CP violation in semileptonic τ decays with unpolarized beams

-

*.__ __ ll!B N-H

ELSEYIER

17 April 1997

PHYSICS

LETTERS B

Physics Letters B 398 (1997) 407-414

CP violation in semileptonic T decays with unpolarized beams J.H. Ktihn, E. Mirkes Institut fir Theoretische Teilchenphysik, Universitiit Karlsruhe, D-76128 Karlsruhe, Germany Received 30 September

1996; revised manuscript received 5 February 1997

Editor: RV. Iandshoff

Abstract CP violating signals in semileptonic r decays are studied. These can be observed in decays of unpolarized single r’s even if their rest frame cannot be reconstructed. No beam polarization is required. The importance of the two meson channel, in particular the KS- final state is emphasized. @ 1997 Elsevier Science B.V.

1. Introduction CP violation has been experimentally observed only in the K meson system. The effect can be explained by a nontrivial complex phase in the CKM flavour mixing matrix [ 11. However, the fundamental origin of this CP violation is still unknown. In particular the CP properties of the third fermion family are largely unexplored. Production and decay of 7 leptons might offer a particularly clean laboratory to study these effects. CP-odd correlations of the F and 7+ decay products, which originate from an electric dipole moment in the Q-pair production, are discussed in [ 2,3]. In this paper, we investigate the effects of possible non-KobayashiMaskawa-type of CP violation, i.e. CP violation effects beyond the Standard Model (SM) on semileptonic 7 decays. Such effects could originate for example from multi Higgs boson models [4]. CP violation effects in the r -+ 2~ decay mode from this scalar sector have recently been discussed in terms of “stage-two spin correlation functions” in [5] and in the case of polarized electron-positron beams at 7 charm factories in [ 61. CP violation in the three pion channel has been also discussed in [ 71

0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00233-5

and in the Kn-n- and KKn- channels in [ 81, where the latter analysis is based on the “T-odd” correlations as derived in [9] and the vector meson dominance parameterizations in [ lo]. In the present paper we show that the structure function formalism in [9] allows for a systematic analysis of possible CP violation effects in the two and three meson cases. Special emphasis is put on the AS = 1 transition 7 -+ Kn-vT where possible CP violating signals from multi Higgs boson models would be signalled by a nonvanishing difference between the structure functions WSF[T- -+ ( K?T) -vT] and WSF [r+ ---f ( K?T) +v7]. Such a measurement is possible for unpolarized single T’S without reconstruction of the 7 rest frame and without polarized incident e+ebeams. It is shown that this difference is proportional to IM (hadronic phases) x IM ( CP-violating phases) , where the hadronic phases arise from the interference of complex Breit-Wigner propagators, whereas the CP violating phases could arise from an exotic charged Higgs boson. An additional independent test of CP violation in the two meson case would require the knowledge of the full kinematics and T polarization. The paper is organized as follows: CP violating

J.H. Kiihn, E. Mirkes/Physics

408

terms in the Hamiltonian for 7 decays are discussed in Section 2. It is shown that CP violating signals induced through the exchange of an exotic intermediate vector boson can only arise, if both vector and axial vector hadronic currents contribute to the same final state, i.e. for final state which are not eigenstates of G parity and involve three or more mesons. The kinematics and the relevant form factors and structure functions for tests of CP violation in the two meson case are presented in Section 3. CP violation effects in three meson final states are briefly discussed in Section 4. Finally, it is shown in the appendix that the reconstruction of the full kinematics is possible for the case where one r decays into one charged hadron and the second r decays into a p or K* with the subsequent decay into a neutral and a charged meson.

