



SLAC-PUB-95-6916 Test of T and CP Violation in Leptonic Decay of o/ \Sigma \Lambda

Yung Su Tsai Stanford Linear Accelerator Center Stanford University, Stanford, California 94309

ABSTRACT The o/ \Sigma , highly polarized in the direction of the incident beam, can be obtained from the e\Sigma collider with the polarized incident e\Gamma (and preferably also the e+)

beam. This polarization vector \Gamma !w i = (w1 + w2)=(1 + w1w2)bez can be used to construct the T odd rotationally invariant product (\Gamma !w i\Theta \Gamma !p _). \Gamma !w _, where w1 and w2 are longitudinal polarization vectors of e\Gamma and e+ respectively; \Gamma !p _ and \Gamma !w _ are the momentum and polarization of the muon in the decay o/ \Gamma ! _\Gamma + *_ + *o/ . T is violated by the existence of such a term. CP can be tested by comparing it with a similar term in o/ + decay. If T violation in such a decay is milliweak or stronger, one can find it using the proposed polarized o/ -charm factory with luminosity of 1 , 3 \Theta 1033=cm2= sec. One can test whether T (and CP ) violation is due to the charged Higgs boson exchange by doing a similar experiment for the _\Sigma decay.

To be presented at the "Workshop on the Tau/Charm Factory,"

21-23 June 1995, Argonne National Laboratory

Submitted to Physical Review Letters. \Lambda Work supported by the Department of Energy, contract DE-AC03-76SF00515.

In the Standard Model of Kobayashi and Maskawa [1] CP violation occurs as a result of a complex phase in the unitary matrix relating gauge eigenstates and mass eigenstates. The leptonic sector does not have CP violation if all neutrinos are massless. Both of these assumptions could be wrong. It is quite possible [2] that CP violation is due to exchange of some new particle such as a heavier W boson or a charged Higgs boson. If CP violation is milliweak or stronger in o/ decay, one should be able to observe it in the proposed o/ -charm factory where it is expected to have 1 , 3 \Theta 108 highly polarized o/ pairs per year [2].

The CP violation in o/ production can be ignored because we are dealing with electromagnetic production. The radiative correction due to CP violation in the weak interaction is of order 10\Gamma 5 if it is weak, but 10\Gamma 8 if it is semiweak [3]. In contrast to the production, the decay of o/ is weak, thus CP violation is of order 1 if it is weak and 10\Gamma 3 if it is milliweak. Up to now, the only CP violation is from KL which is 2 \Theta 10\Gamma 3. In a recent paper [2] we dealt with the CP violation in the semileptonic decay of o/ with two or more final hadrons. For a single hadron in the semileptonic decay or a leptonic decay, the only rotationally invariant quantity we can form is \Gamma !w i \Delta \Gamma !q where \Gamma !q is the momentum of the final visible particle. But this term is T even so we cannot have CP violating effects from this term without violating TCP [4]. It is very desirable to measure CP violation in pure leptonic decay because in the semileptonic decay it is impossible to assign CP violation to the leptonic or hadronic vertex [2]. For the leptonic decay we have to measure the polarization of the muon and construct a rotationally invariant product

(\Gamma !w i \Theta \Gamma !p _) \Delta \Gamma !w _ ; (1) where \Gamma !

w i = w1 + w21 + w

1w2 be

z ; (2)

with w1 and w2 being the longitudinal polarization of the incident electron and positron respectively; \Gamma !p _ is the laboratory momentum, and \Gamma !w _ the polarization of the muon. The muon polarization is measured by the asymmetry in electron distribution coming from the term \Gamma !w _ \Delta \Gamma !q e\Gamma where \Gamma !q e\Gamma is the electron momentum. Thus the correlation in Eq. (1) induces the correlation

c(\Gamma !w i \Theta \Gamma !p _) \Delta \Gamma !q e\Gamma ; (3)

which is also odd under T .

Equation (3) means that if one finds an asymmetry in the perpendicular component of\Gamma ! q e\Gamma with respect to the plane formed by \Gamma !w i and \Gamma !p _, one discovers the existence of T violating effect. Under CP we have w1 $ w2, wi ! wi, \Gamma !p _\Gamma $ \Gamma \Gamma !p _+, \Gamma !q e\Gamma $ \Gamma \Gamma !q e+. Thus if CP is conserved, we have for o/ + decay

c0(\Gamma !w i \Theta \Gamma !p _+ ) \Delta \Gamma !q e+ (4) with c0 = c. But since T is violated by both Eqs. (3) and (4), we better have c0 = \Gamma c in order to preserve T CP invariance. The discussion given above is completely model independent. Later we shall give a model which will illustrate all the above observations.

