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On the anomalous magnetic moment of the μ-meson
Nuclear Physics
1 (t 9r~)
ON THE A N O M A L O U S MAGNETIC M O M E N T OF THE p - M E S O N G. N. F O W L E R
Dept. o/ Theoretical Physics, University o/ Manckester Recived 12 July 1955 A b s t r a c t : A discussion is given of various phenomena involving p-mesons, especially bremsstrahlung and pair production, in which an anomalous magnetic moment of the meson should be important, taking into account finite meson size effects in a relativistic way. I t is found t h a t the empirical evidence rules out the existence of an anomalous magnetic moment of any significance for the /~-meson.
1. Introduction Recent experiments on the scattering of p-mesons by nuclei 1) have disclosed a larger cross section for scattering through large angles than that predicted on the basis of the normal electromagnetic interaction taking into account the finite size of the nucleus. The momentum transfers involved are of the order p0 5 (GeV/c) • deg, where p is the momentum of the incident meson and 0 the angle of deflection. Various explanations of this result have been proposed; in particular, it has been suggested that the #-meson m a y have a sufficiently large anomalous magnetic moment to give the required cross section at large scattering angles 2' 8). We can estimate such an effect by noting the r -s dependence of the magnetic interaction potential. If a~ is the cross section for scattering through an angle 0 due to the anomalous magnetic moment of Bohr magnetons alone, and a~ the usual scattering cross section, then we find for Une ratio of the magnetic to the ordinary scattering .~
~(p0) 2
-- ~ - (p0 in (GeV/c). degrees). ~,~ 30 When p 0 > 6 the experimental results are approximately an order of magnitude larger than the predictions of the usual theory based on the normal electromagnetic interaction. Thus we require AS ~ 10. This is confirmed b y t h e detailed calculations of Gatto a).
126
~ . N. F O W L E R
However, as is mentioned by Gatto, it is necessary to note, that, associated with an anomalous magnetic moment, we must expect a meson cloud which will give rise to a cut off in the momentum transfer when the finite size of the /z-meson is taken into account. Exact evaluation of this effect depends on the type of meson form factor assumed but it should be expected, that the experimentally observed momentum transfers would be excluded by any reasonable choice. In view of this situation it is of interest to investigate the influence of an anomalous magnetic moment on other electromagnetic phenomena, taking into account finite size effects, with a view tO ascertaining the restrictions on the magnitude of )I imposed b y experimental results. A previous discussion of this question has been given b y Peaslee 4) but in that work the finite size effects were not taken into account in a consistent way. In the present note, we use the relativistic procedure given b y Corinaldesi 5). The electromagnetic phenomena of interest besides scattering are the following: (i) Ionisation loss (ii) Bremsstrahhmg (iii) Pair production (iv) Underground star production (v) Fine structure of the /~-mesic atom. So far as the last named is concerned the present experimental results are not sufficiently accurate to say more e) than that ~t < 8 and we will therefore not discuss it further. 2. Relativistic cut-off P r o c e d u r e
The relativistic procedure just referred to consists in writing for the anomalous magnetic part of the /,-meson current density
J'~(x) = where
,f
(2r#~c)4
e"'l®-x')l~6Cf'n~2) amI"(X)ax, d4Xd4p' --
with
=
(r
(1) (2)
r, -
r, rAI2i,
is' the density' tensor of anomalous magnetisation and J=(p*) the Fourier transform o f the form factor describing the assumed finite
os
xnE AnOMaLOUSMA~NErXC ~OMENT OF
rn~ /*-MESON
I~7
size of the meson magnetons; a point particle of anomalous moment 2 is described b y J'(p*)
- - 1.
