state has it happens for example in the reactions 7r + N --.> N + n + 7r) or iniaginary terms due to other intermediate states (such as the intermediary state 7r+N in the reaztions ;+N - N+n+7r). We shall assume that these terms are incorporated info .;o that in this sense fo depends (analytically) on the ever for the r-decay reaction, which will be of interest to us in this investigatio'l', the

imaginary character of the amplitude is entirely conditioned by interaction in the final state and in the following we shall use, without special reservations, the imagirfary part written in eq . (6). The lamas of the kind Uil2 arise from the interaction over small distances . The parameters Y can be considered real . These terms are similar to the terms of the type k 2 ro a,, (r,) being the interaction radius) which appear in the expansion a(k 2) in eq. (0. The functions JA k H E) have singularities when k,l = 0 as well as when E = 0. They are determined by the graphs a the type represented in fig. 2 which differ

Ft. 2

frorn. tho,9e of fig . I in that the for a ta e into account once again the final state interaction .

PRODUCTION nF THREE LOW-ENERGY PARMLES

13.5

he vertices a, b and c correspond tofo and the amplitudes of particle pair scat srin for the zero energy of ail . The fact that only the graphs of fig. 2 Yield a contribution to the singular part, is explained by the following consideration . There are not Feynman graphs with a singularity of the kind k,2k,3 5 ) and therefore the singularities of f2 can be connected only with the graphs having a singularity when E = 0. This singularity has in the general case the form E21nE because the volume of threeparticle phase space is proportional to E 2 near the threshold . The only excpption is the case when in an arbitrary graph of the type of fig . 3 the amplitude A of converting P, P?. P3 three particles into three tends to infinity when q, q2 and q3 are small. If the traph of fig. 2 is represented in the form of fig. 3 the corresponding amplitude A will have a pole in the region of integration over q I, q2 and q 3 , and the order of magnitude E- . This leads to the singularity of the amplitude of fig. 2 having the form EInE. The amplitude A cannot be more singular. It is essential to note that the singular terms in f2 are expressed through the scattering amplitudes, i .e . are connected with long rangy interactions . To understand the structure of the higher terms of the expansion (3) let us first turn to the calculation of the imaginary part of the amplitudef, of a higher order. Using the unitarity conditions these imaginary parts can always be expressed through lower terms of the expansion (3). Besides, in the fourth order in the quantity f4 there will appear the amplitude of conversion of three particles into three at zero energy (the three-particle phase space volume is proportional to 0 near the threshold) . If the functions f. are constructed from the imaginary parts using dispersion relations ! appears that the quantities f contain singuV r terms depending (like the imaginary parts) only on the above amplitudes and the parameters defining lower terms of the expansion and polynomial additions due to short-range interaction . The coefficients of these polynomials cannot be calculated through the amplitudes and appear as subtraction constants in the corresponding dispersion relations. The quantities in eq . (6) furnish the simplest example . Such is the general structure of the expansion (3) . Eq. (6) exemplifies the fact that in the singular part of separate to _s of the expansion (3) there appear complicated funcVons of the ratio ke`it E_' . This circa stance is a curious manifestation of the fact that the amplitude has branch points at k il = 0 and = 0. Indeed any singularity of such a function over its argument is a singularity over the position of which depends on ki-1, or, which amounts to the same, a singul city over kdepending on E . These singularities which the physical amplitude should not have go into the non-physical sheets because of the presence of branch points over k,, and E. The above branch points thus appear the necessary v .2 -1 condition for the existence of functions depending on the ratio k i, E -

M V. AMSOVICH et aL

136

cubic In the present paper we investigate the amplitude f3(k12k13k23)- Since the with respect t o t h e terms, like all odd terms of the expansion, are always singular quantityf3 Wifl depend variables Q it is dear from what has been said above that the only on the same parameters as fl and f2. Similarly it can readily be seen that the odd terms of the expansion (1) do not contain new parameters either than those occurring in the previous terms of the expansion . In the following section we consider neutral zero-spin articles and the results 0 tafined will then be applied to the case of r-decay. Everything that has been said in this section naturally refers to the case of neutral zero-spin particles . In the investigations referred to above a study was made of the reactions of -r-decay, production of two 7r-mesons on a nucleon by y-quanta or n. mesons, and some others . Comparison of the experimental data with the theoretical results obtained makes A possible, in principle, to determine the amplitudes for scattering- of unstable particles on each other at zero energy (such as 7r- n scattering amplitudes). 2.