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In order to measure the angular asymmetry in the decay electrons, we have to slow the muon down to almost at rest. For o/ energy Eo/ equal to 2.087 GeV, where the cross section is maximum [2], the maximum and minimum muon momenta are respectively:

pmax3 = Eo/2 "(1 + fi) \Gamma _

2

M 2 (1 \Gamma fi)# = 1:589 GeV (5)

pmin3 = \Gamma Eo/2 "(1 \Gamma fi) \Gamma _

2

M 2 (1 + fi)# = \Gamma 0:4904 GeV (6)

where

_ = 0:105658 GeV fi = q1:5 \Gamma p1:5 M = 1:777 GeV

pmin3 is negative means that it is going in the opposite direction to the o/ momentum. Approximately 10% of muons are going backward. The asymmetry caused by the detector can be checked by reversing the polarization of the incident beam (or beams).

In this paper we use the same model of T and CP violation as the previous paper [2]. It is shown there that if we limit the weak interaction to be transmitted by exchange of spin 1 and spin 0 particles, then we have only two possible choices of matrix elements denoted by M1 and M2 (see Fig. 1) that can interfere with the Standard Model matrix denoted by M0.

M0 = A u(p2)fl_(1 \Gamma fl5)u(p1)u(p3)fl_(1 \Gamma fl5)v(p4) ; (7) M1 = B u(p2)fl_(1 \Gamma fl5)u(p1)u(p3)fl_(1 \Gamma fl5)v(p4) ; (8) M2 = C u(p2)(1 + fl5)u(p1)u(p3)(1 \Gamma fl5)v(p4) ; (9)

where p1; p2; p3 and p4 are momenta of o/ \Gamma , *o/ , _\Gamma and *_ respectively. We have assumed that m*_=m_ and m*o/ =mo/ to be either zero or too small to be experimentally observable, so that possible terms such as (1 + fl5)u(p4) and (1 + fl5)u(p2) are ignored in M1 and M2. A is chosen to be real while B and C are allowed to be complex. Since there is no final state interaction the imaginary parts of B and C cause T violating effects. If T CP is conserved then B and C for the o/ + decay must be the complex conjugate of B and C:

B = B\Lambda and C = C\Lambda : (10)

Since the Standard Model is good to 10\Gamma 3 to 10\Gamma 2, we can assume M +1 M1 and M +2 M2 to be at most 10\Gamma 2 compared to M +0 M0 and thus we shall ignore them. We shall also ignore M1 completely because its interference with M0 does not depend upon the imaginary part of B that causes the T violation:

M +0 M1 + M +1 M0 = (B + B\Lambda )M +0 M0=A : (11)

2

t-(p1) t-

nt(p2)

nt

u-(p3) nu(p4) W-

u- nu X-

t-

nt

u-

nu H-

(a) M0 (b) M1 (b) M2 6-95fl7971A1 Figure 1: (a) M0: Feynman diagram for o/ \Gamma ! _\Gamma + *_ + _o/ in the Standard Model that conserve T and CP . (b) M1: A possible T violating spin 1 exchange diagram that is shown not to contribute to the T violating effect. (c) M2: A T violating spin 0 exchange diagram with a complex coupling constant.

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Writing M0 = AM0 and M2 = CM2, we have

M +0 M2 + M +2 M0 = A Re C(M+0 M2 + M+2 M0) + iA Im C(M+0 M2 \Gamma M+2 M0) : (12) Only the imaginary part of C contributes to T violation. The real part of B and C should be of order 10\Gamma 2 or less compared with A, thus they will be ignored. We have therefore

(M0 + M1 + M2)+(M0 + M1 + M2) ss A2M+0 M0 + iA Im C(M+0 M2 \Gamma M+2 M0) :

Let \Gamma !w 3 be the polarization vector of the muon in the rest frame of the muon. We are interested in the y component of \Gamma !w 3 defined in Fig. 2(a) whose existence signifies the

violation of T because (\Gamma !w \Theta \Gamma !p 3) \Delta \Gamma !w 3 is odd under T . After averaging over the o/ production angle the polarization vector of o/ \Gamma , \Gamma !w , is replaced by the initial beam polarization \Gamma !w i. We note that since the y direction is perpendicular to \Gamma !p 3 it is invariant under the Lorentz boost along \Gamma !p 3. The y component of the muon polarization can be calculated using the formula

