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A possibility for measuring the S,P,T-type neutral weak current in muon scattering
Volume 89B, number 2
PHYSICS LETTERS
14 January 1980
A POSSIBILITY FOR MEASURING THE S,P,T-TYPE NEUTRAL WEAK CURRENT IN MUON SCATTERING WU Chl-Min 1
CERN, Geneva, Switzerland Received 22 October 1979
A posslblhty for measuring the S,P,T-type neutral weak current in muon scattering is discussed.
Experimental evidence that there are only V - A-type couplings in the neutral weak current is still lacking. The gauge theory and other theories were developed based on the assumption that there are no S,P,T-type couplings. Nevertheless, it IS important to measure the Lorentz structure of the neutral weak current. In this note, we discuss a possibility for measuring the S,P,T-type neutral weak currents in muon scattering. An essential difference between the V - A-type neutral-weak-current coupling and the S,P,T-type coupling is that, in a weak process, say vP scattering, the S,P,T-type couphng flips the heliclty of v, g, whereas the V - A-type coupling preserves it [1 ]. For neutrino neutral-current experiments it is difficult to measure the final neutrino hellcity directly. Considering other lepton-hadron scattering processes which could also involve the weak interaction, e.g., eP ~ eX, there are, although we can get a pure polarized electron beam [2], nevertheless difficulties in the measurement of the final electron heliclty. Here we consider another lepton-hadron scattering process, viz., muon scattering, ~uP ~ gX. In muon scattering, both the electromagnetic and the weak interaction are present. We shall assume that weak current contains S,P,T,V,A-type couplings. Using the quark model, we deduce the relation between the hehcltychanging and the S,P,T couplings and then discuss the possibihties for measuring their presence. For/~P ~/~X, the electromagnetic interaction of the muon and the quarks is
Hie = e~'ru~ "A u + ~ eeoa~tlT, onA, ,
(1 a)
l
and generally, the weak interaction has the following form' Hw = ~(as+ lbs75)~" qtch + ~'1'5 (ap + lbp"/5)/a "qi75q, + ~Th(av + bv75)/a" ~qi3'hqt + Bg'x'r5 (aA + hA3'5 )~u" ~qz%,75 q~ + Box,(aT + ibT75)/~" Ftz°x~qi,
(lb)
where/a and ch are muon field and ith quark field, eqz is the charge of the ith quark, a~ and b~ are the S,P,T,V,Atype coupling constants, respectively. Assuming CP conservation of the weak current, we get bs, bp, b T = 0. The electromagnetic interaction interferes with the V - A weak interaction. They preserve the helicity of the lepton. The cross sections of muon scattering by the left- and right-handed quark are proportional to the following quantities.
1 On leave of absence from the Institute of High Energy Physics, Peking, China. 218
Volume 89B, number 2
PHYSICS LETTERS
14 January 1980
O(/aLL--qL ) ~ [(e2/q2) eqt + a v + b v + a A + bA]264(S - m 2 ) , o(/aLL--qR ) ~ [(eZ/q2)eq~ + av + b v - a A - bA]2 64(s - m2u)(l
y)2 ,
(2)
o (/.tR R --qL ) ~ [(e2/q 2) eq, + a V - b V - a A + b A ] 2 64 (s - m 2) (1 - y )2 , o(/.tRR--qR ) ~ [(e2 /qZ)eq + av - b v + a A - bA] 2 64(s - m 2 ) , where q2 is the square of the momentum transfer between the muon and the quark, s is the total energy squared m the centre-of-mass system, y is the inelasticity./IEE--qL denotes the process in which a left-handed muon interacts with a left-handed quark and becomes a left-handed final muon. The S,P,T-type coupling interactions interfere with each other and they change the helicity of the muon. The relevant cross sections are proportional to the following quantities: O(//LR--qL) ~ ( a s + aP) 2A + aT(aS + ap)B + a2C, O(gtRL--qL) ~ (as - aP) 2A ,
O(PLR--qR ) ~ (a s - ap)ZA,
°(/2RL--qR) ~ (as + aP) 2A + aT(aS + ap)B + a 2 C ,
(3 a)
where
A =16(s-m2)2y2+32m2(s -m2)y,
B =-256(s-m2)2y(1-½y),
C = 1024 [(s - m2)2(1 - y + ~ y 2 ) _ rn u2 l(s _ m 2 ) y ] .