ac

tion of Terms of Third Order

Let us consider in the expansion (3), the terms cubic with respect to the momenta of the particles produced, i .e., the quaKIL . The imaginary part off3 can easily h.. calculated from the unitarity condition if the amplitudes of lower ordersf and f, are known. (As was pointed out we mean the imaginary terms due to the interaction in the final state). Let us see how the real part can be found by directly considering the Feynman graphs for the amplitude of the process . In the case of T- decay (of interest to us in the following) just as in several other cases (such as the prodv ction of two n-mesons by 7r '-mesons on a proton) the cubic terms in the reaction crois section appear only from the interference of the real cubic terms in the amplitude %~i~h unity and the interference of imaginary quadratic and imaginary linear teriTis.. The imaginary cubic terms in the amplitude yield in these cases a contribution only to the terms of the fourth order in the cross section . For reactions of another type, in which there may be states different with respect to the charges the particles produced, and the matrix elements of the production of the particles in these states have dfflérent phases at _7®ro energy, the above grouping of the terms has no meaning. To obtain the terms of third order in the cross section it is necessary in such cases

oo

a

to calculate all cubic terms in the amplitude. The largest terms in the cross s ection depending on the energy are, however, in these cases, linear in the momenta of the particles produced 16 ). Thus the cubic terms are noc-rdy a correction to the corrCC tion given by the quadratic terms. Now, in the case of reactions of the type of the -r-decay the cubic terms are tne first correction to the principal quadratic terms . et us now find the real part off3 in the case of neutral zero-spin particles. S 111ce we are interested in the singular part of the amplitude we should consider the graphs having singularities of the type k1 2 = 0 or E = 0 . The Penera,l fo -In of the graphs

PRODUC71ON OV THREE LOW-ENERGY PA TICLES

137

singularity k,2 = 0 is represented in fig. 4. To obtain the cubic with a root-ty terms in the am litude of the entire process it is necessary to take into account the

Ft. 4 linear and quadratic terms in the amplitudes A and It ca readily be seen that if the linear terms from A and from wc choose those B we arrive at a purely imaginary part ofthe total amplitude. A similar situation arises if the quadratic terms from z, nd the zero-energy ,%,alue of A are substituted into the amplitude. If on the contrary the quadratic terms from A are substituted into the amplitude and B is replaced by the scattering amplitude at zero energy, we have the graphs represented in figs . 5, 6 and 7, among which, in the graphs of figs . 5 and 6, we shall actually find the real

6

F& 5

Fy . 7

cubic terms. The graph of fig . 7 is a "rely imaginary quantity . Below it will also be shown that out of all quadratic terms of A ill the graphs offigs . 5 and 6 only the imaginary terms of A yield a real contribution to the `lanplitude. Let us now consider the graphs with a lo ,"arithmic singularity at E = 0, i.e. the fig . 3. For the amplitude of the process to have the necessary graphs represented in order of magnitude E! it is indispensable that the amplitude B of the conversion of three particles into three should go to infinity as E_ 1 N~ her, E is small . It can readily represented in fig. 8 or fig. 9. be seen that this is possible only if B has the form Substituting B into the graph of fig. 3 we again come to the graphs in fig-, . 5 and 6.

