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Weak effects in e±e± → e±e±
Volume 58B, number 3
PHYSICS LETTERS
15 September
1975
WEAK EFFECTS IN e* e* --f e* e* * R. BUDNY* Institute of Theoretical Physics, Department
of Physics, Stanford University, Stanford, California94305, uSA
Received 12 July 1975 The prediction for the cross Section of e*e* + e*e* is given, assuming one photon and Wo exchange. Pure Wo terms and transverse and longitudinal beam polarizations are included.
A powerful method for learning about the structure of weak neutral currents, or about the W”, is to study their effects in e+e- reactions. There is an extensive literature concerning weak effects in e+e- -f p+p- dating back to 1961 [l] . Recent papers discuss weak effects in other final states such as meson pairs [2,3], baryon pairs [2], e+e- [4-71, and inclusive hadron production [8--l l] . Weak effects become large at resonance (when the CM energy &equals the W” mass), and may even dominate electromagnetic effects. However, the consensus is that the W” is very heavy compared to beam energies available in storage rings, so measurable weak effects are small compared to electromagnetic effects. Since the W”-photon interferences in the cross section grow like G s relative to the one photon terms, large s is advantageous. One way to get higher s with existing machines is to collide accelerator beams against beams stored in storage rings. For instance, a 30 GeV beam against a 5 GeV beam yields s = 600 GeV2, whereas two 5 GeV beams yield s = 100 GeV2. One severe drawback with this is that the luminosity is much smaller, but a way to partially compensate is to observe a fast reaction such as e+e- --, e+e- or e*e* -, e*ef . In the one photon exchange approximation, the t channel amplitude dominates the s channel amplitude in the first (Bhabha) reaction, and the t and u channel amplitudes are comparable in the second (Mdller) reaction, making its rate roughly twice that of the first. Several papers discuss weak effects in e-e- -, e-e- [S, 121. The purpose of this paper is to discuss these effects in greater generality. The cases of transverse and longitudinally polarized beams are considered. The method and notation parallel those of ref. [7] . The cross section is given for the polarization case where the spins of the beams are polarized in a plane containing the beams (in the CM system). The reason for this choice is that it gives the unpolarized case, the “natural” transverse polarization case to which beams in storage rings mature, and the longitudinal polarization case. The amplitude for e*&, sI) + e*(P2, s2) -+ e*(P3) t e’(Pq), assuming one photon and W” exchange, is
1
t ----Is(P3)Y,(gV t-M2
- -
+g*Y5)tQ1,
Sl)@r,)‘y%IV
+g*75)@2,s2)
1
u-M2
ii(P4)7&&
+g*75)u(P1,sl)is(P3)7c((gv
+g*7&02,9).
The beam polarizations s1 and s2 have components in the vertical direction with magnitudes t;I and g2 and in the direction of p1 with magnitudes h and A,. The scattering angle is 0, the azimuthal angle (measured from vertical) - oos 01, u = (Pl - P4) 2 = - ;s(l + cos e), P= u/e2(u - M2), Q is 4, s I (PI + P~)~, t E (PI - p3) 5 =_+(I E t/e2(t - M2), and e 2 = 47~~.The vector and axial coupling constants, gv and gA, are assumed real. z Research supported in part by National Science Foundation grant MPS 74-08916. Address after 1 September 1975: The Rockefeller University, New York, New York 10021, USA.
338
Volume SEB, number 3
PHYSICS LETTERS
When terms involving the electron mass, of order me/ 4 e-e- + e-e- is -$$=(1+h,h,)[(l+COS8)2bf~+(1-COS8)2kf2]
15 September 1975
are neglected, the cross section in the CM system for
-2~,E2sin28COS(2~)M3t(1-hlX2)4Mq+(Al-X2)4M5.
For the case e+e+ + e+e+, the signs of XI and X2 are reversed. The coefficients M, =C;,
M2=C;,
c3 =;r1+
kV
M4 =+{C:
M3 -C r C 2,
+&Q2e1 ++
•t C;},
kv +g,12Pl 7
M, =+{C;
c4 +
+ kV
are de&red as follows:
- C;},
-
g*PQl +; 11-k(&TV -g*)W
All the terms except MS are even under parity, and all except MS are even under charge conjugation. and u are negative, neither P nor Q resonates, and they are real. For the case of transverse polarization (X, = A2 = 0), the cross section is 2s dof’ans -_= .2 dSZ
cot4 t 0
[l+(g+-g:)Q12ttan4
: 0
- 2t1t2 cos(2G) (1 +(g:
-
g;>(e+P)
Since t
[lt(g$-gi)P]2
-cosB)P]
+&{4+4(g;+g;)[(l+cosB)Qt(l
.
t(g$+g;+6g;g;)[(l
tcosB)Qt(l
-cos6’)P]2)
+ (& - gi12PQ}.
