arXiv:0704.0601v4  [hep-ph]  24 Oct 2007
D − D̄ mixing and rare D decays in the Littlest Higgs
model with non-unitarity matrix
Chuan-Hung Chen1,2∗, Chao-Qiang Geng3,4† and Tzu-Chiang Yuan3‡
1Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan
2National Center for Theoretical Sciences, Hsinchu 300, Taiwan
3Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan
4Theory Group, TRIUMF, 4004 Wesbrook Mall,
Vancouver, B.C. V6T 2A3, Canada
(Dated: October 27, 2018)
Abstract
We study the D − D̄ mixing and rare D decays in the Littlest Higgs model. As the new weak
singlet quark with the electric charge of 2/3 is introduced to cancel the quadratic divergence induced
by the top-quark, the standard unitary 3× 3 Cabibbo-Kobayashi-Maskawa matrix is extended to
a non-unitary 4× 3 matrix in the quark charged currents and Z-mediated flavor changing neutral
currents are generated at tree level. In this model, we show that theD−D̄ mixing parameter can be
as large as the current experimental value and the decay branching ratio (BR) of D → Xuγ is small
but its direct CP asymmetry could be O(10%). In addition, we find that the BRs of D → Xuℓ+ℓ−,
D → Xuνν̄ and D → µ+µ− could be enhanced to be O(10−9), O(10−8) and O(10−9), respectively.
∗ Email: physchen@mail.ncku.edu.tw
† Email: geng@phys.nthu.edu.tw
‡ Email: tcyuan@phys.nthu.edu.tw
http://arxiv.org/abs/0704.0601v4
I. INTRODUCTION
As the observation of the Bs − B̄s mixing in 2006 by CDF [1], all neutral pseudoscalar-
antipseudoscalar oscillations (P − P̄ ) in the down type quark systems have been seen. In
the standard model (SM), the most impressive features of flavor physics are the Glashow-
Iliopoulos-Maiani (GIM) mechanism [2] and the large top quark mass. The former results
in the cancellation between the lowest order short-distance (SD) contributions of the first
two generations to the mass difference ∆mK in the K
0 system, while the latter makes ∆mBq
(q = d, s) in the Bq systems dominated by the SD effects [3]. In addition, these features also
lead to sizable flavor changing neutral currents (FCNCs) from box and penguin diagrams,
which contribute to the rare decays, such as K → πνν̄ and B → K(∗)ℓℓ̄. It is known that
these processes could be good candidates to probe new physics effects [4–6]. However, it is
clear that the new physics signals deviated from the SM predications for the P − P̄ mixings
and rare FCNC decays have to wait for precision measurements on these processes.
Unlike K and Bq systems, the SD contributions to charmed-meson FCNC processes,
such as the D − D̄ mixing [7] and the decays of c → uℓ+ℓ− and D → ℓ+ℓ− [8], are highly
suppressed due to the stronger GIM mechanism and weaker heavy quark mass enhancements
in the loops. On the other hand, it is often claimed that the long-distance (LD) effect for
the D−D̄ mixing should be the dominant contribution in the SM. Nevertheless, because the
nonperturbative hadronic effects are hard to control, the result is still inconclusive [9–12].
Recently, BABAR [13] and BELLE [14, 15] collaborations have reported the evidence for
the D − D̄ mixing with
x′2 = (−0.22± 0.30± 0.21)× 10−3 ,
y′ = (9.7± 4.4± 3.1)× 10−3 , (1)
x ≡ ∆mD
= (0.80± 0.29± 0.17)% ,
y ≡ ∆ΓD
= (0.33± 0.24± 0.15)% ,
yCP = (1.31± 0.32± 0.25)% , (2)
respectively, where x′ = x cos δ + y sin δ and y′ = −x sin δ + y cos δ with the assumption
of CP conservation and δ being the relative strong phase between the amplitudes for the
doubly-Cabbibo-suppressed D → K+π− and Cabbibo-favored D → K−π+ decays [16, 17]
and yCP = τ(D → K−π+)/τ(D → K+K−) − 1. Moreover, no evidence for CP violation is
found. The combined results of Eqs. (1) and (2) at the 68% C.L. are [18]
x = (5.5± 2.2)× 10−3 ,
y = (5.4± 2.0)× 10−3 ,
δ = (−38± 46)0 . (3)
Note that the upper bound of x < 0.015 at 95%C.L. can be extracted from the BELLE
data in Eq. (2) [14, 15]. The evidences of the mixing parameters by BABAR and BELLE
collaborations reveal that the era of the rare charmed physics has arrived. The results in
Eq. (3) can not only test the SU(3) breaking effects for the D− D̄ mixing [10, 12], but also
examine new physics beyond the SM [17–21].
It is known that a straightforward way to enhance the rare D processes is to include some
new heavy quarks within the framework of the SM. For instance, if a new heavy quark with
the electric charge of −1/3 is introduced, it could affect the D system since the extra down
type quark violates the GIM mechanism. However, the constraint on this heavy quark is
quite strong as it could also lead to FCNCs for the down type quark sector at tree level,
which are strictly limited by the well measured rare K and B decays, such as KL → µ+µ−
and B → Xsγ [22]. On the other hand, if the charge of the new heavy quark is 2/3, it could
generate interesting tree level FCNCs for the up type quark sector, for which the constraints
are much weaker. In this paper, we will study D physics based on a new weak singlet upper
quark.
It has been known that in the framework of the Littlest Higgs model [23], there exists
a new SU(2)L singlet vector-like up quark [24], hereafter denoted by T . Since the number
of down type quarks is the same as that in the SM, the standard unitary 3 × 3 Cabibbo-
Kobayashi-Maskawa (CKM) matrix is extended to a non-unitary 4×3 matrix in the charged
currents. Moreover, Z-mediated FCNCs for the up quark sector are generated at tree level.
In Ref. [25], it has been shown that the contributions of this new quark to the rare D
processes are small and cannot reach the sensitivities of future experiments [25, 26]. In this
paper, we will demonstrate that by adopting some plausible scenario, the effects could not
only generate a large D − D̄ oscillation but also marginally reach the sensitivity proposed
by BESIII for the rare D decays [27]. We note that the implication of the new data on the
D − D̄ mixing in the Littlest Higgs model with T-parity has been recently studied in Ref.
[21].
The paper is organized as follows. In Sec. II, we investigate that when a gauge singlet T -
quark is introduced in the Littlest Higgs model, how the non-unitary matrix for the charged
current and the tree level Z-mediated FCNC are formed. By using the leading perturbation,
the mixing matrix elements related to the new parameters in the Littlest Higgs model are
derived. In addition, we study how to get the small mixing matrix element for Vu(c)b, which
describes the b → u(c) decays. In Sec. III, we discuss the implications of the non-unitarity
on the D− D̄ mixing and rare D decays by presenting some numerical analysis. Finally, we
summarize our results in Sec. IV.
II. NON-UNITARY MIXING MATRIX IN THE LITTLEST MODEL
To study the new flavor changing effects in the Littlest Higgs model, we start by writing
the Yukawa interactions for the up quarks to be [24, 25]
λabfǫijkǫxyχaiΣjxΣkyu
b + λ0fTT
c + h.c. , (4)
where χT1 = (d1, u1, 0), χ
2 = (s2, c2, 0), χ
3 = (b3, t3, T ), u
b is the weak singlet and Σ =
eiΠ/fΣ0e
iΠT /f with
112×2
112×2
, Π =
2 h∗/
φ hT/
. (5)
The scale f denotes the global symmetry spontaneously breaking scale, which, as usual,
could be around 1 TeV. Consequently, the 4× 4 up-quark mass matrix is given by [25]
iλijv
− − − − −
0 0 λ33f | λ0f
. (6)
We remark that the quadratic divergences for the Higgs mass from one-loop diagrams in-
volving t and T get exactly cancelled as shown in Ref. [28]. Moreover, for other quarks
other than the top quark, the one-loop quadratic divergent contributions do not necessitate
fine-tuning the Higgs potential as the cutoff is around 10 TeV for f ∼ 1 TeV due to the small
corresponding Yakawa couplings. That is why there is no need to introduce extra singlet
states T [28, 29].