Ft. 8

FQ . 9

~ . v . ~~a~~wt~

3

et al.

re detail. ®~ sti~ y t ese g a hs iil n r~~i ai fig. is trivial prcwi e t a t ~ he calc lati®n of t e gra the I~ ~c, ~ is rfe~ ed are n®~~rn . n ee , t e integrati~n ®ver ~ graphs ~ . ti® tain the valt~e ~f the entire inde n erQ~ly ~f ®t er integratip~ s, an t eref~re t® ® c®rres nding gra f sec~n is st~fficie t t® ~ilti l~ t e val e ~ the aph are the elative ~ ent rn and the a r ~i ati®n y â~c 8 ~ ~ât, h~re ~C it and g~ ; ith is reservati~n, ®f c~ rse, a lit de ~f the~ particles scattere at the v~rtu 1 op ressi®r~ f~r the c®ntains real~ terrns that a a~t fr® thz ~~® inear ~:e , t e e~ t~f°~. inca c~ntri ti®n e are i ®~veti~er, ake n® terested ic , ~f a hi~h~r ~rder erel t e i ina s anld ta e art ~f t e Én the rea part `~f the a plit de, e .d a.ti . ci icall y, pr®~ r t e ®f sec®r~ ag~it~ta e~pressi® f®r the a ~r ich t e n s f Iines in fig. , i c~rrt sp®nd t de ~f the graph, represente ave t~ th~se ®f particles, e _ ~ ~12~12~13

~

2

i7B 1 iP1 2 îi1 ~ if1 1 ~-

2 -jr ti13

(' 1( .~1

~2

1~°~®

.~`

2 12

1

e r~~tati®ns are the sarne as in e . (6~. s~rnedvhat ®re c®rn licated sitt~ati®n arises hen c nsi e ing t e graph re . resented in fig. . ~°® calc late it, it is c~nvenient t® ake se f is rsi~n relatic~i~s ~.;~i~h respect t® ~: ~ z an ~ ~ e~nsec tively . °The dispersi~n relati n with res ect t(~ is ®f the f®

~( -hcrc ~~ is t sit~apler gra ti~n regic~n. the q ant:ity

12 ,

2

~ ®

(~,

-~

12 7Z

~

(

®2

11 ~ 1 2 ' 2 2 12( 12 ®

®

~ 2 12

12 s

e ahs® ti~n art ®f ®ver t e variable ~2 . st a,s in t e ca~e ~fthe 6 s c~nsidere ir ref. }, 1 is n®t a rea f nc i®n thr®l~g ~a~t the integra l~e f~Ih ing is rsi®n relati~n ith res ct t~ can e ~vraten f~r t~(Q, } : (

~

1

,®_.

~

rc ®

2(

,

'

'_

In eq . (y) it is actaally necessary t® ake an a e a e n er ® s~tbtractions, the -pc~~~, nc~rnial a diti®ns arising in t e r~cess ein ne ecie since e are interested ~nly in the sing~alar part (0, ~ a . ~. he a, 9~r ti~n p~.rt 2 , ~ ® f t e f ncti~n i e l since it is eter ined ~ ~ ~ ~hc phy ~ical r~gion ~f t e c~nversi® ~f t~v~ particles i t t ee. It can e ¬~ tained ~r~~tp~° fr® the three®particle c~llditi~tl ~f t~r~itarit , s is s ~r in fig. 10 . eat

PRODUCTICN OF THREE LOW-ENERGY PARTICLES

139

particles correspond to the crossed line in the graph . It can easily be shown that both terms contributing to A2 zTe proportional to Because a A 2 being real, thn quantity A(O, E) proves purely imaginary and equil, to i coast. - E~. Thus A(O, E) can be neglected altogether since we are interestsd in the real cubic terms . The absorption put A I J1 2 E) can be obtained by continuing analytically the unitarily condition from the physical region into the channel k 212 (ref Q. If use is made of the real part of the second approximation graph, the quantity proves real 2 < 2,u 12 E and purely imaginary when k 2I2 > 2p, 2 E. After performing the when k12 dispersion integration (8) the corresponding contribution to the amplitude A(k 212 E) proves purely imaginary. We give neither the calculations nor even the final express ~on dace the imaginary toms are of no interest to us. To obtain the real part of A (k 212E) 2 ,A, -, should, in the calculation of Al(k l2E), make use of fhe imaginary part of the second approximation graph . Then from the unitarity condition for the quantity we have

A

A

2

ik,2 a ?.2

a232E

Eq. (10) is illustrated

in

1

I in2 M3

z

2

[-' L 6 (I + 2fl2) + 3(I

2

-4#2)23]

.

