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Right-left asymmetry of inclusive hadron production in electron-positron annihilation
Volume 63B, number 3
PHYSICS LETTERS
RIGHT-LEFT
ASYMMETRY
2 August 1976
OF INCLUSIVE HADRON PRODUCTION
IN ELECTRON-POSITRON
ANNIHILATION
R. SIMARD*
Lawrence Berkeley Laboratory, University of California, Berkeley, Calif. 94720, USA and M. SUZUKI**
Department of Physics, University of California, Berkeley, Claif. 94720, USA Received 11 June 1976 We calculate a possible right-left asymmetry of inclusive hadron production in highly transversely polarized e+eannihilation, which is, ff successfully measured, an unambiguous clean evidence for parity violation. Various tests of weak interactions in e+e - annihilation have been proposed [ 1 , 2 ] , and one of the few promising tests is to measure a forward-backward asymmetry in cos 0 distribution of final hadrons. Unlike the e+e /a+/a- process, strong interaction dynamics prevents us from estimating accurately hadrons due to two-photonannihilation (without e+e - in final states) interfering with one-photon-annihilation. We propose here an alternative method to detect an interference of weak and electromagnetic interactions by making use of high transverse polarization of e+e - beams. Let us write a weak neutral current interaction as Lint = - gl E")'u(1 + ga ~ 5 ) e Zu - gh J(w) Zu ,
(1)
where ,-~)v) is a hadronic weak neutral current, and gl and gh are model-dependent weak couplings. It interferes in cross sections with the electromagnetic interaction Lin t = e ~TueAU - ej(em)Au. The inclusive cross section of onephoton-annihilation is given through a straightforward calculation by Ep d3°dp 3
s]l(e2~ 2\~-~] (F~ e) + ¼/32xF (e) sin20 (1 - p2 cos 2~b')].
(2) +
1
s(s-
(e2gaglgh~ {½/~2xp2 lmF(2ew) sin20 sin 2~b' - ~ Re F(3ew) cos 0},
I
where p is the detected hadron momentum,/3 = Ipl/gp, s is the center-of-mass energy squared, x = 2Ep/sl/2,Mz is the neutral W boson mass, P is the magnitude of transverse polarization of e + e - , ¢' is the azimuthal angle of p measured from the direction of the polarization vector of e - t l and F's are structure functions .2. The interference term of structure function F t ew) is defined by F~ew) = vW~ew)/m, where m is the hadron mass, and W~ew) is the coefficient of the covariant absorptive part Ex(OIJ(vW)lX, p(+k)(X,p(+)lJ(em)lO)(2~r)36(4)(q - p PX)" It is particularly emphasized here that unlike the diagonal structure functions, F~ ew) is generally complex -
* Supported by U.S. Energy Research and Development Administration. ** Supported by National Science Foundation under Grant Number PHY 74-08175-A02. ,1 The spherical angles (0, ¢') are so defined that the polarization vector of e- is along the positive x axis and the e- beam is along the positive z axis. ,2 We have suppressed in (2) contributions of Re F~ew) and Re F~ew) as well as all of the dia~onal weak structure functions, since they are insignificant to the asymmetry parameter A(¢') that we calculate as long as s ~ M z. 304
Volume 63B, number 3
PHYSICS LETTERS
2 August 1976
1.5 1
I,o
0.5
0
io"
20*
30 °
~"
5-o"
Fig. 1. ~(~') for three sets of values ofP 2 and a(x). I for p2 = 0.85 and a(x) = 1.0, II for p2 = 0.70 and a(x) = 0.80, and III for p2 = 0.50 and or(x) = 0.60.
60"
even if time reversal invariance holds. Interference of final interactions in different eigenchannels (I = 0 and 1) gives rise to a nonzero imaginary part since Y.xIX, p(+)>(X, P(+)I ¢ Y-xIX, p(-))(X, p(-)l when p is not summed over •3. Eq. (2) shows a right-left asymmetry in the ~' angular distribution. With a(x) = - ½x F~e)/F~e) for the purely electromagnetic structure functions, the inclusive cross sections for relativistic hadrons (/3 ~ 1) can be written as Epd3O~ 1 (e2~ 2 dP 3 2s----~ \~--~/ {2 - or(x) sin20 - ~ x ) p2 sin20 cos 2~'} F~ e)
+ ~s--~Mz)~ 1 [ ge22!'(}P2x161r aglgh] l lm F(2ew) sin20 sin 2¢' - R e F (ew) cos 0}.
