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A lower bound on heavy-lepton production by neutrinos
Nuclear Physics B56 (1973) 325-332. North-Holland Publishing Company
A LOWER BOUND ON HEAVY-LEPTON PRODUCTION BY NEUTRINOS C.H. LLEWELLYN SMITH CERN, Geneva
Received 22 February 1973 Abstract: A lower bound on o ( v A ~ M . . . ) is given, in terms of the structure functions which can most easily be measured in u + A ~ ~ + ... reactions, for heavy leptons (M) o'f the type required by renormalizable models of weak interactions. 1. Introduction The old idea [1] * that undiscovered heavy leptons may exist has recently attracted renewed interest because t h e y are required in a large class o f renormalizable models o f w e a k i n t e r a c t i o n s * * . The m o s t auspicious reaction for producing charged heavy leptons is e ÷ e - ~ M ÷ M - but if sufficiently energetic e ÷ e - colliding beams are n o t available the reaction u u + A ~ M + + ... (first considered by Gerstein and F o l o m e s h k i n [3]) m a y provide the best hope o f detecting heavy leptons o f the type required in renormalizable models (which are the only ones considered in this paper). In ref. [2] was estimated o ( u + A ~ M + ...) using a simple model. While this estimate m a y be useful for planning experiments, m o d e l - i n d e p e n d e n t - c a l c u l a t i o n s are obviously desirable for interpreting negative e x p e r i m e n t a l results. In principle the hadronic vertex which describes the process v u ~ M + ( ~ u ~ M - ) can be c o m p l e t e l y d e t e r m i n e d f r o m Pu -~/a+ ( v u - ~ / 2 - ) e x p e r i m e n t s * * * . In practice, however, the smallness o f the m u o n ' s mass makes it almost impossible to measure all the relevant structure f u n c t i o n s in " o r d i n a r y " n e u t r i n o experiments. In this n o t e we record a lower b o u n d for heavy-lepton p r o d u c t i o n in terms o f the structure functions which can be measured (relatively) easily in c o n v e n t i o n a l n e u t r i n o reactions. Before stating the bound, we recall the well-known argument which shows w h y heavy leptons are n e e d e d in m a n y models and why (at least in simple cases) it is the * A review of various possible types of heavy leptons and their experimental status has recently been given by Perl [ 1]. See also ref. [2]. The first explicit discussion of heavy leptons in the literature of which I am aware is due to Lipmanov [ 1 ]. ** A discussion of the properties of the heavy leptons required in renormalizable models has been given by Bjorken and Llewellyn Smith 121. Note that the factor 1 + (m2/xs) in eqs. (3.9) and (3.13) of this reference should be 1 + (m2/2xs); this does not change fig. 12 of this reference in which the term multiplying this factor was neglected. *** Assuming (as we do in this paper) that there is only one charged vector boson which couples to ut~ - t~-(~/a-/a +) and ~t - M-(v# -M+) or that it" there are several they couple to u - t~ and v - M in the same combination.
326
C.H. Llewellyn Smith, Lepton production
heavy M+ which has the same lepton number as uu a n d / a - , as we assume below (it is this property which gives M -+ events such a good signature in neutrino reactions). Vp.
~
--
~
~-
-----~
-
-
-
~
W*
W-
Fig. 1. vla .
I~
W*
~
:-
W"
Fig. 2.
f v
k'(massm)
~_
k
Fig. 3. In all models in which the weak Interactions are mediated by vector bosons fig. 1 must exist. On its own this diagram has a " b a d " high-energy behaviour which renders the corresponding foUrth-order amplitude for up ~ up unrenormalizably infinite; in renormalizable models this " b a d " behaviour is cancelled by introducing new particle exchanges in the s-channel (neutral current) or u-channel (heavy lepton) - or both. Note that cancellation can only be achieved using a heavy lepton alone if it is in the u-channel (the lepton mass can be neglected to leading order at high energy and two t-channel contributions obviously make additive contributions); i.e., we need fig. 2 and Pu, # - and M + therefore have the same lepton number. Furthermore, in simple models in which the complete amplitude for vO ~ W+W is given by # and M + exchange alone in lowest order, cancellation requires*: (guUw) 2 = (guMW) 2 .
