CP violation from new quarks in the chiral limit

ELSEVIER

Nuclear Physics B 510 ( 1998) 39-60

CP violation from new quarks in the chiral limit F. de1 Aguila”, J.A. Aguilar-Saavedra”,

G.C. Branco b

’ Departamento de Fisica Teo’rica y de1 Cosmos, Universidad de Granada, 18071 Granada, Spain b Departamento de Fisica and CFIFIUTL, Institute Superior Tknico,

1096 Lisbon, Portugal

Received 26 March 1997; revised 18 September 1997; accepted 24 October 1997

Abstract We characterize CP violation in the SU(2) x U( 1) model due to an extra vector-like quark or sequential family, giving special emphasis to the chiral limit mu,& = 0. In this limit, CP is conserved in the three-generation Standard Model (SM), thus implying that all CP violation is due to the two new CP violating phases whose effects may manifest either at high energy in processes involving the new quark or as deviations from SM unitarity equalities among imaginary parts of invariant quartets (or, equivalently, areas of unitarity triangles). In our analysis we use an invariant formulation, independent of the choice of weak quark basis or the phase convention in the generalized Cabibbo-Kobayashi-Maskawa matrix. We identify the three weak-basis invariants, as well as the three imaginary parts of quartets Bt-3 which, in the chiral limit, give the strength of CP violation beyond the SM. We find that for an extra vector-like quark ]Bi] < 10e4, whereas for an extra sequential family ]Bi] < 10e2. @ 1998 Elsevier Science B.V. PACS: 11.3O.Er; 12.15.Ff; 12.60.-i; 1480.-j Keywords: CP violation; Vector-like quarks; High energy colliders; Top quarks; Chiml symmetries

1. Introduction In the Standard Model (SM) CP violation

is parametrized

by one CP violating

phase

in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [ 11. Although this phase accounts for the observed CP violation in the K”-i(” system [ 21, there is no deep understanding of CP violation. Furthermore, it has been established that the amount of CP violation present in the SM is not sufficient to generate the baryon asymmetry of the universe (see for a review Ref. [ 31). This provides motivation for looking for new sources of CP violation which can lead to deviations from the SM predictions for CP asymmetries in B” decays and/or to new signals of CP violation observable at high energy, in future colliders. 0550-3213/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PIISO550-3213(97)00708-6

40

E de1 Aguila et al./Nuclear

In the SM the CKM matrix Kj are strongly

constrained

parts of all invariant particular, one has

Physics B SlO (1998) 39-60

V&M is a 3 x 3 unitary matrix whose matrix elements

by unitarity

which implies, for example, that the imaginary

quartets, Im KjV$Vk\b$

(i # k, j + 1), are equal up to a sign. In

In the SM one may have CP violation in the limit mu,d = 0, but degeneracy of two quarks of the same charge does imply CP invariance. Hence in the chiral limit mu,d,s = 0, with d and s quark masses degenerate,

CP violation can only originate

in physics beyond the

SM. These CP properties of the SM are summarized through a necessary and sufficient condition for CP invariance, expressed in terms of a weak quark basis invariant [4,5] I

E

det[ MUM:, Md!kfi] = 3 tr[ MUM:, MdMi] 3

= -2i(mf

- m:)(mf

- mi)(m,2 - mi)(m2, - rnz)

x(m~-m~)(m~--m~)lmV,~V,*,V,,Vu~=O,

(2)

where Mu,d are the up and down quark mass matrices and rni the mass of the quark i. Although for three generations the two invariants of Eq. (2) are proportional to each other, tr[ M,Mi, MdMi] 3 = 0 has the advantage of being a nontrivial necessary condition for CP invariance in the SM, for an arbitrary number of generations [ 51. Within the three-generation SM, CP violation in the B system is not suppressed by the factor (m, - md) due to the fact that one is the initial state and kaons from pions in the final crucial in order to detect CP violation effects arising high-energy colliders, the natural asymptotic states

able to distinguish Bd from B, in state. This llavour identification is from gauge interactions [6-g]. At are no longer hadronic states, but

quark jets [ lo]. Now, at high energies, it will be very hard, if not impossible,

to identify

the flavour of light quark jets (d, s, u). In this limit, CP violation can only originate

in

physics beyond the SM. Indeed many simple SM extensions incorporate new sources of CP violation which may be observable at future colliders in the chiral limit [ 111. In this paper we concentrate on the case of extra quarks, with special emphasis on vector-like quarks, i.e. quarks whose left-handed and right-handed components are in the same type of multiplet. Vector-like quarks naturally arise in various extensions of the SM, for example in grand-unified theories based on Es, as well as in other superstring inspired extensions of the SM. In most cases, these vector-like quarks are isosinglets and their mass could be of the order of the electroweak scale [ 21. A new sequential quark family is also allowed [ 161, although precision electroweak data put an upper limit on their square mass difference [2]. We will show in this paper that with the addition of an extra quark, either sequential or vector-like, there is CP violation even in the limit where mU,d,,q,c= 0. Therefore, in these simple extensions of the SM, CP violation effects can be seen even if only the flavour of heavy quarks (b, t, b’) is identified. In both cases, for

R de1 Aguila et al/Nuclear

an isosinglet

quark and for a sequential

bosons is parametrized

Physics B 510 (1998) 39-60

extra family, CP violation

41

mediated

by gauge

by a 4 x 4 unitary matrix V defined up to quark mass eigenstate

phase redefinitions. For a new down (up) vector-like quark the charged couplings are described by the CKM matrix V&I, the first three rows (columns) of V. But these 3 rows (columns) V&

V&

matrix elements

completely

fix V. The neutral couplings

and are not independent.