(3)
Let us define a right-left asymmetry parameter at 0 = 90 ° as A(¢') = {N(¢') - N ( - ¢ ' ) } / { N ( ¢ ' ) + N(-~b')},
(4)
where N(¢') is the number of hadrons coming out in the direction of (0 = 90 °, ¢'). Then
{gaglghi( s A(¢/) -- A(n -- ~') = ~ - - - ~ ] \ ~ ]
~
p2 sin 2q~'
x I m F [ ew)
2 - a(x)(1 + p 2 cos 2~b')
Fie)
(5)
At X/s = 7.4 GeV, a(x) is close to unity for x ~ 0.5, and P is about 70% [3]. It is expected that at higher energies o(x) will be close to unity above even smaller values o f x and P will reach as high as 92.4% [4]. Hadrons produced by the electromagnetic current are depleted at 0 = 90 ° with ~' = 0 ° and 180 °, while the right-left asymmetry shows up maximally at ~b' = ~-cos -1 {oep2/(2 - c0}. In the standard Salam-Weinberg model [5] ~4, the weak couplings are given by gaglgh = ½ e2/sin2 20w' where sin20w ~ 0.30 to fit experimental data. Eq. (5) is written in this model as x Im F~ew) A(¢') - A(~r - ¢') ~ -0.29(x/s-/37 GeV) 2 ~(¢') (6) F~ e)
~(~') = p2 sin 2¢'/'[2 -- oe(x) -- oe(x)P2 cos 2¢'}.
(7)
,3 The superscripts (+) and (-) in the state vectors refer to "outgoing" and "incoming" boundary conditions, respectively. ,4 The asymmetry is zero in models where a leptonic weak neutral current is a pure vector as well as in the model proposed in ref. [61. 305
Volume 63B, number 3
PHYSICS LETTERS
2 August 1976
The function ~(~') is o f the order of unity or less. It is plotted in fig. 1 for a few different values o f P 2 and a(x). A crucial question is how large x Im F~eW)/F~e) would be. If all parton distribution functions are chosen equal, the parton model predicts that in the four-quark model
Ix F(2ew)l/F~e) ~
9
-g - 4 sin20w ~ 0.60.
(8)
Only a product o f currents antisymmetric in isovector and isoscalar currents can contribute to Im F t ew). Estimate o f its imaginary part involves details o f strong interaction dynamics at low energies, which forbids anything bey o n d a mere guess #5. We find from the isospin structure o f currents that the ratio Im T;'(ew)/~(ew) -2 /-2 is about 0.50 at maximum interference in the Salam-Weinberg model. It should be pointed out that if final states consists only o f pions, G parity forbids the interference. Only the events that contain KK, NN, and so forth contribute to the asymmetry. At higher energies such events occupy a large portion o f total events enough to produce a substantial right-left asymmetry. Although this test seems to be rather difficult at SPEAR/DORRIS energies . 6 , it will be quite realistic at PEP/PETRA energies (x/~ = 20 ~ 36 GeV). To conclude, we emphasize that unlike the forward-backward asymmetry the right-left asymmetry is a clean direct test o f parity nonconservation free from backgrounds due to higher order electromagnetic interactions or to weak decays of heavy leptons or charmed hadrons produced electromagneticaUy. If a hadronic weak neutral current is parity-conserving but a leptonic current contains both vector and axial vector currents, there would be no cos 0 asymmetry besides a possible secondary effect due to weak decays in final states and therefore the present m e t h o d would be useful to supplement information obtainable in the e + e - -+/a+~t - process. , s We cannot exclude a pessimistic guess that the phase difference o f / = 0 and I = 1 final state interactions may come out to be negligibly small. ,6 It should be noted here that statistical accuracy of 0.67% has been reached for parity tests using angular correlations of two and three final hadrons at SPEAR energies. G. Goldhaber, private communication.
References [1] N. Cabibbo and R. Gatto, Phys. Rev. 124 (1961) 1577; R. Gatto, Springer Tracts in Modern Physics 39 (1965) 106; R. Gatto and G. Preparata, Riv. Nuovo Cim. 4 (1974) 445; A. McDonald, Nucl. Phys. B75 (1974) 343. [2] V.K. Cung, A.K. Mann and E.A. Paschos, Phys. Lett. 41B (1972) 355; J. Godine and A. Hankey, Phys. Rev. D6 (1972) 3301; A. Love, Lett. Nuovo Cim. 5 (1972) 113; R. Bundy, Phys. Lett 45B (1973) 340; E. Lendvai and G. Posik, Phys. Lett. 56B (1975) 462. [3] R. Schwitters, Proc. 1975 Intern. Symp. on Lepton and photon interactions at high energies ed. W.J. Kirk, Stanford University, 1975, p. 5. [4] R. Schwitters, Nucl. Instr. and Meth. 117 (1974) 331; J.D. Jackson, Lawrence Berkeley Laboratory report, LBL-4232 (August, 1975), to be published in Rev. Mod. Phys. [5] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Satam, in Elementary particle physics, ed. V. Svartholm (Stockholm, 1968) p. 367. [6] V.S. Mathur, S. Okubo and J.E. Kim, Phys. Rev. D10 (1974) 3648.