(3b)
The ratio o f the hehcity-changmg cross section of muon scattering by a nucleon to the heliclty-preserving cross section is obtained b y multiplying eqs. (2), (3) by the probability f~ (x) o f finding a quark o f type i with fractional longitudinal m o m e n t u m in the nucleon x = q2/2Mv. Comparing with the electromagnetic interaction, we neglect the V - A weak interaction in the hehcity-preserving processes (2); then we get the ratio as r=
o(/.ILR--q) °(ktLL--q)
-
O(•RL--q) °(,URR--q)
=
Z i f i ( x ) [2 (a 2 + a2)A + aT(a S + ap)B + a 2C]
(4)
Zzfi(x ) [(e2/q2)eql]264(s- m2) 2 [ l + ( 1 - - y ) 2 ]
If there is only scalar, pseudoscalar or tensor coupling, then
1 r-~21+ r=q41 e4
y2
Z, ft'(x)
a2 '
( l _ y ) 2 Nifl(x)(eq,)2
r_q 4 1
y2
~ift(x)
a2 '
-e4 21+(1--y12Zifz(x)(eqi)2
l _ Y + ¼ y 2 Zif~(x ) a2 , 6-1+(1 _ y ) 2 £ifz(x)(eq,)2
(5)
respectively. As an example, taking y = ½, and t = u, d, s, we obtain the r values with different q2 and ratio R =
aT/X/-2GF, shown in table 1. Next, we examine the observable quantities o f the spin-fhp phenomenon. The initial muon beam comes from kaon or pion decay. We can get a purely polarized initial muon beam b y applying an angle cut on the decay muons Table 1 r-values for different q2 and ratio R = aT/x/~ GF, withy = ~1 and l = u,d,s. R
0.5 0.1 0.01
q2 (GeV/c)2 20
40
100
200
1.2 X 10 -2 4 9 X 10-4 4.9 X 10 -6
4.9 × 10 -2 1.9 × 10 -3 1.9 X 10 -s
3 × 10 -1 1.2 × 10 -2 1.2 X 10-4
1.2 4.9 × 10 -2 4.9 × 10 -4
219
Volume 89B, number 2
PHYSICS LETTERS
14 January 1980
[3]. To measure the hehclty of the final muon, we can examine the stopped muon decay. The corresponding energy and angular spectrum o f the e - (e +) produced in the decay of fully polarized # - (/a+), after neglecting the electromagnetic correction and the electron mass, is given In ref. [4].
I(x) =g2m~5~dg2e 967r3 47r
x2 [3 -
2x + (1 -
2x)P
cos 0] ,
(6)
where P is the degree o f the polarization of the initial muon beam, 0 is the angle between the polarization direction of the muon and the direction of motion of the e - (e+), x = 2Ee/m u A f o r w a r d - b a c k w a r d asymmetry of e - (e +) is expected for the S,P,T-type couplings. From eq. (6), the e - (e +) with maximum energy (½mu) have a distribution ~(1 - P c o s 0). Therefore, after/~+ scattering followed by V+ decay, we get for the positrons the distribution: o(/a[L)(1 - P c o s 0) + o(tl[R ) (1 - P X cos 0) Then, we get for the ratio between the two special quantities' (e + emitted at 0°)/(e + emitted at 180 °) =
+ )/O(laLL + ) O(laLR
(7a)
Similarly, for the electrons with maximum energy m / 1 - decay, we have ( e - emitted at 0°)/(e - emitted at 180 °) = o ( t a ~ , L ) / O ~ R R ) .
(7b)
From eq. (6), by integrating over x, we obtain the asymmetry (backward - forward)/(backward + forward) = gt
[1-
+ + 20(~LR)/O(,ULL)],
( b a c k w a r d - forward)/(backward + forward) = gI [1 - 20~RL)/O(/.t~R)] ,
for
e+
for e - .
Recently, a neutrino experiment using the method described here to determine the upper limit of the S,P,T-type coupling in the charged weak current has been reported [5]. A muon experiment to investigate the Lorentz structure o f the neutral weak current xn the manner discussed here is currently under study [6]. I would like to thank K.W. Chen for a very useful discussion.
References [1] B. Kayser et al., Phys. Lett. 52B (1974) 385, R.L. Kingsley et al., Phys. Rev. D10 (1974) 2216. [2] C.Y. Prescott et al., Phys. Lett. 77B (1978) 347. [3] K.W. Chen, NAL Summer Study, Vol. 4 (1969) p. 387. [4] G. Kallen, Springer Tracts in Modern Physics, Vol. 46 (1968). [5] M. Jonker et al., Phys. Lett. 86B (1979) 229. [6] K.W. Chen, private communication.
