The electric dipole moment of the tau

Volume 252, number 1

PHYSICS LETTERS B

6 December 1990

The electric dipole moment of the tau F. d e l A g u i l a Departamento de Fisica Te6rica, UniversidadAutonoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain and Departamento de Fisica Terrica y del Cosmos, Universidadde Granada, E- 18071 Granada, Spain and Marc Sher Department of Physics, Collegeof William and Mary, Williamsburg, VA 23185, USA Received 1 September 1990

Although stringent limits exist on the electric dipole moments of the neutron, electron and muon, there is very little information on the electric dipole moment of the tau. Such a dipole moment would be observable in the angular distribution of the tau pairs in electron-positron annihilation. We calculate the effects of an electric dipole moment on this angular distribution, and show that the current upper bound on the electric dipole moment is 1.4 × 1016e cm. This bound could be improved significantly at a tau factory. The electric dipole moment of neutrinos is also discussed.

O u r understanding o f the nature a n d origin o f CP violation would be greatly enhanced by detection o f CP violation in some system other than the K system. F o r this reason, extensive searches have been m a d e for the electric dipole m o m e n t ( E D M ) o f the neutron [ 1 ], o f the electron [ 2 ] a n d o f the m u o n [ 3 ]~. In this note, we discuss the possibility o f searching for the electric dipole m o m e n t o f the tau. An E D M for the z can be introduced by adding a term e F ( q 2) ~au~q"ys~u~AU to the lagrangian; the electric dipole m o m e n t is then eF(O). This term is CP violating, and is extremely small in the s t a n d a r d model. H o w can one measure the E D M o f the z? Unlike measurements o f the neutron, electron and m u o n EDMs, one cannot measure the precession o f the x in an electric field. A similar question arose some years ago in connection with the possible magnetic mom e n t o f the z. M o t i v a t e d by the possible presence o f a x magnetic m o m e n t in composite models, Silverm a n and Shaw [4] showed that the e+e - annihilation cross section into x + x - gave the strongest upper limit on the anomalous magnetic moment, which was an order of magnitude larger than the standard model 1 16

value. D o m o k o s et al. [5 ] later showed that the angular distribution o f the x's offered the possibility o f i m p r o v i n g this b o u n d considerably. See the report o f Barish and Stroynowski [ 6 ] for an extensive review and references. The E D M o f the x can be measured by its effect on the x coupling to the virtual p h o t o n in e+e - annihilation. The matrix element, including Z exchange [but neglecting terms o f O ( m 2 / M 2 z ) ], is given by ie 2 ..1[= ~-~ O~TUu~[yu + F( q 2) au,q~Ts ] v~

ig z 1 + - 4 cosZ0w q2-MZz + i F z M z x ~ o T ~ ( C v - C A 7 5 ) uea~7~(Cv-CA75) v , .

(1)

where Cv = - ½+ 2 sin20w and CA = -- 1. In squaring this matrix element, several i m p o r t a n t features are noticed. First, the magnetic m o m e n t contribution would be found by removing the Y5in the E D M term. In that case, there would be interference between the Yu a n d the F ( q 2) au,q ~ terms, leading to a term linear in F (albeit p r o p o r t i o n a l to m~). In our case, however, the presence o f the Y5 eliminates such interfer-

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Volume 252, number 1

PHYSICS LETTERS B

ence. Second, interference would also exist in the magnetic moment case between the new term and the Z-exchange term. This fact was pointed out by Silverman and Shaw [ 4 ], who noted that bounds on the x magnetic moment could be improved by measuring the angular dependence of the x's in Z decay. In our case, such interference will only be proportional to Cv, which is small; since it is also proportional to rnJ Mz as well as Fz/Mz, the interference should be negligible ~t. Finally, any bound obtained will hold if, as expected, the EDM remains pointlike at x/~. If the interaction contains a form factor, the bound will be weakened. The cross section calculated from eq. ( 1 ), ignoring terms of order m~/s, non-leading electroweak corrections and the EDM/Z-interference term (which is sufficient for our purposes), is given by d~

7~O[2

dcosO-

2s [ I + c o s 2 0 + C 2 c o s O ] (2)

+ ½7to~2F2(q 2) s i n 2 0 ,

where 1

C2 = s i n 2 2 ~ Re

s

s_M2z + iFzMz

"

The most recent data on this angular distribution comes from the CELLO detector at PETRA [7 ]. From their data at x/~=35 GeV (which has the smallest errors), we find that F2s=0.03 + 0.03, giving a bound (ltr) on the EDM of the x of 1.4X 10 -16 e cm. This bound is approximately two orders of magnitude larger than the EDM of the muon. Due to the small interference with the Z, it is not likely that this bound can be improved significantly through studies at the Z peak. O f course, more accurate measurements at TRISTAN energies or at a xfactory could improve the bound significantly. Since the EDM interaction is CP-violating, it is of interest to know if, at a x-factory, the CP violation could be observed directly. One would have to observe interference between T-even and T-odd terms, presumably by examining spin-spin correlations; both initial and final state polarizations would have to be #' Since it is proportional to F, and not F 2, eventually the interference term will dominate. We have checked the term explicitly and found that its contribution will be negligible unless the current bound is improved by several orders of magnitude!

