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Right-handed charged currents induced by a strongly interacting Higgs sector
Nuclear Physics B291 (1987) 629-652 North-Holland, Amsterdam
RIGHT-HANDED CHARGED CURRENTS INDUCED BY A STRONGLY INTERACTING HIGGS SECTOR* H. NEUFELD
lnstitut ffir Theoretische Physik, Universitiit Wien, Vienna, Austria
Received 6 February 1987
A strongly interacting Higgs sector could give rise to right-handed charged currents in the quark sector. An analysis of nonleptonic weak K decays strongly restricts the possible size of these right-handed couplings. The experimental value of e can be reproduced even for vanishing = F(b ~ u e u ) / F ( b ~ cev). e'/e may have arbitrarily small values of either sign. The neutron electric dipole moment could be near to its present experimental bound.
1. Introduction
The Higgs sector comprises that part of the standard model [1] of electroweak interactions which has neither been tested experimentally nor really understood theoretically. Therefore, it will be an important task to look for the actual existence of scalar particles with the properties of Higgs bosons. If the Higgs mass Mla is larger than, say, 1 TeV interesting new effects may be expected. In this case the Higgs self-couplings become large such that lowest order calculations of amplitudes involving Higgs particles cease to be reliable. As a consequence strong interaction effects should also show up in the interactions of longitudinally polarized vector bosons [2, 3]. This can be inferred from the so-called equivalence theorem [3,4], which roughly says that matrix elements describing high-energy processes involving longitudinal vector bosons can be calculated (within the R~ gauge) by replacing the vector particles by their corresponding would-be-Goldstone bosons. Phenomenological and theoretical aspects of the standard model in the limit of a large M H have been studied by many authors [5]. Recently it has been conjectured by Casalbuoni, De Curtis, Dominici and Gatto [6] that such a strongly interacting Higgs sector could give rise to new spin-one * Supported by "Jubil~iumsfonds der Osterreichischen Nationalbank", project number 2765. 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
630
H. NeufeM / Right-handed charged currents
bosonic states. The argumentation goes as follows. When one formally takes the limit M H ~ oe [7] in the generating functional for the Higgs sector of the standard model, one obtains the SU(2)L × SU(2)~/SU(2)v nonlinear sigma model [8]. Since the discovery of hidden local symmetries in supergravity theories [9] it is commonly known that any nonlinear sigma model based on a manifold G / H is equivalent to a theory with Ggloba1× Hloc,1 symmetry [10]. At the classical level, the gauge bosons associated with the local group are only dummy fields. In some specific models in two space-time dimensions it has been shown that a kinetic energy term for these gauge fields is generated by quantum corrections [11]. It can be speculated that such an effect might also occur in four dimensions. In the QCD nonlinear sigma model the vector field of the local symmetry has been identified with the p meson, and a series of phenomenological results has been developed [12]. The authors of ref. [6] have applied the idea of dynamically generated gauge bosons to the standard model with a strongly interacting Higgs sector. Thus, they predict an SU(2)v triplet of new physical vector fields. Through mixings with y, W -+, Z ° and unconventional couplings to fermions, these particles significantly modify some phenomenological predictions. It is one of the most interesting features of this model, that right-handed charged currents may be induced in the quark sector. The phenomenological consequences of such currents have been intensively investigated within the framework of left-right (L-R) symmetric gauge theories [13,14]. In particular, the contributions to the parameters of the K ° K ° system [15-18] and to the neutron electric dipole moment [19, 20] could be quite different compared to the standard model. The purpose of the present paper is to show how fight-handed charged currents are obtained from a strongly interacting Higgs sector and to study the corresponding low-energy phenomenology. In sect. 2 the hidden local symmetry of the nonlinear sigma model is discussed. The model of ref. [6] is briefly reviewed in sect. 3. The fermionic sector is presented in sect. 4. A lower bound for the SU(2)v coupling constant g" is derived from the experimental data of M w and M z. The charged current lagrangian for quarks is given, together with a comparison to the situation in L-R symmetric theories. In sect. 5, I collect all the necessary ingredients for an investigation of the K L K s mass difference and the CP-violating parameters 5, 5'. The dominant Feynman diagrams for the effective ]AS1 = 1, 2 hamiltonians are calculated. An analysis of nonleptonic ]ASI = 1 weak decays strongly restricts the size of the right-handed couplings. All relevant hadronic matrix elements are calculated using the vacuum insertion technique and I set up the necessary equations to determine the CP-violating parameters e, e'. The various contributions to the neutron electric dipole moment d n are given in sect. 6. After recalling the experimental situation and the standard predictions for e, e', d n, I present my numerical estimates in sect. 7. It is shown that these observables could be completely determined by non-standard contributions. My conclusions are summarized in sect. 8.
H. Neufeld / Right-handedcharged currents
631
2. The hidden loeal symmetry of the nonlinear sigma model
The Higgs sector of the standard SU(2)L × U(1)v model consists of a single doublet field ¢ = (q'+ ~o )"
(2.1)
The gauge group is broken down to U(1)EM by the vacuum expectation value (~°)vac = v/vr2.
(2.2)
Written in terms of the matrix ¢0, X-
¢+) d:° ,
' (~-
(2.3)
the standard Higgs lagrangian takes the form m 2
£~°i_n~= ½Tr( D~,X)* (D~'X) - ~8~2(Tr X t X - v2) 2
(2.4)
with the Higgs boson mass M n. In the limit M n --* oo the potential in (2.4) gives the constraint X t X = ½v2. 1 and one obtains the nonlinear sigma model. Replacing X by the dimensionless field
U = x/2X/v ~ SU(2),
(2.5)
and omitting the gauge couplings for the moment, we get the following Lagrange density: •~'O1 = l o 2 T r ( oq~U * o'q~U ) .
(2.6)
.L#1is invariant under global SU(2)L and SU(2)R transformations
U(x) ---4gLU(X)g~,
(2.7)
where gL and gR are elements of SU(2)L and SU(2)R, respectively. The nonlinear sigma model possesses a hidden local symmetry. To see this we introduce new SU(2)-matrix valued variables ~L(X) and ~R(X) transforming as ~L(X) --~gL~L(X)h(x),
~R(X) --)gR~R(X)h(x)
(2.8)
under [SU(2)L x SU(2)R]globaI X [SU(2)V]local, where h(x) is an element of the local SU(2)v. The variable U(x) can be realized as U(x) = ~L(X)~(X). Furthermore we
632
H. Neufeld / Right-handed charged currents
introduce the following hermitean and traceless combinations of the fields ~L(X) and ~R(x) 1
~o~-+)(x) = -~(li[(X) O,IiL(X)+_~(X)O,I~R(X)).
(2.9)
Both %(+)(x) and %(-)(x) are SU(2)L X SU(2)~ singlets and transform under the local SU(2)v as ~0~(+)(x) ~ h*(x)~o~+)(x)h(x)
- iht(x) O~,h(x),
¢o(- )( x ) ~ h*( x )co(~-)( x )h ( x ) .
(2.10)
One recovers the nonlinear sigma model by considering the lagrangian ..~O1 =
v2Tr(%(-)(x)~0(-)"(x))= ~v2Tr(O,Ut(x) O"U(x)).
(2.11)
~ 1 ~ (0) We can introduce a gauge boson field ~(°)(x) = 5%VS. (x) (with the Pauli matrices %) for the local group with transformation property
P9)(x) --, h*(x)Y'¢°)(x)h(x) - ih*(x) O,h(x).
(2.12)
N o w it is possible to write down a second invariant £Pa = v2Tr[(%(+)(x) - ~7/°)(x))(co'+)"(x)
- l~'°)"(x))].
(2.13)
£~°1 and ~c~° 2 can also be written in terms of covariant derivatives D,~L,R = 0,(L,k -i ~ L, RI'7/0) :
s°1,2 = - ~v=Tr(~*L D.~L -V ~*~O~)~.
(2.14)
Any linear combination of (2.11) and (2.13)
.~=~, + ~_~
(2.15)
is equivalent to the original lagrangian (2.6). The reason is that in absence of a kinetic term for V9 ), the gauge field is only an auxiliary field and can be eliminated using its equation of motion 17J°l(x) = %(+)(x). Following the suggestion of ref. [6] I will assume that a kinetic energy term is generated by quantum effects. 17¢°~ will thus be regarded as an additional physical vector boson field.
633
H. Neufeld / Right-handed charged currents
3. New vector bosons from a strongly interacting Higgs sector Let us now complete the covariant derivatives by introducing the gauge couplings of the standard SU(2)L × U(1)r gauge fields I,V~(°) and /],, D ~ L = a ~ L + iI,V~(°)~L-- i~LI,7(0), D ~ R = O~ R "-F i / ~ R -- i~Rl?~(°) ,
(3.1)
where the coupling constants g, g', g" of SU(2)L X U ( 1 ) y x SU(2)v have been included by
~(0) = g " I"
~ - ~ V~(°) .
( 3.2 )
Our Higgs lagrangian in the limit of a large Higgs mass is now given by
£~'=--¼vZ[Tr(~tLD~c--~D,~R)Z+KTr(~tLD,~L
+ 8*RD~R)2].
(3.3)
It is not hard to see that one can perform a local S U ( 2 ) c × SU(2)v gauge transformation such that ~c = ~R = ~- In this gauge (3.3) reduces to the mass term for the vector bosons, £P= ¼v2 [Tr(l~(°) - / ~ ) 2 + K Tr(l~(°) +/~ - 21~(°))2] ,
(3.4)
whereas all would-be Goldstone bosons have disappeared. After inserting (3.2) into (3.4) the mass eigenstates and the corresponding mass eigenvalues can be determined. The weak eigenstates of the charged sector (Wl(°~ and V~(°~)are related to the physical states (W1,2 and V1,2) by
cos sin ) V/
sin q~
cos q)
(i = 1,2).
(3.5)
V/(°) J '
In the limit g" >> g (which will turn out to be the relevant one) the mixing angle q9
H. Neufeld / Right-handedchargedcurrents
634
and the masses of W1,2 and V1,2 are given by g g,, ,
-
12 2( g2)
zv •g
1+ ~
.