2. The bound We are interested in the processes, in fig. 3. The hadronic vertex for a virtual
* All the models considered in ref. [2] have this property.
CH. Llewellyn Smith, Lepton production
327
W+(W - ) meson of four-momentum q incident on an unpolarized nucleon is described by the tensor:
+
(p ]Ju• ( 0 ) I F ) (FIJ v+-(0)]p) (27r) 3 6 4(q +p_pF)
1 ~
F
- (guy qlaqv - + [
qtat"\ (pv qvv'~W~
(1) ieuv~eP~q~W~ + quqvW~ - 2M 2
M2
+ (qupv .i. qvpu)W5-+ + i(quPv-qvPu)W~ 2M 2
2M 2
W~ = Wt (q2,v=q'p) , where Y. denotes an average over nucleon spins (the metric issuch that spacelike q2 is negative; the normalization etc., is the same as in ref. [4]*). The differential neutrino cross section is given by contracting I¢,,~ with the spin average lepton tensor and is: d2o
G2
{
-
Wl(m2 q2)
4Evm2 Mq 2
2q 2
m2
m2v2(q2--m2)l +~_3((E(q2+m2)+E,(q 2 m 2 1 + Y 1 q4M2 J
where
m2(m2-q2)W 4
2Em2W5
M2
M
Y=
'
(2)
and E(E') is the lab energy of the neutrino (outgoing lepton), and for vu - ~ -
:
Wi=W +,
~=+1,
v~-~la + :
Wi= WF,
~ =-1,
* Note that the definition of W4,s in ref. [4] is different from that used here, which was chosen to simplify7the inequalities.
328
CH. Llewellyn Smith, Lepton production
V/~ " + M + :
Wi= W [ ,
~= +1 ,
VI~ --->M
Wi-- W+ ,
~=-1
:
;
for p-production 4 1G 2 _
2
gv~w . (q2-M2w)2
for M-production 4
1G2 -
gyM W (q2-M2)2
The physical region for the reaction is most easily stated in terms of the variables W = ~/2v + q2 + M 2- (the missing mass) and q2: M ~< W ~< x / ~ - m , ,
s = (k u +p)2 = 2 M E + M 2 ,
max • (q2)min_ x [2M2m2
(s_M2)(s_m2_W2 )
+ (s M2)x/(s-(W+rn) 2) ( s - ( W - m ) 2 ) ] .
(3)
Note the factor m 2 multiplying I#4,5 in eq. (2), because of which W4,5 may make important contributions to M-production but not to p-production (W4,5 can in fact only be separated from W1,2 by measuring the outgoing lepton's polarization as well as d2o/dq2du). However, the Wi satisfy certain inequalities [5] * from which it is straightforward to find a lower bound for the differential cross section. To see the origin of the bound it is more convenient to use a slightly different notation. We introduce four independent vectors which form a basis for the current (W-meson) helicity states: three of the vectors having spin one, which satisfy qxeR'L'S= 0 and are right handed (R), left handed (L) and longitudinal (S).with respect to q in the lab (and frames connected by Lorentz transforming along q), and one spin-zero (divergence) vector e o = qx/x/Z--q 2. We can then form the W-nucleon cross sections:
* Explicitly the inequalities are 1 ~ I W z I ~ W I 2M 2
M2 W25 + W26 <<.v2 4 M2q2
~<
-
W2
-M~qq 2 W2-W1
(-vWs-q2Wa)..
