Structure functions in τ → 3πντ and the τ polarization measurement

Physics Letters B 308 (1993) 163-173 PHYSICS LETTERS B North-Holland

Structure functions in z 3nv and the z polarization measurement

Paolo Privitera

Instltut fur Expertmentelle Kernphystk, Untversttdt Karlsruhe, W-7500 Karlsruhe, FRG

Received 11 March 1993 E&tor K. Winter

The z polarization measurement with the r ---, 3nut decay has already been performed by LEP experiments. In this note it Is shown that the sensitivity to Pr for this decay mode can be improved by almost a factor two with respect to previous methods. This is obtained by studying new angular distributions. An expliot model for the hadronic structure functions must be used to perform the measurement. The theoretical uncertainty coming from this model dependence is estimated to be (AP~)theor - - - - - +0.015, which will not limit the precision of the Pr measurement.

1. Introduction

The measurement of the z polarization is part of the precision tests of the electroweak interaction cur- rently performed at LEP I l l . The Z ° couples differently to fermlons of opposite heliclty Thus, a z - (r +) lepton produced by a Z ° decay will be found with different probabilities in a negative or in a positive hehclty state, i.e. it will be partially polarized. This is an example of a parity wolatlng process: under the parity operator, which inverts the sign of the hehclty, the z will behave differently. The couplings of the Z ° to fermlons, and thus the fermlon polarization, are related to the fundamental parameter of the electroweak theory, sin 20w. The argument outlined above is valid also for the other Z ° decays. However, only the r offers the experimental possibility of mea- suring such polarization. In fact, the r decay, through the weak interaction, is also parity violating. The r decay product characteristics will depend on the hehcity state of the parent r. Measurements of the z polarization,/Dr, have been performed successfully by the LEP experiments [2], using the different z decay modes.

Up to now, the contribution of the r -~ 3nu~ decay mode to the P~ measurement has been marginal with respect to the other decay channels. It will be shown that a significant improvement is obtained if new kinematical variables are experimentally ex- plored. The new method takes advantage of the most

complete z -~ 3nu~ decay distribution recently determined by Kuhn and Mirkes [3]. The main results of their analysis are reviewed in section 2. A comparison of different model predictions for the hadronic structure functions is presented in section 3 The determination of the r polarization by using the Ktihn

and Mlrkes dlStrlbution and the improved sensitivity to P~ are discussed in section 4. The theoretical uncertainty on the P, measurement arising from the model dependence of the hadromc structure functions is estimated in section 5. Last is a statement of the main conclusion of this work.

2. The z ~ 3nv~ decay distribution

In the process r ~ 3nu, the neutrino escapes de- tectton and only the hadronlc decay products are observed experimentally. Therefore the z rest frame cannot be reconstructed. However, the kinematic config- uration of the decay is still constrained enough to allow at least a partial determination of the kinematical variables. The kinematical properties of r ~ 3nu~

and their relationship to observables have been studied by several authors [4-8 ].

Recently, Kuhn and Mirkes [3] have determined the most general angular distribution that can be observed in the semlleptonlc r decay, opening the possi- bllity of new experimental investigations. In this sec-

0370-2693/93/$ 06.00 (~ 1993-Elsevier Science Pubhshers B.V All rights reserved 163

Volume 308, number 1,2 PHYSICS LETTERS B 24 June 1993

tion, their work is summarized and the definit ion of the experimentally observable kinematical varmbles is introduced. The notat ion ofref. [3 ] will be followed throughout.

The differential decay rate can be wrmen as

d G ~ . n . ~ . - G~ (g2 + g Z ) c o s 2 O c E - E x W x 4m~

X

1 1 ( m 2 - Q2)2 ×

(2n) 5 64 m 2

dQ 2 dy d c o s f l d c o s 0 × - - ~ - dsl ds2 2~r 2 2 (1)

The hadronic structure functions Wx contain the dynamics of the 3rr decay and depend in general on the lnvarlant masses Sl, s2 and 0 2, defined as

s, = (Ej + Ek) 2 - (pj + p k ) 2

( t , j , k = 1,2,3; l ~ J # k ) , (2)

Q2 = E3Zn_ (p3.)2, (3)

wherep, and E, denote the momen tum and the energy of pion t, respectxvely. The index 1 and 2 is assigned to the like-sign pions and the index 3 to the unlike

sign plon. Also P3~ = Pl + P2 --b 103 and E3. i s the sum of the plon energies. All momenta and energms are measured m the laboratory frame.

