Transverse beam polarization and CP-violating triple-gauge-boson couplings in e+e- ->gamma Z

We show that an anomalous CP-violating gamma-gamma-Z vertex gives rise to a novel asymmetry with transversely polarized electron and positron beams in the process e+e- ->gamma Z. This asymmetry, which is odd under naive time reversal, is proportional to the real part of the gamma-gamma-Z CP-violating coupling. This is in contrast to the simple forward-backward asymmetry of the gamma (or Z) with unpolarized or longitudinally polarized beams studied earlier, which is even under naive time reversal, and is proportional to the imaginary part. We estimate the sensitivity of future experiments to the determination of CP-odd gamma-gamma-Z and gamma-Z-Z couplings using these asymmetries and transversely polarized beams.


Introduction
A future linear e + e − collider operating at a centre-of-mass (cm) energy of several hundred GeV would contribute greatly to a precise determination of the parameters of known particles and their interactions, as well as to the constraining of new physics. Longitudinal polarization of the e + and e − beams, which is expected to be feasible at such colliders, would be helpful in reducing background as well as enhancing the sensitivity. It has been realized that spin rotators can be used to convert the longitudinal polarizations of the beams to transverse polarizations. The question has often been asked if such transverse polarization can be put to use to shed light on interactions or parameter ranges not accessible with longitudinal polarization, or to enhance their sensitivity. This question has not been discussed exhaustively in the current context as yet, though there have been some recent studies [1]- [7].
The role of transverse polarization in the context of CP violation has been studied in [6]- [10]. Since transverse beam polarization provides an additional reference coordinate axis in addition to the e + e − beam direction, there is the possibility of studying the azimuthal distribution of a single final-state particle. This has the advantage that the polarization of the produced particle, and hence its decay distribution, need not be measured. In [7] it was pointed out that an azimuthal distribution of a final-state particle A in a semiinclusive process e + e − → A + X arising from the interference between a standard model (SM) contribution and a new-physics contribution arising at some high scale cannot contain a CP-violating part if the new-physics contribution arises from chirality-conserving vector (V) or axial-vector (A) type of interaction, neglecting the electron mass. This result, with the SM contribution restricted to a virtual photon exchange, can be deduced from the work of Dass and Ross [11]. In [7], this was generalized to include virtual Z exchange as well. On the other hand, chirality-violating scalar (S) and tensor (T) interactions can give rise to a simple CP-odd azimuthal asymmetry, as for example, in e + e − → tt [7].
The above results were obtained with the condition that the SM contribution arises only through s-channel exchange of virtual photon and Z. The possibility of t-and u-channel exchange of an electron was not considered. Moreover, since the new physics is supposed to arise at a high scale, no t-or u-channel exchange of new particles was included. The results may get somewhat modified if these effects are taken into account. In particular, the t-or u-channel exchange would introduce an extra dependence on the scattering (polar) angle θ. In a process where A is its own conjugate, there may be a consequent forward-backward asymmetry corresponding to θ → π − θ, which is CP odd. It is well-known that such an asymmetry could arise without transverse polarization (see, for example, [12,13,14]). However, such a forward-backward asymmetry, in the absence of transverse polarization, is even under naive time reversal T ( i.e. , reversal of particle spins and momenta). Hence the CPT theorem implies that the contribution comes only from an absorptive part in one of the interfering amplitudes (see, for example, [15]). Thus, such a symmetry is only sensitive to the imaginary parts of the new-physics couplings.
In this paper we investigate the interesting possibility that if there is transverse polar-ization, a T-odd but CP-even azimuthal asymmetry can be combined with the T-even but CP-odd forward-backward asymmetry to give an asymmetry which is both CP odd as well as T odd. In this case, the CPT theorem dictates that such an asymmetry meaure the real part of the new-physics couplings. The process we have chosen is e + e − → γZ, where the final-state particles are both self-conjugate * . This process occurs at tree level in SM. A CP-violating contribution can arise if anomalous CP-violating γγZ and γZZ couplings are present. The interference of the contributions from these anomalous couplings with the SM contribution gives rise to the expected polar-angle forward-backward asymmetry, as well as new combinations of polar and azimuthal asymmetries. In particular, there is a CP-odd, T-odd asymmetry, which is proportional to the real part of the γγZ coupling. This real part cannot be probed without transverse polarization † . There is an accidental cancellation of a similar contribution arising from real part of the γZZ coupling.