are a function

In the case of an isosinglet

of the

quark, the

neutral couplings are no longer diagonal, i.e. there are flavour changing neutral currents (FCNC) . The 3 x 3 block of the CKM matrix connecting standard quarks is no longer unitary either, but deviations from unitarity are naturally suppressed by powers of m/M, where m is a standard quark mass and M denotes the mass of the isosinglet quark. The strength of FCNC among standard quarks is also suppressed by powers of m/M, since FCNC are proportional to deviations from 3 x 3 unitarity in Vcm. For a new sequential family VC~ = V and the neutral couplings are diagonal and real. It turns out that in both cases there are three CP violating phases in V&M [ 181. In our analysis we will adopt the following strategy: First, we identify a set of weakbasis invariants which completely specify the properties of the model considered in the sense that if any one of the invariants is nonzero there is CP violation, while if all the invariants of the set vanish there is CP invariance. These weak-basis invariants are physically meaningful quantities, and they are the analog of the invariant in Eq. (2) for the class of models we are considering. They can be expressed in terms of quark masses and the imaginary parts of various invariant products of CKM matrix elements. We then study the chiral limit mu,& = 0, starting with the simpler case mu&c = 0. Both are especially interesting since in these cases the three-generation SM conserves CP, thus implying that in the chiral limit CP violation arises exclusively from physics beyond the SM. The chiral limit is not only physically natural for studying CP violation beyond the SM but phenomenologically relevant at high energy, where the light fermion masses are negligible. The above-mentioned weak-basis invariants are especially useful in the analysis of the chiral limit, allowing one to readily identify which Im L$jVGVklvT continue being physically meaningful and nonvanishing when taking these degenerate mass limits. In a 4 x 4 unitary matrix 9 of these imaginary products are independent, and all of them can be made to vanish if CP is conserved. In the chiral limit mu& = 0, there are two CP violating phases and three independent imaginary products physically relevant. If m, is also neglected compared to m,, mu,d,s,c = 0, there is one CP violating phase left and one independent

imaginary

product physically

significant.

These imaginary

products BI = Im &bV$V4btV,*b,= T, - T3, B3 = h

V+v;&,’ Vc+b, = T3,

B2 = hn &bV$k&‘v;, = T2 + T3, (3)

which survive in the chiral limit and which involve mixings in the heavy quark sector (t, b, c and the new quark(s) ) , can also be expressed in terms of the rephasing invariants c defined in Eqs. (1). (We will use for the subindex of the fourth row ‘4’ and ‘t” when referring to the vector-like and sequential cases, respectively, When referring to

F: de1 Apilu

42

et al./Nuclear Physics B 510 (1998) 39-60

both we will also use ‘4’.) These invariants

q only involve mixings

among standard

quarks and they vanish in the SM. Thus, the effects of physics beyond the SM may be seen measuring

B; at high energy in processes involving new quarks or at low energy

through the nonvanishing

of Ti. We shall show that the present bounds on lBil are 10e2

and 10m4 for a fourth family and a new vector-like we will only take into account CP violating

quark, respectively.

In our analysis,

effects arising from gauge interactions

(in

some specific models with isosinglet quarks the Higgs sector is more involved than in the SM, leading to new CP violating contributions from scalar interactions) and we will neglect the Higgs contributions to CP violation. It may be worth to emphasize that the invariant formulation of CP violation requires an educated use of symbolic programs [ 201. However to go beyond the simplest cases is difficult for, as explained in the appendix, the number of invariants needed to get a complete set grows very rapidly with the number of phases. We study the simplest cases of a new vector-like quark or an extra sequential family, deriving limits for observables involving known particles as final states. If there exist more vector-like or sequential quarks, larger CP violating effects than the ones studied here are possible but in observables involving several of these new quarks. This paper is organized as follows. In Section 2 we set our notation and for the case of an extra down (up) quark isosinglet we propose a complete set of weak-basis invariant conditions which are necessary and sufficient for CP invariance. We also give the explicit expressions of the invariants in terms of quark masses and imaginary parts of invariant products. The proof that these invariant conditions form a complete set is given in the appendix. The corresponding set of invariants for the case of a sequential family was discussed in Ref. [ 191. In Section 3 we use these invariants to study the simplest case of two degenerate (massless) up and down quark masses, mu,d,s,c = 0, and the chiral limit, mu,d,s = 0.In Section 4 we discuss the corresponding geometrical description of CP violation with triangles and quadrangles and the bounds on the CP violating effects of these new fermions commenting on the prospects to measure them (at large colliders).

2. Characterization

Section 5 is devoted to our conclusions.

of CP violation for extra quarks

Let us consider the SM with N standard families plus nd down quark isosinglets (the case of n, up quark isosinglets is similar). In the weak eigenstate basis the gauge couplings to quarks and the mass terms are

c ga”ge- -__

5

-(‘) fid’d’ L

UL y

8 Y ( 2cw -cd) UL

P

Wi

+ h.c.

1

cd) _ &d’y@“did’ UL

_ 2&

‘I&

Z,

_

>

-WMuu;) +&d)~~&' +@)mdd;) + h.c., C mass = - UL >

e J&APL,

(4) (5)

E de1 Aguila et al./Nuclear

where uy’,

dLd’ are N SU(2)r_ doublets,

Physics B 510 (1998) 39-60

dr) are nd Sum

are N and N + nd Sum singlets, respectively, the up and down quark mass matrices are

43

singlets and ug’ and dgs)

and JLM = $iiy% - $yj‘d.

Mu = Mu,

Hence,

(6)

with M,, Md and md submatrices of dimension N x N, N x (N+nd) and fld x (N+nd), respectively. The weak quark basis can be transformed by unitary matrices without changing the physics. Under these unitary transformations

with q = u, d, the mass matrices transform M, --f

ULMqUf(t,

whereas the gauge couplings

md +

Ufmd&

as dt

,

remain unchanged.