(3.6)
In the neutral sector we have the photon A, the Z ° boson and an additional vector particle V°: A Zo = Vo
0
cos +
sin qJ
cos ~ sin ~
sin ~ sin ~p - cos ~ sin ~k
- sin ~ cos cos ~ cos
g/G
-g'/G
g'/G 0
g/G 0
o° : ) 1
v#,]
(3.7) with G = g ~ + g,2. Again in the limit of a large g" we have
g2 _ g,2 g"G
M2o= ¼v2(g2+g,2) 1 -
2gg' g"G ' ( g 2 _ g,2)2 (gZ+g,2)g,,2
g2 + g,2 ) M 2 o = 1av 2xg ,2 1 + _ g,,2 _ •
(3.8)
The exact versions of (3.6) and (3.8) can be found in ref. [6]. In the limit g" --->oo the masses of V -+,V ° become infinite and one recovers the standard mass formulae f o r W -+, Z °.
4. The fermionic sector
The physical vector bosons W-+, V-+, A, Z °, V ° couple to fermions through interaction terms which can be directly read off from the gauge couplings of
H. Neufeld / Right-handed charged currents
635
SU(2)L X U(1) y. Let us start with the charged current lagrangian
£#cc
=
i./T,,.~ w-(0)+ + h.c.,
(4.1)
-- 2 V 2 6JCC"/.L
where J~c = Ee= e, ,, ,~eY"( 1 - Ys) d+ J ~ C , h a d r " Using (3.5) yields .Sacc= - ½~/~-~gj~c(COS qoW7 + sin q0V~+) + h.c.,
(4.2)
leading to a low-energy effective lagrangian
l( ,os,
£P~g = - 8
M---~w+
M~
JccJcc~"
(4.3)
From (4.3) we obtain a formula for the Fermi constant GF, GF
g2 = w8M---~(c°s2cp + flvsin2cp) '
(4.4)
with ] ~ v = M w2/ M
v2 .
(4.5)
The electromagnetic and neutral current Lagrangian can be determined in an analogous way. For the electromagnetic coupling constant e one obtains the equation [6] gg' e = --~-cos ~b. (4.6) Using (3.6), (3.8), (4.4) and (4.6) the following relation can be derived [6]:
(1
t2
Together with the experimental values and errors for M w and M z [21] this equation leads to a bound for the coupling constant g " , lfl (@) 2>-(1+g
[ 18.9 7)'[11.5
(UA1 data) t (UA2 data) J '
(4.8)
which justifies a posteriori the preference for large values of g " in the previous formulae. Besides (4.2) and (4.5) one should expect that small direct couplings of V ÷ and V ° to the fermions [61 also appear in the lagrangian. In the following I will
H. Neufeld / Right-handedchargedcurrents
636
restrict myself to the quark sector. The weak eigenstates of the quark fields will be denoted by qR= (P)R.
qL= (P)L ,
(4.9)
With the new field XL = ~tLqL, which is a singlet under SU(2)L and a doublet under SU(2) v, we build up the invariant term g,, ,¢ ) XLiY u 0t, + i-~-- -~ V~(°) + ig'~B~ XL"
(4.10)
)
Quite analogously we define XR = ~tRqR, which again is a singlet with respect to S U ( 2 ) L and a doublet under SU(2)v- The corresponding invariant reads g" 'I" ) ,1 ~ V~(°) + ,g aB~
2RiV ~ O. + i T
(4.11)
.
We can now add (4.10) and (4.11) to the standard SU(2)L X U(1)y terms. In the gauge (L = ~R = ~ and with a proper normalization of the kinetic energy terms we obtain &°ki.
1 nt- ~L
qLiY ~ O~+" tg2
+ . ,1
qL
+"[LqLi~(O~+ig@2V~(O) q-igtl~)qL] +m 1
1
-J-YR
0
-½
B~qR
( gtt'rv(O)+lg__• tlgB~)
+yR~/R/y ~ 3~,+ i--~-- 2
qR] '
(4.12)
where YL,R are new constants of the effective lagrangian measuring the strength of
the additional contributions. The most interesting feature of (4.12) is that it contains right-handed charged currents. Such terms are absent in the usual SU(2)L X U(1)F standard model. Let us now write down the charged current lagrangian in terms of physical fields. It is of the form
-Wcc =
g(a ,cr KCdC + e Rr K a.)ve; +~g(C~LT~'KLdL +d~RY~'KRdR)Vf + h.c.,
(4.13)
H. Neufeld / Right-handedchargedcurrents
637
where 1 a
~
[
tt 1
B
1
b=
")tR g"
l+YR
2
g
1 C - -
-
-
l+y L d=
g
"
1 + ~/L /COSCp- ~ Y L T S l n ¢ p
sin cp,
YL g"
sin + 5 - T c ° s
)
,
1
YR g" - - - - c o s ~. l+y R 2 g
(4.14)
The weak eigenfield PL, R and t/L, R have been related to the mass eigenfields UL, R and dE, R by unitary transformations PL,R = UP, RUL,R,
nL,R = UI~,RdL,R,
(4.15)
leading to the left- (right-) handed mixing matrices KL(R) With ep-follows:
-g/g"
= ITP t
lTn
(4.16)
~ L(R)~"L(R)"
and for small ]tL,R the relations in (4.14) can be simplified as a--1
"YL 2'
g ~L g" c = --+ ----, g" 2 g
b = YR 2' d=
7R g" 2 g
(4.17)
The analogous couplings for the leptonic sector have been discussed in [6]. There, one also has a YL,lept term, but fight-handed charged currents are absent if there are no right-handed neutrinos. Quite reasonable fits of the low-energy phenomenology and of the masses of W -+ and Z ° could be obtained for )tL, lept = 0.1 and rather low values of M v. At this point I would like to compare our model with L-R symmetric gauge theories based on SU(2)L X SU(2)R X U(1) [13]. In the minimal version [14] of such models one has a single (½, ½, 0) scalar multiplet • coupling to quarks. The charged current lagrangian is of the same form as (4.13) with a = c o s ~LR,
b = - e-iXsin ~LR,
c = eiXsin ~LR,
d = COS ~LR,
(4.18)
H. Neufeld / Right-handed charged currents
638
where ~i_l~, )~ are L-R mixing parameters with 21owl ~cR-
ivl2 + 1
ow*
iwl=BV
,
e ix = _ _ _
Ivwl '
0
(~)v~¢ = ~ - ( 0 0 w ) .
(4.19)
It can be demonstrated in specific examples [22] that a natural explanation of the smallness of weak mixing angles requires Iw/vl = O(mb/mt). The natural magnitude is therefore [23] ~LFt--
2m b
flV -- 2" 10 -3 ( M v / T e V ) -2
(4.20)
mt
to be compared with the phenomenological upper bound [24, 25] ]~cRI -< 4.10-3. In models with spontaneous CP violation [17, 26] the mass of the heavy charged vector cR exp leading to boson could be further restricted [18] by requiring IZ~mKLKsl ~~- aJnKLKs, a lower bound M v >_ 2.5 TeV. 5.
IASI
= 1, 2 effective hamiltonians and hadronic matrix elements
There are several contributions to the IAS[ = 1 effective hamiltonian. The first one is due to W ± and V ± exchange: , . ~ S,IV =1 exchange
=
2v~-GF[(lal 2 + BvlCl2)TtL~d-L'¢'UL~LY~,Sc +(IblZ+Bvldl 2 )XRuR -d- e ~ URU¢/R
+ (a*b + BvC, d)X.LRdcY/~u c--u ¢ / R +(ab* + flvCd*)X~Ld-RY"URficT~SL] + h.c.
(5.1)
with
)tAiB=K,~,iaKB,is
(i = U,c,t; A, B = L , R ) .
(5.2)
An analysis of nonleptonic [AS I = 1 weak decays [24] strongly restricts the possible size of the couplings in (5.1):
la*b + flvc*d I]X~"(Re) ] -< 8.10-4, Ibl 2 + Bvldl 2 IXR,RI - -
]al 2 +/~vlCl
2 ]~kLuL]
_< 5" 10 - 2 .
(5.3)
(5.4)
639
H. Neufeld / Right-handed charged currents Inserting (4.17) into (5.3) leads to the inequalities ~L
For small flv, YL, and assuming the left-handed and right-handed mixings to be similar, we find the upper bound
1½vRI<4.10-3.
(5.6)
With the couplings (4.18) of the L-R symmetric model, (5.3) yields the analogous constraint [24] I~LRI < 4 . 1 0 -3 .
(5.7)
Now, let us turn to the second inequality (5.4). Using (4.17) and assuming IXuRR I = IXLLI, (5.4) can be written as
2 ]
1-
"YL ~
5" 10 -2,
(5.8)
which gives no further restriction on the parameters of our model This is quite different in the L-R symmetric model, where (5.4) yields a bound on the heavy vector boson mass (flv -< 5- 10-2). The L-L and R-R penguin diagrams (fig. la, b) lead to effective local hamiltonians
-I- h.c.,
(5.9)
.gf'? R= --v~GFasO~ ~_, 2tRR[Ibl2fp(xi) + flvldlVp(flvX,) ] + h . c .
(5.10)
3~LL = --v~-GFO~so L 37r
3¢r
2tLL[Ja[2fp(xi) -t- flvlCl2fp(flvXi)]
E i=u,c,t
i=u,c,t
I R g
g 91
4
q
q
a)
q
q b)
L(R)
R(L)
I
g(k)
q
q c)
Fig. 1. One-loop penguin diagrams for the [AS] = 1 amplitude.
H. Neufeld / Right-handedchargedcurrents
640
with the operators O L'R
=
_1- /t a -a 4d~/ X SL,R(U'/~)K U-'I- d%Xd+sy~h a as),
(5.11)
the function [27] /p(x) =(1-
X)-4[--
9X + ~ X 2 -
5 X 3 - - 1X4
+ (1 - 4 x + 9x2)lnx]
(5.12)
and
x, = m2/M2w .
