



SLAC-PUB-6685 October 1994 T/E

Production of Polarized o/ Pairs and Tests of CP Violation Using Polarized e\Sigma Colliders Near Threshold ? y

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

ABSTRACT We consider the production of o/ pairs by electron-positron colliding beams at the maximum cross section near the threshold. At this energy o/ pairs are produced mostly in the s-wave which implies that the spin of the o/ pairs are almost always pointing in the beam direction independent of the production angle. When both electrons and positrons are longitudinally polarized in the same direction, for example 90%, one can obtain o/ pairs with 99% polarization in the direction of the polarization vectors of the incident beams. Tests of CP violation and study of the structure of weak interactions using such polarized o/ pairs are discussed.

Submitted to Physical Review D. ? Work supported by the Department of Energy, contract DE-AC03-76SF00515.

y Part of the content of this work was presented at the "Workshop on the Tau-Charm Factory

in the Era of B-Factory and CESR at SLAC", August 15-16, 1994, and the "Workshop on Hadron Physics at e+e

\Gamma Colliders at IEHP", Beijing, China, October 14-18, 1994.

1. Introduction CP violation in the Standard Model is highly ad hoc in the sense that it was invented to explain the decay of k` and it arbitrarily assumes that there is no CP violation in the leptonic and the first generation quark vertices. In this paper we propose to test whether there is CP violation in the o/ decay using the proposed Tau-Charm Factory with longitudinally polarized electron and positron beams.

The Tau-Charm Factory [1] is a proposed electron-positron colliding beam machine operating at around 4 GeV in the center of mass where o/ and charm particles have maximum cross sections. When a pair of spin 12 particles are produced near threshold they are produced mostly in the s-wave, resulting in polarizations of o/ \Sigma both pointing in the same direction [2] along either e+ or e\Gamma depending upon the initial polarization of the incident beams. This is true almost independent of the production angle. We show that at E = 2:087 GeV for the incident electron in the colliding beam the cross section is maximum, and the s-wave production is still dominant. For example, if e+ and e\Gamma are both polarized 90% in the direction of e\Gamma momentum, the o/ pair will be 99% polarized in the direction of e\Gamma momentum.

In Chapter 2 we compute the cross section and the polarization of o/ \Gamma and o/ + using longitudinally polarized e\Gamma and e+ beams. The cross sections and polarization of o/ \Gamma and o/ + from the Tau-Charm Factory are compared with those obtainable from the B-Factory.

In Chapter 3, we discuss how these polarized o/ \Sigma can be used to test CP violation, CPT violation, and conserved vector current theorem in o/ \Sigma decays. We constructed a very generic model of CP violation to investigate many salient features of possible CP violation in the semileptonic decay of o/ into 2ss.

2

In Chapter 4, we generalize the observations made in the previous chapter and devise ways to find CP violation in any CP violating decay mode and any CP violating production mechanism. We also conclude that assuming equal luminosities and initial e\Sigma polarizations, the Tau-Charm Factory is a factor 7.7 better than the B-Factory for checking CP violation in o/ .

2. Production of Polarized o/ \Sigma by

Polarized e\Sigma Colliding Beams

In our problem the mass of the electron can be ignored, the error caused by this approximation can be shown to be O(m2e=E2) by an explicit calculation, which is 10\Gamma 7 in our problem. When the mass is ignored (1 \Gamma fl5)=2 becomes left (right) handed helicity projection operator for an electron (positron), whereas (1 + fl5)=2 becomes right (left) handed helicity projection operator for an electron (positron). fl5 commutes with 1, fl5 and oe_*, but anti-commutes with fl_ and fl_fl5, thus in the electron positron annihilation the helicity of e+ and e\Gamma must be opposite to each other in order to annihilate if the current consists of vector and axial vector. The opposite holds for scalar, pseudo scalar or tensor. The standard electroweak interaction has only vector and axial vector interactions if we ignore the contribution from neutral Higgs exchange and g \Gamma 2 of the electron. The anomalous magnetic moment term is negligible at high energy because its contribution to the cross section is

O ` me2E ffss `n 2Em

e '

2! ,

10\Gamma 15

of the fl_ terms. We shall also ignore the possible existence of electric dipole moment of o/ because many people [3] have worked on this problem already. Thus

3

we assume CP conservation in the production of o/ pairs and we deal only with possible CP violation in o/ decay. In Chapter 4 we point out an all-purpose method for detecting CP violation including the one caused by the existence of an electric dipole moment of o/ . In this paper we shall also ignore the Z0 exchange diagram that contributes 10\Gamma 3 to the polarization. This does not affect the accuracy of our experiment because we cannot measure the polarization of electrons and positrons to this accuracy anyway.

Let H1 and H2 be the helicities of e\Gamma and e+ respectively. Let us write the cross section for e+e\Gamma ! o/ +o/ \Gamma as oe(H1; H2). The argument given above shows that with an accuracy of 10\Gamma 7 we have oe(+; +) = 0 and oe(\Gamma ; \Gamma ) = 0 and only oe(+; \Gamma ) and oe(\Gamma ; +) are not zero.

Suppose there are

N1+ electrons with helicity H1 = +, N1\Gamma electrons with helicity H1 = \Gamma , N2+ positrons with helicity H2 = +, and N2\Gamma positrons with helicity H2 = \Gamma 1.

