The kinematics of inclusive experiments with unstable particles

Nuclear Physics B52 (1973) 1 2 6 - 1 4 0 . North-Holland Publishing C o m p a n y

THE KINEMATICS

OF INCLUSIVE

WITH UNSTABLE

EXPERIMENTS

PARTICLES

G.I. KOPYLOV Joint Institute for Nuclear Research, Dubna, USSR Received 30 August 1972 Abstract: The problem of the reconstruction of the longitudinal or transverse m o m e n t a spcctrum of the 7r° mesons with detection of only one decay 3, is solved. The formulae for the momen ts of this distribution are derived. The results are valid for arbitrary isotropic two- or many-particle decays. The Yen-Berger effect in the decay of resonances is estimated.

1. Introduction The theory o f inclusive experiments predicts the properties of the transverse and longitudinal momenta of produced particles. These predictions have not been verified for all sorts o f particles. The data of inclusive reactions such as n + p (or p + p ) ~ n ° + anything

(1)

I--+7+ 3,

are poor. This is so because usually only one decay 3' is detected, which does not permit one to reconstruct the 4-momentum o f the n °. We want to show that if you know the spectra of components of the 7 momentum you can reconstruct the spectra of the n °. The task reduces to solving some integral equations similar to those which allowed one to discover the n° mesons in cosmic rays in 1949 [1]. It is possible also to determine the moments of the n ° spectra measuring the similar moments of the 7 spectra. These moments can be compared with theoretical predictions [2, 3]. The same problem can be solved for every isotropic two-particle decay. Some properties of these equations are valid for many-particle isotropic systems. The second application o f our formulae is the estimate of the Yen-Berger effect [41 in resonance decays.

G.I. Kopvlov Kinematics of inclusive experiments

127

2. The basic equation Consider a reaction of the general type a + b -+ 0 + anything [-+1+2.

(2)

The decay 0 ~ 1 + 2 is isotropic. The primary particle 0 has massM, energy E, momentum P with the components P,.(i = x, y, z) in the lab. system; the density of the Pi distribution is N(Pi). The same quantities for the secondary particle 1 are rn, co, q and n(qi) respectively. An asterisk denotes the rest system of 0 in which it is convenient to introduce the dimensionless constants a, b characterizing the decay 0-+1+2: a

=

M

-

2342

b

'

_q* - - -

M

-

M

(3)

We are searching for the equation connecting N(P) with n(qi). First we pick out the decays of 0 with the same value of P and calculate the mean value o f q n for these (the nth moment of the spectrum n(qi) ). Let us write the Lorentz transform from the rest system of 0 to the lab. system ,

co+e*

q =q + P - E+M'

co-

Eco* + P .q

M

in the tensor form

qi = (c°* /M)Pi +~ Ri/q~,

(4)

J

where

Ri]

= 6ij + PiPj/M(E +114).

(5)

The average (over all decays with the same P) value of qn is

Because the decay of 0 is isotropic, the axis of a spherical coordinate system may be directed along the vector We immediately obtain that, for even values of 1

Ri].

(odd l's give zero),

<(?Ri]qT) l>=(q*2 ~R2"){/([+1)/]' --

(6)

where

Rg _g2 +# j

M2

(7)

G.L Kopylov, Kinematics of inclusive experiments

128

Denote E i = (/142 +/o2)l, then the fonnula for the nth moment of qi distribution can be written in the form

2 l ~ ( (aPi)n- 21 (b2E2)l

(8)

/=0 Now we can take the average over all possible momenta P. The result of the averaging of over the P's we denote again by .We arrive at the formula connecting the moments of primary and secondary particles = ~ /=0

2~--(<(aPi) n:21 (b2E2i )l>.