This result is consistent with the less general results of refs. [5, 121. Effects of two photons are discussed in ref. [ 121. Weak effects can be studied by comparing measurements with this prediction. Transverse polarization can help separate weak from electromagnetic terms if g+ # gi. For the case where the beams are longitudinally polarized with components X, and h2 along pl , the prediction is
-$g=
+
(Xl -
(1t hlh2){c0t4(e/2)[i
h2) zgvgA s.4e
t(g+
--gi)Q12
{[(I tcose)p-(i
t tm4(e/2)[i
-cose)p]
t(gG
tgi)j(l
t (gt -gi)P]
2}
tc0se)Qt(i
-c0se)p]2).
The one photon part agrees with ref. [ 131. This result implies the following asymmetries reversed: Al_do’“-do
-++ = A$,[(1
do-‘-’ + da-‘+ A, ~ do-+ do-’
+cos8)2Mr
+
(1 + cos8)2Ml +(l
are
(1 - cos8)2M2 - 4M4] - h,4M~ - cosfl)2M2 + 4M4 + h,4M,
’
(Al - X2)4Ms
- do’-+ = + da++
when polarizations
(1 + $h2)[(1
+
COS8)2f!fl
+ (1
-
COS8)211’12]
+ (1
-
h,h,)htf,
The photon-W0 interference part of Ax, with X2 = 0, agrees with ref. [ 121. If s9M2 then Q+-s(1 --z)/(2e2M2)andP + - s( 1 t z)/(2ezM2) where z E cos 0. Within this limit, and neglecting the pure weak term, these asymmetries are 339
Volume 58B, number 3
PHYSICS LETTERS
16A2
(3 + z*)* (7 + 22)(1 - 22)
A, =-h,h,
x {&z2(3+z2)+&2(1 A _ -16(h, * -(3
+Z*)*
+
15 September 1975
S
(3 + z*)*(7 + z*) e*M* +6z2+z4)]
+gVgA[(3tz*)-
x:(7 +z*)(l -z*)]),
- h2)(l - z*)(s/e*M*) -h&(7
+Z*)(l
-Z*)’
The X,X, term in A I is the one photon contribution, gv and gA are expected to be of order M & making A, and the weak correction to A 1 of order Gs/e*. Since A2 violates parity, measurement of it would provide unambiguous evidence for the existence of weak effects. In storage rings it appears feasible, though difficult, to manufacture longitudinally polarized beams by waiting until the beams are transversely polarized, and then rotating the spins with a magnet. In the “natural” transverse case, 51 = f2, so after rotation, h, = A,, and thus A2 vanishes. Since longitudinally polarized accelerator beams are feasible [14], it may become possible to measure A2 some day.
References [l] [2] [3] [4] [S] [6] [7]
N. Cabibbo and R. Gatto, Phys. Rev. 124 (1961) 1577. R. Budny, Phys. Lett. 45B (1973) 340. E. Lendvai and G. Pocsik, Phys. Lett. 56B (1975) 462. D.A. Dicus, Phys. Rev. D8 (1973) 890; Erratum: DlO (1974) 1669. C.H. Llewellyn Smith and D.V. Nanopoulos, Nucl. Phys. B78 (1974) 205; Errata: to be published. R. Budny and A. McDonald, Phys. Rev. DlO (1974) 3107. R. Budny, Phys. Lett. 55B (1975) 227; Errata: Bs and B8 should be BS = Im (Ca - Cs)} and Be = Im {Cz(Ca + C’s)], and the expression for A requires 4Bt instead of B1. [ 81 R. Gatto and G. Preparata, Lettere al Nuovo Cim. 7 (1973) 89. [9] R. Budny and A. McDonald, Phys. Lett. 48B (1974) 423. [lo] A. McDonald, Nucl. Phys. B75 (1974) 343. [ 1 l] G. Kajon and R. Petronzio, Lettere al Nuovo Cim. 10 (1974) 369. [12] R. Gastmans and Y. Van Ham, Phys. Rev. DlO (1974) 3629. [13] A.M. Bincer, Phys. Rev. 107 (1957) 1434. [14] P.S. Cooper et al., Phys. Rev. Lett. 34 (1975) 1589.