2. CP violation in T decays The Hamiltonian responsible for 7 decays is decomposed into the conventional term of the SM, denoted by HIM, a CP violating term of similar structure, induced, e.g. by the exchange of a vector boson, HgG and a CP violating term induced by scalar or pseudo scalar exchange H&F’:

Letters B 398 (1997) 407-414

rived from the ratio I( 7 -+ VT) /I( r 4 YET) using as input f,, and the pion form factor from e+e- annihilation ’ . The study of CP violation will entirely rely on interference terms between HSM and H&j, H&T. Interference terms between the dominant (V - A) leptonic current in HSM and a possible V + A term in the leptonic current of H$’ ‘) are suppressed with the ratio between the mass of the r neutrino and m,. Therefore only contributions from left-handed neutrinos will be included, i.e. gv = gA = dv = gk = gs = gp = 1. The hadronic matrix elements of the currents dy”u (and similarly &*ysu) in HSM and H&,) are of course identical. From this argument it is evident that x and 7 play a fairly different role. Effects from XV and/or XA can only arise if both vector and axial hadronic currents contribute to the same final state and hence only for final states which are not eigenstates of G parity and involve three or more mesons. Conversely, for all two meson decays and, adopting isospin symmetry, even all multipion states, CP violation cannot * arise from a complex x. In contrast, CP violation can arise from a complex v, since J = 0 and J = 1 partial waves are affected differently in this case. For this reason contributions from nonvanishing x will be ignored in the following.

3. Two meson decays: kinematics, and structure functions x

H$

[dy”(l-~+]+h.c., = cos&

x

form factors

&

[F yLy(g: -gays)

[dy”(x$-x~ys)ul

H&i’ = cos&

v5

+h.c.,

[V (gs +gpys)

x [d (7; + q$ys )

Transitions from the vacuum to two pseudoscalar mesons hl and h2 are induced through vector and scalar currents only. Expanding these hadronic matrix elements along the set of independent momenta

~1

ul + h.c.3

(41 - q2)p and Qp = (st + q2)p we define

~1 (1)

plus a similar term with the complex parameters vd, xd replaced by vs. xs for the AS = 1 contribution. The dominance of ( V-A) contributions to the leptonic current in leptonic and semileptonic r-decays has been demonstrated experimentally under fairly mild theoretical assumptions. A pure (V - A) structure of the leptonic current in HSM will therefore be adopted for simplicity. A slight deviation from (V - A) for the hadronic current can in principle be masked by the form factors. However, tight restrictions can be de-

(ht (ql)hz(qz)

I~~&]O)

= (41 - q2Y Tap F(Q*)

(h(ql)Mqdl~+)

+

Qp MQ*>

= MQ*>.

(2) (3)

1The relative sign between hadronic vector and axial vector current is not fixed through this consideration. It can be determined by interference measurements in the Kmr and KKT channels, e.g. of the structure functions WEG,H,I [91. 21n this aspect we disagree with Ref. (61 where it has been claimed that CP violation in the 2~ channel can be induced through the exchange of an exotic intermediate vector boson.

J.H. Kiihn, E. Mirkes/Physics

The general amplitude decay r_(Z,s)

4 v(Z’,s’)

for the strangeness

+

h(q1,m)

+

conserving

x [(St -42)pTapF+e”&]

-ys)u(Z,s)

-‘y5)

dr(T-=

’17s FH mr

-+

(1 fY5).

9s

-+

rls*.

(8)

-+ 2hv,) {&wB

(6)

+ ‘%AwSA

+ ‘%FWSF

+ -hGwSG)

G2 xz

and similarly for the AS = 1 part. In Eq. (5) s denotes the polarization 4-vector of the r lepton satisfying Z,,# = 0 and s&‘ = -P2. P denotes the polarization of the r in the r rest frame with respect to its direction of flight in the laboratory frame and Tap is the projector onto the spin one part T@ = ffi

(1

(5)

with

& = Fs +

discussion we will include both nr and Kr final states. The corresponding T+ decay is obtained from Eq. (5) through the substitutions

Reaction (4) is most easily analyzed in the hadronic rest frame 4t + 42 = 0. After integration over the unobserved neutrino direction, the differential decay rate in the rest frame of ht + h2 is given by [ 9,1 l]

can thus be written as 3 ya(l

409

h2(q2,m2),

(4)

M=cos&ii(Z’,s’) 1/2

Letters B 398 (1997) 407-414

_ -Q"Qp .

