- Email: [email protected]
A polarized τ factory at the Next Linear Collider
Nuclear Instruments and Methods in Physics Research A 472 (2001) 189–194
A polarized t factory at the Next Linear Collider Sekazi K. Mtingwaa,b,*, Mark Strikmanc a
Department of Physics, North Carolina A&T State University, 1601 E. Market Street, Greensboro, NC 27411, USA b Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA 30314, USA c Department of Physics, Pennsylvania State University, University Park, PA 16802, USA
Abstract We discuss the importance of using the Next Linear Collider to study polarization effects in t lepton decay. Our proposal is to Compton backscatter circularly polarized low energy laser pulses off an unpolarized electron beam of energy in the hundreds of GeV. The resulting backscattered photons also will have energies in the hundreds of GeV range and will be circularly polarized as well. Upon striking a fixed target, these polarized hot photons can produce copious polarized t lepton pairs, which could be used to study t decay, CP violation, and a number of other physics issues. r 2001 Elsevier Science B.V. All rights reserved. PACS: 13.60.r; 13.10.+q; 13.88.+e; 14.60.Fg
1. Introduction For some time, there has been an interest in constructing an electron–positron collider of 250 GeV or higher energy per beam. This has become known as the Next Linear Collider (NLC). One of the main reasons for building the NLC is to search for the elusive Higgs boson, whose mass is possibly in the range of a few hundred GeV. Notwithstanding the lofty goal of discovering the Higgs, there is also the possibility of operating a heavy flavor factory at the NLC. Thus, the NLC could be a dual purpose accelerator complex. Previously, we proposed a scheme for photoproducing large quantities of bottom quark pairs and t lepton pairs in a scheme dubbed BACKGAMMON, for BACKscattered GAMMas On Nucleons. The idea has been described at length in previous papers [1–5], and it involves Compton backscattering low energy laser pulses off a hundreds of GeV electron beam from the NLC to produce hot photons of energy roughly equal to that of the initial electron beam. For bottom quark pair production, it would be advantageous for the hot photon beam to have an energy Eg \250 GeV in order to maximize the bottom quark pair production cross-section [6,7]. However, for t pair production, the cross-section varies only as log Eg ; thus, one only needs Eg \100 GeV: There is considerable interest in studying polarized t decay; however, standard t-charm factories (tcF) produce roughly 107 unpolarized t’s per year [8–13]. On the other hand, in BACKGAMMON, there is the possibility of producing \109 polarized t’s per year, with the t’s flying at relatively large angles Bmt =Eg *Corresponding author. Tel.: +1-336-334-7646; fax: +1-336-334-7283. E-mail address: [email protected] (S.K. Mtingwa). 0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 1 8 1 - 0
190
S.K. Mtingwa, M. Strikman / Nuclear Instruments and Methods in Physics Research A 472 (2001) 189–194
and decay lengths B1 cm; thus leaving a rather clean signature in the detector. In a previous paper [3], we noted that, by using polarized electrons and polarized laser pulses, one could produce polarized hot photons, which could then photoproduce polarized t’s on nuclear targets. However, in this paper, we point out that the initial electron beam need not be polarized. As long as one uses the backscattered photons, the circular polarization of the initial laser pulse will be preserved.