(8) Then the mass matrices are defined up

to the unitary transformations in Eq. (8). A set of physical parameters can be defined using the mass eigenstate basis. We assume without loss of generality M, = ‘0, diagonal and & = VDdVt, with V unitary and Dd diagonal (remember that we can always assume M, and Md hermitian with nonnegative eigenvalues by choosing @a appropriately). We define the N x (N + nd) matrix Vim as the first N rows of the (N + nd) x (N + nd) unitary matrix V, and the (N + nd) x (N + nd) matrix X 3 V~mVi~. Then the Lagrangian in Eqs. (4)) (5) reads in the quark mass eigenstate L

- -g

ga”ge - Jz

(iziLy~Vcmd,&

basis (where no superscripts + h.c)

g (fiLy%L - dLypXdL - 2&J&) 2cw

Z, - eJ&Ap,

In this basis one can count the number of CP violating ber of phases in VCKMminus the number of independent ncp=N(N+nd)

- $N(N-

=(N-l)nd++(N-l)(N-2).

are needed)

1) - (2Nfnd

phases. This is equal to the numphase field redefinitions

[ 141,

- 1) (11)

CP is conserved if VCKMcan be made real. This is the case if all ncp phases vanish. In the SM, N = 3, nd = 0, there is only one CP violating phase. For one extra quark, N = 3, nd = 1, there are already three CP violating phases, the same as for an extra family, N = 4, nd = 0. The number of physical parameters, and in particular of CP violating phases, grows rapidly with the addition of more quarks. We will stick to these cases which incorporate many of the new features of the addition of new quark fields.

44

R de1 Aguila

et al./Nuclear

For an extra down quark isosinglet, matrix which can be parametrized

Physics

B 510

(1998)

39-60

Vim consists of the first 3 rows of a 4 x 4 unitary as (we write this explicitly

for later use)

[ 181

(12) Note that the first 3 rows completely fix V. On the other hand, V is the CKM matrix for 4 families. In the limit where the new quark does not mix with the standard quarks (i.e. s4 = ss = s6 = 0), the 3 x 3 block of V becomes just the standard CKM matrix with only one CP violating phase 61. The CP properties of the model with one isosinglet quark can be most conveniently studied by using weak-basis invariants. In the appendix we present the general treatment and provide the proof that the vanishing of the following set of invariants

is necessary

and sufficient to have CP invariance:

11 =ImtrH,Hdhdhfi, 12 =ImtrH,2Hdhdhd, t 13= Im tr( HiHdhdhJ

- HiHdH,hdhi),

14=ImtrH,H~hdh~, Is = Im tr HtHihdhi, 16=Imtr(HE3Hihdh: 17 = Im tr HzHdH,Hi,

- HzH$H,hdhJ), (13)

with H, = MuMi, Hd = MdMi, hd = Mdrni (see Eq. (6)). These invariants, which obviously do not depend on the choice of weak quark basis (see Eq. (8)), can be written in the quark mass eigenstate basis (in the equations below Greek indices run from 1 to 4, Latin indices run from 1 to 3 and a sum over all indices is implicit; rni is the mass of the up quark i and n, the mass of the down quark cy)

E de1 Aguila et al./Nuclear

Notice that X,, that in Eqs.

(14)

= vzI$,

which implies

the sums include

Physics B 510 (1998) 39-60

45

X = Xt and XnpXpr = Xar (note however

also mass factors).

The imaginary

parts involve

invariant quartets and invariant sextets, which can be reduced also to products of moduli squared of t$ elements times imaginary parts of quartets. In the chiral limits there only appear imaginary parts of invariant quartets. For the case of four sequential families, the corresponding set of necessary and sufficient conditions for CP invariance consists of eight invariants which have been given in Ref. [ 191. In both cases these invariant conditions completely characterize the CP properties of the model. If any of the invariants is nonvanishing, there is CP violation and the vanishing of the invariants implies CP invariance. The description of the CP properties of a model through invariants is especially useful when considering limiting cases where some of the quark masses can be considered as degenerate (massless). The invariant approach clearly identifies which ones of the Im KjV$Vkll$; can be nonvanishing in the various limiting cases one considers. The difficult task is, of course, finding a complete set of invariants, but once this is accomplished, the invariant approach is a very convenient method to describe CP violation. In the next section we will illustrate the usefulness by considering some appropriate chiral limits.

of these invariant

conditions,

3. CP violation in the chiral limit In the chiral limit, mu,d,s = 0, CP is conserved within the SM. Hence all CP violating effects are due to new physics. Sizeable CP violation at high energy is expected to have its origin beyond

the SM. Let us discuss in turn the simplest

limit of m, >> m, - 0,

m,,& = 0 and the chiral limit mu& = 0. We will consider the cases of an extra isosinglet quark and of a fourth sequential family.

3.1.

mu,d,&

=

0 limil

In this limit there is only one CP violating phase for an extra down quark isosinglet b’ or for a fourth family b’, t’. The best way to study this limit is to substitute mu&c = 0 in the complete set of invariants characterizing CP The 7 invariants in Eqs. ( 13) reduce to

I? de1 Aguik ef ~~./~~cle~r Physics B 510 ft998) 39-60

46

It is clear that CP

iS

conserved

if and

if Im &b&b’ yb, = 0. Obviously,

Only

we have

made the assumption that b, b’ are nondegenerate, with mb’ > mb. Hence all CP violating effects are proportional to this imaginary product which gives the size of CP violation. Similarly, for 4 families the corresponding 8 invariants [ 191 reduce to (we use a prime to distinguish

them)

In this case CP conservation reduces to requiring Im ~~~~~,~~ t$, = 0, with the implicit assumption rnbr > mb and m,f > m,. Not only the same comments as for an extra isosinglet apply but the observable for SM final states is also the same. Thus using unitarity B2 = Im t&,&\t&l I’$, = - Im &&,‘I’$,

%*,,.

= - Im VtbI$&

(17)

It is clear that B2 measures the strength of CP violation in high-energy processes involving the new quark. On the other hand due to the unitarity constraints, B2 is actually related to the invariants quark mixings.

Ti defined in Eqs. ( I), which only depend on standard

Indeed, one obtains

with T2,3 = 0 in the three-generation

SM as emph~iz~

in the introduction.

All this can also be proven transparent.

using the CKM matrix, although the physics is less 0, the general 4 x 4 unitary matrix in Eq. (12) (up to If @,c = 0, md,s =

quark field phase redefinitions) Cl -sgc2 v=

can be written

SIC3 CIC~C~ + S2S3Cgei6’

-St $2 Ct S2C3 0 !

