Table Of ContentTUM-HEP-573/05
hep-ph/0501230
Non-Decoupling Effects of the Heavy T
5 in the B0 − B¯0 Mixing and
d,s d,s
0
0
Rare K and B Decays
2
n
a
J
5
Andrzej J. Buras, Anton Poschenrieder and Selma Uhlig
2
1
Physik Department, Technische Universit¨at Mu¨nchen, D-85748 Garching, Germany
v
0
3
2
1
0
5 Abstract
0
/
h
p
- We point out that in the case of a heavy top quark T, present in the Littlest
p
e Higgs model (LH), and the t-T mixing parameter x 0.9 the contribution to
L
h ≥
: B0 B¯0 mixing from box diagrams with two T exchanges cannot be neglected.
v d,s− d,s
i Although formally (v4/f4) with v = 246GeV and f > 1 TeV, this contribu-
X O
tion increases linearly with x = m2/M2 and with x = (f2/v2) constitutes
r T T W T O
a effectively an (v2/f2) correction. For x 1, this contribution turns out to be
L
O ≈
more important than the genuine (v2/f2) corrections. In particular it is larger
O
than the recently calculated (v2/f2) contribution of box diagrams with a sin-
O
gle T exchange that increases only logarithmically with x . For x = 0.95 and
T L
f/v = 5,10,15, the short distance function S governing the B0 B¯0 mixing
d,s − d,s
mass differences ∆M receives 56%, 15% and 7% enhancements relative to its
d,s
Standard Model (SM) value, implyinga suppressionof the CKMelement V and
td
| |
an enhancement of ∆M . The short distance functions X and Y, relevant for rare
s
K and B decays, increase only logarithmically with x . With the suppressed V ,
T td
| |
K πνν¯ and B µ+µ− decays are only insignificantly modified with respect to
d
→ →
the SM, while the branching ratio Br(B µ+µ−) receives 66%, 19% and 9% en-
s
→
hancements for x = 0.95 and f/v = 5,10,15, respectively. Similar enhancement
L
is found for Br(B µµ¯)/Br(B µµ¯).
s d
→ →
1 Introduction
It is well known that in the Standard Model (SM), flavour changing neutral current
processes (FCNC) such as B0 B¯0 mixing, CP violation in K ππ and rare K
d,s − d,s →
and B decays are dominated by the contributions of top quark exchanges in box and
penguin diagrams [1, 2]. This dominance originates in the large mass m of the topquark
t
and in its non-decoupling from low energy observables due to the corresponding Yukawa
coupling thatisproportionaltom . Intheevaluationofboxandpenguindiagramsinthe
t
Feynman-t’Hooft gauge this decoupling is realized through the diagrams with internal
fictitious Goldstone boson and top quark exchanges. The couplings of Goldstone bosons
to the top quark, being proportional to m , remove the suppression of the diagrams in
t
question due to top quark propagators so that at the end the box and penguin diagrams
increase with increasing m . In the unitary gauge, in which fictitious Goldstone bosons
t
are absent, this behaviour originates from the longitudinal (k k /M2 ) component of the
µ µ W
±
W –propagators.
In particular, in the case of B0 B¯0 mixing in the SM the relevant m dependent
d,s − d,s t
Inami-Lim function S(x ) S (x ) [1, 3] has the following large m behaviour [2]
t 0 t t
≡
x m2
S(x ) t, x = t . (1.1)
t → 4 t M2
W
Similarly the functions X(x ) and Y(x ), relevant for instance for K πνν¯ and B
t t d,s
→ →
µ+µ−, respectively, have the following large m behaviour
t
x x
t t
X(x ) , Y(x ) . (1.2)
t t
→ 8 → 8
Yet, with x 4.4, these formulae are very poor approximations of the true values
t
≈
S = 2.42, X = 1.54andY = 0.99. Wewill see belowthat inthecaseofthe Littlest Higgs
(LH) model, thevalue ofthecorresponding variablex isat least 400 andtheasymptotic
T
formulae presented below are excellent approximations of the exact expressions.