2. Compton backscattering The theory and applications of Compton backscattering are not new, but they have yet to realize their full potential. The original theory was derived separately by Milburn [14] and Arutyunian et al. [15–17]. The application of Compton backscattering for the construction of photon–photon colliders was proposed by Ginzburg, et al. [18,19], Akerlof [20], and Sens [21], with detailed polariation effects being discussed by Ginzburg et al. [19]. Other discussions of polarization effects in Compton backscattering have been provided by Arutyunyan et al. [16,17], Tsai [22], Kotkin et al. [23,24], Babusci et al. [25], and Serbo [26]. The detailed theory of Compton backscattering, incorporating the accelerator lattice functions of the initial electron beam, was derived by the authors [1]. As for the first application of a Compton backscattered photon beam in an actual experiment, Ballam et al. [27] used highly monochromatic Compton backscattered photon beams of energies 1.44, 2.84, and 4:66 GeV; which were produced at the Stanford Linear Accelerator Center by scattering pulses from a ruby laser off high energy electron beams, to measure hadronic gp cross-sections in a bubble chamber. Subsequently, more experimental work on cross-sections, energies, and polarizations of Compton backscattered photon beams was conducted at Brookhaven National Laboratory’s Laser Electron Gamma Source (LEGS) Facility [28,29]. More recently, Compton backscattered photon beams have been used to measure the polarization of electron beams. Early theoretical work on this application was provided by Baier and Khoze [30] and by Prescott [31]. Gustavson et al. [32], Knudsen et al. [33], and Bardin et al. [34,35] have utilized this concept for experiments. Finally, there has been a number of other applications of Compton backscattered photons to nuclear physics [36].
3. Physics issues Having a t factory that produces \109 polarized t pairs per year would be extremely useful for exploring a variety of crucial physics issues. Some are enumerated as follows: (1) Improve substantially the nt mass limit by studying high multiplicity t decays, such as t -Kþ K p nt : (2) Search for rare and forbidden t decays. (3) Search for CP violation in the lepton sector of the Standard Model. The Standard Model assumes no CP violation in the leptonic sector. By examining t -m þ n% m þ nt and the corresponding tþ decay, Tsai [37,38] has suggested that CP violation could arise due to the exchange of some new particle, such as a spin 1 boson heavier than the Standard Model’s W or a spin 0 charged Higgs boson H : Analyzing the matrix elements for these possibilities, he is able to conclude that only the spin 0 charged Higgs amplitude could contribute to CP violation. To study this effect, he derives an expression for s3y ; where ~ s3 is the polarization vector of the muon in the rest frame of the muon. For t -m þ n% m þ nt ; he obtains s3y ¼
ð~ s~ p3 Þ y ½3M 4E3 þ m2 =MImðC=AÞ ½3M 4E3 2m2 =E3 þ 3m2 =M þ ð~ 8E3 s ~ p3 ÞðM=E3 4 þ 3m2 =ðME3 ÞÞ
ð1Þ
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191
where ~ s is the polarization vector of t ; m; E3 ; and ~ p3 are the mass, energy, and momentum of the muon, M is the mass of the t; and C and A contain the coupling constants of the matrix elements of the spin 0 H ’s and the Standard Model’s W ’s, respectively, contributions to the decay process. Note that A is chosen to be real while C can be complex. By examining the corresponding expression for tþ ; Tsai concludes that the imaginary part of C leads to CP violation. Certainly, these ideas could be tested in BACKGAMMON. (4) Study the Lorentz structure of the charged current in t decay. Pich [39] discusses the process l -nl l0 n% l0 ; where the lepton pair (l; l0 ) may be (t; m), (t; e), or (m; e). As others have done [40–47]. Pich uses the most general local, derivative-free, lepton number conserving, four-lepton interaction Hamiltonian, consistent with locality and Lorentz invariance. For an initial lepton polarization Pl ; he writes the final charged lepton distribution in the decaying lepton rest frame as follows [41,42]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 G ml k 4 2 ¼ 3 G2l0 l x2 x20 xð1 xÞ þ rð4x2 3x x20 Þ þ Zx0 ð1 xÞ dx d cos y 2p 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Pl x x2 x20 cos y 1 x þ dð4x 4 þ 1 x20 Þ ð2Þ 3 3 where the primes refer to the final charged lepton, Gl0 l is the overall Hamiltonian factor, y is the angle between the l spin and the final charged lepton momentum, k ðm2l þ m2l0 Þ=2ml is the maximum l0 energy for massless neutrinos, x El0 =k is the reduced energy and x0 ml0 =k: For unpolarized l’s, the distribution is characterized by the Michel parameter [40] r and the low-energy parameter Z: However, if the initial lepton polarization is known, x and d can be measured. Moreover, if one measures the final charged lepton polarization as well, 5 additional independent parameters x0 ; x00 ; Z00 ; a0 ; b0 [48] can be determined. Thus, BACKGAMMON could help in the determination of these additional parameters.