-

C2S3C@ &

SIS3 CjC2S3

-

s2c3cfjeis’

Cl sg.93 + c2c3cge

f3 S6

-S*i@l i&

c2S6e

-C3%

C6

’

iSI

(19)

)

where we have used the freedom to make unitary transformations in the (u,c) and (d,s) spaces. This freedom results of course from the fact that (u,c) and (d,s) are degenerate in the limit we are considering. As expected, in this limit there is only one CP violating phase and the strength of CP violation is given by B2 = Im vt~v~~v~b/~~, = Ct c2S2C3S3c&

(201

sin 61.

Unitarity allows to recover Eqs. ( 17), ( 18). In this particular fact equal to T3 for 7’1,z = 0 as defined in Eqs. ( 1) .

parametrization

B2 is in

E de1 Aguila et al/Nuclear

Physics B 510 (1998) 39-60

41

3.2. The chiral limit, m”,d,s = 0 In this limit there is no CP violation in the SM, but for one extra quark isosinglet or a fourth sequential family there are two new CP violating phases which remain physical. In the case of the model with an extra down quark isosinglet, CP conservation is equivalent

to the vanishing

of 11-3 in Eqs. (13), because in this limit

It = myI, + rnzl,, I2 = m41t + mzl,, 13 =

m;Z, + mzIC + m:rn~(rn~ - m~>rn$rn2(rn~, - mf) Im VcbV$V&V,*b, + 17,

-

(21) with 1, = m~&m~

(m$ - mi) Im VcbXbb’Vc*b!,

I, = m:m2,,mi(m2 b’ - m’,) Im k&X&’ V$.

(22)

There are only three independent imaginary products, Im V&X&’V$, , Im VcbXbb’Vfcb,and Im V&$,&b’ VC;,, entering in 11-7. Their vanishing guarantees CP conservation. The first one is the only one which survives for m, = m, as proven in the previous subsection. Analogously, in the case of a fourth family CP invariance is equivalent to the vanishing of 4-3 in Ref. [ 191. In this case the complete set of 8 invariants reduces to

Zi = (m: + m,2)Zb, + (m$ + m,2)&

+ (rn;, + m:)Z$,

l,l = (rn: + rnt)lL, + (m;? + rnt)&

+ (m$ + m:)Z$,

Zi = (rn: + mz)Zf, + (mf: + rnz>Z$ + (rn:, + mf)I:,, -mf,m~m~(mf

- m,2)(mf, - rn:>(rnz, - rnF)rn$rnfj

F: de1 Aguila et ul./Nuclear

48 -(

~~&rz;,

-

IVIb1*w$)

Physics B 510 (1998) 39-60

Im VcbVr?bVrfb~Vc~,

+(IVcbfl*rn$ - ~~.h~*rn~)ImV,b~?b&/b~~*b,

z:=(m$ +m;)z;,1:=

1,

(m;S, + mz)Zi,

Zi = (m$ + mz)Z[,

ZA = (m$ + mi)Zi,

(23)

with IL, = -rn:rnz(rnf!

- mz)m2,,m2,(m$

- rn4) Im VchVr*hV,bfV,*h,,

Ii,, = -m:,mz(mf,

- m~)m2,,m~(m$

- rni) Im VcbV,TbVtfbfVc$,

I:,, = -mfimF(mf,

- mf)m$rnfbj(rn$

- rni) Im t&l$h&/b,l$,.

(24)

For four families there are also three independent imaginary products, Im L&V;fD&,b,x V,;,, Im Vch~fhl+pV,*h, and Im V&,~~V&L$~,, entering in Z[_s. Similar to the case of an extra isosinglet their vanishing guarantees CP conservation. There is an interesting connection between these rephasing invariants Bi, which can see CP violation through high-energy

processes

involving

among SM quarks. Using unitarity

the new quark, and T,, which only involve

B2 = Im V&Vq*hV&i$, = - Im &b&b’~~, B 1 = Im &b vq”I&’ yi, B3 = h

mixings

one obtains = T2 + Ts,

= - Im &,,_&h’v,*,, = Tl - T3,

&, KT,K/,’ y.;, = T3,

(25)

where Tl-3 are defined in Eqs. (1) and vanish in the three-generation SM. These results can be also reproduced using explicitly the CKM matrix. freedom one has in the chiral limit to make unitary transformations

Using

the

in the (d, s) space,

The invariants B; can then be expressed in terms of mixing angles and the two physical CP violating phases,

E de1 Aguila et al/Nuclear

The unitary matrices

Physics B 510 (1998) 39-60

49

in Eqs. ( 19), (26) have been used to rederive the relevant

inary parts of invariant

quartets in the chiral limits. Obviously,

to the actual CKM matrix in the physical situation

imag-

they do not correspond

where the quarks are distinguished.

4. Limits on CP violating effects from new quarks In this section we revise the three-generation

SM for which CP violation

is summa-

rized in a unitarity triangle, extending this description to the unitarity quadrangles for an extra vector-like (sequential) quark (family) and their restriction to subtriangles in the chiral limit. Then we estimate the experimental bounds on the three independent invariants Bi characterizing CP violation of the new CP violating effects.

in this case and comment

on the determination

4.1. Triangles, quadrangles and the chiral limit Before considering physics beyond the SM, it is worthwhile reviewing the main features of CP violation in the three-generation SM. The information on CP violation is conventionally summarized in this case in terms of the unitarity triangle [21]. This triangle

is the geometrical

representation

of the unitarity

relations

between

any two

different rows or columns of the 3 x 3 CKM matrix. One can draw different triangles choosing different pairs of rows or columns, but for the three-generation SM unitarity implies

that all these triangles

have the same area (which

equals

i 1Im &V$&ll$I)

because all Im K,iV$Vklq1+have the same modulus, and this modulus gives the strength of CP violation in the SM. The most interesting of the unitarity triangles is the one which results from the orthogonality KdK;

+ vcd&; + &t$

The measurement

of the first and third columns,

= 0.

of CP asymmetries

(28) in B” decays offers the possibility

of measuring

the internal angles of this triangle. Note that these angles are rephasing invariant quantities since they are the arguments of invariant quartets. In the three-generation SM, 1Im b$jVVk+il&bjf 1 is necessarily small, essentially due to the fact that the third generation almost decouples from the other two (in the limit where the third generation decouples there is no CP violation in the SM). An upper limit on 1Im QjV$Vkll$;I is readily obtained since ) Im Vu,VC~V&,Vu>~ 6 IVusllVCsllVC~llVu~l < 5 x 10W5. In the presence of an extra quark, the unitarity relation corresponding to Eq. (28) becomes