329
C.H. Llewellyn Smith, Lepton production + *i i + O~ = C(u,q 2) e u e v W~v ~ O , * u, q2 *D S W + ODS = OSD = C( )e e v vla ,
where C is a flux factor which is rather arbitrary and need not be specified for our purposes. The interference terms satisfy the Schwartz inequality*: O2S + OS2D~ 2OsO D . Suppose the final lepton has a definite spin (denoted by a) with respect to some axis. The lepton current matrix elements can be calculated and written in the form:
c~
Then ~*a~a
*a
a
d a a ~ IC~l 2 o R + [C~l 2 o L + IC~l 2 o S + IC~l 2 o o + c 8 cDOSO + C D CSODS-
The last four terms can be combined in the form ICas eS ( F I J X l N ) + C ~ e D ( F [ J X l N ) I 2 ~ O . F
A trivial lower bound can therefore be obtained by neglecting the longitudinal and divergence contributions altogether. Since the contributions of OR,L do not vanish as mlepton ~ 0 they can be measured in ordinary v-reactions; this bound therefore has the desired form. Furthermore, it obviously remains true when we sum over lepton spins; however, we can do better in this case. The spin averaged cross section is given by
O'a
~-~"S ~"D 7- ~ O C~) (OSD+ ODS) ) .
There is no contribution from the T-violating quantity oSD -- oDS because the lepton vertex conserves T and therefore Z a ( C S*t/ C ~ - c ~ a c ~ ) = 0 automatically. We then (i) put o D = e o s, oSD + oDS = 2rlo S, (ii) use the minimum value o f e allowed by the Schwartz inequality e/> rl2(in the weaker and more obvious form oSD + ODS ~< 2 o ~ S a D ) and (iii) minimize with respect to r/. This gives the bound
* Together with the conditions o a ~ 0 this is equivalent to the inequalities in the footnote p. 328.
CH. LlewellynSmith, Lepton production
330
1 *a (/ *a a ~(IC~l 2 °s +leVI 2 ao +~(Cs Co+Co CS) (~SO+°Ds)) a
oS Ia~ IC~12\
4 ~tC~l 2
/
The bracket is positive semidefinite (this is the Schwartz inequality for the leptonic vertex); this bound is therefore better than that obtained by neglecting Os, crD and uSD + ODS altogether (unless o S = 0). We now restate the bounds explicitly in terms of the structure functions*: A trivial bound is given by neglecting longitudinal and divergence contributions i.e., by putting I~4,5 = 0 and W2(1-
v2
q2M½
)~W1
in eq. (2).
A stronger bound is given by setting Y>~ m 2[u(m2 q2)+ 2EMq212 q2(m2-q2)(u2-M2q 2) [ (1
M2q2 )W2-
In both cases we obtain a lower bound on M-production in terms of the functions WI,2, 3 which can be measured in/~-production.
3. Remarks (i) To bound o(u u ~ M +) [o(Ou ~ M - ) ] we need data for do(~ u ~ U+) [do(v u -+ p - ) ] in general. If we neglect strangeness-changing processes and assume the charge symmetry condition: W+ (proton target) = i (neutron target)
Wt7- (neutron target) (proton target) '
o(u u ~ M +) can be bounded in terms of neutrino data alone. However, renormalizable models generally involve charmed particles and in many of them the charge symmetry condition may be violated. It is therefore dangerous to assume it without further tests. * Given eq. (2), it is easier in practice to obtain the stronger bound by working directly with the Wi and the inequalities which they satisfy (we did it with both notations as a check).