(9)

The left-hand quantities are measured in experiment, those on the right are the object o f theoretical speculations. We solve this equation below, in sect. 5, to find the moments of primaries from it. Now we notice the important property of eq. (9): the dependence on P / ( j :/: i) has vanished; in both sides of the equation there stand the same (ith) components of momenta. The moments of the qi spectrum depend on the moments of the Pi spectrum and do not depend on the spectra of other than i components. As every distribution is determined by all its moments, we conclude that the longitudinal spectrum of secondaries is determined only by the longitudinal spectrum of primaries; the same is true for the momenta projections on an arbitrary axis. We denote this property shortly by the symbol n(qi) ~ N(Pi). For example, two beams of 7r° with the same projection of momenta on the z axis but with different x o r y projections will lead to the same spectra of z components of momenta of 7 quanta. Another consequence of this statement is: let us have two 7r° beams, one with a m o m e n t u m P1, another with a momentum P2; draw the axis perpendicular to P1 - P2 in the (P1, P2) plane; then the spectra of the projections of the 3' m o m e n t o onto this axis are the same for both beams. We can make the same statements about other isotropic decays (K ° ~ 7r + 7r, Z ° -+ A ° + ~, etc.). Moreover, for three-particle decays 0 ~ 1 + 2 + 3 the same statements can be made because they are true for every decay 0 ~ 1 + 2 + 3 with fixed mass m23. Therefore it is true for many-particle decays of spinless or unpolarized particles (such as (co °, 77°, K °) -+ lr+ + 7r- + ir°) and for any many-particle isotropic system (fireballs for example [13]), etc. Keeping in mind this property we will direct the particle 0 as conveniently as possible, i.e. along the i axis, and derive the equation for the projection o f momenta on this axis. Let the particle 0 be emitted along the i axis only (with momentum P = {P., O, 0}), and decay into 1 + 2. Then we have

qi = (~*/M)Pi + (Ei/M)q*" As the decay 0 ~ 1 + 2 is isotropic, all values of q* have equal probability in the interval ( - q * , q*). Therefore the qi spectrum for a fixed Pi will be l I shaped with a

G.L Kopylov, Kinematics of inclusive experiments

129

density M/2q*E i in the interval

- bEi + aPi <~qi ~ bEi + aPi"

(10)

If the momenta Pi are distributed with the density N(Pi) we arrive at the equation

"(qi) :

M

q,J

"N(Pi) dPi

.

(11)

The limits in the integral are m 4= 0:

(M/m 2) (o~*Pi + q*Ei),

(12)

qi - q*2/qi, oo

for qi > 0

_ 0% qi

for qi < O.

m =0:

(13)

q*2/qi

The domains of integration for all qi are shown in figs. 1,2.

Fig. 1. The domain of variation ofqi, Pi for m :~ 0 is between the two hyperbolas with the asymptotes qi = (a +~b)Pi. Though these equations were derived for particle 0 directed along the i axis it is clear from what has been said above that the same equation is valid for an arbitrary distribution of direction of the primary particle. Of course the i axis can be taken in an arbitrary direction, not only as the longitudinal or transverse axis. In order to check the validity of these considerations, we find the average value of the characteristic function exp (qi t) by making use of eqs. (1 1) and (10);

1 FN(Pi) dPi bEi+aPi Eli f exp(qi t) dqi

(exp (qi t)) = ~-~J

- b E i + aP i

= 2b

{exp [(bE i + aPi)t ]

exp [(-bE i + aPi)t ]

.