It can be s h o w n t h a t the matrix dement occurring in the description of a collision process are multiplied b y a factor J~* ((AD)*) where A D is the four momentum transfer. We m a y suppose that J'(p
~) = e
-~- ,
(s).
where ;t~ = ?~/m~,c.The most convenient system in which to express A D is the centre of mass system; we have in this system
AD = ( p - - p ' , O ) . From (8) the largest momentum transfer for which the cross section is appreciable is then APmax = ?~1~.~,or [0 ~ 6 (GeV/c) • deg.
(4)
3. Discussion 8.1.
I O N I S A T I O N LOSS
When the energy of the incident particle is large compare d with its rest energy, the cross section for energy transfer E to an atomic electron neglecting atomic binding is given, in the laboratory system, b y do' =
°~e42-------~• d/(1 -- ]),
(m,,c'p
1
(5)
with / = E/Eo, where E o is the energy of the incident particle r, 8). The four momentum cut off, which reduces to (4) in the centre of mass system, gives an upper limit to i of /max ~-m~cZ/2Eom ,. We then find for the ionisation energy loss per unit length of path when N is the number of electrons per cm s fma]g
a
Eof/go. =dN 2 m . a,
(6)
o
where ru = eg/m~,c2. In deriving this result it has been assumed that
E, > (,n,/~.) ,n,c*.
12S
G . N . FOWLER
This result is the same as that given b y Peaslee, who in fact also uses in this case a cut-off equivalent to (4). However, in the other processes he discusses (bremsstrahlung and pair production) he assumes a formula of type (4) in the rest system of the meson. H a d he used this prescription in the present case he would not have found an upper limit to the energy loss since the two systems considered are the same only in the limit m,/m~,-+ O. The result (6) gives a maximum energy loss per unit path length due to an anomalous magnetic moment of the order 10-* MeV. cmZ/g when A2 ~ 10. The normal ionisation loss is of the order 2 MeV • cm*]g, so that, on this evidence, this value of ~2 cannot be excluded. 3.2.
BREMSSTRAHLUNG
The production of bremsstrahlung using the relativistic procedure adopted to take into account the finite size of the meson way be treated in the following way. With
1
V(X) --
(2=~)'1,Id
d'P'w(P') e''x/*°
~(X)
(2=~y, ] f
d*P"V'(P")e''''x/*°,
and
The expression for the anomalous contribution to the current density becomes, after partial integration with respect to X,
__1
,l
f
d~pd 4p Id~p ll--y,(p I! )a~,W(p t ) •J'~(p') • e'<'-''lx/~° e"('-x):*°(- i p , ) # X .
(v)
Integrating over X and /~ gives
1 /~'(x)=(~c)3 2a f #p'#p",e(F'),~,,,v,(F)e'("-")':"°. • (-
i(p' -
p")),:-((p'
-
~"),).
(8)
The matrix element for bremsstrahlung is of the form oO
K , --
~i*
oO
dr,
dr1
where '
=
-
f
2
(9)
ON THE ANOMALOUS MAGNETIC MOMENT OF THE ~-MESON
129
and e(t 2 -
tx) =
+
=
1, t s > tl
- - 1, t , < t 1.
Using (8) and the Fourier transform of A,(x), this gives -- i
l
K, = 27~' " (2:w~) 9
f
4'
d x,d
4' , ,, , ,,~la41p2d,4~j~2dgpld'4'p"ltd4'/'~la4'~l
°
1 + e(Xo, - - Xox ) e,.,..,+.~.,.,)/i~ 2
• ~(p'z')a~,~Cfi)(- i(p', - P'z')), A,(k,)~?(P'I')%,~I'(P'I) • (--
i(~1
--
~)tl'))xae(kl)/ra((~);
-- ~)'2t)2)/m((~i
--
p',')2)
(]0)
where s, =
+ k, -
f,'.
