# A compact Eulerian representation of axisymmetric inviscid vortex sheet
dynamics

Adriana I. Pesci1, Raymond E. Goldstein1, and Michael J. Shelley2 1Department
of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,
Cambridge CB3 0WA, United Kingdom
2Courant Institute of Mathematical Sciences, New York University
251 Mercer Street, NY 10021, USA

###### Abstract

A classical problem in fluid mechanics is the motion of an axisymmetric vortex
sheet evolving under the action of surface tension, surrounded by an inviscid
fluid. Lagrangian descriptions of these dynamics are well-known, involving
complex nonlocal expressions for the radial and longitudinal velocities in
terms of elliptic integrals. Here we use these prior results to arrive at a
remarkably compact and exact Eulerian evolution equation for the sheet radius
$r(z,t)$ in an explicit flux form associated with the conservation of enclosed
volume. The flux appears as an integral involving the pairwise mutual
induction formula for vortex loop pairs first derived by Helmholtz and
Maxwell. We show how the well-known linear stability results for cylindrical
vortex sheets in the presence of surface tension and streaming flows [A.M.
Sterling and C.A. Sleicher, J. Fluid Mech. 68, 477 (1975)] can be obtained
directly from this formulation. Furthermore, the inviscid limit of the
empirical model of Eggers and Dupont [J. Fluid Mech. 262 205 (1994); SIAM J.
Appl. Math. 60, 1997 (2000)], which has served as the basis for understanding
singularity formation in droplet pinchoff, is derived within the present
formalism as the leading order term in an asymptotic analysis for long slender
axisymmetric vortex sheets, and should provide the starting point for a
rigorous analysis of singularity formation.

##  I introduction

Perhaps the simplest example of finite-time singularities in surface motion is
that which occurs when the two loops supporting a catenoid-shaped soap film
are pulled apart beyond a critical separation, rendering the film unstable.
Extensive experimentation RobinsonSteen  has shown that the collapsing
axisymmetric surface eventually breaks up through self-contact at multiple
points, producing a series of satellite soap bubbles (Fig. 1). Theoretical
work on this dynamical process ranges from Maxwell’s original stability
analysis Maxwell_catenoid  to much more recent computational studies of fluid
dynamical models ListerHinch ; ChenSteen  focused on the nature of the
singularities.

In a broader sense, there have been two schools of thought in the study of
singularity formation by moving surfaces. On the one hand, there is a very
substantial body of rigorous work on the simplest geometrical law, namely
motion by mean curvature mean_curvature_flow . However, while this law is a
physically realistic description of interface motion arising from surface
diffusion, it cannot be applied to the motion of soap films because it can not
account for the dynamics of the surrounding fluid and the conservation laws
that follow. There are modifications of mean curvature flow that conserve
volume enclosed by the surface, but they are not faithful representations of
the dynamics of the surrounding fluid. On the other hand, there is the large
body of more phenomenological work on singularity formation in fluid mechanics
eggers_review , where simplified PDE models have been developed to address
self-similar dynamics near singularities. These models are physically
realistic but have generally lacked rigor.

The two examples where the gap between these two approaches has been bridged
is in the context of interface motion in two dimensions, where a systematic
procedure to derive PDEs for the evolution of asymptotically thin fluid layers
has been developed from the exact boundary-integral formulation GPS_PRL ;
PughShelley ; GPS_PRL2 ; GPS_PF . These analyses have put on a solid
foundation empirical models Constantin ; Dupont ; EggPont  derived within
lubrication theory. Given these results, the question then arises of whether
there is a comparable physical setup in three dimensions in which a PDE can be
derived for the motion of a surface surrounded by an incompressible fluid. The
simplest example of this would naturally be an axisymmetric surface surrounded
by an inviscid fluid. If there is an affirmative answer to this question, such
a PDE should have an explicit flux form.

In the inviscid limit, the moving interface can be represented by a vortex
sheet with surface tension. The predominant approach to this problem has been
through a Lagrangian formulation Moore ; CaflischLi , which is the most
appropriate for computational studies BakerOrzag ; NieBaker ; Nie ; MikeJohn .
However, such an approach does not readily lend itself to the development of
asymptotic models appropriate to thin necks, as would be relevant in the
neighborhood of singularities. In contrast, an Eulerian formulation would not
only be amenable to asymptotic analysis, but would also be subject more easily
to rigorous studies.