E de1 Aguilu et al./Nuclear

SO

Physics B 510 (1998) 39-60

Vuh

Fig. I. The shadowed triangle describes CP violation in the ~d,,~,~ = 0 limit, whereas the complete quadrangle does it in the chiral limit rn,,d,s = 0. We use the same notation 41.3 for the angles of the shadowed triangle as for the quadrangle, although for the (convex) quadrangle they are larger. x,J VU; + &.d v,*h + Vtd v;

In this case, three quadrangles

+ &d v;b = 0.

are required to summarize

(29)

the information

on CP violation

[ 191. The other two quadrangles can be chosen to be the ones obtained by multiplying the first and second columns and the second and third columns, respectively. The vanishing of their areas is a sufficient condition for CP conservation, and in the case of nondegenerate quark masses it is also necessary. Alternatively, one could choose a set of three quadrangles arising from the orthogonality between the rows of V. A relevant observation is that some of the sides of these quadrangles cannot be measured separately in the case of degenerate quark masses. For example, in the limit where mrr,c = 0 the quantities vrdV,*b and VcdV,*bin Eq. (29) cannot be separately measured, due to the freedom to redefine degenerate quark fields. In this case, however, the subtriangle with sides &Vu: + &yb, i&y; and V4dV$bis well-defined. In the chiral limits previously considered, CP violation can be geometrically described as follows. In the mu,d,,s,c = 0 limit, we can define the subtriangle obtained third and fourth columns of V (shadowed region in Fig. I), Vubi$,’ + V&y;’ and considering

+ &bv;t + i&,v;y

= 0,

the

(30)

the sides VUbVU& f VcbV,*h,,&bV,;‘, V&V&, with angles 41.3,

sin41 = 1sinarg(VUbV,*hVrbtVU*J + sin 42

by multiplying

= 1 sin

VcbVt*bVrb~Vc*bOI,

arg &bVq*bVJb’V$I, (31)

The area of this triangle is given by Abb’ = $&I

represents

the strength of CP violation

= ~IImt$bV4+bV4b’~~jI

in this limit. This area

(32)

and vanishes if and only if CP is conserved (see Eqs. ( 15), ( 16) ). Note that under allowed quark mass eigenstate transformations, including those mixing u and c, the length of the sides of the triangle remains constant and therefore they are measurable

E de1 Aguila et al/Nuclear

quantities triangle

Physics B 510 (1998) 39-60

51

even in the limit mU,c = 0, where u and c are indistinguishable. provides

subtriangle

a good description

obtained

by multiplying

of CP violation

Thus this

in the mU,d,s,c = 0 limit.

The

the third and fourth rows,

vdVq*d+ &svq*s+ VtbVq*b + Vtb’&;r = 0,

(33)

. . with sides &V4; + v,,vq”,, vbV;b, &b’V$,, has the SZIIIle iII-f?a A,4 = ;I Im&bV;*bI’&‘I$, 1 and provides an equivalent description of CP violation in this limit. In the chiral limit, mu,d,s = 0, we have to consider the complete quadrangle in Eq. (30)) with sides VubV,*b#,VcbV$, &by;,, hbV&r (see Fig. 1) and angles sin 41 = 1Sin wg Vcb~~~b’ v,*,,I, sin $3 = 1sin arg V&,V$,V&’VU>’ I, The area of this quadrangle

sin 42 = 1sin arg &bV~bV&,‘~*,,1, sin 44 = 1sin arg VUbv$, v&f VU>,1.

(34)

is

(35) We use the same notation as for the angles and area of the triangle in Eqs. (31), (32) because this quadrangle reduces to that subtriangle in the appropriate limit. It is clear that the vanishing of Abbt in Eq. (35) is a necessary and sufficient condition for CP conservation. Alternatively, one can consider adding to the triangle in Eq. (33) the analogous triangles obtained by multiplying the second and third rows and the second and fourth rows respectively, with areas A,, = 1 (B3I = 4 ) Im VcbV,iV& VC$, I and Ac4 = $ IBi I = i) Im &V&&‘ybb’ (. The vanishing sufficient condition for CP conservation. 4.2. Bounds on B1 = Im V&V~bV4bl~~,, B2 = h We now turn to the important CP violating

question

of Act,c4,r4 is also a necessary

b&V~bV4b’~~, and B3 = h

of estimating

the possible

and

k&~~&b’Vfcb,

size of the new

effects when new quarks are added to the SM. In order to establish

upper

bounds on the size of these effects one has to distinguish between the case of an extra isosinglet quark and the case of a sequential fourth family. In both cases, we use the experimental model-independent measurements [ 21 /vu& = 0.9736 f 0.0010, IV,,l = 0.2205 f 0.0018, j&d1 = 0.224 f 0.016, Iv,,1 = 1.01 k 0.18, [Vu&$ = 0.08 f 0.02, lb&l = 0.041 f 0.003. These, together with the unitarity of the 4 x 4 matrix V, give Iv& < 0.079, I&,[ 6 0.516, I&&,dI 6 0.104, (&s,4sl < 0.513, where We have USed the measured lower bounds to obtain these upper limits. Our strategy will be to obtain rigorous upper bounds on IBi( using the previous limits and to check afterwards that they are almost saturated in particular cases fulfilling all present experimental constraints. The fOMler UppeI’ bounds imply 1hl &,V,*b&’ v;b, 1 < I&, 11 &I 1v,,f 11 vu,, 1 < 7.87 X

s2

E de1 Aguila et al/Nuclear

1op6, I&I

1Im&bq$&b’V;tb!I

6

Physics B 510 (1998) 39-60

j&blI&blI&b’IIKb’I

<

1.73

1.11 x lo-*. Then, using unitarity values, these limits translate into

Ivbl I&b’ IIk&t I <

the absolute

x

1oe4,

1 Im&~b%V,*h,I

and the triangular

inequality

<

for

(36) which are rigorous bounds, in particular for a fourth family. These bounds saturated for instance by the 4 x 4 unitary matrix

