Implication of the D^0 Width Difference On CP-Violation in D^0-\bar D^0
  Mixing
IPPP/07/10
DCPT/07/20
Implication of the D0 Width Difference
On CP-Violation in D0-D̄0 Mixing
Patricia Ball
IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK
Abstract
Both BaBar and Belle have found evidence for a non-zero width difference in the D0-D̄0
system. Although there is no direct experimental evidence for CP-violation in D mixing
(yet), we show that the measured values of the width difference y ∼ ∆Γ already imply
constraints on the CP-odd phase in D mixing, which, if significantly different from zero,
would be an unambiguous signal of new physics.
∗Patricia.Ball@durham.ac.uk
http://arxiv.org/abs/0704.0786v2
The highlight of this year’s Moriond conference on electroweak interactions and unified
theories arguably was the announcement by BaBar and Belle of experimental evidence
for D0-D̄0 mixing [1, 2, 3], which was quickly followed by a number of theoretical anal-
yses [4, 5, 6, 7, 8, 9]. While Refs. [4, 7, 8, 9] focused on the constraints posed, by the
experimental results, on various new-physics models, Ref. [5] presented a first analysis
of the implications of these results for the fundamental parameters describing D mixing.
The purpose of this letter is to show that the present experimental results already imply
constraints on a sizeable CP-odd phase in D mixing, which could only be due to new
physics (NP).
To start with, let us shortly review the theoretical formalism of D mixing and the
experimental results, see Refs. [10, 11] for more detailed reviews. In complete analogy to
B mixing, D mixing in the SM is due to box diagrams with internal quarks andW bosons.
In contrast to B, though, the internal quarks are down-type. Also in contrast to B mixing,
the GIM mechanism is much more effective, as the contribution of the heaviest down-type
quark, the b, comes with a relative enhancement factor (m2b −m2s,d)/(m2s −m2d), but also
a large CKM-suppression factor |VubV ∗cb|2/|VusV ∗cs|2 ∼ λ8, which renders its contribution
to D mixing ∼ 1% and hence negligible. As a consequence, D mixing is very sensitive to
the potential intervention of NP. On the other hand, it is also rather difficult to calculate
the SM “background” to D mixing, as the loop-diagrams are dominated by s and d
quarks and hence sensitive to the intervention of resonances and non-perturbative QCD.
The quasi-decoupling of the 3rd quark generation also implies that CP violation in D
mixing is extremely small in the SM, and hence any observation of CP violation will be
an unambiguous signal of new physics, independently of hadronic uncertainties.
The theoretical parameters describing D mixing can be defined in complete analogy to
those for B mixing: the time evolution of the D0 system is described by the Schrödinger
equation
D0(t)
D̄0(t)
M − i Γ
D0(t)
D̄0(t)
with Hermitian matrices M and Γ. The off-diagonal elements of these matrices, M12
and Γ12, describe, respectively, the dispersive and absorptive parts of D mixing. The
flavour-eigenstates D0 = (cū), D̄0 = (uc̄) are related to the mass-eigenstates D1,2 by
|D1,2〉 = p|D0〉 ± q|D̄0〉 (2)
M12 − i2 Γ12
; (3)
|p|2 + |q|2 = 1 by definition.