The lepton factors L x are functions of the angles 0, fl and y. They also depend on the z polar izat ion PT and the chirahty parameter YVA. The explicit expressions for L x can be found in ref. [3]. The angle 0 denotes the angle between the direction of flight of z in the laboratory frame and the direction o f the hadrons as seen in the z rest frame. Its cosine is g~ven by

cos 0 = 2 x m ~ - m~ - Q2 (m 2 - QZ )v /1 - 4rnZ /s '

2E3~ 2 x -- S = Ecm, (4)

w h e r e Ecru is the energy in the centre of mass for the e+e - ~ r+z - reaction.

fl denotes the angle between the normal to the 3n decay plane and the d i recuon of the hadrons in the laboratory system. Its cosine can be wrmen in terms of the plon momenta

COSfl -- 103" ( ~ 1 X P 2 ) (5) LP3,1T '

where

T = k V / - Z ( B I , B ~ , B ~ ) , (6)

;t(x, Y,z)

= X 2 -b y2 + Z 2 _ 2 X Y - 2 X Z - 2 Y Z , (7)

Bt = (E~E3~ --P3n .p , )2_ Q2m2 (8) Q2

The remaining Euler angle y corresponds to a rota- t ion around the normal to the decay plane and deter- mines the orientat ion of the plons within their pro- ductlon plane. The tr igonometric functions of y are given by

A~ (9) c o s y - [P3nlx /~s lnf l '

COS ~ B3 (10) sin y - - T 2 '

where

A , = E3~tP3n " P , -- E,(P3n) 2 (11) Q

Another important angle for the subsequent d~scus- stun ms the angle ~, [5] between the direction of the laboratory and the r as seen from the hadronic rest frame. The cosine of such angle is

x ( m ~ + Q2) _ 2Q2 c o s ~ = (mr - Q 2 ) ~ / x 2 - 4QZ/s " (12)

In the ultrarelatlvist~c hmlt s >> m~, and

cos0(m~ + Q2) + (m 2 _ Q2) cos~u ~ . (13)

cos0(m2 _ Q2) + (rn 2 + Q2)

3. Compar i son of hadronic structure funct ions for

different mode l s

The structure functions Wx depend on the specific choice of the model used for the hadronic current. In fact, they are defined as appropria te s y m m e m c and ant isymmetr ic combinat ions of the hadronic tensor HU~, = j i , ( j , , ) t [3]. The general ansatz to describe the decay into three plons can be written

164


QuQ. ) j~uo) Ju = j(l)~ gu~ Q2 + , (14)

where the current Ju u) corresponds to spin 1 (a l ) and Jut°) corresponds to spin 0 (n ' ) . The general form of the current (14) is de termined by sxmple arguments based on symmetry considerations (Bose statistics, isospln, etc.) [9]. However, the Brel t -Wigner reso- nance parametr lza t lon (for example the a~ and p in the sequence al ~ pn) and the underlying dynamics are not de termined by first principles and specific models must be used.

In the following, numerical predict ions and com- parxsons are presented for &fferent models: - the Kflhn and Santamaraa model [10] which rehes on the assumptmn of chlral dynamics, th~s model ~s used in the KORALZ [ 1 1 ] Monte Carlo program, the e+e - ~ Z ° --+ z+z - generator used by LEP experiments. - the Isgur, Mornlngstar and Reader model [12] based on pre&ctlons of the flux-tube-breaking model, and - the Femdt model [ 13] which uses angular momen- tum elgenstates amphtudes.

These models, which are based on very different theoretical approaches, have been shown to fit successfully the available experimental data. However, only the WA structure function has been studied experimental ly so far, through the Q2 and s, dis tr ibut ions of the decay rate. Approprmte moments [3] could be used to extract the structure functions, and their comparison with the predict ions could distinguish different models. On the other hand, if one wants to use the angular dis tr ibut ions of the lepton functions Lx to measure P~ (or ?VA ), part icular care should be taken in understanding the uncertainties coming from the imperfect knowledge of the hadronic current, since the differentml rate is propor tmnal to ~ x - L x Wx.

One can introduce the s~s2-mtegrated hadronic structure functions as

'IDA,C,SA,SB = f dSl ds2 WA,C,SA,SB, ( 1 5)

WO.e,SO = f dst ds2 sIgn(sl - s 2 ) W D , E , S D . (16)

These structure functions are now only Q2 dependent , and can be visualized. The WE structure function is

1

0 8

0 6

0 4

0 2

0

- 0 2

- 0 4

O) C/COA . . . . . Isgur et ol - ~ - _ ....

Felndt ~.~...~..

CO D/(x) A ........ " .....