To obtain the quark mass hierarchy of mt ≫ mc ≫ mu, we can choose a basis such that
the up-quark mass matrix is [30]
m̂U 0
hf λ0f
 (7)
where m̂Uij = δijλiv/
2 ≡ mi is diagonal matrix and h = (h1, h2, h3). The hi is related to
λ33 by hi = Ṽ
i3 λ33 and hh
† = |λ33|2, in which Ṽ UR is the unitary transformation for the
right-handed up quarks. We note that mi are not the physical masses and in principle their
magnitudes could be as large as the weak scale. In order to preserve the hierarchy in the
quark masses, one expects that m3 > m2 > m1. Furthermore, in terms of this basis, the
charged and neutral currents, defined by
LC = g√
J−µ W
+µ − g√
2 tan θ
J−µ W
H + h.c. ,
cos θW
3 − sin2 θWJµem
tan θ
3 ZHµ + h.c. , (8)
are expressed by
J−µ = ŪLγµṼ
0aVDL ,
µṼ 0aV Ṽ
0†UL −
µDL , (9)
respectively, where UT = (u1, c2, t3, T ), D
T = (d, s, b), aV = diag(1, 1, 1, 0) and
Ṽ 0 =
(V 0CKM)3×3 0
 (10)
with V 0CKMV
CKM = 113×3. The null entry in aV denotes the new T -quark being a weak
singlet; and without the new T -quark, V 0CKM is just the CKM matrix. Since the down quark
sector is the same as that in the SM, we have set the unitary transformation UDL to be an
identity matrix.
For getting the physical eigenstates, the mass matrix in Eq. (7) can be diagonalized by
unitary matrices V UL,R so that we have
†diag
U = V
ULMUM
L (11)
m̂Um̂
f (|λ33|2 + |λ0|2)f 2
 . (12)
Since (|λ33|2 + |λ0|2)f 2 is much larger than other elements, we can take the leading order
of the perturbation in himi/f (i = 1, 2, 3). According to Eq. (11), the leading expansion is
given by
†diag
U = V
ULMUM
L ≈ (1 + ∆L)MUM †U (1−∆L) . (13)
By looking at the off-diagonal terms (M
†diag
U )i4(4i), we can easily get
∆Li4 ≈ −∆L4i = −
himif
(|λ33|2 + |λ0|2)f 2 −m2i
with i 6= 4. From the diagonal entries, if we set the light quark masses to be mu ≈ mc ≈ 0,
we obtain
0 ≈ m2uj ≈ m
j + 2∆Lj4(MUM
U)4j ,
∆Lj4 ≈ −
with j = 1, 2. To be consistent with Eq. (14), at the leading expansion the relation
2h2j = (|λ33|2 + |λ0|2) (16)
should be satisfied. We emphasize that the choice of Eq. (16) is somewhat fine-tuned in
order to have Eqs. (14) and (15) simultaneously. Since the top-quark is much heavier than
other ordinary quarks, we have 2h23 ≈ (1−m2t/m23)(|λ33|2+ |λ0|2) if f > m3 > mt. Similarly,
one obtains the flavor mixing effects for i 6= j 6= 4 to be
∆Lij =
hihjmimj
m2i −m2j
f 2[2(|λ33|2 + |λ0|2)f 2 − (m2i +m2j )]
(|λ33|2 + |λ0|2)f 2 −m2j
((|λ33|2 + |λ0|2)f 2 −m2i )
. (17)
After diagonalization, the currents become
J−µ = ŪLγµV
ULṼ 0aVDL = ŪLγµV
ULV 0DL ,
3 = ŪLγ
µV ULṼ 0aV Ṽ
0†V †ULUL = ŪLγ
µV ULV 0V 0†V †ULUL , (18)
where UT = (u, c, t, T ),
V 0 = Ṽ 0aV =
(V 0CKM)3×3 0
 (19)
and diag(V 0V 0†) = aV . Since the 4-th component of aV is different from the first 3 ones, it
is obvious that the matrix V ≡ V ULV 0 associated with the charged current does not satisfy
unitarity. In addition, V ULaV V
L, which is associated with the neutral current, is not the
identity matrix. As a result, Z-mediated FCNCs at tree level are induced. According to
Eq. (18), we see that
V V † = V ULaV V
L (20)
which is just the same as the effects of the Z-mediated FCNCs. Due to V being a non-unitary
matrix, one finds
(V V †)ij = Vi4V
j4 . (21)
Consequently, the interesting phenomena arising from non-unitary matrix elements are al-
ways related to Vi4V
j4 = ∆Li4∆j4. We note that as we do not particularly address CP
problem, in most cases, we set the parameters to be real numbers.
It has been known that enormous data give strict bounds on the flavor changing effects.
In particular, the pattern describing the charged current has been fixed quite well. Any
new parametrization should respect these constraints. It should be interesting to see the
relationship with and without the new vector-like T -quark. From Eq. (18), we know that
the new flavor mixing matrix for the charged current is given by V = V ULV 0. At the leading
order perturbation, one gets
V = V ULV 0 ≈ (1 + ∆L)V 0 = V 0 +∆LV 0 . (22)
If V 0tb ∼ 1 is taken, one finds that Vub ≈ V 0ub+∆L13 and Vcb ≈ V 0cb+∆L23. In terms of Eq. (17)
and h1 = h2 ≈ h3, the relations ∆L13 ≈ −m1/m3 and ∆L23 ≈ −m2/m3 are obtained. Hence,
in our approach, we have
Vus ≈ V 0us −
, Vub ≈ V 0ub −
, Vcb ≈ V 0cb −
. (23)
From these results, it is clear that when the T -quark decouples from ordinary quarks,
Vus → V 0us, Vub → V 0ub and Vcb → V 0cb, while m1/m2 → mu/mc, m1/m3 → mu/mt and
m2/m3 → mc/mt, respectively. According to the observations in the decays of b → uℓν̄ℓ and
b → cℓν̄ℓ, the corresponding values have been determined to be |Vub| = 3.96 ± 0.09 × 10−3
and |Vcb| = 42.21+0.10−0.80 × 10−3, respectively [16]. Since V 0ij and mi are free parameters, to
satisfy the experimental limits with interesting phenomena in low energy physics, it is rea-
sonable to set the orders of magnitude for m1/m3 and m2/m3 (m1/m2) to be O(10
−2) and
O(10−1), respectively. Consequently, the non-unitary effects on the rare charmed meson
decays governed by V14V
24 could be as large as ∆14∆24 ∼ O(10−4), which could be one order
of magnitude larger than those in Ref. [25].
III. D − D̄ MIXING AND RARE D DECAYS
A. D − D̄ mixing
It is well known that the GIM mechanism has played an important role in the K − K̄
oscillation in the SM. In addition, due to the top-quark in the box and penguin diagrams,
Bq − B̄q mixings are dominated by the SD effects, which are consistent with the data [16].
On the contrary, for the D− D̄ mixing the GIM cancellation further suppresses the mixing
effect to be ∆mD ∼ O(m4s/m2Wm2c) [7] and the bottom quark contribution actually is a
subleading effect due to the suppression of (VubV
2. In the SM, the SD contribution to the
mixing parameter is O(10−7) [34]. However, the LD contribution to the mixing is believed
to be dominant. Due to the nonperturbative hadronic effects, the result is still uncertain
with the prediction on the mixing parameter ranging from O(10−3) [9] to O(10−2) [10–12].