The total number of events is proportional to

N1+N2\Gamma oe(+; \Gamma ) + N1\Gamma N2+oe(\Gamma ; +) : (2:1) The longitudinal polarizations (not helicities) of electrons and positrons are by definition:

w1 = N1+ \Gamma N1

\Gamma

N1 where N1 = N1+ + N1

\Gamma :

w2 = \Gamma N2+ \Gamma N2

\Gamma

N2 where N2 = N2+ + N2

\Gamma :

4

From these four equations we have

N1+

N1 =

1 + w1

2 ;

N1\Gamma

N1 =

1 \Gamma w1

2 ;

N2+

N2 =

1 \Gamma w2

2 ;

N2\Gamma

N2 =

1 + w2

2 : (2:2)

Substituting Eq. (2.2) into Eq. (2.1) we obtain

N1N2

4 [(1 + w1w2) foe(+; \Gamma ) + oe(\Gamma ; +)g + (w1 + w2) foe(+; \Gamma ) \Gamma oe(\Gamma ; +)g] :

(2:3)

From Eq. (2.3), we observe the following:

1. When both electrons and positrons are unpolarized, the cross section is by

definition

1 4 foe(+; \Gamma ) + oe(\Gamma ; +)g : (2:4)

When only the electron beam is polarized, the cross section is

1 4 foe(+; \Gamma ) + oe(\Gamma ; +)g +

w1

4 foe(+; \Gamma ) \Gamma oe(\Gamma ; +)g : (2:5)

Where both the electron and positron beams are polarized, the cross section is

1 + w1w2

4 foe(+; \Gamma ) + oe(\Gamma ; +)g +

w1 + w2

4 foe(+; \Gamma ) \Gamma oe(\Gamma ; +)g : (2:6)

2. Comparison of Eqs. (2.4), (2.5) and (2.6) shows that no new physics is

obtained by polarizing both beams. However when both beams are polarized and when the polarization of e+ is in the same direction as that of e\Gamma , the total number of counts is increased by a factor (1 + w1w2) and the effective polarization is increased from w1 to (w1 + w2)=(1 + w1w2). We shall often

5

assume that only the electron is polarized in order to simplify the calculation and discussion. When both e\Sigma are polarized all we need to do is to multiply the whole expression by a factor (1 + w1w2) and change w1 to (w1 + w2)=(1 + w1w2).

3. The w1 and w2 dependence of the cross section given here is applicable also

to e+e\Gamma ! Z0 ! o/ + + o/ \Gamma .

4. If we let w1 = w2 = 0:9, we obtain (w1 + w2)=(1 + w1w2) = 0:994.

In this paper we shall not assume the existence of the electric dipole moment of o/ , thus T is conserved in the production. When T is not violated, the polarization of o/ \Sigma cannot have components perpendicular to the production plane, i.e. terms proportional to (\Gamma !p 1 \Theta \Gamma !p \Gamma ) \Delta \Gamma !w must be zero, where \Gamma !p 1 and \Gamma !p \Gamma are momenta of e\Gamma and o/ \Gamma respectively, and \Gamma !w is the polarization vector of o/ \Gamma , because\Gamma !

p 1; \Gamma !p \Gamma and \Gamma !w all change signs under T . There is no complex phase associated with the interaction to allow the existence of such a T violating term. Under CP transformation the polarization of o/ \Gamma turns into polarization of o/ + denoted by \Gamma !w

0,

w1 ! w2, \Gamma !p \Gamma ! \Gamma \Gamma !p +, and \Gamma !p 1 ! \Gamma \Gamma !p 2. Thus

\Gamma !w = \Gamma !w 0 : (2:7)

This statement is true even when Z0 is exchanged.

In this paper we use the convention of my 1971 paper (see Section IV of that paper). We use the three-dimensional vectors \Gamma !s and \Gamma !w in the rest frame of o/ \Gamma to represent its spin and polarization vectors respectively. \Gamma !s is an unit vector

6

whereas \Gamma !w is defined as

wi =

Number of o/ \Gamma with \Gamma !s = bei \Gamma Number of o/ \Gamma with \Gamma !s = \Gamma bei

Number of o/ \Gamma with \Gamma !s = bei +Number of o/ \Gamma with \Gamma !s = \Gamma bei

(2:8)

(s\Gamma )_ is the four vector which becomes (0; \Gamma !s ) in the rest frame of o/ \Gamma . We define similar vectors \Gamma !s

0, (s

+)_ and \Gamma !w

0 for o/ +. The cross section for producing o/ \Gamma with

spin \Gamma !s and o/ + with spin \Gamma !s

0 with initial polarization w

1 for e\Gamma and w2 for e+ can

be written as:

doe d\Omega (w1; w2; \Gamma !s ; \Gamma !s

0) =

= e

4

(2ss)2

1 4(p1 \Delta p2) Z

d3p+

2E Z

d3p\Gamma

2E ffi

4(p1 + p2 \Gamma p

\Gamma \Gamma p+)

\Theta 14 Tr(1 + fl5w1)p=1fl_(1 + fl5w2)p=2fl* \Theta 14 Tr(1 + fl5s=\Gamma )(p=\Gamma + M )fl* (1 + fl5s=+)(p=+ \Gamma M )fl_

= ff

2

16E2 fi(1 + w1w2)" ae1 + cos

2 ` + sin2 `

fl2 oe + ( `1 \Gamma

1 fl2 ' sin

2 `(s

\Gamma \Delta s+)

+ 1E2 h2(p1 \Delta s\Gamma )(p1 \Delta s+) \Gamma (p1 \Delta s\Gamma )(p\Gamma \Delta s+)(1 + fix) \Gamma (p1 \Delta s+)(p+ \Delta s\Gamma )(1 \Gamma fix)i)

+ w1 + w21 + w

1w2

1 flE f2(p1 \Delta s

\Gamma ) + 2(p1 \Delta s\Gamma ) \Gamma (p\Gamma \Delta s+) \Gamma (p+ \Delta s\Gamma g # ;

(2:9) where x = cos `, fl = E=M , and fi = (1 \Gamma fl\Gamma 2)0:5. We notice that w1, w2, (pi \Delta s\Gamma ) and (pi \Delta s+) are pseudoscalars, therefore these quantities have to occur

7

an even number of times in our expression because we are dealing with parity conserving electromagnetic interactions in the production. Parity conservation is violated when Z0 exchange is included. at our energy the correction due to weak interaction is O(4E2=M 2z ) = 10\Gamma 3. The first curly bracket in Eq. (2.9) represents the cross section when the final polarizations are not measured, the second curly bracket represents the spin correlation and it was first discussed by the author [2] in 1971 and treated subsequently by many people, so we shall not discuss it here. The third curly bracket contains terms which produce polarization. Since we do not have to observe both polarizations at the same time we let s+ = 0. We can obtain the polarization vector \Gamma !w for o/ \Gamma using Eqs. (2.8) and (2.9). For this calculation we shall use the coordinate system shown in Fig. 1. In this frame, for \Gamma !s = bez 0, we have

x'

t- P-

P1

z' z e-

y' b q

a

w

9-94fl7816A1 Figure 1. Coordinate system used in calculating the polarization vector \Gamma !w for o/

\Gamma . w

y = 0

because of T invariance in the production of o/ pairs.