G.L Kopylov, Kinematics of inclusive experiments

130

Comparing equal powers of t we obtain
1 /;(bEi+aPi) n+l n + i\ ~S~

( - b E i + a P i ) n+l ) "

(14)

The direct calculation shows that eqs. (14) and (9) coincide. It is interesting that the equations for all four components of 4-momenta have a similar form. The equation for the energy component [5, 6, 9] can be obtained from the equations for the momentum components by replacing Pi -->E, qi -->co, Ei -> P: M ( ' N ( E ) dE

.(co) =2-~j ~ ,

(15)

with the integration limits (M/m 2) (co co* + qq*). In the same way the moments of the energy spectrum can be obtained: I

"+I)

-

(co">= n + l \

2bP

"

(16)

However, this symmetry is not obvious at first glance: it is observed only after the averaging over other components. It is clear that the above mentioned many-particle decays have the same property n(qi) +-N(Pi) (for other generalizations of the two-particle formula see ref. [6]). The theory of inclusive experiments often predicts the properties of spectra taken with the energy E as the weight [7] (not with the weight 1 as above). The reconstruction of these "weighted spectra" of primaries, detecting only one secondary particle, is a more difficult task. Itowever, all moments of the primary weighted spectrum can be reconstructed easily. For this it is necessary to average coqin" (coqy> :

aE +

e/q[/M

Pi +

Ri/q

"

1 In order to simplify the calculations, it is convenient to take the second axis of the spherical coordinate system in the (P, Ri: ) plane. Then the weighted moments ~ (coqi )/(co) are expressed by the weighted moments Pi =- (EPi ) / (L ) ( l ~ n):

1½nl (coqn) = ~_j D 1 ((L/M) (aPi)n 2l (b2E2)/),

(17)

/=0 wherc for all I and n D/n = co* 1

¢)

1 + 21 + l co*

2l + 3

except l = ~n, when Dn/ = co*/(n + 1). Solving these equations for n = O, 1,2 ..... it is possible to reconstruct all weighted moments of primaries (see sect. 5 below).

G.L Kopylov, Kinematics of inclusive experiments

131

3. The reconstruction of the spectrum of primaries

3.1. n ° -+ 3' + "7 decay We consider in detail this important case with m = 0 (fig. 2). There is only one value ofqi , qi = O, which can be reached for every Pi" Therefore there must be a peak at the point qi = 0 in the spectrum n(qi). There can be a plateau around this point (not wider than M~r) bczause the spectrum N(Pi) has a finite length (fig. 2b, c).

__

rn=0

(a)

I N (P~)

1

I

i

I~

Fig. 2. (a) The same domain as in fig. 1 but for m = 0. (b) An example of the Pi spectrum. (c) An example of the qi spectrum.

To solve eq. (11) in the case n ° --, 7 + % we must differentiate it:

N(Pi)=-

q dn(qi)l i dq i Jqi = ~2(Pi+El)"

(18)

In order to obtain the spectrum N(Pi) in the intervals Pi >~0 and Pi <~O, it is sufficient to differentiate the spectrum n(qi) in the intervals qi ~ ½Mr and qi <~ - ½Mr respectively. The spectrum n(qi) in the interval ( - 5M~r, ~ ' ~-M~r ) is not needed. We can reconstruct the longitudinal and transverse spectra of the n ° with the help of (18); therefore some predictions of the theory of inclusive experiments can be tested in the reaction (1) by detecting only one 7'in each event. It is possible also to measure the n ° anisotropy in experiments similar to that of ref. [ 1]. We should refer here to Sternheimer's formula [5] which is similar to eq. (11) (used, e.g., in ref. [8]). The difference between the two formulae is as follows. The old formula uses the energy spectrum of secondaries which are detected in a given direction. Instead of it we assume that secondaries are detected in all directions but their momenta are projected onto a given direction. As a result, the new formulae are exact for any isotropic decay, and any energies and angles of primaries. The formulae from ref. [5] are approximate and are not appropriate for low energy or for-

132

G.I. Kopylov,

Kinematics of inclusive experiments

ward emitted rr’. However, in the high-energy domain the two sorts of formulae must practically coincide. Some theories make predictions about the properties of rapidity spectra. This spec trum for rr” can be reconstructed from the 7~’+ y + y decay also. Let us calculate for every y its rapidity along the i-axis given by the formula qi = 4” exp yi (4i > q*). Let the density of the yi spectrum be WDi). Analogously for 7~~‘s Pi = M sinh Yi (Pi > 0), the density of the spectrum is W(Yi). Then, from (1 l),

sm

W(Yi) dYi

WcVi) exp (-Yi) = t

cash Yi ’

Yi

Yi>O,

(19)

from where [W(Yi)

W(Yi)= [l + eXp(-2Yi)]

W’(Yi)]

3

Yi ~ 0.