340
PHYSICS LETTERS
15 September
1975
WEAK EFFECTS IN e* e* --f e* e* * R. BUDNY* Institute of Theoretical Physics, Department
of Physics, Stanford University, Stanford, California94305, uSA
Received 12 July 1975 The prediction for the cross Section of e*e* + e*e* is given, assuming one photon and Wo exchange. Pure Wo terms and transverse and longitudinal beam polarizations are included.
A powerful method for learning about the structure of weak neutral currents, or about the W”, is to study their effects in e+e- reactions. There is an extensive literature concerning weak effects in e+e- -f p+p- dating back to 1961 [l] . Recent papers discuss weak effects in other final states such as meson pairs [2,3], baryon pairs [2], e+e- [4-71, and inclusive hadron production [8--l l] . Weak effects become large at resonance (when the CM energy &equals the W” mass), and may even dominate electromagnetic effects. However, the consensus is that the W” is very heavy compared to beam energies available in storage rings, so measurable weak effects are small compared to electromagnetic effects. Since the W”-photon interferences in the cross section grow like G s relative to the one photon terms, large s is advantageous. One way to get higher s with existing machines is to collide accelerator beams against beams stored in storage rings. For instance, a 30 GeV beam against a 5 GeV beam yields s = 600 GeV2, whereas two 5 GeV beams yield s = 100 GeV2. One severe drawback with this is that the luminosity is much smaller, but a way to partially compensate is to observe a fast reaction such as e+e- --, e+e- or e*e* -, e*ef . In the one photon exchange approximation, the t channel amplitude dominates the s channel amplitude in the first (Bhabha) reaction, and the t and u channel amplitudes are comparable in the second (Mdller) reaction, making its rate roughly twice that of the first. Several papers discuss weak effects in e-e- -, e-e- [S, 121. The purpose of this paper is to discuss these effects in greater generality. The cases of transverse and longitudinally polarized beams are considered. The method and notation parallel those of ref. [7] . The cross section is given for the polarization case where the spins of the beams are polarized in a plane containing the beams (in the CM system). The reason for this choice is that it gives the unpolarized case, the “natural” transverse polarization case to which beams in storage rings mature, and the longitudinal polarization case. The amplitude for e*&, sI) + e*(P2, s2) -+ e*(P3) t e’(Pq), assuming one photon and W” exchange, is
1
t ----Is(P3)Y,(gV t-M2
- -
+g*Y5)tQ1,
Sl)@r,)‘y%IV
+g*75)@2,s2)
1
u-M2
ii(P4)7&&
+g*75)u(P1,sl)is(P3)7c((gv
+g*7&02,9).
The beam polarizations s1 and s2 have components in the vertical direction with magnitudes t;I and g2 and in the direction of p1 with magnitudes h and A,. The scattering angle is 0, the azimuthal angle (measured from vertical) - oos 01, u = (Pl - P4) 2 = - ;s(l + cos e), P= u/e2(u - M2), Q is 4, s I (PI + P~)~, t E (PI - p3) 5 =_+(I E t/e2(t - M2), and e 2 = 47~~.The vector and axial coupling constants, gv and gA, are assumed real. z Research supported in part by National Science Foundation grant MPS 74-08916. Address after 1 September 1975: The Rockefeller University, New York, New York 10021, USA.
338
Volume SEB, number 3
PHYSICS LETTERS
When terms involving the electron mass, of order me/ 4 e-e- + e-e- is -$$=(1+h,h,)[(l+COS8)2bf~+(1-COS8)2kf2]
15 September 1975
are neglected, the cross section in the CM system for
-2~,E2sin28COS(2~)M3t(1-hlX2)4Mq+(Al-X2)4M5.
For the case e+e+ + e+e+, the signs of XI and X2 are reversed. The coefficients M, =C;,
M2=C;,
c3 =;r1+
kV
M4 =+{C:
M3 -C r C 2,
+&Q2e1 ++
•t C;},
kv +g,12Pl 7
M, =+{C;
c4 +
+ kV
are de&red as follows:
- C;},
-
g*PQl +; 11-k(&TV -g*)W
All the terms except MS are even under parity, and all except MS are even under charge conjugation. and u are negative, neither P nor Q resonates, and they are real. For the case of transverse polarization (X, = A2 = 0), the cross section is 2s dof’ans -_= .2 dSZ
cot4 t 0
[l+(g+-g:)Q12ttan4
: 0
- 2t1t2 cos(2G) (1 +(g:
-
g;>(e+P)
Since t
[lt(g$-gi)P]2
-cosB)P]
+&{4+4(g;+g;)[(l+cosB)Qt(l
.
t(g$+g;+6g;g;)[(l
tcosB)Qt(l
-cos6’)P]2)
+ (& - gi12PQ}.