8

s\Gamma = (fifl; 0; 0; fl) : (2:10) For \Gamma !s = bex 0, we have

s\Gamma = (0; 1; 0; 0) : (2:11)

p1 = E(1; sin `; 0; cos `) : (2:12) p\Gamma = E(1; 0; 0; fi) : (2:13) p+ = E(1; 0; 0; \Gamma fi) : (2:14)

The magnitude of the polarization can be obtained readily from Eqs. (2.8) through (2.14)

j\Gamma !w j = (w2x 0 + w2z 0 )1=2 = fifififi w1 + w21 + w

1w2 fifififi

2Epp2 cos2 ` + M 2 E2 + M 2 + p2 cos2 ` ; (2:15)

where p2 = E2 \Gamma M 2. The component of \Gamma !w along the o/ \Gamma direction is

wz 0 j j\Gamma !w j cos ff = j\Gamma !w j E cos `pp2 cos2 ` + M 2 : (2:16) The component of \Gamma !w along the incident electron direction is

wz j j\Gamma !w j cos fi = j\Gamma !w j E cos

2 ` + M sin2 `p

p2 cos2 ` + M 2 : (2:17) Equation (2.15) shows that at ` = 0 or 180ffi, the magnitude of the polarization is always maximum independent of energy:

j\Gamma !w jmax = fifififi w1 + w21 + w

1w2 fifififi : (2:18)

In Fig. 2a, the magnitudes of the o/ \Sigma polarization are plotted assuming j\Gamma !w jmax = 1 for the Tau-Charm Factory energy E = 2:087 GeV and the B-Factory energy

9

E = 6:0 GeV . It is seen that at energy E = 2:087 GeV where the cross section is maximum, the polarization is almost complete but at the B-Factory energy the polarization is less complete even if the incident electron is completely polarized.

10-94fl7816A02 -1.0 -0.5 0 0.5

|w|

-0.5 0.5

1.0 cosq

0.7 0.8 0.9 1.0 E = 2.087 GeV

E = 6 GeV

a)

-1.0

-1

1.0

1

E = 2.087 GeV

E = 6 GeV

cosq

cosa

b)

0.6

Figure 2. (a) Magnitude of o/ polarization j\Gamma !w j as a function of cos ` assuming completely polarized electron beam. (b) cos ff versus cos `, where ff is the angle between \Gamma !w (polarization of o/ ) and \Gamma !p

\Gamma .

In Fig. 2b, the cosine of angle between o/ \Gamma and its direction of polarization is plotted for E = 2:087 and 6.0 GeV . \Gamma !w is almost parallel to the e\Gamma direction if w1 is positive for E = 2:087 GeV whereas for E = 6:0 GeV \Gamma !w is no longer so parallel to the initial electron polarization because the production is no longer dominated

10

cosb

Wz

E = 2.087 GeV

E = 6 GeV

E = 2.087 GeV 0.6 0.7 0.8 0.9 1.0

E = 6 GeV

-1.0 -0.5 0

0.84

0.88

cosq 0.5 1.0

-1.0 -0.5 0

cosq

0.5 1.0

1.00 a)

b) 10-94 7816A3

0.92

0.86

Figure 3. (a) cos fi versus cos `; (b) wz is the component of o/ polarization vector along the electron beam direction.

by the s wave.

In Fig. 3a we plot the cosine of the angle between the o/ \Sigma polarization vector and the incident electron assuming it to have positive helicity. At ` = 0ffi; 90ffi and 180ffi, o/ \Sigma polarization is always parallel to the electron polarization at all energies. cos fi is almost equal to 1 for E = 2:087 GeV but not quite so for the B-Factory energy.

In Fig. 3b we plot components of o/ \Sigma polarization along the electron direction assuming the electron to be completely right-handed polarized.

11

2.1 Total Cross Section and Production Rate

The first curly bracket in Eq. (2.9) gives the differential cross section summed over the final spins. Integrating it with respect to solid angle we obtain the total cross section:

oe(e+e\Gamma ! o/ +o/ \Gamma ) = r

2ess

6 i

me

M j

2 fi(1 \Gamma fi2)(3 \Gamma fi2)(1 + w

1w2) : (2:19)

The cross section has a maximum at fi = p1:5 \Gamma p1:5 = 0:5246 or E = 2:087 GeV for M = 1:777 GeV . When fi = 0:5246 we have fi(1 \Gamma fi2)(3 \Gamma fi2) = 1:036: Let us therefore write f (fi) = (1=1:036) fi(1 \Gamma fi2)(3 \Gamma fi2) and oe(e+e\Gamma ! o/ +o/ \Gamma ) = oemaxf (fi)(1 + w1w2) where

oemax = r

2ess

6 i

me

M j

2 1:036 = 3:562 \Theta 10\Gamma 33cm2 :

Table 1 Energy dependence of the cross section for e+e\Gamma ! o/ +o/ \Gamma .

fi E(GeV ) f (fi) 0.1 1.786 0.2857 0.3 1.863 0.7668 0.4 1.939 0.9210 0.5 2.052 0.9953 0.5246 2.087 1.0000

0.55 2.128 0.9988

0.6 2.221 0.9785 0.9951 6.0 0.1688

12

Table I gives the numerical value of f (fi). We notice that at the B-Factory energy the cross section is 1/6 that of the maximum cross section at E = 2:087 GeV .