0 4 0 8 I 2 16 2 '24 '28' Q2 (GeV 2)

Fig. 1 The spin 1 hadronac structure functions, normalized to w.t, as functions of Q2. Kuhn-Mlrkes (solid lines), Isgur et al (dashed hnes), Felndt (dotted lines; f D / f S = --0.11)

closely related to the pan ty violating asymmetry proposed in refs. [5,10,13] which has been used by AR- GUS [ 14] to measure yva.

Kuhn and Mlrkes [3] presented numerical predictions for the Wx using the Kfihn and Santamaria model for the spin 1 part of the hadromc current. The corresponding structure functions normalized to WA are shown in fig. 1.

They also studied the effect of a scalar current (d(0)) corresponding to ~ 5% contr ibut ion on the total rate F3,. The predict ions for the structure functions arising from the presence of such scalar compo- nent are shown in fig. 2 One observes that the scalar contr ibut ions are not negligible ( ~ 10-15%), and thus could be observable experimentally Notice that no direct experimental measurement of the hadronie structure functions has been performed so far.

Some relevant aspects of the hadronic structure functions have been investigated.

( 1 ) Dependence on al mass parameters A change of the al mass and width is not expected to mo&fy fig. 1, since wx ~ B W m. The al Bre l t -Wlgner contribution will thus cancel in the ratio wx/wA. However,

165


0 3

0.25

0 2

0 1 5

01

0 05

0

-0 05

- 0 1

-0 15

- 0 2

Kuhn-M~rkes

. . . . . ~v / ( # , Isgur et al

(.~ SD/(-~) A

(x~ SB / (-,c) A

0 '4 ' ' 0 '8 ' ' '1 . . . . . ' . . . . ' . . . . . . . . . . . . . . 2 1 6 2 24 2 8

Q~ (OeVb

2400

2000

1600

1200

800

400

/, a)

b)" .............

0 4 0 8 1 2 1 6 2 2 4 2 8

Q2 (CeV ~)

Fig. 2. The spin 0 hadronic structure functions, normalized to wA, as functions of Q2. Kuhn-Mirkes (sohd hnes), Isgur et al. (dashed, dashed-dotted and dotted lines).

the Q2 dependence of the structure functions is affected, as it is clearly visible for WA in fig. 3, where the Kiihn and Santamaria model has been used. In particular, the model predict ion for the total decay rate can change significantly. In table 1 the Kiihn and San- tamar ia model predict ions for the r ~ alu decay rate normalized to the leptonic decay rate ( F ~ - a l , / F ~ - e ~ ) are reported for various values of the a~ mass and width. Notice that the max imum difference in the predictions of table 1 corresponds to a variat ion ABr ,~ 1% of the z ~ 3rrv~ branching ratio. The present experimental error [ 15 ] on the z ~ 3rru~ branching ra- t io is 0.4%.

(2) Model dependence. The spin 1 hadronic structure functions normal ized to w~ for the Isgur et al. and Feindt models are shown in fig. 1, together with the predict ions of Kuhn and Mirkes. A value f o / f s = -0 .11 of the ratio of D- over S-wave decay ampli- tudes is used for the Feindt model, based on the re- cent A R G U S measurement [14]. The predict ions of the models for the ratio w c / w ~ are very close to each other. The wD and we form factors show a different behaviour for Q2 > 1.4 GeV 2. In fact, these structure functions contain interference terms where, in partlc-

Fig. 3. The hadromc structure function WA as a funcUon of Q2. (a) rna I = 1.251 GeV, Fal = 0.599 GeV, (b) real = 1.200 GeV, Fal = 0.549 GeV

Table 1 Kuhn and Santamana predictions for the al total decay rate for different values of the a[ mass and width.

ma I Fa I Fr--al L, / Fz~et,'~

1.251 0 599 0.355 1.251 0.549 0.398 1.251 0.649 0.319 1.200 0.599 0.322 1 300 0.599 0.380 1.200 0.549 0.362

ular for high Q2, the differences in the parametnza- tions of the Brei t -Wigner terms become important .

The scalar structure functions calculated with the Isgur et al. model are compared to those of Ktihn and Mirkes in fig. 2. There are clear differences in the predictions. The scalar contr ibution to the total rate F3n is only ~ 1.6% in the Isgur et al. model, compared with 5% of Kuhn and Mirkes.

The differences in the model predict ions are size- able enough to be experimentally accessible. However, a large sample of z ~ 3~uT is needed, since all the variat ions are at large Q2. On the other hand, it is im- por tant to est imate the possible biases to the P, mea-

166


surement due to these theoretical uncertainties. A de- tailed discussion on the corresponding systematic error for the measurement of P~ will be presented in section 5.

