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Theory of peripheral interactions of fast particles
7.C
[
Nuclear Physisc 31 (1962) 308--314; ~) North-Holland Publishing Co., Amsterdam
I
Not to be reproduced b y photoprint or microfilm without written permission from the publisher
T H E O R Y OF P E R I P H E R A L I N T E R A C T I O N S OF FAST P A R T I C L E S V. S. BARASHENKOV, H. J. KAISER AND VAN-PEI Department of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, USSR
Received 9 May 1961 Abstract: The expression for the energy spectrum of peripheral mesons in a fast moving nucleon is obtained. This spectrum corresponds to the expansion of the mesic field of a moving nucleon into plane waves and can be used for the calculation of inelastic fast particle nucleon collisions. Within distant peripheral collisions the theory under consideration coincides with the peripheral collision theory of Heitler and Peng.
1. Introduction
At present there is no consistent theory of strong interactions for high energies and the calculation of them is based on approximate semi-phenomenological theories. The Fermi statistical theory has long been used for calculating inelastic interactions (the details concerning these calculations can be found in refs. 1, 3)). Gradually, however, experimental data which cannot be interpreted in the framework of this theory 3, 6) have accumulated. The model of central and peripheral collisions was suggested in refs. 7-8) to account for these facts. This model takes account of the internal structure of the colliding particles and incorporates the Fermi statistical theory as a particular case. Analysis of the experimental data on inelastic interactions at energies T > 1 GeV points to a very large contribution made by the peripheral interactions 3, 4). In refs. 9 - 1 2 ) , the peripheral interactions were studied by the pole method, and in refs. 3, 5, 6) with the aid of the expansion of the meson fields of the colliding particles into plane waves which were then considered as real "peripheral mesons". This approach was for the first time worked out by Weizs~icker and Williams in electrodynamics and later was applied to the nucleon-nucleon interactions by Heitler and Peng 13). As has been shown by comparative calculations the pole method and plane wave resolution method, when applied to inelastic interactions at large energies lead to similar results in the more important cases. The merit of the latter method is simplicity and transparence. The purpose of the present paper is a more elaborate treatment and improvement of this method. t We indicate merely the pioneer works; at present there are very many papers concerned with these problems. 398
THEORY OF PERIPHERAL INTERACTIONS
309
2. Expansion of Nucleon Mesic Field into Plane Waves The energy spectrum of peripheral mesons in the nucleon q(r; ~) was used in refs. s-7). The method had been obtained by Heitler and Peng aa) under the assumption that the mesic field in the nucleon is described by the classical expression ~o(r) oc ~,~ • ," - -
d
e -r
--
rdr r (here and further on h = c = # = 1, where # is the mass of the n-meson). This expression, however, is a very poor approximation in the region r < 1 (ref. 8)). A more accurate expression for q(r; e) can be obtained and the limits on the applicability of the Heilter-Peng theory can be established. First of all let us bear in mind that the spectrum of peripheral mesons at a distance r from the centre of a fast moving nucleon can, in the general case, be determined from comparing two expressions for the density of the total energy flux of the nucleon mesic field in a layer of radius r:
~
oo
I(r) =
8q(r; ~)d~,
(1)
S(r;
(2)
1
~(r)
-- 2nr
Odt,
where S = ~2 [(1 + f l 2 ) T l ° - f l ( T °° + T l t ) ] is the x-component of the Poynting vector at x = 0 at the instant t and T uv -= (NITer(r; x ) l N ) is the average value, over the state of the physical nucleon, of a component of the energy-momentum tensor of the mesic field in the nucleon-bound system of coordinates; furthermore, x' = y ( x - f l t ) (below we assume that x = 0 in the nucleon-bound system of coordinates), y = (1 -f12)-~ and fl is the velocity parallel to the axis x' of the nucleon in the laboratory system. In the tensor T ~v only the terms due to the energy of the mesons themselves, and not the meson-nucleon coupling energies, should be taken into account. Correspondingly, one h a s T ~ = T ~. It can readily be shown that
is
= 2(2n)a
TOO+T11 =
1
da(pq)
f da(pq)
2(2n) 3J
l
ple'(V-")'=<(ap+ a_* p)(a,• -- a_q)),
(3)
ei(V-,).x
x/~v ~q
x {p, q~<(a*_~+ a~)(a* + a_~)>- ~ % < ( a ~ - a~)(~'- ~_.)>},
(4)
where oq 2 = 1 + q2 and the b r a c k e t s < . . . > designate the average values of the normal products of mesic field operators.