Figure 1: Collapse of a catenoid.

Here, as a first step, and starting from the Lagrangian formulation, we derive
an exact Eulerian dynamics for axisymmetric vortex sheets with surface
tension, and show that it has an explicit flux form. Naturally, because the
problem is deeply nonlocal due to the Biot-Savart interactions between distant
elements of the vortex sheet, the flux is an integral over the entire sheet
whose kernel is precisely the mutual induction between two coaxial loops as
derived by Helmholtz for fluids Helmholtz_loops  and by Maxwell for electrical
currents Maxwell_loops . As the mutual induction is, by definition, the flux
through one loop due to the circulation in the other, this is a very intuitive
result. We show that known stability results for cylindrical vortex sheets can
be recovered by direct calculation. While nonlocal, the mutual induction
between two coaxial loops of the same radius is sharply peaked as the distance
between the loops vanishes, and this feature suggests a natural asymptotic
analysis to reduce systematically the dynamics to a local PDE. In this limit
we recover at first order the inviscid version EggSIAM  of the empirical model
first proposed by Eggers and Dupont EggPont  for this problem.

##  II Flux Equation

We consider an infinite three-dimensional inviscid fluid of density $\rho$
within which is a vortex sheet that is axisymmetric about the $z$ axis, and
whose time-dependent radius is $r(z,t)$. In this inviscid limit there is in
general a discontinuity in the tangential fluid velocity across the sheet, and
this jump defines the vortex sheet strength $\gamma$. In the same way that we
use a minimal surface as an idealization of a static soap film, here the
vortex sheet is the idealized representation of a moving film endowed with
surface tension $\sigma$. As is well-known CaflischLi , the problem of the
self-induced motion of the sheet reduces to the coupled dynamics of $r(z,t)$
and $\gamma(z,t)$, for which the vortex sheet strength evolves as

${\partial\gamma\over\partial
t}=-{\sigma\over\rho}{\partial\kappa\over\partial\alpha},$ (1)

where $\kappa$ is the mean curvature of the surface,

$\kappa=\frac{r_{zz}}{\left(1+r_{z}^{2}\right)^{3/2}}-\frac{1}{r\left(1+r_{z}^{2}\right)^{1/2}}~{},$
(2)

and the radial and axial Lagrangian velocities are the principal value
integrals

$\displaystyle z_{t}\equiv{dz\over dt}$ $\displaystyle=-{1\over
4\pi}\mathchoice{{\vbox{\hbox{$\textstyle=$}}\kern-7.3611pt}}{{\vbox{\hbox{$\scriptstyle=$}}\kern-4.80557pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-3.40279pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-2.98611pt}}\\!\int\gamma^{\prime}\left[{r^{\prime}}^{2}J_{2}-rr^{\prime}J_{1}\right]d\alpha^{\prime},$
(3a) $\displaystyle r_{t}\equiv{dr\over dt}$ $\displaystyle=-{1\over
4\pi}\mathchoice{{\vbox{\hbox{$\textstyle=$}}\kern-7.3611pt}}{{\vbox{\hbox{$\scriptstyle=$}}\kern-4.80557pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-3.40279pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-2.98611pt}}\\!\int\gamma^{\prime}r^{\prime}(z-z^{\prime})J_{1}d\alpha^{\prime},$
(3b)

where $\alpha^{\prime}$ is a Lagrangian label, $r^{\prime}\equiv
r(\alpha^{\prime},t)$, $\gamma^{\prime}\equiv\gamma(\alpha^{\prime},t)$, and

$\displaystyle J_{1}$
$\displaystyle=\int_{0}^{2\pi}\cos\theta^{\prime}\left[(z-z^{\prime})^{2}+r^{2}+{r^{\prime}}^{2}-2rr^{\prime}\cos\theta^{\prime}\right]^{-3/2}d\theta^{\prime},$
(4a) $\displaystyle J_{2}$
$\displaystyle=\int_{0}^{2\pi}\left[(z-z^{\prime})^{2}+r^{2}+{r^{\prime}}^{2}-2rr^{\prime}\cos\theta^{\prime}\right]^{-3/2}d\theta^{\prime}.$
(4b)