(5.13)
With the couplings (4.17) of our model we obtain [al 2 = 1 - "/L and flv lc[ 2 = f12 K __ flv'/e + (½YL)Z/x" So we see that the second term in (5.9) is negligible for small flv, 3'L. The corresponding expressions for the L-R symmetric model are lal 2= 1 and Bvlc[ 2 =flv~ER<< 1. From (4.17) we also find Ibl 2--- (1'~R)2 a n d B v l d l 2 - (17R)2/K. Because of (5.6), ~ R R can be completely neglected in our model. This is a little bit different in the L-R model, where [b[ 2 -- ~2LRbut flvldl 2 -- flv and so the second term in (5.11) is not completely negligible in this case. The L-R penguin graph in fig. lc gives a nonlocal operator in coordinate space. In momentum space, one obtains v~-GFOt s
.~#R
L m,([ab*fLR(X,) + flvcd , fLR(flvXi)]A Ri L OLR.P
~. i=u,c,t
R + [a*bfLg(xi) +flv c . dfLR(flvxi)l)ti LROLa.P} + h.c. (5.14)
with 1
OLLkm.p --
4£~dio~,k')taSL,R(Fty~aU + d ~ a d + gy~XaS)
(5.15)
and fLR(X) = (1 -- X)-3(1 -- 3X-- ¼X3 + -32xlnx ).
(5.16)
Using (4.17) we get ab = 1 y R ( 1 -- g~L) ' and flvcd = ½YR(--flV+ ~' Y L / K ) . Again we compare with the L-R model, where we have ab* = - e i ~ L R and flvcd * = e i ~ f l V ~ L R. Let us now discuss the contributions to the IASI = 2 effective hamiltonian. The standard L-L box diagram (fig. 2a) gives [28, 29] ~,oIASI=2 ~'eff,2a
-
2 GFM w2 ~ dL~IV'SLdLY,~SL E
LL XLL i Xj
i,j~u,c,t
X[(1 + ¼xixj)I2(xi, xj,1 ) - 2xixjIt(xi,xj,1)]
+h.c.,
(5.17)
H. NeufeM / Right-handedchargedcurrents s
d
L
s
D
dL
s
L
Ld
641
R d
s L
{ { w
dE
R s
a)
P
R s
dL c)
b)
Fig. 2. Box graphs for the I A S I = 2 amplitude.
where
xiln x i Ii(xi,
xj, fl) =
(1
-- xi)(1
-
fl lnfi
flxi)(x i - xj) + (i ~ j )
x21n x i I 2 ( x , , xj, fl) = (1 - x i ) ( 1 - flxi)(x
-
(1 - fl)(1 - fix,)(1 -
flxj) '
lnfl
i - xj) + (i ~ j ) - (1 - / 3 ) ( 1
-
j~xi)(1
-
~xj) " (5.18)
Because of the small "/R [eq. (5.6)], only diagrams with a single right-handed vertex (fig. 2b) could lead to an appreciable additional contribution. The corresponding IASI = 2 hamiltonian is given by :/,,olaSl= 2 J%ff,2b --
2 2 GFMw 2¢r2
X
YR, Xf~-~ 2-v--s
E ~jxLL{xLR[dLSRJLSRgl(Xi'Xj) i,j=u,c,t + dLgt*SLdLYtLSLg2( Xi, Xj) + do~Vsdot~vsRg3( xi, xj) ] +•RL[aLSRd--RSLgl(xi, xj) + d2Y~SLd-RY~SRg2(xi, xj)]} , (5.19)
where
gl(xi,xs) = 2xiIl(xi, Xs,1 ) + I2(xi, xj,1 ) - 3xiI2(xi, xj,1 ) -2XiKl(Xi, xj) + xiK2(x i, xj), g2(xi, xj) = XiIl(Xi,Xj,1 ) - ¼xiI2(xi, xj,1 ) + 2Kl(Xi, Xj), g3(xi,
Xj) = l I2( xi, Xj,1)
--
Kl( Xi, xj)
H. NeufeM / Right-handed charged currents
642
with
{
1,,[(1)
(1)]) ( 1)]
Ka(xi, xj)= ½ Ii(xi, x j , 1 ) + x , ( 1 -
I 1 1,1,--xi - x j I 1 xj, xj, -ffi
K2(xi'xj)=12(xi'xj'l)+
I1 1,1,
2 x i ( 1 - xj)
- x ~ I 1 xj, xj,
'
.
Here, only the leading term in x ~ has been taken into account. All box graphs with two right-handed vertices are multiplied with a factor (½~,R)2 < 1.6. 10 -s and are thus completely negligible. This is again in contrast to the L-R symmetric model, where the relevant factor for the diagram of fig. 2c is Bv [18], leading to a noticeable contribution. The appropriate hadronic matrix elements of the four-quark operators contained in the effective hamiltonians of the preceding formulae were calculated using the vacuum insertion technique [28]. In view of possible QCD corrections the operators appearing in (5.1) can be expressed in terms of multiplicatively renormalizable operators /,-) LL(RR) =
--
--
~ +
dy~UL(R)fi'Y~SL(R) q- dTI~SL(R)UypUL(R) ,
O LR(RL)=
d'y~UL(R)~t'Y~SR(L) + ~dSR(L)UUL(R) ,
O L_R(RL) =
2-
-
2
dS R(L) UU L(R)
-
--
(5.20)
•
have to consider matrix elements of the type (~r~r(I=0,2)lOlK °) where I = 0, 2 denotes the isospin of the two pions in the final state. One finds We
( I = 01oL+L(RR)IK0) = (+)~v, S3O , ( I = 21oLC(RR)IK0) = (-)5v + 2,/2--2 3 0,
(I = 01 oL_L(RR)IK.O) = (_+)½~-Q, ( I = 210 LL(RR)IK °) = 0, (I=0IoL+R(RL)]~.~0)= ( -+- ) _4,fi-_t3 9V 3 Y~' ( I = 21OL+R(RL)I~O) =
z,Fc)
( -+- ) 9 1 / 3 ~--'
1 1 + ~qr2+ g 1 r 1)Q, = (+)¢~-(~ -
-
!,U_2( ! ( I = 21OL_R(mlF~° > = (+)6 I/3 K6
--
rl)Q,
(5.21)
H. Neufeld /
Right-handed charged currents
643
where Q = iv/2 f,~( mZK-- m~ ) , m2 rl = (m s +
m~
md) 2 m
fK(l_ m21-2(1 r2= f~ k
( 1 _ m~o) -2
2K- -- -- -m,~ 2
mZ~]
t
m2) 2 --~
'
with f ¢ = 1.22f~ and f~ = 93 MeV. The scalar matrix elements are corrected [30] for momentum dependence and I take m o = 700 MeV. For the L-L and R-R penguin operators (5.11) we have (I=01oL(R)IF0)__ ( -+) 3V 2f!-(1 3 '2 -- rlr2 + rl)O"
(5.22)
In the L-R penguin operators (5.15) we replace - k 2 by the square of a typical hadronic mass /~h involved in the problem (in our case #h = mK). For the matrix elements we obtain [31] (I=0]t3L(R) I~0\= VLR-PI'~ /
+ ~2mc°nst ( - ) V ~ - / . ~ ; r1(1 + r2)Q,
(5.23)
where m¢onst = 300 MeV is a constituent quark mass. Finally we need the matrix elements of the operators occurring in the JAS I = 2 hamiltonians (5.17) and (5.19): OVLL = dLYgScdLYoSL ,
0 LR = dLYt'SLdRYl, SR,
OU = dLS GSL, OSRR = dLSRdLS R
OT = do"V sd%~sR .
(5.24)
One finds (KOloLLIKO > = ~4-'2 2 JKmK
,
(K01oLRIK0 > = -- (1 + x2 r ) f K2m K2,
= ( 6' + r)fam (KOlO~RIKO > =
,
- - g5..~2 r J K r r_t 2 K ,
(K°I OTIK.°> = 2rfi~m 2 ,
(5.25)
644
H. NeufeM / Right-handedcharged currents
with r=
( m s + r o d ) 2" We are now in a position to calculate the amplitudes M12 =
(K°l~[~sl=2l~°)/2mK,
A'0,2 = (~r~r( I = 0 , 2 ) ] - i ~ s l = l l K ° ) ,
(5.26)
which determine the CP-violating observables e, g. Let us also define )~12
2M12 AmKLKs
,
(5.27)
with ArnKLKs = m K L - rnKs. Normalizing A0, 2 by the experimental values of their real parts -4o = A o / R e A~oxp ,
-~2 = A 2 / R e A~2xp ,
(5.28)
we get for the phases ~o,2 = - Im A*0 , 2 "
(5.29)
The CP-violating parameters e, e' can then be written as 1 [~-~i~r/4[T__ e = 2V2 c ~un M12 -~-2~o), .
R,,~ A exp
e' = ~e'(~/2+'2-n°) R;A2Xp (~2 - ~o),
(5.30)
o
where 6o,2 are the strong interaction rr~r phase shifts and ReAe2Xp/ReA~oXp-- 1/22.3. Experimentally, 3o - 32 --- ~r 1 and the ratio e'/e is therefore approximately real.
6. T h e electric dipole m o m e n t of the neutron
Together with the quark loop diagrams (fig. 3), one must also consider the exchange diagrams (fig. 4) where a vector particle is exchanged within a diquark system in the neutron. The relevant formulae for a general gauge theory can be found in ref. [20].
H. Neufeld / Right-handedchargedcurrents
645
w,v v
D-
q
q
q
o)
b)
/
q Y
Fig. 3. Gauge boson loop diagrams for the electric dipole moment of quark q.
d
~
d
~
d ~"
U
~
d
I W,V u
.~-
~
Fig. 4. Exchange diagram contributing to d n. The photon must be appended to quark lines in all possible ways.
The electric dipole moments of up and down quarks are given by d~
eg2yR 64¢tEMZw E
i=d,s,b
milm(KL,uiK~.,u,)
(6.1) dd
eg2yR 64~r2M~
Y'- miIm(KL, i=u,c,t
iaK~,id )
><{(1--½~L)[hl(x,)+2h2(x,)l--(Bv-- E1 TL][ h,(~vX,)+ }h2(~vX,)]}. (6.2) T h e k i n e m t i c a l functions
hk(x ) are 2
defined as
{
hi(x)-
(l-x) 2 1
h2(x )
(l--x)
2
(
3x21n x )
~x + a1x 2 - 2(1-x) '
12
3x In x
2 l + ¼x + ax + -2(1- - x )
(6.3)
H. Neufeld / Right-handed charged currents
646
where the masses of external quarks have been neglected. The standard way to relate d u and d a to the electric dipole moment of the neutron d~ is to appeal to the non-relativistic quark model which gives d loop = l(4da _ du)
(6.4)
for the one-loop contribution d- - nl°°p• The exchange contribution to d~ (fig. 4), calculated in a non-relativistic harmonic oscillator quark model [32], reads [20]
-n
48,n.3/2M2
1 - f l v + 5 L -K - 1
Im(KL,~aK¢,~d ).