310
v . s . BARASHENKOVe t al.
If one considers sufficiently distant peripheral collisions, the nucleon recoil can be neglected and the fixed nucleon theory used for the calculation of the averages in (3) and (4) 14). The expression for the vector-component S in this case is
S(x) =
72 [ d3(pq) e'°'-')'*(og,+ogq)-' E (e~+c%)-l(en+c°q) -1 (27r)3d~ x ((1 + fl2)copogqpl[Tq(n)Tp(n ) - T~(n)Tq(n)]
(5)
+ fl[E~(pl q l - cop ¢o,) + P l q l(a)p + ~q)] [ Tp(n)T~(n) + Tq(n)Tp(n)] }, where T,(n) =
~i J_- ~V.. ( N + nnlp " trTIN).
(6)
The quantity f, w i t h f 2 = 0.08, is the renormalized meson-nucleon coupling constant, z is the isobaric spin operator, and Vp is the form factor of the mesic field source. Summation is performed over the total system of orthogonal states with n = 0, 1, 2, •.., mesons. From comparing eqs. (2), (5) and (6) it can be seen that the fraction S(x) proportional to (1 +f12) gives no contribution to I(r) and can henceforth be neglected. In the one-nucleon approximation eq. (5) becomes 27 fl So(x),= o = ~ f
2
3 Vp Vq d (Pq)~PiqlP"
qe~(p_q),
x
(7)
(Dp ~Oq
for neutral mesons, and S±(x),=0 = (1 +%)So(x),= o
(8)
for charged mesons. The mesic corrections (n > 0) can be expressed through the integral of the total ± cross s'ections for meson-nucleon interactions a± = atot(Tr°p)17). To do this let us note that o
Tp(n)T~(n)+ Tq(n)Tp(n) = 2 pq VpV~ cok k 2 Vff ~/o),o)q T•(n)Tk(n), and
(½(l+%)a++½(1-%)a_ 2rCk.>o ~ ~(E"--c°k)Ttk(n)Tk(n)= / ½ ( 1 - % ) a + + ½ ( l + % ) a ~a o Hence it follows
27 fl So(x). >o -
for a n+ spectrum, for a n - spectrum, for a n ° spectrum.
(9)
3 Vp Vqp" q e~(p-~), x d (pq) cop¢o,(o9t,+ to,) f"dk a°(k)°)k°)'°gq-plql(°)k+°9"+°)') × Jo
v2
o , k ) ( o , , + o,k)
(10) '
where a is the characteristic dimension of the form factor; the expression for S±(x),> o
THEORY OF PERIPHERAL INTERACTIONS
311
is obtained from eq. (10) by substituting the cross section tro(k) in accordance with (9). F r o m c o m p a r i n g eqs. (1), (2) a n d eqs. ( 7 ) - ( 1 0 ) we shall obtain, after integrating in polar coordinates, the peripheral m e s o n energy spectrum in a layer o f radius r:
2 r l f ~ f°°d(pq ) V,,Vq, [(y~)2 )] qo(r; e) = ~-~ -e.)o Jo ~ pq J°(Pr)J°(qr)+pqJl(Pr)Jl(qr COp,COq,
X
(I
-r~e )- 2 ( 1-"-~ - ' C 3 ) J t - 2 n
co~ co,, co~, X
(.D~,o)q,
fadk
:
l~za a_(k)
)
(11)
(cok + cop' + co¢) 1
(co,,+ co~)(coq,+ co~)
Here p '2 = p2+(e/~fl)2 and q,2 = q2+(e/~fl)2 ' with e > 1, while Jo and J1 are the well-known Bessel functions. The total energy spectrum determining peripheral collisions with impact parameters r > ro is obtained by the integration Q o ( r o ; e) = +
I;
o
qo(r;
e)dr.
