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Vector-like fermion contributions to ε′
Nuclear Physics B271 (1986) 61-79 © North-Holland Publishing Company
VECTOR-LIKE FERMION
CONTRIBUTIONS
TO e'*
F. DEL AGUILA1, M.K. CHASE and J. CORTES: CERN, Geneva, Switzerland Received 15 August 1985 (Revised 14 October 1985) We consider extensions to the standard electroweak model which contain new vector-like fermions at the Fermi scale as a source of CP violation. After a detailed calculation of the QCD corrections to leading order, we find that the new contributions to le'/e[ range typically from 10 3 to 10 2 when we require that the corresponding contributions to e saturate its experimental value. If, for instance, a new vector-like down-quark exists at the Fermi scale le'/el is predicted to be ~2 x 10-3. These results are within the range of sensitivity of presently planned experiments.
1. Introduction The m i n i m a l v e r s i o n o f the s t a n d a r d m o d e l [1], so successful in d e s c r i b i n g all l o w - e n e r g y p h e n o m e n a , m a y a p p e a r in the future to be i n a d e q u a t e for d e s c r i b i n g CP v i o l a t i o n in the k a o n system. All CP v i o l a t i o n in this m o d e l , w h i c h is explicitly i n c o r p o r a t e d in the Y u k a w a couplings, can be r e l a t e d to a u n i q u e p h a s e in the K o b a y a s h i - M a s k a w a ( K M ) m a t r i x [2]. W h e n this p h a s e is a d j u s t e d to r e p r o d u c e e, e ' can t h e n be p r e d i c t e d to the extent t h a t one c a n estimate the c o r r e s p o n d i n g m a t r i x e l e m e n t s i n c o r p o r a t i n g strong interactions. In p a r t i c u l a r e' is r e l a t e d to the size o f the p e n g u i n d i a g r a m c o n t r i b u t i o n a n d if this h a p p e n s to b e small then so is A t p r e s e n t there is no o b v i o u s d i s a g r e e m e n t b e t w e e n the m i n i m a l v e r s i o n o f the s t a n d a r d m o d e l a n d e x p e r i m e n t . H o w e v e r , the l a c k o f a d e e p u n d e r s t a n d i n g o f C P v i o l a t i o n in t h e s t a n d a r d m o d e l a n d s o m e r e c e n t e x p e r i m e n t a l results [3]**, have r e n e w e d t h e interest in p o s s i b l e e x t e n s i o n s o f the s t a n d a r d m o d e l , w h e r e C P v i o l a t i o n c o u l d b e b e t t e r u n d e r s t o o d o r w h o s e p r e d i c t i o n s c o u l d d e v i a t e f r o m the s t a n d a r d m o d e l in a definite way. T h e s e e x t e n s i o n s c o u l d i n c o r p o r a t e new scalars [5], new f e r m i o n s [6, 7]***, o r n e w v e c t o r b o s o n s [9]. In this p a p e r w e discuss in detail the c o n s e q u e n c e s o f a d d i n g n e w vector-like f e r m i o n s . T h e s e are f e r m i o n s w h o s e lefta n d r i g h t - h a n d e d p a r t s t r a n s f o r m e q u a l l y u n d e r the e l e c t r o w e a k g a u g e group. T w o k i n d s o f f e r m i o n s , a f o u r t h g e n e r a t i o n o f f e r m i o n s [6] o r vector-like f e r m i o n s [10, 11], c a n be a d d e d to the s t a n d a r d m o d e l with a n a t u r a l s u p p r e s s i o n o f flavour * Work partially supported by the CAICYT. 1 On leave of absence from Universitat Autonoma de Barcelona, Spain. 2 On leave of absence from Universidad de Zaragoza, Spain. ** Presently planned experiments [4] can clarify the situation if the expected accuracy of order 10 3 for ]e'/e] is reached. ***For supersymmetric models, see ref. [8] and references therein. 61
62
F. del Aguila et al. / Fermion contributions to e'
changing neutral currents (FCNC) and without introducing new symmetries. In the first case the G I M mechanism applies [12]. In the second case the G I M suppression is replaced by new and natural suppression factors of order some power of m~ M, where m is an ordinary fermion mass and M the mass of the new vector-like fermions o f order 10z-103 GeV*. These factors of re~M, which must typically be smaller than 10-2-10 -3 to suppress F C N C as required by experiment, do not exclude, however, sizeable CP-violating effects, given the smallness of the observed CP violation. In this paper we calculate the new contributions to e' induced by these fermions using e as an input parameter. In a recent letter [7] by some of the present authors a new model of weak CP violation, incorporating a new vector-like down quark and two new scalar doublets, was considered as a minimal extension of the standard model where the observed CP violation in the kaon system is due to the presence of the new vector-like fermion and CP is spontaneously broken. Now we consider different and more general cases and we give full details. We add to the minimal standard model only new vector-like fermions, singlets and doublets under S U ( 2 ) L , distinguishing different cases depending on which are the iightest vector-like fermions; CP is explicitly broken by Yukawa and mass terms. The KM phase can be present, as well as the new phases associated with the new fermions. A priori we will assume that the effects of the KM phase are small and that the new phases induce the observed e. Then we calculate e' and find that le'/e] ranges typically from 10-3 to 10 -2. In the particular model considered previously [7] it was estimated that the new contributions to e' could be up to an order of magnitude smaller than those of the standard model. However, QCD corrections to e and e' are important and reduce that estimate by a factor of 3. Thus, if a new vector-like down-quark exists at the Fermi scale a value of le'/el 2 x 10 3 is predicted. Any other value can be accommodated if for instance new vector-like up- and down-quarks in a SU(2)L doublet are considered. The value of [e' I is the result of many new contributions in these models and, in contrast with the standard model, it is essentially independent of the size of the penguin terms. Finally we must note that if the KM phase is large, its effects on e' can be comparable to the new vector-like fermion contributions which do not have a definite sign because the new contributions to e are quadratic in the new phases whereas the new contributions to e' are linear. Thus, a cancellation between both terms cannot be excluded. Also, in some models with more than one relevant new phase e' can have a wide range of values. Implications of these n e w p h a s e s for electric dipole moments are not restrictive at present [7]. Departures from the standard predictions on the bottom system deserve a separate discussion. In Sect. 2 we present models which result from adding to the minimal standard model new vector-like fermions transforming as singlets and doublets under SU(2)L. Details are given in appendix A. In sect. 3 we obtain the A S = 1 hamiltonian * This is the natural scale in an effective theory of weak interactions and is the natural range of masses required to reproduce the experimental value of e, as we will see later.
F. d e l A g u i l a et al.
/
F e r m i o n contributions
to e'
63
incorporating Q C D corrections in the leading logarithm approximation; some of the details are relegated to appendix B. Sect. 4 is devoted to the phenomenological analysis of e and e' in these models. We give our conclusions in sect. 5.
2. Minimal extensions of the standard model We consider the simplest possibilities explicitly. To the minimal standard model we add new up- and down-quarks with their left-handed (LH) and right-handed (RH) parts transforming a s S U ( 2 ) L singlets (model 1) or a s S U ( 2 ) L doublets (model 2). We will distinguish later two subcases of model 1, depending on whether the up or down vector-like quark is the lightest one. In model 2 the two new fermions are nearly degenerate as they have a c o m m o n large bare mass [ 11]. The reader not interested in the details needs only to look at eqs. (4) and (8), where the corresponding lagrangians to second order in the small parameters m / M are given in the mass eigenstate basis. They contain all necessary information for the rest of the paper. The important result is that non-standard couplings among standard fermions are suppressed by powers of ( m / M ) 2. ,t i U,I i i Model I. This model contains the three standard fermion families ~'L = (~mL, UR, D~, i = 1, 2, 3 plus four S U ( 2 ) L singlets containing a new up- and a new down-quark U~, U~, D4L, D~. The Yukawa couplings and mass terms are 5fv+ M = -~/-~ (d/~_¢D~eL + d~LgU~eL) AAd r~4 r-~a _
u -4
ot
i=1,2,3, a=1,...,4,
Mo, U L U R + h . c . ,
-- zv~ a x J EL,- R
(1)
where ~b is the standard Higgs field and ¢ = i~'2¢ *. The lepton sector is the same as in the standard model. When the neutral scalar gets a vacuum expectation value ~d(ul -_ . ~ 2 c~ _O(u),r (¢o) = V, the charge ~ and - ~ quarks get masses ,,,i, ,. Without loss of generality we can assume that in eq. (1) M id= M i = O , i = I , 2 , 3 , M O = M d and M ~ = M4u are real, and that rn~d = m~d3u, i, j = 1, 2, 3 but with rnj4 d = mj,d e ~d~, i.e. in general complex. In order to discuss the predictions of the model it is more convenient to work in the mass eigenstate basis. Down- and up-quark eigenstates result from diagonalizing the matrices u
r d /ml
I td i~ d "] i ml e /
/ ,/~d
/
I
'1~2
~ I I
[ 0
L- ....
/
-
/ i
d
rn3dl re;de ~ [
-6
.
.
.
.
.
(2)
.
(3)
V~u=
.......
o
U---:l
,MJ
F. del Aguila et al. / Fermion contributions to e'
64
respectively, where m ~ are in general complex, d~ ~ and j//a have two types of entries: m entries are standard quark masses and M entries are large bare masses taken to be of order 102-103 GeV. Thus, eqs. (2) and (3) can be diagonalized order by order in m/M. In appendix A we give the eigenvalues and the diagonalizing matrices to order ( m / M ) 2. We note, however, that ~/~ can be diagonatized more easily in two steps. First rotate the 3 x 3 upper left block to bring it to diagonal u form, then the resulting matrix is similar to ~ d (but with entries m~, mjlu e i4~~ and M ~) and is diagonalized in the same way. The 3 x 3 (LH) matrix introduced in the first step coincides to order ( m / M ) 2 with t h e K M matrix and we will denote it by (KM)~j. It turns out that to order (re~M) 2 (see appendix A) rn~, M d and m~, u M u are the physical charge- -½((1, 2, 3) = (d, s, b)) and charge-~ ((1, 2, 3) = (u, c, t)) quark masses respectively. After diagonalizing the mass matrices the gauge interactions can be written ~.~z=l(g2 ~.,2]l/27 -
6
a~ = aj~
[ L Ga
p..,,8
hL ~a
l~d3
0
z - ' ~ , a a ~ - - L 3 / U L - - u a J ~ t ~ L ' Y " L --'¢"
,'
m t U m TM 'J '"~ , ~ ' ( ~ - ~ ) MU2 ~
aL
44 =
E, m'? 2 ~/---~
,
aL4 -
aL*
--
m~ u
4j--'~'-ff
sin z 0 w J E ,~ M)
,
.
e'4j ,
bLt3 = a L ~ ( u ~ d ) ,
(4a)
+ L -or /x fl ~ w = ~/~1 gW.c~¢uL 7 dL+h.c.,
M u2 td
+ (KM)jt
rn~2 m ; 2 e i(6~ ¢~)(KM)/k (-1)%m~ ' 2 -
td
ml mk - Md - 2
m~ 2 (_l)Gm~2_
m Cd
m~
c~4 = ( K M ) j k - ~ e ' ~ , L
mj
C4 4 = ~
m ~ 2 e "'~%~2) ,
tu
caLj='~--~ e ' ~ ( K M ) k j , mid
(KM)jk
~
e i(~-+~)
(4b)
The photon couplings are standard, whereas, the Higgs couplings (which can be written in a similar way) will not be used explicitly. Model 2. In this model the extra vector-like up- and down-quarks are in SU(2)L doublets 0 4 = ( ~ ) 4 , OR = (~)R. The Yukawa couplings and mass terms now read
- M~0LOR+h.c.,
a = l . . . . ,4, i = 1 , 2 , 3 .
