

 31 Jan 1995

LA PLATA 94-10

NUB-3113

A Method to Determine the TauNeutrino Helicity Using Polarized

Taus M.-T.Dova, L.N.Epele, H.Fanchiotti,

C.A.Garc'ia Canal, P.E.Lacentre

Departamento de F'isica Universidad Nacional de La Plata

1900 La Plata, Argentina

J.D.Swain Department of Physics Northeastern University Boston, Massachusetts 02115, USA

Abstract A method is presented to extract the tau neutrino helicity, or equivalently, the chirality parameter flVA, independent of any tau polarization which may be present. The method is thus well-suited to measurements using taus produced from the Z0 and is complementary to analyses using tau correlations since it provides the sign of the chirality parameter which is otherwise unavailable without recourse to lower energy experiments where taus are unpolarized. Results of Monte Carlo studies and comments regarding the use of the technique in experiments are also included.

1 Introduction In the Standard Model the l \Gamma W \Gamma *l vertex is supposed to have the V-A structure for any lepton. This fact has been extensively checked for electrons and muons [1].

Moreover, measurements by the ARGUS Collaboration [2] at energies near the production threshold of o/ + o/ \Gamma supported this fact also for the o/ lepton. At higher energies such direct tests have not been made, though the neutral couplings of the o/ lepton to the Z0 have been measured (for a recent summary see [3]). The ALEPH collaboration [4] has performed an analysis suggesting that the o/ charged current is either pure V-A or pure V+A. (Studies of correlations alone do not suffice to determine the sign of flV A.)

We analyze here the decay distributions of e+e\Gamma ! o/ +o/ \Gamma and o/ \Gamma ! a\Gamma 1 *o/ at the Z0 peak with the purpose of obtaining an estimator of the coupling constant of the o/ \Gamma W \Gamma *o/ vertex. We present an observable for the determination of this constant, which is independent of the value of the o/ polarization.

We use here the single-tau decay modes. Other methods using o/ +o/ \Gamma spin correlation observables are under study and will be presented in a separate publication [5].

Lepton pairs o/ +o/ \Gamma are created at LEP from the electron-positron annihilation at energies of the Z0 resonance.

In the Standard Model the neutral current J_ has the form:

J_ = _u(o/ +) fl_ (vo/ \Gamma ao/ fl5) u(o/ \Gamma ) (1) where the coupling constants vo/ and ao/ are given by:

vo/ = \Gamma 1 + 4 sin2 `W (2) ao/ = 1 (3)

The LEP beams are unpolarized but the inequality of the Z couplings to left-handed and right-handed leptons induces a polarization of the taus. The

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longitudinal polarization of the o/ \Gamma averaged over all the production angles is related to the o/ + \Gamma Z0 \Gamma o/ \Gamma vertex coupling constants, at the Z0 peak, by:

P = \Gamma 2 vo/ ao/v2

o/ + a

2o/ (4)

Subsequently, as it is well known, taus decay via weak interactions where parity is not conserved. The *o/ cannot be observed experimentally and the measurable quantities are the energies and momenta of the hadrons or leptons in the final state.

The vertex o/ \Gamma W \Gamma *o/ is supposed to be V \Gamma A in the Standard Model. To take into account a possible deviation from this form we introduce coupling constants gV and gA for the vector and axial vector tau currents, namely:

J_ = _u(p0; s0) fl_ (gV + gA fl5) u(p; s) (5) It is usual to define a quantity analogous to the polarization, which characterizes the handedness of the charged leptonic current. This quantity is called the chirality parameter and is given by:

flVA = 2 gV gAg2

V + g

2A (6)

In the Standard Model, as it was stated above, flVA = \Gamma 1.

2 o/

\Gamma ! a\Gamma

1 *o/ Decays

The a1 is a pseudovector resonance decaying into three pions. The decay process is as follows:

o/ \Gamma ! a\Gamma 1 *o/ ; a1 ! ae0ss\Gamma ; ae0 ! ss+ ss\Gamma

One can only measure the energies and momenta of the three charged pions in the final state. Only the neutrino escapes the detection so the kinematics of the system is well enough constrained to allow a partial reconstruction of the events, despite the fact that we cannot reconstruct the o/ -rest frame. The appropriate frame here is the a1-rest frame where the pions are coplanar as illustrated in Figure 1.