(12)
+
At high energies o f colliding particles the average energy of peripheral mesons is sufficiently large and the cross sections for their interactions do not depend on the isobaric spins. In this case the cross section for peripheral interactions is determined by the total spectrum (Qo+Q+ + Q - ) t . At lower energies account should be taken o f the difference in the cross sections for the interaction o f n +, n - and n ° mesons. This leads in particular to different energy dependences for the angular a s y m m e t r y produced in pp- and pn-collisions (as well as in np- and nn-collisions). This circumstance can be used for an experimental check-up o f the peripheral collision theory. 3. Numerical Calculations and Discussion
I f one neglects the mesic corrections and puts Vk = a2/(k2+a2), integration in q(r; e) and Q(r o ; e) can be p e r f o r m e d analytically, and the expression for the spectrum becomes m u c h simpler: f 2 e / a2 \ 2 t
+ a2(g~(ro b ) - K2(ro b))] + 2ro[sro(ro s)gl(ro s) + bro(ro b)rl(ro b) 2b
+ ---
a2--1
Ko(ros)K~(ro b ) -
__ 2a2s a2_1
(13)
}
Ko(ro b)K~(ro b)] ,
t Outside the mesic field source (r > 1/a) the one-nucleon part of the total spectrum coincides with the expression given in ref. a) (accurately to the coefficient 2a-rromitted in ref. a). At smaller impact parameters the difference is due to the fact that in ref. a) account was taken of the terms traceable to Ti~n~t. These terms are due to the meson-nucleon couplingenergy, and are not included in eq. (11). This approach seems to us more corret.
v. $. BARASHENKOV et
312
q (ro;
al.
= (1__+ 3)Qo(ro;
(14)
Here we used quantities b and s defined by b 2 = a 2 + (e/?fl) 2 and s 2 = 1 + (e/?fl) 2, while K o and K~ are modified Hankel functions. Fig. 1 represents the values o f the function (?fl3/f2)Qo(ro; e) calculated for different values o f the minimum collision parameter r o (at a = 5.6). Fig. 2 indicates the corre3 2Q
/ !
o.3
/ I
I
!
\
\ \ \ \
o.~ O o.
\ \
\
\
\ \\\
o.I
i
\
,
I
Fig. 1. Spectrum o f p e r i p h e r a l ~°-mesons in a nucleon. Solid a nd da s he d curves designate the values calcu lated by eq. (15) and by the Heitler-Peng f o r m u l a 13) respectively. N u m b e r s by the curves i n d i c a t e the values of r0 in nucleon C o m p t o n wave-lengths ro = n / M ~ 0.21 × n fm (n = 3; 4 ; 5; 6). Vertical dash ed lines m a r k the integration limits #/tzyfl) and e*/(l~yfl) for N N - c o l l i s i o n s at T = 10 GeV (? = 11.64, lab), # being the mass o f the zt-meson.
sponding cross sections for peripheral N N - c o l l i s i o n s
ap = 2
a~N(e)Q(ro; a)ds,
313
THEORY OF PERIPHERAL INTERACTIONS
w h e r e e* =
E-x/½M(E+M) is
in the laboratory
system
the maximum
e n e r g y loss o f t h e i n c i d e n t n u c l e o n
u n d e r t h e c o n d i t i o n t h a t its r e c o i l e n e r g y is n e g l e c t e d ,
E = yM being the total energy of this nucleon in the laboratory system, and Mits mass, while Q = Qo+Q++Q-. T h e c u r v e s c a l c u l a t e d b y t h e H e i t l e r - P e n g f o r m u l a 13) a r e a l s o g i v e n f o r c o m p a r i s o n . I t is c l e a r t h a t a t ro > 1 f m t h e c a l c u l a t i o n s w i t h t h e H e i t l e r - P e n g s p e c t r u m y i e l d p r a c t i c a l l y t h e s a m e r e s u l t s as t h e c a l c u l a t i o n s b y t h e m o r e a c c u r a t e f o r m u l a (14) *.