(10)

y fig. It . The quantity z is the cosine of the angle between

pg.

II

X' 2 of particles in the intermedlate state and P3 . The quantity 23 can easily be expressed through X2 2 = X 122 and Xe 2 _ X2 + 2zipIl )X2 (11) 12(l '%C2 _P2 _- 12)* 12 -L P20 12) 23 = in accordance Substituting eq. (11) into eq . (10) and integrating the quantity

the momenturrn., k'

2

with eq. (8) we obtain

in, in 2 n13 I 2 E) Q 1'2) I 2 A(k 1 2 2 L Jakll -t' = P2A P2_ --Sk I k12al 2a23 E in

2

, + m2 +m 3

-

lp N 2 -1 tj ,)~ -*, 12 ,

To find all real cubic terms it is necessary to collect all graphs of the type represented in fig. 5 (six graphs) and the graphs of the type of fig. 6 (twelve graphs) . The final expression for the real parQ is then of the carrn

Reh=

a2,2a l3 k, 2 +a 213a23k,3 +a l2al3a23

&(k l 2o+a 2,2a23k 1 2 2 3(k,3Q+a23a12k23

Q,01020

2 (k 1 2p

n( k23 E)

al3a 1 1 2k 1 3 2 + a23a13 k 23

1 (k 1 3E) 3Q 23

E)

(13)

~t ~l.

° . ~. A~d~~S~~l~

il

~1

~

~~

i ~dl

°~

t3~d 1 + tßß 2 °~- i'ïß 3 tßß

:

~`~12

13

23,8

tiß

t1ß

_

~t

tßß 1 -~° t12 2 °}- tîß 3

3~`°~

~l °~~ z~

~r the ter:~ s of t%e ti~.ird ~rd~r in the cr®ss secti®~ f®r tl~~ r~~~~~~~ ~e h~v~ ~i~e f~~~c~~~i g e~.l ic lP~ y~®rt~ia il~ tiZC rn.c~ ez'ta ® t e artic es i`®r~°eed~ 'd~

i° :~L -

~,,d~

3 ) ~ 1"

~3 ~2

lg(2~

2 2

23t~`12~°13+

~12+

1 :~ `~

2

12

3

a tfid 2 tl 13- ~ ~ _

tt1 1 + tiß 2 -~- t?ß 3 I,,° _

3

ß 1_ ti12 nß 3 _ ~

iFß 1 3~d 9 ii'! 3

.___

-__

_ __._

tB11 -f- itß ; -}- tîß3

~i

®

9

12~13~23'

3 ~ ~_

tßß 1 .._. ~ ~_ __ 2 ~

t'1~

t'tß t3ß tß13 _ _ _ 4 __2 _ ____ ~ ~ .~ tï1 1 -i - YFß 2 + ttß 3

~~___

tß3 2 + Ftß 3

1 + ttZ 3

-`

~ 9

~

~ tTâ 1 - tid 2 ~t'1d ~ _ ___~ ~ _______-___ s ~iéï 1 °$- tiß 3 tT81 2

it11 -r tß2 2 ~- tiß 3 __

~

L2~1s~23

2

4~ 1

~

23~ 3

313 ~23~12+

12

iy?