Performing the integrations in $\theta^{\prime}$ yields the following
expressions for the velocities in terms of $K$ and $E$, the complete elliptic
integrals of the first and second kind, respectively, with argument
$k^{2}=4rr^{\prime}/\left[(z-z^{\prime})^{2}+(r+r^{\prime})^{2}\right]$,

$\displaystyle z_{t}$
$\displaystyle={1\over\pi}\mathchoice{{\vbox{\hbox{$\textstyle=$}}\kern-7.3611pt}}{{\vbox{\hbox{$\scriptstyle=$}}\kern-4.80557pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-3.40279pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-2.98611pt}}\\!\int
d\alpha^{\prime}\gamma^{\prime}r^{\prime}\left({k^{2}\over
4rr^{\prime}}\right)^{3/2}\left[{r^{\prime}-r\over 1-k^{2}}E(k)-{2r\over
k^{2}}\left[E(k)-K(k)\right]\right]\equiv\mathchoice{{\vbox{\hbox{$\textstyle=$}}\kern-7.3611pt}}{{\vbox{\hbox{$\scriptstyle=$}}\kern-4.80557pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-3.40279pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-2.98611pt}}\\!\int
d\alpha^{\prime}\gamma^{\prime}P(\alpha,\alpha^{\prime})$ (5a) $\displaystyle
r_{t}$
$\displaystyle={1\over\pi}\mathchoice{{\vbox{\hbox{$\textstyle=$}}\kern-7.3611pt}}{{\vbox{\hbox{$\scriptstyle=$}}\kern-4.80557pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-3.40279pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-2.98611pt}}\\!\int
d\alpha^{\prime}\gamma^{\prime}r^{\prime}(z-z^{\prime})\left({k^{2}\over
4rr^{\prime}}\right)^{3/2}\left[{2\over k^{2}}\left[E(k)-K(k)\right]+{1\over
1-k^{2}}E(k)\right]\equiv\mathchoice{{\vbox{\hbox{$\textstyle=$}}\kern-7.3611pt}}{{\vbox{\hbox{$\scriptstyle=$}}\kern-4.80557pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-3.40279pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-2.98611pt}}\\!\int
d\alpha^{\prime}\gamma^{\prime}Q(\alpha,\alpha^{\prime}).$ (5b)

In order to obtain an Eulerian equation of motion from these Lagrangian
velocities we make the change of variables from $\alpha$ to $z=z(\alpha,t)$.
This yields PughShelley

$\displaystyle{\partial r\over\partial t}$ $\displaystyle=r_{t}-z_{t}{\partial
r\over\partial z}$ (6a) $\displaystyle{\partial\gamma\over\partial
t}+z_{t}{\partial\gamma\over\partial z}$
$\displaystyle=-{\sigma\over\rho}z_{\alpha}{\partial\kappa\over\partial z}.$
(6b)

The quantity $\tilde{\gamma}=\gamma/s_{\alpha}$ represents the actual jump in
tangential velocity Peng ; Herrmann , where $s$ is the arclength, whereas it
is the unnormalized vortex sheet strength $\gamma$ that appears in Eqs. 5 and
6. For an Eulerian description in which we make the change of variables from
$\alpha$ to $z=z(\alpha,t)$, it is natural to consider instead the quantity
$\eta=\gamma/z_{\alpha}$ , so
$d\alpha^{\prime}\gamma^{\prime}=dz^{\prime}\eta^{\prime}$.