IVI=

arg V =

0.973

0.220

0.0035

0.070

0.230 0.082 0.082

0.918 0.254 0.212

0.041 0.655 0.755

0.321 0.712 0.621 i

are mostly

(37)

’

0

00 r 0.007

0 -1.08

-0.057

r

0.095

0.873

-3.00

i T -1.31

2.38

1.62

(38)

’ 1

for which IBt I = 6.1 x 10m3, I&] = 6.3 x 10P3, I&] = 6.1 x 10m3. We have also required in these matrices that the imaginary parts of the quartets involving the first two columns and entering in the calculation of ek for four sequential families are N low4 not to rely on large cancellations. Without this requirement the bounds in Eqs. (36) can be almost completely saturated. Hence ]Bi] < lo-* for four families. In the case of an extra isosinglet quark the size of the CKM matrix elements is further constrained by existing bounds on FCNC [ 221. The constraint on j V&l 1V,,l = IX,,1 is rather severe due to the experimental upper bound on strangeness changing neutral currents.

The strongest

Br(K+

+ ~‘vV)

Comparison

limit on lXds I arises from the experimental =

T(K+

-+ 7~+z+)

T(K+

t

all)

bound

[ 21

< 2.4 x 10-9.

with the process K+ -+ ?r’e+v leads to

Br(K+ 4 7.r+vfi)_ I&sl2 x 3 Br(K+

--f ?roe+v)

’

2]V,,]*

(40)

where the factor 3 takes into account the three different vV pairs. Then the observed Br(K+ -+ n-“e+v) = (4.82 f 0.06)% [2] gives IXd,] < 4.08 x 10e5. The limits on bound [2] Ix& = ~~d~~~b~~ l&bl = Ihsllv 4b 1 arise from the experimental r(B

+

P+P-W

T(B 4 ,~u.yX) which leads to [22]

< 4 6 x

’

1o-4

(41)

E de1 Aguila et al./Nuclear Physics B 510 (1998) 39-60

53

(42)

’ where R N 0.5 is a phase space factor, giving the experimental

IX&$bl < 1.91 x 10v3. Using this limit,

bounds above and the triangular

in~uality

for the absolute values, we

obtain

lBll = (h&,Vq*bb&‘k$,#j

< 1.02 x 1os4,

I&/ =lIm~b~~~bf~~,

+hb’&vd,*bk&‘b$,f/

< 1.12 x 1ow4,

(&I=lh&&Vfrb&‘V~bf

fh~~bV_bbV4b’~j,l)

< 1.11 x 1om4-

(44)

limits for an extra down quark isosinglet.

(We have not made

These are the rigorous

explicit use of the /Xds/ bound.)

IV=

argv=

The 4 x 4 unitary matrix

0,975

0.222

0.0033

0.0007

0.222 0.011 0.0022

0.974 0.039 0.0093

0.039 0.978 0.204

0.013 0.204 0.979

00 n- 0.0006 T -0.300

0 -1.97 0,832

0 -2.09 1.59

T -0.234

2.56

0.142

(45)

(46)

gives lBr / = 7.6 X 10m5, /&I = 7.7 X 10e5, 16131= 7.6 x 10T5 which are near the upper bounds in Eqs. (44). We also require that the dominant contributions to ek are the same as in the three-generation level diagrams [ 13,221.

SM without large cancellations,

and not mediated

by 2 tree

4.3. CP violation from new quarks Vector-like and sequential quark contributions to CP violating observables not distinguishing between d and s quarks are proportions to &. We have shown that for vector-like and sequential quarks lBil 6 10v4 and ]BJ < 10h2, respectively. These values are relatively large. For instance, the maximum of j Im !+V$V&V;:rIfor an arbitrary 4 x 4 unitary matrix is l/66 N 0.096, which is the same as for a 3 x 3 unitary matrix [23]. On the other hand, ) Im b&V~Vk~l$l < 5 x 10M5 in the three-generation SM. In spite of the relatively large values allowed for Bi, it is clear that observing direct CP violation from gauge couplings of new quarks will not be an easy task. It is worth emphasizing that Bi can also be obtained indirectly, by measuring Ti and using

E de1 Aguila et al./Nuclear

54

Eqs. (25) unitarity

which give Bi as functions relations

Physics B 510 (1998) 39-60

of 7;. The failure of the three-generation

would point out to new (CP violating)

physics, in particular

SM to new

quarks if Bi in Eqs. (25) are of the correct size. The study of CP violation at high energies would thus complement the information of CP asymmetries in B meson decays, at B factories. The effects of vector-like B” decays have been extensively quarks, the most important

or sequential

quarks on the CP asymmetries

studied in the literature

in

[ 221. In the case of vector-like

effect results from a new contribution

to Bd - Bd and B, -B,

mixings, arising from tree level FCNC Z exchange diagrams. It has been shown [22] that even for relatively small FCNC couplings, the prediction for the CP asymmetries in Bz -+ J/$Ka and Bz -+ 7r+7~- decays can differ significantly from the predictions of the SM. At this point, it should be emphasized that although the observation of CP asymmetries at B factories may lead to unambiguous evidence for new physics, it will not be easy to identify

the origin of the new physics by studying

CP asymmetries

alone [ 171. The study of CP violation at high energies, together with the study of rare B decays, will play an important role in identifying the origin of the new sources of CP violation.