!, _

13t `~12

~t1ß 1 `$- tßß ,3~tfi3 2 ~tiß 1 `~- t3ß2~tßß 1 ttZ3 t?ß2~~21 -$°

__

2

1

3

tT31 -~

2 -~- tfiß 3

2~~

~ s2 ~~~ e~~~rcs~~®~z ® tai~le

~r~t it d~~s d~~t c~ntai ~~rr~e~t~®

c~acti

~~

f~r

a ~ ~~~ f®

Éer

~;~d s

nd i

)3

iir~ear i

'~. ~`23

2®

k=~

t ~ case

~t~.t~: L~~ c~r~ ~-_~ °.:heciCe direct~y ir~ ~~~ ~ a.~ r~~~ ---~ ~jl a

®ssesses

a

~

cr~

i$ is

f t¬~ ;

`®te that t e a

~;e

r

f t le ~ac cc

t e ~t%cr t`~'o

ess t

~

s~~~

se

i

rcpcrtie~ .

t~w~~rti~y

c ®f

t

ea

.~.

ti~~l ~~ . ~his

at ~~he r

k lz -~ ~,

s ir~ ti~cg cr~~s

~~~~ct~~~ ~;r~i~t~ f~r ~ y ~,al e c~f the t~ta er ¬~r~ , si t~tai er~er~Y itré~r a , ~~ ~ ~ ~.~~i ~~~.i~e ~fkl2 ~®~ the ed~e ~

~~~~~

i .~

t e s rctr

r~~~~ t e

atr

e e te

t~

th~; rc~c®

~i` th~; fc~r

e

a~gti y ~

i

~. f~~~~ction

PRODUCTI)N OF THREE LOW-ENERGY PARTICLES

141

and two momentum trans&& The quantity Mo depends on one of k12, kTL 3, k23 the momentum transfer and, the total energy Z since k13 and k23 are expressed in physical region through E when k, 2 ~ 0lt should be emphasised that eq. (16) is true only in the physical region, i .e. for the values k12 close to the point A on the Mandelstam plane of the three variables k12, 1 2 k 13 andd Mg . 12); this relation does not hold in the neighbourhood of other Points with k 1.2 = 0.

Ag 12

The latter is evident from the derivation of eq . (16) which can be performed, for example, in the following manner. If we want to calculate a k, 2 -linear term it is sufficieiit to consider only the Feynman graphs represented symbolically in fig. 13, since no ocher graph has the necessary singularity when k, ? = 0.

Fig. 13

e coefficient of k,2 can then be calculated according to ref. ') ifwe put q, 0, q 10 = m, and q20 = m2 in the centre-of-mass system of particles I and 2 in the amplitudes A and B in the integration over q, and q2 . ob-1ously, the amplitude )2 will remain dependent only on the two invariants (k, + k 2 )2 and (P3 -)2k2 while the determined through (k, +k2 so that they invariants (q, +,)2 and ( aU, )2 are After this the integrations over q, and q2 are correspond to the point A in fig. 11 easily performed and we obtain eq. (16). Eq. (16) can in principle be used for obtaining information on the amplitude of the charge exchange of particles in the study of the spectrum edge of the reaction with the formation of three particles . If a'-, 2 is the amplitude of the elastic scattering of particles

~r . ~ . ~r~~s®~t~

142

~~r ~l.

af t eir ~° ar ~ I an 2 at zer® energy and i 2 is the a~n litu ~~er 1 ~ is ®f t e f r° ele ent ~ the e ge ®f t e spectru 1 ®( °Fi

i2~iz)

i

i ~

dT° P

, t e r`~~atri

i~

e~°e ® is the matrix ale ant o the reaction in the exchanged particles 1 and 2 ~~~ en , ~ = . ® e . X17) f~~llod~~s the expression of t e crass sectic~ d~

a

a nel with t e theme® ® t'e re cti~

_ ~ ~,~®~ z~ 1® ~ ~ , c~ i z ~~ sin ~), = I~

~/~1 ® I,

~P = arg ~

~/t~ ®) .