C.H. Llewellyn Smith, Lepton production
331
(ii) If we assume scale invariance lira u -+ °°
( W 1 -+ FI (x) ,
q2 {uW2 3 x = -- 2~ fixed ( M2' --+F2,3(x ) , then the bound for do takes a simple form in the region where the limit is reached: d2o
~dy
rr(yxs+M2) 2 { Fl(X )
+ [F2(x ) - 2 x F l ( x ) ]
+x(y 2 -2y)
(l-y
whe re 0 <~y
-
u ~ 2p ME - - s <~
m2 sx
1 - - - ,
m2 --<~x<~l s
'
and terms of order M/E have been dropped. The simple bound is obtained by neglecting the last term, i.e., by putting F 2 -~ 2xF 1 (this term is positive semidefinite in the physical region since F L = F 2 - 2xF 1 ) 0 ) . Putting M W ~ oo and integrating over y we obtain: 1
o >1 G 2ME f lt m2[s
dx{ x l'X'I'
O+1Im 411
m2t2 ( m2_41 + ~ ~xF3(x ) ( 1 - ~ ] \ sx ] + [F2(x)
2XFl(X)] [ ( 1 - ~ ) ( ½ +
m2~ m 2 2~]-~-log
xs (~-)]
A.simpler bound may again be obtained by neglecting the last term. If we do this and also assume the charge symmetry condition, we obtain.a bound which coincides with the result of the model calculation in ref. [2] * [eqs. (3.8) and (3.9)], when we make the identification FI(X ) = f ( x ) +)~(x) ,
F3(x ) = 2 (f(x) -f(x)),
* Apart from the misprint in ref. [2] mentioned above.
332
CH. Llewellyn Smith, Lepton production
with the notation used there i.e., the model of ref. [2] gives a lower bound in all models in which scaling and the charge symmetry condition hold, provided it is a good approximation to put M w ~ ~. However, whenever this bound is needed it
shouM be recalculated using the best data currently available and the complete formula if there is then evidence against the charge symmetry condition or if it turns out that F L does not vanish. 1 am grateful to K.Winter for urging me to work out this bound in detail.
References [1] M. Perl, SLAC-PUB-1062(1972); E.M. Lipmanov, JETP (Sov. Phys.) 16 (1963) 634; ZhETF 43 (1962) 893. [2] J.D. Bjorken and C.H. Llewellyn Smith, Phys. Rev. D7 (1973) 887. [3] S.S. Gerstein and V.N. Folomeshkin, Sov. J. Nucl. Phys. 8 (1969) 447; Yad. Fiz. 8 (1968) 768. [4] C.H. Llewellyn Smith, Phys. Reports 3C (1972) 263. [5] T.D. Lee and C.N. Yang, Phys. Rev. 126 (1962) 2239; M.G. Doncel and E. de Rafael, Nuovo Cimento 4A (1971) 363.
A LOWER BOUND ON HEAVY-LEPTON PRODUCTION BY NEUTRINOS C.H. LLEWELLYN SMITH CERN, Geneva
Received 22 February 1973 Abstract: A lower bound on o ( v A ~ M . . . ) is given, in terms of the structure functions which can most easily be measured in u + A ~ ~ + ... reactions, for heavy leptons (M) o'f the type required by renormalizable models of weak interactions. 1. Introduction The old idea [1] * that undiscovered heavy leptons may exist has recently attracted renewed interest because t h e y are required in a large class o f renormalizable models o f w e a k i n t e r a c t i o n s * * . The m o s t auspicious reaction for producing charged heavy leptons is e ÷ e - ~ M ÷ M - but if sufficiently energetic e ÷ e - colliding beams are n o t available the reaction u u + A ~ M + + ... (first considered by Gerstein and F o l o m e s h k i n [3]) m a y provide the best hope o f detecting heavy leptons o f the type required in renormalizable models (which are the only ones considered in this paper). In ref. [2] was estimated o ( u + A ~ M + ...) using a simple model. While this estimate m a y be useful for planning experiments, m o d e l - i n d e p e n d e n t - c a l c u l a t i o n s are obviously desirable for interpreting negative e x p e r i m e n t a l results. In principle the hadronic vertex which describes the process v u ~ M + ( ~ u ~ M - ) can be c o m p l e t e l y d e t e r m i n e d f r o m Pu -~/a+ ( v u - ~ / 2 - ) e x p e r i m e n t s * * * . In practice, however, the smallness o f the m u o n ' s mass makes it almost impossible to measure all the relevant structure f u n c t i o n s in " o r d i n a r y " n e u t r i n o experiments. In this n o t e we record a lower b o u n d for heavy-lepton p r o d u c t i o n in terms o f the structure functions which can be measured (relatively) easily in c o n v e n t i o n a l n e u t r i n o reactions. Before stating the bound, we recall the well-known argument which shows w h y heavy leptons are n e e d e d in m a n y models and why (at least in simple cases) it is the * A review of various possible types of heavy leptons and their experimental status has recently been given by Perl [ 1]. See also ref. [2]. The first explicit discussion of heavy leptons in the literature of which I am aware is due to Lipmanov [ 1 ]. ** A discussion of the properties of the heavy leptons required in renormalizable models has been given by Bjorken and Llewellyn Smith 121. Note that the factor 1 + (m2/xs) in eqs. (3.9) and (3.13) of this reference should be 1 + (m2/2xs); this does not change fig. 12 of this reference in which the term multiplying this factor was neglected. *** Assuming (as we do in this paper) that there is only one charged vector boson which couples to ut~ - t~-(~/a-/a +) and ~t - M-(v# -M+) or that it" there are several they couple to u - t~ and v - M in the same combination.