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The angular distribution for the decay can be written following reference [6].

d\Gamma = N [h\Gamma 1 WA \Gamma (h\Gamma 1 cos ` \Gamma h2 sin `) WA flVA Po/ + h3 cos fi WE Po/

+3 Q2 cos cos fi WE flVA] (m

2o/ \Gamma Q2)2

Q2 dQ2

Q2 ds1 ds2

d cos `

2

d cos fi

2 (7)

where q1, q2 and q3 are the final pion 4-momenta and Q = q1 + q2 + q3. The o/ rest frame decay angle ` and the angle between the direction of the o/ and the laboratory as seen from the a1 rest frame, can be reconstructed from the energy of the hadronic system,

cos ` = 2 x m

2o/ \Gamma m2o/ \Gamma Q2

(m2o/ \Gamma Q2) p1 \Gamma m2o/ =E2beam ; ` 2 [0; ss] (8)

cos = x (m

2o/ + Q2) \Gamma 2 Q2

(m2o/ \Gamma Q2) px2 \Gamma Q2=E2beam (9) with

x = E1 + E2E

beam (10)

fi denotes the angle between the normal n? to the three pion plane and the three pions laboratory line of flight. cos fi is obtained from the measured pion momenta using the analytic approximation of reference [7]

cos fi = 8 Q

2 ~p1 \Delta ( ~p2 \Theta ~p3)=j ~p1 + ~p2 + ~p3j

[\Gamma *(*(Q2; s1; m2ss); *(Q2; s2; m2ss); *(Q2; s3; m2ss))]

1 2 (11)

where *(x; y; z) = x2 + y2 + z2 \Gamma 2xy \Gamma 2yz \Gamma 2zx.

h\Sigma 1 = m2o/ \Sigma 2 Q2 \Gamma (m2o/ \Upsilon Q2) 3 cos

2 \Gamma 1

2

3 cos2 fi \Gamma 1

2 (12)

h2 = 3 mo/ pQ2 sin 2 2 3 cos

2 fi \Gamma 1

2 (13) h3 = \Gamma 3 Q2 (cos ` cos + mo/pQ2 sin ` sin ) (14)

N = G

2

8 m3o/ (g

2 V + g

2 A) cos

2 `C 1

64 (2 ss)5 (15)

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and

WA = jF1j2 ^(s2 \Gamma 4 m2ss) + (s3 \Gamma s1)

2

4 Q2 *

+jF2j2 ^(s1 \Gamma 4 m2ss) + (s3 \Gamma s2)

2

4 Q2 *

+ ^(Q2 \Gamma 2 s3 \Gamma m2ss) + (s3 \Gamma s1) (s3 \Gamma s2)2 Q2 * Re(F1 F \Lambda 2 ) (16)

WE = 3 ^ s1 s2 s3 \Gamma m

2ss (Q2 \Gamma m2ss)2

q2 *

1=2

Im(F1 F \Lambda 2 ) (17)

The Dalitz variables s1 and s2 are defined by

si = (qj + q+)2 ; i 6= j = 1; 2 (18) where q+ is the momentum of the positive pion. The model dependence is contained in the functions WA and WE. The discussion presented by J.H.Kuhn and E.Mirkes [6] is based on a hadronic current of the form:

J_ = F1(s1; s2; Q2) V _1 + F2(s1; s2; Q2) V _2 (19) with

V _1 = q_1 \Gamma q_3 \Gamma Q_ Q (q1 \Gamma q3)Q2 (20) V _2 = q_2 \Gamma q_3 \Gamma Q_ Q (q2 \Gamma q3)Q2 (21)

F1 = F (s1; s2; Q2) ; F2 = F (s2; s1; Q2) and a choice for F as follows: F (s1; s2; Q2) = \Gamma 2p2i3 f

ss BWa(Q

2) Bae(s2),

where BWa and Bae denote Breit - Wigner resonances. This model for the current has previously been worked out by J.H.Kuhn

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and F.Wagner [8] and is implemented in the KORALZ event generator [9] widely used to simulate o/ production and decays. Given that the two negative pions are not distinguishable, there are two possible ways to form the ae-meson. The interference between them is contained in the function WE through the imaginary part of the structure functions F1 and F2. Notice that the only term in the angular distribution that contain flVA without the presence of Po/ is proportional to WE. This is the interference that makes the o/ ! a1 *o/ the unique hadronic channel from which we can disentangle the dependence on the chirality parameter.