(5)
. d(u) ~d(u) w The quark masses are m ~ = v. f~i c~ v. We can assume that M~ = 0 and M = M4 is d d real as well as rn~j = m~ 6~i; i, j = (1, 2, 3). The other masses are in general complex.
F. del Aguila et aL / Fermion contributions to e'
65
The corresponding mass matrices are written
ma
0
?2
~d= ___o_ . . . . . . .
,]
i
]
t ~,,td o-ich~
__,d =-i~b~----~,--fd-'--i-ch'-~-r-~-,g--[
L~tl
***2
~"
~"
rrt 3
(6)
'
i~v, 3
e
I
m qu
I
0
(7)
+I . . . . .
.....
M
with m~j complex. To write the lagrangmn in the mass eigenstate basis we need to diagonalize eqs. (6) and (7), but this is similar to the previous model. The LH (RH) diagonalizing matrix of model 1 is the RH (LH) diagonalizing matrix of model 2 (see appendix A). Using these matrices the gauge interactions are written in the mass eigenstate basis:
~z=½(g2+g
,2"d/2 7 [-~ l L"/x\UL'Y
,~
a~_,~R -c~ I'/L--~C~BURT
~
13 -R --~x UR-d~T~d'~ - b ~ d R y
m (° . M
m j ,umk, u
M 2
a j kR=
ei(*~-4~) '
/~
~8
"
dR--2sln
1"
+
L
-~
~
fl
R
-a
,u.
0wJ~M)
ram2 '
IVI
b ~R -_ a R ( u ~ d ) --
2
'
(8a)
/3
~ w - - x/~gWt~(Cc~ULT dL + Ca~URT dR) + h . c . , c~ = (KM)ik + ( 6 ) t - 1) mj"rn~" m~m~ ei(,¢,), ,~)(KM)~k M z ( - 1 ) % m ) ~2- m~ 2 d
m ;d m 'd
+ (KM)jl(61k -- 1) - M2 m d m td
c~ = - my m~U eie~ + (KM)jk ~ M2 M d
L
mj mj
c4j = -
¢d
.
u
lu
-
( _ 1 ) ~ .
M 2
L -C441 ,
cR = I - y i
rn( u R _ _ "'-, e , ~ cj 4 M ' .
,
mi,u2 + ~ i mi,d2 2M 2
e_i(~_~)
tu
mi, mk-
td
.
md2
e i'~ ,
M 2 e-'~+~e-'~(KM)kj,
c j ~ - m j rnk ei(e~_e~ )
d
mt rnk
R_ cni--
m 1d L e-i~ M ' (8b)
We will finish this section with a few comments on the gauge couplings. Looking at eqs. (4) and (8) one sees explicitly how the suppression factors proportional to some power of ( m / M ) appear in the gauge couplings. In particular we see that in
66
F. del Aguila et al. / Fermion contributions to e'
both models any correction to the standard model couplings involving only ordinary fermions with or without new phases is proportional to (re~M) 2 while the complex couplings involving a standard fermion and a vector-like fermion are at least of order (re~M). 3. Strong interaction corrections to the A S = 1 hamiitonian
From eqs. (4) and (8) one can read off the effective AS = 1 hamiltonian in the absence of strong interactions [taking only the dominant terms in an expansion in (re~M) involving only standard quarks]
HaS eli =1
-4x/~ GFbLd(R)( T 3 -- sin 20wQ) L(R)(g~y~d~)L(R)(CT~%,q~)L(R)
+44
L(R)*C~d L(R)(S,~'y - ~q,~)L(R)(qt~'y~dt3)L(R), GFCCas
(9)
where we use standard notation [13] and a summation on L, R = (1 q: 3'5)/2, on colour a,/3 and on light quarks q is understood. The first term in eq. (9) corresponds to the interchange of a Z-boson and q stands for any standard quark whereas the second term results from the interchange of a W-boson and q stands only for the three light charge -2 quarks. Now we must include strong interaction effects. Using the operator product expansion the short-distance effects due to the W- and Z-boson masses can be factorized out o f the matrix elements and can then be calculated in perturbative QCD following the methods discussed in refs. [14] and [15]. Similarly, the effects of heavy quarks can be factorized and calculated [ 14, 15]. The net result is that each operator in eq. (9) is replaced by a linear combination ~iOi where ~ are calculable strong interaction correction factors taking into account short distance effects and Oi are operators involving only the light quarks u, d and s. The corresponding hamiltonian reads ~ aefs[= l : C ,tat
/L(R)L(R) zL(R)L(R) __TL(R)L(R) u:L(R)L(R) wL(R)L(R) f~ wL(R)L(R) q ~i , ( ) i~ q "~C - q ~i " ~i q ,
(10)
where C are the coefficients in eq. (9), with self-explanatory notation, and a summation in all the indices is understood. For each operator in the original hamiltonian L(R)L(R)
-
/~
d~)L(R)(qo%,qt3)L~R), W L(R)L(m= (g,,y~'q~,)L(R)(Fl~Ywd~)L(R), Zq
= (s~y
(11)
we find a set of operators O~ where i runs from 1 to 6, which is the maximum required to close under renormalization. This set of operators contains the initial operator 7L(R)L(R)I L(R)L(R)I • ~q or Wq (l = 1), the same operator but with the colour contractions interchanged --qZL(R)L(R)2 =
(g,~yJ'd/3)L(R)(Ot3 Y~,q,,)L(R)
wL(R)L(R)2
(g,,Y~'qa)L(R)(qI3%,d,,)L(R)
=
(i = 2 ) ,
and the four penguin operators (for initial RL and RR, operators L and R are
F. del Aguila et al. / Fermion contributions to e'
67
interchanged)
pLL1 = (~a,yttda)L(I~/,yp~U¢3 + . . . )L
(i=3) ;
pLL2 = (g~yt, d/3)L(t~ ,yt~u,~+ . . ")L
(i=4) ;
p L a l = ( g,~y " d,~ )L( ~ y~U~ +" " ")R
(i=5);
pLR2
(i=6).
=
( g,~y'~d~ )L( a~ V,~u,~ +" • ")R
This section is devoted to the calculation o f the strong interaction correction factors rli in the leading logarithm approximation. Although the method is standard [14, 15], some results are new. The details are collected in appendix B. We point out that our calculation includes all possible correction factors ~7i required by any extension of the standard model where the interchange of a vector boson dominates. The essential point is that the absence of the G I M mechanism and the presence of RH couplings in the standard quark sector forces us to consider all possible operators at the vector boson scale. Table 1 contains the strong correction factors 7/ for a particular choice of parameters. In particular, we take the renormalization scale/z of the operators to be 0.5 GeV, which represents the limit of validity of the leading logarithm approximation. A blank in the table means that the operator does not appear in the hamiltonian at the hadronic scale. This is because it involves quarks heavier than 0.5 GeV or because it does not mix under renormalization. The ~7 which are not given explicitly in the table can be obtained from the ~ quoted there as follows: ZsL L - Z LL, z L R -- 7 L R - - 7 L R a -~ - ~ u - Also ZqRL "- Z q LR , ZqRR = Z qLL, WqRL ---W q LR and WqRR -__W q LL but with interchange of L and R everywhere in the operators (including the penguin operators). Z LL'LR are zero up to the third decimal place and Wc,t LR= 0. With eq. (10) we can now discuss the e' contributions in any model. Before doing so let us mention that Higgs interchanges among standard quarks, which should in TABLE 1 N o n - z e r o a n d n o n - r e l a t e d s t r o n g i n t e r a c t i o n f a c t o r s * / f o r A 2 = 0.03, be = 0.5, m c = 1.5, m b = 4.5, m t = 40, m w = 81 a n d m z = 93, all in G e V 1
2
Z LL
1.438
-0.774
Z LL Z LL Z LL Z La Z La Z LR W LL W EL
0.665
0.815
1.434
1.372
-0.703
W LL
W LR
0.818
1.390
3
4
0.012 0.034 -0.006 -0.002 0.018 0.007 0.003 -0.033" 0.030 0.004
-0.056 0.009 0.003 -0.037 -0.011 -0.004 -0.044 -0.005
5
6
-0.007
0.037
0.012 -0.002 0.0 0.010 0.002 0.001 0.018 0.008 0.0
-0.108 0.017 0.008 -0.053 -0.021 -0.009 -0.141 -0.098 -0.015
IF. del Aguila et al. / Fermion contributions to e'
68
principle be added to eq. (9), are extra suppressed by Yukawa couplings in models with extra vector-like fermions. Penguin operators resulting from the exchange of these heavy fermions are extra suppressed by powers of some small mass ratio and are next-to-leading order in a~. 4. Vector-like fermion contributions to e v e
Following standard conventions [16] e'_ e
1
ReA__~2(ImA2
x/21elReAo
Re 32
ImA~) Re
(12) "
Given our ignorance of non-perturbative effects involved in the matrix elements o f the operators appearing in eq. (10) we take Re A~ from experiment, as well as [e[, whereas Im AI is calculated from eq. (10). The result depends on the specific model we consider, that is to say on the coefficients C. Uncertainties in the operator matrix elements, which we evaluate using the vacuum saturation method [17], appear to be rather unimportant because, in general, large contributions not involving penguin operators are present. Let us discuss in detail two specific models. They are realistic in the sense that they agree with present phenomenotogy and can be tested in the future [4] and they provide enough information to discuss other possibilities. Also, they are probably the most attractive possibilities. The first model is a particular case of model 1 in sect. 2. We assume that there is only one extra vector-like down quark at the Fermi scale [MU~oo in eq. (4)], and we assume that the KM phase is negligible*. For definiteness we will assume that the three new parameters m l d a r e comparable. With these assumptions e' can be predicted when e is adjusted to reproduce the experimental value. The essential point is that only a definite unknown factor appears in the calculations, ms,oms,a sin (4)do-~bff)/(Ma) 2, and e fixes it. To calculate e' we first read from eq. (9) the effective hamiltonian in the absence of QCD corrections pd
/./AS=I
Im , , e~
rd
GF me m~ sin (~b~- ~bd)
= 4 x/2
X [_(Ta_sin
M d2 2
EL ] , 0wQ)qL,a ZqLL,LR +cos 2 O~Wu
(13)
other terms are absent or are suppressed by extra mass ratios in this model. We then use eq. (10) to write eq. (13) at p. =0.5 GeV, incorporating QCD corrections in the leading logarithm approximation: v
..aS=l
lmme~
GF
ict
fd
md ms d LLI ..~ 1.567ZuLL2+ 0.282Z LL = 4 x / ~ - M- a2 s i n ( q ~ - q~s)(-1.164Zu
* This is e s s e n t i a l l y the m o d e l o f ref. [7]. In that case a n e x t e n s i o n o f the s c a l a r sector o f the s t a n d a r d m o d e l as well as a n a d d i t i o n a l discrete s y m m e t r y was i n c o r p o r a t e d in o r d e r to enforce a real K M m a t r i x naturally.