(6.5)
In this formula, mqtO is a parameter of the harmonic oscillator model. In the literature [32] one finds values for m~qto ranging approximately from 200 to 450 MeV. For the numerical discussion I shall use m~q~0 = 300 MeV. 7. Numerical estimates
The K ° K ° system has been a traditional testing ground for theoretical mechanisms of mass mixing and CP violation. In the standard model with three fermion generations all relevant information is contained in the Kobayashi-Maskawa mixing matrix [33]. So far, the standard model has passsed all tests in the K ° K ° system. However, recent measurements of the bottom lifetime [34], of upper bounds for the semileptonic branching ratio R = F(b ~ ue~)/F(b ~ ceu) [34, 36] and of the ratio e'/e in K ° ---, 2~r decays [37, 38] have put some pressure on the Kobayashi-Maskawa mechanism for CP violation [34, 39]. In the future, two possible problems for the standard model may be anticipated: (i) e problem: the only experimentally established CP-violating parameter I~1 = 2.28.10 .3 [40] comes out too small for m t < 50 GeV [41] and R < 0.01; (ii) e'/e problem: if the penguin operators are responsible for the entire AI-- 1 enhancement, the standard model predicts [35] a positive lower bound e'/e >_.10 -3 to be compared with the present experimental values: Et/E
= f (1.7 ___8.2). 10 -3 ( - 4 . 6 + 5.3 + 2.4). 10 -3
BNL-Yale [37]
(7.1)
Chicago-Saclay [38].
Besides the parameters of the K ° K ° system the electric dipole moment of the neutron has been investigated intensively as another possible manifestation of CP violation. The present experimental limit [42] is Id,~XPl < 2.6.10 -25 e - c m (95% c.1.).
(7.2)
H. Neufeld / Right-handedchargedcurrents
647
Such a small value can be understood in the standard model with the KobayashiMaskawa mechanism which leads to Idn I < 10-30 e. cm as a generous upper bound. This small bound is essentially due to the fact that the electric dipole moments of individual quarks vanish up to a second-loop order [43] in the standard model. This fact makes d n an ideal observable to distinguish between the standard model and alternative theories because d n is in general non-vanishing already at the one-loop level [20]. Next I want to derive some useful approximations to the results of the previous sections in order to demonstrate the relevance of additional contributions from the model with a strongly interacting Higgs sector. For all numerical estimates the parameters Yc, flv of (4.17) will be neglected. Including QCD corrections [18, 44], one obtains the following approximate equations for the various contributions to ~o and ~2: ~o (W-exchange) _- _ 2707Rim( ~kLu R -- ~kRu L), ~2 (W-exchange) = 570YRIm (xLuR -- ~RuC), ~o(LL-penguin) = - 1.7 Im ? ~ C l n ( x t / x c ) , ~o (LR-penguin) = - Y,a [4.2 Im( XLuR- ~RuL) + 18 Im( ~ R _ ?tRY) + 8.5(m t/GeV)Im(XLtR -- xRtC)].
(7.3)
The standard contribution (fig. 2a) to M12 is approximately )1~12(standard) = 13-5 [ ~/1(?tLcC)2+-'q2 ZXt t~,?~l:L']~2t + 2~31n__ cxcxt 2~LL~ktLL]] ,
(7.4)
with QCD correction factors [45] ~/a = 0.8, 72 = 0.6, ~s = 0.4. From the diagram of fig. 2b one obtains 3412(non-standard ) - - 7R(28XLcCX½R + 23X½C)tRC).
(7.5)
For the e x t e r n a l s-quark mass in (5.19) I have used m ...... t = 500 MeV. Additional terms in (5.19) (with XAtB) are negligible if one assumes ~kLR c,t -- IXc,RLt[ = xLL c,t • The imaginary part of 3412 can now be written as
Im)lTI12-----27s2s2s3s~( --'lJant-'02 s2Xt 2Z
q- r/31n/t Xc ]]
- 7RIm(28xLLx LR + 23xLLXRcL),
(7.6)
where s i = sin Oi, sn = sin& are the parameters of the left-handed mixing matrix
K L•
H. Neufeld / Right-handedchargedcurrents
648
With h l ( X ) = h 2 ( . x ) = 2 for small quark masses (see (6.3)), one gets approximate equations for the electric dipole moments of u and d quarks: du = --~'R [3.2-10-23Im(Ke,udK~,ud) + 6.2.10-22Im(KL,usK~,us) +2.9.10
20Im(KL,ubK~,~b)]e.cm, (7.7)
dd = --'YR [2.3" 10-23Im(KL,udKl~,ud) + 8.2.10-21Im(KL,cdKp~,cd) + 1 . 9 . 1 0 - 2 1 ( m t / G e V ) ( h l ( X t ) + ] h2( xt) )Im( KL,tdK ~,td) ] e ' c m . (7.8) The exchange contribution is given by (7.9)
--nt'/exch---- - 2 . 1 0 - 21,{RI m ( K L, udK R, ud ) e- cm
Summing up all contributions we arrive at the final result 4 d n = ~d d - s1 d u +
dexch
= --TR [2-10-21Im(KL,~dK~,ud) + 1.1.10 20Im(KL,cdK~,¢d) + 2 . 6 - l O - 2 1 ( m t / G e V ) ( h x ( x t ) + ~h2( xt) )Im( gL,tdg~,td) - 2 . 1 . 1 0 - 22Im(KLusK~,us) - 9.6-10-2qm(KL,ubK~,ub)] e . c m . (7.10) If we want to perform more detailed numerical estimates, we encounter the problem that the angles and phases of the right-handed mixing matrix K R are completely unknown. Nevertheless it is clear from the formulae of this section that sizable non-standard contributions to the parameters of the K°g, ° system and to the neutron electric dipole moment can be expected. I will now discuss the interesting special case s 3 = 0, where e, g and d n are completely determined by non-standard effects. For the right handed mixing matrix I will assume IKR,~jl = [KL, ij[- In this case, mixings with heavy quarks can be neglected in (7.3) and (7.10), and it is sufficient to consider
L:(c s) -s
c '
R:lce °
\se '°2
seia+o3,) ce i(a2+aD
'
s023 "
H. Neufeld / Right-handedchargedcurrents
649
Then we have e --
-- ei~r/4y R
{ 10( CS )2sin( o 2 + 03 ) + 8( cs )2sin o 2 190cs [sin(o 1 + 03) - sin ol]
-
- 13cs
[sin(o 2 + o 3) - sin 02] },
(7.12)
e ' = - e i<~/2+a2-a°)TRCS {27[sin(o I + o3) -- sin ol] + 0.57[sin(oz + o3) -- sin 02] },
(7.13)
d n - --'/R[--2-10-2Xc2sin Ol + 1.1 • 10-2°s2sin o2 - 2 . 1 . 1 0 - 2 2 s 2 s i n ( o l + o3)]e, cm.
(7.14)
We want to reproduce the experimental value for e by (7.12) and simultaneously have a small g from (7.13). Therefore, 03 must be very small. In this case (7.12) and (7.13) can be simplified as follows: t~ = -- ei~r/418YR (
cs)2sin o2,
e' = - e ' ¢ ' / 2 + n 2 - a o ) y r t c s
(7.15)
sin o3(27 cos o I + 0.57 cos 02).
(7.16)
Eq. (7.15) can now be used to fix the product 7Rsino2 by the experimental value f o r £~
7Rsin O2 = -- 2.6" 10 - 3
(7.17)
which, together with (5.6), implies the following range for I~'RI: 2.6.10 -3 _< I~'RI -< 8 . 1 0
3.
(7.18)
Also sin 02 is restricted by [sino2l >__0.33.
(7.19)
Inserting (7.17) into (7.14) yields dn=(1.4.10-24-4.8.10
-24sin°l) o-----~ sin e-cm,
(7.20)
and the experimental bound (7.2) restricts the possible values of sin ox, sin o 1 0.24 _< - _.<0.35. sin o 2
(7.21)
650
H. Neufeld / Right-handed charged currents
The ratio e ' / e is now given by cos olsin o 3 e ' / e = 6.8
sin 0 2
(7.22)
and the whole experimental range (7.1) may be covered with appropriately chosen values of sin 03 . Thus we have seen that our CP-violating observables could be completely explained b y non-standard terms, if I~RI is within the bounds of (7.18). What happens now, if I'~RI is smaller than the lower bound of (7.18)? In this case, (7.12) is not sufficient to reproduce the experimental value of e, and we have to allow for the standard contribution in (7.6) with s 3 :g 0. For possible effects on e ' / e and d n I will give a simple numerical example. Let us assume sin(o 1 + 0 3 ) = - s i n 0"3 = 7, ~ 2 : 1, in (7.11). Then we have .
-- = E
+ 3.6- 103rlyR,
(7.23)
standard
d n -- 2.10-21~FYRe • cm.
(7.24)
With ['YRI --< 0 ( 1 0 - 6 ) we would obtain the values of (7.1) and Idnl < O(10-27)e • cm.
8. Conclusions In this p a p e r I have considered the standard model in the limit of a large Higgs mass. In this case the Higgs lagrangian is a nonlinear sigma model based on the manifold SU(2)L × SU(2)p,/SU(2)v. The assumption of physical gauge bosons for the local SU(2) v group leads to a model with additional vector particles V +, V 0 [6]. It is an interesting feature of this approach that it allows for right handed charged currents in the quark sector. I have studied the phenomenological consequences of such right-handed currents induced by a strongly interacting Higgs sector. An analysis of nonleptonic [AS[ = 1 weak decays strongly restricts the possible size of the right-handed couplings. In contrast to L-R symmetric models no bound for the mass of the heavy charged vector boson can be derived. I have calculated the CP-violating parameters e, e' and d n. The experimental value for e can be reproduced even for arbitrarily small = F(b ~ u e u ) / F ( b ~ cep). At the same time the model can accommodate small values for e ' / t of either sign. Furthermore, d n can be very near to its present experimental bound. I wish to thank R. K~Sgerler and W. Grimus for m a n y useful discussions. I am grateful to G. Ecker for profitable conversations and a careful reading of the manuscript.