3 Determination of the Chirality Parameter A method to obtain an estimator of the chirality parameter, which is model dependent though, consists in taking appropriate moments using the distribution function given in equation (7). To go further along this new method, let us introduce the following notation

d\Gamma = \Omega (Q2; s1; s2; cos `; cos fi) d5x (22) d5x = dQ2 ds1 ds2 d cos `2 d cos fi2 (23)

Then for any quantity m, we define a moment hmi by

hmi = R m \Omega (Q

2; s1; s2; cos `; cos fi) d5xR

\Omega (Q2; s1; s2; cos `; cos fi) d5x (24)

The important observation is that it is possible to eliminate the dependence on the o/ polarization by taking the moment of the quantity

M = cos fi cos ` sgn(s1 \Gamma s2)cos ` cos + m

o/p

Q2 sin ` sin

(25)

The function sgn of (s1 \Gamma s2) is introduced in order to take into account the ambiguity in the direction of the normal to the decay plane, due to the Bose symmetry of the two negative pions. Then one hasZ

M \Omega ds1 ds2 d cos `2 d cos fi2

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= N flVA (m

2o/ \Gamma Q2)2

Q2 Z

cos ` cos cos ` cos + m

o/p

Q2 sin ` sin

WE sgn(s1 \Gamma s2) ds1 ds2 d cos `2 (26)

while the normalization constant N is determined by the condition

1 = Z \Omega ds1 ds2 d cos `2 d cos fi2

= N (m

2o/ \Gamma Q2)2

Q4 (m

2 o/ + 2 Q

2) Z WA ds

1 ds2 (27)

Finally one can write

hMi = \Gamma flVA ALR(Q2) T (Q2) (28) where we have introduced the functions

ALR = \Gamma Q

2

(m2o/ + 2 Q2) R

WE sgn(s1 \Gamma s2) ds1 ds2R

WA ds1 ds2 (29)

T (Q2) = \Gamma 1(m2

o/ \Gamma Q2) f[Q

2 + m2

o/ [1 + 3 m

2o/ + Q2

3 K(Q2) log

3 m2o/ \Gamma Q2 \Gamma K(Q2) 3 m2o/ \Gamma Q2 + K(Q2)

+ log mo/pQ2 ]g (30)

with

K(Q2) = [(9 m2o/ \Gamma Q2)(m2o/ \Gamma Q2)]1=2 (31)

4 Monte Carlo Studies We have performed a Monte Carlo study using the Koralz [9] program to generate samples of 200,000 events with a1 decays assuming pure V-A and pure V+A charged current couplings, as well as 200,000 events with nonstandard values of flVA to represent a hypothetical data sample with gV = 0:6 and g2V + g2A unchanged from its standard model value, giving flVA = \Gamma 0:768. The calculated values of the moments and their errors are shown in Figure 2 for each of the three data samples. A O/2 fit for the best linear combination of V-A and V+A samples to match the flVA = \Gamma 0:768 sample gave a

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statistical error of 0.049, which includes errors due to the finite Monte Carlo V-A and V+A samples as well as those due to the finite number of events with nonstandard couplings. Monte Carlo studies using samples of fully right or left-handedly polarized taus give consistent answers, verifying that the method of this papers gives a method for the determination of the tau neutrino chirality parameter which is independent of the tau polarization.