0.4
1
o.,~
o.~
o.1
o.o
0.,
'
'
0.e
r:
Fig. 2. The portion of the peripheral collisions in the total cross section for inelastic NN-collisions at T = I 0 GeV (lab.). A separate graph represents the ratio of the cross sections ap calculated by eq. ( 15 ): R = (ap)H.-P./(%)eq, x5>. The values of r 0 are given in fm. At smaller collision parameters the use of the Heitler-Peng formula leads to considerable errors. * The cross sections ap = crp (r0) given in ref. 7) should be trebled, since the spectrum Q . was not taken into account in this investigation. To obtain the same values ¢rp as in ref. 7) it is necessary to choose smaller values of the collision parameter r0. Since the quantity r0 was regarded as an adjustable parameter, the conclusions of ref. 7) do not change. The relevant changes of the quantity r0 should be also made in refs. 5, 6). The change of the experimental value aln(NN) from 22 mb to 32 mb ref. xs)) should also be taken into account. As a result the ratio crp(NN)/aln(NN) roughly doubles for each given value of r0.
314
V. S. BARASHENKOVet aL
It should be noted that for the values r o < 2 h / M c a considerable c o n t r i b u t i o n is made by mesons with energies of the order o f the n u c l e o n energy itself: e ~ e* ~ M. A very large m o m e n t u m can be transferred i n a collision i n this case a n d the Weizs/icker-Williams m e t h o d proves to be inacceptable. F r o m this viewpoint it is more c o n v e n i e n t to divide " c e n t r a l " a n d "peripheral" collisions n o t by the impact parameter ro b u t by the m e s o n energy e determining the m o m e n t u m transfer. The estimates o f the mesic corrections i n eq. (14) prove to be a considerably more complex p r o b l e m for calculation. I n the cases o f the a n o m a l o u s m a g n t i c m o m e n t 17), m e a n square radii is), electric a n d magnetic n u c l e o n polarizabilities 19), these corrections yield a c o n t r i b u t i o n which does n o t exceed 20 to 30 ~ of tha.t of the o n e - n u c l e o n a p p r o x i m a t i o n . It m a y be hoped that in the case u n d e r consideration the mesic corrections will c o n t r i b u t e appreciably only at small collision parameters. The authors are indebted to D. I. Blokhintsev for valuable discussions.
References 1) S. Z. Belenky et al., Uspekhi Fiz. Nauk. 62 (1957) 1 2) R. Hagedorn, Fortschritte der Phys. 9 (1961) 1 3) V. S. Barashenkov, Fortschritte der Phys. 9 (1961) 29 4) V. S. Barashenkov et aL, Nuclear Physics 9 (1958) 74 5) V. S. Barashenkov, V. M. Maltsev and E. K. Mihul, Nuclear Physics 13 (1958) 583; 17 (1960) 377 6) V. S. Barashenkov, Nuclear Physics 15 (1960) 486 7) D. I. Blokhintsev, Proc. CERN Symposium 2 (1958) 155 8) D. I. Blokhintsev, V. S. Barashenkov and B. M. Barbashov, Uspekhi Fiz. Nauk. 68 (1959) 417 9) I. M. Dremin and D. S. Chernavsky, JETP 38 (1960) 29 10) I. M. Dremin, JETP 39 (1960) 122 11) F. Salzman and G. Salzman, Phys. Rev. 120 (1960) 599; Phys. Rev. Lett. 5 (1960) 377 12) D. I. Blokhintsev and Wang Yung, Nuclear Physics 22 (1961) 410 13) W. Heitler and H. W. Peng, Proc. RIA 49 (1943) 101 14) G. E. Chew and F. E. Low, Phys. Rcv. 101 (1956) 1570 15) V. S. Barashenkov, Uspekhi Fiz. Nauk 72 (1960) 53 16) V. S. Barashenkov, Uspekhi Fiz. Nauk. (in the press); preprint JINR D-630, 1960 17) H. Miyazawa, Phys. Rev. 101 (1956) 1564 18) S. B. Treiman and R. G. Sacks, Phys. Rev. 103 (1965) 435 19) V. S. Barashenkov, H. J. Kaiser and A. A. Ogreba, Nuovo Cim. (in the press)
[
Nuclear Physisc 31 (1962) 308--314; ~) North-Holland Publishing Co., Amsterdam
I
Not to be reproduced b y photoprint or microfilm without written permission from the publisher
T H E O R Y OF P E R I P H E R A L I N T E R A C T I O N S OF FAST P A R T I C L E S V. S. BARASHENKOV, H. J. KAISER AND VAN-PEI Department of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, USSR
Received 9 May 1961 Abstract: The expression for the energy spectrum of peripheral mesons in a fast moving nucleon is obtained. This spectrum corresponds to the expansion of the mesic field of a moving nucleon into plane waves and can be used for the calculation of inelastic fast particle nucleon collisions. Within distant peripheral collisions the theory under consideration coincides with the peripheral collision theory of Heitler and Peng.