4. Polarized s’s from BACKGAMMON For our purposes, the most instructive discussion of the polarization of Compton backscattered photons is that due to Babusci et al. [25]. They calculate the azimuthal f-averaged values of the Stokes parameters for backscattered photons in the case of an unpolarized electron beam, giving in the electron rest frame (ERF) ð1 þ jcos yjÞ2 2/A0 S cos y n n0 f i /s2 S ¼ s2 þ /A0 S n0 n
/sf1 S7si1
/sf3 S ¼ si3
ð1 þ jcos yjÞ2 2/A0 S
ð3Þ
where y is the scattering angle of the photon. In the notation of Babusci et al., the first and third Stokes parameters determine the amount of two linear polarizations which make an angle of p=4 with each other while the second parameter determines the circular polarization. A Stokes parameter s ¼ 71 (0) corresponds to completely polarized (unpolarized) photons. Also, /A0 S ¼
n n0 þ sin2 y n0 n
ð4Þ
192
S.K. Mtingwa, M. Strikman / Nuclear Instruments and Methods in Physics Research A 472 (2001) 189–194 Table 1 Backscattered photon Stokes parameters in ERF vs. Ee Ee (GeV)
sf2 (circular)
sf3 (linear)
2.8 (LEGS) 250 (JLC, TESLA) 350 (NLC in USA) 500 (NLC in USA) 1500 (CLIC)
1:0 1:0 1:0 1:0 1:0
0.98 0.11 0.08 0.05 0.02
with n (n0 ) being the magnitude of the initial (final) photon momentum divided by the mass of the electron and i and f refer to the initial and final photons’ Stokes parameters. Backscattered photons correspond to scattering angle y ¼ p: Examining the Stokes parameters in the electron rest frame for a linearly polarized initial laser pulse (si3 ¼ 1) on an unpolarized electron beam, one finds that as one goes from a few GeV to hundreds of GeV for the initial electron beam energy Ee in the lab frame, the backscattered photons in the electron rest frame become more poorly polarized. On the other hand, a circularly polarized initial laser pulse on an unpolarized electron beam yields completely circularly polarized backscattered photons. In Table 1, we give specific values of the ERF Stokes parameters for ultraviolet photons from a frequency quadrupled Nd– YAG laser with l ¼ 266 nm ðolaser ¼ 4:66 eVÞ in the lab frame and the following lab frame electron beam energies: 2:8 GeV (LEGS) [28,29], 250 GeV (Japan Linear Collider (JLC) [49] and TESLA [50]), 350 and 500 GeV (NLC in the United States) [51], and 1500 GeV (CLIC) [52]. In the Babusci et al. formulae, it is useful to note, in the electron rest frame, that the ratio of the initial laser photon energy (o) to the backscattered photon energy (o0 )is given by o n 2o ¼ ¼1þ o0 n0 m
ð5Þ
where o¼2
E e olaser m
ð6Þ
with m being the electron mass. As one boosts from the electron rest frame to the laboratory frame, one obtains similar results for the Stokes parameters. Thus, by scattering circularly polarized laser pulses off unpolarized ultra high energy electron beams from the NLC, BACKGAMMON could produce highly polarized photons of 250 GeV or higher for photoproducing \109 polarized t lepton pairs per year in a manner similar to that proposed by Tsai for lighter leptons [22].
5. Conclusion We have examined an important feature of BACKGAMMON, namely that by Compton backscattering circularly polarized laser pulses on even an unpolarized ultra high energy electron beam at the Next Linear Collider, completely circularly polarized hot photons at roughly the electron energy will be produced which can be used to photoproduce \109 polarized t lepton pairs per year. This t factory should be an excellent laboratory for studying a variety of important physics issues, such as mass limits on nt ; rare and forbidden t decays, CP violation in the leptonic sector of the Standard Model, and the Lorentz structure of the charged current in t decay.