4. Sensitivity to P,

In the decay r ~ a~v ~ nnnv , , the hadronic system helicity can assume the values 0 or - 1. Summing over the hehcity states, the expected a~ angular distribution in the z rest frame is

1 dN I(1 + P~aat cos0) (17) N d c o s 0 - 2

where cos 0 was defined in expression (4) and Oral 0.12. In the case of r --, nu decay, an = 1. As a conse- quence of the two possible helicity states, the sensitivity to the polarization of the d N / d cos 0 distribution is significantly reduced. New distributions should be studied in order to improve the sensitivity.

In the following, several distributions sensitive to P, will be shown. The definition of sensitivity given by Roug6 [16] is used. Consider a general distribution which can be written in a form which shows explicitly the P, dependence:

W ( x ) = f (x) + P , g ( x ) , (18)

where x is a decay observable (or a set o f observables) and the functions f and g satisfy the normal- lzation and posltivity condition f f ( x ) dnx = 1, f g ( x ) d"x = O, f >1 0 and Igl ~< f . Given a sample of N measurements {x,} of the decay observables, the measured polarization P~ which maximizes the likelihood function L (Pr, {x,}) is determined by

logL = Z l o g [ f ( x , ) + PTg(x , ) ] , (19) 1

0 g ( x , ) OPt logL = Z f ( x , ) + P~g(x,) ' (20)

l

02 g 2 ( x ' ) (21) OEpr logL = Z [ f (x~) + P ~ g ( x , ) ] 2 "

l

The fitted value of P~ is the solution of the equation 0 logL/OP~ = 0 and its error ae~ is determined by

1 0 2 a~, - -O-~P~ 1ogL

= N f + p---------~ dx = N S 2 , (22)

where S is the sensitivity. The procedure which will be used to determine the sensitivity of a given dis- tnbution is the following. The f and g functions of (18) are explicitly calculated for the distribution of interest. Then, N events are generated with the KO- RALZ [ 11 ] Monte Carlo program. For each generated event f and g are calculated, and after summing over all events, a2 is obtained from expression (21). The

sensitivity is then S = 1/NV/-~e2,. To be sure that the

asymptotic approximation (22) for the sensitivity is valid, a large number of Monte Carlo events has been used (N = 200 000). Notice that the sensmvity and the distributions which will be discussed below take into account the effects of radiative corrections, since these are included m the KORALZ Monte Carlo program. Also, the Standard Model value YVA = 1 has been assumed.

4.1. The two-dlmenstonal (cos O, cos fl) dlstrlbutton

The sensitivity can be improved by measunng the helicity of the a~ through the decay distribution of the hadronic system [ 5-8 ]. However, the r rest frame hehclty cannot be measured because of the missing neutrino, and only the laboratory frame helicity is accessible. To compute the decay angular distribution, the components o f the spin state must be rotated by the angle ~t between the direction of the laboratory, and the r both as seen from the hadronic rest frame (cf. expression (12) )

Following Roug6 [7], the general decay distribution for a spin 1 hadromc system reads

W ( cos O , cos fl ) = 3 8(m 2 + 2Q 2)

x[ (1 + P~)W+ (cosO, cosfl)

+ (1 - P~)W-(eosO, cos f l )] , (23)

where fl is the angle between the normal to the decay plane and the laboratory direction (cf. expression (5)). The explicit form of the W :L functions can be found m ref. [7].

167


A fit to the two-dimensional dis t r ibut ion (cos 0, cos fl) will maximize the sensitivity to P~. Using the procedure previously described, a sensit ivity S = 0.24 is obtained, which corresponds to an improvement of more than a factor 2 with respect to d N / d cos 0.

Notice that the der ivat ion of expression (23) rests only on spin-pari ty considerat ions and not on the de- tails of the hadronic system structure. This means that the determinat ion of P, using the (cos 0, cos fl) distri- but ion does not depend on the specific model of the hadromc current. This s tatement can be clarified by comparing Roug6 expressions with the differential decay rate de termined by Kuhn and Mirkes, which has been discussed in section 2. In fact, the (cos 0, cos fl) dis t r ibut ion can be derived by integrating the decay rate (1) over the a and y angles. After the integration, the decay rate is of the form