2 The process e + e − → γZ We now describe the details of our work. The process considered is The most general effective CP -violating Lagrangian for γγZ and γZZ interactions, consistent with Lorentz invariance and electromagnetic gauge invariance, and retaining terms upto dimension 6, can be written as where c W = cos θ W and s W = sin θ W and θ W is the weak mixing angle. Terms involving divergences of the vector fields have been dropped from the Lagrangian as they would not contribute when the corresponding particle is on the mass shell, or is virtual, but coupled to a conserved fermionic current. Since we will neglect the electron mass, the corresponding current can be assumed to be conserved. We have not tried to impose full SU(2) L × U(1) invariance, but only electromagnetic gauge invariance, as this is more general.
The SM diagrams contributing to the process (1) are shown in Figs. 1 (a) and 1 (b), which correspond to t-and a u-channel electron exchange, while the extra piece in the Lagrangian (2) introduces two s-channel diagrams with γ-and Z-exchange respectively, shown in Figs. 1 (c) and 1 (d). The corresponding matrix element is then given by where (4) We have used q = k 1 + k 2 , and the tensors V (1) and V (2) corresponding to the three-vector vertices are given by In the above, the vector and axial vector Z couplings of the electron are given by For compactness, we introduce the notation : Using eqns. (3 -7), we obtain the differential cross section for the process (1) to be where θ is the angle between photon and the e − directions, and φ is the azimuthal angle of the photon, with e − direction chosen as the z axis and the direction of its transverse polarization chosen as the x axis. The e + polarization direction is chosen parallel to the e − polarization direction. P e and P e are respectively the degrees of polarization of the e − and e + . We have kept only terms of leading order in the anomalous couplings, since they are expected to be small. The above expression may be obtained either by using standard trace techniques for Dirac spinors with a transverse spin four-vector, or by first calculating helicity amplitudes and then writing transverse polarization states in terms of helicity states [16]. We will assume a cut-off θ 0 on the polar angle θ of the photon in the forward and backward directions. This cut-off is needed to stay away from the beam pipe. It can further be chosen to optimize the sensitivity. The total cross section corresponding to the cut θ 0 < θ < π − θ 0 can then be easily obtained by integrating the differential cross section above.
It is interesting to note that the contribution of the interference between the SM amplitude and the anomalous amplitude vanishes for s = m 2 Z . The reason for this is that for s = m 2 Z the photon in the final state is produced with zero energy and momentum. As can be seen from eq. (5), the anomalous couplings vanish for k 1 = 0, leading to a vanishing intereference term.
In order to understand the CP properties of various terms in the differential cross section, we note the following relations: where P = 1 2 ( p − − p + ). Observing that the vector P is C and P odd, that the photon momentum k 1 is C even but P odd, and that the spin vectors s ± are P even, and go into each other under C, we can immediately check that only the left-hand side (lhs) of eq. (9) is CP odd, while the lhs of eqs. (10) and (11) are CP even. Of all the above, only the lhs of (10) is odd under naive time reversal T.
We now define the following CP-odd asymmetries, which combine a forward-backward asymmetry with an appropriate asymmetry in φ, so as to isolate appropriate anomalous couplings: with These are easily evaluated to be In the above equations, we have defined .
(20) We now make some observations on the above expressions which justify the choice of our asymmetries and highlight the novel features of our work. It can be seen that A 1 (θ 0 ) is proportional to Reλ 2 , and the other two asymmetries depend on Imλ 1 and Imλ 2 . Moreover, the latter two measured simultaneously can be used to get limits on the two couplings Imλ 1 and Imλ 2 . It is interesting that A 1 does not depend on λ 1 , which is the result of an accidental cancellation. This would not be the case, for example, if the Z in the s-channel exchange in Fig. 1(a) were different from the Z produced in the final state, so that their couplings to the the electron were different.

Numerical Results
We now present our numerical results. The cross section with a cut-off θ 0 on θ is plotted in Fig. 2 as a function of θ 0 . Figs. 3, 4 show the asymmetries as a function of the cut-off when the values of the anomalous couplings are taken to be nonzero one at a time. All the asymmetries vanish not only for θ 0 = 0, by definition, but also for θ 0 = 90 • , because they are proportional to cos θ 0 . They peak at around 45 • . We have calculated 90% CL limits that can be obtained with a linear collider with √ s = 500 GeV, Ldt = 500 fb −1 , P e = 0.8, and P e = 0.6 making use of the asymmetries A i .