306
PHYSICS LETTERS
RIGHT-LEFT
ASYMMETRY
2 August 1976
OF INCLUSIVE HADRON PRODUCTION
IN ELECTRON-POSITRON
ANNIHILATION
R. SIMARD*
Lawrence Berkeley Laboratory, University of California, Berkeley, Calif. 94720, USA and M. SUZUKI**
Department of Physics, University of California, Berkeley, Claif. 94720, USA Received 11 June 1976 We calculate a possible right-left asymmetry of inclusive hadron production in highly transversely polarized e+eannihilation, which is, ff successfully measured, an unambiguous clean evidence for parity violation. Various tests of weak interactions in e+e - annihilation have been proposed [ 1 , 2 ] , and one of the few promising tests is to measure a forward-backward asymmetry in cos 0 distribution of final hadrons. Unlike the e+e /a+/a- process, strong interaction dynamics prevents us from estimating accurately hadrons due to two-photonannihilation (without e+e - in final states) interfering with one-photon-annihilation. We propose here an alternative method to detect an interference of weak and electromagnetic interactions by making use of high transverse polarization of e+e - beams. Let us write a weak neutral current interaction as Lint = - gl E")'u(1 + ga ~ 5 ) e Zu - gh J(w) Zu ,
(1)
where ,-~)v) is a hadronic weak neutral current, and gl and gh are model-dependent weak couplings. It interferes in cross sections with the electromagnetic interaction Lin t = e ~TueAU - ej(em)Au. The inclusive cross section of onephoton-annihilation is given through a straightforward calculation by Ep d3°dp 3
s]l(e2~ 2\~-~] (F~ e) + ¼/32xF (e) sin20 (1 - p2 cos 2~b')].
(2) +
1
s(s-
(e2gaglgh~ {½/~2xp2 lmF(2ew) sin20 sin 2~b' - ~ Re F(3ew) cos 0},
I
where p is the detected hadron momentum,/3 = Ipl/gp, s is the center-of-mass energy squared, x = 2Ep/sl/2,Mz is the neutral W boson mass, P is the magnitude of transverse polarization of e + e - , ¢' is the azimuthal angle of p measured from the direction of the polarization vector of e - t l and F's are structure functions .2. The interference term of structure function F t ew) is defined by F~ew) = vW~ew)/m, where m is the hadron mass, and W~ew) is the coefficient of the covariant absorptive part Ex(OIJ(vW)lX, p(+k)(X,p(+)lJ(em)lO)(2~r)36(4)(q - p PX)" It is particularly emphasized here that unlike the diagonal structure functions, F~ ew) is generally complex -
* Supported by U.S. Energy Research and Development Administration. ** Supported by National Science Foundation under Grant Number PHY 74-08175-A02. ,1 The spherical angles (0, ¢') are so defined that the polarization vector of e- is along the positive x axis and the e- beam is along the positive z axis. ,2 We have suppressed in (2) contributions of Re F~ew) and Re F~ew) as well as all of the dia~onal weak structure functions, since they are insignificant to the asymmetry parameter A(¢') that we calculate as long as s ~ M z. 304
Volume 63B, number 3
PHYSICS LETTERS
2 August 1976
1.5 1
I,o
0.5
0
io"
20*
30 °
~"
5-o"
Fig. 1. ~(~') for three sets of values ofP 2 and a(x). I for p2 = 0.85 and a(x) = 1.0, II for p2 = 0.70 and a(x) = 0.80, and III for p2 = 0.50 and or(x) = 0.60.
60"
even if time reversal invariance holds. Interference of final interactions in different eigenchannels (I = 0 and 1) gives rise to a nonzero imaginary part since Y.xIX, p(+)>(X, P(+)I ¢ Y-xIX, p(-))(X, p(-)l when p is not summed over •3. Eq. (2) shows a right-left asymmetry in the ~' angular distribution. With a(x) = - ½x F~e)/F~e) for the purely electromagnetic structure functions, the inclusive cross sections for relativistic hadrons (/3 ~ 1) can be written as Epd3O~ 1 (e2~ 2 dP 3 2s----~ \~--~/ {2 - or(x) sin20 - ~ x ) p2 sin20 cos 2~'} F~ e)
+ ~s--~Mz)~ 1 [ ge22!'(}P2x161r aglgh] l lm F(2ew) sin20 sin 2¢' - R e F (ew) cos 0}.