A o4

~0.08

I N~"~ 0 . 0 4

0 0

0 V

-0.04

-0.08

-0.12

- 0 16

- 02

+ _ _ Q _ _ . - o ~ - o - - - - o - - + +

- - - i 1 ~ +

--il.- +

i -

- t -

- - I -

--oi-

l , , L I , f , i i i , , , I i i , , - 05 0 05

cosq

dF, . . . . . . cx -LA WA d cos 0 d cos fl dQ z dsL ds2

= I K, + K2 + ½(3cos2/ - 1)1

× WAd cos 0 d cos fl dQ 2 dSl ds2,

where the K functions are given by [3]

mE (1 + 7vAP, cosO) Kl = 1 - 7vAP~ COS 0 -- ~ -

m~ K2 = ~ - ( 1 + 7vaP, eosO),

K3 = 7VA -- P, cosO,

KI = KI -½(3cos2 V - 1) - 3 K 4 s i n 2 v ,

K2 = K 2 c o s ~ + K4s lne / ,

K3 = K3 cos ~ / - Ks sin ~,,

g ~ 2 m , c

K4 = ----~YvAP~SlnO,

Ks = V ~ ~sin0"

(24)

(25)

Notice that the angular dependence is contained only in the leptomc function LA. Since WA factorizes in expression (24), the specific form of the hadronic current does not affect the angular distr ibutions, and the P, de terminat ion through (cos0, cosf l ) is thus

Fig. 4. The (½ (3 cos 2 f l - 1 )) (cos 0) distribution for negative (dots) and posmve (squares) r hehclty.

model- independent . Introducing the explicit expressions (25) for KI, K2 and KI in (24), the Roug6 decay dis tr ibut ion (23) is obtained.

F rom the cos 0 and cosfl dependence of expression (24) one can see that all the information is contained in the two one-dimensional distributions: d N / d c o s O and ( ½ ( 3 c o s 2 f l - 1 ) ) ( cos0 ) . The (½ (3 cos 2 fl - 1 )) (cos 0 ) dis tr ibut ion for negative and posit ive z helicity is shown in fig. 4.

Most of the sensitivity to P, is contained in the (L (3cos 2 fl - 1 )) (cos0) distr ibution. Thus a one-

2

dimensional combined fit to the d N / d c o s O and (½ (3 cos 2 fl - 1 )) (cos 0) distr ibutions can be performed to determine P,, without any loss in the sensitivity.

4.2. Improved sensmvtty to PT usmg all the decay

variables

In section 2 the most general decay distr ibution for z ---, 3nu~ determined by Ktihn and Mirkes was presented. The authors emphasized that the use of all the decay variables cos 0, y, cos fl, Sh S2, and Q2 allows a determinat ion of the hadronic structure functions.

In this section it will be shown that a major ira-

168


provement of the sensit ivity to P~ is also achieved. The decay rate can be written (cf. section 2)

d ~ l t g T t v z

vc ~_a-LxWxdTdcosOdcos f ldQEds ldS2 . (26) x

A mult i -dimensional fit can thus be performed in order to determine P~. Notice that all the P~ dependence is contained in the leptonic functions Lx . However, the hadronic structure functions do not factorize in expression (26). Hence, for a given set of cos0, y, cosfl , Sl, s2, and Q2, the Wx(sl,s2, Q2) must be calculated in order to perform the fit. In general, the result will depend on the part icular model assumed for the hadronic current. The model dependence of the P~ measurement with the method proposed here is discussed in detail in section 5.

The improvement in the sensitivity to P~ is most easily understood in terms of one-dimensional distributions. Given the form of the L x functions, an appropr ia te set of moments can be used.

1 ) (cos 0) = . / ( 2 K l + 3K2)wA dQ 2 , (27) (

- l ) ) ( c o s 0 ) = 1 . f~- lZOadQ2, <½ (3cos2 fl (28)

(cos 0 ) = - ½ / -Kl Wc dQ 2 , (29) <cos 2~,)

(sin 2y sign (st - s2 ) ) (cos O)

½ . /KILOD dQ 2 , (30)

( c o s f l s i g n ( s l - s 2 ) ) ( c o s 0 ) = J ' K 3 w e d Q 2. (31)

The contr ibut ion from a scalar current, which is expected to be small and affects only the moment (27), is neglected (WsA = WsB = Wsc = Wsn = WsE = o)

The first two moments are equivalent to the d N / d c o s 0 and (3 (3 cos / f l - 1 )) (cos 0) dis tr ibut ions previously discussed.