ita ity c®nditiutt . ®r~~e infor ation ®n the e.~at~tity ~ sin ears be abtained r®t the ese data, ®ever, are nc~t s® simple as in the case of ic~ t® al e er y ' ) . ()~~ the same the dependence ~f ®ther hand, va~hen the t®tal energy is n®t a® small, e can ps;nç:~ on the irecti®n ®f the incident e~ and analyse t e si :g larity in a crass sect ~®n al®n t e lines ®f az' pa er ~ ). n conclusi®n ®f this secti®n e w®ul like t® enti~n ang er ra rty of e . (1 ~) irectly, fcr for (d~/ I')3, vii. the fact that ( ~/rii")~ ~ , as call easily e c ee e i enticay eutral articles, i.e. vv en all ~r;~ an t e article asses eaincide . This fact will e essential in t e analysis ~~ t e ~®decay. 3~. et us apply t e results ® taine three ~r® es®ns :

y t® the analysis of the

stay of a t ~

~ : s~~ra

ints~

~~'e shill designate t e a plit~ e ®f 'the first reaction y °~ an that f the" sc~c~,a, by = ~` . The particles charged it t e sa e may are esi nets y the su ers~°ruts 1 and 2. The ex ressi®ns for the arnpaitu es and r~ba ilities the g ave reactions can be obtained si i arly to the ~vay t ey gars ~btaine in t e neutral cas~1. All possible a s ~rit r~s act t® es®n themes ®f t e t e rese to in figs . 5 and ~ should be c®nsrdered. e ex ressi®n fc~r t e areas®seetian ~s, li e eq . ßi5), a polynomial c~f the third degree ®ver t e relative c~ ant of e esans roduce . The polynomial contains three cub+es c~f ~a ante k~ 2 , fl~ a cross an six ~ i~ tar s fl ~k~ 3 ,k fl ~ 23' fl~ f ese las ternns iz' fl3 ~:~ " ~~ g~an x~ ~~® stag 'e shall use, as Bn e endent co r~atic~ns, ex resigns f t e t 12( t3`~ ~3l 1S 9S CC1nv'~n1~ :gt eca~se when the fl 2 a~'e small, for ~°x le, tlZe ea inati ~ero 9 s that the tar ~ z ( ;3 -~ i 3) is nc~t in 1°act li ear i h e ct ta 2 . Skip in t the talc lations, e ire the final result far t t e cu ~e tern s i e ex ressic~n ("or the m~

P 0 UCTIIJN OF T

143

LOW-ENERGY PARTICLES

ilities f the reactions d ilv' -

pr

dW - )

dF

3

jw)

df

1

jMj0 PQk ;3+k 32+ + ~ 2 [k IA 22 3) 1

2

+~-Akl#12 - k23)+ k23(k,2 -kl3)1)1

= IM+12 3

0

2

2

(19a)

3 2 2 2 2 2 2 {qlkl2+q2kl2(kl3+k23)+q3[k,3(kl2-k23)+k23(kl2-kl3)11,

4 243,v/3 4 24Q ~3

k 22 3)+ k23(k 2i 2. + k 2I

j[ - 2(a 2 - 7a.) - 3(5a 2 + ajp + 2(4a 2 - a0)~02]~ j[

-

2(7a 2 + 5a o) - 3(5a 2 + a,)p + (11 a 2 + 4a 0)p2]~

(19b)

(20a)

(a - a o # 2 + 2a o #2(a 2 + 2a o ) + (a 2 - 4a o )p - (a 2 - a O)p 2 ],

813 0 8 (a 243, 8 243%/3

,) Z [ -

2(4a 2 - ao) + 3(5a, + ao)p - ' + 2(a 2 + 7a, )E) - 2],

, ) [-(I

2 - a,)(a

1a 2 +4a o)+ 3(5a 2 +a o )p - ' +2(7a : +5a o )p - 2 ],

2 + 2a,,)a 2 (1 - 2p - 1),

(20b)

P = and ao are the amplitudes for scattering of 7r- mesons at zero energy in the 1 states sirh isobaric spins equal to 2 and 0 respectively, and 1 r 2 ,; the ratio of the Here

a2

goba bili6es for the decays in the two channel,,,.

