November 17, 2014

CKM studies in the charm sector

M.J. Charles111Work supported by CNRS/IN2P3

LPNHE, Université Pierre et Marie Curie
Université Paris Diderot, CNRS/IN2P3, Paris, France

> A summary of recent progress in charm mixing and CP violation is presented,
> with a heavy bias towards experimental results.

> PRESENTED AT
>
>
>
>
> The 8th International Workshop on the CKM Unitarity Triangle (CKM 2014),
> Vienna, Austria, September 8–12, 2014

##  1 Introduction

Famously, CKM studies of charm began even before its experimental discovery
[1]. The experimental and theoretical context has changed a great deal since
then, but the goal today remains the same: to reveal or constrain New Physics
(NP) through study of flavour observables. For charm physics this means making
precise measurements of CP violation (CPV) and mixing*** The branching
fractions of very rare decays such as $D^{0}\rightarrow\mu^{+}\mu^{-}$ are
also sensitive to NP, but are outside the scope of these Proceedings. . CPV in
charm is typically divided into two types according to its mechanism: indirect
(mixing-induced) and direct. Although both are ultimately driven by CKM
physics—at least, within the Standard Model (SM)—they are studied in different
ways at both theoretical and experimental levels, and will be discussed
separately.

##  2 Mixing

Charm is far from the only arena where flavour observables can be used to
search for NP. In particular, numerous studies of mixing and CPV have been
carried out in the $b$ and $s$ sectors. The differences are stark. Mixing
occurs at a much higher rate in ${K}^{0}$, ${B}^{0}$, and ${B}^{0}_{s}$
mesons, as shown in Table 1. The normalised mixing parameters quoted, $x$ and
$y$, are defined via

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{2}+\Gamma_{1}}{2}$ $\displaystyle x$
$\displaystyle=$ $\displaystyle\frac{m_{2}-m_{1}}{\Gamma}$ $\displaystyle y$
$\displaystyle=$ $\displaystyle\frac{\Gamma_{2}-\Gamma_{1}}{\Gamma}$

where $m_{i}$ and $\Gamma_{i}$ are the masses and widths of the eigenstates of
the Hamiltonian. The overall mixing rate $R_{M}$ is then given by
$R_{M}=(x^{2}+y^{2})/2$. For each of the mesons able to mix, this rate is of
order 1 or more—except for charm, where it is approximately $3\times 10^{-5}$.
This small rate is why it took about three decades from the discovery of charm
to the first evidence of mixing [2, 3]. Even so, the mixing parameters are at
the upper end of what is plausible within the SM: long-distance effects can
generate values of $x$ and $y$ up to $\mathcal{O}(10^{-2})$ [4, 5]. Short-
distance effects would be much smaller, with a simple estimate indicating they
should enter at the $10^{-5}$ level [6]; more sophisticated calculations give
numbers that are larger but not dramatically so [7]. It should be noted that
all calculations of the SM mixing rate to date are at best order-of-magnitude:
for all we know, the mixing rates observed could actually be dominated by NP
effects. Inverting the argument, the observed level of mixing in charm can be
used to constrain models of NP by requiring that they at most saturate it [8].

Meson | $x$ | $y$
---|---|---
${K}^{0}$ | $9.5$ | almost 1
$D^{0}$ | $0.0041\pm 0.0015$ | $0.0063\pm 0.0008$
${B}^{0}$ | $0.774\pm 0.006$ | $0.0005\pm 0.005$
${B}^{0}_{s}$ | $26.85\pm 0.13$ | $0.069\pm 0.006$
Table 1: Mixing parameters $x$ and $y$ for neutral mesons. Data from HFAG [9]
and the PDG [10].