H. Neufeld / Right-handed charged currents
651
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RIGHT-HANDED CHARGED CURRENTS INDUCED BY A STRONGLY INTERACTING HIGGS SECTOR* H. NEUFELD
lnstitut ffir Theoretische Physik, Universitiit Wien, Vienna, Austria
Received 6 February 1987
A strongly interacting Higgs sector could give rise to right-handed charged currents in the quark sector. An analysis of nonleptonic weak K decays strongly restricts the possible size of these right-handed couplings. The experimental value of e can be reproduced even for vanishing = F(b ~ u e u ) / F ( b ~ cev). e'/e may have arbitrarily small values of either sign. The neutron electric dipole moment could be near to its present experimental bound.
1. Introduction
The Higgs sector comprises that part of the standard model [1] of electroweak interactions which has neither been tested experimentally nor really understood theoretically. Therefore, it will be an important task to look for the actual existence of scalar particles with the properties of Higgs bosons. If the Higgs mass Mla is larger than, say, 1 TeV interesting new effects may be expected. In this case the Higgs self-couplings become large such that lowest order calculations of amplitudes involving Higgs particles cease to be reliable. As a consequence strong interaction effects should also show up in the interactions of longitudinally polarized vector bosons [2, 3]. This can be inferred from the so-called equivalence theorem [3,4], which roughly says that matrix elements describing high-energy processes involving longitudinal vector bosons can be calculated (within the R~ gauge) by replacing the vector particles by their corresponding would-be-Goldstone bosons. Phenomenological and theoretical aspects of the standard model in the limit of a large M H have been studied by many authors [5]. Recently it has been conjectured by Casalbuoni, De Curtis, Dominici and Gatto [6] that such a strongly interacting Higgs sector could give rise to new spin-one * Supported by "Jubil~iumsfonds der Osterreichischen Nationalbank", project number 2765. 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
630
H. NeufeM / Right-handed charged currents
bosonic states. The argumentation goes as follows. When one formally takes the limit M H ~ oe [7] in the generating functional for the Higgs sector of the standard model, one obtains the SU(2)L × SU(2)~/SU(2)v nonlinear sigma model [8]. Since the discovery of hidden local symmetries in supergravity theories [9] it is commonly known that any nonlinear sigma model based on a manifold G / H is equivalent to a theory with Ggloba1× Hloc,1 symmetry [10]. At the classical level, the gauge bosons associated with the local group are only dummy fields. In some specific models in two space-time dimensions it has been shown that a kinetic energy term for these gauge fields is generated by quantum corrections [11]. It can be speculated that such an effect might also occur in four dimensions. In the QCD nonlinear sigma model the vector field of the local symmetry has been identified with the p meson, and a series of phenomenological results has been developed [12]. The authors of ref. [6] have applied the idea of dynamically generated gauge bosons to the standard model with a strongly interacting Higgs sector. Thus, they predict an SU(2)v triplet of new physical vector fields. Through mixings with y, W -+, Z ° and unconventional couplings to fermions, these particles significantly modify some phenomenological predictions. It is one of the most interesting features of this model, that right-handed charged currents may be induced in the quark sector. The phenomenological consequences of such currents have been intensively investigated within the framework of left-right (L-R) symmetric gauge theories [13,14]. In particular, the contributions to the parameters of the K ° K ° system [15-18] and to the neutron electric dipole moment [19, 20] could be quite different compared to the standard model. The purpose of the present paper is to show how fight-handed charged currents are obtained from a strongly interacting Higgs sector and to study the corresponding low-energy phenomenology. In sect. 2 the hidden local symmetry of the nonlinear sigma model is discussed. The model of ref. [6] is briefly reviewed in sect. 3. The fermionic sector is presented in sect. 4. A lower bound for the SU(2)v coupling constant g" is derived from the experimental data of M w and M z. The charged current lagrangian for quarks is given, together with a comparison to the situation in L-R symmetric theories. In sect. 5, I collect all the necessary ingredients for an investigation of the K L K s mass difference and the CP-violating parameters 5, 5'. The dominant Feynman diagrams for the effective ]AS1 = 1, 2 hamiltonians are calculated. An analysis of nonleptonic ]ASI = 1 weak decays strongly restricts the size of the right-handed couplings. All relevant hadronic matrix elements are calculated using the vacuum insertion technique and I set up the necessary equations to determine the CP-violating parameters e, e'. The various contributions to the neutron electric dipole moment d n are given in sect. 6. After recalling the experimental situation and the standard predictions for e, e', d n, I present my numerical estimates in sect. 7. It is shown that these observables could be completely determined by non-standard contributions. My conclusions are summarized in sect. 8.
H. Neufeld / Right-handedcharged currents
631
2. The hidden loeal symmetry of the nonlinear sigma model
The Higgs sector of the standard SU(2)L × U(1)v model consists of a single doublet field ¢ = (q'+ ~o )"
(2.1)
The gauge group is broken down to U(1)EM by the vacuum expectation value (~°)vac = v/vr2.
(2.2)
Written in terms of the matrix ¢0, X-
¢+) d:° ,
' (~-
(2.3)
the standard Higgs lagrangian takes the form m 2
£~°i_n~= ½Tr( D~,X)* (D~'X) - ~8~2(Tr X t X - v2) 2
(2.4)
with the Higgs boson mass M n. In the limit M n --* oo the potential in (2.4) gives the constraint X t X = ½v2. 1 and one obtains the nonlinear sigma model. Replacing X by the dimensionless field
U = x/2X/v ~ SU(2),
(2.5)
and omitting the gauge couplings for the moment, we get the following Lagrange density: •~'O1 = l o 2 T r ( oq~U * o'q~U ) .
(2.6)
.L#1is invariant under global SU(2)L and SU(2)R transformations
U(x) ---4gLU(X)g~,
(2.7)
where gL and gR are elements of SU(2)L and SU(2)R, respectively. The nonlinear sigma model possesses a hidden local symmetry. To see this we introduce new SU(2)-matrix valued variables ~L(X) and ~R(X) transforming as ~L(X) --~gL~L(X)h(x),
~R(X) --)gR~R(X)h(x)
(2.8)
under [SU(2)L x SU(2)R]globaI X [SU(2)V]local, where h(x) is an element of the local SU(2)v. The variable U(x) can be realized as U(x) = ~L(X)~(X). Furthermore we
632
H. Neufeld / Right-handed charged currents
introduce the following hermitean and traceless combinations of the fields ~L(X) and ~R(x) 1
~o~-+)(x) = -~(li[(X) O,IiL(X)+_~(X)O,I~R(X)).
(2.9)
Both %(+)(x) and %(-)(x) are SU(2)L X SU(2)~ singlets and transform under the local SU(2)v as ~0~(+)(x) ~ h*(x)~o~+)(x)h(x)
- iht(x) O~,h(x),
¢o(- )( x ) ~ h*( x )co(~-)( x )h ( x ) .
(2.10)
One recovers the nonlinear sigma model by considering the lagrangian ..~O1 =
v2Tr(%(-)(x)~0(-)"(x))= ~v2Tr(O,Ut(x) O"U(x)).
(2.11)
~ 1 ~ (0) We can introduce a gauge boson field ~(°)(x) = 5%VS. (x) (with the Pauli matrices %) for the local group with transformation property
P9)(x) --, h*(x)Y'¢°)(x)h(x) - ih*(x) O,h(x).
(2.12)
N o w it is possible to write down a second invariant £Pa = v2Tr[(%(+)(x) - ~7/°)(x))(co'+)"(x)
- l~'°)"(x))].
(2.13)
£~°1 and ~c~° 2 can also be written in terms of covariant derivatives D,~L,R = 0,(L,k -i ~ L, RI'7/0) :
s°1,2 = - ~v=Tr(~*L D.~L -V ~*~O~)~.
(2.14)
Any linear combination of (2.11) and (2.13)
.~=~, + ~_~
(2.15)
is equivalent to the original lagrangian (2.6). The reason is that in absence of a kinetic term for V9 ), the gauge field is only an auxiliary field and can be eliminated using its equation of motion 17J°l(x) = %(+)(x). Following the suggestion of ref. [6] I will assume that a kinetic energy term is generated by quantum effects. 17¢°~ will thus be regarded as an additional physical vector boson field.
633
H. Neufeld / Right-handed charged currents
3. New vector bosons from a strongly interacting Higgs sector Let us now complete the covariant derivatives by introducing the gauge couplings of the standard SU(2)L × U(1)r gauge fields I,V~(°) and /],, D ~ L = a ~ L + iI,V~(°)~L-- i~LI,7(0), D ~ R = O~ R "-F i / ~ R -- i~Rl?~(°) ,
(3.1)
where the coupling constants g, g', g" of SU(2)L X U ( 1 ) y x SU(2)v have been included by
~(0) = g " I"
~ - ~ V~(°) .
( 3.2 )
Our Higgs lagrangian in the limit of a large Higgs mass is now given by
£~'=--¼vZ[Tr(~tLD~c--~D,~R)Z+KTr(~tLD,~L
+ 8*RD~R)2].
(3.3)
It is not hard to see that one can perform a local S U ( 2 ) c × SU(2)v gauge transformation such that ~c = ~R = ~- In this gauge (3.3) reduces to the mass term for the vector bosons, £P= ¼v2 [Tr(l~(°) - / ~ ) 2 + K Tr(l~(°) +/~ - 21~(°))2] ,
(3.4)
whereas all would-be Goldstone bosons have disappeared. After inserting (3.2) into (3.4) the mass eigenstates and the corresponding mass eigenvalues can be determined. The weak eigenstates of the charged sector (Wl(°~ and V~(°~)are related to the physical states (W1,2 and V1,2) by
cos sin ) V/
sin q~
cos q)
(i = 1,2).
(3.5)
V/(°) J '
In the limit g" >> g (which will turn out to be the relevant one) the mixing angle q9
H. Neufeld / Right-handedchargedcurrents
634
and the masses of W1,2 and V1,2 are given by g g,, ,
-
12 2( g2)
zv •g
1+ ~
.