Nonetheless, the mixing parameters shown in Eq. (3) could arise from the LD contribution.
Thus, it is important to have a better understanding of the LD effect. On the other hand, it
is also possible that the mixings in Eq. (3) could result from new physics. In the following,
we will concentrate on the Littlest Higgs model.
In the quark sector of the Littlest Higgs model due to the introduction of a new weak
singlet, a direct impact on the low energy physics is the FCNCs at tree level. According to
Eq. (18), the most attractive process with |∆C| = 2 via the Z-mediated c−u−Z interaction,
illustrated in Fig. 1, is given by
H(|∆C| = 2) = g
(V14V
ūγµPLc ūγ
µPLc ,
(V14V
ūγµPLc ūγ
µPLc . (24)
In terms of the hadronic matrix element, defined by
〈D̄|(ūc)V−A (ūc)V−A|D〉 =
D , (25)
FIG. 1: Z-mediated flavor diagram with |∆C| = 2.
the mass difference for the D meson is [25]
∆mD ≈
DmDBD|(V14V ∗24)2| . (26)
If we assume no cancellation between new physics and SM contributions, by taking τD =
1/ΓD = 6.232 × 1011 GeV−1, fD
BD = 200 MeV [31, 32] and mD = 1.86 GeV and using
Eq. (3), we obtain
ζ0 ≡ |V14V ∗24| = |∆L14∆L24| = (1.47± 0.29)× 10−4 , (27)
which is in the desirable range. In other words, the result in Eq. (27) demonstrates that the
non-unitarity in the Littlest Higgs model could enhance the D − D̄ mixing at the observed
level. We note that the limit of x < 0.015 (95%C.L.) leads to
ζ0 < 2.5× 10−4. (28)
In addition, we note that cancellation between the LD effect in the SM and the SD one from
new physics could happen. In this case, the values in Eqs. (27) and (28) could be relaxed.
B. D → Xuγ decay
In the SM, the D-meson FCNC related processes are all suppressed since the internal
fermions in the loops are all much lighter than mW . For the decay of D → Xuγ, without
QCD corrections, the branching ratio is O(10−17); and it becomes O(10−12) when one-loop
QCD corrections are included [8]. However, it is found that the two-loop QCD corrections
can boost the BR to be as large as 3.5× 10−8 [33]. It should be interesting to see how large
the non-unitarity effect on c → uγ is in the Littlest Higgs model.
To study the radiative decay of c → uγ, we write the effective Lagrangian to be
Lc→uγ = −
mcūσµνPRcF
µν , (29)
where C7 = C
7 + C
7 and C
7 ≈ −(0.007 + i0.02) = 0.021eiδs with δs = 70.7◦ [33]
being the strong phase induced by the two-loop QCD corrections. In the extension of the
SM by including a weak singlet particle, the flavor mixing matrix in the charged current
is not unitary and the Z-mediated FCNC at tree level is generated as well. For c → uγ,
besides the QED-penguin diagrams induced by the W -boson displayed in Figs. 2a and 2b,
the Z-mediated QED-penguin one in Fig. 2c will also give contributions. We note that the
d, s, b
W d, s, b
u, c, T
(a) (c)(b)
FIG. 2: Flavor diagrams for c → uγ.
contributions from WH and ZH can be ignored as m
and m2Z/m
are much less
than one.
At the first sight, due to the light quarks in the loops, the contributions from Figs. 2a
and 2b could be negligible. However, due to the non-unitarity of (V V †)uc = V14V
24 6= 0,
even in the limits of md, s, b → 0, the contributions from the mass independent terms do not
vanish anymore and can be sizable. In terms of unitary gauge [22], we obtain
CW7 =
(V V †)12
VusV ∗cs
VusV ∗cs
. (30)
Furthermore, if we set mu ≈ mc = 0, the contributions from Fig. 2c are given by
CZ7 = (f
u + f
c + f
T )/VusV
fZc + f
− eu sin2 θW
[4ξ0(0)− 6ξ1(0) + 2ξ2(0)]
+eu sin
2 θW [4ξ0(0)− 4ξ1(0)]
fZT =
eu [2ξ0(yT )− 3ξ1(yT ) + ξ2(yT )] , (31)
where the functions ξn(x) are defined by
ξn(x) ≡
zn+1dz
1 + (x− 1)z
and yT = mT/mZ . Numerically, the total contribution in Fig. 2 is
C7 = C
7 + C
V ∗csVus
24 . (33)
If we regard V14V
24 as an unknown complex parameter, i.e. V14V
24 = ζ0e
iθ with θ being
the CP violating phase, one can study the decay BR and direct CP asymmetry (CPA) of
D → Xuγ defined by
BR(D → Xuγ) =
6αem|C7|2
π|Vcd|2
BR(D → Xdeν̄e) ,
ACP =
BR(c̄ → ūγ)− BR(c → uγ)
BR(c̄ → ūγ) + BR(c → uγ)
, (34)
as functions of ζ0 and θ. In Fig. 3, the BR and CPA as functions of ζ0 are presented,
where the solid, dotted, dashed and dash-dotted lines represent the CP violating phase at
θ = 0, 45◦, 90◦ and 135◦, respectively. From these results, it is interesting to see that
0 1 2 3 4 5 6
0 1 2 3 4 5 6
(a) (b)
FIG. 3: BR (in units of 10−8) and CPA (in units of 10−2) for D → Xuγ as functions of ζ0, where
the solid, dotted, dashed and dash-dotted lines represent the CP violating phase at θ = 0, 45◦, 90◦
and 135◦, respectively.
BR(D → Xuγ) is insensitive to the new physics effects, whereas the direct CPA could be as
large as O(10%). Explicitly, if we take θ = 90◦ and ζ0 = 1.5 × 10−4, the CPA is about 3%.
Note that this CPA vanishes in the SM.
C. D → Xuℓℓ̄ and D0 → ℓ+ℓ− decays
Because the current experimental measurements in K and Bq decays are all consistent
with the SM predictions, it is inevitable that if we want to observe any deviations from the
SM, we have to wait for precision measurements for K and Bq. SuperB factories or LHCb
could provide a hope. However, the situation in D physics is straightforward. As stated
before, unlike K and Bq systems, due to no heavy quark enhancement in the D system, the
rare D-meson decays, such as D → Xuℓℓ̄ (ℓ = e, µ, ν), are always suppressed. Even by
considering the long-distance effects, the related decays, such as D → µ+µ− and D → Xuνν̄,
get small corrections to the SD predictions on the BRs [35]. Therefore, these rare decays
definitely could be good candidates to probe the new physics effects. Since the values in the
SM are hardly reachable at D factories [27], if any exotic event is found, it must be a strong
evidence for new physics. In the following analysis, we are going to discuss the implication
of the Littlest Higgs model on the rare D decays involving di-leptons.
To study these decays, we first write the effective Hamiltonian for c → uℓ+ℓ− (ℓ = e, µ)
H(c → uℓ+ℓ−) = −
GFαem√
V ∗csVus
Ceff9 O9 + C7O7 + C10O10
, (35)
O7 = −
ūiσµνq
νPRcℓ̄γ
O9 = ūγµPLc ℓ̄γ
O10 = ūγµPLc ℓ̄γ
µγ5ℓ , (36)
where the effective Wilson coefficients are given by
Ceff9 =
(V V †)14
V ∗csVus
cℓV + (h(zs, s)− h(zd, s)) (C2(mc) + 3C1(mc)) ,
C10 = −
(V V †)14
V ∗csVus
cℓA , (37)
with s = q2/m2c , zi = mi/mc, c
V = −1/2 + 2 sin2 θW , cℓA = −1/2 and
h(z, s) = −4
ln z +
(2 + x)
|1− x|
1−x+1√
1−x−1
− i π, for x ≡ 4z2/s < 1 ,
2 arctan 1√
x−1 , for x ≡ 4z
2/s > 1
. (38)
Here, we have neglected the small contributions from the penguin and box diagrams. We
note that in the SM, the SD contributions are mainly from the term with h(z, s), induced
by the insertion of O2 = ūLγµqLq̄Lγ
µcL and mixing with O9 at one-loop level [35, 36].