5. Conclusions Understanding

the origin of CP violation

will probably

require obtaining

new experi-

mental information on CP violating observables outside the kaon system and the possible identification of new sources of CP violation. One of the simplest ways of obtaining new sources of CP violation

consists of adding extra quarks to the SM. The addition of extra

vector-like fermions is especially attractive, since they naturally arise in grand-unified theories, like Eg. We have derived a complete set of weak-basis invariants which constitute necessary and sufficient conditions for CP invariance. These weak-basis invariants are physical quantities which can be expressed in terms of quark masses and imaginary parts of rephasing invariant quartets. For simplicity we have restricted ourselves to the case of one additional

isosinglet

quark, since it is sufficient to illustrate

of new quarks for observables involving At this point, it is worth emphasizing

only known fermions. the usefulness of weak-basis

the implications invariants

in the

study of CP violation: ( I ) The invariant approach can be very useful in model building. At the moment, there is no standard theory of flavour and for example in the SM the Yukawa couplings are arbitrary free parameters. As a result, one does not have in the SM any insight into the pattern of fermion masses and mixings. In the literature, there have been various attempts of introducing additional family symmetries in the Lagrangian leading to Yukawa couplings which are no longer arbitrary but are expressed in terms of a fewer number of parameters, with the Yukawa couplings exhibiting some texture zeros [ 241. Of course the quark mass matrices are no longer arbitrary, being constrained by the family symmetries. One has to check whether in spite of the additional family symmetries the model leads to genuine CP violation mediated

E de1 Aguila et a/./Nuclear

by W interactions.

The usual method

becomes rather inadequate,

especially

Physics B 510 (1998) 39-60

of diagonalizing

55

the quark mass matrices

in models with vector-like

quarks, where the

CKM matrix is no longer a unitary matrix. The simplest way of checking whether CP violation occurs in models with additional family symmetries consists of directly evaluating

the weak-basis

conditions

for CP invariance

non-vanishing, (2)

invariants

which constitute

in the model considered.

the necessary

and sufficient

If any of these invariants

is

one is sure to have CP violation.

Weak-basis invariants are also very useful for studying CP violation in various physical limits, especially those involving degenerate and vanishing masses. Here we discuss two chiral limits, the extreme one m, > m, N 0, m,,& = 0 and the standard chiral limit m”,d,s = 0. These limits are especially relevant at highenergy colliders, where the natural asymptotic states are quark jets. In this chiral limit, d and s quark jets are either very difficult or impossible to distinguish from each other and there are no CP violation effects in the three-generation SM. In the case of one extra vector-like quark or an extra sequential family, we have shown, using weak-basis invariants, that there is CP violation even in this chiral limit. We have shown that CP violation can be characterized by two CP violating phases which are proportional

to Bt = Im &,l$,V&,/V~, , Bz = Im &bV&V&/l$, and

B3 = Im V&V$&b! l$b!. In the eXtKme chiral limit (mu,d,s,c = 0) we have shown that there is one CP violating phase and one weak-basis invariant which controls the strength of CP violation in this limit and is proportional to B2. In conclusion, extra quarks lead to new sources of CP violation which can manifest themselves in various phenomena, including CP asymmetries in B” decays, rare B decays as well as in CP violating observables at high energy. Weak-basis invariants, together with the imaginary part of rephasing invariant quartets, like Bi and Ti which we have introduced, are useful tools to study CP violation in these minimal extensions of the SM.

Acknowledgements We thank J. Bernabeu

and M. Zralek for discussions.

This work was partially

ported by CICYT under contract AEN96-1672, by the Junta de Andalucfa European Union under contract CHRX-CT92-0004.

sup-

and by the

Appendix A The Lagrangian 13sauge+ C,,, there exist unitary transformations

qLd) --)

ULcqL

in Eqs. (4)) (5) is invariant UL, Ue, U;” such that [5]

Cd)*

(A.1)

7

with C the Dirac charge-conjugation

under CP if and only if

matrix, satisfying

F: de1 Aguilu et al. /Nuclear Physics B 510 (I 998) 39-60

lgM,Uf: = M;,

U~tmJJ~ = m(;.

As 172” are unobservable M,

(if Higgs mediated interactions

and Md in Eq. (6) hermitian lJlH&_

t uLH,&_

= H;,

(A.2)

with nonnegative t ULh&

= HL;,

are neglected)

eigenvalues. = hi,

we can assume

Then, (_ft h’ U” = A’* L dL d’

with h: = mdmi, are equivalent conditions for CP conservation. Eq. (A.3).) From these equalities independent

to Eqs. (A.2) (The condition

new constraints

and are also necessary and sufficient Ut’hiUL = hz follows trivially from

for CP invariance

of the choice of weak basis, with no reference

Eqs. (A.l)-(A.3).

(A.3)

They result from the observation

can be derived which are to unitary

matrices

that any combination

as in

of products

‘I’ t with p arbitrary, has invariant trace and determinant. Then CP of H,, Hd, &h&d, invariance, Eq. (A.3), requires that their imaginary part vanishes. This also holds for any combination of products of h: and h;Hhd, with H any of the former combinations, but there is no need to consider these combinations because they do not give new constraints. Which subsets of these constraints are also sufficient has to be determined case by case. The explicit proof can be done in a simple, convenient basis. For the search of necessary and sufficient constraints and in general for parametrizing the model, we find it convenient to consider the basis where M, = M, is diagonal with nonnegative eigenvalues and

J&=(;;)=(: !?),

(A.4)

with liid upper triangular with real, nonnegative diagonal elements, %d diagonal with nonnegative eigenvalues and Nd arbitrary (ad could also be chosen to be hermitian). The proliferation of invariant constraints required to guarantee CP invariance makes necessary the use of a symbolic algebraic program to write down the expressions and to solve explicitly the constraints. This is done with Mathematics [25] and a set of routines

analogous

to those in Ref. [ 201.

N = 3, nd = 1. In this case we shall show that 11.7 = 0 in Eqs. ( 13) is a set of necessary

and sufficient conditions for CP conservation. In the proof we assume M, diagonal with ( Mu),i = mi6ij and Md upper triangular with matrix elements (Md)i
m~)In4412(Imn12n;2n24n;4 -

m~)ln44121mn13n~3n34n;4

+ Imnl3n;,n24n;h) +

(rns -

m~)In44121m~23n;3n34~2*4

= 0.