F. del Aguila et al, / Fermion contributions to e '
+0.282Z~LL +0.125Z~LR1 +0.220Z~LP-,2 --O.062zLRZ--O.110zLR2--O.O62Z
69
TM
- 0.110Z L ~ - 0.005 pLEa -- 0.051 P LL2+ 0.031 P LR~_ 0.243 pLR2),
(14)
where we have used sin 2 0w = 0.23 and cos 2 01 = 0.947. The operators Z with an upper-index 1 correspond to those given in eq. (11), while those with an upper-index 2 stand for the same operator but with the colour contractions interchanged, pLLX, pLL2, pLR~ and pLR2 correspond to the penguin operators 3, 4, 5 and 6 respectively in the previous section. Finally, using the vacuum saturation method [17] with 2 m~/ms(mu+ rod) = 1.7 and (fv,Jf,~)/(1 rn 2K/rn~) 2 = 2.6 to evaluate the corresponding matrix elements we find -
ImAo=(l=0
md,d ms,a a d 2 M sin (~bd- ~bs)f~mK(-- 1.03),
lIra Heias=l r I Ko) = 4 ~
metmm s,a I m A z = ( I = 2 1 I m H ~as=l IKo) = 4 GF / ~ - M- d2 sin ( t had - 6s)f~.mK(0.24), a 2
(15)
Then eq. (12) gives
-
-
= 2.1 × 10_3/ma ms sin (~b~- q~)
10 -5 '
(16)
with lel = 2.3 x 10 -3, Re Ao = 4.7 × 10 -7 GeV, Re A2 = ~o Re Ao and f= = 130 MeV taken from experiment. It must be noted that the dominant contribution comes from I m A2, due essentially to the relative size of the corresponding real parts and to the Q C D corrections. This makes the final result rather stable under possible uncertainties in the matrix element calculations. Eq. (16) is a prediction of the model once • (~bd - Cb~)/(Ma) 2 has been fixed to reproduce e. Following ref. the factor m amm~,d sm [7], with standard conventions [17], e is given by f~ ~irr/4
Im
M12
2 Re Mz2'
(17)
where experiment gives I m M12 = 6.48 x 10 -3 . Re M~2
(18)
In this model I m M~2 is dominated by the interchange of a single Z. Using eq. (4) /
;d
td\ 2
I m M ~ z = - ~ / - ~o v ~2,,2 J ~ c m K~ Jl Jmd ~ ) ms 't sin 2(~bda - ~b~a)r/,
(19a)
where B is defined by
--~f KmKB
(19b)
F. del Aguila et al. / Fermion contributions to e'
70
and rl is a strong correction factor [18] for the AS = 2 effective hamiltonian
(as(me)~6/27
(ors(Mz) ~ 6/21( as ( mt ) ~6/23 (O~s( mb ) ~6/25 7/= \ as(mt) ] \as(mb)] \a~(m¢)] \ as(tz) ]
"
(19c)
Taking Re M12=½AM from experiment, AM being the K ° mass splitting, B = 1" and ~ = 0.665, where the same values for A, ~, rnc. • • as in the table have been used to evaluate eq. (19c), we find
['/tdtd\2
XmM12--2"7x 10 --3 L t M Re M 1 2
]/
) sin 2/6
-C)
lO -1°
(20)
) < 2 x 10 -5 from Relating eqs. (20) and (18) and remembering that ma,a ms/~1vl ,d/( .,a,2 0 + /z -- [11], we obtain KL~IX s . 6.4 x 10-6<~ ~ ds m
which implies
[see
d
d
1.9x10 - 5 ,
(21)
eq. (16)] ]e'/e[ = (1.3 - 3 . 9 ) x 10 .3 .
(22)
The sign o f e ' / e is not determined but the size is a prediction of the model. The range of variation comes from the different functional dependence o f the different parameters in e and e'. We will discuss our conclusions later. Let us now consider another model, the second model of sect. 2. As before we assume that the K M phase is negligible. We also assume that the new parameters m' are all of the same order. In this case eq. (9) reads
I./zlS=I
In,,~n
=
4GF
,f~
F [ m~' d m ' d
sin
~ f)$ L,RTRL, RR
(q~dd--6sd)(T3--sin 2 vW~lq
~q
rd pu rnd m~ sin (~b,~- cb~)(KM)*~W~R M2 m std m utu
1
M------T - sin ( 6 ~ - 6~) cos 01W~ L •
(23)
At the scale tz = 0.5 GeV and in leading logarithm approximation it becomes (using the table) l_l a s = 1
Im--~
GF Fm'am'a L ~
=4~
d ~bs)(-0.221Z~ d RRI + 0.119Z~RR2 sin (~bd--
+ 0.051Z RR + 0.051Z RR + 0.282Z~RL1+ 0.497ZuRL2 -
0.345Z RL1- 0.607Z RLz
0.007P RR2-
0 . 0 0 2 P RL1 - 0 . 0 0 1 p R L 2 )
-- 0 . 3 4 5 Z RL1 - 0 . 6 0 7 / R L 2 -- 0 . 0 0 4 P
TM
+
mstd mutU d RL1 wRL2 1 M~sin(~b~-~b~)(0.796W~ +1.353 ~ ) . • In this m o d e l B c a n n o t b e s m a l l e r t h a n 0.6 if e has no other contributions.
(24)
71
F. del Aguila et al. / Fermion contributions to e '
Evaluating the matrix elements with the vacuum saturation method gives ImA0=(l=01Im
as=l n~
o G F r| m d,a ra~,d sin (~b~- ~b~)(2.35) IK)=a~-~L M2
msIdmutu M : sin (6u~ - 6 ~ ) ( - 5 . 0 4 ) 1 f~m~,
lmA::(I=2llmneT:llK°):
[-mrdmrd
L M :'s sin(6~-~b~)(-0.67)
mtdmtu
]
M ~ ' ~TM sin ( 6 ~ - 6~)(-0.87) f ~ m ~ . Then, eq. (12) gives [with
[el,
(25)
Re A o , . . . as for eq. (16)]
--=
sm (~b~- ~b
E
x 10-3[~ M
sin (4~:- 4~)
10-'+4.3
10-s '
(26)
to be compared with eq. (16) in the former model. Before discussing the size of the two contributions we must note that the parameters in eq. (26), in particular those entering the second term, are poorly known. Thus e ' / e can have a wide range of values due to possible cancellations between the terms. In fact only the first term in eq. (26) has a definite size. It comes from the interchange of a Z-boson, which gives also the main contribution to e, just as in the previous model. Then eq. (21) applies and up to the sign the first term ranges from 3.5 x 10-3 to 10 -2, but the second term is in principle comparable and if it cancels the first term the model can fit any experimental value. We can now discuss the contributions of vector-like fermions in general. In any model there are always new mass parameters: small rn' masses, large ones M and new phases ~b. The new CP-violating effects always have a factor m'/M to some power and a sin ~b. There are two extreme possibilities concerning m' - that they are of the same order or that they are in the same hierarchy as for the known fermion masses. We will always make the first assumption. The second has consequences for the bottom system and will be discussed elsewhere. One must distinguish different models to make non-trivial statements. We see three different cases- when only a new vector-like down quark transforming as a singlet under SU(2)L exists at the Fermi scale, when the only vector-like fermion is an up quark or when both quarks transforming as a doublet of SU(2)I. exist. The first two cases are subcases of model 1 in sect. 2 and the third case is model 2 in that same section. Within these three cases one must distinguish if the standard contribution to e due to the KM phase is sizeable or not. If it is large, then e' is never a prediction because standard and
72
F.
del Aguila et aL / Fermion contributions to e'
new contributions are comparable and the relative signs are undertermined. If the KM phase is negligible then the three cases are different. In the first case [see eq. (22)] l e ' / e I ~ 2 x 10 -3 but the sign is undetermined, with e reproducing experiment. This model can then be tested in the near future. The second case we have not analyzed in detail. The dominant contribution to e' is due to the interchange of a W because the neutral couplings of the down quarks are diagonal. Also, e comes from the box diagram for the same reason. As the box contribution is suppressed relative to the tree level one typically by at least one order of magnitude, the CP-violating factor ( m ' / M ) 2 sin $ must be larger than in the first model to reproduce e. This implies a larger contribution to e', a priori in disagreement with experiment. The third case, discussed in detail above [see eq. (26)], is an example that when more than one phase is available no prediction is possible. The saturation of e implies a typical contribution to e' of order 7 × 10 -3. However, other contributions to e' are o f the same order of magnitude and large cancellations can be present.
5. Conclusions The presence of vector-like fermions at the Fermi scale is experimentally possible and theoretically natural [10, 11]. In this paper we have discussed the consequences of their existence for the description of CP violation in the kaon system. We have studied in detail several models, including a complete calculation o f the strong interaction correction factors (in leading logarithm approximation) entering the AS = 1 hamiltonian. The result of our analysis is that vector-like CP non-conserving contributions are milliweak. That is to say that if the observed CP violation, e, is due to the existence of these fermions then I '/el is typically of order 10-3-10 -2, independently of the size of the penguin diagram contribution. If there is only a new vector-like down quark then 10-3, with the possibility o f accommodating any sign. If there is a vector-like doublet of quarks, le'/e[ can have a large range of values because large cancellations among contributions of similar size but involving different phases are possible. If the standard CP-violating contributions due to the presence of a phase in the KM matrix are not suppressed, le'/el is not a prediction in any model. Given the milliweak character of both types of contributions, they could add but could also cancel. In order to further clarify our results let us write down e and e ' / e as a sum of the standard contribution [19] plus the new vector-like one. Following our previous conventions [16] (see (17), (18) and (20)) Im M~2= 6.48 x 10-3(exp.)-- 0.9s2s3s6 -1.8 + 1.35s~ 3 + 0 . 9 Re M12 mc /
~d
¢d\ 2
ms ] sin 2 ( ~ +2.7 x 107/m~ \---M-~-
In
(standard)
"1
J
~b~)(vector-like) B
(27)
F. del Aguila et al. / Fermion contributions to ~'
73
and e'
m'aa m'~a ~ 6s2s3s6P(standard) + 210 ~ sin (~b~- 4~a)(vector-like)
(28a)
or ~'
m rdd m srd
--e ~ 6s2s3s6P(standard) - 550 ~ td
•
sm (~b~- ~a)
ru
ms mu . + 430 ~ sm (~b~- ~b~)(vector-like),
(28b)
where (28a) corresponds to a new vector-like down-quark (see (16)) and (28b) to new vector-like up- and down-quarks in a SU(2)L doublet (see (26)) at the Fermi scale. The contribution to e is the same in both cases (see (27)), but with M d replaced by M in case (28b). As usual s2, s3, s6 stand for sin 02, sin 03, sin 6, with 02, 03, 6 the standard K M angles. The matrix element B is defined in (19b), whereas the penguin P = 4(1 = 0[(~T~d/3)L(at~O/~,ua + . . . ) R [ K ° ) . The vector-like contributions in (28) are the result of many terms (see (14) and (24)) and they are rather insensitive to particular matrix elements. To obtain a numerical insight let us take B = I , P = 0 . 7 and mr~me=40~1.5, as before, and $2=0.05, $3=0.03. We also assume m,a d m ,sa // M a2 = 2 × 1 0 -5, saturating the bound from K ° ~ / x + / z - , and that the same value applies to msmmu'U'/M" .2, although this mass ratio can be larger. Then, defining 6v = ~ba a - ~ , 6 v' --~b~-~b~, we obtain for (27) and (28) 6.48 x 10 3 _ 8.8 × 10-3s6(standard) + 10.8 x 10-3S28v(vector-like), E ~
- - ~ 6.3 x 10-3sS(standard) + 4.2 x 10-3SSv(vector-like),
(29) (30a)
E or Et
- - ~ 6.3 x 10-3s6(standard) - 11 x 10-3S6v + 8.6 8
× 10-3sS~/(vector-like).
(30b)
I f the forthcoming C E R N experiment measures e ' / e = 5 x 10 -3, (29) and (30) will be compatible with the standard model predictions, s 8 - 0.74, s6~, ~ 0. This would not exclude vector-like fermions which could exist but with small new phases (or mass ratios m ' 2 / M 2 ) . I f on the other hand e ' / e happens to be < 2 x 10 -3, then we can conclude that either the value used for the penguin P was too high and P < 0.3 should be used in which case s6 ~0.74, s6~,~0 is still allowed or that P =0.7 was correct in which case vector-like fermions would explain the smallness of e'/e. In fact for any value of e ' / e there are 6, 6~ ) values fulfilling (29) and (30).