In the Little Higgs models [4]-[8], that offer an attractive and a rather simple solution
to the gauge hierarchy problem, there is a new very heavy top quark T. In the LH model
[7] its mass is given by
f m λ2
t 1
m = , x = . (1.3)
T v x (1 x ) L λ2 +λ2
L L 1 2
−
p
Here λ parametrize the Yukawa interactions of the top quark and v = 246GeV is the
i
vacuum expectation value of the SM Higgs doublet. The parameter x enters the sine of
L
the t-T mixing which is simply given by x v/f. The new scale f > 1 TeV is related to
L
Λ 4πf (10 TeV) at which the gauge group of the LH model [SU(2) U(1) ]
1 1
∼ ∼ O ⊗ ⊗
[SU(2) U(1) ] is broken down to the SM gauge group. The SM results for various
2 2
⊗
observables of interest receive (v2/f2) corrections that originate in new heavy gauge
O
boson and scalar exchanges and in particular in the diagrams with the heavy T. The
constraints from various processes, in particular from electroweak precision observables
and direct new particles searches, have been extensively analyzed in [9]-[16].
As already discussed in [9, 17], the parameter x describes together with v/f the size
L
of the violation of the three generation CKM unitarity and is also crucial for the gauge
interactions of the heavy T quark with the ordinary down quarks. x can in principle
L
vary in the range 0 < x < 1. For x 1, the mass m becomes large and its coupling
L L T
≈
to the ordinary W± bosons and the down quarks, W±T¯d , being (x v/f), is only
L L j O L
suppressed by v/f. In Fig. 1 we show the dependence of m on x for three values of
T L
f/v.
We are aware of the fact that with increasing m also one-loop corrections to the
T
SM Higgs mass increase. Typically for m 6 TeV a fine-tuning of at least 1% has to
T
≥
be made in order to keep m below 200GeV [11, 18]. As roughly f/v 8 is required
H
≥
by electroweak precision studies [9]-[16], the non-decoupling effects of T considered here
can be significant and simultaneously consistent with these constraints only in a narrow
range of f/v. But these bounds are clearly model dependent and we will consider the
range 5 f/v 15 and x 0.95 for completeness.
L
≤ ≤ ≤
12
10
]
V 8
e
T
[ 6 f/v=15
T
m
4 f/v=10
f/v=5
2
0
0.5 0.6 0.7 0.8 0.9
xL
Figure 1: The mass of T as a function of x for different values of f/v.
L
As with f/v 5 the T quark is by an order of magnitude heavier than the ordinary
≥
top quark, it is of interest to ask what are its effects in the FCNC processes. In the
present letter we summarize the main results of an analysis of T contributions to the
function S for x close to unity, pointing out that the existing (v2/f2) calculations
L
O
2
of the function S in the LH model [17, 19] are inadequate for the description of this
parameter region. The (v4/f4) contributions involving the T quark have to be taken
O
into account as well and in fact for x > 0.85 these (v4/f4) corrections turn out to be
L
O
as important as the genuine (v2/f2) corrections. For higher values of x they become
L
O
even dominant. The details of these investigations will appear in an update of [17].
Ontheotherhandouranalysis ofthefunctionsX andY [20]shows thattheyincrease
with x only as logx and that an (v2/f2) analysis is sufficient for the study of the
T T
O
large x behaviour. Below we will summarize the main results of this analysis. The
T
details will be presented in [20].
2 The Function S and ∆M in the LH Model
d,s
In [17] we have calculated the (v2/f2) contributions to the function S in the LH model.
O
Our results have been recently confirmed in [19]. The (v2/f2) correction ∆S can be
O
written as follows
∆S = (∆S)T +(∆S)W± +(∆S)Rest (2.1)
H
±
with the three contributions representing the T, W (new heavy charged gauge bosons)
H
and the remaining contributions that result from (v2/f2) corrections to the vertices
O
involving only SM particles.
As demonstrated in [17] the three contributions in question are given to a very good
± ±
approximation as follows (W W )
L ≡
1v2 m2
(∆S) = x2 x (logx 1.57), x = T , (2.2)
T (cid:20)2f2 L(cid:21) t T − T M2
±
W
L
c2 m2 v2
(∆S)WH± = 2s2M2t± = 2f2c4xt , (2.3)
W
H
v2
4
(∆S) = 2 c S (x ), (2.4)
Rest − f2 SM t
where x has been defined in (1.1), S(x ) = S (x ) in [17] and the numerical factor in
t t SM 0 t
(2.2) corresponds to m (m ) = 168.1GeV. The mixing parameters s and c [9] are related
t t
through s = √1 c2 with 0 < s < 1.