(A.5)

F: de1 Aguila er al./~uc~~ar

Physics B 510 f1998j39-60

57

The expression of It suggests that we consider products with higher powers of H,, to obtain independent linear combinations of the imaginary factors. In this way we find Z2 =

(~721 + mi>(m: +(mf

f

-

m~)In4412(Imn12n~2n24n;b

+ Imnl3&24nB>

m:>

(m:

-

$)1m412

Imv3n;3n34$.4

+(ml + ~$1

(ms

-

mz)ln4412

Imn23n3*3n34$4

Z3=(m;l+rf&(m~ +(m;t

i-

-m~)In4412(Imn12n;2n24n;b m$

(m:

-

&

1md2

= 0, +Imm3n%n24nT4)

Im m3ni3n34$4

--~)ln44j21mn23nhn34~34

+(~~~~~)(~~

=O.

(A.61

We will assume for the moment no # 0 and nondegenerate masses. Then Eqs. (AZ?), (A.6) imply Im n12rZ;2YZ24$4

Im n13$3R~4$4

+

=O,

Im. n13n;~n34nT~

= h

n23n;3n34$4

= 0.

(A.7) These conditions do not guarantee CP conservation, hence we go on considering the products with increasing number of factors and giving independent conditions. The next lowest order invariant 14 has 10 mass submatrix factors and after substituting (A.7) it can be written 14 = (m$

-

mT)jn33/21n44j2fmni2~~2~24n;4

-t-(m:

-

m4)

ln44121mn12n;2n2421;4n33nr3

+(m:

-

m$

jn4412 Imn12n;2n23n;3n34n;

= 0.

(A.81

We again look for products with higher powers of H, to obtain independent linear combinations of the imaginary factors, finding 15= (m! +mi:>(mi +(mt

+ m$

+(m:

+ r.+> tmf

16 = (mt

+ m$

-m:)ln3312f~44121mn~2nz*2n24zn~4

($

(ms

-

-

&

-

~.$)/n441~

m:)

[ml2

Im ~,2~~2~24~~4~33~~3 Irn~l2~~2~23~~3~34~~4

l~3312~~~J2

fmnl2&n24n;k

+(m”;

d- m$

(m:

-

ml>ln4412

Imn12n;2n24ni4n33nT3

-t-(my

+ rn$

(m:

-

~2:)

Im

[ml2

= 0,

n12n2*2m3n53n34n;4

= 0.

(A.91

These equations imply for no # 0 and nondegenerate masses

A tedious calculation shows that Fqs. (A-7)) (A.lO) do imply CP conservation. First we find all the solutions to Eqs. (A-7), (A.lO) with all nij # 0, then with one nij = 0, with two, etc. In all cases we can redefine the quark eigenstate phases to make Nd real.

E de1 Aguila et al. /Nuclear

58

Physics B 510 (I 998) 39-60

When two up quark masses are degenerate, of generality

say ml = m2, we can assume without loss

1112= 0. Then, Z2,3 are proportional

to Ii and 15,6 to 14, Whereas

11 = (m: - mi) In4412(Im1213123*3123411;4 + Imn23n;3n34n;4> 14= Cm: - m~)ln4412(1~~~121mn~3n;3~34n;4

= 0,

+ ln22121mn23n;3n34n;4>

= 0.

(A.ll) If ]ni I/ f

112221, these equations

Im n13n;3n34n;4

are independent

= Im n23$3n34$4

and (A.12)

= 0.

If llzi t I = In22], we can assume ni3 = 0 and Eqs. (A. 12) still hold. A long and tedious calculation shows that Eqs. (A. 12) imply CP conservation. We look for all their solutions and check that we can redefine the weak quark basis conveniently and make Md real for each solution. (In most cases it is only necessary to redefine the phases of the eigenstates.) If the three up quark masses are degenerate, CP is conserved. When 1244 = 0, 11-6 = 0 and we need to introduce more constraints on the mass matrices to ensure CP conservation. In this case we can assume ntt = n22 = nss = 0 by properly choosing the weak basis. There is only one CP violating phase, and the vanishing of the generalization of the SM invariant, f7 =

(rni - m:)(mT

- mi)(rn:

- m~)ln34121mn13n;3n24nr4

=0

(A.13)

does ensure CP conservation. This completes the proof. N = 3, nd > 1. In these cases the number of necessary and sufficient invariant conditions for CP invariance is too large to be in general manageable. Let us argue the fast growth of the number of these constraints by deriving lower bounds for nd = 2 and nd = 3. These bounds are general and based on cycle counting. A k-cycle is a product of k matrix elements nij of Md: C(il, . . . , ik) = Eiliziii~i~ . . . fiikil, where the indices i,j are all different and fiij can be nij or nTi. The number pk,, of invariant constraints obtained by this method

is‘smaller

than the actual number

p for (i) we consider

only in this

counting nondegenerate up masses, and (ii) we assume that only one invariant is needed to ensure the reality of a cycle (although we know that often this is not the case, see Refs. [ 19,201 for examples). Then comparing with the exact result for the simplest case we expect pmj, < p - 2pti”. For nd = 2 there are seven 3-cycles C(1,2,3), C(1,2,4), C(1,2,5), C(1,3,4), C(1,3,5), C(2,3,5), C(2,3,4). We work in a convenient basis where n45 = ~254= 0, so the cycles with 245 are zero (for instance, the 3-cycles C( 1,4,5), C(2,4,5) and C( 3,4,5) ). To ensure the reality of these seven cycles we need seven constraints. In addition, there are situations in which all the 3-cycles are real but not necessarily the 4-cycles. This happens when some Md matrix elements vanish. The maximum number of nonreal 4-cycles is achieved for instance if niz = ni3 = n23 = 0. We have in this case three 4-cycles not necessarily real C(1,4,2,5), C(1,4,3,5), C(2,4,3,5), and to ensure their reality we need three more constraints. Their reality then implies the reality of the 5-cycles. Thus, ok,, = 10 for nd = 2. The analogous computation gives ptin = 20

E de1 Aguila et al/Nuclear

for nd = 3. Finally have shown that

Physics B 510 (1998) 39-60

if we perform the computation

p =

59

for nd = 1, we find phn = 4 and we

7. It must be noted that for nd = 1 the 3-cycle C( 1,2,3)

appear in the expressions of the invariants in this appendix. due to the basis redefinition it is replaced by C( 1,3,2,4).

does not

It should appear in 17 but

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60

F: de1 Aguila et al/Nuclear

Physics B 510 (1998) 39-60

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