(3)
Let us define a right-left asymmetry parameter at 0 = 90 ° as A(¢') = {N(¢') - N ( - ¢ ' ) } / { N ( ¢ ' ) + N(-~b')},
(4)
where N(¢') is the number of hadrons coming out in the direction of (0 = 90 °, ¢'). Then
{gaglghi( s A(¢/) -- A(n -- ~') = ~ - - - ~ ] \ ~ ]
~
p2 sin 2q~'
x I m F [ ew)
2 - a(x)(1 + p 2 cos 2~b')
Fie)
(5)
At X/s = 7.4 GeV, a(x) is close to unity for x ~ 0.5, and P is about 70% [3]. It is expected that at higher energies o(x) will be close to unity above even smaller values o f x and P will reach as high as 92.4% [4]. Hadrons produced by the electromagnetic current are depleted at 0 = 90 ° with ~' = 0 ° and 180 °, while the right-left asymmetry shows up maximally at ~b' = ~-cos -1 {oep2/(2 - c0}. In the standard Salam-Weinberg model [5] ~4, the weak couplings are given by gaglgh = ½ e2/sin2 20w' where sin20w ~ 0.30 to fit experimental data. Eq. (5) is written in this model as x Im F~ew) A(¢') - A(~r - ¢') ~ -0.29(x/s-/37 GeV) 2 ~(¢') (6) F~ e)
~(~') = p2 sin 2¢'/'[2 -- oe(x) -- oe(x)P2 cos 2¢'}.
(7)
,3 The superscripts (+) and (-) in the state vectors refer to "outgoing" and "incoming" boundary conditions, respectively. ,4 The asymmetry is zero in models where a leptonic weak neutral current is a pure vector as well as in the model proposed in ref. [61. 305
Volume 63B, number 3
PHYSICS LETTERS
2 August 1976
The function ~(~') is o f the order of unity or less. It is plotted in fig. 1 for a few different values o f P 2 and a(x). A crucial question is how large x Im F~eW)/F~e) would be. If all parton distribution functions are chosen equal, the parton model predicts that in the four-quark model
Ix F(2ew)l/F~e) ~
9
-g - 4 sin20w ~ 0.60.
(8)
Only a product o f currents antisymmetric in isovector and isoscalar currents can contribute to Im F t ew). Estimate o f its imaginary part involves details o f strong interaction dynamics at low energies, which forbids anything bey o n d a mere guess #5. We find from the isospin structure o f currents that the ratio Im T;'(ew)/~(ew) -2 /-2 is about 0.50 at maximum interference in the Salam-Weinberg model. It should be pointed out that if final states consists only o f pions, G parity forbids the interference. Only the events that contain KK, NN, and so forth contribute to the asymmetry. At higher energies such events occupy a large portion o f total events enough to produce a substantial right-left asymmetry. Although this test seems to be rather difficult at SPEAR/DORRIS energies . 6 , it will be quite realistic at PEP/PETRA energies (x/~ = 20 ~ 36 GeV). To conclude, we emphasize that unlike the forward-backward asymmetry the right-left asymmetry is a clean direct test o f parity nonconservation free from backgrounds due to higher order electromagnetic interactions or to weak decays of heavy leptons or charmed hadrons produced electromagneticaUy. If a hadronic weak neutral current is parity-conserving but a leptonic current contains both vector and axial vector currents, there would be no cos 0 asymmetry besides a possible secondary effect due to weak decays in final states and therefore the present m e t h o d would be useful to supplement information obtainable in the e + e - -+/a+~t - process. , s We cannot exclude a pessimistic guess that the phase difference o f / = 0 and I = 1 final state interactions may come out to be negligibly small. ,6 It should be noted here that statistical accuracy of 0.67% has been reached for parity tests using angular correlations of two and three final hadrons at SPEAR energies. G. Goldhaber, private communication.
References [1] N. Cabibbo and R. Gatto, Phys. Rev. 124 (1961) 1577; R. Gatto, Springer Tracts in Modern Physics 39 (1965) 106; R. Gatto and G. Preparata, Riv. Nuovo Cim. 4 (1974) 445; A. McDonald, Nucl. Phys. B75 (1974) 343. [2] V.K. Cung, A.K. Mann and E.A. Paschos, Phys. Lett. 41B (1972) 355; J. Godine and A. Hankey, Phys. Rev. D6 (1972) 3301; A. Love, Lett. Nuovo Cim. 5 (1972) 113; R. Bundy, Phys. Lett 45B (1973) 340; E. Lendvai and G. Posik, Phys. Lett. 56B (1975) 462. [3] R. Schwitters, Proc. 1975 Intern. Symp. on Lepton and photon interactions at high energies ed. W.J. Kirk, Stanford University, 1975, p. 5. [4] R. Schwitters, Nucl. Instr. and Meth. 117 (1974) 331; J.D. Jackson, Lawrence Berkeley Laboratory report, LBL-4232 (August, 1975), to be published in Rev. Mod. Phys. [5] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Satam, in Elementary particle physics, ed. V. Svartholm (Stockholm, 1968) p. 367. [6] V.S. Mathur, S. Okubo and J.E. Kim, Phys. Rev. D10 (1974) 3648.
306