1. Introduction
At present there is no consistent theory of strong interactions for high energies and the calculation of them is based on approximate semi-phenomenological theories. The Fermi statistical theory has long been used for calculating inelastic interactions (the details concerning these calculations can be found in refs. 1, 3)). Gradually, however, experimental data which cannot be interpreted in the framework of this theory 3, 6) have accumulated. The model of central and peripheral collisions was suggested in refs. 7-8) to account for these facts. This model takes account of the internal structure of the colliding particles and incorporates the Fermi statistical theory as a particular case. Analysis of the experimental data on inelastic interactions at energies T > 1 GeV points to a very large contribution made by the peripheral interactions 3, 4). In refs. 9 - 1 2 ) , the peripheral interactions were studied by the pole method, and in refs. 3, 5, 6) with the aid of the expansion of the meson fields of the colliding particles into plane waves which were then considered as real "peripheral mesons". This approach was for the first time worked out by Weizs~icker and Williams in electrodynamics and later was applied to the nucleon-nucleon interactions by Heitler and Peng 13). As has been shown by comparative calculations the pole method and plane wave resolution method, when applied to inelastic interactions at large energies lead to similar results in the more important cases. The merit of the latter method is simplicity and transparence. The purpose of the present paper is a more elaborate treatment and improvement of this method. t We indicate merely the pioneer works; at present there are very many papers concerned with these problems. 398
THEORY OF PERIPHERAL INTERACTIONS
309
2. Expansion of Nucleon Mesic Field into Plane Waves The energy spectrum of peripheral mesons in the nucleon q(r; ~) was used in refs. s-7). The method had been obtained by Heitler and Peng aa) under the assumption that the mesic field in the nucleon is described by the classical expression ~o(r) oc ~,~ • ," - -
d
e -r
--
rdr r (here and further on h = c = # = 1, where # is the mass of the n-meson). This expression, however, is a very poor approximation in the region r < 1 (ref. 8)). A more accurate expression for q(r; e) can be obtained and the limits on the applicability of the Heilter-Peng theory can be established. First of all let us bear in mind that the spectrum of peripheral mesons at a distance r from the centre of a fast moving nucleon can, in the general case, be determined from comparing two expressions for the density of the total energy flux of the nucleon mesic field in a layer of radius r:
~
oo
I(r) =
8q(r; ~)d~,
(1)
S(r;
(2)
1
~(r)
-- 2nr
Odt,
where S = ~2 [(1 + f l 2 ) T l ° - f l ( T °° + T l t ) ] is the x-component of the Poynting vector at x = 0 at the instant t and T uv -= (NITer(r; x ) l N ) is the average value, over the state of the physical nucleon, of a component of the energy-momentum tensor of the mesic field in the nucleon-bound system of coordinates; furthermore, x' = y ( x - f l t ) (below we assume that x = 0 in the nucleon-bound system of coordinates), y = (1 -f12)-~ and fl is the velocity parallel to the axis x' of the nucleon in the laboratory system. In the tensor T ~v only the terms due to the energy of the mesons themselves, and not the meson-nucleon coupling energies, should be taken into account. Correspondingly, one h a s T ~ = T ~. It can readily be shown that
is
= 2(2n)a
TOO+T11 =
1
da(pq)
f da(pq)
2(2n) 3J
l
ple'(V-")'=<(ap+ a_* p)(a,• -- a_q)),
(3)
ei(V-,).x
x/~v ~q
x {p, q~<(a*_~+ a~)(a* + a_~)>- ~ % < ( a ~ - a~)(~'- ~_.)>},
(4)
where oq 2 = 1 + q2 and the b r a c k e t s < . . . > designate the average values of the normal products of mesic field operators.