On the experimental side, there are four main sources of information on charm
mixing††† This list is not exhaustive. For example, it omits time-integrated
measurements of wrong-sign semileptonic decays. : the time-dependent ratio of
wrong-sign (WS) to right-sign (RS) decays, primarily in $D^{0}\rightarrow
K^{\pm}\pi^{\mp}$, measuring $y_{CP}$; the ratio of the mean lifetime seen in
CP eigenstates vs. CP-mixed states, primarily
$D^{0}\rightarrow{{K}^{-}}{{K}^{+}},{{\pi}^{-}}{{\pi}^{+}}$ vs.
${{K}^{-}}{{\pi}^{+}}$, measuring $x^{\prime}$ and $y^{\prime}$‡‡‡
$x^{\prime}$ and $y^{\prime}$ are related to $x$ and $y$ by rotation through
the strong phase $\delta$ between the DCS and CF decays to that final state. ;
time-dependent amplitude analyses of Dalitz plots of self-conjugate final
states, primarily $D^{0}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$, measuring $x$ and $y$; and measurements of
relative strong phases $\delta$ between doubly Cabibbo-suppressed (DCS) and
Cabibbo-favoured (CF) decays. In most cases these do not grant direct access
to $(x,y)$ but rather combinations of physics parameters ($x$, $y$, relative
strong phases, indirect CPV), such that the world-average is not dominated by
any one type of measurement but rather by their combination. The time-
dependent amplitude analyses are the exception: they allow simultaneous
measurement of the mixing rate and of the strong phases of the contributing
amplitudes, thereby granting direct access to $x$ and $y$.

The most recent world average results on $x$ and $y$ from HFAG are illustrated
in Fig. 1(a). Allowing for CPV, the central values are:

$\displaystyle x$ $\displaystyle=$
$\displaystyle\left(0.41^{+0.14}_{-0.15}\right)\%$ $\displaystyle y$
$\displaystyle=$ $\displaystyle\left(0.63^{+0.07}_{-0.08}\right)\%$

This represents a major improvement in precision from the status as of CKM
2012. Several new or updated experimental results have been released since
then, from Belle [11, 12], CDF [13] LHCb [14, 15], and BESIII [16, 17]. In
particular, after many years in which the overall statistical significance of
mixing was high but no individual measurement exceeded $5\sigma$, Belle, CDF,
and LHCb have now all passed that threshold. The most significant of these
results by some way is a measurement with WS $D^{0}\rightarrow K^{+}\pi^{-}$
decays by LHCb: this is an analysis which can capitalise on the large charm
cross-section and the boost.

|
---|---
(a) | (b)
Figure 1: World-average results from HFAG [9], as of 30 June 2014.

As discussed above, the size of mixing in charm is a powerful constraint on
models of NP. However, there is not much more to be learned on this front from
experiment: improving the precision on the mixing parameters will not move the
upper bound significantly. It is important for another reason, though:
relating measurements of time-dependent CP asymmetries to the underlying
physics parameters of indirect CP violation requires knowledge of $x$ and $y$.

##  3 Indirect CPV

Indirect CPV in charm may be characterised by the magnitude and phase of
$q/p$, with deviation from $|q/p|=1$ and $\phi\equiv\mathrm{arg}(q/p)=0$
indicating CPV. In the HFAG phase convention [9], these are related to
experimental observables as follows:§§§ Direct CPV is neglected in these
expressions. This is a good approximation at present, but when the precision
on $A_{\Gamma}$ improves it may need to be taken into account [18].

$\displaystyle
2y_{CP}=\left(|q/p|+|p/q|\right)y\cos\phi-\left(|q/p|-|p/q|\right)x\sin\phi$
$\displaystyle
2A_{\Gamma}=\left(|q/p|-|p/q|\right)y\cos\phi-\left(|q/p|+|p/q|\right)x\sin\phi$
$\displaystyle x^{\prime\pm}=\left(\frac{1\pm A_{M}}{1\pm
A_{M}}\right)^{1/4}\,\left(x^{\prime}\cos\phi\pm y^{\prime}\sin\phi\right)$
$\displaystyle y^{\prime\pm}=\left(\frac{1\pm A_{M}}{1\pm
A_{M}}\right)^{1/4}\,\left(y^{\prime}\cos\phi\mp x^{\prime}\sin\phi\right)$

where

$\displaystyle A_{M}=\frac{|q/p|^{2}-|p/q|^{2}}{|q/p|^{2}+|p/q|^{2}}.$

Here $y_{CP}$ is the mixing parameter obtained from the ratio of CP-even¶¶¶
CP-odd final states can also be used, with suitable changes to the formulae.
to CP-mixed lifetimes, discussed in the previous section. $A_{\Gamma}$ is a
related asymmetry:

$\displaystyle A_{\Gamma}=\frac{\tau({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow
f_{CP,+})-\tau(D^{0}\rightarrow f_{CP,+})}{\tau({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow
f_{CP,+})+\tau(D^{0}\rightarrow f_{CP,+})},$

with the most precise results coming from $D^{0}\rightarrow{{K}^{+}}{{K}^{-}}$
and ${{\pi}^{+}}{{\pi}^{-}}$. The observables $x^{\prime\pm}$ and
$y^{\prime\pm}$ are simply the values of $x^{\prime}$ and $y^{\prime}$
measured with $D^{0}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ events, respectively; a
difference between $x^{\prime+}$ and $x^{\prime-}$ or $y^{\prime+}$ and
$y^{\prime-}$ would indicate CPV.