F. del Aguila et al. / Fermion contributions to e'
74
In conclusion the first observable effect induced by new vector-like fermions at the Fermi scale should be CP violation in the kaon system. We have shown that their contribution to e ' / e is milliweak and that they may be an alternative to the standard model description of CP violation. In order to disentangle this new source of CP violation from the standard model contribution more experimental and theoretical work is required. A better understanding of B-meson decays and a more precise determination of the top mass should be accompanied by a more accurate measurement of e'/e. On the theoretical side a drastic reduction of the present uncertainties in the estimates of the matrix elements involved in kaon decays (particularly the penguin) is needed. Perhaps lattice calculations could provide the required accuracy for those matrix elements. We thank A. De Rfijula and J. Ellis for useful comments.
Appendix A As explained in the text the diagonalization of the different mass matrices appearing in the models considered reduces to the diagonalization of the mass matrix I I I
~ =
m/~s~ ',, ms'e<
j, k = 1, 2, 3,
I "1" . . . . . . . . . I
. . . . . . . . . .
0
~ I
(A.1)
M
where m and m ' are small entries and M a large one. This matrix is diagonalized as usual by two unitary transformations UL,~U ~ =
D.
(A.2)
To second order in ( m ' / M ) 2, the eigenvalues are [ mn \ Aj = r n j ~ l - ~ ) , =M
1 + ~
and the diagonalizing matrices are
ULSk = e i%-~k) 6jk 4 mJnk g 2
ms (-1)~km~-m
ULj4=-U* .=e %{-mJ'~ L40 \ MJ' Ugjk = ei(%-e'k)[ 8Sk+ (1 - 6Sk) L
'
UL44 = 1 2k re'k2 2M 2' m~mtk
I'?lSl'Hk
] "~
2 2 M 2 ( - 1 ) 8,~mk--msJ
F. del Aguila et aL / Fermion contributions to e'
URj4 = -- U*4j = e ~% -
,
UR44 = 1.
75
(A.4)
Appendix B In this appendix we outline the calculation of the strong interaction correction factors ~7 for the AS = 1 hamiltonian, involving four-fermion operators which result from the interchange of a vector boson. The method is standard [14, 15] but new cases are considered. We work in the leading logarithm approximation. Each particle is considered as massless for energy scales above its mass and as infinitely heavy for energy scales below it. Then the QCD correction factors ~ can be written as the product of four steps, corresponding to the integrating out of the vector boson and the three heavy quarks t, b and c. The evolution between two mass scales is described by a factor Vrs for the coeffÉcient of the operator r expanded in the basis s. The matrices Vr~ are functions of ratios of strong interaction coupling constants y and anomalous dimension matrices y:
Vrs('Y, y) = Xslya'S~r I ,
(B.1)
where X is (up to a factor of g2/8'rr2) the matrix diagonalizing the corresponding anomalous dimension matrix 3': -1 T Xtm ym, X,r = 81rzr,
(B.2)
zr being the eigenvalues, and z,
(B.3)
with / 3 o = - 1 1 + 2 n being the first-order coefficient of the t3 function and n the number o f massless quarks. Mso, projection operators must sometimes be inserted to take into account the appearance of linear relations among operators. Indices vary from 1 to 6 as a maximum. Given that QCD preserves flavour and chirality, an operator mixes at most under renormalization with the same operator but with the colour index contractions interchanged and with the usual four penguin operators in the leading logarithm approximation. Let us now discuss all the particular cases. The relations given in sect. 3 relating different sets of '1 factors are due to the fact that QCD does not distinguish chirality nor flavour. ZuLL factors are obtained from [ (6)a~(M~)~ ( ( 5 ) a ~ ( m 2 t ) ' ~ / {~(4)a~(m~)~ r/zkCr= V,t~TLL, a~(m2t) ] Vtk_ TeL, a,(rn~)] "kj\ rLL, as(m~)] "
(~(3)%(mc2)~ "YLL,
(B.4)
F del Aguila et al. / Fermion contributions to e'
76
where -1 3
3 -1
-~1
~I
-~1
1
Ii 9
H 9
2 --9
2 3
3-1n
-l+In
-in
in
1
-3
-in
In
-~n
-8+~n
(B.5)
with l, k, j = 1, 2 , . . . , 6 corresponding to Z LL~, Z LL2, pLL,, pLL2, pLR, and pLR2, respectively, and 1 -1 ] 1
1
1
(BT)5× 6 =
1
,
(B.6)
1 1
where j, 7 = 1, 2. . . . ,5 correspond to Z LL1, Z LL2, pLL1, pLR~ and pL~, respectively, and (L3L) [ =
[
--1 _~,
3
u3 229 -~2
-1
1
~2
1 -~
~
(B.7)
-3 -7
Z LL factors are given by ,, / (6) as(M2)~v / (5> as(m{)]V ~ { . ( 4 ) a~(mbz)'~ nz~L~= V,,~'rLL, adm 2) ] ",k~ 3'LL, adm2)] k~i rLL,---;--~_,l
.
(.
a~(m:)~
where l, k, j are as before but changing Z~ r~'2 to Z~ Ll'2 (this change to be done in each case will not be state explicitly in what follows),
~v
(n)4×6
'
=
1
|
1 ,
1
(B.9)
with ], /' = 1 , . . . , 4 corresponding to the four penguin operators pLL1,2 pLRL2and 11 --9-
11 3
2 --9
2 3
'
t -~n
~n
-~n
]n
1
-3
-~n
-8+~n
(B.10)
F. del Aguila et al. / Fermioncontributionsto e'
77
Analogously
,,.(~(4) as(rn~)~V
X--kj~I'LL, OIS(m2)]
[ ~ ( 3 ) a~(m¢:)'~ 3f~ /'LL, ~ ) •
(B.11)
For the down-quarks the strong correction factors can be obtained using ~TZ~LV=
a:(m2t)~ a~(m~)~ Vrk,i y~) ~(m~)J Vk'j'~ -- { ")ILL t(4) , ~(m3c)J
• r { ,<6> ° ~ s ( M 2 ) ~ F I / ' ~ ' Y L L , a~(mZt) j O~
2
× V.j , \{ .LL, ,,(3) ~(m¢___~)~ .~( ~)) ,
(B.12)
where 1', k', j' and i' run from 1 to 5 and stand for Z~ L, pLL1,2 peru,2 and 2
-} 11 9
3-~n
g1
-~1
1
1~ 3
2 --9
2
-l+In
-~n
½n
1
-3
(B.13) In this case the operator Z~ L but with the colour indices interchanged is the same operator by a Fierz transformation, so there are only five operators mixing under renormalization. The Z~ L factors follow from
~TzkL~=
v,,[..,, ~ ,~(M~)] / (. o,~(m~t)~ ~. BkC, \ .LL, ,~(m~) ) V,k~ WL, Vr/~(4) a~(rn2)~V {C(3) %(m2)~ x k S t r L L%, ~t lm\ d / J * k r L L , ~ ) .
(B.14)
Z~ R factors come from //
(6,
as(M~)'~ V~,( y(~, a,(mt:)'~
rlzk% = V, tk YLR, a~(m2) /
a~(m~)/
( as(m~)~ ,/ {. ,3, %(m:)~ x Vk,\ y[~, a~(m~)] ,,,\rrR, as(p2)].
(B.15)
where 1
-3 -8
1 --9 11
1 3 lj
1 --9 2
! 3 2
3-~n
-l+½n
-~n
In
1
-3
-~,,
-8+~n
),(n) _ LR--
-9',,
I,,
(B.16)
78
F. del Aguila et aL / Fermion contributions to e'
with l, k, j, i = 1, 2, . . . , 6 corresponding to --~zLR1,ZuLR2, pLL1, pLL2, pLR1, pLl~, respectively. Factors of Z~ R are given by
--[ (6)~(M~)] Vllk('YLR, a sOts(m2t)~Vkj(3/(L4)R, ( m 3 ) ~
nZ~R~ Vll~ ~/LR' Ods(tnt2) ]
( 5 )
a~(m~)]
=
a,(m~)]
" . (,'(3) as(m~)'~ x B,j• Vj,^_YLL, ot~(~?))'
(B.17)
where YLR "(") = ~"~, whereas ~,f {^ (6) °~s(g2)~
m2
T~zLRi~= ll~YLR, oe~(m2) ] Dlf V ~ \| 'YLL,~c~.~t/ x V~j(~(4)
b)/
as(m~)~V [^(3)a~(rn~)~
(B.18)
.LL. ,,s(r,OJ ~;V ~L' ~ ) "
Finally // (6)°~s(M2)~
[ (5) °~(mt2)'~
nz~.~ = v,,t ~L., ,~s(~t~) ] v,~k VLR. '~('~b)] ^
(,,(4, oq(n~)'~,, {*(3)~(m~)~
XBkkV,~y\ YLL, as~rnc)/" 2\/v 3~£|~LL,\. O~s(/,/2) /
•
(B.19)
The factors for WuLL, W~LL and W EE are equal to eqs. (B.4), (B.8) and (B.11), 2 by respectively but replacing 1 by 2 in the initial matrix elements and a~( M .z) a~(M2) in their arguments. The factors for W LR are given by
.wv,= V,~(~,~.,"'a,(m~) (M~v)~ ( ,,,(,,,2),~ / Vr~x~LR, oq(m~,)J (K2o) where (B.21) and I,/~, y and ~ = 1, 2 correspond to W LRl and W,LR2, respectively.
References [1] S.L Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in Elementary particle theory, ed. N. Svartholm (Almqvist and Wiksell, Stockholm, 1968) p. 367 [2] M. Kobayashi and T. Maskawa, Progr. Theor. Phys. 49 (1973) 652 [3] J.K. Black et al., Phys. Rev. Lett. 54 (1985) 1628;
F. del Aguila et al. / Fermion contributions to e'
79
R.H. Bernstein et al., Phys. Rev. Lett. 54 (1985) 1631 [4] CERN experiment NA31, CERN/SPSC/P174; FNAL experiment E731 [5] T.D. Lee, Phys. Rev. D8 (1973) 1226; S. Weinberg, Phys. Rev. Lett. 37 (1976) 657 [6] A.A. Anselm, J.L. Chkareuli, N.G. Vraltsev and T.A. Zhukovskaya, Phys. Lett. 156B (1985) 102; X.G. He and S. Pakvasa, Phys. Lett. 156B (1985) 236 [7] F. del Aguila and J. Cortes, Phys. Lett. 156B (1985) 243 [8] M. Dugan, B. Grinstein and L. Hall, Nucl. Phys. B255 (1985) 413; J.M. Grrard, W. Grimus, A. Masiero, D.V. Nanopoulos and A. Raychaydhuri, Nucl. Phys. B253 (1985) 93 and references therein [9] M.G. Gavela and H. Georgi, Phys. Lett. l19B (1982) 141; R. Decker, J.M. Grrard and G. Zoupanos, Phys. Lett. 137B (1984) 83 and references therein; R. Mohapatra, Maryland preprint 85-124 (1985) and references therein [10] P. Ramond, Proc. 4th Kyoto Summer Institute on Grand unified theories and related topics, Kyoto, Japan (1981), eds. M. Konuma and T. Maskawa (World Sicence, Singapore, 1981) [11] F. del Aguila and M.J. Bowick, Nucl. Phys. B224 (1983) 107 [12] S.L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285 [13] G.G. Wohl et al. (Particle Data group) Rev. Mod. Phys. 56 (1984) 51 [14] A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, Nucl. Phys. B120 (1977) 316; F.J. Gilman and M.B. Wise, Phys. Rev. D20 (1979) 2392 [15] R.D.C. Miller and B.H.J. McKellar, Phys. Reports 106 (1984) 169 [16] L.L. Chau, Phys. Reports 95 (1983) 1 [17] A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, Sov. JETP 45 (1977) 670 [18] F.J. Gilman and M.B. Wise, Phys. Rev. D27 (1983) 1128 [ 19] A.J. Buras, MPI-PAE/PTh 46/84 (1984), invited talk at Workshop on the future of medium physics in Europe (Freiburg, April, 1984) and references therein
VECTOR-LIKE FERMION
CONTRIBUTIONS
TO e'*
F. DEL AGUILA1, M.K. CHASE and J. CORTES: CERN, Geneva, Switzerland Received 15 August 1985 (Revised 14 October 1985) We consider extensions to the standard electroweak model which contain new vector-like fermions at the Fermi scale as a source of CP violation. After a detailed calculation of the QCD corrections to leading order, we find that the new contributions to le'/e[ range typically from 10 3 to 10 2 when we require that the corresponding contributions to e saturate its experimental value. If, for instance, a new vector-like down-quark exists at the Fermi scale le'/el is predicted to be ~2 x 10-3. These results are within the range of sensitivity of presently planned experiments.