The errors are, admittedly, large when scaled to realistic numbers of events at LEP, but two points are worth bearing in mind : 1) studies of correlated tau decays at LEP can give quite accurate determinations of the absolute value of the tau neutrino chirality parameter, but with absolutely no information about its sign. The information from the correlated decays and the method in this paper are statistically independent, and thus the likelihood distributions for the tau neutrino chirality parameter can be multiplied. The final likelihood will be quite sensitive to both the sign and magnitude of flVA. 2) Future accelerators (and perhaps higher luminosity options for LEP) may provide large enough data samples so that the intrinsic interest of this method (without recourse to studies of correlated tau decays) will be greater. In particular, the method can be used to study singly-produced tau decays from hadron machines.

Clearly in taking moments to remove the tau polarization dependence we have assumed a perfect detector. In any real experiment, some polarization dependence in the calculated moments is bound to appear and must be studied. In addition, we are studying the possibility of using a related method fixing the product of the tau polarization and tau neutrino chirality parameters [10].

One final comment of interest for experiments is that a sample of o/ \Gamma decays with a V+A charged current interaction can be obtained simply by using o/ + decays with a V-A interaction and simply reversing the signs of the charges of all particles.

5 Summary In summary, we have found an observable for the determination of flVA which is independent of the o/ polarization. The function ALR(Q2) is the parity vi8

olating asymmetry measured by the ARGUS collaboration to determine the chirality at low energies where Po/ = 0. The method has been checked with simulated events generated by Monte Carlo and the sensitivity estimated.

This method can also be applied to the process in which a\Gamma 1 ! ss0 ss0 ss\Gamma . To this end one has only to change q+ ! q\Gamma in equation (18) and qj ; j = 1; 2 are now the neutral pion momenta. Clearly, in this case one is dealing with a negative ae-meson.

6 Acknowledgements This work was supported in part by Consejo Nacional de Investigaciones Cient'ificas y T'ecnicas (CONICET), the National Science Foundation, and the World Laboratory Project. M.-T.D. would like to thank CERN for its hospitality while part of this work was being done. J.D.S. would like to thank the Universidad Nacional de La Plata for its kind hospitality when this work was being finished.

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References

[1] K. Mursula, M. Roos, F. Scheck, Nucl. Phys. B219 (1983) 321. W.

Fetscher, H.J. Gerber, K.F. Johnson, Phys. Lett. B173 (1986) 102.

[2] ARGUS Collaboration, H. Albrecht et al., Phys. Lett. B250 (1990)

164. ARGUS Collaboration, H. Albrecht et al., DESY preprint 94-120 (1994).

[3] J. Swain, "Weak Couplings of the o/ Lepton : Results from LEP", invited summary talk at the American Physical Society, DPF Meeting, Albuquerque, New Mexico, August 1-6, 1994 (to appear in the proceedings). Available as Northeastern University preprint NUB-3100.

[4] D. Buskulic et al., Phys. Lett. B321 (1994) 168. [5] M.-T. Dova et al., "Correlated Angular Distributions in o/ +o/ \Gamma Semileptonic Decays", La Plata preprint, submitted for publication.

[6] J.H. K"uhn and E. Mirkes, Phys. Lett. B286 (1992) 381. [7] A. Roug'e, Workshop on Tau Lepton Physics Orsay 1990, Editions Frontiers (1991) 213. A. Roug'e, Z. Phys. C48 (1990) 75.

[8] J.H.K"uhn and F.Wagner, Nucl. Phys. B236 (1984) 16. [9] S. Jadach and Z. Was, Comput. Phys. Commun. 35 (1985);

R. Kleiss, "Z Physics at LEP", CERN-8908 (1989), Vol. III, p. 1.

[10] M.-T. Dova et al., in preparation.

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Figure Captions Figure 1: Kinematics for the a1 decay. Figure 2: Moments vs. Q2 for various values of flVA.

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n q

n

t -a 1-

p 1-

b

a

3

2 (p )2 (p )3p t

a1- 1(p )1p direction in lab

in rest framet-decay plane of in its rest frame

a 1-

in lab

y

Figure 1.

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Q2(GeV2) gVA=-1.0 gVA=-0.768 gVA=+1.0

-0.08 -0.06 -0.04 -0.02

0 0.02 0.04 0.06 0.08

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Figure 2.

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