We note that the resonant decays of D → XuV → Xuℓ+ℓ− (V = φ, ρ, ω) would have
large corrections to c → uℓ+ℓ− at the resonant regions. However, in this paper we do
not discuss these contributions as we only concentrate on the SD contributions. Moreover,
these resonance contributions can be removed by imposing proper cuts in the phase space
in dedicated searches.
From Eq. (35), the decay rate for D → Xuℓ+ℓ− as a function of the invariant mass
s = q2/m2c can be found to be
768π5
|VusV ∗cs|2(1− s)2R(s) ,
R(s) =
|Ceff9 |2 + |C10|2
(1 + 2s) + 12Re(C∗7C
9 ) + 4
|C7|2 . (39)
In addition, by utilizing the lepton angular distribution, we can also study the forward-
backward asymmetry (FBA), given by
−1 d cos θdΓ/dsd cos θ sgn(cos θ)
−1 d cos θdΓ/dsd cos θ
Ceff9 +
, (40)
where θ is the angle of ℓ+ related to the momentum of the D meson in the ℓ+ℓ− invariant
mass frame. Since C10 is small in the SM, AFB is negligible. With mc = 1.4 GeV and the
mixing parameter in Eq. (27), we get
BR(D → Xue+e−) = (4.18± 0.91)× 10−10 ,
BR(D → Xuµ+µ−) = (2.51± 0.86)× 10−10 , (41)
comparing with the SM predictions of BR(D → Xue+e−)SM = 2.1 × 10−10 and BR(D →
+µ−)SM = 0.5 × 10−10, respectively. Clearly, if some cancellation occurs between new
physics and SM contributions in the D − D̄ mixing, a larger value of ζ0 can be allowed. In
Fig. 4, we show the tendency of the decay as a function of ζ0, where the negative horizontal
values correspond to -ζ0. In addition, we present the differential decay BR [FBA] of
D → Xue+e− as a function of s = q2/m2c in Fig. 5a [b], where the thick solid, dotted and
dashed lines correspond to ζ0 = 1.5, 3.0 and 5.0, while the thin ones denote the cases for
−ζ0 except ζ0 = 0 for the thin solid line in Fig. 5a. From Fig. 5b, we see that the FBA
-6 -4 -2 0 2 4 6
FIG. 4: BR(in units of 10−9) for D → Xue+e− as a function of ζ0.
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
(a) (b)
FIG. 5: (a)[(b)] Differential BR (in units of 10−9) [FBA] for D → Xue+e− as a function of s, where
the thick solid, dotted and dashed lines correspond to ζ0 = 1.5, 3.0 and 5.0, while the thin ones
denote the cases for −ζ0 except ζ0 = 0 for the thin solid line in (a).
is only at percent level. In the Littlest Higgs model, this is because the Z coupling to the
charged lepton cℓV = −1/2 + 2 sin2 θW appearing in Ceff9 is much smaller than one. This is
quite different from that in b → sℓ+ℓ− where the dominant effect in the SM for the FBA is
from the box and QED-penguin diagrams.
Next, we discuss the decay of D → Xuνν̄. In the SM, the BR for D → Xuνν̄ is estimated
to be O(10−16) − O(10−15) [35], which is vanishing small. In the Littlest Higgs model, by
taking C7 = 0, C
9 = −C10 = −π(V V †)14/(αemVusV ∗cs), the effective Hamiltonian in Eq. (35)
can be directly applied to c → uνν̄. The decay rate forD → Xuνν̄ as a function of s = q2/m2c
can be obtained as
768π5
(1− s)2(1 + 2s)
2π2|(V V †)12|2
, (42)
where the factor of 3 stands for the neutrino species. With ζ0 = 1.5×10−4, we get BR(D →
Xuνν̄) = 1.31 × 10−9. However, if we relax the constraint on V14V †24, the BR as a function
of ζ0 is shown in Fig. 6a. For a larger value of ζ0, BR(D → Xuνν̄) could be as large as
O(10−8).
0 1 2 3 4 5 6
0 1 2 3 4 5 6
(a) (b)
FIG. 6: (a) BR (in units of 10−8) for D → Xuνν̄ and (b) BR (in units of 10−9) for D → µ+µ−.
Finally, we study the decays of D → ℓ+ℓ−. It has been known that, in the SM, the SD
contributions to D → µ+µ− are O(10−18), while the LD ones are O(10−13) [35]. It is clear
that any signal to be observed at the sensitivity of the proposed detector, such as BESIII,
will indicate new physics effects. Since the effective interactions at quark level are the same
as those in Eq. (35), one finds that
BR(D → ℓ+ℓ−) =
τDmDm
|πV14V ∗24|2 . (43)
Here we have used equation of motion for the charged lepton so that ℓ̄/pDℓ = 0. We also
note that operators O7,9 make no contributions. With |V14V ∗24| = ζ0 = 1.5 × 10−4, the
predicted BR for D → µ+µ− is 1.17× 10−10. In Fig. 6b, we present the BR as a function of
ζ0. We see that BR(D → µ+µ−) in the Littlest Higgs model could be as large as O(10−9).
IV. CONCLUSIONS
We have studied the D − D̄ mixing and rare D decays in the Littlest Higgs model. In
the model, as the new weak singlet vector-like quark T with the electric charge of 2/3
is introduced to cancel the quadratic divergence induced by the top-quark, the standard
unitary 3× 3 CKM matrix is extended to a non-unitary 4× 3 matrix in the quark charged
currents and Z-mediated flavor changing neutral currents are generated at tree level. We
have shown that the effects on |∆C| = 2 and |∆C| = 1 processes are all related to V14V ∗24 in
Eq. (21).
To avoid the scenario adopted by Ref. [25], in which λ0 ∼ λ33 ≫ λij was assumed, we
choose the basis such that the effective mass matrix for u1, c2 and t3 is diagonal, while
the corresponding masses m1, m2 and m3 are free parameters and can be as large as the
weak scale v. Since the global symmetry breaking scale f is larger than v, the mixing
matrix relating physical and unphysical states could be extracted by taking the leading
perturbative expansion. Accordingly, by using the approximation of mu ≈ mc ≈ 0, the
explicit expressions for V14 and V24 have been obtained. In terms of the data for Vub and
Vcb, we have found that the natural value for ζ0 ≡ |V14V ∗24| is O(10−4), which agrees with the
observed parameter in the D − D̄ mixing but it is one order of magnitude larger than that
in Ref. [25].
For the rare D decays, due to the non-unitarity effects in the model, BR(D → Xuℓ+ℓ−),
BR(D → Xuνν̄) and BR(D → µ+µ−) could be enhanced to be O(10−9), O(10−8) and
O(10−9), respectively, which could marginally reach the sensitivity proposed by BESIII [27].
Acknowledgments
This work is supported in part by the National Science Council of R.O.C. under Grant
#s: NSC-95-2112-M-006-013-MY2, NSC-95-2112-M-007-001, NSC-95-2112-M-007-059-MY3
and NSC-96-2918-I-007-010.
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  We study the $D-\bar D$ mixing and rare D decays in the Littlest Higgs model.