The factor (1 + w1w2) is the spin dependence of the total cross section. When either w1 = 0 or w2 = 0 this factor is one. When w1 = w2 = \Sigma 1 this factor is 2. When w1 = \Gamma w2 = \Sigma 1 this factor is zero. In the circular ring if one waits long enough, positrons (electrons) will be polarized parallel (antiparallel) to the magnetic field, reaching the value 0.924 if the guiding field is uniform. These transverse polarizations can be rotated 90ffi so that polarizations become longitudinal. In the ideal case we have w1 = w2 = \Sigma 0:924. In this case we have (1 + w1w2) = 1:85: The time necessary to reach this maximum possible radiative beam polarization is too long with the existing design of the Tau-Charm Factory. The time dependence of the polarization is [4,5] p(t) = 0:924 (1 \Gamma e\Gamma t=Tpol ) ; where Tpol in sec is given by

Tpol(sec) = 98:7 r

2R

E5

where

E = 2:087 GeV , is the beam energy r = 12 meters, is the bending radius R = 60 meters, is the mean radius of the machine.

Tpol is approximately 6 hours (which is too long). One can reduce this time by reducing r and R and also by inserting wigglers. Another way to obtain the polarized beam is to inject a polarized electron beam which reaches about 80% polarization at SLAC now but eventually may reach almost [6] 100%. Polarized positrons [7] can be obtained by pair production using high energy circularly polarized photons

13

produced by back scattering of polarized laser beams on high energy electrons.

The design luminosity in 1989 was 1033cm\Gamma 2=sec, but now it probably could [8] reach 3 \Theta 1033cm\Gamma 2=sec. Using 1033 we obtain a rate of 3.56 (1 + w1w2) o/ pairs/sec. Thus we obtain (1 , 6)\Theta 108 o/ pairs/year. This means with several years of running one can obtain a sensitivity of 10\Gamma 4 for testing CP violation in the o/ decay. If CP violation in o/ decay is of order 10\Gamma 3, similar to the neutral kaon decay, we should be able to investigate the structure of CP violation in o/ decay using the Tau-Charm Factory.

3. Tests of CP and CPT Violations in o/ Decay In quantum mechanics, the time reversal operator, T , is the least intuitive among T , C, and P operators, because under T i must become \Gamma i in addition to changing t into \Gamma t. The requirement of i going into \Gamma i can be seen by applying T to the most important commutators in quantum mechanics:

[xi; pj] = iffiij : (3:1) T [xi; pj] T \Gamma 1 = \Gamma [xi; pj] : Thus the commutation relation, Eq. (3.1), will not be true unless T iT \Gamma 1 = \Gamma i.

In order to construct a T noninvariant model, we first construct a T invariant interaction with a real coupling constant and then make this real coupling constant complex with a nonvanishing imaginary part.

Let A = jAjeiffi

w be such a coupling constant with ffiw 6= 0 or ss for o/ \Gamma decay.

We have T AT \Gamma 1 = A\Lambda 6= A, thus T is violated in the theory. Testing the existence of ffiw in the o/ decay is the purpose of this chapter. In quantum mechanics, the

14

overall phase of the matrix element of any process is undetectable because the transition probability is square of the matrix element. Thus the complex coupling constant must be defined with respect to some other coupling constant whose phase is known. Only the interference between the two will produce a T violating effect.

The weak Hamiltonian responsible for o/ \Sigma decay can be written in general as

Hweak = X

i `

j+i J \Gamma i q2 \Gamma M 2i +

j\Gamma i J +i q2 \Gamma M 2i ' ; (3:2)

where j+i represents a leptonic current whose final charge \Gamma initial charge is positive, i.e. o/ \Gamma ! *o/ , i represents different particles exchanged such as left-handed W 's, right-handed W 's, charged Higgs, etc. J \Gamma i is the hadronic or leptonic current whose final charge \Gamma initial charge is negative. The first term in Eq. (3.2) gives the decay of o/ \Gamma , whereas the second term gives the decay of o/ +. One of the requirements of TCP theorem is that Hweak be Hermitian, and thus the second term is the Hermitian conjugate of the first. Therefore if there is any complex coupling constant in the decay of o/ \Gamma , the corresponding coupling constant for the o/ + decay must be the complex conjugate of the former.

Let A = jAjeiffi

w be the complex coupling constants responsible for the T noninvariant decay of o/ \Gamma , then TCP invariance demands that the coupling constant A responsible for the T noninvariant o/ + decay must be

A j jAj eiffi

w = jAj e\Gamma iffiw ; (3:3)

which implies jAj = jAj and ffiw = \Gamma ffiw. If either of these is violated, TCP is violated.

15

In the semileptonic decay mode of o/ with more than one hadron in the final state, for example o/ \Gamma ! *o/ ss\Gamma ss0, we have complex phase due to final state interactions given by Breit-Wigner's formula for the p wave resonance (ae). Because the strong interaction is invariant under charge conjugation this phase is not changed when going from o/ \Gamma ! *o/ + ss\Gamma + ss0 to o/ + ! *o/ + ss+ + ss0. Let the phase shift due to strong interaction be ffis, we have then for o/ \Gamma decay the phase factor ei(ffi

s+ffiw),

but for o/ + decay we have ei(ffi

s\Gamma ffiw), if TCP is conserved but T is violated. The

existence of the strong phase makes it possible to detect the existence of ffiw even from seemingly T invariant term such as \Gamma !w \Delta \Gamma !q 1, where \Gamma !w is the polarization of o/ \Gamma and \Gamma !q 1 is the momentum of ss\Gamma .

In the previous chapter we showed that o/ can be polarized almost 100% and its direction of polarization is almost along the beam direction independent of the production angle (see Figs. 2 and 3) at E = 2:087 GeV . We have also shown that the polarization vector for o/ \Gamma and o/ + are parallel to each other and equal in magnitude as long as CP invariance holds in the production. This extra polarization vector \Gamma !w of o/ \Gamma enables us to construct rotationally invariant dot products such as c1\Gamma !w \Delta \Gamma !q 1 or c2(\Gamma !w \Theta \Gamma !q 1)\Delta \Gamma !q 2 where \Gamma !q 1 and \Gamma !q 2 are the momenta of decay product of o/ \Gamma and similar quantities c 01\Gamma !w

0 \Delta \Gamma !q 0

1, c

02(\Gamma !w 0 \Theta \Gamma !q 01) \Delta \Gamma !q 02 where

\Gamma !w 0 is the polarization vector of o/ + and \Gamma !q 01 and \Gamma !q 02 are the momenta of the charge

conjugates of \Gamma !q 1 and \Gamma !q 2 respectively.