For this system, conservation of fluid volume should be expressible as a flux
form involving the cross-sectional area $\pi r^{2}$, that is,

${\partial r^{2}\over\partial t}=-{\partial F\over\partial z},$ (7)

for some function $F$ which we now seek. The fact that $2r\partial r/\partial
t=\partial r^{2}/\partial t$, implies that $2r(r_{t}-z_{t}\partial r/\partial
z)=-\partial F/\partial z$. However, to infer $F$ by direct substitution of
(5a) and (5b) into this expression is cumbersome and non-trivial. A much
better way to find $F$ is to make use of Eqs. (3) and (4). After some algebra
and integrations by parts, we find

$F={1\over
2\pi}\mathchoice{{\vbox{\hbox{$\textstyle=$}}\kern-7.3611pt}}{{\vbox{\hbox{$\scriptstyle=$}}\kern-4.80557pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-3.40279pt}}{{\vbox{\hbox{$\scriptscriptstyle=$}}\kern-2.98611pt}}\\!\int\eta^{\prime}rr^{\prime}dz^{\prime}\int_{0}^{2\pi}{\cos\theta^{\prime}\over\left[(z-z^{\prime})^{2}+r^{2}+{r^{\prime}}^{2}-2rr^{\prime}\cos\theta^{\prime}\right]^{1/2}}d\theta^{\prime}.$
(8)

The integral over the variable $\theta^{\prime}$ can be calculated in terms of
elliptic integrals, with the result

${\partial r^{2}\over\partial t}=-{1\over\pi}{\partial\over\partial z}\int
dz^{\prime}\eta^{\prime}{\sqrt{rr^{\prime}}}M(k)~{},$ (9)

where $M(k)$ is Maxwell’s function first derived for the mutual induction of a
pair of circular current loops of radii $r$ and $r^{\prime}$ at locations $z$
and $z^{\prime}$,

$M={2\over k}\left(K-E-{k^{2}\over 2}K\right).$ (10)

That this remarkably compact result has not been previously obtained in this
context may be a consequence of the natural emphasis on the Lagrangian
formulation and its computational applications. The results in (9) and (10)
are intuitive, in that the mutual induction $M$ is, by definition, the flux
passing through one loop due to another, and so $F$ is simply the sum of all
the individual fluxes through the loop at position $z$. Moreover, in the case
of a vortex loop, the integrand in $F$ was identified by Helmoltz
Helmholtz_loops  as the stream function of the flow.

In closing this section, we note that if we rewrite the pair (5) as $z_{t}=W$
and $r_{t}=V$, then $\eta_{t}+W_{z}\eta=-(\sigma/\rho)\kappa_{z}$. Considering
$\eta(\alpha,t)=\tilde{\eta}(z(\alpha,t),t)$, and similarly
$r(\alpha,t)=R(z(\alpha,t),t)$, the Eulerian evolution equations for $\eta$
and $R$ that parallel (6) are

$\displaystyle{\partial R\over\partial t}$ $\displaystyle=V-W{\partial
R\over\partial z}$ (11a) $\displaystyle{\partial\tilde{\eta}\over\partial
t}+{\partial\over\partial z}\left(W\tilde{\eta}\right)$
$\displaystyle=-{\sigma\over\rho}{\partial\kappa\over\partial z}.$ (11b)

Multiplying (11b) by $R$ and integrating the incompressibility relationship
$(1/r)(rV)_{r}+W_{z}=0$, one can easily show

$\frac{\partial}{\partial
t}\left(\frac{1}{2}R^{2}\right)=-\frac{\partial}{\partial
z}\int_{0}^{R(z)}\\!\\!dr^{\prime}r^{\prime}W(z,r^{\prime}).$ (12)

Introducing the Stokes stream function $\Psi$, which we identify as the
integral on the rhs of (12), with $rV=-\partial\Psi/\partial z$ and
$rW=\partial\Psi/\partial r$, we obtain

$\frac{\partial}{\partial
t}\left(\frac{1}{2}R^{2}\right)=-\frac{\partial\Psi}{\partial z},$ (13)

thus confirming the connection between the functions $F$ and $\Psi$.

##  III Hamiltonian Structure

In this section we discuss some issues regarding a possible Hamiltonian
formulation of the present system, for which there is a vortex sheet strength
with nontrivial dynamical evolution. It is useful to contrast this case with
that of a system of discrete vortex rings and its continuous limit Meleshko ,
for despite some fundamental distinctions involving conservations laws, there
are common mathematical structures involved. As first shown by Dyson Dyson ,
the Lagrangian dynamics of a discrete set of coaxial vortex rings with centers
$Z_{i}$ on the $z$-axis, with radii $R_{i}$ and circulation $\Gamma_{i}$, can
be written as