−
For x 0.8 considered in [17, 19] and s < 0.4 the most important contribution
L
≤
turns out to be (∆S)W±. However, for higher values of s and xL > 0.7, the contribution
H
(∆S) becomes more important.
T
We observe that all three contributions have a characteristic linear behaviour in x
t
that signals the non-decoupling of the ordinary top quark. However, the corresponding
non-decoupling of T is only logarithmic. This is related to the fact that with the W±T¯d
L j
3
couplingbeing (v/f)onlyboxdiagramswithasingleT exchange(seeFig.2)contribute
O
at (v2/f2). Similarly to the SM box diagrams with a single t exchange, that increase
O
as logx , the T contribution in the LH model increases only as logx .
t T
d b d b
W W
L L
u,c,t T T T
W W
L L
(cid:1)b d (cid:1)b d
Figure 2: Single and double heavy Top contributions to the function S in the LH model
at (v2/f2) and (v4/f4), respectively.
O O
Here we would like to point out that for x 0.85 , the term (∆S) in (2.2) does
L T
≥
not give a proper description of the non-decoupling of T. Indeed in this case also the
box diagram with two T exchanges given in Fig. 2 has to be considered. Although
formally (v4/f4), this contribution increases linearly with x and with x = (f2/v2)
T T
O O
constitutes effectively an (v2/f2) contribution.
O
Calculating the diagram with two T exchanges in Fig. 2 and adding those (v4/f4)
O
corrections from box diagrams with t, T and u quark exchanges [17], that have to be
taken into account in order to remove the divergences characteristic for a unitary gauge
calculation and for the GIM mechanism [21] to become effective, we find
v4 x v2 x3 x
(∆S) x4 T = L t . (2.5)
TT ≈ f4 L 4 f21 x 4
L
−
Formula (2.5) represents for x > 0.85 and f/v 5 the exact expression given in [17] to
L
≥
within 3% and becomes rather accurate for x > 0.90 and f/v 10.
L
≥
In fact the result in (2.5) can easily be understood. (∆S) has a GIM structure [17]
TT
v4
4
(∆S) = x [F(x ,x ;W )+F(x ,x ;W ) 2F(x ,x ;W )] (2.6)
TT f4 L T T L t t L − t T L
with the function F(x ,x ;W ) resulting up to an overall factor from box diagram with
i j L
±
two W and two quarks (i,j) exchanges. This GIM structure is identical to the one of
L
S in the SM that depends on x and x and is given by
t u
S(x ) = F(x ,x ;W )+F(x ,x ;W ) 2F(x ,x ;W ). (2.7)
t t t L u u L u t L
−
4
For large x it turns out to be a good approximation to evaluate (∆S) with x = 0.
T TT t
In this case (2.6) reduces to S(x ) in (2.7) with x replaced by x and x by x . The
t t T u t
factor x /4 in (1.1) is then replaced by x /4 as seen in (2.5).
t T
0.8 f/v =5 (∆S)TT 0.2 f/v =10 (∆S)TT
0.6 0.15
(∆S)T
∆S)i 0.4 (∆S)WH± (∆S)T ∆S)i 0.1 (∆S)WH±
( (
0.2 0.05
0 0
(∆S)Rest (∆S)Rest
0.2 0.05
− 0.5 0.6 0.7 0.8 0.9 − 0.5 0.6 0.7 0.8 0.9
xL xL
Figure 3: The contributions (∆S) as functions of x for f/v = 5, f/v = 10 and
i L
s = √1 c2 = 0.2.
−
In Fig. 3 we show the four contributions (∆S) as functions of x for f/v = 5 and
i L
f/v = 10 and s = 0.2. We observe that for x < 0.85 the new contribution is smaller
L
than (∆S) but for x > 0.90 it becomes a significant new effect in S.