Q2 As stated before, terms proportional to x do not contribute to CP violation in the two meson case and have therefore been neglected. The representation of the hadronic amplitude (hth$WpdlO) = (41 - qzYT,p F + QpFs corn--Ð sponds to a decomposition into spin one and spin zero contributions, e.g. the vector form factor F( Q2) corresponds to the Jp = 1- component of the weak charged current, and the scalar form factor FS ( Q2) to the Jp = O+ component. Up to the small isospin breaking terms, induced for example by the small quark mass difference, CVC implies the vanishing of FS for the two pion case. The small u and d quark masses enter presumably the couplings from charged Higgs exchange and thus lead to a small contribution from ~],FH. The perspectives are more promising for the AS = 1 transition r -+ Kn-u. The J = l- form factor F is dominated by the K*(892) vector resonance contribution. However, in this case the scalar form factor FS is expected to receive a sizable resonance contribution (- 5% to the decay rate) from the Ko ( 1430) with Jp = O+ [ 111. In the subsequent 3We suppress the superscript

d (or s) in 7s in the following.

dQ2 dcose x 1411 ZTG2’

da dcosp (9)

The coefficients LX contain all a,/3 and B angular and r-polarization dependence and will be specified below. The hadronic structure functions W,, X E {B,SA,SF;SG}, depend only on Q2 and the form factors F and F, of the hadronic current. The dependence can be obtained from Eq. (34) in [ 9] with the replacements x4 -+ 2 q1 , Fs -+ -iF, F4 --+ Fs. One has

WB[T-I = 4(4t)2

IFI 7

wSA[T-1

= Q2 I&/‘,

WSF[T-I

= 4@lqII

wSG[T-1

=

-4&74,IIm

(10) (11)

Re [FF’]

,

.

[FF;]

where /qt I = qf is the momentum frame of the hadronic system: 412 = $

([Q2 -mf

-rnz12

(12)

(13) of ht in the rest

-4m:mi)“2. (14)

The hadronic structure functions WX [ @] are obtained by the replacement 7s + 7; in Fs in Eqs. ( 11) -( 13), (6). CP conservation implies that all four structure functions are identical for rf and r-. With the ansatz for the form factors formulated in Eq. (5) CP violation can be present in WSF and WSG only and requires complex 7~s. As will be shown in the subsequent discussion CP violation in WSG in maximal for fixed r~s

J.H. Kiihn, E. Mirkes/Physics

410

Letters B 398 (1997) 407-414

in the absence of hadronic phases whereas WS,Vin contrast requires complex vs and hadronic phases simultaneously. We wiil now demonstrate that WS,Fcan be measured in efe- annihilation experiments in the study of single unpolarized r decays even if the r rest frame cannot be reconstructed. In this respect the result differ from earlier studies of the two meson modes where either polarized beams and reconstruction of the full kinematics [6] or correlated fully reconstructed rand rf decays were required [ 51. The determination of W,,, however, requires the knowledge of the full r kinematics and r polarization. For the definition and discussion of the angles and leptonic coefficients 15~ in Eq. (9) we therefore consider two different situations, depending on whether the direction of flight of the r in the hadronic rest frame [denoted by n,] can be measured or not.

I) The T direction in the hadronic rest frame is not known In this case, the angle p in Eq. (9) denotes the angle between the direction of hl (41 = q1/ 1q1I) and the direction of the laboratory rz~ viewed from the hadronic rest frame (see Fig. 1) cosP=nL-41.

(l-9

IZL is obtained from no. = -nQ, where nQ denotes the direction of the two-meson-system in the laboratory. The azimuthal angle (Yis not observable and has to be averaged out. The angle 0 (0 < 0 < n) in Eq. (9) which is only relevant for the analysis of polarized r’s, is the angle between the direction of flight of the r in the laboratory frame and the direction of the hadrons as seen in the C-rest frame. The cosine of the angle 8 is observable even in experiments, where the 7 direction cannot be measured experimentally. This is because cos 8 can be calculated [ 13,12,9] from the energy Eh of the hadronic system in the laboratory frame

case =

(2xm: - rn: - Q’) (mf - Q2)dw’

s = 4EEe,.

x=2-,

Fig. 1. Definition of the angles cx and p in two meson decays. and the r as seen from the hadronic rest frame. Again the cosine of this angle can be calculated from the hadronic energy Eh [ 13,12,9]. One has

coslj =

The angular coefficients &s~,s~,so Fq. (9) are given by [91

(17) for r-

decay in

&=2/3K,+K2-2/3&(3cos2+1)/2, GA = K2, GF

= -&

LSG

=o,

cosp, (18)

with

Kl = 1 - y~,PcosB KZ = (mz/Q2> (1 K 1 =

El2 fi

x(m: + Q2) - 2Q2 (ms-Q2)dv’