$\displaystyle R_{i}{\dot{Z}}_{i}$
$\displaystyle=\frac{\Gamma_{i}}{4\pi}\left(\log\frac{8R_{i}}{a_{i}}-\frac{1}{4}\right)+\sum_{j=1,j\neq
i}^{N}\frac{\Gamma_{j}}{2\pi}\frac{\partial I_{ij}}{\partial R_{i}}$ (14a)
$\displaystyle R_{i}{\dot{R}}_{i}$ $\displaystyle=-\sum_{j=1,j\neq
i}^{N}\frac{\Gamma_{j}}{2\pi}\frac{\partial I_{ij}}{\partial Z_{i}},$ (14b)

where $I_{ij}=\sqrt{R_{i}R_{j}}M(k_{ij})$ is the mutual inductance between the
two loops $i$ and $j$. The first term on the rhs of (14a) is the approximate
self-induced velocity of ring $i$, where $a_{i}$ is the core radius, which
serves as a cutoff for the localized induction approximation.

The evolution equation (9) can now be seen as the continuum limit of (14b),
expressed in an Eulerian form. A direct calculation shows a less obvious
result, namely that the axial velocity $z_{t}$ in (5a) is the equivalent of
(14a), without the self-induction term. In fact,

$rz_{t}={1\over\pi}{\partial\over\partial
r}\int_{-\infty}^{\infty}d\alpha^{\prime}\gamma^{\prime}{\sqrt{rr^{\prime}}M(k)}={\partial
F\over\partial r}.$ (15)

By analogy to the discrete case, in which one can introduce an energy that is
quadratic in the circulations, $\Gamma_{i}$, namely

$U=\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\frac{\Gamma_{i}\Gamma_{j}}{2\pi}I_{ij},$ (16)

and from which one obtains the equations of motion

$\displaystyle\Gamma_{i}R_{i}{\dot{Z}}_{i}$
$\displaystyle=\frac{\Gamma_{i}^{2}}{4\pi}\left(\log\frac{8R_{i}}{a_{i}}-\frac{1}{4}\right)+\frac{\partial
U}{\partial R_{i}}$ (17a) $\displaystyle\Gamma_{i}R_{i}{\dot{R}}_{i}$
$\displaystyle=-\frac{\partial U}{\partial Z_{i}},$ (17b)

one can take the continuum limit to obtain Novikov

${\cal U}=\frac{1}{2\pi}\int\\!d\alpha\int
d\alpha^{\prime}\gamma(\alpha)\gamma(\alpha^{\prime})\sqrt{rr^{\prime}}M(k),$
(18)

which, generalizing to functional derivatives, yields

$\gamma rz_{t}={\delta{\cal U}\over\delta r}\ \ \ \ {\rm and}\ \ \ \ \gamma
rr_{t}=-{\delta{\cal U}\over\delta z}.$ (19)

This is of the same form as the discrete dynamics (17), but with the important
distinction that the vortex sheet strength $\gamma$ itself depends on time,
whereas the individual circulations $\Gamma_{i}$ in the discrete case do not.
Moreover, in the discrete case it is possible to rewrite the dynamics so the
left-hand-sides are total time derivatives, rendering them truly Hamiltonian.
In contrast, the dynamics (19) do not obviously have this feature. The fact
that (1) can be throught of as a nonholonomic constraint may offer a path to
obtain an “almost-Hamiltonian” dynamics Fernandez .

##  IV Stability Analysis

In this section we show how the vortex sheet dynamics in the Eulerian form
reproduces known stability results Alterman ; Sterling  for capillary jets,
both with and without a streaming velocity within the fluid enclosed by the
sheet. Note that our assumption that the fluids inside and outside the sheet
have the same density precludes recovering the original stability result of
Rayleigh Rayleigh , which assumed vacuum outside.