W3y =

i Im C R "(M+0 M2\Gamma M+2 M0)\Gamma !s

=bey

\Gamma (M+2 M2\Gamma M+2 M0)\Gamma !s

=\Gamma bey #

\Theta d

3p2

2E2

d3p4

2E4 ffi

4(p1\Gamma p2\Gamma p3\Gamma p4)

A R PSpin of _(M+0 M0) d

3p2

2E2

d3p4

2E4 ffi4(p1 \Gamma p2 \Gamma p3 \Gamma p4)

: (13)

The second term inside the square bracket of the numerator is the negative of the first, thus the bracket is equal to twice the first term. The phase space integration with respect to the two undetected neutrons is carried out in the rest frame of u = p2 + p4. We denote quantities in this frame by b.Z

d3p2

2E2

d3p4

2E4 ffi

4(p1 \Gamma p2 \Gamma p3 \Gamma p4) = Z d b\Omega 4 bE4

2 ffi(u

2 \Gamma 2u bE4) d bE4

= 18 Z d b\Omega 4 ; (14)

(m+0 m2 \Gamma m+2 m0)

= Tr4 (1 + fl5 6 w)(6 p1 + M )(1 + fl5)fl_ 6 p2 Tr4 6 p4(1 + fl5)fl_(1 + fl5 6 s)(6 p3 + m)

\Gamma Tr4 (1 + fl5 6 w)(6 p1 + M )(1 \Gamma fl5) 6 p2fl_ Tr4 6 p4(1 + fl5)(1 + fl5 6 s)(6 p3 + m)fl_ = 4i" \Gamma (s \Delta p4)EP S(w; p1; p3; p4) + (p3 \Delta p4)EP S(s; w; p1; p4)

+ 12 (p1 \Delta p4)EP S(s; w; p1; p3) \Gamma 12 n(p1 \Delta p3) \Gamma m2o EP S(s; w; p1; p4)# : (15)

4

q

f x

y

p3

p4W

y z p4

x

W

p2

p1=p3

6-95fl7971A2

(a) (b)

z

Figure 2: (a) The rest frame of o/ \Gamma , the coordinate system used in Eq. (18). (b) The rest frame of u = p2 + p4 = p1 \Gamma p3, the coordinate system used in integrating out the two unobserved neutrinos in Eqs. (13) and (14). This frame is obtained from the above diagram

by boosting against the direction of \Gamma !p 3.

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In the above we have dropped all those terms that are odd in bp4x, bp4y, and bp4x because they yield zero after integration with respect to d b\Omega 4. The Levi-Civita EPS's are evaluated in the rest frame of o/ and then p4_ is Lorentz transformed to bp4_ before the angular integration shown in Eq. (14). In the rest frame of u = p2 + p4 we have bp1 = bp3, thus the Lorenz boost is in the z direction [5]: E4 = fl bE4 \Gamma fifl bp4z, p4x = bp4x, p4y = bp4y and p4z = \Gamma flfi bE4 + fl bp4z with fl = (M \Gamma E3)=u, fi = p3=u, u = pM 2 + m2 \Gamma 2M E3. The result of the angular integration is Z

(M+0 M2 \Gamma M+2 M0)\Gamma !s =b

ey d b\Omega 4 =

4ssi

3 h3M

2 \Gamma 4E3M + m2i EP S(bey; w; p1; p3) ; (16)

with

EP S (bey; w; p1; p3) = M (\Gamma !w \Theta \Gamma !p 3)y :

The denominator in Eq. (13) can be obtained similarly:Z X

spin of_

(M+0 M0) d b\Omega 4 = 32ss M

2E3

3 "3M \Gamma 4E3 \Gamma

2m2

E3 +

3m2

M + (\Gamma !w \Delta \Gamma !p 3)

M E3 \Gamma 4 +

3m2 E3M !# :

(17) Equation (17) agrees with the result of my previous paper [6] written several years before the discovery of the o/ .