Pairing off the wave functions representing emission and absorption of virtual particles and taking the expectation value for the meson vacuum introduces, instead of y~(p',) ~(P'I'), the expression (Heitler*), p. 285) t --
.
e
4,
t
tt
t2
S<% (p,.)%, (ibl)>0 = T., } (] +e(p~o)e(Xo,--Xoa)) t~ (Pg. -- Pl )(~(~)o. "JI- /.~l), where S refers to a summation over spin states of the virtual particles and T~, = r ~ " P', + i/~. The factor e (x0s -- xoa) arises when we take account of the permutation of the emission and absorption operators of the v i r t u ~ particles. The factor (1 + e~0)eCxos
- - XO1))(~(~2 2
-3C /t$')
has the same effect under the integral sign (in the present case) as 8+(p~2 + ps) (Heitlerg), p. 286) and when we sum over all permutations of the points xx, xz, using ½ Z P(1
+
e(Xo,
--
xOl))
=
],
P
the integrand of (10) depends on the: x~ only through the factor exp(is~ • x~/?i,c). The integrations over the x~ may now be carried out giving the usual Dirac functions (2a~c)4'~4'(s~). If now we take plane waves for the incident and outgoing mesons: ~0#(~'1) =
~4'(P -- ~ ; )
a,(P),
~,(p~')
~'(P' _ p;') %(P'),
130
G.N. FOWLER I
It
II
we m a y c a r r y out the integrations over Pl, Pl and p=. The result m a y be written, before integrating over kl, i 1 dK~ = 21~ --~ " 2 ~
fdep,~ 4 ' zd k~(5 (p~ + k s - - p , ) ~ 4 ( p + kl _ P2)
_
• a(P') av.TA~,(k,)aQ,a(P).
(p~ - - P ' ) , ( P
- - P'2),
(11)
ne(kl) J " ( k [ ) J " (k~). The t functions in
(11)
imply
ks=P'--P--kl and p~ = P + kl.
The remaining integrations m a y be carried out to give i 1 dK2 = - - 2/~ --~ " 2z#~ " a ( P ' ) a ~ ' T A ~ ( P ' - - P - - kl)a°~a(P) " (kl + P - - P ' ) , ( k z ) ~ "
AQ(kl) ~ " ( ( P ' - - P - - k z ) ~ )
(12)
~"(k~).
Possible restrictions on four momentum transfers are contained in the terms J " ( ( P ' - - P - - k I ) ~ ) , fl;"(k~) and also in A ~ ( P ' - - P - - k x ) . This last term sets an upper limit to momentum transfer since it includes a cut off introduced b y the finite nuclear size. This leads to
[(P'
-
P -- kz) I <
~/R,
(13)
w h e r e R is t h e nuclear radius. T h e t e r m J - ( ( e ' - P - - kl)=) introd u c e s an effect due to the finite size of the meson when the meson
and nucleus exchange a quantum. This obviously yields a less restrictive condition than (13) due to the fact that R > ;t,. For the remaining term we have
There is thus no cut off in radiation loss other than that introduced b y the finite nuclear size. We can therefore simply use the formula derived b y Powell s0) for this case. The ratio of total radiation loss due to anomalous effects to the normal radiation loss is, according to Powell, ....
9~
A~Eo
~n
m/, c~"
-- ~ 0.1-
(14)
O n t h e other hand Peaslee's cut off prescription led him to the essentially different expression
ON THE ANOMALOUS MAGNETIC MOMENT OF THE ~-MESON
l~t
The observations on cosmic ray bursts (soft showers initiated b y high energy bremsstrahlung produced b y mesons with energies of the order 108 Gev) agree approximately with the results predicted b y the usual theory 11). Thus (14) implies ks<< 1 in 'contrast to Peaslee's result and this conclusion is independent of the form factor specifying the finite size of the meson, 3.4.
PAIR
PRODUCTION
Without going into the details of the calculation it is easy to see that, as the photon responsible for the pair production is virtual, i~ follows that k~ =# 0 a n d < 1.