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Acknowledgements One of the authors (SKM) would like to thank A. Msezane and C. Handy for the hospitality extended to him while visiting the Center for Theoretical Studies of Physical Systems at Clark Atlanta University, where part of this work was completed.
References [1] S. Mtingwa, M. Strikman, Phys. Rev. Lett. 64 (1990) 1522. [2] S. Mtingwa, M. Strikman, in: D. Cline, A. Fridman (Eds.), CP Violation and Beauty Factories and Related Issues in Physics, Ann. New York Acad. Sci. Vol. 619 (1991) 211. [3] S. Mtingwa, M. Strikman, in: D. Cline (Eds.), Rare and Exclusive B & K Decays and Novel Flavor Factories, Conference Proceedings, Vol. 261, American Institute of Physics Publication, AIP, New York, 1992, p. 236. [4] S. Mtingwa, M. Strikman, in: D. Axen, D. Bryman, M. Comyn (Eds.), The Vancouver Meeting, Particles & Fields ’91, World Scientific, Singapore, 1992, p. 1106. [5] S. Mtingwa, M. Strikman, Nucl. Instr. and Meth. A 455 (2000) 50. [6] J.J. Aubert, et al., Nucl. Phys. B 213 (1983) 1. [7] J.J. Aubert, et al., Phys. Lett. B 89 (1980) 267. [8] J. Kirkby, in: Workshop on the Tau=Charm Factory, Conference Proceedings, Vol. 349, American Institute of Physics Publication, AIP, New York, 1996, p. 11. [9] J. Kirkby, A t-charm Factory at CERN, CERN-EP=87-210, 1987. [10] J. Kirkby, in: Procceedings of the International School of Physics with Low-Energy Antiprotons: Spectroscopy of Light and Heavy Quarks, Erice, Sicily, Plenum Press, New York, 1989, p. 401. [11] J.M. Jowett, Initial design of a t-charm Factory at CERN, CERN LEP-TH=87-56, 1987. [12] J.M. Jowett, The t-charm factory storage ring, CERN-LEP=88-22, 1988. [13] J.M. Jowett, in: Proceedings of the First European Particle Accelerator Conference, Rome, World Scientific, Singapore, 1989, p. 368. [14] R. Milburn, Phys. Rev. Lett. 10 (1963) 75. [15] F. Arutyunian, V. Tumanian, Phys. Lett. 4 (1963) 176. [16] F. Arutyunyan, V. Tumanyan, Sov. Phys. Usp. 83 (1964) 339. [17] F.R. Arutyunyan, I.I. Gol’dman, V.A. Tumanyan, Sov. Phys. JETP 18 (1964) 218. [18] I.F. Ginzburg, G.L. Kotkin, V.G. Serbo, V.I. Telnov, Nucl. Instr. and Meth. 205 (1983) 47. [19] I.F. Ginzburg, G.L. Kotkin, S.L. Panfil, V.G. Serbo, V.I. Telnov, Nucl. Instr. and Meth. 219 (1984) 5. [20] C. Akerlof, Preprint UM HE 81-59, University of Michigan, 1981. [21] J.C. Sens, CERN-EP=88-99, 1988. [22] Y.S. Tsai, Phys. Rev. D 48 (1993) 96. [23] G.L. Kotkin, H. Perlt, V.G. Serbo, Nucl. Instr. and Meth. A 404 (1998) 430. [24] G.L. Kotkin, S.I. Polityko, V.G. Serbo, Nucl. Instr. and Meth. A 405 (1998) 30. [25] D. Babusci, G. Giordano, G. Matone, Phys. Lett. B 355 (1995) 1. [26] V.G. Serbo, Nucl. Instr. and Meth A 472 (2001) 260, these