As the new weak singlet quark with the electric charge of 2/3 is introduced to
cancel the quadratic divergence induced by the top-quark, the standard unitary
$3\times 3$ Cabibbo-Kobayashi-Maskawa matrix is extended to a non-unitary
$4\times 3$ matrix in the quark charged currents and Z-mediated flavor changing
neutral currents are generated at tree level. In this model, we show that the
$D-\bar D$ mixing parameter can be as large as the current experimental value
and the decay branching ratio (BR) of $D\to X_u \ga$ is small but its direct CP
asymmetry could be $O(10%)$. In addition, we find that the BRs of $D\to X_u
\ell^{+} \ell^{-}$, $D\to X_u\nu \bar \nu$ and $D\to \mu^{+} \mu^{-}$ could be
enhanced to be $O(10^{-9})$, $O(10^{-8})$ and $O(10^{-9})$, respectively.

arXiv:0704.0601v4  [hep-ph]  24 Oct 2007
D − D̄ mixing and rare D decays in the Littlest Higgs
model with non-unitarity matrix
Chuan-Hung Chen1,2∗, Chao-Qiang Geng3,4† and Tzu-Chiang Yuan3‡
1Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan
2National Center for Theoretical Sciences, Hsinchu 300, Taiwan
3Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan
4Theory Group, TRIUMF, 4004 Wesbrook Mall,
Vancouver, B.C. V6T 2A3, Canada
(Dated: October 27, 2018)
Abstract
We study the D − D̄ mixing and rare D decays in the Littlest Higgs model. As the new weak
singlet quark with the electric charge of 2/3 is introduced to cancel the quadratic divergence induced
by the top-quark, the standard unitary 3× 3 Cabibbo-Kobayashi-Maskawa matrix is extended to
a non-unitary 4× 3 matrix in the quark charged currents and Z-mediated flavor changing neutral
currents are generated at tree level. In this model, we show that theD−D̄ mixing parameter can be
as large as the current experimental value and the decay branching ratio (BR) of D → Xuγ is small
but its direct CP asymmetry could be O(10%). In addition, we find that the BRs of D → Xuℓ+ℓ−,
D → Xuνν̄ and D → µ+µ− could be enhanced to be O(10−9), O(10−8) and O(10−9), respectively.
∗ Email: physchen@mail.ncku.edu.tw
† Email: geng@phys.nthu.edu.tw
‡ Email: tcyuan@phys.nthu.edu.tw
http://arxiv.org/abs/0704.0601v4
I. INTRODUCTION
As the observation of the Bs − B̄s mixing in 2006 by CDF [1], all neutral pseudoscalar-
antipseudoscalar oscillations (P − P̄ ) in the down type quark systems have been seen. In
the standard model (SM), the most impressive features of flavor physics are the Glashow-
Iliopoulos-Maiani (GIM) mechanism [2] and the large top quark mass. The former results
in the cancellation between the lowest order short-distance (SD) contributions of the first
two generations to the mass difference ∆mK in the K
0 system, while the latter makes ∆mBq
(q = d, s) in the Bq systems dominated by the SD effects [3]. In addition, these features also
lead to sizable flavor changing neutral currents (FCNCs) from box and penguin diagrams,
which contribute to the rare decays, such as K → πνν̄ and B → K(∗)ℓℓ̄. It is known that
these processes could be good candidates to probe new physics effects [4–6]. However, it is
clear that the new physics signals deviated from the SM predications for the P − P̄ mixings
and rare FCNC decays have to wait for precision measurements on these processes.
Unlike K and Bq systems, the SD contributions to charmed-meson FCNC processes,
such as the D − D̄ mixing [7] and the decays of c → uℓ+ℓ− and D → ℓ+ℓ− [8], are highly
suppressed due to the stronger GIM mechanism and weaker heavy quark mass enhancements
in the loops. On the other hand, it is often claimed that the long-distance (LD) effect for
the D−D̄ mixing should be the dominant contribution in the SM. Nevertheless, because the
nonperturbative hadronic effects are hard to control, the result is still inconclusive [9–12].
Recently, BABAR [13] and BELLE [14, 15] collaborations have reported the evidence for
the D − D̄ mixing with
x′2 = (−0.22± 0.30± 0.21)× 10−3 ,
y′ = (9.7± 4.4± 3.1)× 10−3 , (1)
x ≡ ∆mD
= (0.80± 0.29± 0.17)% ,
y ≡ ∆ΓD
= (0.33± 0.24± 0.15)% ,
yCP = (1.31± 0.32± 0.25)% , (2)
respectively, where x′ = x cos δ + y sin δ and y′ = −x sin δ + y cos δ with the assumption
of CP conservation and δ being the relative strong phase between the amplitudes for the
doubly-Cabbibo-suppressed D → K+π− and Cabbibo-favored D → K−π+ decays [16, 17]
and yCP = τ(D → K−π+)/τ(D → K+K−) − 1. Moreover, no evidence for CP violation is
found. The combined results of Eqs. (1) and (2) at the 68% C.L. are [18]
x = (5.5± 2.2)× 10−3 ,
y = (5.4± 2.0)× 10−3 ,
δ = (−38± 46)0 . (3)
Note that the upper bound of x < 0.015 at 95%C.L. can be extracted from the BELLE
data in Eq. (2) [14, 15]. The evidences of the mixing parameters by BABAR and BELLE
collaborations reveal that the era of the rare charmed physics has arrived. The results in
Eq. (3) can not only test the SU(3) breaking effects for the D− D̄ mixing [10, 12], but also
examine new physics beyond the SM [17–21].
It is known that a straightforward way to enhance the rare D processes is to include some
new heavy quarks within the framework of the SM. For instance, if a new heavy quark with
the electric charge of −1/3 is introduced, it could affect the D system since the extra down
type quark violates the GIM mechanism. However, the constraint on this heavy quark is
quite strong as it could also lead to FCNCs for the down type quark sector at tree level,
which are strictly limited by the well measured rare K and B decays, such as KL → µ+µ−
and B → Xsγ [22]. On the other hand, if the charge of the new heavy quark is 2/3, it could
generate interesting tree level FCNCs for the up type quark sector, for which the constraints
are much weaker. In this paper, we will study D physics based on a new weak singlet upper
quark.
It has been known that in the framework of the Littlest Higgs model [23], there exists
a new SU(2)L singlet vector-like up quark [24], hereafter denoted by T . Since the number
of down type quarks is the same as that in the SM, the standard unitary 3 × 3 Cabibbo-
Kobayashi-Maskawa (CKM) matrix is extended to a non-unitary 4×3 matrix in the charged
currents. Moreover, Z-mediated FCNCs for the up quark sector are generated at tree level.
In Ref. [25], it has been shown that the contributions of this new quark to the rare D
processes are small and cannot reach the sensitivities of future experiments [25, 26]. In this
paper, we will demonstrate that by adopting some plausible scenario, the effects could not
only generate a large D − D̄ oscillation but also marginally reach the sensitivity proposed
by BESIII for the rare D decays [27]. We note that the implication of the new data on the
D − D̄ mixing in the Littlest Higgs model with T-parity has been recently studied in Ref.
[21].
The paper is organized as follows. In Sec. II, we investigate that when a gauge singlet T -
quark is introduced in the Littlest Higgs model, how the non-unitary matrix for the charged
current and the tree level Z-mediated FCNC are formed. By using the leading perturbation,
the mixing matrix elements related to the new parameters in the Littlest Higgs model are
derived. In addition, we study how to get the small mixing matrix element for Vu(c)b, which
describes the b → u(c) decays. In Sec. III, we discuss the implications of the non-unitarity
on the D− D̄ mixing and rare D decays by presenting some numerical analysis. Finally, we
summarize our results in Sec. IV.