Under CP we have \Gamma !q 1 ! \Gamma \Gamma !q

0

1, \Gamma !q 2 ! \Gamma \Gamma !q

0 2 and \Gamma !w ! \Gamma !w

0. Thus \Gamma !w \Delta \Gamma !q

1 !

\Gamma \Gamma !w

0 \Delta \Gamma !q 0

1, \Gamma !w \Delta \Gamma !q 2 = \Gamma \Gamma !w

0 \Delta \Gamma !q 0

2, (\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2 ! (\Gamma !w

0 \Theta \Gamma !q 0

1) \Delta \Gamma !q

0 2, \Gamma !p 1 ! \Gamma \Gamma !p 2,

and w1 ! w2 under CP operation. Thus if CP holds we have

c1 = \Gamma c 01 and c2 = c 02 (3:4)

16

and violation of Eq. (3.4) is violation of CP invariance. \Gamma !w \Delta \Gamma !q 1 is T even and CP odd, thus c1 + c 01 6= 0 means not only CP violation but also CPT violation for any process which does not have a strong interaction phase such as pure leptonic decay mode and any semileptonic decay with only one hadron, such as *o/ + ss and *o/ + k. In the leptonic decay of o/ there is only one visible final state, thus one cannot construct the triple product (\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2. Only when the polarization of the final e or _ is measured, one can test the CP violation from the pure leptonic decay of o/ unless CPT is violated. Similarly any CP violating effect in the decay o/ ! *o/ + ss or o/ ! *o/ + k means CPT is also violated.

We conclude that only the semileptonic decay modes of o/ with two or more hadronic final particles can exhibit CP violation without violating CPT at the same time. The best candidate is the decay mode o/ \Sigma ! *o/ + ss\Sigma + ss0. Let us investigate this mode in detail and learn several interesting lessons. The lessons learned can obviously be applied to other decay modes.

3.1 o/ \Gamma ! *o/ + ss\Gamma + ss0 and o/ + ! *o/ + ss+ + ss0.

The energy angle distributions of these two decay modes from polarized o/ 's had been worked out in detail in my 1971 [2] paper several years before the discovery of the o/ . The investigation of possible CP violation using these two decays had been carried out by C. A. Nelson et al. [9] using spin correlation methods first proposed in my 1971 paper [2]. Since in the Tau-Charm Factory o/ \Sigma can be made highly polarized we do not need to use the spin correlation which requires the detection of twice the number of particles and thus is more complicated. We also note that in our method s and p wave interference in the two ss state is crucial in untangling the CP violation whereas Nelson el al.'s paper does not seem to have any s wave.

17

The two ss decay modes have two distinguished advantages. 1. They have the largest branching ratio (25%). 2. It has a two-body (detectable) hadronic final state which has a large phase shift (ae resonance). This makes it possible to have a coefficient of \Gamma !w \Delta \Gamma !q 1 violating CP invariance without violating TCP invariance. It also enables one to construct a triple product term (\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2 to test CP invariance. Our investigation is exploratory. We want to know how different types of CP violating terms in various Lagrangians manifest themselves as the CP violating effect in the experiment.

We shall assume that the o/ neutrino mass is either zero or so small that anything that is of order (m*=mo/ )2 is unobservable experimentally. With this assumption 1\Gamma fl5 and 1+fl5 are good helicity projection operators for the o/ neutrino states and the matrix element containing (1 \Gamma fl5)u(*o/ ) and that containing (1 + fl5) u(*o/ ) do not interfere. As mentioned previously the complex coupling constant responsible for CP violation can manifest itself only through interference with other terms which have a real coupling constant. This consideration shows that one cannot obtain a CP nonconserving effect through interference of right-handed current with the left-handed current by assuming that the coupling constant of the former has a weak phase compared with the latter.

The consideration given above also shows that if we limit the weak interaction to be transmitted only 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. 4) that can interfere with the Standard Model matrix denoted by M0:

M0 = u(p2)(q=1 \Gamma q=2)(1 \Gamma fl5) u(p1) L (3:5) M1 = u(p2) fP (q=1 \Gamma q=2) + S(q=1 + q=2)g (1 \Gamma fl5) u(p1) (3:6)

18

9-94fl7816A4

po (q2) p- (P1)

nt (P2) p- (q

1)

L gu(1-g5) (

q1-q2)u W-

(a)

po (q2) t- (P1)

nt p- (q

1)

gu(1-g5) P (q1-q2)u+S(q1+q2)uX-

(b)

po (q2) t-

nt p- (q

1)

(1+g5)

HH-

(c)

Figure 4. Feynman diagrams for M0, M1, M2 defined in Eqs. (3.5), (3.6) and (3.7); (a) M0: W

\Gamma exchange; (b) M1: X\Gamma exchange; (c) M2: H\Gamma exchange.

M2 = u(p2)(1 + fl5) u(p1) H (3:7)

p1; p2; q1; q2 are momenta of o/ \Gamma , *o/ , ss\Gamma and ss0 respectively and [2]

L = go/ae* gaessss \Gamma 1(q

1 + q2)2 \Gamma M 2ae + i\Gamma Mae

= go/ae* gaessss e

iffis1q

((q1 + q2)2 \Gamma M 2ae )2 + \Gamma 2M 2ae

:

(3:8)

ffis1 is the strong interaction phase shift for the ss\Gamma ss0 system in p wave (ae resonance). Notice that the conserved vector current theorem [2] in the Standard Model says that ss\Gamma ss0 cannot be in the S state.

19

M1 is another left-handed current due to exchange of a higher mass spin 1 particle called X. For this current, both s and p waves are allowed for the ss\Gamma ss0 system because there is no CVC theorem here and we allow T violating complex coupling constants in M1. The vector particle X couples to all leptons and quarks, probably obeying some yet to be discovered symmetry principle. In our problem X is coupled to both o/ *o/ and the first generation quarks ud. Thus we will be seeing the combined effect of CP violation in both the o/ *o/ and ud sectors. Let the complex weak phase for the o/ *o/ X vertex be exp(i ffiwo/X) and that for the udX vertex be exp(iffiw1X). Then in our problem only the combination

ffiwX j ffiwo/X + ffiw1X (3:9) will appear.