proceedings. [27] J. Ballam, et al., Phys. Rev. Lett. 23 (1969) 498. [28] A.M. Sandorfi, et al., IEEE Trans. Nucl. Sci. NS-30 (1983) 3083. [29] C.E. Thorn, et al., Nucl. Instr. and Meth. A 285 (1989) 447. [30] V.N. Baier, V.A. Khoze, Sov. J. Nucl. Phys. 9 (1969) 238. [31] C. Prescott, SLAC Internal Report, SLAC-TN-73-1, 1973. [32] D.B. Gustavson, et al., Nucl. Instr. and Meth. 165 (1979) 177. [33] L. Knudsen, et al., Phys. Lett. B 270 (1991) 97. [34] G. Bardin et al, SACLAY Report, DAPNIA-SPhN-96-14, 1996. [35] G. Bardin, C. Cavata, J.-P. Jorda, Compton polarimeter studies for TESLA, SACLAY Internal Report, 1997. [36] A. D’Angelo, in: Proceedings of the Symposium, New Visions in Laser-Beam Interactions: Fundamental Problems and Applications of Laser-Compton Scattering, Tokyo Metropolitan University and High Energy Accelerator Research Organization (KEK), Tokyo, Japan, 1999. [37] Y.S. Tsai, Phys. Rev. D 51 (1995) 3172.
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[38] Y.S. Tsai, in: Workshop on the Tau=Charm Factory, Conference Proceedings, Vol. 349, American Institute of Physics Publication, AIP, New York, 1996, p. 104. [39] A. Pich, in: Workshop on the Tau=Charm Factory, Conference Proceedings, Vol. 349, American Institute of Physics Publication, AIP, New York, 1996, p. 45. [40] L. Michel, Proc. Phys. Soc. A 63 (1950) 514, 1371. [41] C. Bouchiat, L. Michel, Phys. Rev. 106 (1957) 170. [42] T. Kinoshita, A. Sirlin, Phys. Rev. 107 (1957) 593; T. Kinoshita, A. Sirlin, Phys. Rev. 108 (1957) 844. [43] F. Scheck, Leptons, Hadrons and Nuclei, North-Holland, Amsterdam, 1983. [44] F. Scheck, Phys. Rep. 44 (1978) 187. [45] W. Fetscher, H.-J. Gerber, K.F. Johnson, Phys. Lett. B 173 (1986) 102. [46] W. Fetscher, H.-J. Gerber, in: P. Langacker (Ed.), Precision Tests of the Standard Electroweak Model, Advanced Series on Directions in High Energy Physics, Vol. 14, World Scientific, Singapore, 1995, p. 657. [47] A. Pich, J.P. Silva, Phys. Rev. D 52 (1995) 4006. [48] M. Aguilar-Ben!itez, et al., Phys. Rev. D 50 (1994) 1173. [49] T. Hirose, in: Proceedings of the Symposium, New Visions in Laser-Beam Interactions: Fundamental Problems and Applications of Laser-Compton Scattering, Tokyo Metropolitan University and High Energy Accelerator Research Organization (KEK), Tokyo, Japan, 1999. [50] O. Napoly, News from TESLA, in: Proceedings of the Sixth Workshop on Second ECFA=DESY Study on Physics and Detectors for a Linear Electron–Positron Collider, University of Padova, Padova, Italy, 2000. [51] J. Gronberg, Photon collider at NLC, in: Proceedings of the International Workshop on High Energy Photon Colliders, DESY, Hamburg, Germany, 2000. [52] H. Burkhardt, Nucl. Instr. and Meth. A 472 (2001) 67, these proceedings.