(3.6)
In the neutral sector we have the photon A, the Z ° boson and an additional vector particle V°: A Zo = Vo
0
cos +
sin qJ
cos ~ sin ~
sin ~ sin ~p - cos ~ sin ~k
- sin ~ cos cos ~ cos
g/G
-g'/G
g'/G 0
g/G 0
o° : ) 1
v#,]
(3.7) with G = g ~ + g,2. Again in the limit of a large g" we have
g2 _ g,2 g"G
M2o= ¼v2(g2+g,2) 1 -
2gg' g"G ' ( g 2 _ g,2)2 (gZ+g,2)g,,2
g2 + g,2 ) M 2 o = 1av 2xg ,2 1 + _ g,,2 _ •
(3.8)
The exact versions of (3.6) and (3.8) can be found in ref. [6]. In the limit g" --->oo the masses of V -+,V ° become infinite and one recovers the standard mass formulae f o r W -+, Z °.
4. The fermionic sector
The physical vector bosons W-+, V-+, A, Z °, V ° couple to fermions through interaction terms which can be directly read off from the gauge couplings of
H. Neufeld / Right-handed charged currents
635
SU(2)L X U(1) y. Let us start with the charged current lagrangian
£#cc
=
i./T,,.~ w-(0)+ + h.c.,
(4.1)
-- 2 V 2 6JCC"/.L
where J~c = Ee= e, ,, ,~eY"( 1 - Ys) d+ J ~ C , h a d r " Using (3.5) yields .Sacc= - ½~/~-~gj~c(COS qoW7 + sin q0V~+) + h.c.,
(4.2)
leading to a low-energy effective lagrangian
l( ,os,
£P~g = - 8
M---~w+
M~
JccJcc~"
(4.3)
From (4.3) we obtain a formula for the Fermi constant GF, GF
g2 = w8M---~(c°s2cp + flvsin2cp) '
(4.4)
with ] ~ v = M w2/ M
v2 .
(4.5)
The electromagnetic and neutral current Lagrangian can be determined in an analogous way. For the electromagnetic coupling constant e one obtains the equation [6] gg' e = --~-cos ~b. (4.6) Using (3.6), (3.8), (4.4) and (4.6) the following relation can be derived [6]:
(1
t2
Together with the experimental values and errors for M w and M z [21] this equation leads to a bound for the coupling constant g " , lfl (@) 2>-(1+g
[ 18.9 7)'[11.5
(UA1 data) t (UA2 data) J '
(4.8)
which justifies a posteriori the preference for large values of g " in the previous formulae. Besides (4.2) and (4.5) one should expect that small direct couplings of V ÷ and V ° to the fermions [61 also appear in the lagrangian. In the following I will
H. Neufeld / Right-handedchargedcurrents
636
restrict myself to the quark sector. The weak eigenstates of the quark fields will be denoted by qR= (P)R.
qL= (P)L ,
(4.9)
With the new field XL = ~tLqL, which is a singlet under SU(2)L and a doublet under SU(2) v, we build up the invariant term g,, ,¢ ) XLiY u 0t, + i-~-- -~ V~(°) + ig'~B~ XL"
(4.10)
)
Quite analogously we define XR = ~tRqR, which again is a singlet with respect to S U ( 2 ) L and a doublet under SU(2)v- The corresponding invariant reads g" 'I" ) ,1 ~ V~(°) + ,g aB~
2RiV ~ O. + i T
(4.11)
.
We can now add (4.10) and (4.11) to the standard SU(2)L X U(1)y terms. In the gauge (L = ~R = ~ and with a proper normalization of the kinetic energy terms we obtain &°ki.
1 nt- ~L
qLiY ~ O~+" tg2
+ . ,1
qL
+"[LqLi~(O~+ig@2V~(O) q-igtl~)qL] +m 1
1
-J-YR
0
-½
B~qR
( gtt'rv(O)+lg__• tlgB~)
+yR~/R/y ~ 3~,+ i--~-- 2
qR] '
(4.12)
where YL,R are new constants of the effective lagrangian measuring the strength of
the additional contributions. The most interesting feature of (4.12) is that it contains right-handed charged currents. Such terms are absent in the usual SU(2)L X U(1)F standard model. Let us now write down the charged current lagrangian in terms of physical fields. It is of the form
-Wcc =
g(a ,cr KCdC + e Rr K a.)ve; +~g(C~LT~'KLdL +d~RY~'KRdR)Vf + h.c.,
(4.13)
H. Neufeld / Right-handedchargedcurrents
637
where 1 a
~
[
tt 1
B
1
b=
")tR g"
l+YR
2
g
1 C - -
-
-
l+y L d=
g
"
1 + ~/L /COSCp- ~ Y L T S l n ¢ p
sin cp,
YL g"
sin + 5 - T c ° s
)
,
1
YR g" - - - - c o s ~. l+y R 2 g
(4.14)
The weak eigenfield PL, R and t/L, R have been related to the mass eigenfields UL, R and dE, R by unitary transformations PL,R = UP, RUL,R,
nL,R = UI~,RdL,R,
(4.15)
leading to the left- (right-) handed mixing matrices KL(R) With ep-follows:
-g/g"
= ITP t
lTn
(4.16)
~ L(R)~"L(R)"
and for small ]tL,R the relations in (4.14) can be simplified as a--1
"YL 2'
g ~L g" c = --+ ----, g" 2 g
b = YR 2' d=
7R g" 2 g
(4.17)
The analogous couplings for the leptonic sector have been discussed in [6]. There, one also has a YL,lept term, but fight-handed charged currents are absent if there are no right-handed neutrinos. Quite reasonable fits of the low-energy phenomenology and of the masses of W -+ and Z ° could be obtained for )tL, lept = 0.1 and rather low values of M v. At this point I would like to compare our model with L-R symmetric gauge theories based on SU(2)L X SU(2)R X U(1) [13]. In the minimal version [14] of such models one has a single (½, ½, 0) scalar multiplet • coupling to quarks. The charged current lagrangian is of the same form as (4.13) with a = c o s ~LR,
b = - e-iXsin ~LR,
c = eiXsin ~LR,
d = COS ~LR,
(4.18)
H. Neufeld / Right-handed charged currents
638
where ~i_l~, )~ are L-R mixing parameters with 21owl ~cR-
ivl2 + 1
ow*
iwl=BV
,
e ix = _ _ _
Ivwl '
0
(~)v~¢ = ~ - ( 0 0 w ) .
(4.19)
It can be demonstrated in specific examples [22] that a natural explanation of the smallness of weak mixing angles requires Iw/vl = O(mb/mt). The natural magnitude is therefore [23] ~LFt--
2m b
flV -- 2" 10 -3 ( M v / T e V ) -2
(4.20)
mt
to be compared with the phenomenological upper bound [24, 25] ]~cRI -< 4.10-3. In models with spontaneous CP violation [17, 26] the mass of the heavy charged vector cR exp leading to boson could be further restricted [18] by requiring IZ~mKLKsl ~~- aJnKLKs, a lower bound M v >_ 2.5 TeV. 5.
IASI
= 1, 2 effective hamiltonians and hadronic matrix elements
There are several contributions to the IAS[ = 1 effective hamiltonian. The first one is due to W ± and V ± exchange: , . ~ S,IV =1 exchange
=
2v~-GF[(lal 2 + BvlCl2)TtL~d-L'¢'UL~LY~,Sc +(IblZ+Bvldl 2 )XRuR -d- e ~ URU¢/R
+ (a*b + BvC, d)X.LRdcY/~u c--u ¢ / R +(ab* + flvCd*)X~Ld-RY"URficT~SL] + h.c.
(5.1)
with
)tAiB=K,~,iaKB,is
(i = U,c,t; A, B = L , R ) .
(5.2)
An analysis of nonleptonic [AS I = 1 weak decays [24] strongly restricts the possible size of the couplings in (5.1):
la*b + flvc*d I]X~"(Re) ] -< 8.10-4, Ibl 2 + Bvldl 2 IXR,RI - -
]al 2 +/~vlCl
2 ]~kLuL]
_< 5" 10 - 2 .
(5.3)
(5.4)
639
H. Neufeld / Right-handed charged currents Inserting (4.17) into (5.3) leads to the inequalities ~L
For small flv, YL, and assuming the left-handed and right-handed mixings to be similar, we find the upper bound
1½vRI<4.10-3.
(5.6)
With the couplings (4.18) of the L-R symmetric model, (5.3) yields the analogous constraint [24] I~LRI < 4 . 1 0 -3 .
(5.7)
Now, let us turn to the second inequality (5.4). Using (4.17) and assuming IXuRR I = IXLLI, (5.4) can be written as
2 ]
1-
"YL ~
5" 10 -2,
(5.8)
which gives no further restriction on the parameters of our model This is quite different in the L-R symmetric model, where (5.4) yields a bound on the heavy vector boson mass (flv -< 5- 10-2). The L-L and R-R penguin diagrams (fig. la, b) lead to effective local hamiltonians
-I- h.c.,
(5.9)
.gf'? R= --v~GFasO~ ~_, 2tRR[Ibl2fp(xi) + flvldlVp(flvX,) ] + h . c .
(5.10)
3~LL = --v~-GFO~so L 37r
3¢r
2tLL[Ja[2fp(xi) -t- flvlCl2fp(flvXi)]
E i=u,c,t
i=u,c,t
I R g
g 91
4
q
q
a)
q
q b)
L(R)
R(L)
I
g(k)
q
q c)
Fig. 1. One-loop penguin diagrams for the [AS] = 1 amplitude.
H. Neufeld / Right-handedchargedcurrents
640
with the operators O L'R
=
_1- /t a -a 4d~/ X SL,R(U'/~)K U-'I- d%Xd+sy~h a as),
(5.11)
the function [27] /p(x) =(1-
X)-4[--
9X + ~ X 2 -
5 X 3 - - 1X4
+ (1 - 4 x + 9x2)lnx]
(5.12)
and
x, = m2/M2w .