(20)

Analogously the spectrum W(Yi) for Yi < 0 can be obtained from the interval 4i < -q* (turning the i axis in the opposite direction). If the spectrum N(Pi) is symmetrical, N(Pi) =N(-Pi), or if N(Pi < 0) E 0 (i.e. if the part Pi 3 0 of these spectra determines all the spectrum) then the even moments of the Yi spectrum can be calculated from the moments of theyi spectrum. It is easy to obtain from eq. (19) for the characteristic function the equation (exPO/i t)) =~~fo~~)~~sfh+~j from where

3.2. Decays

(YY,

(n even)

(Y; coth Yr,

(n odd).

I

CyY)+ n Cy;-’ ) =

(21)

with m # 0

In order to analyse the spectra of the decay of particles with m # 0, it is convenient to introduce the rapidities into eq. (11): 4i = m sinhyj, Pi =M sinh Yi, q* = m sinh y*. Designate N(M sinh Yi) E iV(Yi), n(m sinh yi) = E(ji). Then _Vi+_V * 2bn~i)

=

s

N(Yi)

dYi.

(22)

Yj-.v* It

follows from this that 2bn’Cvi)

Substitute identity

= NOli + y *) - NOli - y*).

(23)

here yi = yi + 2y*, yi + 4y*, etc., and add these formulae. We arrive at the

G.L Kopylov, Kinematics of inclusive experiments

B(y i + 2ky*) = const

133

(24)

k~_~

for every Yi" If we introduce the new variable Yi = [(Yi - y*)/2y*] ([a] denotes a fractional part of a) then (24) can be written in the form n(Yi) = const for every Yi. It means that the curve n(Yi) is always II shaped independent of N(Pi) (fig. 3).

(a) -2-~

..1,/

y~/

~ ~,

"

(/~)

l i. '

-~ -i

+t

(c) '

k-k2y

J

Fig. 3. (a) The primary spectrum recalculated to the rapidities Pi" (b) The recalculated secondary spectrum. (c) The same spectrum if the variable Yi is used. In order to reconstruct N(Pi) we can use either the interval Yi >/ Yi +Y* or the interval Yi ~ Yi - Y*" It follows from (23) that _N(Y/) = - 2 b

~ n'(Yi)yi= =2b ~ n'0ei) . h = 1,3,... Yi + xy* ~.= 1,3,... Yi =Yi - by*

(25)

Hence one can deduce the formulae similar to those for the overall energy spectrmn [6, 9] or for the energy spectrum in a given direction [5]:

N(Pi) = T- 2b h=1,3,5~.... (exE i +-phPi) [n'(qi) ] qi=exPi±oxEi

(26)

where

e x = (m/M) cosh Xy*,

Ox = (m/M) sinh Xy*.

As the spectrum of 0 determines the spectra of both secondary particles 1 and 2, the spectrum of one of these particles can be, point by point, recalculated into the similar spectrum of another particle following the pattern of ref. [9].