310
v . s . BARASHENKOVe t al.
If one considers sufficiently distant peripheral collisions, the nucleon recoil can be neglected and the fixed nucleon theory used for the calculation of the averages in (3) and (4) 14). The expression for the vector-component S in this case is
S(x) =
72 [ d3(pq) e'°'-')'*(og,+ogq)-' E (e~+c%)-l(en+c°q) -1 (27r)3d~ x ((1 + fl2)copogqpl[Tq(n)Tp(n ) - T~(n)Tq(n)]
(5)
+ fl[E~(pl q l - cop ¢o,) + P l q l(a)p + ~q)] [ Tp(n)T~(n) + Tq(n)Tp(n)] }, where T,(n) =
~i J_- ~V.. ( N + nnlp " trTIN).
(6)
The quantity f, w i t h f 2 = 0.08, is the renormalized meson-nucleon coupling constant, z is the isobaric spin operator, and Vp is the form factor of the mesic field source. Summation is performed over the total system of orthogonal states with n = 0, 1, 2, •.., mesons. From comparing eqs. (2), (5) and (6) it can be seen that the fraction S(x) proportional to (1 +f12) gives no contribution to I(r) and can henceforth be neglected. In the one-nucleon approximation eq. (5) becomes 27 fl So(x),= o = ~ f
2
3 Vp Vq d (Pq)~PiqlP"
qe~(p_q),
x
(7)
(Dp ~Oq
for neutral mesons, and S±(x),=0 = (1 +%)So(x),= o
(8)
for charged mesons. The mesic corrections (n > 0) can be expressed through the integral of the total ± cross s'ections for meson-nucleon interactions a± = atot(Tr°p)17). To do this let us note that o
Tp(n)T~(n)+ Tq(n)Tp(n) = 2 pq VpV~ cok k 2 Vff ~/o),o)q T•(n)Tk(n), and
(½(l+%)a++½(1-%)a_ 2rCk.>o ~ ~(E"--c°k)Ttk(n)Tk(n)= / ½ ( 1 - % ) a + + ½ ( l + % ) a ~a o Hence it follows
27 fl So(x). >o -
for a n+ spectrum, for a n - spectrum, for a n ° spectrum.
(9)
3 Vp Vqp" q e~(p-~), x d (pq) cop¢o,(o9t,+ to,) f"dk a°(k)°)k°)'°gq-plql(°)k+°9"+°)') × Jo
v2
o , k ) ( o , , + o,k)
(10) '
where a is the characteristic dimension of the form factor; the expression for S±(x),> o
THEORY OF PERIPHERAL INTERACTIONS
311
is obtained from eq. (10) by substituting the cross section tro(k) in accordance with (9). F r o m c o m p a r i n g eqs. (1), (2) a n d eqs. ( 7 ) - ( 1 0 ) we shall obtain, after integrating in polar coordinates, the peripheral m e s o n energy spectrum in a layer o f radius r:
2 r l f ~ f°°d(pq ) V,,Vq, [(y~)2 )] qo(r; e) = ~-~ -e.)o Jo ~ pq J°(Pr)J°(qr)+pqJl(Pr)Jl(qr COp,COq,
X
(I
-r~e )- 2 ( 1-"-~ - ' C 3 ) J t - 2 n
co~ co,, co~, X
(.D~,o)q,
fadk
:
l~za a_(k)
)
(11)
(cok + cop' + co¢) 1
(co,,+ co~)(coq,+ co~)
Here p '2 = p2+(e/~fl)2 and q,2 = q2+(e/~fl)2 ' with e > 1, while Jo and J1 are the well-known Bessel functions. The total energy spectrum determining peripheral collisions with impact parameters r > ro is obtained by the integration Q o ( r o ; e) = +
I;
o
qo(r;
e)dr.