(5.13)
With the couplings (4.17) of our model we obtain [al 2 = 1 - "/L and flv lc[ 2 = f12 K __ flv'/e + (½YL)Z/x" So we see that the second term in (5.9) is negligible for small flv, 3'L. The corresponding expressions for the L-R symmetric model are lal 2= 1 and Bvlc[ 2 =flv~ER<< 1. From (4.17) we also find Ibl 2--- (1'~R)2 a n d B v l d l 2 - (17R)2/K. Because of (5.6), ~ R R can be completely neglected in our model. This is a little bit different in the L-R model, where [b[ 2 -- ~2LRbut flvldl 2 -- flv and so the second term in (5.11) is not completely negligible in this case. The L-R penguin graph in fig. lc gives a nonlocal operator in coordinate space. In momentum space, one obtains v~-GFOt s
.~#R
L m,([ab*fLR(X,) + flvcd , fLR(flvXi)]A Ri L OLR.P
~. i=u,c,t
R + [a*bfLg(xi) +flv c . dfLR(flvxi)l)ti LROLa.P} + h.c. (5.14)
with 1
OLLkm.p --
4£~dio~,k')taSL,R(Fty~aU + d ~ a d + gy~XaS)
(5.15)
and fLR(X) = (1 -- X)-3(1 -- 3X-- ¼X3 + -32xlnx ).
(5.16)
Using (4.17) we get ab = 1 y R ( 1 -- g~L) ' and flvcd = ½YR(--flV+ ~' Y L / K ) . Again we compare with the L-R model, where we have ab* = - e i ~ L R and flvcd * = e i ~ f l V ~ L R. Let us now discuss the contributions to the IASI = 2 effective hamiltonian. The standard L-L box diagram (fig. 2a) gives [28, 29] ~,oIASI=2 ~'eff,2a
-
2 GFM w2 ~ dL~IV'SLdLY,~SL E
LL XLL i Xj
i,j~u,c,t
X[(1 + ¼xixj)I2(xi, xj,1 ) - 2xixjIt(xi,xj,1)]
+h.c.,
(5.17)
H. NeufeM / Right-handedchargedcurrents s
d
L
s
D
dL
s
L
Ld
641
R d
s L
{ { w
dE
R s
a)
P
R s
dL c)
b)
Fig. 2. Box graphs for the I A S I = 2 amplitude.
where
xiln x i Ii(xi,
xj, fl) =
(1
-- xi)(1
-
fl lnfi
flxi)(x i - xj) + (i ~ j )
x21n x i I 2 ( x , , xj, fl) = (1 - x i ) ( 1 - flxi)(x
-
(1 - fl)(1 - fix,)(1 -
flxj) '
lnfl
i - xj) + (i ~ j ) - (1 - / 3 ) ( 1
-
j~xi)(1
-
~xj) " (5.18)
Because of the small "/R [eq. (5.6)], only diagrams with a single right-handed vertex (fig. 2b) could lead to an appreciable additional contribution. The corresponding IASI = 2 hamiltonian is given by :/,,olaSl= 2 J%ff,2b --
2 2 GFMw 2¢r2
X
YR, Xf~-~ 2-v--s
E ~jxLL{xLR[dLSRJLSRgl(Xi'Xj) i,j=u,c,t + dLgt*SLdLYtLSLg2( Xi, Xj) + do~Vsdot~vsRg3( xi, xj) ] +•RL[aLSRd--RSLgl(xi, xj) + d2Y~SLd-RY~SRg2(xi, xj)]} , (5.19)
where
gl(xi,xs) = 2xiIl(xi, Xs,1 ) + I2(xi, xj,1 ) - 3xiI2(xi, xj,1 ) -2XiKl(Xi, xj) + xiK2(x i, xj), g2(xi, xj) = XiIl(Xi,Xj,1 ) - ¼xiI2(xi, xj,1 ) + 2Kl(Xi, Xj), g3(xi,
Xj) = l I2( xi, Xj,1)
--
Kl( Xi, xj)
H. NeufeM / Right-handed charged currents
642
with
{
1,,[(1)
(1)]) ( 1)]
Ka(xi, xj)= ½ Ii(xi, x j , 1 ) + x , ( 1 -
I 1 1,1,--xi - x j I 1 xj, xj, -ffi
K2(xi'xj)=12(xi'xj'l)+
I1 1,1,
2 x i ( 1 - xj)
- x ~ I 1 xj, xj,
'
.
Here, only the leading term in x ~ has been taken into account. All box graphs with two right-handed vertices are multiplied with a factor (½~,R)2 < 1.6. 10 -s and are thus completely negligible. This is again in contrast to the L-R symmetric model, where the relevant factor for the diagram of fig. 2c is Bv [18], leading to a noticeable contribution. The appropriate hadronic matrix elements of the four-quark operators contained in the effective hamiltonians of the preceding formulae were calculated using the vacuum insertion technique [28]. In view of possible QCD corrections the operators appearing in (5.1) can be expressed in terms of multiplicatively renormalizable operators /,-) LL(RR) =
--
--
~ +
dy~UL(R)fi'Y~SL(R) q- dTI~SL(R)UypUL(R) ,
O LR(RL)=
d'y~UL(R)~t'Y~SR(L) + ~dSR(L)UUL(R) ,
O L_R(RL) =
2-
-
2
dS R(L) UU L(R)
-
--
(5.20)
•
have to consider matrix elements of the type (~r~r(I=0,2)lOlK °) where I = 0, 2 denotes the isospin of the two pions in the final state. One finds We
( I = 01oL+L(RR)IK0) = (+)~v, S3O , ( I = 21oLC(RR)IK0) = (-)5v + 2,/2--2 3 0,
(I = 01 oL_L(RR)IK.O) = (_+)½~-Q, ( I = 210 LL(RR)IK °) = 0, (I=0IoL+R(RL)]~.~0)= ( -+- ) _4,fi-_t3 9V 3 Y~' ( I = 21OL+R(RL)I~O) =
z,Fc)
( -+- ) 9 1 / 3 ~--'
1 1 + ~qr2+ g 1 r 1)Q, = (+)¢~-(~ -
-
!,U_2( ! ( I = 21OL_R(mlF~° > = (+)6 I/3 K6
--
rl)Q,
(5.21)
H. Neufeld /
Right-handed charged currents
643
where Q = iv/2 f,~( mZK-- m~ ) , m2 rl = (m s +
m~
md) 2 m
fK(l_ m21-2(1 r2= f~ k
( 1 _ m~o) -2
2K- -- -- -m,~ 2
mZ~]
t
m2) 2 --~
'
with f ¢ = 1.22f~ and f~ = 93 MeV. The scalar matrix elements are corrected [30] for momentum dependence and I take m o = 700 MeV. For the L-L and R-R penguin operators (5.11) we have (I=01oL(R)IF0)__ ( -+) 3V 2f!-(1 3 '2 -- rlr2 + rl)O"
(5.22)
In the L-R penguin operators (5.15) we replace - k 2 by the square of a typical hadronic mass /~h involved in the problem (in our case #h = mK). For the matrix elements we obtain [31] (I=0]t3L(R) I~0\= VLR-PI'~ /
+ ~2mc°nst ( - ) V ~ - / . ~ ; r1(1 + r2)Q,
(5.23)
where m¢onst = 300 MeV is a constituent quark mass. Finally we need the matrix elements of the operators occurring in the JAS I = 2 hamiltonians (5.17) and (5.19): OVLL = dLYgScdLYoSL ,
0 LR = dLYt'SLdRYl, SR,
OU = dLS GSL, OSRR = dLSRdLS R
OT = do"V sd%~sR .
(5.24)
One finds (KOloLLIKO > = ~4-'2 2 JKmK
,
(K01oLRIK0 > = -- (1 + x2 r ) f K2m K2,
,
- - g5..~2 r J K r r_t 2 K ,
(K°I OTIK.°> = 2rfi~m 2 ,
(5.25)
644
H. NeufeM / Right-handedcharged currents
with r=
( m s + r o d ) 2" We are now in a position to calculate the amplitudes M12 =
(K°l~[~sl=2l~°)/2mK,
A'0,2 = (~r~r( I = 0 , 2 ) ] - i ~ s l = l l K ° ) ,
(5.26)
which determine the CP-violating observables e, g. Let us also define )~12
2M12 AmKLKs
,
(5.27)
with ArnKLKs = m K L - rnKs. Normalizing A0, 2 by the experimental values of their real parts -4o = A o / R e A~oxp ,
-~2 = A 2 / R e A~2xp ,
(5.28)
we get for the phases ~o,2 = - Im A*0 , 2 "
(5.29)
The CP-violating parameters e, e' can then be written as 1 [~-~i~r/4[T__ e = 2V2 c ~un M12 -~-2~o), .
R,,~ A exp
e' = ~e'(~/2+'2-n°) R;A2Xp (~2 - ~o),
(5.30)
o
where 6o,2 are the strong interaction rr~r phase shifts and ReAe2Xp/ReA~oXp-- 1/22.3. Experimentally, 3o - 32 --- ~r 1 and the ratio e'/e is therefore approximately real.
6. T h e electric dipole m o m e n t of the neutron
Together with the quark loop diagrams (fig. 3), one must also consider the exchange diagrams (fig. 4) where a vector particle is exchanged within a diquark system in the neutron. The relevant formulae for a general gauge theory can be found in ref. [20].
H. Neufeld / Right-handedchargedcurrents
645
w,v v
D-
q
q
q
o)
b)
/
q Y
Fig. 3. Gauge boson loop diagrams for the electric dipole moment of quark q.
d
~
d
~
d ~"
U
~
d
I W,V u
.~-
~
Fig. 4. Exchange diagram contributing to d n. The photon must be appended to quark lines in all possible ways.
The electric dipole moments of up and down quarks are given by d~
eg2yR 64¢tEMZw E
i=d,s,b
milm(KL,uiK~.,u,)
(6.1) dd
eg2yR 64~r2M~
Y'- miIm(KL, i=u,c,t
iaK~,id )
><{(1--½~L)[hl(x,)+2h2(x,)l--(Bv-- E1 TL][ h,(~vX,)+ }h2(~vX,)]}. (6.2) T h e k i n e m t i c a l functions
hk(x ) are 2
defined as
{
hi(x)-
(l-x) 2 1
h2(x )
(l--x)
2
(
3x21n x )
~x + a1x 2 - 2(1-x) '
12
3x In x
2 l + ¼x + ax + -2(1- - x )
(6.3)
H. Neufeld / Right-handed charged currents
646
where the masses of external quarks have been neglected. The standard way to relate d u and d a to the electric dipole moment of the neutron d~ is to appeal to the non-relativistic quark model which gives d loop = l(4da _ du)
(6.4)
for the one-loop contribution d- - nl°°p• The exchange contribution to d~ (fig. 4), calculated in a non-relativistic harmonic oscillator quark model [32], reads [20]
-n
48,n.3/2M2
1 - f l v + 5 L -K - 1
Im(KL,~aK¢,~d ).