T L
A general upper bound on the function S in models with minimal flavour violation
(MFV) can be obtained from the usual analysis of the unitarity triangle [22]. It is valid
also in the LH model considered here. A recent update [23] of this bound gives
S 3.3, (95% C.L.) (2.8)
≤
to be compared with S = 2.42 in the SM [2]. In Fig. 4 we plot
SM
SLH = SSM(xt)+(∆S)TT +(∆S)T +(∆S)W± +(∆S)Rest (2.9)
H
as a function of x for different values of f/v and s = 0.2. We also show there the SM
L
value and the upper bound in (2.8). For a comparison we recall that from the studies
of the ρ parameter an upper bound on x of 0.95, almost independently of f/v, has
L
been obtained [13]. We observe that for f/v = 5 a bound x 0.90 at 95%C.L. can
L
≤
be obtained. Weaker bounds are found for 5 < f/v < 10 and in order to find a bound
stronger than in [13] for f/v > 10, S should be within 10% of the SM value. This
max
would require significant reduction of the theoretical uncertainties in the analysis of the
unitarity triangle.
With the precisely measured ∆M and V V , the observed enhancement of the
d ts cb
| | ≈ | |
5
f/v = 5
4.0
6
3.5 Smax
H 8
SL3.0 10
15
2.5 SM
0.6 0.7 0.8 0.9
xL
Figure 4: S as a function of x for different values of f/v, s = 0.2. The dashed line is
LH L
the bound in (2.8).
function S implies
( V ) S (∆M ) S
td LH SM s LH LH
| | = , = , (2.10)
( V ) rS (∆M ) S
td SM LH s SM SM
| |
that is the suppression of V and the enhancement of ∆M . As we will see below in
td s
| |
the context of K πνν¯ and B µ+µ− decays, the suppression of V compensates
d td
→ → | |
to a large extent the enhancements of the functions X and Y in the relevant branching
ratios. On the other hand, ∆M involving V and being proportional to S , is for
s ts LH
| |
x = 0.95 enhanced by 56%, 15% and 7% for f/v = 5,10,15, respectively.
L
3 Rare K and B Decays in the LH Model
The analysis of the functions X and Y is much more involved due to many box and
penguin diagrams with T and new heavy gauge boson Z0, A0 and W± exchanges. The
H H H
details of this analysis in the full space of the parameters involved will be presented in
[20]. Here we present only the results for x > 0.7, where the diagrams with the heavy
L
T and ordinary SM particles shown in Fig. 5 become dominant.
Due to a different topology of penguin diagrams and the fact that in the relevant
box diagrams with νν¯ and µµ¯ in the final state only a single T can be exchanged, the
(v2/f2) diagrams in Fig. 5 give an adequate description of the x > 0.7 region. The
L
O
diamonds in this figure indicate (x2v2/f2) corrections to the SM vertices that are
O L
explicitely given in [9, 17]. Many of the diagrams in this figure give in the unitary gauge
contributions to X and Y that grow as x and x logx but due to the GIM mechanism
T T T
all these contributions cancel each other in the sum. In particular the penguin diagram
involving two T propagators, being proportionalto sin2θ , is canceled by other diagrams
w
6
involving sin2θ so that sin2θ does not appear in the final result for X and Y as it
w w
should be.
s d s d s d
W t W
L L
t
t t W W
L L d
ZL ZL ZL
(cid:1)ν ν (cid:1)ν ν (cid:1)ν ν
d ν s d s d
W T
L
W
L
T T W W
L L
t l
−
Z Z
L L
W
L
(cid:1)s ν (cid:1)ν ν (cid:1)ν ν
s d s d d ν
W W
L L
W
L
T
T t
d
T l
−
Z Z
L L
W
L
(cid:1)ν ν (cid:1)ν ν (cid:1)s ν
Figure 5: Top and heavy top quark contributions to the function X in the LH model
at (v2/f2) which are proportional to x2.
O L
We find then that to a very good approximation the corrections to X and Y in the
LH model are given in the x > 0.7 region as follows
L
v2 x 3
2 t
∆X = x ( + )logx 3.32 , (3.1)
(cid:20)f2 L(cid:21)(cid:20) 4 8 T − (cid:21)
v2 x 3
2 t
∆Y = x ( + )logx 3.53 , (3.2)
(cid:20)f2 L(cid:21)(cid:20) 4 8 T − (cid:21)
with the numerical factors corresponding to m (m ) = 168.1GeV. Exact formulae will
t t
be presented in [20]. A characteristic x -decoupling is observed with a logarithmic de-
t
pendence on x . The combined dependence on x and x in this leading contribution
T t T
7
makes it clear from which diagrams in Fig. 5 this leading contribution comes from. This
is the Z0-penguin diagram with both T and t exchanges and the corresponding diagram
with t and T.
f/v = 5 1.3 f/v = 5
1.8
1.2
XLH 1.7 10 YLH1.1 10
15
1.6 15 1.0 SM
SM
0.6 0.7 0.8 0.9 0.6 0.7 0.8 0.9
xL xL
Figure 6: X and Y as functions of x for different values of f/v.