However, when the energy of the incident meson is sufficiently high the four momentum of the virtual photon will satisfy ~ = 0 to order r,%c/p, where p is the momentum of the incident meson. There will then be no cut off in electron pair energy. B y comparison with the bremsstrahlung case we may infer that the ratio of the cross sections for pair production will be again of order ~t4Eo/mt,c2. Now, the experimental work of Goldsack and Kannangara 13) has shown that the observed pair production is in reasonable agreement with the usual theory, so that again 42 << 1 independently of the form factor. 3.5.
STAR
PRODUCTION
In this case we have to do with collisions between the incident meson and a nuclear proton, and so we have as an upper limit to the possible energy transfers =
~ 10 MeV.
Since nuclear stars have energies larger than 300 MeV we m a y say that such processes are unlikely to give any information on the value of ~t.
4. Conclusions The conclusion from this discussion is that 4 2 << 1 independently of the finite meson size effects, so that there should be no significant contribution to the scattering of p-mesons b y nuclei from an anomalous magnetic moment.
132
o.N.
FOWLER
References 1) 2) 3) 4 5 6 7 8 9 10 11 12
G. D. Rochester and A. W. Wolfendale, Phil. Mag. 45 (1954) 980 D. A. Tidman, Proc. Phys. Soc. A b7 (1954) 559 R. Gatto, Nuovo Cimento 12 (1955) 613 D. C. Peaslee, Nuovo Cimento 9 (1952) 56 E. Corinaldesi, Nuovo Cimento 8 (1951) 62 L. N. Cooper and E. M. Henley, Phys. Rev. 92 (1953) 801 H. C. Corben and J. Schwinger, Phys. Rev. 58 (1940) 953 W. Pauli, Rev. rood. Phys. 13 (1941) 203 W. Heitler, Q u a n t u m Theory of Radiation, 3rd Edition, Oxford (1954) J. L. Powell, Phys. t{~v. 75 (1949) 32 C. N. Chou and M. Schein, Phys. Rev. 97 (1955) 206 S. J. Goldsack and M. L. T. Kannangaxa, Phil. !V~ag. 44 (1953) 811
1 (t 9r~)
ON THE A N O M A L O U S MAGNETIC M O M E N T OF THE p - M E S O N G. N. F O W L E R
Dept. o/ Theoretical Physics, University o/ Manckester Recived 12 July 1955 A b s t r a c t : A discussion is given of various phenomena involving p-mesons, especially bremsstrahlung and pair production, in which an anomalous magnetic moment of the meson should be important, taking into account finite meson size effects in a relativistic way. I t is found t h a t the empirical evidence rules out the existence of an anomalous magnetic moment of any significance for the /~-meson.
1. Introduction Recent experiments on the scattering of p-mesons by nuclei 1) have disclosed a larger cross section for scattering through large angles than that predicted on the basis of the normal electromagnetic interaction taking into account the finite size of the nucleus. The momentum transfers involved are of the order p0 5 (GeV/c) • deg, where p is the momentum of the incident meson and 0 the angle of deflection. Various explanations of this result have been proposed; in particular, it has been suggested that the #-meson m a y have a sufficiently large anomalous magnetic moment to give the required cross section at large scattering angles 2' 8). We can estimate such an effect by noting the r -s dependence of the magnetic interaction potential. If a~ is the cross section for scattering through an angle 0 due to the anomalous magnetic moment of Bohr magnetons alone, and a~ the usual scattering cross section, then we find for Une ratio of the magnetic to the ordinary scattering .~
~(p0) 2
-- ~ - (p0 in (GeV/c). degrees). ~,~ 30 When p 0 > 6 the experimental results are approximately an order of magnitude larger than the predictions of the usual theory based on the normal electromagnetic interaction. Thus we require AS ~ 10. This is confirmed b y t h e detailed calculations of Gatto a).