(6.5)
In this formula, mqtO is a parameter of the harmonic oscillator model. In the literature [32] one finds values for m~qto ranging approximately from 200 to 450 MeV. For the numerical discussion I shall use m~q~0 = 300 MeV. 7. Numerical estimates
The K ° K ° system has been a traditional testing ground for theoretical mechanisms of mass mixing and CP violation. In the standard model with three fermion generations all relevant information is contained in the Kobayashi-Maskawa mixing matrix [33]. So far, the standard model has passsed all tests in the K ° K ° system. However, recent measurements of the bottom lifetime [34], of upper bounds for the semileptonic branching ratio R = F(b ~ ue~)/F(b ~ ceu) [34, 36] and of the ratio e'/e in K ° ---, 2~r decays [37, 38] have put some pressure on the Kobayashi-Maskawa mechanism for CP violation [34, 39]. In the future, two possible problems for the standard model may be anticipated: (i) e problem: the only experimentally established CP-violating parameter I~1 = 2.28.10 .3 [40] comes out too small for m t < 50 GeV [41] and R < 0.01; (ii) e'/e problem: if the penguin operators are responsible for the entire AI-- 1 enhancement, the standard model predicts [35] a positive lower bound e'/e >_.10 -3 to be compared with the present experimental values: Et/E
= f (1.7 ___8.2). 10 -3 ( - 4 . 6 + 5.3 + 2.4). 10 -3
BNL-Yale [37]
(7.1)
Chicago-Saclay [38].
Besides the parameters of the K ° K ° system the electric dipole moment of the neutron has been investigated intensively as another possible manifestation of CP violation. The present experimental limit [42] is Id,~XPl < 2.6.10 -25 e - c m (95% c.1.).
(7.2)
H. Neufeld / Right-handedchargedcurrents
647
Such a small value can be understood in the standard model with the KobayashiMaskawa mechanism which leads to Idn I < 10-30 e. cm as a generous upper bound. This small bound is essentially due to the fact that the electric dipole moments of individual quarks vanish up to a second-loop order [43] in the standard model. This fact makes d n an ideal observable to distinguish between the standard model and alternative theories because d n is in general non-vanishing already at the one-loop level [20]. Next I want to derive some useful approximations to the results of the previous sections in order to demonstrate the relevance of additional contributions from the model with a strongly interacting Higgs sector. For all numerical estimates the parameters Yc, flv of (4.17) will be neglected. Including QCD corrections [18, 44], one obtains the following approximate equations for the various contributions to ~o and ~2: ~o (W-exchange) _- _ 2707Rim( ~kLu R -- ~kRu L), ~2 (W-exchange) = 570YRIm (xLuR -- ~RuC), ~o(LL-penguin) = - 1.7 Im ? ~ C l n ( x t / x c ) , ~o (LR-penguin) = - Y,a [4.2 Im( XLuR- ~RuL) + 18 Im( ~ R _ ?tRY) + 8.5(m t/GeV)Im(XLtR -- xRtC)].
(7.3)
The standard contribution (fig. 2a) to M12 is approximately )1~12(standard) = 13-5 [ ~/1(?tLcC)2+-'q2 ZXt t~,?~l:L']~2t + 2~31n__ cxcxt 2~LL~ktLL]] ,
(7.4)
with QCD correction factors [45] ~/a = 0.8, 72 = 0.6, ~s = 0.4. From the diagram of fig. 2b one obtains 3412(non-standard ) - - 7R(28XLcCX½R + 23X½C)tRC).
(7.5)
For the e x t e r n a l s-quark mass in (5.19) I have used m ...... t = 500 MeV. Additional terms in (5.19) (with XAtB) are negligible if one assumes ~kLR c,t -- IXc,RLt[ = xLL c,t • The imaginary part of 3412 can now be written as
Im)lTI12-----27s2s2s3s~( --'lJant-'02 s2Xt 2Z
q- r/31n/t Xc ]]
- 7RIm(28xLLx LR + 23xLLXRcL),
(7.6)
where s i = sin Oi, sn = sin& are the parameters of the left-handed mixing matrix
K L•
H. Neufeld / Right-handedchargedcurrents
648
With h l ( X ) = h 2 ( . x ) = 2 for small quark masses (see (6.3)), one gets approximate equations for the electric dipole moments of u and d quarks: du = --~'R [3.2-10-23Im(Ke,udK~,ud) + 6.2.10-22Im(KL,usK~,us) +2.9.10
20Im(KL,ubK~,~b)]e.cm, (7.7)
dd = --'YR [2.3" 10-23Im(KL,udKl~,ud) + 8.2.10-21Im(KL,cdKp~,cd) + 1 . 9 . 1 0 - 2 1 ( m t / G e V ) ( h l ( X t ) + ] h2( xt) )Im( KL,tdK ~,td) ] e ' c m . (7.8) The exchange contribution is given by (7.9)
--nt'/exch---- - 2 . 1 0 - 21,{RI m ( K L, udK R, ud ) e- cm
Summing up all contributions we arrive at the final result 4 d n = ~d d - s1 d u +
dexch
= --TR [2-10-21Im(KL,~dK~,ud) + 1.1.10 20Im(KL,cdK~,¢d) + 2 . 6 - l O - 2 1 ( m t / G e V ) ( h x ( x t ) + ~h2( xt) )Im( gL,tdg~,td) - 2 . 1 . 1 0 - 22Im(KLusK~,us) - 9.6-10-2qm(KL,ubK~,ub)] e . c m . (7.10) If we want to perform more detailed numerical estimates, we encounter the problem that the angles and phases of the right-handed mixing matrix K R are completely unknown. Nevertheless it is clear from the formulae of this section that sizable non-standard contributions to the parameters of the K°g, ° system and to the neutron electric dipole moment can be expected. I will now discuss the interesting special case s 3 = 0, where e, g and d n are completely determined by non-standard effects. For the right handed mixing matrix I will assume IKR,~jl = [KL, ij[- In this case, mixings with heavy quarks can be neglected in (7.3) and (7.10), and it is sufficient to consider
L:(c s) -s
c '
R:lce °
\se '°2
seia+o3,) ce i(a2+aD
'
s023 "
H. Neufeld / Right-handedchargedcurrents
649
Then we have e --
-- ei~r/4y R
{ 10( CS )2sin( o 2 + 03 ) + 8( cs )2sin o 2 190cs [sin(o 1 + 03) - sin ol]
-
- 13cs
[sin(o 2 + o 3) - sin 02] },
(7.12)
e ' = - e i<~/2+a2-a°)TRCS {27[sin(o I + o3) -- sin ol] + 0.57[sin(oz + o3) -- sin 02] },
(7.13)
d n - --'/R[--2-10-2Xc2sin Ol + 1.1 • 10-2°s2sin o2 - 2 . 1 . 1 0 - 2 2 s 2 s i n ( o l + o3)]e, cm.
(7.14)
We want to reproduce the experimental value for e by (7.12) and simultaneously have a small g from (7.13). Therefore, 03 must be very small. In this case (7.12) and (7.13) can be simplified as follows: t~ = -- ei~r/418YR (
cs)2sin o2,
e' = - e ' ¢ ' / 2 + n 2 - a o ) y r t c s
(7.15)
sin o3(27 cos o I + 0.57 cos 02).
(7.16)
Eq. (7.15) can now be used to fix the product 7Rsino2 by the experimental value f o r £~
7Rsin O2 = -- 2.6" 10 - 3
(7.17)
which, together with (5.6), implies the following range for I~'RI: 2.6.10 -3 _< I~'RI -< 8 . 1 0
3.
(7.18)
Also sin 02 is restricted by [sino2l >__0.33.
(7.19)
Inserting (7.17) into (7.14) yields dn=(1.4.10-24-4.8.10
-24sin°l) o-----~ sin e-cm,
(7.20)
and the experimental bound (7.2) restricts the possible values of sin ox, sin o 1 0.24 _< - _.<0.35. sin o 2
(7.21)
650
H. Neufeld / Right-handed charged currents
The ratio e ' / e is now given by cos olsin o 3 e ' / e = 6.8
sin 0 2
(7.22)
and the whole experimental range (7.1) may be covered with appropriately chosen values of sin 03 . Thus we have seen that our CP-violating observables could be completely explained b y non-standard terms, if I~RI is within the bounds of (7.18). What happens now, if I'~RI is smaller than the lower bound of (7.18)? In this case, (7.12) is not sufficient to reproduce the experimental value of e, and we have to allow for the standard contribution in (7.6) with s 3 :g 0. For possible effects on e ' / e and d n I will give a simple numerical example. Let us assume sin(o 1 + 0 3 ) = - s i n 0"3 = 7, ~ 2 : 1, in (7.11). Then we have .
-- = E
+ 3.6- 103rlyR,
(7.23)
standard
d n -- 2.10-21~FYRe • cm.
(7.24)
With ['YRI --< 0 ( 1 0 - 6 ) we would obtain the values of (7.1) and Idnl < O(10-27)e • cm.
8. Conclusions In this p a p e r I have considered the standard model in the limit of a large Higgs mass. In this case the Higgs lagrangian is a nonlinear sigma model based on the manifold SU(2)L × SU(2)p,/SU(2)v. The assumption of physical gauge bosons for the local SU(2) v group leads to a model with additional vector particles V +, V 0 [6]. It is an interesting feature of this approach that it allows for right handed charged currents in the quark sector. I have studied the phenomenological consequences of such right-handed currents induced by a strongly interacting Higgs sector. An analysis of nonleptonic [AS[ = 1 weak decays strongly restricts the possible size of the right-handed couplings. In contrast to L-R symmetric models no bound for the mass of the heavy charged vector boson can be derived. I have calculated the CP-violating parameters e, e' and d n. The experimental value for e can be reproduced even for arbitrarily small = F(b ~ u e u ) / F ( b ~ cep). At the same time the model can accommodate small values for e ' / t of either sign. Furthermore, d n can be very near to its present experimental bound. I wish to thank R. K~Sgerler and W. Grimus for m a n y useful discussions. I am grateful to G. Ecker for profitable conversations and a careful reading of the manuscript.
H. Neufeld / Right-handed charged currents
651
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