A polarized t factory at the Next Linear Collider Sekazi K. Mtingwaa,b,*, Mark Strikmanc a
Department of Physics, North Carolina A&T State University, 1601 E. Market Street, Greensboro, NC 27411, USA b Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA 30314, USA c Department of Physics, Pennsylvania State University, University Park, PA 16802, USA
Abstract We discuss the importance of using the Next Linear Collider to study polarization effects in t lepton decay. Our proposal is to Compton backscatter circularly polarized low energy laser pulses off an unpolarized electron beam of energy in the hundreds of GeV. The resulting backscattered photons also will have energies in the hundreds of GeV range and will be circularly polarized as well. Upon striking a fixed target, these polarized hot photons can produce copious polarized t lepton pairs, which could be used to study t decay, CP violation, and a number of other physics issues. r 2001 Elsevier Science B.V. All rights reserved. PACS: 13.60.r; 13.10.+q; 13.88.+e; 14.60.Fg
1. Introduction For some time, there has been an interest in constructing an electron–positron collider of 250 GeV or higher energy per beam. This has become known as the Next Linear Collider (NLC). One of the main reasons for building the NLC is to search for the elusive Higgs boson, whose mass is possibly in the range of a few hundred GeV. Notwithstanding the lofty goal of discovering the Higgs, there is also the possibility of operating a heavy flavor factory at the NLC. Thus, the NLC could be a dual purpose accelerator complex. Previously, we proposed a scheme for photoproducing large quantities of bottom quark pairs and t lepton pairs in a scheme dubbed BACKGAMMON, for BACKscattered GAMMas On Nucleons. The idea has been described at length in previous papers [1–5], and it involves Compton backscattering low energy laser pulses off a hundreds of GeV electron beam from the NLC to produce hot photons of energy roughly equal to that of the initial electron beam. For bottom quark pair production, it would be advantageous for the hot photon beam to have an energy Eg \250 GeV in order to maximize the bottom quark pair production cross-section [6,7]. However, for t pair production, the cross-section varies only as log Eg ; thus, one only needs Eg \100 GeV: There is considerable interest in studying polarized t decay; however, standard t-charm factories (tcF) produce roughly 107 unpolarized t’s per year [8–13]. On the other hand, in BACKGAMMON, there is the possibility of producing \109 polarized t’s per year, with the t’s flying at relatively large angles Bmt =Eg *Corresponding author. Tel.: +1-336-334-7646; fax: +1-336-334-7283. E-mail address: [email protected] (S.K. Mtingwa). 0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 1 8 1 - 0
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and decay lengths B1 cm; thus leaving a rather clean signature in the detector. In a previous paper [3], we noted that, by using polarized electrons and polarized laser pulses, one could produce polarized hot photons, which could then photoproduce polarized t’s on nuclear targets. However, in this paper, we point out that the initial electron beam need not be polarized. As long as one uses the backscattered photons, the circular polarization of the initial laser pulse will be preserved.
2. Compton backscattering The theory and applications of Compton backscattering are not new, but they have yet to realize their full potential. The original theory was derived separately by Milburn [14] and Arutyunian et al. [15–17]. The application of Compton backscattering for the construction of photon–photon colliders was proposed by Ginzburg, et al. [18,19], Akerlof [20], and Sens [21], with detailed polariation effects being discussed by Ginzburg et al. [19]. Other discussions of polarization effects in Compton backscattering have been provided by Arutyunyan et al. [16,17], Tsai [22], Kotkin et al. [23,24], Babusci et al. [25], and Serbo [26]. The detailed theory of Compton backscattering, incorporating the accelerator lattice functions of the initial electron beam, was derived by the authors [1]. As for the first application of a Compton backscattered photon beam in an actual experiment, Ballam et al. [27] used highly monochromatic Compton backscattered photon beams of energies 1.44, 2.84, and 4:66 GeV; which were produced at the Stanford Linear Accelerator Center by scattering pulses from a ruby laser off high energy electron beams, to measure hadronic gp cross-sections in a bubble chamber. Subsequently, more experimental work on cross-sections, energies, and polarizations of Compton backscattered photon beams was conducted at Brookhaven National Laboratory’s Laser Electron Gamma Source (LEGS) Facility [28,29]. More recently, Compton backscattered photon beams have been used to measure the polarization of electron beams. Early theoretical work on this application was provided by Baier and Khoze [30] and by Prescott [31]. Gustavson et al. [32], Knudsen et al. [33], and Bardin et al. [34,35] have utilized this concept for experiments. Finally, there has been a number of other applications of Compton backscattered photons to nuclear physics [36].