326
C.H. Llewellyn Smith, Lepton production
heavy M+ which has the same lepton number as uu a n d / a - , as we assume below (it is this property which gives M -+ events such a good signature in neutrino reactions). Vp.
~
--
~
~-
-----~
-
-
-
~
W*
W-
Fig. 1. vla .
I~
W*
~
:-
W"
Fig. 2.
f v
k'(massm)
~_
k
Fig. 3. In all models in which the weak Interactions are mediated by vector bosons fig. 1 must exist. On its own this diagram has a " b a d " high-energy behaviour which renders the corresponding foUrth-order amplitude for up ~ up unrenormalizably infinite; in renormalizable models this " b a d " behaviour is cancelled by introducing new particle exchanges in the s-channel (neutral current) or u-channel (heavy lepton) - or both. Note that cancellation can only be achieved using a heavy lepton alone if it is in the u-channel (the lepton mass can be neglected to leading order at high energy and two t-channel contributions obviously make additive contributions); i.e., we need fig. 2 and Pu, # - and M + therefore have the same lepton number. Furthermore, in simple models in which the complete amplitude for vO ~ W+W is given by # and M + exchange alone in lowest order, cancellation requires*: (guUw) 2 = (guMW) 2 .
2. The bound We are interested in the processes, in fig. 3. The hadronic vertex for a virtual
* All the models considered in ref. [2] have this property.
CH. Llewellyn Smith, Lepton production
327
W+(W - ) meson of four-momentum q incident on an unpolarized nucleon is described by the tensor:
+
(p ]Ju• ( 0 ) I F ) (FIJ v+-(0)]p) (27r) 3 6 4(q +p_pF)
1 ~
F
- (guy qlaqv - + [
qtat"\ (pv qvv'~W~
(1) ieuv~eP~q~W~ + quqvW~ - 2M 2
M2
+ (qupv .i. qvpu)W5-+ + i(quPv-qvPu)W~ 2M 2
2M 2
W~ = Wt (q2,v=q'p) , where Y. denotes an average over nucleon spins (the metric issuch that spacelike q2 is negative; the normalization etc., is the same as in ref. [4]*). The differential neutrino cross section is given by contracting I¢,,~ with the spin average lepton tensor and is: d2o
G2
{
-
Wl(m2 q2)
4Evm2 Mq 2
2q 2
m2
m2v2(q2--m2)l +~_3((E(q2+m2)+E,(q 2 m 2 1 + Y 1 q4M2 J
where
m2(m2-q2)W 4
2Em2W5
M2
M
Y=
'
(2)
and E(E') is the lab energy of the neutrino (outgoing lepton), and for vu - ~ -
:
Wi=W +,
~=+1,
v~-~la + :
Wi= WF,
~ =-1,
* Note that the definition of W4,s in ref. [4] is different from that used here, which was chosen to simplify7the inequalities.