We first consider the case with a quiescent fluid on both sides of a vortex
sheet of radius $R$, and linearize the equations of motion for small
perturbations in $\gamma$ and $r$ of the form

$\gamma={\hat{\gamma}}e^{iqz+\beta t}\ \ \ \ {\rm and}\ \ \ \
r=R+{\hat{\zeta}}e^{iqz+\beta t}.$ (20)

At this order the mean curvature has the simplified form $\kappa\simeq
r_{zz}-r^{-1}$. The resulting vortex sheet evolution equation is

$\beta{\hat{\gamma}}=iP(1-P^{2}){\sigma\over\rho R^{3}}~{}{\hat{\zeta}},$ (21)

where $P=qR$. Since, in the absence of background fluid motion, $\gamma$ is
first order in the perturbation, the remaining factors on the rhs of (9) are
those corresponding to a cylinder. Let

$k_{R}^{2}={4R^{2}\over(z-z^{\prime})^{2}+4R^{2}},\ \ \ \ {\rm and}\ \ \ \
x={(z^{\prime}-z)\over 2R}.$ (22)

Then, the linearization of (9) is

$\beta{\hat{\zeta}}=-{4\over\pi}iP{\hat{\gamma}}\int_{0}^{\infty}M(k_{R})\cos(2Px)~{}dx$
(23)

After calculating the integral (see Appendix), we substitute $\hat{\zeta}$
from (23) into (21) to obtain, in agreement with previous results Alterman ;
Sterling , the growth rate

$\beta^{2}={\sigma\over\rho
R^{3}}~{}P^{2}\left(1-P^{2}\right)I_{1}(P)K_{1}(P)$ (24)

where $I_{n}$ and $K_{n}$ are the $n$-th order modified Bessel functions of
the first and second kind, respectively.

Demonstrating that the flux-form PDE reproduces the stability results in the
presence of streaming flows, obtained by Alterman Alterman  and by Sterling
and Sleicher Sterling , follows the same procedure as the calculation above,
but requires a more delicate analysis. This complexity is related to the
limiting procedure of vanishing sheet thickness $\delta$ and vanishing
viscosity used to arrive at the evolution equation for the vortex sheet
strength (6). In particular, these limits preclude the determination of a a
unique value of the tangential sheet velocity $z_{t}$. For concreteness,
consider the situation in which the streaming velocities inside and outside of
the cylindrical vortex sheet are $U$ and $0$, respectively. In the Lagrangian
formalism, this ambiguity is eliminated by setting $z_{t}=U/2$ CaflischLi .
However, this choice cannot satisfy the boundary conditions on either side of
the sheet in the Eulerian frame. It is therefor necessary to reinstate the
appropriate boundary conditions, which is equivalent to taking the limit
$\delta\to 0$ at the end of the calculation.

The calculation proceeds by systematic perturbation of the terms within the
flux integral. We expand theseparately the two, obtaining

$M(k)=M(k_{R})+{dM\over dk}\bigg{|}_{k_{R}}(k-k_{R})+\cdots\ \ \ \ {\rm and}\
\ \ \ \gamma^{\prime}{\sqrt{rr^{\prime}}}\simeq
R(\gamma_{0}+{\tilde{\gamma}})\left(1+{\zeta^{\prime}+\zeta\over
2R}\right)+\cdots,$ (25)

where $k_{R}$ is defined in (22). Using

$k-k_{R}\simeq k_{R}(1-k_{R}^{2})\left({\zeta^{\prime}+\zeta\over 2R}\right),$
(26)

and collecting terms, we obtain (see Appendix)

$\gamma^{\prime}{\sqrt{rr^{\prime}}}M(k)\simeq\gamma_{0}RM(k_{R})+R{\tilde{\gamma}^{\prime}}M(k_{R})-\gamma_{0}Rk_{R}[E(k_{R})-K(k_{R})]\left({\zeta^{\prime}+\zeta\over
2R}\right).$ (27)

Substituting into the equations of motion and retaining only first order
terms, we find

$\beta{\hat{\zeta}}=-{1\over\pi}\left[2iP{\hat{\gamma}}\int_{0}^{\infty}M(k_{R})\cos(2Px)~{}dx-i{P\gamma_{0}{\hat{\zeta}}\over
R}\int_{0}^{\infty}\left[1+\cos(2Px)\right]k_{R}\left[E(k_{R})-K(k_{R})\right]~{}dx\right]$
(28)