The basic observables in D mixing are the mass and lifetime difference of D1,2, which
are usually normalised to the average lifetime Γ = (Γ1 + Γ2)/2:
x ≡ ∆M
M2 −M1
, y ≡ ∆Γ
Γ2 − Γ1
. (4)
In this letter we follow the sign convention of Ref. [5], according to which x is positive
by definition. The sign of y then has to be determined from experiment. In addition, if
there is CP-violation in the D system, one also has
6= 1, φ ≡ arg(M12/Γ12) 6= 0. (5)
While previously only bounds on x and y were known, both BaBar and Belle have
now found evidence for non-vanishing mixing in the D system. BaBar has obtained this
evidence from the measurement of the doubly Cabibbo-suppressed decay D0 → K+π−
(and its CP conjugate), yielding
y′ = (0.97± 0.44(stat)± 0.31(syst))× 10−2,
x′2 = (−0.022± 0.030(stat)± 0.021(syst))× 10−2, (6)
while Belle obtains
yCP = (1.31± 0.32(stat)± 0.25(syst))× 10−2 (7)
from D0 → K+K−, π+π− and
x = (0.80±0.29(stat)±0.17(syst))×10−2, y = (0.33±0.24(stat)±0.15(syst))×10−2 (8)
from a Dalitz-plot analysis of D0 → K0Sπ+π−. Here yCP → y in the limit of no CP
violation in D mixing, while the primed quantities x′, y′ are related to x, y by a rotation
by a strong phase δKπ:
y′ = cos δKπ − x sin δKπ, x′ = x cos δKπ + y sin δKπ. (9)
Limited experimental information on this phase has been obtainted at CLEO-c [12]:
cos δKπ = 1.09± 0.66 , (10)
which can be translated into δKπ = (0 ± 65)◦. An analysis with a larger data-set is
underway at CLEO-c, with an expected uncertainty of ∆ cos δKπ ≈ 0.1 in the next couple
of years [13]; BES-III is expected to reach ∆ cos δKπ ≈ 0.04 after 4 years of running
[14]. The experimental result (10) agrees with theoretical expectations, δKπ = 0 in the
SU(3)-limit and |δKπ| <∼ 15
◦ from a calculation of the amplitudes in QCD factorisation
[15]. Based on these experimental results, a preliminary HFAG-average was presented at
the 2007 CERN workshop “Flavour in the Era of the LHC” [13]:
x = (8.5+3.2
−3.1)× 10−3, y = (7.1+2.0−2.3)× 10−3. (11)
Adding errors in quadrature, this implies
= 1.2± 0.6. (12)
The exact relations between ∆M , ∆Γ, M12 and Γ12 are given by
(∆M)2 − 1
(∆Γ)2 = 4|M12|2 − |Γ12|2,
(∆M)(∆Γ) = 4Re(M∗
Γ12) = 4|M12||Γ12| cosφ . (13)
Eq. (13) implies x/y > 0 for |φ| < π/2 and x/y < 0 for π/2 < |φ| < 3π/2. In view of the
above experimental results, we assume |φ| < π/2 from now on.
As for the CP-violating observables, |q/p| 6= 1 characterises CP-violation in mixing
and can be measured for instance in flavour-specific decays D0 → f , where D̄0 → f is
possible only via mixing. The prime example is semileptonic decays with
ASL =
Γ(D0 → ℓ−X)− Γ(D̄0 → ℓ+X)
Γ(D0 → ℓ−X) + Γ(D̄0 → ℓ+X)
|q/p|2 − |p/q|2
|q/p|2 + |p/q|2
. (14)
Although the B factories may have some sensitivity to this asymmetry, its measurement
is severely impaired by the fact that D mixing proceeds only very slowly, resulting in a
large suppression factor of the mixed vs. the unmixed rate:
Γ(D0 → ℓ−X)
Γ(D0 → ℓ+X)
x2 + y2
2 + x2 + y2
≈ 6× 10−5. (15)
Both in the K and the B system the quantity
− 1 (16)
is very small, which however need not necessarily be the case forD’s. From (3) one derives
the general expression
4 + r2 + 4r sin φ
4 + r2 − 4r sinφ
with r = |Γ12/M12| and the weak phase φ defined in (5). In the B system, one has r ≪ 1
(the current up-to-date numbers are r ≈ 7 × 10−3 for Bd and r ≈ 5 × 10−3 for Bs [16]),
so that upon expansion in r
= 1 +
sin φ+O(r2). (18)
Note that this formula refers to the definition φ = arg(M12/Γ12), which differs by +π
from the one used in Ref. [16], φ = arg(−M12/Γ12). For the K system, one finds r ≈
|∆Γ/∆M | ≈ 2 from experiment, but now the phase φ turns out to be small, so that
= 1 +
4 + r2
φ+O(φ2) ≈ 1 + φ. (19)
In both cases, |q/p| ≈ 1 to a very good approximation. In the D system, however, there
is no natural hierarchy r ≪ 1, and of course one hopes that NP-effects induce |φ| ≫ 0. In
-1.5 -1. -0.5 0. 0.5 1. 1.5
Figure 1: |q/p|2, Eq. (20), as a function of the CP-odd phase φ for the central experimental
value r̃ = 7.1/8.5. Solid line: full expression, dashed line: first order expansion around φ = 0.
this case, and because x and y have been measured, while |M12| and |Γ12| are difficult to
calculate, it is convenient to express |q/p|D in terms of x, y, φ, using the exact relations
(13). From (3), and defining r̃ = y/x, we then obtain
2(1 + r̃2)
2(1 + r̃2)2 + 16r̃2 tan2 φ
+8r̃ tanφ secφ
(1 + r̃2)2 − (1− r̃2)2 sin2 φ
. (20)
Note that for finite xy and φ = ±π/2, |q/p| diverges because xy → 0 for φ → ±π/2 from
(13). In Fig. 1 we plot |q/p|2 as function of φ, for the central experimental value from
HFAG, r̃ = 7.1/8.5, Eq. (11). It is obvious that even for moderate values of φ the small-φ
expansion is not really reliable.