Apart from the obvious symmetry with respect to the indices I and 2, leading, for example, to tbe absence of the term k I 2(ki1 3 - Q 2 &eqs. (19) and (20) possess several additional Properties . first d W - and d W - tend to zero when a2 = ao i .e., when there ii no charge exchange of the n-mesons . This fact is in full accordance with the cubic terms vanishing altogether in the expression for the probability of the process in the case of neutral identical particles, as was indicated above. For the samz- reason there are no terms of the in d ki3 2 and k, 2(k12 3 + k223) in the f irst reaction (%%, here particles 2 3 +A,3 and k2 and 2 cannot exchange charge) nor terms k,3 23 k,3( k,2+_')(ßi-t° 23) in the second reaction (where particles 1 . 3 and 2, 3 cannot exchange.k1 charge) . Furthermore the coetficients of the cubes of the momenta and the li ear 2 2 are proportional to (a2 -ao ) 2 . Besides, these coterms of the type k,2(k,3+k23) e dents in the first and second reactions are connected by the relations 2 ~2 + p 2t1 2. = 0. 2~j j)+ 2 rjj = ()~ The latter pro Indeed, using e

rties can be interpreted specifically at least for the linear ternis, (16) Or the behaviour of the amplitude on the edge of the spectrum .

1-~4

v.

v.

~wfl~r~~rt~

.s

lit de

e can rite the ex ressic~ f~r t e ~ + ~~ en fl Z is s. all : ~ 3 and

1

t

la~ f~r t e rcr a ilities ® t e rea. ti

I-Ience we ave the f®r s ectr

s

t ee

e ~f t ~

(23~)

c 1c late with i~ the c ic ter s are ®f interest f~r d racy . a ratic a ®rn e s. ( 3a~ an (2 ~ it fall® s t at, irst, t e c®e icients f e li re r ~~~re~°s ~ z ( ~ st act a y e r® ®rti®nal t~ Z ~ ~) si ce ) tends ®f kfl 3 an k a a 3ae e . ( ~ c~r e c e cients w en z = ~zo ; sec®n , fr® t ese c~r t® z~rr®l a ~ s1, 1 carried c~~~t an ~ irectly ® l®~rs (t~ see this t e s s it ti~ s fl ~ l ~ ( ) . e c®ef icie ts ~ a ~ ca c ca. c late r e . (2 1, in ®ne ®f f an s® ~nre maro ® tain e . ( ). 7` e c ic te s ~btaine in the resent a er are a secc~ a r i. atic~n fi~r :.i~; ~m ecay re ~ti®ns . e first a pr®xi ati®n c®rres ra s t t e ad atic te rrrs ~leri~~c+ in rel: 5 ~. T°~ r~ n ® ® ex ®siti® we ive . el®~rr the c res ~ in r~e~~~srlttiw th~~ t®tal pr® a ility / I' is t e s ~ t?~e e rcssic~ns i~e ab~~~e ar d ~~~, " (~ ) : o+z

®1~{1

.~ .

a

~gz{1--~°yfl ~- "a'Z

w ere

~ ~

®1

flz

flz~ fl~

k fla)

..3) -~

~. fl3 -~k, .

fl3

ka3)°~°

~

aZ)~°

pa~)`~ .

~ka~)°~

~3~4-~~ka3)~~ ka3) ~- y~2

(cx, -~ ~~~ )~(2 ~ ~~ j "

"(

~ )

~ ~~)

~kflz)°~

~.ki2i'

~? bl

,

UCTION OF THREE LOW--N'--'-GY

he detailed investigotion of the quadratic terms is presente in ref. 5 ) i conclusion the autiors express their deep gratitude t yatiov for stimulating discussions. 'SIC 0, ribov, Nuclear Physics 5 (19-58) 633 nselm and V.N . Gribov, JMP M (1959) 18W; P (1959) 501 ribo v,3ETP 38 (1960) 553 V. 1. T. yatlov, JETP 37 (1959) 1330 v, JETP 341 (1961) 122 V. V. V. nisovich, A. A. AnArn and 4 N. Gribov, JETP 42 (1962) 224 Landau, Nuclear Physics 13 (1959) 181 a, 3 ETP 33 0957) 923 81

1) 2) 3) 4) 5) Q

V.

anflov