126
~ . N. F O W L E R
However, as is mentioned by Gatto, it is necessary to note, that, associated with an anomalous magnetic moment, we must expect a meson cloud which will give rise to a cut off in the momentum transfer when the finite size of the /z-meson is taken into account. Exact evaluation of this effect depends on the type of meson form factor assumed but it should be expected, that the experimentally observed momentum transfers would be excluded by any reasonable choice. In view of this situation it is of interest to investigate the influence of an anomalous magnetic moment on other electromagnetic phenomena, taking into account finite size effects, with a view tO ascertaining the restrictions on the magnitude of )I imposed b y experimental results. A previous discussion of this question has been given b y Peaslee 4) but in that work the finite size effects were not taken into account in a consistent way. In the present note, we use the relativistic procedure given b y Corinaldesi 5). The electromagnetic phenomena of interest besides scattering are the following: (i) Ionisation loss (ii) Bremsstrahhmg (iii) Pair production (iv) Underground star production (v) Fine structure of the /~-mesic atom. So far as the last named is concerned the present experimental results are not sufficiently accurate to say more e) than that ~t < 8 and we will therefore not discuss it further. 2. Relativistic cut-off P r o c e d u r e
The relativistic procedure just referred to consists in writing for the anomalous magnetic part of the /,-meson current density
J'~(x) = where
,f
(2r#~c)4
e"'l®-x')l~6Cf'n~2) amI"(X)ax, d4Xd4p' --
with
=
(r
(1) (2)
r, -
r, rAI2i,
is' the density' tensor of anomalous magnetisation and J=(p*) the Fourier transform o f the form factor describing the assumed finite
os
xnE AnOMaLOUSMA~NErXC ~OMENT OF
rn~ /*-MESON
I~7
size of the meson magnetons; a point particle of anomalous moment 2 is described b y J'(p*)
- - 1.
It can be s h o w n t h a t the matrix dement occurring in the description of a collision process are multiplied b y a factor J~* ((AD)*) where A D is the four momentum transfer. We m a y suppose that J'(p
~) = e
-~- ,
(s).
where ;t~ = ?~/m~,c.The most convenient system in which to express A D is the centre of mass system; we have in this system
AD = ( p - - p ' , O ) . From (8) the largest momentum transfer for which the cross section is appreciable is then APmax = ?~1~.~,or [0 ~ 6 (GeV/c) • deg.
(4)
3. Discussion 8.1.
I O N I S A T I O N LOSS
When the energy of the incident particle is large compare d with its rest energy, the cross section for energy transfer E to an atomic electron neglecting atomic binding is given, in the laboratory system, b y do' =
°~e42-------~• d/(1 -- ]),
(m,,c'p
1
(5)
with / = E/Eo, where E o is the energy of the incident particle r, 8). The four momentum cut off, which reduces to (4) in the centre of mass system, gives an upper limit to i of /max ~-m~cZ/2Eom ,. We then find for the ionisation energy loss per unit length of path when N is the number of electrons per cm s fma]g
a
Eof/go. =dN 2 m . a,
(6)
o
where ru = eg/m~,c2. In deriving this result it has been assumed that
E, > (,n,/~.) ,n,c*.
12S
G . N . FOWLER
This result is the same as that given b y Peaslee, who in fact also uses in this case a cut-off equivalent to (4). However, in the other processes he discusses (bremsstrahlung and pair production) he assumes a formula of type (4) in the rest system of the meson. H a d he used this prescription in the present case he would not have found an upper limit to the energy loss since the two systems considered are the same only in the limit m,/m~,-+ O. The result (6) gives a maximum energy loss per unit path length due to an anomalous magnetic moment of the order 10-* MeV. cmZ/g when A2 ~ 10. The normal ionisation loss is of the order 2 MeV • cm*]g, so that, on this evidence, this value of ~2 cannot be excluded. 3.2.