The term P in Eq. (3.6) contains the same strong interaction phase factor exp(iffis1) defined in Eq. (3.8) and thus in the interference between M0 and M1 given by M +0 M1 + M +1 M0 this strong interaction phase factor cancels out. Thus the term P does not contribute to the CP violating effect, only the term S in Eq. (3.6) does. The s wave part contains the I = 2; J = 0 ss\Gamma ss0 phase factor eiffi

s0

which is different from the p wave one.

M2 is the matrix element for charged Higgs exchange [10]. The part proportional to (1 \Gamma fl5) in Eq. (3.7) does not interfere with M0, so we left it out. It has s wave interaction phase factor exp(iffis0) and the weak phase factor exp(iffiwH), where

ffiwH j ffiwo/H + ffiw1H ; (3:10) where ffiwo/H is the T violating weak phase associated with o/ *o/ H vertex, while ffiw1H is the similar phase for the first generation quarks. In summary the phases

20

associated with L, P , S and H are

L = jLj exp(iffis1) (3:11) P = jP j exp(iffis1 + iffiwX) (3:12)

S = jSj exp(iffis0 + iffiwX) (3:13) H = jHj exp(iffis0 + iffiwH) : (3:14)

If CPT invariance holds we have for o/ + decay

L = jLj exp(iffis1) (3:15) P = jP j exp(iffis1 \Gamma iffiwX) (3:16)

S = jSj exp(iffis0 \Gamma iffiwX) (3:17) H = jHj exp(iffis0 \Gamma iffiwH) : (3:18)

Since strong interaction is C invariant, the strong interaction phase shifts ffis1, ffis0 are not changed when going from o/ \Gamma to o/ + whereas the weak phases ffiwX and ffiwH change sign because of Hermiticity of the Lagrangian, which results in the TCP Theorem. The decay energy-angle distribution of the decay, polarized o/ \Gamma ! *o/ + ss\Gamma + ss0 can be written as:

\Gamma = 12M

o/

1 (2ss)5 Z

d3p2

2E2 Z

d3q1

2w1 Z

d3q2

2w2 ffi

4(p1 \Gamma p2 \Gamma q1 \Gamma q2) jM0 + M1 + M2j2 :

(3:19) We assume M1 and M2 to be much smaller than M0, therefore we compute: [11]

(M0+M1+M2)+(M0+M1+M2) ,= M +0 M0+(M +0 M1+M +1 M0)+(M +0 M2+M +2 M0)

21

M +0 M0 = 2jLj2 Tr4 (1 + fl5w=)(p=1 + M )(1 + fl5)(q=1 \Gamma q=2)p=2(q=1 \Gamma q=2)(1 \Gamma fl5)

= 2jLj2"4(w \Delta q1)M \Phi (q1 \Delta q2) \Gamma (p1 \Delta q1) + (p1 \Delta q2) \Gamma m2ss\Psi

+ 4(w \Delta q2)M \Phi (q1 \Delta q2) + (p1 \Delta q1) \Gamma (p1 \Delta q2) \Gamma m2ss\Psi + 4n \Gamma (q1 \Delta q2)(p1 \Delta q1 + p1 \Delta q2 + m2ss) + (p1 \Delta q1)2 + (p1 \Delta q2)2

\Gamma 2(p1 \Delta q1)(p1 \Delta q2) + m2ss(p1 \Delta q1 + p1 \Delta q2) \Gamma m2ssM 2o# :

(3:20) This gives the energy-angle distribution of ss\Gamma and ss0 in the Standard Model which was treated in detail in my 1971 paper [2].

22

M +0 M1 + M +1 M0

= 2jLje\Gamma iffi

s1 Tr4 (1 + fl5w=)(p=1 + M )(1 + fl5)(q=1 \Gamma q=2)p=2

\Theta fP (q=1 \Gamma q=2) + S(q=1 + q=2)g (1 \Gamma fl5) + 2jLjeiffi

s1 Tr

4 (1 + fl5w=)(p=1 + M )(1 + fl5)

\Theta fP \Lambda (q=1 \Gamma q=2) + S\Lambda (q=1 \Gamma q=2)g p=2(q=1 + q=2)(1 \Gamma fl5)

= 4jLj"4(w \Delta q1)M cos ffiwX(q1 \Delta q2 \Gamma p1 \Delta q1 + p1 \Delta q2 \Gamma m2ss) jP j

+ 4(w \Delta q1)M cos(ffis0 \Gamma ffis1 + ffiwX)(q1 \Delta q2 \Gamma p1 \Delta q1 + m2ss) jSj + 4(w \Delta q2)M cos ffiwX(q1 \Delta q2 + p1 \Delta q1 \Gamma p1 \Delta q2 \Gamma m2ss) jP j + 4(w \Delta q2)M cos(ffis0 \Gamma ffis1 + ffiwX)(\Gamma q1 \Delta q2 + p1 \Delta q2 \Gamma m2ss) jSj + (\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2M 2 sin(ffis0 \Gamma ffis1 + ffiwX) jSj + 4 cos ffiwXn \Gamma (q1 \Delta q2)(p1 \Delta q1) \Gamma (q1 \Delta q2)(p1 \Delta q2) + (q1 \Delta q2)m2ss

+ (p1 \Delta q1)2 \Gamma 2(p1 \Delta q1)(p1 \Delta q2) + (p1 \Delta q1)m2ss + (p1 \Delta q2)2 + (p1 \Delta q2)m2ss \Gamma m2ssM 2o jP j + 4 cos(ffis0 \Gamma ffis1 + ffiwX)n \Gamma (q1 \Delta q2)(p1 \Delta q1) + (q1 \Delta q2)(p1 \Delta q2)

+ (p1 \Delta q1)2 \Gamma (p1 \Delta q1)m2ss \Gamma (p1 \Delta q2)2 + (p1 \Delta q2)m2sso jSj#

(3:21)

23

M +0 M 2 + M +2 M0 = 2jLj Tr4 (1 + fl5w=)(p=1 + M )(1 + fl5)(q=1 \Gamma q=2)p=2(1 + fl5)H