1. Introduction The m i n i m a l v e r s i o n o f the s t a n d a r d m o d e l [1], so successful in d e s c r i b i n g all l o w - e n e r g y p h e n o m e n a , m a y a p p e a r in the future to be i n a d e q u a t e for d e s c r i b i n g CP v i o l a t i o n in the k a o n system. All CP v i o l a t i o n in this m o d e l , w h i c h is explicitly i n c o r p o r a t e d in the Y u k a w a couplings, can be r e l a t e d to a u n i q u e p h a s e in the K o b a y a s h i - M a s k a w a ( K M ) m a t r i x [2]. W h e n this p h a s e is a d j u s t e d to r e p r o d u c e e, e ' can t h e n be p r e d i c t e d to the extent t h a t one c a n estimate the c o r r e s p o n d i n g m a t r i x e l e m e n t s i n c o r p o r a t i n g strong interactions. In p a r t i c u l a r e' is r e l a t e d to the size o f the p e n g u i n d i a g r a m c o n t r i b u t i o n a n d if this h a p p e n s to b e small then so is A t p r e s e n t there is no o b v i o u s d i s a g r e e m e n t b e t w e e n the m i n i m a l v e r s i o n o f the s t a n d a r d m o d e l a n d e x p e r i m e n t . H o w e v e r , the l a c k o f a d e e p u n d e r s t a n d i n g o f C P v i o l a t i o n in t h e s t a n d a r d m o d e l a n d s o m e r e c e n t e x p e r i m e n t a l results [3]**, have r e n e w e d t h e interest in p o s s i b l e e x t e n s i o n s o f the s t a n d a r d m o d e l , w h e r e C P v i o l a t i o n c o u l d b e b e t t e r u n d e r s t o o d o r w h o s e p r e d i c t i o n s c o u l d d e v i a t e f r o m the s t a n d a r d m o d e l in a definite way. T h e s e e x t e n s i o n s c o u l d i n c o r p o r a t e new scalars [5], new f e r m i o n s [6, 7]***, o r n e w v e c t o r b o s o n s [9]. In this p a p e r w e discuss in detail the c o n s e q u e n c e s o f a d d i n g n e w vector-like f e r m i o n s . T h e s e are f e r m i o n s w h o s e lefta n d r i g h t - h a n d e d p a r t s t r a n s f o r m e q u a l l y u n d e r the e l e c t r o w e a k g a u g e group. T w o k i n d s o f f e r m i o n s , a f o u r t h g e n e r a t i o n o f f e r m i o n s [6] o r vector-like f e r m i o n s [10, 11], c a n be a d d e d to the s t a n d a r d m o d e l with a n a t u r a l s u p p r e s s i o n o f flavour * Work partially supported by the CAICYT. 1 On leave of absence from Universitat Autonoma de Barcelona, Spain. 2 On leave of absence from Universidad de Zaragoza, Spain. ** Presently planned experiments [4] can clarify the situation if the expected accuracy of order 10 3 for ]e'/e] is reached. ***For supersymmetric models, see ref. [8] and references therein. 61
62
F. del Aguila et al. / Fermion contributions to e'
changing neutral currents (FCNC) and without introducing new symmetries. In the first case the G I M mechanism applies [12]. In the second case the G I M suppression is replaced by new and natural suppression factors of order some power of m~ M, where m is an ordinary fermion mass and M the mass of the new vector-like fermions o f order 10z-103 GeV*. These factors of re~M, which must typically be smaller than 10-2-10 -3 to suppress F C N C as required by experiment, do not exclude, however, sizeable CP-violating effects, given the smallness of the observed CP violation. In this paper we calculate the new contributions to e' induced by these fermions using e as an input parameter. In a recent letter [7] by some of the present authors a new model of weak CP violation, incorporating a new vector-like down quark and two new scalar doublets, was considered as a minimal extension of the standard model where the observed CP violation in the kaon system is due to the presence of the new vector-like fermion and CP is spontaneously broken. Now we consider different and more general cases and we give full details. We add to the minimal standard model only new vector-like fermions, singlets and doublets under S U ( 2 ) L , distinguishing different cases depending on which are the iightest vector-like fermions; CP is explicitly broken by Yukawa and mass terms. The KM phase can be present, as well as the new phases associated with the new fermions. A priori we will assume that the effects of the KM phase are small and that the new phases induce the observed e. Then we calculate e' and find that le'/e] ranges typically from 10-3 to 10 -2. In the particular model considered previously [7] it was estimated that the new contributions to e' could be up to an order of magnitude smaller than those of the standard model. However, QCD corrections to e and e' are important and reduce that estimate by a factor of 3. Thus, if a new vector-like down-quark exists at the Fermi scale a value of le'/el 2 x 10 3 is predicted. Any other value can be accommodated if for instance new vector-like up- and down-quarks in a SU(2)L doublet are considered. The value of [e' I is the result of many new contributions in these models and, in contrast with the standard model, it is essentially independent of the size of the penguin terms. Finally we must note that if the KM phase is large, its effects on e' can be comparable to the new vector-like fermion contributions which do not have a definite sign because the new contributions to e are quadratic in the new phases whereas the new contributions to e' are linear. Thus, a cancellation between both terms cannot be excluded. Also, in some models with more than one relevant new phase e' can have a wide range of values. Implications of these n e w p h a s e s for electric dipole moments are not restrictive at present [7]. Departures from the standard predictions on the bottom system deserve a separate discussion. In Sect. 2 we present models which result from adding to the minimal standard model new vector-like fermions transforming as singlets and doublets under SU(2)L. Details are given in appendix A. In sect. 3 we obtain the A S = 1 hamiltonian * This is the natural scale in an effective theory of weak interactions and is the natural range of masses required to reproduce the experimental value of e, as we will see later.
F. d e l A g u i l a et al.
/
F e r m i o n contributions
to e'
63
incorporating Q C D corrections in the leading logarithm approximation; some of the details are relegated to appendix B. Sect. 4 is devoted to the phenomenological analysis of e and e' in these models. We give our conclusions in sect. 5.
2. Minimal extensions of the standard model We consider the simplest possibilities explicitly. To the minimal standard model we add new up- and down-quarks with their left-handed (LH) and right-handed (RH) parts transforming a s S U ( 2 ) L singlets (model 1) or a s S U ( 2 ) L doublets (model 2). We will distinguish later two subcases of model 1, depending on whether the up or down vector-like quark is the lightest one. In model 2 the two new fermions are nearly degenerate as they have a c o m m o n large bare mass [ 11]. The reader not interested in the details needs only to look at eqs. (4) and (8), where the corresponding lagrangians to second order in the small parameters m / M are given in the mass eigenstate basis. They contain all necessary information for the rest of the paper. The important result is that non-standard couplings among standard fermions are suppressed by powers of ( m / M ) 2. ,t i U,I i i Model I. This model contains the three standard fermion families ~'L = (~mL, UR, D~, i = 1, 2, 3 plus four S U ( 2 ) L singlets containing a new up- and a new down-quark U~, U~, D4L, D~. The Yukawa couplings and mass terms are 5fv+ M = -~/-~ (d/~_¢D~eL + d~LgU~eL) AAd r~4 r-~a _
u -4
ot
i=1,2,3, a=1,...,4,
Mo, U L U R + h . c . ,
-- zv~ a x J EL,- R
(1)
where ~b is the standard Higgs field and ¢ = i~'2¢ *. The lepton sector is the same as in the standard model. When the neutral scalar gets a vacuum expectation value ~d(ul -_ . ~ 2 c~ _O(u),r (¢o) = V, the charge ~ and - ~ quarks get masses ,,,i, ,. Without loss of generality we can assume that in eq. (1) M id= M i = O , i = I , 2 , 3 , M O = M d and M ~ = M4u are real, and that rn~d = m~d3u, i, j = 1, 2, 3 but with rnj4 d = mj,d e ~d~, i.e. in general complex. In order to discuss the predictions of the model it is more convenient to work in the mass eigenstate basis. Down- and up-quark eigenstates result from diagonalizing the matrices u
r d /ml
I td i~ d "] i ml e /
/ ,/~d
/
I
'1~2
~ I I
[ 0
L- ....
/
-
/ i
d
rn3dl re;de ~ [
-6
.
.
.
.
.
(2)
.
(3)
V~u=
.......
o
U---:l
,MJ
F. del Aguila et al. / Fermion contributions to e'
64
respectively, where m ~ are in general complex, d~ ~ and j//a have two types of entries: m entries are standard quark masses and M entries are large bare masses taken to be of order 102-103 GeV. Thus, eqs. (2) and (3) can be diagonalized order by order in m/M. In appendix A we give the eigenvalues and the diagonalizing matrices to order ( m / M ) 2. We note, however, that ~/~ can be diagonatized more easily in two steps. First rotate the 3 x 3 upper left block to bring it to diagonal u form, then the resulting matrix is similar to ~ d (but with entries m~, mjlu e i4~~ and M ~) and is diagonalized in the same way. The 3 x 3 (LH) matrix introduced in the first step coincides to order ( m / M ) 2 with t h e K M matrix and we will denote it by (KM)~j. It turns out that to order (re~M) 2 (see appendix A) rn~, M d and m~, u M u are the physical charge- -½((1, 2, 3) = (d, s, b)) and charge-~ ((1, 2, 3) = (u, c, t)) quark masses respectively. After diagonalizing the mass matrices the gauge interactions can be written ~.~z=l(g2 ~.,2]l/27 -
6
a~ = aj~
[ L Ga
p..,,8
hL ~a
l~d3
0
z - ' ~ , a a ~ - - L 3 / U L - - u a J ~ t ~ L ' Y " L --'¢"
,'
m t U m TM 'J '"~ , ~ ' ( ~ - ~ ) MU2 ~
aL
44 =
E, m'? 2 ~/---~
,
aL4 -
aL*
--
m~ u
4j--'~'-ff
sin z 0 w J E ,~ M)
,
.
e'4j ,
bLt3 = a L ~ ( u ~ d ) ,
(4a)
+ L -or /x fl ~ w = ~/~1 gW.c~¢uL 7 dL+h.c.,
M u2 td
+ (KM)jt
rn~2 m ; 2 e i(6~ ¢~)(KM)/k (-1)%m~ ' 2 -
td
ml mk - Md - 2
m~ 2 (_l)Gm~2_
m Cd
m~
c~4 = ( K M ) j k - ~ e ' ~ , L
mj
C4 4 = ~
m ~ 2 e "'~%~2) ,
tu
caLj='~--~ e ' ~ ( K M ) k j , mid
(KM)jk
~
e i(~-+~)
(4b)
The photon couplings are standard, whereas, the Higgs couplings (which can be written in a similar way) will not be used explicitly. Model 2. In this model the extra vector-like up- and down-quarks are in SU(2)L doublets 0 4 = ( ~ ) 4 , OR = (~)R. The Yukawa couplings and mass terms now read
- M~0LOR+h.c.,
a = l . . . . ,4, i = 1 , 2 , 3 .