220
PHYSICS LETTERS
14 January 1980
A POSSIBILITY FOR MEASURING THE S,P,T-TYPE NEUTRAL WEAK CURRENT IN MUON SCATTERING WU Chl-Min 1
CERN, Geneva, Switzerland Received 22 October 1979
A posslblhty for measuring the S,P,T-type neutral weak current in muon scattering is discussed.
Experimental evidence that there are only V - A-type couplings in the neutral weak current is still lacking. The gauge theory and other theories were developed based on the assumption that there are no S,P,T-type couplings. Nevertheless, it IS important to measure the Lorentz structure of the neutral weak current. In this note, we discuss a possibility for measuring the S,P,T-type neutral weak currents in muon scattering. An essential difference between the V - A-type neutral-weak-current coupling and the S,P,T-type coupling is that, in a weak process, say vP scattering, the S,P,T-type couphng flips the heliclty of v, g, whereas the V - A-type coupling preserves it [1 ]. For neutrino neutral-current experiments it is difficult to measure the final neutrino hellcity directly. Considering other lepton-hadron scattering processes which could also involve the weak interaction, e.g., eP ~ eX, there are, although we can get a pure polarized electron beam [2], nevertheless difficulties in the measurement of the final electron heliclty. Here we consider another lepton-hadron scattering process, viz., muon scattering, ~uP ~ gX. In muon scattering, both the electromagnetic and the weak interaction are present. We shall assume that weak current contains S,P,T,V,A-type couplings. Using the quark model, we deduce the relation between the hehcltychanging and the S,P,T couplings and then discuss the possibihties for measuring their presence. For/~P ~/~X, the electromagnetic interaction of the muon and the quarks is
Hie = e~'ru~ "A u + ~ eeoa~tlT, onA, ,
(1 a)
l
and generally, the weak interaction has the following form' Hw = ~(as+ lbs75)~" qtch + ~'1'5 (ap + lbp"/5)/a "qi75q, + ~Th(av + bv75)/a" ~qi3'hqt + Bg'x'r5 (aA + hA3'5 )~u" ~qz%,75 q~ + Box,(aT + ibT75)/~" Ftz°x~qi,
(lb)
where/a and ch are muon field and ith quark field, eqz is the charge of the ith quark, a~ and b~ are the S,P,T,V,Atype coupling constants, respectively. Assuming CP conservation of the weak current, we get bs, bp, b T = 0. The electromagnetic interaction interferes with the V - A weak interaction. They preserve the helicity of the lepton. The cross sections of muon scattering by the left- and right-handed quark are proportional to the following quantities.
1 On leave of absence from the Institute of High Energy Physics, Peking, China. 218
Volume 89B, number 2
PHYSICS LETTERS
14 January 1980
O(/aLL--qL ) ~ [(e2/q2) eqt + a v + b v + a A + bA]264(S - m 2 ) , o(/aLL--qR ) ~ [(eZ/q2)eq~ + av + b v - a A - bA]2 64(s - m2u)(l
y)2 ,
(2)
o (/.tR R --qL ) ~ [(e2/q 2) eq, + a V - b V - a A + b A ] 2 64 (s - m 2) (1 - y )2 , o(/.tRR--qR ) ~ [(e2 /qZ)eq + av - b v + a A - bA] 2 64(s - m 2 ) , where q2 is the square of the momentum transfer between the muon and the quark, s is the total energy squared m the centre-of-mass system, y is the inelasticity./IEE--qL denotes the process in which a left-handed muon interacts with a left-handed quark and becomes a left-handed final muon. The S,P,T-type coupling interactions interfere with each other and they change the helicity of the muon. The relevant cross sections are proportional to the following quantities: O(//LR--qL) ~ ( a s + aP) 2A + aT(aS + ap)B + a2C, O(gtRL--qL) ~ (as - aP) 2A ,
O(PLR--qR ) ~ (a s - ap)ZA,
°(/2RL--qR) ~ (as + aP) 2A + aT(aS + ap)B + a 2 C ,
(3 a)
where
A =16(s-m2)2y2+32m2(s -m2)y,
B =-256(s-m2)2y(1-½y),
C = 1024 [(s - m2)2(1 - y + ~ y 2 ) _ rn u2 l(s _ m 2 ) y ] .