Putting everything together we have finally:

W3y = \Gamma (\Gamma !w \Theta \Gamma !p 3)y8E

3 h

3M \Gamma 4E3 + m

2

M i Im (C=A)

3M \Gamma 4E3 \Gamma 2m

2

E3 +

3m2

M + (\Gamma !w \Delta \Gamma !p 3) i

M E3 \Gamma 4 +

3m2 ME3 j :

(18)

For o/ + decay we use the substitution \Gamma !p 3 ! \Gamma \Gamma !p

0

3, \Gamma !w ! \Gamma !w

0, E

3 ! E

03, C ! C. C is equal

to C\Lambda if TCP is conserved [2]. Under CP we have w ! w0, \Gamma !p 3 = \Gamma \Gamma !p

0

3, thus it is oppositeto the TCP conserved case. Thus CP must be violated in order to conserve TCP.

The polarization of the muon is analyzed by the decay electron momentum \Gamma !q . Thus the measurement of the existence of the T violating term (\Gamma !w \Theta \Gamma !p 3) \Delta \Gamma !w 3 can be done by

measuring the existence of the T violating correlation (\Gamma !w \Theta \Gamma !p 3) \Delta \Gamma !q , where \Gamma !p 3 and \Gamma !q are momenta of muon and decaying electron in the rest frame of o/ . Exactly at threshold such a correlation can be calculated using Eq. (18), but as energy is increased one must integrate over the o/ production angle. The result must be proportional to the only T noninvariant correlation in the center-of-mass system (\Gamma !w i \Theta \Gamma !p _) \Delta \Gamma !q e, where \Gamma !w i is the initial beam

polarization defined in Eq. (2); \Gamma !p _ and \Gamma !q e are center-of-mass momenta of the muon and electron respectively.

We have shown above that only the spin 0 exchange can produce T violating leptonic decay of the o/ . By measuring a similar effect in _\Sigma ! e\Sigma + *e + *_ one should be able to decipher if the exchanged particle is the Higgs boson discussed by T. D. Lee [7] and S. Weinberg [8]. The test of T , CP , and charged Higgs boson exchange in the leptonic decay

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of o/ proposed in this paper as well as the test of CP , T CP and CV C in the semileptonic decay of o/ proposed in my previous paper [2] are mostly for the proposed tau-charm factory. However they can also be carried out in the B factories being constructed at SLAC, KEK and Cornell provided they add a capability to longitudinally polarize their initial electron (and preferably also positron) beam. It is regrettable that none of the B factories mentioned above have any plans to polarize their incident beam (or beams). At the B factory energy the cross section is about 1/6 that of the tau-charm factory for producing o/ pairs and the polarization of produced o/ is about 23% less favorable due to the reduced s wave dominance in the production [see Eq. (4.11) of Ref. 2]. However the luminosity of the machine is supposed to be roughly proportional to the energy that is a factor of three in favor of the B factory. Thus the tests proposed in this paper and Ref. [2] are still do-able with the B factories if they polarize their incident beams.

Acknowledgments The author wishes to thank Professor W.K.H. Panofsky for reviving my interest in the Tau-Charm Factory. I also would like to thank Bill Dunwoodie, Charles Prescott and Francesco Villa for consultation on measurement of transverse polarization of the muon and electron.

References

[1] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652, (1973). [2] Y. S. Tsai, Phys. Rev. D51, 3172 (1995). [3] The effect due to the electric dipole moment of o/ can be regarded as the weak or

milliweak radiative corrections to the electromagnetic vertex of o/ and thus the correction must be of order (mo/ =mw)2ff = 3 \Theta 10\Gamma 6 if it is weak and another factor of 10\Gamma 3 if it is milliweak. The interference of one fl exchange and one Higgs boson exchange is (me=E)(E2=m2H ) mo/ me=m2w ss (m2em2o/ )=(m2H m2w) = 1:23 \Theta 10\Gamma 10GeV2=m2H .

[4] T. D. Lee made this valuable remark. [5] For readers interested in the technical aspects of the calculation, the choice of the rest

frame of u = (p1 \Gamma p3) = (p2 + p4) to do the angular integration, the choice of the

direction of \Gamma !p 3 as the z axis to facilities Lorenz transformation and the choice of the rest frame of o/ to express the final result are very important in expediting the whole calculation.

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[6] Y. S. Tsai, Phys. Rev. 4D, 2821 (1971). See Eq. (2.10). [7] T. D. Lee, Phys. Rev. D8, 1226 (1973); Phys. Rep. 9C, 143 (1974). [8] S. Weinberg, Phys. Rev. Lett. 37, 657 (1976).

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