- (mz/Q2) (1 +yv,~PcosfY) , $_yVAPcos8),

Kl (3~0~~44 - 1)/2-3/2Kq

sin2$,

??2=K2cos+i-K4sinq, (16)

Of particular importance for the subsequent discussion is @, the angle between the direction of the laboratory

Kd= dGYvnPsin8, K5 = drn:/Q2

P sin 6,

(19)

J.H. Kiihn, E. Mirkes/Physdcs

Letters B 398 (19971407-414

411

where

(20) (KS in Eq. ( 19) is needed in the later discussion.) In our case of purely left-handed leptonic currents yVA = 1. For P = 0, the case relevant in the low energy region fi M 10 GeV, Eq. ( 18) simplifies to

-- ; (1-S)

?I2

&A = 7

(3cos2~-l)(3cos2p-1),

,

Q2 LSF = -2

rr12 cos$ cosp. Q2

(21)

Note that the angular coefficient &d vanishes if the hadronic rest frame is experimentally not known and only the distribution in /3 is considered. The differential rate (Eq. (9)) for the CP conjugated process can be obtained from the previous results by reversing all momenta p + -p, the 7 spin vector s -+ -s, the polarization P -+ -P, and the transition yVA + -yvA. CP therefore relates the differential decay rates for r+ and r- as follows:

and one obtains for the only nonvanishing spin-one term LSFWSF AWSF

=4@1qr]

$

Im(FFg)

Im(rls)

spin-zero

.

(27)

In essence this measurement analyses the difference in the correlated energy distribution of the mesons hr and h2 from r+ and r- decay in the laboratory. As already mentioned, AW.~F is observable for single r+ and rdecays without knowledge of the r rest frame. Any nonvanishing experimental result for A WSF would be a clear signal of CP violation. Note that a nonvanishing A W~F requires nontrivial hadronic phases (in addition to the CP violating phases TS) in the form factors F and FH. Such hadronic phases in F ( FH) originate in the Kkv, decay mode from complex Breit-Wigner propagators for the K* (K;) resonance. Sizable effects of these hadronic phases are expected in this decay mode [ 111. Once the r rest frame is known and a preferred direction of polarization exists one may proceed further, determine also WSG and thus perform a second independent test for CP violation. The subsequent analysis is performed for a r with spin direction s. IZ) The Q-direction in the hadronic rest frame is known

drb-l(%‘A,P,WX[~-l)

Note that the coefficients Lx contain the full yVAand P dependence. From the interference between the spinzero spin-one terms (denoted by the subscript X = Sl;; SG) one can construct the following CP-violating quantities:

As discussed in the appendix, the r direction can be determined in a large number of cases with the help of vertex detectors. In this case, the vector rz~ is replaced by n7 and hence + + 0, whereas cos 6 = --sn7. The angles LYand ,6 are in this case defined by (see Fig. 1 with the replacement no -+ n,; the X’ - Z/-plane is defined by the spin vector s and n,):

Ax = ;

cosp

(22)

-dr[7+l(-YVA,-~WX[7+1),

(LXhA,

-EX(-YVAT

=

zX(yVA,p)

f zxA Wx.

p) -p>

wx[T-l

(23)

wXb+l)

4 (WXb-1

-

WXb+l)

= n,. 41,

coscy =

(G x s>. (4 x

(24) (25)

As mentioned before the hadronic structure functions WX [ 7-1 and WX [ T+] differ only in the complex parameter 7s in

sina

(28)

= -

41)

I% x SI In, x $11 ’ s. (n, x 61)

I%xsl l~,Xcill’

and the coefficients

= K2.

(30)

Lx for the T- decay are given by

& = K1 sin’ /3 + K2 - K4 sin 2p cos a, &A

(29)

412

J.H. Kiihn, E. Mirkes/Physics

Lp

= - K2 cm p - K4 sin p cos LY,

Letters B 398 (1997) 407-414

AWSG

LSG = KS sin /3 sin a!.