II. NON-UNITARY MIXING MATRIX IN THE LITTLEST MODEL
To study the new flavor changing effects in the Littlest Higgs model, we start by writing
the Yukawa interactions for the up quarks to be [24, 25]
λabfǫijkǫxyχaiΣjxΣkyu
b + λ0fTT
c + h.c. , (4)
where χT1 = (d1, u1, 0), χ
2 = (s2, c2, 0), χ
3 = (b3, t3, T ), u
b is the weak singlet and Σ =
eiΠ/fΣ0e
iΠT /f with
112×2
112×2
, Π =
2 h∗/
φ hT/
. (5)
The scale f denotes the global symmetry spontaneously breaking scale, which, as usual,
could be around 1 TeV. Consequently, the 4× 4 up-quark mass matrix is given by [25]
iλijv
− − − − −
0 0 λ33f | λ0f
. (6)
We remark that the quadratic divergences for the Higgs mass from one-loop diagrams in-
volving t and T get exactly cancelled as shown in Ref. [28]. Moreover, for other quarks
other than the top quark, the one-loop quadratic divergent contributions do not necessitate
fine-tuning the Higgs potential as the cutoff is around 10 TeV for f ∼ 1 TeV due to the small
corresponding Yakawa couplings. That is why there is no need to introduce extra singlet
states T [28, 29].
To obtain the quark mass hierarchy of mt ≫ mc ≫ mu, we can choose a basis such that
the up-quark mass matrix is [30]
m̂U 0
hf λ0f
 (7)
where m̂Uij = δijλiv/
2 ≡ mi is diagonal matrix and h = (h1, h2, h3). The hi is related to
λ33 by hi = Ṽ
i3 λ33 and hh
† = |λ33|2, in which Ṽ UR is the unitary transformation for the
right-handed up quarks. We note that mi are not the physical masses and in principle their
magnitudes could be as large as the weak scale. In order to preserve the hierarchy in the
quark masses, one expects that m3 > m2 > m1. Furthermore, in terms of this basis, the
charged and neutral currents, defined by
LC = g√
J−µ W
+µ − g√
2 tan θ
J−µ W
H + h.c. ,
cos θW
3 − sin2 θWJµem
tan θ
3 ZHµ + h.c. , (8)
are expressed by
J−µ = ŪLγµṼ
0aVDL ,
µṼ 0aV Ṽ
0†UL −
µDL , (9)
respectively, where UT = (u1, c2, t3, T ), D
T = (d, s, b), aV = diag(1, 1, 1, 0) and
Ṽ 0 =
(V 0CKM)3×3 0
 (10)
with V 0CKMV
CKM = 113×3. The null entry in aV denotes the new T -quark being a weak
singlet; and without the new T -quark, V 0CKM is just the CKM matrix. Since the down quark
sector is the same as that in the SM, we have set the unitary transformation UDL to be an
identity matrix.
For getting the physical eigenstates, the mass matrix in Eq. (7) can be diagonalized by
unitary matrices V UL,R so that we have
†diag
U = V
ULMUM
L (11)
m̂Um̂
f (|λ33|2 + |λ0|2)f 2
 . (12)
Since (|λ33|2 + |λ0|2)f 2 is much larger than other elements, we can take the leading order
of the perturbation in himi/f (i = 1, 2, 3). According to Eq. (11), the leading expansion is
given by
†diag
U = V
ULMUM
L ≈ (1 + ∆L)MUM †U (1−∆L) . (13)
By looking at the off-diagonal terms (M
†diag
U )i4(4i), we can easily get
∆Li4 ≈ −∆L4i = −
himif
(|λ33|2 + |λ0|2)f 2 −m2i
with i 6= 4. From the diagonal entries, if we set the light quark masses to be mu ≈ mc ≈ 0,
we obtain
0 ≈ m2uj ≈ m
j + 2∆Lj4(MUM
U)4j ,
∆Lj4 ≈ −
with j = 1, 2. To be consistent with Eq. (14), at the leading expansion the relation
2h2j = (|λ33|2 + |λ0|2) (16)
should be satisfied. We emphasize that the choice of Eq. (16) is somewhat fine-tuned in
order to have Eqs. (14) and (15) simultaneously. Since the top-quark is much heavier than
other ordinary quarks, we have 2h23 ≈ (1−m2t/m23)(|λ33|2+ |λ0|2) if f > m3 > mt. Similarly,
one obtains the flavor mixing effects for i 6= j 6= 4 to be
∆Lij =
hihjmimj
m2i −m2j
f 2[2(|λ33|2 + |λ0|2)f 2 − (m2i +m2j )]
(|λ33|2 + |λ0|2)f 2 −m2j
((|λ33|2 + |λ0|2)f 2 −m2i )
. (17)
After diagonalization, the currents become
J−µ = ŪLγµV
ULṼ 0aVDL = ŪLγµV
ULV 0DL ,
3 = ŪLγ
µV ULṼ 0aV Ṽ
0†V †ULUL = ŪLγ
µV ULV 0V 0†V †ULUL , (18)
where UT = (u, c, t, T ),
V 0 = Ṽ 0aV =
(V 0CKM)3×3 0
 (19)
and diag(V 0V 0†) = aV . Since the 4-th component of aV is different from the first 3 ones, it
is obvious that the matrix V ≡ V ULV 0 associated with the charged current does not satisfy
unitarity. In addition, V ULaV V
L, which is associated with the neutral current, is not the
identity matrix. As a result, Z-mediated FCNCs at tree level are induced. According to
Eq. (18), we see that
V V † = V ULaV V
L (20)
which is just the same as the effects of the Z-mediated FCNCs. Due to V being a non-unitary
matrix, one finds
(V V †)ij = Vi4V
j4 . (21)
Consequently, the interesting phenomena arising from non-unitary matrix elements are al-
ways related to Vi4V
j4 = ∆Li4∆j4. We note that as we do not particularly address CP
problem, in most cases, we set the parameters to be real numbers.
It has been known that enormous data give strict bounds on the flavor changing effects.
In particular, the pattern describing the charged current has been fixed quite well. Any
new parametrization should respect these constraints. It should be interesting to see the
relationship with and without the new vector-like T -quark. From Eq. (18), we know that
the new flavor mixing matrix for the charged current is given by V = V ULV 0. At the leading
order perturbation, one gets
V = V ULV 0 ≈ (1 + ∆L)V 0 = V 0 +∆LV 0 . (22)
If V 0tb ∼ 1 is taken, one finds that Vub ≈ V 0ub+∆L13 and Vcb ≈ V 0cb+∆L23. In terms of Eq. (17)
and h1 = h2 ≈ h3, the relations ∆L13 ≈ −m1/m3 and ∆L23 ≈ −m2/m3 are obtained. Hence,
in our approach, we have
Vus ≈ V 0us −
, Vub ≈ V 0ub −
, Vcb ≈ V 0cb −
. (23)
From these results, it is clear that when the T -quark decouples from ordinary quarks,
Vus → V 0us, Vub → V 0ub and Vcb → V 0cb, while m1/m2 → mu/mc, m1/m3 → mu/mt and
m2/m3 → mc/mt, respectively. According to the observations in the decays of b → uℓν̄ℓ and
b → cℓν̄ℓ, the corresponding values have been determined to be |Vub| = 3.96 ± 0.09 × 10−3
and |Vcb| = 42.21+0.10−0.80 × 10−3, respectively [16]. Since V 0ij and mi are free parameters, to
satisfy the experimental limits with interesting phenomena in low energy physics, it is rea-
sonable to set the orders of magnitude for m1/m3 and m2/m3 (m1/m2) to be O(10
−2) and
O(10−1), respectively. Consequently, the non-unitary effects on the rare charmed meson
decays governed by V14V
24 could be as large as ∆14∆24 ∼ O(10−4), which could be one order
of magnitude larger than those in Ref. [25].
III. D − D̄ MIXING AND RARE D DECAYS
A. D − D̄ mixing
It is well known that the GIM mechanism has played an important role in the K − K̄
oscillation in the SM. In addition, due to the top-quark in the box and penguin diagrams,
Bq − B̄q mixings are dominated by the SD effects, which are consistent with the data [16].