LH LH L
In Fig. 6 we show
X = X +∆X, Y = Y +∆Y (3.3)
LH SM LH SM
asfunctionsofx forthreevaluesoff/v. Weobservethatduetothefactthat∆X ∆Y
L
≈
but X 1.6 Y , the relative corrections to Y are larger.
SM SM
≈
The branching ratios for K π0νν¯ and B µ+µ− are proportional to X2 and
L s,d LH
→ →
Y2 , respectively. The branching ratio for K+ π+νν¯ grows slower with X due to
LH LH
→
an additive charm contribution that is essentially unaffected by the LH contributions.
Now, K π0νν¯, K+ π+νν¯ and B µ+µ− involve the CKM element V , whose
L d td
→ → →
value is decreased in the LH model as discussed in the previous section. Consequently
the enhancement of X and Y in the LH model is compensated by the suppression of V
td
at the level of the branching ratios. The situation is quite different in B µ+µ−, where
s
→
the relevant CKM element V is approximately equal to V and the enhancement of
ts cb
| | | |
Y is fully visible in the branching ratio. Explicitly we have
Br(K π0νν¯) S X2
L LH SM LH
→ = , (3.4)
Br(K π0νν¯) S X2
L SM LH SM
→
Br(B µ+µ−) S Y2
d LH SM LH
→ = , (3.5)
Br(B µ+µ−) S Y2
d SM LH SM
→
Br(B µ+µ−) Y2
s LH LH
→ = . (3.6)
Br(B µ+µ−) Y2
s SM SM
→
8
This pattern is clearly seen in Table 1. The values given there have been obtained
by using the MFV formulae for branching ratios given in [24] with F = 203MeV,
Bd
F = 238MeV and setting the three CKM parameters V , V and the angle β in
Bs | us| | cb|
the unitarity triangle to
◦
V = 0.224, V = 0.0415, β = 23.3 . (3.7)
us cb
| | | |
The fourth parameter, the UT side R is then calculated according to
t
S
SM
(R ) = (R ) , (R ) = 0.89 (3.8)
t LH t SMrS t SM
LH
with (R ) = 0.89 being the central value of the UT fit in [25].
t SM
We observe that Br(K+ π+νν¯) and Br(K π0νν¯) are very close to the SM
L
→ →
value and with increasing x they are even slightly suppressed. Br(B µ+µ−) is
L d
→
slightly enhanced with respect to the SM, while the enhancement of Br(B µ+µ−)
s
→
for f/v 10 is significant.
≤
Table 1: Branching ratios for rare decays in the LH model and the SM for f/v = 5. In the
last two rows the results for f/v = 10 are given.
x 0.80 0.90 0.95 SM
L
Br(K+ π+νν¯) 1011 7.91 7.78 7.34 7.88
→ ×
Br(K π0νν¯) 1011 3.07 3.00 2.76 3.05
L
→ ×
Br(B µ+µ−) 1010 1.32 1.34 1.27 1.20
d
→ ×
Br(B µ+µ−) 109 5.17 5.81 6.38 3.86
s
→ ×
R 39.2 43.4 50.4 32.2
sd
Br(B µ+µ−) 109 4.27 4.44 4.58 3.86
s
→ ×
R 34.3 35.4 37.2 32.2
sd
The branching ratios Br(B µµ¯) and ∆M are subject to uncertainties in V
s,d s,d td
→ | |
and the meson decay constants F . On the other hand the ratio
Bs,d
Br(B µµ¯)
s
R = → (3.9)
sd
Br(B µµ¯)
d
→
is theoretically cleaner. We show this ratio in Table 1.
4 Conclusions
We have calculated the dominant (v4/f4) corrections to the function S in the LH
O
model. Due to a large value of m , this contribution can compete with the genuine
T
9