After some laborious calculation of these non-trivial integrals we obtain

$\beta{\hat{\zeta}}=-iPI_{1}(P)K_{1}(P){\hat{\gamma}}-i{\gamma_{0}\over
2R}P\left[1+P\left(I_{0}(P)K_{1}(P)-I_{1}(P)K_{0}(P)\right)\right]~{}{\hat{\zeta}}.$
(29)

Using the Bessel function identity $P(I_{0}K_{1}+I_{1}K_{0})=1$, we further
reduce this expression to

$\left(\beta+i{P\gamma_{0}\over
R}K_{1}(P)I_{0}(P)\right){\hat{\zeta}}=-2iPI_{1}(P)K_{1}(P)~{}{\hat{\gamma}}$
(30)

This equation is the evolution for perturbations to the mean radius, i.e. the
mean location of the vortex sheet: $r=(r^{+}+r^{-})/2)$, where $r^{+}$ and
$r^{-}$ are the outer and inner radius of the infinitesimaly thin sheet,
respectively ($r^{+}-r^{-}=\delta\to 0$). To reinstate the proper boundary
conditions we note that (30) should be split into two parts, the contribution
from $r^{+}$ and the one from $r^{-}$. To do so we rewrite (30) as

$\left(P\left(\beta+i{P\gamma_{0}\over
R}\right)K_{1}(P)I_{0}(P)+P(\beta-0)K_{0}(P)I_{1}(P)\right){\hat{\zeta}}=-{iP\over
2}{\hat{\gamma}}K_{1}(P)I_{1}(P)-{iP\over 2}{\hat{\gamma}}K_{1}(P)I_{1}(P)$
(31)

In this equation all terms involving $I_{0}$ are connected to the inner
region, that is $r^{-}$, while those involving $I_{1}$ correspond to the outer
one ($r^{+}$) yielding two equations, one for each region:

$P\left(\beta+i{P\gamma_{0}\over
R}\right)K_{1}(P)I_{0}(P){\hat{\zeta}}=-{iP\over
2}{\hat{\gamma}}K_{1}(P)I_{1}(P)$ (32)

and

$P\beta{\hat{\zeta}}K_{0}(P)I_{1}(P)=-{iP\over
2}{\hat{\gamma}}K_{1}(P)I_{1}(P)$ (33)

A similar expansion of the vortex sheet evolution equation (1) yields for the
inner region

$\left(\beta+i{P\gamma_{0}\over
R}\right){\hat{\gamma}}={\sigma\over\rho}{iP(1-P^{2})\over
R^{3}}{\hat{\zeta}},$ (34)

and for the outer one

$\beta{\hat{\gamma}}={\sigma\over\rho}{iP(1-P^{2})\over R^{3}}{\hat{\zeta}}.$
(35)

Substituting (34) and (35) into (32) and (33), respectively, and adding the
contributions to obtain the total expansion for $r=(r^{-}/2)+(r^{+}/2)$ yields

$\beta^{2}+2i\beta\frac{\gamma_{0}}{R}P^{2}I_{0}(P)K_{1}(P)=\frac{\sigma}{\rho
R^{3}}P^{2}\left(1-P^{2}\right)I_{1}(P)K_{1}(P)+\frac{\gamma_{0}^{2}}{R^{2}}P^{3}I_{0}(P)K_{1}(P)~{},$
(36)

and which coincides with previous results Sterling ; Alterman  for the case
where the inner fluid velocity is $U=\gamma_{0}$ and the outer fluid is
stationary.

##  V Derivation of a local PDE

To obtain an approximate PDE that describes the dynamics of slender necks, it
is necessary to find a suitable approximation to $M(k)$ that would make it
possible to calculate the integral in the flux equation in a controlled
manner. With this purpose in mind, it is very useful to note GradRyz  that
$M(k)=Q_{1/2}(\chi)$, with $Q_{1/2}(\chi)$ the associated Legendre function of
index $1/2$ with variable $\chi=(2/k^{2})-1$, which obeys the differential
equation

$(1-\chi^{2}){d^{2}Q\over d\chi^{2}}-2\chi{d^{2}Q\over d\chi^{2}}+{3\over
4}Q=0.$ (37)

This identity makes it possible to obtain a uniform approximation for $M(k)$
by matching the inner and outer solutions, ($\chi\to 1$ and $\chi\to\infty$,
respectively). In the inner region the limiting behavior is