This result is consistent with the less general results of refs. [5, 121. Effects of two photons are discussed in ref. [ 121. Weak effects can be studied by comparing measurements with this prediction. Transverse polarization can help separate weak from electromagnetic terms if g+ # gi. For the case where the beams are longitudinally polarized with components X, and h2 along pl , the prediction is
-$g=
+
(Xl -
(1t hlh2){c0t4(e/2)[i
h2) zgvgA s.4e
t(g+
--gi)Q12
{[(I tcose)p-(i
t tm4(e/2)[i
-cose)p]
t(gG
tgi)j(l
t (gt -gi)P]
2}
tc0se)Qt(i
-c0se)p]2).
The one photon part agrees with ref. [ 131. This result implies the following asymmetries reversed: Al_do’“-do
-++ = A$,[(1
do-‘-’ + da-‘+ A, ~ do-+ do-’
+cos8)2Mr
+
(1 + cos8)2Ml +(l
are
(1 - cos8)2M2 - 4M4] - h,4M~ - cosfl)2M2 + 4M4 + h,4M,
’
(Al - X2)4Ms
- do’-+ = + da++
when polarizations
(1 + $h2)[(1
+
COS8)2f!fl
+ (1
-
COS8)211’12]
+ (1
-
h,h,)htf,
The photon-W0 interference part of Ax, with X2 = 0, agrees with ref. [ 121. If s9M2 then Q+-s(1 --z)/(2e2M2)andP + - s( 1 t z)/(2ezM2) where z E cos 0. Within this limit, and neglecting the pure weak term, these asymmetries are 339
Volume 58B, number 3
PHYSICS LETTERS
16A2
(3 + z*)* (7 + 22)(1 - 22)
A, =-h,h,
x {&z2(3+z2)+&2(1 A _ -16(h, * -(3
+Z*)*
+
15 September 1975
S
(3 + z*)*(7 + z*) e*M* +6z2+z4)]
+gVgA[(3tz*)-
x:(7 +z*)(l -z*)]),
- h2)(l - z*)(s/e*M*) -h&(7
+Z*)(l
-Z*)’
The X,X, term in A I is the one photon contribution, gv and gA are expected to be of order M & making A, and the weak correction to A 1 of order Gs/e*. Since A2 violates parity, measurement of it would provide unambiguous evidence for the existence of weak effects. In storage rings it appears feasible, though difficult, to manufacture longitudinally polarized beams by waiting until the beams are transversely polarized, and then rotating the spins with a magnet. In the “natural” transverse case, 51 = f2, so after rotation, h, = A,, and thus A2 vanishes. Since longitudinally polarized accelerator beams are feasible [14], it may become possible to measure A2 some day.
References [l] [2] [3] [4] [S] [6] [7]
N. Cabibbo and R. Gatto, Phys. Rev. 124 (1961) 1577. R. Budny, Phys. Lett. 45B (1973) 340. E. Lendvai and G. Pocsik, Phys. Lett. 56B (1975) 462. D.A. Dicus, Phys. Rev. D8 (1973) 890; Erratum: DlO (1974) 1669. C.H. Llewellyn Smith and D.V. Nanopoulos, Nucl. Phys. B78 (1974) 205; Errata: to be published. R. Budny and A. McDonald, Phys. Rev. DlO (1974) 3107. R. Budny, Phys. Lett. 55B (1975) 227; Errata: Bs and B8 should be BS = Im (Ca - Cs)} and Be = Im {Cz(Ca + C’s)], and the expression for A requires 4Bt instead of B1. [ 81 R. Gatto and G. Preparata, Lettere al Nuovo Cim. 7 (1973) 89. [9] R. Budny and A. McDonald, Phys. Lett. 48B (1974) 423. [lo] A. McDonald, Nucl. Phys. B75 (1974) 343. [ 1 l] G. Kajon and R. Petronzio, Lettere al Nuovo Cim. 10 (1974) 369. [12] R. Gastmans and Y. Van Ham, Phys. Rev. DlO (1974) 3629. [13] A.M. Bincer, Phys. Rev. 107 (1957) 1434. [14] P.S. Cooper et al., Phys. Rev. Lett. 34 (1975) 1589.
340