+ 2jLj Tr4 (1 + fl5w=)(p=1 + M )(1 \Gamma fl5)p=2(q=1 \Gamma q=2)(1 \Gamma fl5)H\Lambda

= 4jHj"2(w \Delta q1) cos(ffis0 \Gamma ffis1 + ffiwH)(2(p1 \Delta q2) \Gamma m2ss)

+ 2(w \Delta q2) cos(ffis0 \Gamma ffis1 + ffiwH)(\Gamma 2(p1 \Delta q1) + m2ss) + 4(\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2M sin(ffis0 \Gamma ffis1 + ffiwH) + 2M (p1 \Delta q1 \Gamma p1 \Delta q2)#

(3:22)

3.2 Observations

1. The decay energy angle distribution of o/ + ! *o/ + ss+ + ss0 can be obtained

by reversing all momenta of the particles p1 ! \Gamma p 01, p2 ! \Gamma p 02, q1 ! \Gamma q 01, q2 ! \Gamma q 02 and reverse the signs of all weak phases ffiwX ! \Gamma ffiwX, ffiwH ! \Gamma ffiwH. When CP is conserved, i.e. ffiwX = ffiwH = 0, the coefficients of w \Delta q1 and w \Delta q2 change sign but the coefficients of (\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2 remain the same under CP operation in agreement with Eq. (3.4). If ffiwX 6= 0 or ffiwH 6= 0, then Eq. (3.4) is violated thus CP is violated.

2. Only the interference between s wave in M1;2 and p wave in M0 contributes

to the triple product term (\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2. Experimentally the existence of this term manifests itself as the asymmetry of ss0 distribution with respect to the plane formed by \Gamma !w and ss\Gamma momenta. CVC is an exact statement in the Standard Model, thus the existence of the triple product term shows the existence of weak interaction mechanisms other than the Standard Model.

24

CP is violated if the asymmetry in o/ \Gamma ! *o/ + ss\Gamma + ss0 is different from that for o/ + decay. 3. The P wave part of M1 does not contribute to the observable CP violation

because cos ffiwX = cos(\Gamma ffiwX). From this example we can make a very interesting conclusion: Unless two diagrams have two different strong interaction phases, we cannot observe the existence of weak phase using terms involving w \Delta q1 or w \Delta q2. This is because w \Delta q1 and w \Delta q2 are T even in the absence of strong interaction phase differences. Thus we cannot have CP violation without violating CPT using these terms.

4. When the strong interaction phases in M0 and M1 are different the CP

violation is proportional to

cos(ffis0 \Gamma ffis1 + ffiwX) \Gamma cos(ffis0 \Gamma ffis1 \Gamma ffiwX) = 2 sin(ffis1 \Gamma ffis0) sin ffiwX (3:23) for the coefficients of w \Delta q1 and w \Delta q2, but

sin(ffis0 \Gamma ffis1 + ffiwX) \Gamma sin(ffis0 \Gamma ffis1 \Gamma ffiwX) = 2 cos(ffis1 \Gamma ffis0) sin ffiwX (3:24) for the coefficients of (\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2. We notice that when ffis1 \Gamma ffis0 = 0, Eq. (3.23) is zero whereas Eq. (3.24) is maximum. The physical reason for the former is already explained in point 3 and the reason for the latter is that (\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2 is T odd. Thus CP violation in this term does not cause violation of CPT even in the absence of strong interactions. 5. Exactly the same observation as point 4 can be made for Eq. (3.22). 6. All observable effects in CP violation can only be produced by the interference

between the p wave in M0 and the s wave in M1 and M2 in our model. Our model is generic, so it must be true in general.

25

4. Discussions and Concluding Remarks Since o/ + and o/ \Gamma are not observable directly, we have to integrate the production angles and obtain energy-angle distribution of ss\Gamma (q1) and ss0(q2) for o/ \Gamma decay and ss+(q 01) and ss0(q 02) distributions for o/ + decay. Since we are not doing spin correlation experiments, they do not have to come from the same event. We investigate here features of these energy-angle distributions which will exhibit the CP violation after integrating over o/ \Sigma momenta. To simplify the argument let us assume that only the incident electron is polarized. As mentioned in Chapter 2, this does not change any physics. All we need to change is to increase the overall cross section by a factor (1 + w1w2) and replace the electron polarization w1 by (w1 + w2)=(1 + w1w2) when positron has a polarization w2.

Let us choose the direction of polarization of e\Gamma as well as its momentum as the z axis and ss\Gamma (q1) lies on the xz plane as shown in Fig. 5.

f x

e- (P1) and W1

po (q2) p - (q1)

z

y q1 q2

7816A5

9-94

Figure 5. Coordinate system used in Eqs. (4.1) through (4.10).

26

There are 6 rotationally invariant products involving \Gamma !w 1:

\Gamma !w 1 \Delta \Gamma !q 1 = w1q1z (4:1) \Gamma !w 1 \Delta \Gamma !q 2 = w1q2z (4:2) \Gamma !w 1 \Delta \Gamma !q 01 = w1q 01z (4:3) \Gamma !w 1 \Delta \Gamma !q 02 = w1q 02z (4:4)

(\Gamma !w 1 \Theta \Gamma !q 1) \Delta \Gamma !q 2 = w1q1xq2y (4:5) (\Gamma !w 1 \Theta \Gamma !q

0

1) \Delta \Gamma !q

0 2 = w1q

01xq 02y : (4:6)