(3b)
The ratio o f the hehcity-changmg cross section of muon scattering by a nucleon to the heliclty-preserving cross section is obtained b y multiplying eqs. (2), (3) by the probability f~ (x) o f finding a quark o f type i with fractional longitudinal m o m e n t u m in the nucleon x = q2/2Mv. Comparing with the electromagnetic interaction, we neglect the V - A weak interaction in the hehcity-preserving processes (2); then we get the ratio as r=
o(/.ILR--q) °(ktLL--q)
-
O(•RL--q) °(,URR--q)
=
Z i f i ( x ) [2 (a 2 + a2)A + aT(a S + ap)B + a 2C]
(4)
Zzfi(x ) [(e2/q2)eql]264(s- m2) 2 [ l + ( 1 - - y ) 2 ]
If there is only scalar, pseudoscalar or tensor coupling, then
1 r-~21+ r=q41 e4
y2
Z, ft'(x)
a2 '
( l _ y ) 2 Nifl(x)(eq,)2
r_q 4 1
y2
~ift(x)
a2 '
-e4 21+(1--y12Zifz(x)(eqi)2
l _ Y + ¼ y 2 Zif~(x ) a2 , 6-1+(1 _ y ) 2 £ifz(x)(eq,)2
(5)
respectively. As an example, taking y = ½, and t = u, d, s, we obtain the r values with different q2 and ratio R =
aT/X/-2GF, shown in table 1. Next, we examine the observable quantities o f the spin-fhp phenomenon. The initial muon beam comes from kaon or pion decay. We can get a purely polarized initial muon beam b y applying an angle cut on the decay muons Table 1 r-values for different q2 and ratio R = aT/x/~ GF, withy = ~1 and l = u,d,s. R
0.5 0.1 0.01
q2 (GeV/c)2 20
40
100
200
1.2 X 10 -2 4 9 X 10-4 4.9 X 10 -6
4.9 × 10 -2 1.9 × 10 -3 1.9 X 10 -s
3 × 10 -1 1.2 × 10 -2 1.2 X 10-4
1.2 4.9 × 10 -2 4.9 × 10 -4
219
Volume 89B, number 2
PHYSICS LETTERS
14 January 1980
[3]. To measure the hehclty of the final muon, we can examine the stopped muon decay. The corresponding energy and angular spectrum o f the e - (e +) produced in the decay of fully polarized # - (/a+), after neglecting the electromagnetic correction and the electron mass, is given In ref. [4].
I(x) =g2m~5~dg2e 967r3 47r
x2 [3 -
2x + (1 -
2x)P
cos 0] ,
(6)
where P is the degree o f the polarization of the initial muon beam, 0 is the angle between the polarization direction of the muon and the direction of motion of the e - (e+), x = 2Ee/m u A f o r w a r d - b a c k w a r d asymmetry of e - (e +) is expected for the S,P,T-type couplings. From eq. (6), the e - (e +) with maximum energy (½mu) have a distribution ~(1 - P c o s 0). Therefore, after/~+ scattering followed by V+ decay, we get for the positrons the distribution: o(/a[L)(1 - P c o s 0) + o(tl[R ) (1 - P X cos 0) Then, we get for the ratio between the two special quantities' (e + emitted at 0°)/(e + emitted at 180 °) =
+ )/O(laLL + ) O(laLR
(7a)
Similarly, for the electrons with maximum energy m / 1 - decay, we have ( e - emitted at 0°)/(e - emitted at 180 °) = o ( t a ~ , L ) / O ~ R R ) .
(7b)
From eq. (6), by integrating over x, we obtain the asymmetry (backward - forward)/(backward + forward) = gt
[1-
+ + 20(~LR)/O(,ULL)],
( b a c k w a r d - forward)/(backward + forward) = gI [1 - 20~RL)/O(/.t~R)] ,
for
e+
for e - .
Recently, a neutrino experiment using the method described here to determine the upper limit of the S,P,T-type coupling in the charged weak current has been reported [5]. A muon experiment to investigate the Lorentz structure o f the neutral weak current xn the manner discussed here is currently under study [6]. I would like to thank K.W. Chen for a very useful discussion.
References [1] B. Kayser et al., Phys. Lett. 52B (1974) 385, R.L. Kingsley et al., Phys. Rev. D10 (1974) 2216. [2] C.Y. Prescott et al., Phys. Lett. 77B (1978) 347. [3] K.W. Chen, NAL Summer Study, Vol. 4 (1969) p. 387. [4] G. Kallen, Springer Tracts in Modern Physics, Vol. 46 (1968). [5] M. Jonker et al., Phys. Lett. 86B (1979) 229. [6] K.W. Chen, private communication.
220