(12)
+
At high energies o f colliding particles the average energy of peripheral mesons is sufficiently large and the cross sections for their interactions do not depend on the isobaric spins. In this case the cross section for peripheral interactions is determined by the total spectrum (Qo+Q+ + Q - ) t . At lower energies account should be taken o f the difference in the cross sections for the interaction o f n +, n - and n ° mesons. This leads in particular to different energy dependences for the angular a s y m m e t r y produced in pp- and pn-collisions (as well as in np- and nn-collisions). This circumstance can be used for an experimental check-up o f the peripheral collision theory. 3. Numerical Calculations and Discussion
I f one neglects the mesic corrections and puts Vk = a2/(k2+a2), integration in q(r; e) and Q(r o ; e) can be p e r f o r m e d analytically, and the expression for the spectrum becomes m u c h simpler: f 2 e / a2 \ 2 t
+ a2(g~(ro b ) - K2(ro b))] + 2ro[sro(ro s)gl(ro s) + bro(ro b)rl(ro b) 2b
+ ---
a2--1
Ko(ros)K~(ro b ) -
__ 2a2s a2_1
(13)
}
Ko(ro b)K~(ro b)] ,
t Outside the mesic field source (r > 1/a) the one-nucleon part of the total spectrum coincides with the expression given in ref. a) (accurately to the coefficient 2a-rromitted in ref. a). At smaller impact parameters the difference is due to the fact that in ref. a) account was taken of the terms traceable to Ti~n~t. These terms are due to the meson-nucleon couplingenergy, and are not included in eq. (11). This approach seems to us more corret.
v. $. BARASHENKOV et
312
q (ro;
al.
= (1__+ 3)Qo(ro;
(14)
Here we used quantities b and s defined by b 2 = a 2 + (e/?fl) 2 and s 2 = 1 + (e/?fl) 2, while K o and K~ are modified Hankel functions. Fig. 1 represents the values o f the function (?fl3/f2)Qo(ro; e) calculated for different values o f the minimum collision parameter r o (at a = 5.6). Fig. 2 indicates the corre3 2Q
/ !
o.3
/ I
I
!
\
\ \ \ \
o.~ O o.
\ \
\
\
\ \\\
o.I
i
\
,
I
Fig. 1. Spectrum o f p e r i p h e r a l ~°-mesons in a nucleon. Solid a nd da s he d curves designate the values calcu lated by eq. (15) and by the Heitler-Peng f o r m u l a 13) respectively. N u m b e r s by the curves i n d i c a t e the values of r0 in nucleon C o m p t o n wave-lengths ro = n / M ~ 0.21 × n fm (n = 3; 4 ; 5; 6). Vertical dash ed lines m a r k the integration limits #/tzyfl) and e*/(l~yfl) for N N - c o l l i s i o n s at T = 10 GeV (? = 11.64, lab), # being the mass o f the zt-meson.
sponding cross sections for peripheral N N - c o l l i s i o n s
ap = 2
a~N(e)Q(ro; a)ds,
313
THEORY OF PERIPHERAL INTERACTIONS
w h e r e e* =
E-x/½M(E+M) is
in the laboratory
system
the maximum
e n e r g y loss o f t h e i n c i d e n t n u c l e o n
u n d e r t h e c o n d i t i o n t h a t its r e c o i l e n e r g y is n e g l e c t e d ,
E = yM being the total energy of this nucleon in the laboratory system, and Mits mass, while Q = Qo+Q++Q-. T h e c u r v e s c a l c u l a t e d b y t h e H e i t l e r - P e n g f o r m u l a 13) a r e a l s o g i v e n f o r c o m p a r i s o n . I t is c l e a r t h a t a t ro > 1 f m t h e c a l c u l a t i o n s w i t h t h e H e i t l e r - P e n g s p e c t r u m y i e l d p r a c t i c a l l y t h e s a m e r e s u l t s as t h e c a l c u l a t i o n s b y t h e m o r e a c c u r a t e f o r m u l a (14) *.