G.I. Kopylov, Kinematicsof inclusive experiments

134

4. The reconstruction of the transverse spectra

Having obtained the spectraN(Px) and N(Py) (x andy are the transverse axes) with the help of the formulas from sects. 2, 3, one can reconstruct the spectrum N(PT) of transverse momenta PT = (p2 + p~2)~. Assume that~ all azimuthal angles of particle 0 have equal probability. Then the spectrum N(PT) can be obtained from N(Px). To achieve this we change the variables (PT, ~) ~ (PT, Px) in the integral

o

f N(PT)d/'Tdv--~f aPx f N(PT)dPT 2 2 1'

0

0

Px (PT-- Px)2

Hence

N(Px) =1 f

N(-~PT-)dP~T

7

(p2

Px2)1 "

(27)

IPxl This is the Abel equation. Any half of the Px spectrum (either Px > 0 or Px < 0) can be used for the reconstruction of the spectrum N(PT). For example,

d fN(Px)PxdPx

N(PT) = - 2 dP~ a ~ PT (PxZ-- PT )T

•

(28)

Hence, the reconstruction of the transverse spectrum requires the solution of an additional (as compared with the longitudinal case) integral equation. in order to avoid this, one can write a new equation which directly connects the transverse spectra of primaries and secondaries. We shall deduce this for 7r° --, 3' + 7 decay. Firstly we write the equation for ~'(qT) similar to (28): 2 d @T qfT n(q~x)--qxdq~x 2 2} ' (qx - qT )

n'(qT):

and substitute here eq. (11) for n(qx): ~(qT) =

--2 d ~ qx dqx ~ X(Px) dPx dqT ~/T 2 2(qx - qT)2 f(qx ) E x -

-

-

-

1

J

'

where

f(qx) = qx

q*2/qx.

Then we change the order of integration (fig. 4) and integrate over qx :

n"(qT) = --2 ~

d

ff(qT) N(Px)i¼(Px+Ex)2q21½1~ldPx.

G.I. Kopylov, Kinematics of inclusive experiments

135

Fig. 4. The domain of variation (qi, Pi ) for a given qT (right); (Pi' PT) for a given qT (left).

Now we can differentiate ~'(qT) = 2qT

f

N(Px)[¼(Px+ Ex)2 - q2] -½ Ex I dPx.

(29)

f(qy )

Finally substitute N(Px) from eq. (27) and then change the order of integration (fig. 4). We come for qT ~ q* to an equation with variable limits of integration

f N(PT)K(t(PT), s(qT))dP T

~'(qT) = (2/Tr)qT

(30)

f(qT )

where t(PT) = In [(PT + ET)/M]' s(qT) = ln(2qT/M)'

t ~> s,

(31)

t

K(t,

s) = 2M -2 f d u [ ( s h 2 t - sh2u) (e 2u

e2S)l@.

(32)

$

For qT ~< q* the lower limit in the integral (30) is zero and

K(t, -t)

f o r P T ~< I)'(qT)l

K(t, s) =

(33)

tK(t, s)

f o r P T/> [f(qy)l.

Eq. (30) solves the problem of the reconstruction N(PT) +- n'(qT) for the process 7r° --' 7 + 7- We want to emphasize that it is better to solve numerically eqs. (11 ), (22), (30) than to use the analytic formulae (18), (28). The numerical methods permit one to take into account the experimental errors [10]. Some other results can be derived from eq. (27). We list two of them without deriving them. The first is the connection between N(PT) and n(qx) (for 7r° decay):

G.I. Kopylov, Kinematicsof inclusiveexperiments

136

n(qx) = (1/rr) f N~(PT)ET1L(PT, qx)dPT, f(qx ) The kernel L is the elliptic function F(k, ¢), k = PT/ET,

qx > q * "

cos ¢ = f(qx)/PT.