BREMSSTRAHLUNG
The production of bremsstrahlung using the relativistic procedure adopted to take into account the finite size of the meson way be treated in the following way. With
1
V(X) --
(2=~)'1,Id
d'P'w(P') e''x/*°
~(X)
(2=~y, ] f
d*P"V'(P")e''''x/*°,
and
The expression for the anomalous contribution to the current density becomes, after partial integration with respect to X,
__1
,l
f
d~pd 4p Id~p ll--y,(p I! )a~,W(p t ) •J'~(p') • e'<'-''lx/~° e"('-x):*°(- i p , ) # X .
(v)
Integrating over X and /~ gives
1 /~'(x)=(~c)3 2a f #p'#p",e(F'),~,,,v,(F)e'("-")':"°. • (-
i(p' -
p")),:-((p'
-
~"),).
(8)
The matrix element for bremsstrahlung is of the form oO
K , --
~i*
oO
dr,
dr1
where '
=
-
f
2
(9)
ON THE ANOMALOUS MAGNETIC MOMENT OF THE ~-MESON
129
and e(t 2 -
tx) =
+
=
1, t s > tl
- - 1, t , < t 1.
Using (8) and the Fourier transform of A,(x), this gives -- i
l
K, = 27~' " (2:w~) 9
f
4'
d x,d
4' , ,, , ,,~la41p2d,4~j~2dgpld'4'p"ltd4'/'~la4'~l
°
1 + e(Xo, - - Xox ) e,.,..,+.~.,.,)/i~ 2
• ~(p'z')a~,~Cfi)(- i(p', - P'z')), A,(k,)~?(P'I')%,~I'(P'I) • (--
i(~1
--
~)tl'))xae(kl)/ra((~);
-- ~)'2t)2)/m((~i
--
p',')2)
(]0)
where s, =
+ k, -
f,'.
Pairing off the wave functions representing emission and absorption of virtual particles and taking the expectation value for the meson vacuum introduces, instead of y~(p',) ~(P'I'), the expression (Heitler*), p. 285) t --
.
e
4,
t
tt
t2
S<% (p,.)%, (ibl)>0 = T., } (] +e(p~o)e(Xo,--Xoa)) t~ (Pg. -- Pl )(~(~)o. "JI- /.~l), where S refers to a summation over spin states of the virtual particles and T~, = r ~ " P', + i/~. The factor e (x0s -- xoa) arises when we take account of the permutation of the emission and absorption operators of the v i r t u ~ particles. The factor (1 + e~0)eCxos
- - XO1))(~(~2 2
-3C /t$')
has the same effect under the integral sign (in the present case) as 8+(p~2 + ps) (Heitlerg), p. 286) and when we sum over all permutations of the points xx, xz, using ½ Z P(1
+
e(Xo,
--
xOl))
=
],
P
the integrand of (10) depends on the: x~ only through the factor exp(is~ • x~/?i,c). The integrations over the x~ may now be carried out giving the usual Dirac functions (2a~c)4'~4'(s~). If now we take plane waves for the incident and outgoing mesons: ~0#(~'1) =
~4'(P -- ~ ; )
a,(P),
~,(p~')
~'(P' _ p;') %(P'),
130
G.N. FOWLER I
It
II
we m a y c a r r y out the integrations over Pl, Pl and p=. The result m a y be written, before integrating over kl, i 1 dK~ = 21~ --~ " 2 ~
fdep,~ 4 ' zd k~(5 (p~ + k s - - p , ) ~ 4 ( p + kl _ P2)
_
• a(P') av.TA~,(k,)aQ,a(P).
(p~ - - P ' ) , ( P
- - P'2),
(11)
ne(kl) J " ( k [ ) J " (k~). The t functions in
(11)
imply
ks=P'--P--kl and p~ = P + kl.
The remaining integrations m a y be carried out to give i 1 dK2 = - - 2/~ --~ " 2z#~ " a ( P ' ) a ~ ' T A ~ ( P ' - - P - - kl)a°~a(P) " (kl + P - - P ' ) , ( k z ) ~ "
AQ(kl) ~ " ( ( P ' - - P - - k z ) ~ )
(12)
~"(k~).