3. Physics issues Having a t factory that produces \109 polarized t pairs per year would be extremely useful for exploring a variety of crucial physics issues. Some are enumerated as follows: (1) Improve substantially the nt mass limit by studying high multiplicity t decays, such as t -Kþ K p nt : (2) Search for rare and forbidden t decays. (3) Search for CP violation in the lepton sector of the Standard Model. The Standard Model assumes no CP violation in the leptonic sector. By examining t -m þ n% m þ nt and the corresponding tþ decay, Tsai [37,38] has suggested that CP violation could arise due to the exchange of some new particle, such as a spin 1 boson heavier than the Standard Model’s W or a spin 0 charged Higgs boson H : Analyzing the matrix elements for these possibilities, he is able to conclude that only the spin 0 charged Higgs amplitude could contribute to CP violation. To study this effect, he derives an expression for s3y ; where ~ s3 is the polarization vector of the muon in the rest frame of the muon. For t -m þ n% m þ nt ; he obtains s3y ¼
ð~ s~ p3 Þ y ½3M 4E3 þ m2 =MImðC=AÞ ½3M 4E3 2m2 =E3 þ 3m2 =M þ ð~ 8E3 s ~ p3 ÞðM=E3 4 þ 3m2 =ðME3 ÞÞ
ð1Þ
S.K. Mtingwa, M. Strikman / Nuclear Instruments and Methods in Physics Research A 472 (2001) 189–194
191
where ~ s is the polarization vector of t ; m; E3 ; and ~ p3 are the mass, energy, and momentum of the muon, M is the mass of the t; and C and A contain the coupling constants of the matrix elements of the spin 0 H ’s and the Standard Model’s W ’s, respectively, contributions to the decay process. Note that A is chosen to be real while C can be complex. By examining the corresponding expression for tþ ; Tsai concludes that the imaginary part of C leads to CP violation. Certainly, these ideas could be tested in BACKGAMMON. (4) Study the Lorentz structure of the charged current in t decay. Pich [39] discusses the process l -nl l0 n% l0 ; where the lepton pair (l; l0 ) may be (t; m), (t; e), or (m; e). As others have done [40–47]. Pich uses the most general local, derivative-free, lepton number conserving, four-lepton interaction Hamiltonian, consistent with locality and Lorentz invariance. For an initial lepton polarization Pl ; he writes the final charged lepton distribution in the decaying lepton rest frame as follows [41,42]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 G ml k 4 2 ¼ 3 G2l0 l x2 x20 xð1 xÞ þ rð4x2 3x x20 Þ þ Zx0 ð1 xÞ dx d cos y 2p 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Pl x x2 x20 cos y 1 x þ dð4x 4 þ 1 x20 Þ ð2Þ 3 3 where the primes refer to the final charged lepton, Gl0 l is the overall Hamiltonian factor, y is the angle between the l spin and the final charged lepton momentum, k ðm2l þ m2l0 Þ=2ml is the maximum l0 energy for massless neutrinos, x El0 =k is the reduced energy and x0 ml0 =k: For unpolarized l’s, the distribution is characterized by the Michel parameter [40] r and the low-energy parameter Z: However, if the initial lepton polarization is known, x and d can be measured. Moreover, if one measures the final charged lepton polarization as well, 5 additional independent parameters x0 ; x00 ; Z00 ; a0 ; b0 [48] can be determined. Thus, BACKGAMMON could help in the determination of these additional parameters.