A significant improvement in the sensitivity to P, is expected with the use of the moment (cos 2y)(cos 0). In fact, its P, dependence ( K l ) is identical to that of (½ (3 cos 2 fl - 1 )) (cos 0) and its hadronic structure function Wc/WA is about unity, as was shown in

A O.4

04

0 O.3

V

0.2

0.1

- 0 1

-0 .2

-0 .3

0 - -

- - I - -

+

U

- - 0 - - I - -

+ +

-M-- - - 0 - - ~ 0 - - - - 0 - - ~ _ 0 ~ _ _ 0 _

a i t i I i , i L I i i i , 1 , A i i

-05 0 05

cos~

Fig. 5. The (cos 2y) (cos 0) &strlbutlon for negative (dots) and posmve (squares) z hehoty

Table 2 The sensitivity to Pr for the &fferent moments discussed in the text

Moment Sensitivity

A = (l)(eos0) 0.10 B = (½(3cos2fl - 1))(cos0) 0.22 C = (cos 2~) (cos 0) 0.32 D = (sm2yslgn(sl - s2))(cos0) 0.13 E = (cosf ls lgn(s l - s2))(cos0) 0 13

A + B + C + D + E 0.44

section 3. A sensitivity of the same order of that of (3 (3 cos 2 fl - 1 )) (cos 0) then should be achieved. In fig. 5 the (cos 2y) (cos 0) distr ibution for negative and posit ive z hellClty is shown.

In the case of the last two moments IWDI/WA <~ 0.2 and ]WEI/WA <~ 0.3. Thus, their contributions to the sensitivity are expected to be smaller.

The sensitivities to P~ for the different moments are summarized in table 2

Combining all the moments, the expected sensitivity is given by S = ( ~ S ~ ) 1/2 = 0.44. Therefore, an improvement of ~ 2 is achieved with respect to the usual (cos 0, cos fl) analysis i f all the decay van-

169


Table 3 Sensmvlty (Sx), branching ratio (B x ) and relative weight (S2Bx) for the different z decay modes. For ~ --. alu, the values determined with the method presented in this section are reported. For comparison, the corresponding values for the usual two-dimensional (cos 0, cos fl) fit are quoted in parentheses. The relative weights are normalized with respect to the r --~ 7tv channel.

Decay mode "r ~ X Sx = (APrx/N)- t Bx Relative weight S2xBx

z --* xu 0.6 0.11 1 z ~ pv 0 52 0.23 1.6 z ~ air 0.44 (0.24) 0.08 0.4 (0.1) z ~ ev-Y 0.22 0.18 0.2 z --*/zuP 0.22 0.18 0.2

ables are used. To clarify this point, a comparison of the sensitivity to PT for the different decay channels is presented in table 3. The contr ibut ion of the z ---, a lv channel to the precision on P, given by the combinat ion of the different decay modes (AP, = 1 / ( ~ x SZx BxN* ) 1/2 ) is improved by a factor of about 4 (0.1 ~ 0.4), and in both sensitivity and relahve weight is equivalent to the leptomc channels r ~ eu-ff and r --+ ItuF combined.

Another possible approach to the determinat ion of P, is the method proposed by Roug6 [16] which re- duces the problem of a mult i -dimensional fitting to a one-dimensional one. Notice that expressions (20), (21) can be rewrit ten as

0 X-" g ( x , ) / f ( x , ) l ogL (32)

OP, = /---., 1 + P , g ( x , ) / f (x , ) ' l

02 ~ [ g ( x ~ ) / f (xt) ] 2 l ogL (33)

OzPT = / ' ~ [1 ~ P ~ ~ x ~ ) ] 2" l

Given a set of decay observables x, the varmble ~ = g ( x ) / f ( x ) can be computed. Notice that a fit to P, using W (~) is equivalent to a fit using W (x) . In fact, substi tuting in (32) g ( x ) / f ( x ) = ~, it is clear that m both cases the fi t ted value of P, is a solution of ~ , ~ , / ( 1 + P ~ , ) = 0. Similarly, the error on the es t imated P~ ~s the same since ~t is g~ven by (33).

The r ~ 3rtu~ decay rate dastr ibuuon (26) can be written in a form like (18), where x = (cos 0, y, cos B, Sl, s2, Q2). The exphcit expressions for f (x) and g (x) are

f (cosO, y, cosfl, sbs2, Q 2)

= 5 2 + ~ - ) + f l - WA

-- ½sm z fl cos 2 y U W c + ½ sin 2,8 sin 2yUWD

+ yv~ cos ~u cos flWE, (34)

g (cos 0, y, cos fl, Sh S2, Q2 )

= 7va[3_ c o s 0 \ ~ - ~ - f m ~ - 2 ) - ½ ( 3 c o s 2 f l - 1 ) V ] W A

+ ½ YVA sin 2 fl cos 2~ V Wc

- ½7va sin 2 fl sin 2y V Wo

( - cos fl cos 0 cos ~' + sin 0 sin ~u V Q-~) wE,

(35)

where

U 3cos2 q / - I (1 - m~'~ - 2 Q - - ~ J ' ( 3 6 )

V - 3C°S2 ~ - 2 1 c o s 0 ( 1 + ~-)m~

m 2 + 3 s m 2 ~ V ~ s in0" (37)

One can verify that g (x) in (35) satisfies f g (x) d"x = 0. Notice that the f (x) m (34) is not normalized

to unity. The reason is that the normahzat ion factor ~s the same for both f (x) and g (x) , and thus it cancels in their ratio ~. The Standard Model predict ion for the ch~rality parameter 7va = 1 ~s assumed.