Two key points follow directly from these expressions. First, the observable
$A_{\Gamma}$ is scaled down by a factor of $x$ or $y$, such that even if
indirect CPV is maximal $A_{\Gamma}$ will be limited in magnitude to
approximately $x$ or $y$, i.e. $\mathcal{O}(\frac{1}{2}\%)$. We have only
recently entered a regime where the uncertainty on $A_{\Gamma}$ is smaller
than this. Second, while a non-zero value of $A_{\Gamma}$ would necessarily
imply CPV, extracting values or limits on the physics parameters $\phi$ and
$|q/p|$ would require $x$ and $y$ as inputs. Similar considerations apply to
$x^{\prime\pm}$ and $y^{\prime\pm}$. The relative uncertainties on the mixing
parameters are still substantial, although the situation is much better now
than a few years ago. Thus, improving the precision with which $x$ and $y$ is
known translates directly to improved constraints on the underlying CPV
parameters.

Some of the recent mixing measurements noted in Section 2 included
measurements of CP asymmetries [11, 15], and in addition LHCb has published a
measurement of $A_{\Gamma}$ with 1/invfb of data [19]. Including these, the
current world-average results with CPV fully allowed are shown in Fig. 1(b),
and correspond to:

$\displaystyle|q/p|$ $\displaystyle=$ $\displaystyle 0.93^{+0.09}_{-0.08}$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\left(-8.7^{+8.7}_{-9.1}\right)^{\circ}.$

Those values allow for the presence of direct CPV in suppressed decays. If it
is assumed that there is no direct CPV in DCS decays∥∥∥ This is certainly true
for charm in the SM, and is also true in many but not all models of NP on the
market. However, we should be a little cautious simply because there have been
few experimental tests of this.  [20], these constraints are strengthened
considerably:

$\displaystyle|q/p|$ $\displaystyle=$ $\displaystyle 1.007^{+0.015}_{-0.014}$
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle(-0.30^{+0.58}_{-0.60})^{\circ}.$

As with the mixing parameters, these represent a great improvement on the
state of play as of CKM 2012. It is now clear that we do not have
$\mathcal{O}(1)$ indirect CPV in charm. If direct CPV is negligible in DCS
decays—and the excellent agreement with the null hypothesis seems to back this
up—then $\mathcal{O}(10^{-1})$ is also excluded and even
$\mathcal{O}(10^{-2})$ is not far away. We have not yet reached the SM level,
which is estimated to be $\mathcal{O}(10^{-4})$ to $\mathcal{O}(10^{-3})$ [21,
22, 23], but it looms ahead in the distance. On the plus side, it is dominated
by short-distance physics so is a more theoretically tractable problem than
direct CP violation.

##  4 Direct CPV

While indirect CPV is not quite universal in the SM—CPV in the interference
between mixing and decay in general depends on the final state—it can be
expressed in terms of a small number of parameters with modest assumptions.
The story is very different for direct CPV, where the asymmetry can vary
greatly from one decay mode to another. Both within the SM and in general
models of NP, the largest asymmetries are expected in SCS modes [24] where
interference between penguin and tree diagrams can be substantial, and this is
where both experimental and theoretical interest has been focused. Direct CPV
in DCS charm decays is possible in certain models of NP but negligible in the
SM; the experimental precision is necessarily poorer than for SCS decays
simply because of the reduced branching fractions. In CF decays, direct CPV is
negligible even in the presence of NP since the favoured tree diagram
dominates.