Possible restrictions on four momentum transfers are contained in the terms J " ( ( P ' - - P - - k I ) ~ ) , fl;"(k~) and also in A ~ ( P ' - - P - - k x ) . This last term sets an upper limit to momentum transfer since it includes a cut off introduced b y the finite nuclear size. This leads to
[(P'
-
P -- kz) I <
~/R,
(13)
w h e r e R is t h e nuclear radius. T h e t e r m J - ( ( e ' - P - - kl)=) introd u c e s an effect due to the finite size of the meson when the meson
and nucleus exchange a quantum. This obviously yields a less restrictive condition than (13) due to the fact that R > ;t,. For the remaining term we have
There is thus no cut off in radiation loss other than that introduced b y the finite nuclear size. We can therefore simply use the formula derived b y Powell s0) for this case. The ratio of total radiation loss due to anomalous effects to the normal radiation loss is, according to Powell, ....
9~
A~Eo
~n
m/, c~"
-- ~ 0.1-
(14)
O n t h e other hand Peaslee's cut off prescription led him to the essentially different expression
ON THE ANOMALOUS MAGNETIC MOMENT OF THE ~-MESON
l~t
The observations on cosmic ray bursts (soft showers initiated b y high energy bremsstrahlung produced b y mesons with energies of the order 108 Gev) agree approximately with the results predicted b y the usual theory 11). Thus (14) implies ks<< 1 in 'contrast to Peaslee's result and this conclusion is independent of the form factor specifying the finite size of the meson, 3.4.
PAIR
PRODUCTION
Without going into the details of the calculation it is easy to see that, as the photon responsible for the pair production is virtual, i~ follows that k~ =# 0 a n d < 1.
However, when the energy of the incident meson is sufficiently high the four momentum of the virtual photon will satisfy ~ = 0 to order r,%c/p, where p is the momentum of the incident meson. There will then be no cut off in electron pair energy. B y comparison with the bremsstrahlung case we may infer that the ratio of the cross sections for pair production will be again of order ~t4Eo/mt,c2. Now, the experimental work of Goldsack and Kannangara 13) has shown that the observed pair production is in reasonable agreement with the usual theory, so that again 42 << 1 independently of the form factor. 3.5.
STAR
PRODUCTION
In this case we have to do with collisions between the incident meson and a nuclear proton, and so we have as an upper limit to the possible energy transfers =
~ 10 MeV.
Since nuclear stars have energies larger than 300 MeV we m a y say that such processes are unlikely to give any information on the value of ~t.
4. Conclusions The conclusion from this discussion is that 4 2 << 1 independently of the finite meson size effects, so that there should be no significant contribution to the scattering of p-mesons b y nuclei from an anomalous magnetic moment.
132
o.N.
FOWLER
References 1) 2) 3) 4 5 6 7 8 9 10 11 12
G. D. Rochester and A. W. Wolfendale, Phil. Mag. 45 (1954) 980 D. A. Tidman, Proc. Phys. Soc. A b7 (1954) 559 R. Gatto, Nuovo Cimento 12 (1955) 613 D. C. Peaslee, Nuovo Cimento 9 (1952) 56 E. Corinaldesi, Nuovo Cimento 8 (1951) 62 L. N. Cooper and E. M. Henley, Phys. Rev. 92 (1953) 801 H. C. Corben and J. Schwinger, Phys. Rev. 58 (1940) 953 W. Pauli, Rev. rood. Phys. 13 (1941) 203 W. Heitler, Q u a n t u m Theory of Radiation, 3rd Edition, Oxford (1954) J. L. Powell, Phys. t{~v. 75 (1949) 32 C. N. Chou and M. Schein, Phys. Rev. 97 (1955) 206 S. J. Goldsack and M. L. T. Kannangaxa, Phil. !V~ag. 44 (1953) 811