(5)
. d(u) ~d(u) w The quark masses are m ~ = v. f~i c~ v. We can assume that M~ = 0 and M = M4 is d d real as well as rn~j = m~ 6~i; i, j = (1, 2, 3). The other masses are in general complex.
F. del Aguila et aL / Fermion contributions to e'
65
The corresponding mass matrices are written
ma
0
?2
~d= ___o_ . . . . . . .
,]
i
]
t ~,,td o-ich~
__,d =-i~b~----~,--fd-'--i-ch'-~-r-~-,g--[
L~tl
***2
~"
~"
rrt 3
(6)
'
i~v, 3
e
I
m qu
I
0
(7)
+I . . . . .
.....
M
with m~j complex. To write the lagrangmn in the mass eigenstate basis we need to diagonalize eqs. (6) and (7), but this is similar to the previous model. The LH (RH) diagonalizing matrix of model 1 is the RH (LH) diagonalizing matrix of model 2 (see appendix A). Using these matrices the gauge interactions are written in the mass eigenstate basis:
~z=½(g2+g
,2"d/2 7 [-~ l L"/x\UL'Y
,~
a~_,~R -c~ I'/L--~C~BURT
~
13 -R --~x UR-d~T~d'~ - b ~ d R y
m (° . M
m j ,umk, u
M 2
a j kR=
ei(*~-4~) '
/~
~8
"
dR--2sln
1"
+
L
-~
~
fl
R
-a
,u.
0wJ~M)
ram2 '
IVI
b ~R -_ a R ( u ~ d ) --
2
'
(8a)
/3
~ w - - x/~gWt~(Cc~ULT dL + Ca~URT dR) + h . c . , c~ = (KM)ik + ( 6 ) t - 1) mj"rn~" m~m~ ei(,¢,), ,~)(KM)~k M z ( - 1 ) % m ) ~2- m~ 2 d
m ;d m 'd
+ (KM)jl(61k -- 1) - M2 m d m td
c~ = - my m~U eie~ + (KM)jk ~ M2 M d
L
mj mj
c4j = -
¢d
.
u
lu
-
( _ 1 ) ~ .
M 2
L -C441 ,
cR = I - y i
rn( u R _ _ "'-, e , ~ cj 4 M ' .
,
mi,u2 + ~ i mi,d2 2M 2
e_i(~_~)
tu
mi, mk-
td
.
md2
e i'~ ,
M 2 e-'~+~e-'~(KM)kj,
c j ~ - m j rnk ei(e~_e~ )
d
mt rnk
R_ cni--
m 1d L e-i~ M ' (8b)
We will finish this section with a few comments on the gauge couplings. Looking at eqs. (4) and (8) one sees explicitly how the suppression factors proportional to some power of ( m / M ) appear in the gauge couplings. In particular we see that in
66
F. del Aguila et al. / Fermion contributions to e'
both models any correction to the standard model couplings involving only ordinary fermions with or without new phases is proportional to (re~M) 2 while the complex couplings involving a standard fermion and a vector-like fermion are at least of order (re~M). 3. Strong interaction corrections to the A S = 1 hamiitonian
From eqs. (4) and (8) one can read off the effective AS = 1 hamiltonian in the absence of strong interactions [taking only the dominant terms in an expansion in (re~M) involving only standard quarks]
HaS eli =1
-4x/~ GFbLd(R)( T 3 -- sin 20wQ) L(R)(g~y~d~)L(R)(CT~%,q~)L(R)
+44
L(R)*C~d L(R)(S,~'y - ~q,~)L(R)(qt~'y~dt3)L(R), GFCCas
(9)
where we use standard notation [13] and a summation on L, R = (1 q: 3'5)/2, on colour a,/3 and on light quarks q is understood. The first term in eq. (9) corresponds to the interchange of a Z-boson and q stands for any standard quark whereas the second term results from the interchange of a W-boson and q stands only for the three light charge -2 quarks. Now we must include strong interaction effects. Using the operator product expansion the short-distance effects due to the W- and Z-boson masses can be factorized out o f the matrix elements and can then be calculated in perturbative QCD following the methods discussed in refs. [14] and [15]. Similarly, the effects of heavy quarks can be factorized and calculated [ 14, 15]. The net result is that each operator in eq. (9) is replaced by a linear combination ~iOi where ~ are calculable strong interaction correction factors taking into account short distance effects and Oi are operators involving only the light quarks u, d and s. The corresponding hamiltonian reads ~ aefs[= l : C ,tat
/L(R)L(R) zL(R)L(R) __TL(R)L(R) u:L(R)L(R) wL(R)L(R) f~ wL(R)L(R) q ~i , ( ) i~ q "~C - q ~i " ~i q ,
(10)
where C are the coefficients in eq. (9), with self-explanatory notation, and a summation in all the indices is understood. For each operator in the original hamiltonian L(R)L(R)
-
/~
d~)L(R)(qo%,qt3)L~R), W L(R)L(m= (g,,y~'q~,)L(R)(Fl~Ywd~)L(R), Zq
= (s~y
(11)
we find a set of operators O~ where i runs from 1 to 6, which is the maximum required to close under renormalization. This set of operators contains the initial operator 7L(R)L(R)I L(R)L(R)I • ~q or Wq (l = 1), the same operator but with the colour contractions interchanged --qZL(R)L(R)2 =
(g,~yJ'd/3)L(R)(Ot3 Y~,q,,)L(R)
wL(R)L(R)2
(g,,Y~'qa)L(R)(qI3%,d,,)L(R)
=
(i = 2 ) ,
and the four penguin operators (for initial RL and RR, operators L and R are
F. del Aguila et al. / Fermion contributions to e'
67
interchanged)
pLL1 = (~a,yttda)L(I~/,yp~U¢3 + . . . )L
(i=3) ;
pLL2 = (g~yt, d/3)L(t~ ,yt~u,~+ . . ")L
(i=4) ;
p L a l = ( g,~y " d,~ )L( ~ y~U~ +" " ")R
(i=5);
pLR2
(i=6).
=
( g,~y'~d~ )L( a~ V,~u,~ +" • ")R
This section is devoted to the calculation o f the strong interaction correction factors rli in the leading logarithm approximation. Although the method is standard [14, 15], some results are new. The details are collected in appendix B. We point out that our calculation includes all possible correction factors ~7i required by any extension of the standard model where the interchange of a vector boson dominates. The essential point is that the absence of the G I M mechanism and the presence of RH couplings in the standard quark sector forces us to consider all possible operators at the vector boson scale. Table 1 contains the strong correction factors 7/ for a particular choice of parameters. In particular, we take the renormalization scale/z of the operators to be 0.5 GeV, which represents the limit of validity of the leading logarithm approximation. A blank in the table means that the operator does not appear in the hamiltonian at the hadronic scale. This is because it involves quarks heavier than 0.5 GeV or because it does not mix under renormalization. The ~7 which are not given explicitly in the table can be obtained from the ~ quoted there as follows: ZsL L - Z LL, z L R -- 7 L R - - 7 L R a -~ - ~ u - Also ZqRL "- Z q LR , ZqRR = Z qLL, WqRL ---W q LR and WqRR -__W q LL but with interchange of L and R everywhere in the operators (including the penguin operators). Z LL'LR are zero up to the third decimal place and Wc,t LR= 0. With eq. (10) we can now discuss the e' contributions in any model. Before doing so let us mention that Higgs interchanges among standard quarks, which should in TABLE 1 N o n - z e r o a n d n o n - r e l a t e d s t r o n g i n t e r a c t i o n f a c t o r s * / f o r A 2 = 0.03, be = 0.5, m c = 1.5, m b = 4.5, m t = 40, m w = 81 a n d m z = 93, all in G e V 1
2
Z LL
1.438
-0.774
Z LL Z LL Z LL Z La Z La Z LR W LL W EL
0.665
0.815
1.434
1.372
-0.703
W LL
W LR
0.818
1.390
3
4
0.012 0.034 -0.006 -0.002 0.018 0.007 0.003 -0.033" 0.030 0.004
-0.056 0.009 0.003 -0.037 -0.011 -0.004 -0.044 -0.005
5
6
-0.007
0.037
0.012 -0.002 0.0 0.010 0.002 0.001 0.018 0.008 0.0
-0.108 0.017 0.008 -0.053 -0.021 -0.009 -0.141 -0.098 -0.015
IF. del Aguila et al. / Fermion contributions to e'
68
principle be added to eq. (9), are extra suppressed by Yukawa couplings in models with extra vector-like fermions. Penguin operators resulting from the exchange of these heavy fermions are extra suppressed by powers of some small mass ratio and are next-to-leading order in a~. 4. Vector-like fermion contributions to e v e
Following standard conventions [16] e'_ e
1
ReA__~2(ImA2
x/21elReAo
Re 32
ImA~) Re
(12) "
Given our ignorance of non-perturbative effects involved in the matrix elements o f the operators appearing in eq. (10) we take Re A~ from experiment, as well as [e[, whereas Im AI is calculated from eq. (10). The result depends on the specific model we consider, that is to say on the coefficients C. Uncertainties in the operator matrix elements, which we evaluate using the vacuum saturation method [17], appear to be rather unimportant because, in general, large contributions not involving penguin operators are present. Let us discuss in detail two specific models. They are realistic in the sense that they agree with present phenomenotogy and can be tested in the future [4] and they provide enough information to discuss other possibilities. Also, they are probably the most attractive possibilities. The first model is a particular case of model 1 in sect. 2. We assume that there is only one extra vector-like down quark at the Fermi scale [MU~oo in eq. (4)], and we assume that the KM phase is negligible*. For definiteness we will assume that the three new parameters m l d a r e comparable. With these assumptions e' can be predicted when e is adjusted to reproduce the experimental value. The essential point is that only a definite unknown factor appears in the calculations, ms,oms,a sin (4)do-~bff)/(Ma) 2, and e fixes it. To calculate e' we first read from eq. (9) the effective hamiltonian in the absence of QCD corrections pd
/./AS=I
Im , , e~
rd
GF me m~ sin (~b~- ~bd)
= 4 x/2
X [_(Ta_sin
M d2 2
EL ] , 0wQ)qL,a ZqLL,LR +cos 2 O~Wu
(13)
other terms are absent or are suppressed by extra mass ratios in this model. We then use eq. (10) to write eq. (13) at p. =0.5 GeV, incorporating QCD corrections in the leading logarithm approximation: v
..aS=l
lmme~
GF
ict
fd
md ms d LLI ..~ 1.567ZuLL2+ 0.282Z LL = 4 x / ~ - M- a2 s i n ( q ~ - q~s)(-1.164Zu
* This is e s s e n t i a l l y the m o d e l o f ref. [7]. In that case a n e x t e n s i o n o f the s c a l a r sector o f the s t a n d a r d m o d e l as well as a n a d d i t i o n a l discrete s y m m e t r y was i n c o r p o r a t e d in o r d e r to enforce a real K M m a t r i x naturally.