(34)

(35)

For 0 ~< qx <~q* the similar equation holds but the lower limit in (34) should be substituted by zero, and kernel L by 2K(k) for PT <~Jf(qx)[ and by K(k) + F(k ~) for PT ~> ]J(qx)[ (where sin ~ =f(qx)ET/P (M 2 ~J2(qx))r ) Maybe, eq (34) will be more convenient for the reconstruction of N(PT). The second result is the connection between the even moments of the spectraP x and PT (or qx and qT):

(p2n)_ (2n(~n) --~.~l)!! (P2Tn)"

(36)

5. The m o m e n t s o f primaries

It is sufficient sometimes, instead of reconstruction of the primary spectra, to calculate its lowest moments. Formulae (9) give for the lowest moments of the longitudinal spectra of the secondaries (qk) = a(P L) (q2L) = fb2M 2 +(a 2 + } b 2 ) ( p 2 ) ,

f=}

(q3) = ab2M2(PL) + a(a2 + b 2) (p3)

(37)

= gb4M4 + hb2(Sa 2 + b2)M 2 +(a 4 + 2 a 2 b 2 + { b 4)(/~L) '

g=ks,h=2.

To obtain the formulae for (even) moments of transverse spectra from these formulae, it is sufficient (taking into account (36)) to replace the index L by T and take f = 2 ~-, g = h = ~s. Solving these equations for n = 1, "~.,..., one by one, we express 0"i~>, by
1

2

4(qL)2 -- 4Mrr' (38)


The lowest moments of "weighted spectra" can be found by combining formulae

G.L Kopylov, Kinematics of inclusive experiments

137

(16) and (17). Wc denote p.n=

_



l

(E)

_

'

n_

q



(co)

'

and obtain for rr° -+ 7 + 7 p3_S

3

9

2--

--ffqL -- 7gMnqL ' /~L = 3 ~ L - g/VlTrqL 6 ..2 ~-2-.+ igM4, 1 PT

= 2q2

(39)

l 2 - 3-MTr'

_ ( ~ T ) 2 = g q ~ T _ 4 ( ~ T ) 2 _ T g 4M, , 2 7qT ~ + 1-~" The odd moments of the transverse spectra cannot be calculated without direct solution of eqs. (27) or (30). The comparison with theoretical predictions (as in [3]) should be based on even moments.

6. T h e Y e n - B e r g e r m e c h a n i s m

the

It follows from eq. (1 O) that if the qi spectrum has to be cut by

Pi spectrum is extended t o / ~ -~ Pi max then

qii = aPi + bffi.

(40)

I f m 1 and O = M - m 1 - m 2 are small (as compared with M) then the narrower than the Pi spectrum:

~]~ml+m2Q+F2mlm2Q

qi spectrum is

(1-~g-)]M2 ½

This effect observed in transverse momenta of ~r mesons was explained for the first time by Yen and Berger [4]. The decays of isobars with small Q into 7r mesons and lighter nucleon resonances manifest itself in the existence of the peak e x p ( - 1 5 q 2) in the q2 spectrum [11 ]. Formula (40) estimates the strength of the effect. As it deals with the extreme possible width of spectra, this estimate holds for anisotropic decays also. The effect can be seen also in the longitudinal spectra of secondaries and in the energy spectra as well. One can estimate the mean square (not extreme) width of secondaries spectra. It follows from (37) for isotropic two-particle decays (q2)= 2A + B(P2.), (co2) = - A + B(E2),

(q2L)=A + B(p2),

(41)

G.I. Kopylov, Kinematics of inclusive experiments

138 where A = .~q,2

B = (co .2 + ~q*2)M-2.

(42)

For weighted moments (see (17)) the term ~ q , 2 in B should be replaced by q , 2 . If m, Q -~ 0, the terms A, B -~ 0, the spectra become narrower. One of the consequences of (41) is convenient for experimental test: (2q2 _ q 2 ) _ (q2 + 602) B = const. (43) (2p2 p2) (/o2 + E2) -

The formulae (41), (43) will be valid for any multi-particle system (isotropic in its c.m.s.) if we average the coefficients A, B over the phase space available (i.e. over the MM of detected particle). They hold, e.g., for isotropic groups of particles in inclusive reactions (if there are any) or for three-particle decays of unpolarized resonances. For example, we have for r~° -+ rr+ + rr- + rr° (averaging (41 ) over the Dalitz plot) (q27rT ) = 0.56 m2rr + 0.138 (P2T).