The limiting value λ lim ( i.e. the respective real or imaginary part of the coupling) is related to the value A of the asymmetry for unit value of the coupling constant by where N SM is the number of SM events. A 1 depends on Re λ 2 alone, and can therefore place an independent limit on Re λ 2 . We emphasize once again that information on Re λ 2 cannot be obtained without transverse polarization. Fig. 5 shows the 90% CL limit on Re λ 2 as a function of the cut-off. The asymmetries A 2 and A 3 depend on both Im λ 1 and Im λ 2 . Fig. 6 shows the 90% CL limits on Im λ i taken to be nonzero one at a time, using the asymmetries A 2 and A 3 . It can be seen from these figures that the limits are relatively insensitive to the cut-off at least for small values Coupling Individual limit from Simultaneous limits 1.38 × 10 −2 Im λ 1 6.22 × 10 −3 3.82 × 10 −3 7.05 × 10 −3 Im λ 2 9.10 × 10 −2 3.01 × 10 −2 6.74 × 10 −2 Table 1: 90 % CL limits on the couplings from asymmetries A i for a cut-off angle of 26 • , √ s = 500 GeV, and integrated luminosity of 500 fb −1 . The electron and positron transverse polarizations are assumed to be respectively 0.8 and 0.6.
of the cut-off. We find that the best limits are obtained for θ 0 = 26 • , though any nearby value of θ 0 would give very similar results. These correspond to Re λ 2 = 0.0138 (from A 1 ), Im λ 1 = 0.00622 (from A 2 ), Im λ 1 = 0.00382 (from A 3 ), Im λ 2 = 0.0910 (from A 2 ), and Im λ 2 = 0.0301 (from A 3 ). As stated earlier, because of g V being numerically small, the limits on Im λ 2 , which appears in the expressions for the asymmetries multiplied by g V , are worse than those on Im λ 1 . However, it can also be seen that A 3 fares better than A 2 so far as Im λ 2 is concerned.
Finally, we have also evaluated the simultaneous 90% CL limits that can be obtained on Im λ 1 and Im λ 2 by measurement of A 2 and A 3 . For this we have chosen θ 0 = 26 • . The corresponding contour for allowed values of the couplings for a null result of the measurement of A 2 and A 3 is shown in Fig. 7. This contour is obtained by equating the asymmetry obtained simultaneously from nonzero Im λ 1 as well as nonzero Im λ 2 to 2.15/ √ N SM . It can be seen that the simultaneous limits that can be obtained are weaker than individual limits, with numerical values Im λ 1 = 0.00705 and Im λ 2 = 0.0674. The best limits are summarized in Table 1.

Conclusions
In conclusion, we have studied a novel CP-violating asymmetry (A 1 ) in e + e − → γZwith anomalous neutral gauge boson couplings, which is special to a neutral final state, the observation of which needs both electron and positron transverse polarizations. This is of special interest in the context of the negative result stated in [7], that the observation of CP violation in a two-particle final state, without measuring the polarization of the finalstate particles, is not possible with transversely polarized beams, unless there are chiralityviolating couplings of the electron and positron. That result depended on an analysis where t-and u-channel particle exchanges were not taken into account.
Forward-backward asymmetry of a neutral particle with unpolarized or longitudinally polarized beams as a signal of CP violation has been studied before. However, the CPT theorem implies that in such a case the asymmetry is proportional to the absorptive part of the amplitude. The asymmetry A 1 that we study in the presence of transverse polarizations includes also an azimuthal angle asymmetry, which makes it odd under naive time reversal.
It is thus proportional to the real part of the anomalous coupling. This real part cannot be studied without transverse polarization.
We have also made a numerical study of the limits on various couplings that could be obtained at a future linear collider with √ s = 500 GeV and an integrated luminosity of 500 fb −1 assuming realistic transverse polarizations of 80% and 60% respectively for e − and e + , respectively. The best limits are summarized in Table 1. We thus see that transverse polarization would provide a sensitive test of anomalous couplings, particularly, Re λ 2 .