F. del Aguila et al, / Fermion contributions to e '
+0.282Z~LL +0.125Z~LR1 +0.220Z~LP-,2 --O.062zLRZ--O.110zLR2--O.O62Z
69
TM
- 0.110Z L ~ - 0.005 pLEa -- 0.051 P LL2+ 0.031 P LR~_ 0.243 pLR2),
(14)
where we have used sin 2 0w = 0.23 and cos 2 01 = 0.947. The operators Z with an upper-index 1 correspond to those given in eq. (11), while those with an upper-index 2 stand for the same operator but with the colour contractions interchanged, pLLX, pLL2, pLR~ and pLR2 correspond to the penguin operators 3, 4, 5 and 6 respectively in the previous section. Finally, using the vacuum saturation method [17] with 2 m~/ms(mu+ rod) = 1.7 and (fv,Jf,~)/(1 rn 2K/rn~) 2 = 2.6 to evaluate the corresponding matrix elements we find -
ImAo=(l=0
md,d ms,a a d 2 M sin (~bd- ~bs)f~mK(-- 1.03),
lIra Heias=l r I Ko) = 4 ~
metmm s,a I m A z = ( I = 2 1 I m H ~as=l IKo) = 4 GF / ~ - M- d2 sin ( t had - 6s)f~.mK(0.24), a 2
(15)
Then eq. (12) gives
-
-
= 2.1 × 10_3/ma ms sin (~b~- q~)
10 -5 '
(16)
with lel = 2.3 x 10 -3, Re Ao = 4.7 × 10 -7 GeV, Re A2 = ~o Re Ao and f= = 130 MeV taken from experiment. It must be noted that the dominant contribution comes from I m A2, due essentially to the relative size of the corresponding real parts and to the Q C D corrections. This makes the final result rather stable under possible uncertainties in the matrix element calculations. Eq. (16) is a prediction of the model once • (~bd - Cb~)/(Ma) 2 has been fixed to reproduce e. Following ref. the factor m amm~,d sm [7], with standard conventions [17], e is given by f~ ~irr/4
Im
M12
2 Re Mz2'
(17)
where experiment gives I m M12 = 6.48 x 10 -3 . Re M~2
(18)
In this model I m M~2 is dominated by the interchange of a single Z. Using eq. (4) /
;d
td\ 2
I m M ~ z = - ~ / - ~o v ~2,,2 J ~ c m K~ Jl Jmd ~ ) ms 't sin 2(~bda - ~b~a)r/,
(19a)
where B is defined by
--~f KmKB
(19b)
F. del Aguila et al. / Fermion contributions to e'
70
and rl is a strong correction factor [18] for the AS = 2 effective hamiltonian
(as(me)~6/27
(ors(Mz) ~ 6/21( as ( mt ) ~6/23 (O~s( mb ) ~6/25 7/= \ as(mt) ] \as(mb)] \a~(m¢)] \ as(tz) ]
"
(19c)
Taking Re M12=½AM from experiment, AM being the K ° mass splitting, B = 1" and ~ = 0.665, where the same values for A, ~, rnc. • • as in the table have been used to evaluate eq. (19c), we find
['/tdtd\2
XmM12--2"7x 10 --3 L t M Re M 1 2
]/
) sin 2/6
-C)
lO -1°
(20)
) < 2 x 10 -5 from Relating eqs. (20) and (18) and remembering that ma,a ms/~1vl ,d/( .,a,2 0 + /z -- [11], we obtain KL~IX s . 6.4 x 10-6<~ ~ ds m
which implies
[see
d
d
1.9x10 - 5 ,
(21)
eq. (16)] ]e'/e[ = (1.3 - 3 . 9 ) x 10 .3 .
(22)
The sign o f e ' / e is not determined but the size is a prediction of the model. The range of variation comes from the different functional dependence o f the different parameters in e and e'. We will discuss our conclusions later. Let us now consider another model, the second model of sect. 2. As before we assume that the K M phase is negligible. We also assume that the new parameters m' are all of the same order. In this case eq. (9) reads
I./zlS=I
In,,~n
=
4GF
,f~
F [ m~' d m ' d
sin
~ f)$ L,RTRL, RR
(q~dd--6sd)(T3--sin 2 vW~lq
~q
rd pu rnd m~ sin (~b,~- cb~)(KM)*~W~R M2 m std m utu
1
M------T - sin ( 6 ~ - 6~) cos 01W~ L •
(23)
At the scale tz = 0.5 GeV and in leading logarithm approximation it becomes (using the table) l_l a s = 1
Im--~
GF Fm'am'a L ~
=4~
d ~bs)(-0.221Z~ d RRI + 0.119Z~RR2 sin (~bd--
+ 0.051Z RR + 0.051Z RR + 0.282Z~RL1+ 0.497ZuRL2 -
0.345Z RL1- 0.607Z RLz
0.007P RR2-
0 . 0 0 2 P RL1 - 0 . 0 0 1 p R L 2 )
-- 0 . 3 4 5 Z RL1 - 0 . 6 0 7 / R L 2 -- 0 . 0 0 4 P
TM
+
mstd mutU d RL1 wRL2 1 M~sin(~b~-~b~)(0.796W~ +1.353 ~ ) . • In this m o d e l B c a n n o t b e s m a l l e r t h a n 0.6 if e has no other contributions.
(24)
71
F. del Aguila et al. / Fermion contributions to e '
Evaluating the matrix elements with the vacuum saturation method gives ImA0=(l=01Im
as=l n~
o G F r| m d,a ra~,d sin (~b~- ~b~)(2.35) IK)=a~-~L M2
msIdmutu M : sin (6u~ - 6 ~ ) ( - 5 . 0 4 ) 1 f~m~,
lmA::(I=2llmneT:llK°):
[-mrdmrd
L M :'s sin(6~-~b~)(-0.67)
mtdmtu
]
M ~ ' ~TM sin ( 6 ~ - 6~)(-0.87) f ~ m ~ . Then, eq. (12) gives [with
[el,
(25)
Re A o , . . . as for eq. (16)]
--=
sm (~b~- ~b
E
x 10-3[~ M
sin (4~:- 4~)
10-'+4.3
10-s '
(26)
to be compared with eq. (16) in the former model. Before discussing the size of the two contributions we must note that the parameters in eq. (26), in particular those entering the second term, are poorly known. Thus e ' / e can have a wide range of values due to possible cancellations between the terms. In fact only the first term in eq. (26) has a definite size. It comes from the interchange of a Z-boson, which gives also the main contribution to e, just as in the previous model. Then eq. (21) applies and up to the sign the first term ranges from 3.5 x 10-3 to 10 -2, but the second term is in principle comparable and if it cancels the first term the model can fit any experimental value. We can now discuss the contributions of vector-like fermions in general. In any model there are always new mass parameters: small rn' masses, large ones M and new phases ~b. The new CP-violating effects always have a factor m'/M to some power and a sin ~b. There are two extreme possibilities concerning m' - that they are of the same order or that they are in the same hierarchy as for the known fermion masses. We will always make the first assumption. The second has consequences for the bottom system and will be discussed elsewhere. One must distinguish different models to make non-trivial statements. We see three different cases- when only a new vector-like down quark transforming as a singlet under SU(2)L exists at the Fermi scale, when the only vector-like fermion is an up quark or when both quarks transforming as a doublet of SU(2)I. exist. The first two cases are subcases of model 1 in sect. 2 and the third case is model 2 in that same section. Within these three cases one must distinguish if the standard contribution to e due to the KM phase is sizeable or not. If it is large, then e' is never a prediction because standard and
72
F.
del Aguila et aL / Fermion contributions to e'
new contributions are comparable and the relative signs are undertermined. If the KM phase is negligible then the three cases are different. In the first case [see eq. (22)] l e ' / e I ~ 2 x 10 -3 but the sign is undetermined, with e reproducing experiment. This model can then be tested in the near future. The second case we have not analyzed in detail. The dominant contribution to e' is due to the interchange of a W because the neutral couplings of the down quarks are diagonal. Also, e comes from the box diagram for the same reason. As the box contribution is suppressed relative to the tree level one typically by at least one order of magnitude, the CP-violating factor ( m ' / M ) 2 sin $ must be larger than in the first model to reproduce e. This implies a larger contribution to e', a priori in disagreement with experiment. The third case, discussed in detail above [see eq. (26)], is an example that when more than one phase is available no prediction is possible. The saturation of e implies a typical contribution to e' of order 7 × 10 -3. However, other contributions to e' are o f the same order of magnitude and large cancellations can be present.
5. Conclusions The presence of vector-like fermions at the Fermi scale is experimentally possible and theoretically natural [10, 11]. In this paper we have discussed the consequences of their existence for the description of CP violation in the kaon system. We have studied in detail several models, including a complete calculation o f the strong interaction correction factors (in leading logarithm approximation) entering the AS = 1 hamiltonian. The result of our analysis is that vector-like CP non-conserving contributions are milliweak. That is to say that if the observed CP violation, e, is due to the existence of these fermions then I '/el is typically of order 10-3-10 -2, independently of the size of the penguin diagram contribution. If there is only a new vector-like down quark then 10-3, with the possibility o f accommodating any sign. If there is a vector-like doublet of quarks, le'/e[ can have a large range of values because large cancellations among contributions of similar size but involving different phases are possible. If the standard CP-violating contributions due to the presence of a phase in the KM matrix are not suppressed, le'/el is not a prediction in any model. Given the milliweak character of both types of contributions, they could add but could also cancel. In order to further clarify our results let us write down e and e ' / e as a sum of the standard contribution [19] plus the new vector-like one. Following our previous conventions [16] (see (17), (18) and (20)) Im M~2= 6.48 x 10-3(exp.)-- 0.9s2s3s6 -1.8 + 1.35s~ 3 + 0 . 9 Re M12 mc /
~d
¢d\ 2
ms ] sin 2 ( ~ +2.7 x 107/m~ \---M-~-
In
(standard)
"1
J
~b~)(vector-like) B
(27)
F. del Aguila et al. / Fermion contributions to ~'
73
and e'
m'aa m'~a ~ 6s2s3s6P(standard) + 210 ~ sin (~b~- 4~a)(vector-like)
(28a)
or ~'
m rdd m srd
--e ~ 6s2s3s6P(standard) - 550 ~ td
•
sm (~b~- ~a)
ru
ms mu . + 430 ~ sm (~b~- ~b~)(vector-like),
(28b)
where (28a) corresponds to a new vector-like down-quark (see (16)) and (28b) to new vector-like up- and down-quarks in a SU(2)L doublet (see (26)) at the Fermi scale. The contribution to e is the same in both cases (see (27)), but with M d replaced by M in case (28b). As usual s2, s3, s6 stand for sin 02, sin 03, sin 6, with 02, 03, 6 the standard K M angles. The matrix element B is defined in (19b), whereas the penguin P = 4(1 = 0[(~T~d/3)L(at~O/~,ua + . . . ) R [ K ° ) . The vector-like contributions in (28) are the result of many terms (see (14) and (24)) and they are rather insensitive to particular matrix elements. To obtain a numerical insight let us take B = I , P = 0 . 7 and mr~me=40~1.5, as before, and $2=0.05, $3=0.03. We also assume m,a d m ,sa // M a2 = 2 × 1 0 -5, saturating the bound from K ° ~ / x + / z - , and that the same value applies to msmmu'U'/M" .2, although this mass ratio can be larger. Then, defining 6v = ~ba a - ~ , 6 v' --~b~-~b~, we obtain for (27) and (28) 6.48 x 10 3 _ 8.8 × 10-3s6(standard) + 10.8 x 10-3S28v(vector-like), E ~
- - ~ 6.3 x 10-3sS(standard) + 4.2 x 10-3SSv(vector-like),
(29) (30a)
E or Et
- - ~ 6.3 x 10-3s6(standard) - 11 x 10-3S6v + 8.6 8
× 10-3sS~/(vector-like).
(30b)
I f the forthcoming C E R N experiment measures e ' / e = 5 x 10 -3, (29) and (30) will be compatible with the standard model predictions, s 8 - 0.74, s6~, ~ 0. This would not exclude vector-like fermions which could exist but with small new phases (or mass ratios m ' 2 / M 2 ) . I f on the other hand e ' / e happens to be < 2 x 10 -3, then we can conclude that either the value used for the penguin P was too high and P < 0.3 should be used in which case s6 ~0.74, s6~,~0 is still allowed or that P =0.7 was correct in which case vector-like fermions would explain the smallness of e'/e. In fact for any value of e ' / e there are 6, 6~ ) values fulfilling (29) and (30).