=

42/e2lq1 I $ Re (FG)

Im (ml .

(41)

(31)

The coefficients for KI , Kz, Ks , K4, K5 for + Z 0 are given in ( 19). For P = 0, the case relevant in the low energy region Js M 10 GeV, Eq. (3 1) simplifies to (32)

Any observed nonzero value of these quantities would signal a true CP violation. Eqs. (40) and (4 1) show that the sensitivity to CP violating effects in AW,F and AWso can be fairly different depending on the hadronic phases, Whereas A WSF requires nontrivial hadronic phases AWso is maximal for fixed Q in the absence of hadronic phases.

(33) LSF =

4. Three meson decays

nl2 -? cosp, Q2

ESG= 0.

(35)

Evidently W,G cannot be measured in this case and thus no additional test of CP violation is possible. However, longitudinally polarized incident beams [ 61 or the study of r+ and Q-- correlations [ 51 could allow to recover polarization. The distributions in [ 61 are in fact equivalent to the product of &W~G if the angle a is defined with respect to the 7 spin vector through Eqs. (29)) (30). Correlations between the r+ and rdecay products may even allow to define the angle cy without r polarization. The differential rate (Eq. (9)) for the CP conjugated process can be obtained from the previous results by reversing all momenta p 4 -p, the r spin vector s -+ -s, the polarization P -+ -P, since --+ - sin a and the transition yVA + -yV.. CP therefore relates the differential decay rates for r+ and r- as dr[7-](sina,YVA,P,WX[7-]) +

(36)

dr[7’](-sina,-YVA,-P,WX[7+]).

From the interference between the spin-zero spin-one terms one can construct the following CP-violating quantities: Ax = 4 (tx(sina,YV..

P) WX[T-]

-Lx(-sina,-YVA,-P) = Ex(sinff, x;

WX[r’])

(37)

yv,4, P)

(WXFI

- wxb+l)

(38)

= L,A W,, AWSF

=

4@1q,

(39) I$

Im (Fq)

Im (ml

.

(40)

The structure function formalism [9] allows also for a systematic analysis of possible CP violation effects in the three meson case. Some of these effects have already been briefly discussed in [ 141. The Kn-r and KKr decay modes with nonvanishing vector and axial vector current are of particular importance for the detection of possible CP violation originating from exotic intermediate vector bosons. This would be signalled by a nonvanishing difference between the structure functions WX(T-) and WX( T+) with X E {F, G, H, I}. A difference in the structure functions with X E {SB, SC, SD, SE, SE SG) can again be induced through a CP violating scalar exchange. More details will be presented in a subsequent publication.

Appendix A. Tau kinematics in one-prong decays from impact parameters The reconstruction of the full kinematics is evidently a significant advantage in the analysis of r decays. It has been shown in [ 1.51 that this is possible for the r+(--+ rr+ii)r-( -+ r-v) final state if the tracks of rr+ and 7~~ are experimentally measured - even if the production vertex is not known. If the production vertex is known, the knowledge of one track of 7~+ or rTT- is sufficient. Also three-prong decays with measured tracks allow the full kinematic reconstruction. In this appendix the discussion of [ 151 is generalized to the case where one r decays into one charged hadron and the second T decays into p or K* with the subsequent decay into a neutral and a charged meson. For definiteness assume the r- decays into r-v and the r+ decays into p+(--+ n-+rrO)c. This is a generalization of the situation discussed in [ 151 where both r’s

J.H. Kiihn, E, Mirkes/Physics

were assumed to decay into one charged hadron each. In fact, the method is also applicable in the multibody one-prong decay, if the momenta of all neutral decays are determined experimentally. Assume that the momenta of all hadrons are measured and, furthermore, the tracks of & and V- are also measured. We shall demonstrate that also in this case the impact vector allows to reconstruct the original direction of r+ and r- in most of the cases. We use the following notation: n+ and n_ denote unit vectors into the directions of the p+ and the rTTmomentum. r denotes the direction of the V+ track. The cosines of the angles 19: between the r* directions and the hadronic systems in the lab frame are denoted by ck = cos S$ = n,* n*. c+ and c- can be calculated from the energy and mass of the hadronic system as giveninEq. (1) of [15]. In a first step, we determine the vector d which connects the r+ and r- decay vertex. As shown in [ 1.51 d can be geometrically be reconstructed from L up to a two-fold ambiguity. Only the c+,c-,n+, directions, but not the length of d can be obtained in this way, therefore we define D = d/ Idl. From EQ (9) in [ 1.51 we find the unit vector D=Dtiin+

1 1 - (n+n_)Z

I-- (c+(n+n-)

+

(c-(n+n-> + c+) n+l

E

Dtii, + Do.