On the contrary, for the D− D̄ mixing the GIM cancellation further suppresses the mixing
effect to be ∆mD ∼ O(m4s/m2Wm2c) [7] and the bottom quark contribution actually is a
subleading effect due to the suppression of (VubV
2. In the SM, the SD contribution to the
mixing parameter is O(10−7) [34]. However, the LD contribution to the mixing is believed
to be dominant. Due to the nonperturbative hadronic effects, the result is still uncertain
with the prediction on the mixing parameter ranging from O(10−3) [9] to O(10−2) [10–12].
Nonetheless, the mixing parameters shown in Eq. (3) could arise from the LD contribution.
Thus, it is important to have a better understanding of the LD effect. On the other hand, it
is also possible that the mixings in Eq. (3) could result from new physics. In the following,
we will concentrate on the Littlest Higgs model.
In the quark sector of the Littlest Higgs model due to the introduction of a new weak
singlet, a direct impact on the low energy physics is the FCNCs at tree level. According to
Eq. (18), the most attractive process with |∆C| = 2 via the Z-mediated c−u−Z interaction,
illustrated in Fig. 1, is given by
H(|∆C| = 2) = g
(V14V
ūγµPLc ūγ
µPLc ,
(V14V
ūγµPLc ūγ
µPLc . (24)
In terms of the hadronic matrix element, defined by
〈D̄|(ūc)V−A (ūc)V−A|D〉 =
D , (25)
FIG. 1: Z-mediated flavor diagram with |∆C| = 2.
the mass difference for the D meson is [25]
∆mD ≈
DmDBD|(V14V ∗24)2| . (26)
If we assume no cancellation between new physics and SM contributions, by taking τD =
1/ΓD = 6.232 × 1011 GeV−1, fD
BD = 200 MeV [31, 32] and mD = 1.86 GeV and using
Eq. (3), we obtain
ζ0 ≡ |V14V ∗24| = |∆L14∆L24| = (1.47± 0.29)× 10−4 , (27)
which is in the desirable range. In other words, the result in Eq. (27) demonstrates that the
non-unitarity in the Littlest Higgs model could enhance the D − D̄ mixing at the observed
level. We note that the limit of x < 0.015 (95%C.L.) leads to
ζ0 < 2.5× 10−4. (28)
In addition, we note that cancellation between the LD effect in the SM and the SD one from
new physics could happen. In this case, the values in Eqs. (27) and (28) could be relaxed.
B. D → Xuγ decay
In the SM, the D-meson FCNC related processes are all suppressed since the internal
fermions in the loops are all much lighter than mW . For the decay of D → Xuγ, without
QCD corrections, the branching ratio is O(10−17); and it becomes O(10−12) when one-loop
QCD corrections are included [8]. However, it is found that the two-loop QCD corrections
can boost the BR to be as large as 3.5× 10−8 [33]. It should be interesting to see how large
the non-unitarity effect on c → uγ is in the Littlest Higgs model.
To study the radiative decay of c → uγ, we write the effective Lagrangian to be
Lc→uγ = −
mcūσµνPRcF
µν , (29)
where C7 = C
7 + C
7 and C
7 ≈ −(0.007 + i0.02) = 0.021eiδs with δs = 70.7◦ [33]
being the strong phase induced by the two-loop QCD corrections. In the extension of the
SM by including a weak singlet particle, the flavor mixing matrix in the charged current
is not unitary and the Z-mediated FCNC at tree level is generated as well. For c → uγ,
besides the QED-penguin diagrams induced by the W -boson displayed in Figs. 2a and 2b,
the Z-mediated QED-penguin one in Fig. 2c will also give contributions. We note that the
d, s, b
W d, s, b
u, c, T
(a) (c)(b)
FIG. 2: Flavor diagrams for c → uγ.
contributions from WH and ZH can be ignored as m
and m2Z/m
are much less
than one.
At the first sight, due to the light quarks in the loops, the contributions from Figs. 2a
and 2b could be negligible. However, due to the non-unitarity of (V V †)uc = V14V
24 6= 0,
even in the limits of md, s, b → 0, the contributions from the mass independent terms do not
vanish anymore and can be sizable. In terms of unitary gauge [22], we obtain
CW7 =
(V V †)12
VusV ∗cs
VusV ∗cs
. (30)
Furthermore, if we set mu ≈ mc = 0, the contributions from Fig. 2c are given by
CZ7 = (f
u + f
c + f
T )/VusV
fZc + f
− eu sin2 θW
[4ξ0(0)− 6ξ1(0) + 2ξ2(0)]
+eu sin
2 θW [4ξ0(0)− 4ξ1(0)]
fZT =
eu [2ξ0(yT )− 3ξ1(yT ) + ξ2(yT )] , (31)
where the functions ξn(x) are defined by
ξn(x) ≡
zn+1dz
1 + (x− 1)z
and yT = mT/mZ . Numerically, the total contribution in Fig. 2 is
C7 = C
7 + C
V ∗csVus
24 . (33)
If we regard V14V
24 as an unknown complex parameter, i.e. V14V
24 = ζ0e
iθ with θ being
the CP violating phase, one can study the decay BR and direct CP asymmetry (CPA) of
D → Xuγ defined by
BR(D → Xuγ) =
6αem|C7|2
π|Vcd|2
BR(D → Xdeν̄e) ,
ACP =
BR(c̄ → ūγ)− BR(c → uγ)
BR(c̄ → ūγ) + BR(c → uγ)
, (34)
as functions of ζ0 and θ. In Fig. 3, the BR and CPA as functions of ζ0 are presented,
where the solid, dotted, dashed and dash-dotted lines represent the CP violating phase at
θ = 0, 45◦, 90◦ and 135◦, respectively. From these results, it is interesting to see that
0 1 2 3 4 5 6
0 1 2 3 4 5 6
(a) (b)
FIG. 3: BR (in units of 10−8) and CPA (in units of 10−2) for D → Xuγ as functions of ζ0, where
the solid, dotted, dashed and dash-dotted lines represent the CP violating phase at θ = 0, 45◦, 90◦
and 135◦, respectively.
BR(D → Xuγ) is insensitive to the new physics effects, whereas the direct CPA could be as
large as O(10%). Explicitly, if we take θ = 90◦ and ζ0 = 1.5 × 10−4, the CPA is about 3%.
Note that this CPA vanishes in the SM.
C. D → Xuℓℓ̄ and D0 → ℓ+ℓ− decays
Because the current experimental measurements in K and Bq decays are all consistent
with the SM predictions, it is inevitable that if we want to observe any deviations from the
SM, we have to wait for precision measurements for K and Bq. SuperB factories or LHCb
could provide a hope. However, the situation in D physics is straightforward. As stated
before, unlike K and Bq systems, due to no heavy quark enhancement in the D system, the
rare D-meson decays, such as D → Xuℓℓ̄ (ℓ = e, µ, ν), are always suppressed. Even by
considering the long-distance effects, the related decays, such as D → µ+µ− and D → Xuνν̄,
get small corrections to the SD predictions on the BRs [35]. Therefore, these rare decays
definitely could be good candidates to probe the new physics effects. Since the values in the
SM are hardly reachable at D factories [27], if any exotic event is found, it must be a strong
evidence for new physics. In the following analysis, we are going to discuss the implication
of the Littlest Higgs model on the rare D decays involving di-leptons.