F. del Aguila et al. / Fermion contributions to e'
74
In conclusion the first observable effect induced by new vector-like fermions at the Fermi scale should be CP violation in the kaon system. We have shown that their contribution to e ' / e is milliweak and that they may be an alternative to the standard model description of CP violation. In order to disentangle this new source of CP violation from the standard model contribution more experimental and theoretical work is required. A better understanding of B-meson decays and a more precise determination of the top mass should be accompanied by a more accurate measurement of e'/e. On the theoretical side a drastic reduction of the present uncertainties in the estimates of the matrix elements involved in kaon decays (particularly the penguin) is needed. Perhaps lattice calculations could provide the required accuracy for those matrix elements. We thank A. De Rfijula and J. Ellis for useful comments.
Appendix A As explained in the text the diagonalization of the different mass matrices appearing in the models considered reduces to the diagonalization of the mass matrix I I I
~ =
m/~s~ ',, ms'e<
j, k = 1, 2, 3,
I "1" . . . . . . . . . I
. . . . . . . . . .
0
~ I
(A.1)
M
where m and m ' are small entries and M a large one. This matrix is diagonalized as usual by two unitary transformations UL,~U ~ =
D.
(A.2)
To second order in ( m ' / M ) 2, the eigenvalues are [ mn \ Aj = r n j ~ l - ~ ) , =M
1 + ~
and the diagonalizing matrices are
ULSk = e i%-~k) 6jk 4 mJnk g 2
ms (-1)~km~-m
ULj4=-U* .=e %{-mJ'~ L40 \ MJ' Ugjk = ei(%-e'k)[ 8Sk+ (1 - 6Sk) L
'
UL44 = 1 2k re'k2 2M 2' m~mtk
I'?lSl'Hk
] "~
2 2 M 2 ( - 1 ) 8,~mk--msJ
F. del Aguila et aL / Fermion contributions to e'
URj4 = -- U*4j = e ~% -
,
UR44 = 1.
75
(A.4)
Appendix B In this appendix we outline the calculation of the strong interaction correction factors ~7 for the AS = 1 hamiltonian, involving four-fermion operators which result from the interchange of a vector boson. The method is standard [14, 15] but new cases are considered. We work in the leading logarithm approximation. Each particle is considered as massless for energy scales above its mass and as infinitely heavy for energy scales below it. Then the QCD correction factors ~ can be written as the product of four steps, corresponding to the integrating out of the vector boson and the three heavy quarks t, b and c. The evolution between two mass scales is described by a factor Vrs for the coeffÉcient of the operator r expanded in the basis s. The matrices Vr~ are functions of ratios of strong interaction coupling constants y and anomalous dimension matrices y:
Vrs('Y, y) = Xslya'S~r I ,
(B.1)
where X is (up to a factor of g2/8'rr2) the matrix diagonalizing the corresponding anomalous dimension matrix 3': -1 T Xtm ym, X,r = 81rzr,
(B.2)
zr being the eigenvalues, and z,
(B.3)
with / 3 o = - 1 1 + 2 n being the first-order coefficient of the t3 function and n the number o f massless quarks. Mso, projection operators must sometimes be inserted to take into account the appearance of linear relations among operators. Indices vary from 1 to 6 as a maximum. Given that QCD preserves flavour and chirality, an operator mixes at most under renormalization with the same operator but with the colour index contractions interchanged and with the usual four penguin operators in the leading logarithm approximation. Let us now discuss all the particular cases. The relations given in sect. 3 relating different sets of '1 factors are due to the fact that QCD does not distinguish chirality nor flavour. ZuLL factors are obtained from [ (6)a~(M~)~ ( ( 5 ) a ~ ( m 2 t ) ' ~ / {~(4)a~(m~)~ r/zkCr= V,t~TLL, a~(m2t) ] Vtk_ TeL, a,(rn~)] "kj\ rLL, as(m~)] "
(~(3)%(mc2)~ "YLL,
(B.4)
F del Aguila et al. / Fermion contributions to e'
76
where -1 3
3 -1
-~1
~I
-~1
1
Ii 9
H 9
2 --9
2 3
3-1n
-l+In
-in
in
1
-3
-in
In
-~n
-8+~n
(B.5)
with l, k, j = 1, 2 , . . . , 6 corresponding to Z LL~, Z LL2, pLL,, pLL2, pLR, and pLR2, respectively, and 1 -1 ] 1
1
1
(BT)5× 6 =
1
,
(B.6)
1 1
where j, 7 = 1, 2. . . . ,5 correspond to Z LL1, Z LL2, pLL1, pLR~ and pL~, respectively, and (L3L) [ =
[
--1 _~,
3
u3 229 -~2
-1
1
~2
1 -~
~
(B.7)
-3 -7
Z LL factors are given by ,, / (6) as(M2)~v / (5> as(m{)]V ~ { . ( 4 ) a~(mbz)'~ nz~L~= V,,~'rLL, adm 2) ] ",k~ 3'LL, adm2)] k~i rLL,---;--~_,l
.
(.
a~(m:)~
where l, k, j are as before but changing Z~ r~'2 to Z~ Ll'2 (this change to be done in each case will not be state explicitly in what follows),
~v
(n)4×6
'
=
1
|
1 ,
1
(B.9)
with ], /' = 1 , . . . , 4 corresponding to the four penguin operators pLL1,2 pLRL2and 11 --9-
11 3
2 --9
2 3
'
t -~n
~n
-~n
]n
1
-3
-~n
-8+~n
(B.10)
F. del Aguila et al. / Fermioncontributionsto e'
77
Analogously
,,.(~(4) as(rn~)~V
X--kj~I'LL, OIS(m2)]
[ ~ ( 3 ) a~(m¢:)'~ 3f~ /'LL, ~ ) •
(B.11)
For the down-quarks the strong correction factors can be obtained using ~TZ~LV=
a:(m2t)~ a~(m~)~ Vrk,i y~) ~(m~)J Vk'j'~ -- { ")ILL t(4) , ~(m3c)J
• r { ,<6> ° ~ s ( M 2 ) ~ F I / ' ~ ' Y L L , a~(mZt) j O~
2
× V.j , \{ .LL, ,,(3) ~(m¢___~)~ .~( ~)) ,
(B.12)
where 1', k', j' and i' run from 1 to 5 and stand for Z~ L, pLL1,2 peru,2 and 2
-} 11 9
3-~n
g1
-~1
1
1~ 3
2 --9
2
-l+In
-~n
½n
1
-3
(B.13) In this case the operator Z~ L but with the colour indices interchanged is the same operator by a Fierz transformation, so there are only five operators mixing under renormalization. The Z~ L factors follow from
~TzkL~=
v,,[..,, ~ ,~(M~)] / (. o,~(m~t)~ ~. BkC, \ .LL, ,~(m~) ) V,k~ WL, Vr/~(4) a~(rn2)~V {C(3) %(m2)~ x k S t r L L%, ~t lm\ d / J * k r L L , ~ ) .
(B.14)
Z~ R factors come from //
(6,
as(M~)'~ V~,( y(~, a,(mt:)'~
rlzk% = V, tk YLR, a~(m2) /
a~(m~)/
( as(m~)~ ,/ {. ,3, %(m:)~ x Vk,\ y[~, a~(m~)] ,,,\rrR, as(p2)].
(B.15)
where 1
-3 -8
1 --9 11
1 3 lj
1 --9 2
! 3 2
3-~n
-l+½n
-~n
In
1
-3
-~,,
-8+~n
),(n) _ LR--
-9',,
I,,
(B.16)
78
F. del Aguila et aL / Fermion contributions to e'
with l, k, j, i = 1, 2, . . . , 6 corresponding to --~zLR1,ZuLR2, pLL1, pLL2, pLR1, pLl~, respectively. Factors of Z~ R are given by
--[ (6)~(M~)] Vllk('YLR, a sOts(m2t)~Vkj(3/(L4)R, ( m 3 ) ~
nZ~R~ Vll~ ~/LR' Ods(tnt2) ]
( 5 )
a~(m~)]
=
a,(m~)]
" . (,'(3) as(m~)'~ x B,j• Vj,^_YLL, ot~(~?))'
(B.17)
where YLR "(") = ~"~, whereas ~,f {^ (6) °~s(g2)~
m2
T~zLRi~= ll~YLR, oe~(m2) ] Dlf V ~ \| 'YLL,~c~.~t/ x V~j(~(4)
b)/
as(m~)~V [^(3)a~(rn~)~
(B.18)
.LL. ,,s(r,OJ ~;V ~L' ~ ) "
Finally // (6)°~s(M2)~
[ (5) °~(mt2)'~
nz~.~ = v,,t ~L., ,~s(~t~) ] v,~k VLR. '~('~b)] ^
(,,(4, oq(n~)'~,, {*(3)~(m~)~
XBkkV,~y\ YLL, as~rnc)/" 2\/v 3~£|~LL,\. O~s(/,/2) /
•
(B.19)
The factors for WuLL, W~LL and W EE are equal to eqs. (B.4), (B.8) and (B.11), 2 by respectively but replacing 1 by 2 in the initial matrix elements and a~( M .z) a~(M2) in their arguments. The factors for W LR are given by
.wv,= V,~(~,~.,"'a,(m~) (M~v)~ ( ,,,(,,,2),~ / Vr~x~LR, oq(m~,)J (K2o) where (B.21) and I,/~, y and ~ = 1, 2 correspond to W LRl and W,LR2, respectively.
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R.H. Bernstein et al., Phys. Rev. Lett. 54 (1985) 1631 [4] CERN experiment NA31, CERN/SPSC/P174; FNAL experiment E731 [5] T.D. Lee, Phys. Rev. D8 (1973) 1226; S. Weinberg, Phys. Rev. Lett. 37 (1976) 657 [6] A.A. Anselm, J.L. Chkareuli, N.G. Vraltsev and T.A. Zhukovskaya, Phys. Lett. 156B (1985) 102; X.G. He and S. Pakvasa, Phys. Lett. 156B (1985) 236 [7] F. del Aguila and J. Cortes, Phys. Lett. 156B (1985) 243 [8] M. Dugan, B. Grinstein and L. Hall, Nucl. Phys. B255 (1985) 413; J.M. Grrard, W. Grimus, A. Masiero, D.V. Nanopoulos and A. Raychaydhuri, Nucl. Phys. B253 (1985) 93 and references therein [9] M.G. Gavela and H. Georgi, Phys. Lett. l19B (1982) 141; R. Decker, J.M. Grrard and G. Zoupanos, Phys. Lett. 137B (1984) 83 and references therein; R. Mohapatra, Maryland preprint 85-124 (1985) and references therein [10] P. Ramond, Proc. 4th Kyoto Summer Institute on Grand unified theories and related topics, Kyoto, Japan (1981), eds. M. Konuma and T. Maskawa (World Sicence, Singapore, 1981) [11] F. del Aguila and M.J. Bowick, Nucl. Phys. B224 (1983) 107 [12] S.L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285 [13] G.G. Wohl et al. (Particle Data group) Rev. Mod. Phys. 56 (1984) 51 [14] A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, Nucl. Phys. B120 (1977) 316; F.J. Gilman and M.B. Wise, Phys. Rev. D20 (1979) 2392 [15] R.D.C. Miller and B.H.J. McKellar, Phys. Reports 106 (1984) 169 [16] L.L. Chau, Phys. Reports 95 (1983) 1 [17] A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, Sov. JETP 45 (1977) 670 [18] F.J. Gilman and M.B. Wise, Phys. Rev. D27 (1983) 1128 [ 19] A.J. Buras, MPI-PAE/PTh 46/84 (1984), invited talk at Workshop on the future of medium physics in Europe (Freiburg, April, 1984) and references therein