328
CH. Llewellyn Smith, Lepton production
V/~ " + M + :
Wi= W [ ,
~= +1 ,
VI~ --->M
Wi-- W+ ,
~=-1
:
;
for p-production 4 1G 2 _
2
gv~w . (q2-M2w)2
for M-production 4
1G2 -
gyM W (q2-M2)2
The physical region for the reaction is most easily stated in terms of the variables W = ~/2v + q2 + M 2- (the missing mass) and q2: M ~< W ~< x / ~ - m , ,
s = (k u +p)2 = 2 M E + M 2 ,
max • (q2)min_ x [2M2m2
(s_M2)(s_m2_W2 )
+ (s M2)x/(s-(W+rn) 2) ( s - ( W - m ) 2 ) ] .
(3)
Note the factor m 2 multiplying I#4,5 in eq. (2), because of which W4,5 may make important contributions to M-production but not to p-production (W4,5 can in fact only be separated from W1,2 by measuring the outgoing lepton's polarization as well as d2o/dq2du). However, the Wi satisfy certain inequalities [5] * from which it is straightforward to find a lower bound for the differential cross section. To see the origin of the bound it is more convenient to use a slightly different notation. We introduce four independent vectors which form a basis for the current (W-meson) helicity states: three of the vectors having spin one, which satisfy qxeR'L'S= 0 and are right handed (R), left handed (L) and longitudinal (S).with respect to q in the lab (and frames connected by Lorentz transforming along q), and one spin-zero (divergence) vector e o = qx/x/Z--q 2. We can then form the W-nucleon cross sections:
* Explicitly the inequalities are 1 ~ I W z I ~ W I 2M 2
M2 W25 + W26 <<.v2 4 M2q2
~<
-
W2
-M~qq 2 W2-W1
(-vWs-q2Wa)..
329
C.H. Llewellyn Smith, Lepton production + *i i + O~ = C(u,q 2) e u e v W~v ~ O , * u, q2 *D S W + ODS = OSD = C( )e e v vla ,
where C is a flux factor which is rather arbitrary and need not be specified for our purposes. The interference terms satisfy the Schwartz inequality*: O2S + OS2D~ 2OsO D . Suppose the final lepton has a definite spin (denoted by a) with respect to some axis. The lepton current matrix elements can be calculated and written in the form:
c~
Then ~*a~a
*a
a
d a a ~ IC~l 2 o R + [C~l 2 o L + IC~l 2 o S + IC~l 2 o o + c 8 cDOSO + C D CSODS-
The last four terms can be combined in the form ICas eS ( F I J X l N ) + C ~ e D ( F [ J X l N ) I 2 ~ O . F
A trivial lower bound can therefore be obtained by neglecting the longitudinal and divergence contributions altogether. Since the contributions of OR,L do not vanish as mlepton ~ 0 they can be measured in ordinary v-reactions; this bound therefore has the desired form. Furthermore, it obviously remains true when we sum over lepton spins; however, we can do better in this case. The spin averaged cross section is given by
O'a
~-~"S ~"D 7- ~ O C~) (OSD+ ODS) ) .
There is no contribution from the T-violating quantity oSD -- oDS because the lepton vertex conserves T and therefore Z a ( C S*t/ C ~ - c ~ a c ~ ) = 0 automatically. We then (i) put o D = e o s, oSD + oDS = 2rlo S, (ii) use the minimum value o f e allowed by the Schwartz inequality e/> rl2(in the weaker and more obvious form oSD + ODS ~< 2 o ~ S a D ) and (iii) minimize with respect to r/. This gives the bound
* Together with the conditions o a ~ 0 this is equivalent to the inequalities in the footnote p. 328.
CH. LlewellynSmith, Lepton production
330
1 *a (/ *a a ~(IC~l 2 °s +leVI 2 ao +~(Cs Co+Co CS) (~SO+°Ds)) a
oS Ia~ IC~12\
4 ~tC~l 2
/
The bracket is positive semidefinite (this is the Schwartz inequality for the leptonic vertex); this bound is therefore better than that obtained by neglecting Os, crD and uSD + ODS altogether (unless o S = 0). We now restate the bounds explicitly in terms of the structure functions*: A trivial bound is given by neglecting longitudinal and divergence contributions i.e., by putting I~4,5 = 0 and W2(1-
v2
q2M½
)~W1
in eq. (2).