What is the currently available experimental information on CP-violating in D mixing,
i.e. |q/p| and φ? As already mentioned, the semileptonic CP-asymmetry (14) has not
been measured yet. What has been measured, though, is the effect of CP-violation on the
time-dependent rates of D0 → K+π− and D̄0 → K−π+. The BaBar collaboration has
parametrised these rates as
Γ(D0(t) → K+π−) ∝ e−Γt
x′2+ + y
(Γt)2
Γ(D̄0(t) → K−π+) ∝ e−Γt
+ y′2
(Γt)2
and fit the D0 and D̄0 samples separately. They find [2]
= (9.8± 6.4(stat)± 4.5(syst))× 10−3,
= (9.6± 6.1(stat)± 4.3(syst))× 10−3. (22)
Adding errors in quadrature, this means y′
= 1.0±1.1. BaBar also obtains values for
which we do not quote here, because the sensitivity to the quadratic term in (21) is
-1.5 -1. -0.5 0. 0.5 1.
-1.5 -1. -0.5 0. 0.5 1. 1.5
Figure 2: Left: y′+/y
as function of φ for x/y = 1.2 (solid line) and x/y = {0.6, 1.8} (dashed
lines), from Eq. (11). δKπ = 0. Right: y
as function of φ for x/y = 1.2 for δKπ = 0 (solid
line) and δKπ = ±65◦ (dashed lines).
less than that to the linear term in y′
D is the ratio of the doubly Cabbibo-suppressed
to the Cabibbo-favoured amplitude, R
D = |A(D0 → K+π−)/A(D0 → K−π+)|. δKπ is
the relative strong phase in the Cabibbo-favoured and suppressed amplitudes:
A(D0 → K+π−)
A(D̄0 → K+π−)
−iδKπ ; (23)
the minus-sign comes from the relative sign between the CKM matrix elements Vcd and
Vus. In the limit of no CP-violation in the decay amplitude, one has |A(D0 → K−π+)| =
|A(D̄0 → K+π−)|, which is expected to be a very good approximation, in view of the fact
that the decay is solely due to a tree-level amplitude. Then the relation of y′
to x, y and
φ is given by
{(y cos δKπ − x sin δKπ) cosφ+ (x cos δKπ + y sin δKπ) sinφ} ,
{(y cos δKπ − x sin δKπ) cosφ− (x cos δKπ + y sin δKπ) sinφ} . (24)
Presently, the experimental result for y′+/y
is compatible with 1, although with con-
siderable uncertainties. Any significant deviation from 1 would be a sign for new physics.
In Fig. 2 we plot y′+/y
as function of φ, for different values of x/y and δKπ. The figures
clearly show that the value of y′
is very sensitive to the phase φ, at least if δKπ is not
too close to −65◦, which corresponds to the nearly constant dashed line in Fig. 2b. The
reason for this dependence on δKπ becomes clearer if y
is expanded to first order in
= 1− 2φ x(x
2 + 2y2) cos δKπ + y
3 sin δKπ
(x2 + y2)(x sin δKπ − y cos δKπ)
+O(φ2) . (25)
For the central values of x and y, Eq. (11), this amounts to 1+3.4φ for δKπ = 0, 1− 3.3φ
for δKπ = +65
◦ and 1 + 0.45φ for δKπ = −65◦, which explains the shape of the curves in
Fig. 2b. Evidently it is important to reduce the uncertainty of δKπ, which, as mentioned
x/ySM
Figure 3: Plot of |∆Γ/∆ΓSM|, Eq. (26), as a function of x/ySM and φ.
earlier, will be achieved within the next few years. On the other hand, as shown in
Fig. 2a, y′
, which depends only on the ratio x/y, but not x and y separately, is not
very sensitive to the precise value of that ratio, but very much so to φ. The conclusion
is that, even if x/y itself cannot be determined very precisely, y′
will nonetheless be
a powerful tool to constrain φ, at least once δKπ will be known more precisely. Already
now very large values φ ∼ π/2 are excluded.
Another, more theory-dependent constraint on φ can be derived from the value of y.