Under CP we have

\Gamma !w 1 ! \Gamma !w 2 ; \Gamma !q 1 ! \Gamma \Gamma !q 01 ; \Gamma !q 2 ! \Gamma \Gamma !q 02 ; \Gamma !p 1 $ \Gamma \Gamma !p 2 ; (4:7)

where \Gamma !p 1 and \Gamma !p 2 are momenta of electron and positron respectively. We note that (w1 + w2)=(1 + w1w2) is symmetric with respect to w1 $ w2. Let f1(q1z), f2(q2z), f 1(q 01z), f 2(q 02z) be the longitudinal distribution of ss\Gamma , ss0 (from o/ \Gamma ), ss+, and ss0 (from o/ +) respectively. Let f3(q1x; q2y) and f 3(q 01x; q 02y) be the transverse momentum distributions of ss\Gamma ss0 for o/ \Gamma and those of ss+ss0 for o/ + respectively. If CP is invariant, we have

f1(q1z) = f 1(\Gamma q 01z) ; (4:8) f2(q2z) = f 2(\Gamma q 02z) ; and (4:9) f3(q1x; jq2yj) \Gamma f3(q1x; \Gamma jq2yj) = f 2(q 01x; jq 02yj) \Gamma f 2(q 01x; \Gamma jq 02yj) : (4:10)

Violation of any one of the equalities in Eqs. (4.8), (4.9) and (4.10) signifies the violation of CP. Nonvanishing of either side of Eq. (4.10) signifies the violation of

27

CVC but does not imply the violation of CP unless the equality is violated. The difference in the detection efficiencies of ss+ and ss\Gamma may make Eq. (4.8) rather difficult to verify, but Eq. (4.9) does not have this problem.

As mentioned previously, for leptonic decays or o/ ! *o/ + ss (or k) we cannot have violation of equality like Eq. (4.8) without violating CPT. Thus observation of violation of equality like Eq. (4.8) for these modes is evidence of violation of CPT in these decay modes.

For decays such as o/ ! *o/ + ss + k, o/ ! *o/ + 3ss we do not have CVC, thus observation of nonvanishing of either side of Eq. (4.10) does not imply violation of the Standard Model. However violation of equality in any one of Eqs. (4.8), (4.9) or (4.10) signifies CP violation in these modes.

Since the derivations of Eqs. (4.8), (4.9) and (4.10) are independent of detail mechanisms of CP violation, they are applicable to all decay channels, as well as all possible CP violations in production of o/ 's such as the existence of o/ electric dipole moment. Experimentalists can go ahead and measure the differences between the left and right hand sides of Eqs. (4.8)-(4.10), while theorists can figure out how different models of CP violation will affect the behavior of these functions.

The applications of colliding beams with polarized e\Sigma in the production of other particles have not been fully investigated. When hadrons are produced instead of o/ 's, their production angles can usually be reconstructed because their decays usually do not involve neutrinos. The method used in Chapter 3 can be used for example in the analysis of \Lambda \Lambda and ,, production and their decays. The discussions on physics involved in using the transversly polarized e\Sigma machine can be found in my 1975 paper [12].

28

4.1 B-Factory versus Tau-Charm Factory for Testing CP in o/ Decay

Let us compare the B-Factory and Tau-Charm Factory for testing CP violation as described in this chapter. Since we are going to integrate with respect to the production angle of o/ , we expect the z component of the o/ polarization wz given by Eq. (2.17) averaged over the differential cross section to give the effective polarization. We obtain from Eqs. (2.17) and (2.9):

wz =

1Z

\Gamma 1

wz doedx dxOEoe = w1 + w21 + w

1w2

1 + 2a

2 + a2 j

w1 + w2 1 + w1w2 F (a) (4:11)

where a = M=E. a = 0:8514 and 0.2961 respectively for E = 2:087 and 6:0 GeV ; and for E = 2:087 GeV we have F (0:8514) = 0:992, and for E = 6:0 GeV we have F (0:2961) = 0:763. We note F (1) = 1 and F (0) = 0:5. The total cross section is given by Eq. (2.19) which has a factor (1 + w1w2) that cancels out with the denominator in Eq. (4.11).

Finally the overall merit factor for each machine is

Merit = Luminosity \Theta wz\Theta total cross section

/ Luminosity \Theta (w1 + w2) \Theta p1 \Gamma a2 a2(1 + 2a);

where a = M=E :

(4:12)

Thus if electron and positron are unpolarized, i.e. w1 = w2 = 0, it has zero value. Assuming the luminosity and the initial beam polarization to be the same for the two machines, the merit factor is determined by the function fm(a) =p

1 \Gamma a2 a2(1 + 2a). For the Tau-Charm Factory we have fm(0:8514) = 1:0276 whereas for the B-Factory we have fm(0:2961) = 0:1333. Thus the Tau-Charm Factory is better than the B-Factory by a factor 7.7 if both have the same luminosity and the initial beam polarizations.

29

ACKNOWLEDGMENTS The author wishes to thank Professor T. D. Lee for making a critical remark, and J. D. Bjorken, Richard Prepost, and David Hitlin for useful discussions.

REFERENCES [1] John Jowett, in Proceedings of the Tau-Charm Factory Workshop (1989), p.

7; SLAC Report 343 (1989). [2] Y. S. Tsai, Phys. Rev. 4D, 2821 (1971). [3] All earlier papers on the electric dipole moment of o/ can be traced back from

B. Aranthanarayan and S. D. Rindani, Phys. Rev. Lett. 73, 1215 (1994). [4] A. A. Sokolov and I. M. Ternov, Sov. Phys.-Dokl. 8 (1964). [5] J. R. Johnson, R. Prepost, D. E. Wiser, J. J. Murray, R. F. Schwitters and

C. K. Sinclare, Nuclear Instruments and Methods 204, 261 (1983). [6] R. Prepost, private communication. [7] Y. S. Tsai, Phys. Rev. D48, 96 (1993). [8] J. Kirkby, private communication. [9] Charles A. Nelson, et al., SUNY-BING-8-16-93 (1993). [10] CP violation from Higgs exchange has been considered by: T. D. Lee, Phys.

Rev. D8, 1226 (1973), and Phys. Rep. 9C, 143 (1974) and S. Weinberg, Phys. Rev. Lett. 37, 657 (1976). [11] Throughout this paper our metric is p1 \Delta p2 = p10p20 \Gamma \Gamma !p 1 \Delta \Gamma !p 2. The term

(\Gamma !w \Theta \Gamma !q 1) \Delta \Gamma !q 2 is Eps (w; q1; q2; p1)=M evaluated in the rest frame of o/ \Gamma (p1 = (M; 0; 0; 0)). [12] Y. S. Tsai, Phys. Rev. 12D, 3533 (1975).

30