A stronger bound is given by setting Y>~ m 2[u(m2 q2)+ 2EMq212 q2(m2-q2)(u2-M2q 2) [ (1
M2q2 )W2-
In both cases we obtain a lower bound on M-production in terms of the functions WI,2, 3 which can be measured in/~-production.
3. Remarks (i) To bound o(u u ~ M +) [o(Ou ~ M - ) ] we need data for do(~ u ~ U+) [do(v u -+ p - ) ] in general. If we neglect strangeness-changing processes and assume the charge symmetry condition: W+ (proton target) = i (neutron target)
Wt7- (neutron target) (proton target) '
o(u u ~ M +) can be bounded in terms of neutrino data alone. However, renormalizable models generally involve charmed particles and in many of them the charge symmetry condition may be violated. It is therefore dangerous to assume it without further tests. * Given eq. (2), it is easier in practice to obtain the stronger bound by working directly with the Wi and the inequalities which they satisfy (we did it with both notations as a check).
C.H. Llewellyn Smith, Lepton production
331
(ii) If we assume scale invariance lira u -+ °°
( W 1 -+ FI (x) ,
q2 {uW2 3 x = -- 2~ fixed ( M2' --+F2,3(x ) , then the bound for do takes a simple form in the region where the limit is reached: d2o
~dy
rr(yxs+M2) 2 { Fl(X )
+ [F2(x ) - 2 x F l ( x ) ]
+x(y 2 -2y)
(l-y
whe re 0 <~y
-
u ~ 2p ME - - s <~
m2 sx
1 - - - ,
m2 --<~x<~l s
'
and terms of order M/E have been dropped. The simple bound is obtained by neglecting the last term, i.e., by putting F 2 -~ 2xF 1 (this term is positive semidefinite in the physical region since F L = F 2 - 2xF 1 ) 0 ) . Putting M W ~ oo and integrating over y we obtain: 1
o >1 G 2ME f lt m2[s
dx{ x l'X'I'
O+1Im 411
m2t2 ( m2_41 + ~ ~xF3(x ) ( 1 - ~ ] \ sx ] + [F2(x)
2XFl(X)] [ ( 1 - ~ ) ( ½ +
m2~ m 2 2~]-~-log
xs (~-)]
A.simpler bound may again be obtained by neglecting the last term. If we do this and also assume the charge symmetry condition, we obtain.a bound which coincides with the result of the model calculation in ref. [2] * [eqs. (3.8) and (3.9)], when we make the identification FI(X ) = f ( x ) +)~(x) ,
F3(x ) = 2 (f(x) -f(x)),
* Apart from the misprint in ref. [2] mentioned above.
332
CH. Llewellyn Smith, Lepton production
with the notation used there i.e., the model of ref. [2] gives a lower bound in all models in which scaling and the charge symmetry condition hold, provided it is a good approximation to put M w ~ ~. However, whenever this bound is needed it
shouM be recalculated using the best data currently available and the complete formula if there is then evidence against the charge symmetry condition or if it turns out that F L does not vanish. 1 am grateful to K.Winter for urging me to work out this bound in detail.
References [1] M. Perl, SLAC-PUB-1062(1972); E.M. Lipmanov, JETP (Sov. Phys.) 16 (1963) 634; ZhETF 43 (1962) 893. [2] J.D. Bjorken and C.H. Llewellyn Smith, Phys. Rev. D7 (1973) 887. [3] S.S. Gerstein and V.N. Folomeshkin, Sov. J. Nucl. Phys. 8 (1969) 447; Yad. Fiz. 8 (1968) 768. [4] C.H. Llewellyn Smith, Phys. Reports 3C (1972) 263. [5] T.D. Lee and C.N. Yang, Phys. Rev. 126 (1962) 2239; M.G. Doncel and E. de Rafael, Nuovo Cimento 4A (1971) 363.