4. Polarized s’s from BACKGAMMON For our purposes, the most instructive discussion of the polarization of Compton backscattered photons is that due to Babusci et al. [25]. They calculate the azimuthal f-averaged values of the Stokes parameters for backscattered photons in the case of an unpolarized electron beam, giving in the electron rest frame (ERF) ð1 þ jcos yjÞ2 2/A0 S cos y n n0 f i /s2 S ¼ s2 þ /A0 S n0 n
/sf1 S7si1
/sf3 S ¼ si3
ð1 þ jcos yjÞ2 2/A0 S
ð3Þ
where y is the scattering angle of the photon. In the notation of Babusci et al., the first and third Stokes parameters determine the amount of two linear polarizations which make an angle of p=4 with each other while the second parameter determines the circular polarization. A Stokes parameter s ¼ 71 (0) corresponds to completely polarized (unpolarized) photons. Also, /A0 S ¼
n n0 þ sin2 y n0 n
ð4Þ
192
S.K. Mtingwa, M. Strikman / Nuclear Instruments and Methods in Physics Research A 472 (2001) 189–194 Table 1 Backscattered photon Stokes parameters in ERF vs. Ee Ee (GeV)
sf2 (circular)
sf3 (linear)
2.8 (LEGS) 250 (JLC, TESLA) 350 (NLC in USA) 500 (NLC in USA) 1500 (CLIC)
1:0 1:0 1:0 1:0 1:0
0.98 0.11 0.08 0.05 0.02
with n (n0 ) being the magnitude of the initial (final) photon momentum divided by the mass of the electron and i and f refer to the initial and final photons’ Stokes parameters. Backscattered photons correspond to scattering angle y ¼ p: Examining the Stokes parameters in the electron rest frame for a linearly polarized initial laser pulse (si3 ¼ 1) on an unpolarized electron beam, one finds that as one goes from a few GeV to hundreds of GeV for the initial electron beam energy Ee in the lab frame, the backscattered photons in the electron rest frame become more poorly polarized. On the other hand, a circularly polarized initial laser pulse on an unpolarized electron beam yields completely circularly polarized backscattered photons. In Table 1, we give specific values of the ERF Stokes parameters for ultraviolet photons from a frequency quadrupled Nd– YAG laser with l ¼ 266 nm ðolaser ¼ 4:66 eVÞ in the lab frame and the following lab frame electron beam energies: 2:8 GeV (LEGS) [28,29], 250 GeV (Japan Linear Collider (JLC) [49] and TESLA [50]), 350 and 500 GeV (NLC in the United States) [51], and 1500 GeV (CLIC) [52]. In the Babusci et al. formulae, it is useful to note, in the electron rest frame, that the ratio of the initial laser photon energy (o) to the backscattered photon energy (o0 )is given by o n 2o ¼ ¼1þ o0 n0 m
ð5Þ
where o¼2
E e olaser m
ð6Þ
with m being the electron mass. As one boosts from the electron rest frame to the laboratory frame, one obtains similar results for the Stokes parameters. Thus, by scattering circularly polarized laser pulses off unpolarized ultra high energy electron beams from the NLC, BACKGAMMON could produce highly polarized photons of 250 GeV or higher for photoproducing \109 polarized t lepton pairs per year in a manner similar to that proposed by Tsai for lighter leptons [22].
5. Conclusion We have examined an important feature of BACKGAMMON, namely that by Compton backscattering circularly polarized laser pulses on even an unpolarized ultra high energy electron beam at the Next Linear Collider, completely circularly polarized hot photons at roughly the electron energy will be produced which can be used to photoproduce \109 polarized t lepton pairs per year. This t factory should be an excellent laboratory for studying a variety of important physics issues, such as mass limits on nt ; rare and forbidden t decays, CP violation in the leptonic sector of the Standard Model, and the Lorentz structure of the charged current in t decay.
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Acknowledgements One of the authors (SKM) would like to thank A. Msezane and C. Handy for the hospitality extended to him while visiting the Center for Theoretical Studies of Physical Systems at Clark Atlanta University, where part of this work was completed.
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