+ CL) iz-

Letters B 398 (1997) 407-414

With In+ x n-1 = dw

Dei, denotes the direction of the vector which connects the points of closest approach between the straight lines through the decay vertex of r+ and rdirections n+ and n-, respectively, and hence is normal to rz+ and n-. DO is the remaining piece of D which by construction lies in the plane spanned by n+ and n_. Note that DC,, I DO by construction and hence

on obtains

(n+xn-) D = * (1 - (n+n_)2) x

1 - [c$ + cY_+ (n+n_)2

- (1 _ (nyn_)2) - (c-(n+n-)

[(c+(n+n->

+ c+)

02,

andD2=1,

(-4.4)

[((Dr)

+ ((Dn-)

(m_) (n-r>

- (DL)) - (Dr))

rl 1 _ c:n_j2 .

Up to now no use has been made of track measurements. At this point r is given by the momentum of the &. Inserting the two solutions Dk = fD1 + DO one obtains two predictions for v. vi = D+ + [((D+r)

+ ((D-n-)

= 1 - 0; =

l

_

f(n+Q2

(nyn_)2 {I- [c: +c: + 2c+c_(n+n_)]}

.

n-

(A.51

(n-p)

v2 = D_ + [((D-r)

Dki,-Do=Oonehas

n-

n+l

(m-)

- (D+n-))

- (D+r))

(A.21 (A.l)

+c-1

No information on the tracks has been used. In a second step one predicts from measured momenta the vector ct of minimal distance between the positive and negative track with the help of r, It-, and D. Depending on the two choices for D one will obtain two predictions for the direction of v. Frequently, only one of them will be compatible with the actual measurement. In order to calculate the vector v we choose a coordinate frame where the r- decay point lies at the origin and the negative and positive tracks are parameterized as An_ and D + hr respectively. [In reality one should scale D with ldl, the actual decay length of r+ plus r-. However, since we are only interested in directions, this factor drops out.] A straightforward geometrical calculation predicts

+ ((D+n-)

FromEq.

+ 2c+c_(n+n_)]

=~DJ_+Do.

v= D+ (A.1)

413

(n-r)

(m_)

rl

1 1 - (rn_)2 ’ (A.61

- (DAL))

- (D-r))

It-

iz_

rl 1 _ cin_)Z. (A.7)

(A.3)

Note that both solutions vi and v2 are normal to n- and parallel or antiparallel to each other, i.e. vi x v:! = 0.

J.H. Kiihn, E. Mirkes/Physics

414

Letters B 398 (1997) 407-414

Acknowledgements The work of E.M. was supported Contract Ku .502/5-l.

in part by DFG

References 60 -

Ill M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 ( 1973) 652.

121 W. Bemreuther and 0. Nachtmann, Phys. Rev. Lett. 63 40 -

20 -

0’.

’

0.5

’

.

’

1.0 mm,,

’

’ .

1.5

,”

[GeVl

Fig. A.l. Percentage of events where the difference in the direction of q and ~2 allows the full reconstruction.

The next step is to compare the measured u with the calculated solutions ut , v2 and select the proper one. In order that this strategy can work we have to restrict ourselves to the case where the two results for v in Eqs. (A.6), (A.7) correspond to two opposite directions for u (and not only to different lenghts which would not allow to resolve the ambiguity). Using Monte Carlo methods we have checked for the r-n-O configuration that this is possible for 100% of the events with rr-rrO at threshold, about 50% for events with m, N m,, (the dominant configuration!) and zero at the kinematical endpoint rnTT= m,, as illustrated in Fig. A.1. On the other hand, in the limit m, -+ m, the difference between the two solutions per se shrinks to zero, such that an approximate reconstruction seems possible in this case again.

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