0.4
1
o.,~
o.~
o.1
o.o
0.,
'
'
0.e
r:
Fig. 2. The portion of the peripheral collisions in the total cross section for inelastic NN-collisions at T = I 0 GeV (lab.). A separate graph represents the ratio of the cross sections ap calculated by eq. ( 15 ): R = (ap)H.-P./(%)eq, x5>. The values of r 0 are given in fm. At smaller collision parameters the use of the Heitler-Peng formula leads to considerable errors. * The cross sections ap = crp (r0) given in ref. 7) should be trebled, since the spectrum Q . was not taken into account in this investigation. To obtain the same values ¢rp as in ref. 7) it is necessary to choose smaller values of the collision parameter r0. Since the quantity r0 was regarded as an adjustable parameter, the conclusions of ref. 7) do not change. The relevant changes of the quantity r0 should be also made in refs. 5, 6). The change of the experimental value aln(NN) from 22 mb to 32 mb ref. xs)) should also be taken into account. As a result the ratio crp(NN)/aln(NN) roughly doubles for each given value of r0.
314
V. S. BARASHENKOVet aL
It should be noted that for the values r o < 2 h / M c a considerable c o n t r i b u t i o n is made by mesons with energies of the order o f the n u c l e o n energy itself: e ~ e* ~ M. A very large m o m e n t u m can be transferred i n a collision i n this case a n d the Weizs/icker-Williams m e t h o d proves to be inacceptable. F r o m this viewpoint it is more c o n v e n i e n t to divide " c e n t r a l " a n d "peripheral" collisions n o t by the impact parameter ro b u t by the m e s o n energy e determining the m o m e n t u m transfer. The estimates o f the mesic corrections i n eq. (14) prove to be a considerably more complex p r o b l e m for calculation. I n the cases o f the a n o m a l o u s m a g n t i c m o m e n t 17), m e a n square radii is), electric a n d magnetic n u c l e o n polarizabilities 19), these corrections yield a c o n t r i b u t i o n which does n o t exceed 20 to 30 ~ of tha.t of the o n e - n u c l e o n a p p r o x i m a t i o n . It m a y be hoped that in the case u n d e r consideration the mesic corrections will c o n t r i b u t e appreciably only at small collision parameters. The authors are indebted to D. I. Blokhintsev for valuable discussions.
References 1) S. Z. Belenky et al., Uspekhi Fiz. Nauk. 62 (1957) 1 2) R. Hagedorn, Fortschritte der Phys. 9 (1961) 1 3) V. S. Barashenkov, Fortschritte der Phys. 9 (1961) 29 4) V. S. Barashenkov et aL, Nuclear Physics 9 (1958) 74 5) V. S. Barashenkov, V. M. Maltsev and E. K. Mihul, Nuclear Physics 13 (1958) 583; 17 (1960) 377 6) V. S. Barashenkov, Nuclear Physics 15 (1960) 486 7) D. I. Blokhintsev, Proc. CERN Symposium 2 (1958) 155 8) D. I. Blokhintsev, V. S. Barashenkov and B. M. Barbashov, Uspekhi Fiz. Nauk. 68 (1959) 417 9) I. M. Dremin and D. S. Chernavsky, JETP 38 (1960) 29 10) I. M. Dremin, JETP 39 (1960) 122 11) F. Salzman and G. Salzman, Phys. Rev. 120 (1960) 599; Phys. Rev. Lett. 5 (1960) 377 12) D. I. Blokhintsev and Wang Yung, Nuclear Physics 22 (1961) 410 13) W. Heitler and H. W. Peng, Proc. RIA 49 (1943) 101 14) G. E. Chew and F. E. Low, Phys. Rcv. 101 (1956) 1570 15) V. S. Barashenkov, Uspekhi Fiz. Nauk 72 (1960) 53 16) V. S. Barashenkov, Uspekhi Fiz. Nauk. (in the press); preprint JINR D-630, 1960 17) H. Miyazawa, Phys. Rev. 101 (1956) 1564 18) S. B. Treiman and R. G. Sacks, Phys. Rev. 103 (1965) 435 19) V. S. Barashenkov, H. J. Kaiser and A. A. Ogreba, Nuovo Cim. (in the press)