There was considerable excitement at the time of CKM 2012: LHCb had seen
indications of direct CPV at $3.5\sigma$ in $\Delta
A_{CP}=A_{CP}(D^{0}\rightarrow{{K}^{+}}{{K}^{-}})-A_{CP}(D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}})$
with 0.6$\mbox{\,fb}^{-1}$ of prompt charm data [25]. CDF and Belle reported
similar central values after analysing their full data sets [26, 27], giving a
world average of $\Delta A_{CP}^{\mathrm{dir}}=(-0.678\pm 0.147)\%$. The
effect was studied in numerous theory papers (e.g. [28, 29, 30, 31]), both in
the context of the SM and of various models of NP. The consensus was that
direct CPV of order $0.5\%$ in these final states could be accommodated within
the SM if the penguin amplitude is large, which is allowed and would be
consistent with the unusual pattern of branching fractions
[$\mathcal{B}(D^{0}\rightarrow{{K}^{+}}{{K}^{-}})>\mathcal{B}(D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}})$].
Since CKM 2012, LHCb has released three new results which have brought the
world average much closer to zero: an update of the prompt charm measurement
to 1$\mbox{\,fb}^{-1}$ [32], a measurement with charm produced in semileptonic
$B$ decays with 1$\mbox{\,fb}^{-1}$ [33], and an update of the latter
measurement to 3$\mbox{\,fb}^{-1}$ [34]. With these new results, the HFAG
average [9] is $\Delta A_{CP}^{\mathrm{dir}}=(-0.253\pm 0.104)\%$, consistent
with zero CPV.

As well as updating the $\Delta A_{CP}$ measurement, the 3$\mbox{\,fb}^{-1}$
LHCb paper mentioned above [34] also quoted values for the individual
asymmetries $A_{CP}(D^{0}\rightarrow{{K}^{+}}{{K}^{-}})$ and
$A_{CP}(D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}})$. Because the $pp$ initial
state at LHCb is not charge antisymmetric (in the way that an $e^{+}e^{-}$ or
$p\overline{p}$ collision is), measuring these asymmetries requires correcting
for production and efficiency asymmetries. These were obtained from a careful
combination of control modes, extending a technique that had been used
previously at LHCb [35, 15]. This is a promising development for future
measurements at LHCb.

Numerous other results on CPV have been reported or published since CKM 2012,
from CLEO [36], BaBar [37, 38], Belle [39, 40] and LHCb [35, 41, 42, 43, 44].
None report a significant CP asymmetry. While there is not space here to
discuss all of these measurements, there are a couple of points worth noting.
One is that the analysis technology used to study CP asymmetries in multi-body
is advancing rapidly. Model-independent techiques were pioneered in 3-body
decays with studies of binned Dalitz plots [45, 46] and have proven their
sensitivity in $B$ decay modes to similar final states [47]. However, there is
always a question as to how to choose the binning: the optimal choice depends
on the model of CP violation. Unbinned methods [48] are now being applied.
These are especially interesting for decays to four-body (or more!) final
states, where a tradiational binned analysis (e.g. [42]) runs swiftly into the
curse of dimensionality. Another item worth singling out is the new Belle
result on $A(D^{0}\rightarrow\pi^{0}\pi^{0})$ [39]. This is an analysis that
LHCb cannot hope to do, and is an excellent example of the complementarity
between hadron and $e^{+}e^{-}$ experiments that we will see in the Belle-
II/LHCb upgrade era.

All of this experimental progress is very good news: multiple two-body decay
modes reaching precisions well below 1% and closing in on $10^{-3}$, and
searches for CPV in the phase space of multi-body decays with signal yields in
the millions. The theoretical side, however, remains unclear: one key lesson
from $\Delta A_{CP}$ is that direct CP asymmetries can arise at the
$\mathcal{O}(10^{-3})$ level in the SM, and as much as
$\mathcal{O}(\mathrm{few}\times 10^{-3})$ can be accommodated. We have already
reached this level of precision, so in the absence of a major advance in
theoretical engineering it seems like we would not be able to say whether some
future observation of direct CPV in a generic SCS mode at the $0.1\%$-level is
SM physics or not. A path forward has been suggested: to combine information
from $SU(3)$-related final states, testing whether the overall pattern is
consistent with the SM [49]. Alternatively, effort can be focused on modes
where the SM contribution is suppressed (e.g. SCS modes with $\Delta I=3/2$
[30]).

One last note: charmed baryons are also produced at significant rates at LHCb
[50], and good-sized samples have been accumulated at the $B$-factories. Their
decays are also sensitive to direct CPV, but measurements of this are sorely
lacking.