6 December 1990

measured. This would require very high statistics, and it is certain that the asymmetry of eq. (2) would be seen first. We also briefly comment on the EDM of neutrinos. As pointed out by Voloshin, Vysotsky and Okun [ 8 ], it is not necessary for a particle to have mass in order to have a magnetic and/or electric dipole moment; it is only necessary that a right-handed helicity state exists. They point out that one cannot distinguish between the EDM, e, and the magnetic moment, g, of relativistic neutrinos, since the phase of the righthanded state can be rotated so that only (#2+ E2)i/2 appears in the lagrangian. As a result, bounds on the magnetic moment of neutrinos will also give bounds on their EDM. Very stringent bounds on the magnetic moments of the ve and the v, can be obtained from laboratory experiments on ve scattering [ 9 ], and from astrophysical bounds (which arise from not allowing too much energy loss from y*--,vg); see ref. [ 8 ] for a complete list of references. These bounds all give (approximately)/~ < 10 - 1o/za = 2 × 10- 21 e cm (where/za is the Bohr magneton) for the magnetic moments of the electron and muon neutrinos. The identical bound will then apply to the EDM of the ve and the v~. Guidice [ l0 ] has very recently pointed out that these bounds do not apply to a v~ with a mass between 1 and 35 MeV. He argued that such a v,, with f l = 1 0 - 6 fiB, is a viable dark matter candidate, and showed that the strongest bound on the magnetic moment (and thus, the EDM) ofv~ arises from e + e - ~ Tvg, and is given b y # < 8 X 1 0 - 6 f i b ~ 1.6X 10-16 ecm. Such a moment will induce through a loop, an EDM for the tau. Calculation of the loop, involving a Wexchange in the standard model, is straightforward and gives (in a renormalizable gauge)

F~=_Fv,(.128zr~in20 w m,~m,.~ m2 ]

(3)

for the contribution of the EDM of a neutron, v~, to the EDM of the lepton, ~. Because of the smallness of the v~ mass, this is negligible (although the effect could be much larger if a heavy fourth generation neutrino exists). Of course, should such a v~ magnetic or electric dipole moment exist, then it is quite possible that whatever mechanism gives such a moment, will give a similar size magnetic or electric dipole moment to the tau. 117

Volume 252, number 1

PHYSICS LETTERS B

In summary, we have shown that the strongest b o u n d to date on the electric dipole m o m e n t of the tau comes from measurements of the angular distrib u t i o n in e+e - a n n i h i l a t i o n into tau pairs, a n d is given by 1.4)< 10 -16 ecm. This b o u n d could be improved significantly at a tau-factory. We thank Steven Gasiorowicz, Ryszard Stroynowski and G i a n Guidice for useful discussions, and are grateful to the Aspen Center for Physics, where this work was carried out.

Note added. After this work was submitted, we learned of a calculation by Barr a n d Marciano, which has just appeared in a book [ 11 ]. They considered the effect of an electric dipole m o m e n t for the tau on the total cross section, a n d by assuming the cross section was measured to 3%, they found an upper b o u n d which is similar to ours. O f course, more i n f o r m a t i o n can be obtained from an angular distribution. We also learned of a discussion on how to detect CP-violating effects at a tau factory by Bernreuther a n d

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N a c h t m a n n [ 12 ]. As discussed above, however, the effect would first be seen in the angular distribution.

References [ 1] K.F. Smith et al., Phys. Lett. B 234 (1990) 191. [2] S.A. Murphy et al., Phys. Rev. Lett. 63 (1989) 965. [3] J. Baileyet al., Nucl. Phys. B 150 (1987) 1. [4] D.J. Silverman and G.L. Shaw, Phys. Rev. D 27 (1983) 1196. [5 ] G. Domokos, S. Kovesi-Domokos,C. Vaz and D. Wormser, Phys. Rev. D 32 (1985) 247. [6 ] B. Barish and R. Stroynowski,Phys. Rep. 157 ( 1988) 1. [7] CELLO Collab., H.J. Behrend et al., Phys. Lett. B 222 (1989) 163. [8] M.B. Voloshin,M.I. Vysotskyand L.B. Okun, Sov. J. Nucl. Phys. 44 (1988) 440. [9] A.V. Kyoldjiev,Nucl. Phys. B 243 (1984) 387. [ 10 ] G.F. Guidice, Fermilab preprint FERMILAB-PUB-90/143T (July 1990). [ 11 ] S. Barr and W. Marciano, in: CP violation, ed. C. Jarlskog (World Scientific,Singapore, 1990). [ 12] W. Bernreuther and O. Nachtmann, Phys. Rev. Lett. 63 (1989) 2787.