170


The sensitivity corresponding to the ~ distribution is S -- 0.44, as expected from the combined moments result.

The method has the advantage of keeping the full sensitivity m only one distribution, which is easy to display and compare with the data with respect to the multi-dimensional one. However, particular care should be taken when dealing with real data, since possible biases could be difficult to recognize on the ba- sis of the ~ distribution alone. To be confident of the fitted result, independent distributions like the moments presented also should be fitted and compared to the data.

5. Theoretical uncertainties from the model dependence

The use o f all the decay variables has improved significantly the sensitivity to P~. The method shown here, either through the combined fit of the moments or of the ~ variable, relates the various components of the hadronic current. Thus, particular care should be taken in understanding possible biases in the measurement coming from the model dependence of the hadronic structure functions Wx. An important difference exists between the moments method and the

variable method. To calculate the ~ variable for a given event, an explicit model for Wx must be chosen. Thus, the ~ histogram depends on the particular model used for the calculation of ~. The moments, instead, are calculated only from the decay variables and no assumption on a specific model is needed to produce the corresponding histograms. It is only when the moments are fitted that a particular model must be chosen.

In the following, the model dependence for both and the moments is analyzed, since the systematic effects in the two methods are different, and this allows useful cross-checks. The procedure used to estimate the uncertainties is the following. Distributions of the moments (and ~) for positive and negative r helicity, which will be referred in general as ~u+ and ~u_, have been generated with the KORALZ Monte Carlo program using the standard values for the Kiahn and San- tamaria model. Then, distributions of the moments (and ~) for a fixed value of the r polarization (PT = - 0 16) are produced with a modified version of KO-

RALZ containing the variation under study (see below). A linear combination of ~u+ and ~_ is used to fit P~ from these distributions, and the observed shift from the nominal P~ value is taken as an estimate of the uncertainty.

Several possibilities have been considered: (1) Vartatlon of the al parameters. The mass and

width of the al are not very well determined and depend on the specific model used to fit the 3n invanant mass distribution (see for example ref. [14]). Also, it has been shown in section 3 that the three pion Qz distribution and the total rate can significantly change depending on mal and F~ 1 . As far as P~ is concerned, no effect is expected in the case of the ~ variable. In fact, ~ can be written

~ x gx Wx - ~ x f x W x ' X = A , C , D , E . (38)

The al Breit-Wigner factor is the same for all Wx and can be factorized both in the numerator and denominator of (38), and thus it cancels. The correctness of this argument has been checked producing ~ distributions for different values of mal and F~ as input to the KORALZ Monte Carlo, and no difference has been found.

On the other hand, the moments could be affected, since they are proportional to Wx (cf. expressions (27 ) - (31 ) ) and thus to the al Breit-Wigner. Mo- ment distributions obtained changing ma, and Fal by +0.05 GeV have been produced to estimate the effect. In order to give a quantitative estimate of the possible bias, a combined fit of the moments obtained for each roam and F~, variation is performed, following the procedure described above. Also, Kiahn and San- tamaria model contains a contribution of the p' res- onance to the al decay. Distributions produced without the p ' contribution in the p Brelt-Wigner have been fitted. The maximum shift observed in all the described checks is AP~ = 0.01.

(2) Dtfferent theorettcal models. The models of Kuhn and Santamaria, Isgur et al. and Felndt have been compared in section 3 in their predictions for the hadronic structure functions. It is important to understand if the differences between the model predictions affect the PT measurement significantly. For this purpose, the models of Isgur et al. and Feindt have been introduced into the KORALZ Monte Carlo program, substituting for the Kiahn and San-

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tamar ia model, and the distr ibut ions sensitive to P~ (moments and ~) have been produced following the procedure described above. Notice that ~ is calculated using always the Kiahn and Santamaria model. The max imum observed shift in the fits performed is APt = 0.01. The dependence of P~ on rna~ and Fa~ has also been investigated for the predict ions of the Isgur et al. and Feindt models, following the procedure of point (1). Similar results have been obtained.