(44)

For such decays the Yen-Berger effect can be appreciable (compare (44) with (q2T) = 1.75m2 + 0.06 (PIT) for A -+ 7r + p).

A nisotropic decays. Formulae (41) assume that the 0 particle decays isotropically. In order to find the correction for these formulae owing to the decay anisotropy, one must calculate (q2) taking into account the density matrix for a decay 0 -+ 1 + 2. We have done this for spins 1 and 3 of the 0 particle produced by unpolarized particles. Calculating ( q 2 ) = ( q 2 + q~) 9 as in . (8) we . averaged . . It. with .the density matrix weight (see, e.g., ref. [12]). The result for spin 3 takes the form

(q2) = a2(p2) + fb2((2M 2 + p2) (gp, + hp")

(45)

a')v2V2(F.+M)-2>,

+

2 2 9 (qL) = a (P~) + fb2(E2T(gp ' + hp")

(46)

+ k(p" - p') [M + p 2 ( E + M) -1 ] 2), where

p'=033_,

P"=PLL,

22

f'=~,

g=6,

h=4,

k=3.

22

To obtain the formula for spin 1, it is necessary to take !

P =Pll,

i¢

P =P00,

f=~',

g=4,

h =1,

k=2.

G.L Kopylov, Kinematics of inclusive experiments

139

In order to calculate the corrections for the formulae (41), we assume that the density matrix elements weakly depend on the emission angle of 0 and that correlation between longitudinal and transverse momenta (more exactly between P2T and PQ(E + M) -2) can be neglected. Then the last term in (45) is equal to (p2) ( P [ ( E + M)-2). When the particle 0 is emitted in the backward direction (in the c.m.s. of the reaction (2)) we can take PL = 0 and neglect this term; when the particle is emitted in the forward direction we can approximately assume that PL ~ E and that this term is equal to (p2). Furthermore, it is convenient to isolate the term coincident with (41) from (45). We obtain for the spin (q2) =

2 (qT) isotropic

(47)

+ 2fb2(p 3-3- - 3) [ 2/142 + (e2) (1 - 3(p2(E + M)-2))]. 2 2 1 For spin 1 it is necessary to exchange f = ~ by f = ~ and 03 3_- -~ by 011 - 31 in this formula. 22 The correction term reaches maximum for P , = 0 and p3 3 = 0 or t But even un21 ~] -2" der these extreme conditions the ratio (qT)-f/(P~)-~ for A 27rz + p changes less than by 10 per cent compared with its isotropic value (41); for the forward direction of A the corrections are somewhat less; the value of p3_3 taken from experiment leads to the correction (narrowing of spectra) of the ord~rZof 3 - 4 per cent only. The de2-1 2-1 2-1 pendence of qLmax/PLmax on PLmax and the dependence of (qT)2/(PT)2 on (PT~2 for resonances 2x (1236) and N' (1470) can be seen in fig. 5; the maximum anisotropic corrections for the isobar are drawn also. Therefore the main responsibility for

Ge~

Q2

0.6

Fig. 5. The dependence qimax/Pi m a x on Pi m a x and ~-T/PT on fiT for resonances ~ and N' (~ = (g2>t/2); the dotted lines represent the m a x i m u m corrections for decays due to the anisotropy calculated by eq. (47).

140

G.L Kopylov, Kinematics o f inclusive experiments

the Yen-Berger effect (for not very high spins - 1, -~) is borne out by the low decay energy of resonance and not by the spin effects. (Yen and Berger considered their effect for fixed qL' But, as we have seen, the qT spectrum for isotropic decays does not depend on the qL value. The dependence of anisotropics is far below these 10% variations.) Thus, formula (41) is suitable for more detailed estimates of the effect. The author thanks V.G. Grishin who raised the problem discussed in the paper, and M.I. Podgoretsky and V.L. Liuboshitz for valuable advice.

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