To study these decays, we first write the effective Hamiltonian for c → uℓ+ℓ− (ℓ = e, µ)
H(c → uℓ+ℓ−) = −
GFαem√
V ∗csVus
Ceff9 O9 + C7O7 + C10O10
, (35)
O7 = −
ūiσµνq
νPRcℓ̄γ
O9 = ūγµPLc ℓ̄γ
O10 = ūγµPLc ℓ̄γ
µγ5ℓ , (36)
where the effective Wilson coefficients are given by
Ceff9 =
(V V †)14
V ∗csVus
cℓV + (h(zs, s)− h(zd, s)) (C2(mc) + 3C1(mc)) ,
C10 = −
(V V †)14
V ∗csVus
cℓA , (37)
with s = q2/m2c , zi = mi/mc, c
V = −1/2 + 2 sin2 θW , cℓA = −1/2 and
h(z, s) = −4
ln z +
(2 + x)
|1− x|
1−x+1√
1−x−1
− i π, for x ≡ 4z2/s < 1 ,
2 arctan 1√
x−1 , for x ≡ 4z
2/s > 1
. (38)
Here, we have neglected the small contributions from the penguin and box diagrams. We
note that in the SM, the SD contributions are mainly from the term with h(z, s), induced
by the insertion of O2 = ūLγµqLq̄Lγ
µcL and mixing with O9 at one-loop level [35, 36].
We note that the resonant decays of D → XuV → Xuℓ+ℓ− (V = φ, ρ, ω) would have
large corrections to c → uℓ+ℓ− at the resonant regions. However, in this paper we do
not discuss these contributions as we only concentrate on the SD contributions. Moreover,
these resonance contributions can be removed by imposing proper cuts in the phase space
in dedicated searches.
From Eq. (35), the decay rate for D → Xuℓ+ℓ− as a function of the invariant mass
s = q2/m2c can be found to be
768π5
|VusV ∗cs|2(1− s)2R(s) ,
R(s) =
|Ceff9 |2 + |C10|2
(1 + 2s) + 12Re(C∗7C
9 ) + 4
|C7|2 . (39)
In addition, by utilizing the lepton angular distribution, we can also study the forward-
backward asymmetry (FBA), given by
−1 d cos θdΓ/dsd cos θ sgn(cos θ)
−1 d cos θdΓ/dsd cos θ
Ceff9 +
, (40)
where θ is the angle of ℓ+ related to the momentum of the D meson in the ℓ+ℓ− invariant
mass frame. Since C10 is small in the SM, AFB is negligible. With mc = 1.4 GeV and the
mixing parameter in Eq. (27), we get
BR(D → Xue+e−) = (4.18± 0.91)× 10−10 ,
BR(D → Xuµ+µ−) = (2.51± 0.86)× 10−10 , (41)
comparing with the SM predictions of BR(D → Xue+e−)SM = 2.1 × 10−10 and BR(D →
+µ−)SM = 0.5 × 10−10, respectively. Clearly, if some cancellation occurs between new
physics and SM contributions in the D − D̄ mixing, a larger value of ζ0 can be allowed. In
Fig. 4, we show the tendency of the decay as a function of ζ0, where the negative horizontal
values correspond to -ζ0. In addition, we present the differential decay BR [FBA] of
D → Xue+e− as a function of s = q2/m2c in Fig. 5a [b], where the thick solid, dotted and
dashed lines correspond to ζ0 = 1.5, 3.0 and 5.0, while the thin ones denote the cases for
−ζ0 except ζ0 = 0 for the thin solid line in Fig. 5a. From Fig. 5b, we see that the FBA
-6 -4 -2 0 2 4 6
FIG. 4: BR(in units of 10−9) for D → Xue+e− as a function of ζ0.
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
(a) (b)
FIG. 5: (a)[(b)] Differential BR (in units of 10−9) [FBA] for D → Xue+e− as a function of s, where
the thick solid, dotted and dashed lines correspond to ζ0 = 1.5, 3.0 and 5.0, while the thin ones
denote the cases for −ζ0 except ζ0 = 0 for the thin solid line in (a).
is only at percent level. In the Littlest Higgs model, this is because the Z coupling to the
charged lepton cℓV = −1/2 + 2 sin2 θW appearing in Ceff9 is much smaller than one. This is
quite different from that in b → sℓ+ℓ− where the dominant effect in the SM for the FBA is
from the box and QED-penguin diagrams.
Next, we discuss the decay of D → Xuνν̄. In the SM, the BR for D → Xuνν̄ is estimated
to be O(10−16) − O(10−15) [35], which is vanishing small. In the Littlest Higgs model, by
taking C7 = 0, C
9 = −C10 = −π(V V †)14/(αemVusV ∗cs), the effective Hamiltonian in Eq. (35)
can be directly applied to c → uνν̄. The decay rate forD → Xuνν̄ as a function of s = q2/m2c
can be obtained as
768π5
(1− s)2(1 + 2s)
2π2|(V V †)12|2
, (42)
where the factor of 3 stands for the neutrino species. With ζ0 = 1.5×10−4, we get BR(D →
Xuνν̄) = 1.31 × 10−9. However, if we relax the constraint on V14V †24, the BR as a function
of ζ0 is shown in Fig. 6a. For a larger value of ζ0, BR(D → Xuνν̄) could be as large as
O(10−8).
0 1 2 3 4 5 6
0 1 2 3 4 5 6
(a) (b)
FIG. 6: (a) BR (in units of 10−8) for D → Xuνν̄ and (b) BR (in units of 10−9) for D → µ+µ−.
Finally, we study the decays of D → ℓ+ℓ−. It has been known that, in the SM, the SD
contributions to D → µ+µ− are O(10−18), while the LD ones are O(10−13) [35]. It is clear
that any signal to be observed at the sensitivity of the proposed detector, such as BESIII,
will indicate new physics effects. Since the effective interactions at quark level are the same
as those in Eq. (35), one finds that
BR(D → ℓ+ℓ−) =
τDmDm
|πV14V ∗24|2 . (43)
Here we have used equation of motion for the charged lepton so that ℓ̄/pDℓ = 0. We also
note that operators O7,9 make no contributions. With |V14V ∗24| = ζ0 = 1.5 × 10−4, the
predicted BR for D → µ+µ− is 1.17× 10−10. In Fig. 6b, we present the BR as a function of
ζ0. We see that BR(D → µ+µ−) in the Littlest Higgs model could be as large as O(10−9).
IV. CONCLUSIONS
We have studied the D − D̄ mixing and rare D decays in the Littlest Higgs model. In
the model, as the new weak singlet vector-like quark T with the electric charge of 2/3
is introduced to cancel the quadratic divergence induced by the top-quark, the standard
unitary 3× 3 CKM matrix is extended to a non-unitary 4× 3 matrix in the quark charged
currents and Z-mediated flavor changing neutral currents are generated at tree level. We
have shown that the effects on |∆C| = 2 and |∆C| = 1 processes are all related to V14V ∗24 in
Eq. (21).
To avoid the scenario adopted by Ref. [25], in which λ0 ∼ λ33 ≫ λij was assumed, we
choose the basis such that the effective mass matrix for u1, c2 and t3 is diagonal, while
the corresponding masses m1, m2 and m3 are free parameters and can be as large as the
weak scale v. Since the global symmetry breaking scale f is larger than v, the mixing
matrix relating physical and unphysical states could be extracted by taking the leading
perturbative expansion. Accordingly, by using the approximation of mu ≈ mc ≈ 0, the
explicit expressions for V14 and V24 have been obtained. In terms of the data for Vub and
Vcb, we have found that the natural value for ζ0 ≡ |V14V ∗24| is O(10−4), which agrees with the
observed parameter in the D − D̄ mixing but it is one order of magnitude larger than that
in Ref. [25].
For the rare D decays, due to the non-unitarity effects in the model, BR(D → Xuℓ+ℓ−),
BR(D → Xuνν̄) and BR(D → µ+µ−) could be enhanced to be O(10−9), O(10−8) and
O(10−9), respectively, which could marginally reach the sensitivity proposed by BESIII [27].
Acknowledgments
This work is supported in part by the National Science Council of R.O.C. under Grant
#s: NSC-95-2112-M-006-013-MY2, NSC-95-2112-M-007-001, NSC-95-2112-M-007-059-MY3
and NSC-96-2918-I-007-010.
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