This argument centers around the fact that (a) the experimental result (11) is at the top
end of theoretical predictions ySM ∼ 1% [17] and (b) new physics indicated by a non-zero
value of φ always reduces the lifetime difference, independently of the value of x. This
observation is similar to what was found, some time ago, for the Bs system [18]. In order
to derive it, we assume that new physics does not affect Γ12,
1 so that Γ12 = Γ
12 . We then
have 2|Γ12| = ∆ΓSM and hence |ySM| = |Γ12|/Γ. Using the relations (13), we can then
express the ratio |∆Γ/∆ΓSM| in terms of ySM, x and φ:
+ x2/ cos2 φ
. (26)
This implies that new physics always reduces the lifetime difference, independently of the
value of x (and any new physics in the mass difference). In particular one has y = 0
for φ = ±π/2 and x 6= 0, which follows from the 2nd relation (13). Eq. (26) is the
manifestation of the fact that one does not need to observe CP-violation in order to
constrain it. A famous example for this is the unitarity triangle in B physics, whose
sides are determined from CP-conserving quantities only, but nonetheless allow a precise
measurement of the size of CP-violation in the SM, via the angles and the area of the
triangle. In Fig. 3, we plot |∆Γ/∆ΓSM| as a function of r = x/ySM. The zero at φ = ±π/2
is clearly visible. The experimental value |y/ySM| = O(1) then excludes phases φ close
to ±π/2. In order to make more quantitative statements, apparently a more precise
calculation of ySM is needed.
1See, however, Ref. [19] for a discussion of the effect of tiny NP admixtures to Γ12.
-1.5 -1. -0.5 0. 0.5 1. 1.5
5.yCP
-1.5 -1. -0.5 0. 0.5 1. 1.5
Figure 4: Left: yCP/y as function of φ, for x/y = 1.2 (solid line) and x/y = {0.6, 1.8} (dashed
lines), see Eq. (12). Right: AΓ/yCP as function of φ.
Two more CP-sensitive observables related to D0 → K+K− have been measured by
the Belle collaboration [3]:
yCP =
[Γ(D0 → K+K−) + Γ(D̄0 → K+K−)]− 1
y cosφ+
x sin φ, (27)
[Γ(D0 → K+K−)− Γ(D̄0 → K+K−)]− 1
y cos φ+
x sin φ. (28)
The present experimental value of yCP is given in (7), that for AΓ is (0.01± 0.30(stat)±
0.15(syst)) × 10−2. Again, we can study the dependence of these observables on φ. In
Fig. 4a we plot the ratio yCP/y, which is a function of x/y and φ, in dependence on φ. As
it turns out, this quantity is far less sensitive to φ than y′
, the reason being that its
deviation from 1 is only a second-order effect in φ:
yCP = y
1 + φ2
x4 + x2y2 − y4
2(x2 + y2)2
+O(φ4)
. (29)
Hence, unless the experimental accuracy is dramatically increased, and because the results
on y′
and y/ySM already exclude a large CP-odd phase φ ≈ ±π/2, it is safe to interpret
yCP as measurement of y. In Fig. 4b we plot the quantity AΓ/yCP. Also here there is a
distinctive dependence on φ, with AΓ/y ∝ φ for small φ, but the effect is less dramatic
than that in y′
In conclusion, we find that the experimental results on D mixing reported by BaBar
and Belle already exclude extreme values of the CP-odd phase φ close to ±π/2. This
follows from the result for y, which is close to the top end of theoretical predictions and
can only be reduced by new physics, and from y′
∼ 1. While y′
− 1 vanishes in
the limit of no CP-violation, y ∼ ∆Γ is a CP-conserving observable, which demonstrates
the usefulness of such quantities in constraining CP-odd phases. Also yCP, AΓ and the
ratio AΓ/yCP can be useful in constraining φ. As long as there is no major breakthrough
in theoretical predictions for D mixing, which are held back by the fact that the D meson
is at the same time too heavy and too light for current theoretical tools to get a proper
grip on the problem, the long-distance SM contributions to x will completely obscure any
NP contributions and their detection. The observation of CP violation, however, presents
a theoretically clean way for NP to manifest itself and it is to be hoped that in the near
future, i.e. at the B factories or the LHC, at least one of the plentiful opportunities for
NP to show up in CP violation [20] will be realised.
Acknowledgments
This work was supported in part by the EU networks contract Nos. MRTN-CT-2006-
035482, Flavianet, and MRTN-CT-2006-035505, Heptools.
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