##  5 The future

From an experimental point of view, the coming years will be a very exciting
time for charm physics. At the time of writing, the LHC is near the end of a
long shutdown during which the machine is upgraded to operate at
$\sqrt{s}\approx 13$ TeV. Data-taking is scheduled to resume in 2015, with Run
2 lasting around three years. During this time LHCb will integrate about
5–6$\mbox{\,fb}^{-1}$. This will give the charm statistics a major boost: the
new data should correspond to about twice the current integrated luminosity,
at a higher energy (i.e. nearly twice the production cross-section), with an
improved software trigger. Indirect CPV measurements in $D^{0}\rightarrow
h^{+}h^{\prime-}$ with this sample will have an exciting level of sensitivity
to NP. In parallel, Belle II will begin commissioning in 2015 and should start
to integrate luminosity in earnest around 2018. Another LHC shutdown is
planned around 2018–2019, during which the LHCb detector will be upgraded to
allow a factor 5–10 increase in instantaneous luminosity. In total, LHCb
intends to integrate about 50$\mbox{\,fb}^{-1}$, and Belle II
50$\mbox{\,ab}^{-1}$. (Note that the cross-sections at the two colliders are
different, such that the LHCb sample corresponds to more $c\overline{c}$ pairs
produced.)

Both experiments have published estimates of their long-run physics
sensitivity [51, 52]. While such estimates should be treated with a degree of
caution****** This in a spirit of optimism: “more data makes us smarter”. , we
can expect an improvement of at least an order of magnitude in the
uncertainties on charm CP asymmetries. There is good complementarity between
the two experiments: the sheer statistics at LHCb will pin down key
observables in decays to $D^{0}\rightarrow h^{+}h^{\prime-}$ and
$D_{(s)}^{+}\rightarrow h^{+}h^{\prime+}h^{\prime\prime-}$ final states; Belle
II will give us depth, opening up many modes that LHCb struggles with; and
redundancy between the two will validate our control over systematic
uncertainties.

Last but not least, we can look forward to new results from charm-at-threshold
experiments such as BESIII. These are one of the only sources of information
on the strong phases $\delta$ that feed into the mixing average discussed in
Section 2. In particular, separate measurements of the strong phase in Dalitz
plot bins for $D^{0}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$
allows for model-independent measurements of $x$ and $y$ [53, 54] at LHCb and
Belle II.

##  6 Measurements reported during CKM 2014

These proceedings are based on a talk given in the opening session. Several
new experimental results, listed below, were reported during the conference.
For more details please consult the proceedings from those talks.

  * •

Measurements of $A_{\Gamma}$ in $D^{0}\rightarrow{{K}^{+}}{{K}^{-}}$,
${{\pi}^{+}}{{\pi}^{-}}$ at CDF [55]. Combining the two modes, an average
value of $A_{\Gamma}=(-0.12\pm 0.12)\%$ was reported. Presented by L. Sabato.

  * •

A time-integrated search for CPV in $D^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ at
LHCb [56]. An unbinned, model-independent method was used with a sample of
approximately 0.6M signal. No evidence for CPV was reported, with a $p$-value
for consistency with the null hypothesis of $(2.6\pm 0.5)\%$. Presented by E.
Gersabeck.

  * •

Preliminary results from BaBar on $P$, $C$, and $CP$ violation in 4-body $D$
decays, following a method proposed recently [57] to reinterpret existing
measurements of $T$-odd correlations. Presented by M. Martinelli.

##  7 Conclusions

There has been an enormous amount of progress in the past two years, with new
results from several experiments. After a ramp-up, the LHCb charm machine is
now in high gear, producing a steady stream of measurements. We can look
forward to a similar flow from Belle II in the not-too-distant future.

After a burst of excitement, $\Delta A_{CP}$ now looks firmly consistent with
the Standard Model once again. The same is true across the board in charm: we
find no evidence of anything amiss. The game is far from over, however: we can
expect a jump in precision for indirect CPV in the next few years, followed by
steady progress as the LHCb upgrade takes data. We are still operating in a
regime where the SM is—at least in principle—safely below current sensitivity
to indirect CPV, so that any observation would point to NP. For direct CPV the
situation is less clear, so all we can do is plough ahead, measure as much as
we can in as many modes as we can, and try to make sense of whatever nature
decides to give us.

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