Another difference between the models is the treat- ment of the f O i l S ratio. Kuhn and Santamarm current implies a fixed value of the f D / f s ratio, which can be calculated from expressions (B 12) of ref. [ 12 ] taking gap~ = 0:

f o E? - - mp - -X/2Ep --~ 2-m-a "~ - 0 . 0 5 . (39) f s

The A R G U S Collaborat ion has measured [14] f D / f s = -0 .11 + 0.02. However, this result is ob- ta ined from a fit to the Isgur et al. model, and it is not excluded that the model of Kuhn and San tamana could fit the Dali tz plot projections for different in- tervals of Q2 which are used to determine f o / f s .

In the A R G U S paper it is also stated that the results are consistent with those obtained using the model of Femdt . The f D / f S ratio has been var ied by 4-0.06 in both Feindt and Isgur et al. models. The shifts found in the fi t ted value of P~ have been always smaller than 0.01.

(3) Effect o f a scalar contrtbutwn In the Kuhn and Santamaria model the effect of a possible scalar contr ibut ion (n ' ) is not considered (Ws~ = WSB =

Wsc = WSD = Wse = 0). In case such a contribu- t ion exists, one can ask what is the effect of neglecting it in the P~ measurement. It is clear that the combined fit of the moments will be only marginally affected by the presence of such contribution, since the only moment which should be modif ied is d N / d c o s O

which is not very sensitive to Pr. The ~ variable is modif ied by the presence of a scalar part in the sense that terms g x W x ( f x W x ) X = WsA .... should be added in the numerator (denominator ) of expression (38). Notice that the Isgur et al. model contains a scalar contr ibut ion, and thus this effect already is contained in the comparisons between the models previously performed. As a further check, the Ktihn and Mirkes parametr izat lon for the n ' has been in- t roduced into KORALZ and the procedure described

before has been applied. The observed variat ion is again negligible.

The various checks which have been performed in this section allow an est imate of the theoretical systematic error of the P~ measurement performed with the increased sensitivity method. The conclusion is that such an error is very small. A conservative estimate (APt)theor = 4- (0.0154-0.011 ) is obtained combining in quadrature the max imum shifts (0.01) from the variat ion of the a~ parameters and the comparison of the different models. The error on this estimate comes from the Monte Carlo statistics used for the various fits described in this section.

6. Conclusions

The r -+ 3nv~ decay allows the experimental in- vestigation of both electroweak and hadronlc interac- t ion properties, through P~ and Yv~ and the hadronlc structure functions, respectively.

It has been shown that different models for the hadronic current give different predict ions for the hadronic structure functions. However, the differences are significant only at large Q2. Thus, a large sample of z ~ 3nv~ decays is needed in order to distinguish between the models.

A new method to measure the z polarization, through the study of the z ~ 3nu~ decay rate determined by Kuhn and Mlrkes [3], has been investigated. When all the kinematical variables available in the decay are exploited, the sensitivity to Pr becomes S = 0.44, which is almost a factor two better than that obtained with the usual (cos 0, cos fl) distribution. Appropr ia te moments of the angular variables are well suited to perform the measurement. This result improves significantly the contr ibution of the r ~ 3nv~ decay mode to the P~ measurement with respect to the other decay channels. In particular, in both sensitivity and relative weight r --~ 3nu~ is

equivalent to the leptonic channels z ~ ev'Y and r ~ # v F combined. Prel iminary measurements of Pr with the improved sensitivity method described in this article have already been performed [ 17].

The use of all the decay variables has improved significantly the sensitivity to P~. However, the method discussed relates the various components of the hadronlc current. Thus, a specific study of

172


possible biases in the measurement coming from the model dependence of the hadronlc structure func- tmns has been performed. A conservative est imate (Aer)theor = + 0 . 0 1 5 has been obtained. This theoretical uncertainty will not l imit the P~ measurement, even in case of large statistics.

After complet ion of th~s paper, the author received information regarding related work by Davier et al. [ 18 ], where the improved sensmvity with the use of~ is discussed. A measurement of P~ [ 19 ] based on the work by Davier et al. [ 18] also has been performed.

Acknowledgement

Several subtle points in the theoretlcal aspects of this study have been clarified by discussions with E. Mirkes and M. Feindt. Also, the possibil i ty of using Monte Carlo programs written by J. H. Kuhn and E. Mirkes and by M. Femdt st imulated the study of the model comparison and allowed useful cross-checks of the results. The continuous support of W. de Boer is gratefully acknowledged.

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Structure functions in τ → 3πντ and the τ polarization measurement