$Q_{1/2}\simeq A\ln\left(\sqrt{{2\over\chi-1}}\right)\ \ \ {\rm for}\ \ \
\chi\to 1,$ (38)

while in the outer one is

$Q_{1/2}\simeq B\sqrt{{(\chi+1)^{3}(\chi-1)\over 16}}\ \ \ {\rm for}\ \ \
\chi\to\infty,$ (39)

where $A$ and $B$ are constants to be adjusted by matching the inner and outer
solutions, with the final result in terms of $k$ given by

$M(k)\simeq M_{u}(k)={1\over 4}\left[3\ln\left({1\over(1-k^{2})}\right)-\pi
k^{3/2}{\sqrt{1-k^{2}}}\right].$ (40)

The fit of $M(k)$ provided by the uniform approximation $M_{u}(k)$ is
remarkable; $M(k)-M_{u}(k)\gtrapprox 0$ only in a small neighbourhood of
$k^{2}=1/2$; everywhere else it is almost impossible to distinguish $M_{u}(k)$
from the exact solution (see Fig. 2).

Figure 2: Uniform approximation to the Maxwell function. Solid black line is
the exact expression $M(k(x))$. Dash-dot and dashed blue lines are the inner
and outer approximations, respectively. Dashred red line is the uniform
approximation $M_{u}(k)$.

We can now approximate $M(k)$ by $M_{u}(k)$ in the evolution equation for
$r^{2}(t)$ (9). In order to expression $M_{u}(k)$ in a form that allows
calculation of the relevant integral, we Further, using the expression for $k$
in terms of $z$, $z^{\prime}$, $r$, and $r^{\prime}$ it is possible to rewrite
in a way that allows to calculate the integrals. In particular, using

$k^{2}={4rr^{\prime}\over{(z^{\prime}-z)^{2}+(r+r^{\prime})^{2}}},$ (41)

the denominator inside the logarithm in (40) is expandable in Taylor series in
the limit $z\to z^{\prime}$ because the ratio
$(r^{\prime}-r)/(z^{\prime}-z)\to r_{z}$. In this limit, $r\to r^{\prime}$,
and rescaling $r=\epsilon R$ we find

$\displaystyle\ln\left({1\over(1-k^{2})}\right)$
$\displaystyle\simeq\ln\left[{(z^{\prime}-z)^{2}+\epsilon^{2}(2R)^{2}\over{(z^{\prime}-z)^{2}\left(1+\epsilon^{2}R_{z}^{2}\right)}}\right]=\ln\left[{x^{2}+\epsilon^{2}\over
x^{2}}\right]-\ln\left(1+\epsilon^{2}R_{z}^{2}\right),$ (42a) $\displaystyle
k^{2}{\sqrt{1-k^{2}}}$ $\displaystyle\simeq
8\epsilon^{3}R^{3}{|z^{\prime}-z|\left(1+\epsilon^{2}R_{z}^{2}\right)^{1/2}\over{\left[(z^{\prime}-z)^{2}+\epsilon^{2}(2R)^{2}\right]^{2}}}={\epsilon^{3}|x|\over(x^{2}+\epsilon^{2})^{2}}\left(1+\epsilon^{2}R_{z}^{2}\right)^{1/2},$
(42b)

where $x$ is the same as defined in (22). The contribution to the flux $F$
arising from the two terms in the uniform approximation (40) involves $I_{1}$
and $I_{2}$, where, by the symmetry of the integrand,

$\displaystyle I_{1}$ $\displaystyle={3\over 2}\epsilon\gamma
R^{2}\int_{0}^{\infty}\ln\left[{x^{2}+\epsilon^{2}\over
x^{2}}\right]dx+O(\epsilon^{3}RR_{z}^{2})$ (43a) $\displaystyle I_{2}$
$\displaystyle=\pi\epsilon^{4}\gamma
R^{2}\int_{0}^{\infty}{x\over\left[x^{2}+\epsilon^{2}\right]^{2}}dx+O(\epsilon^{3}RR_{z}^{2})$
(43b)

Performing the integrations we obtain

$I_{1}={3\pi\over 2}\epsilon^{2}\gamma R^{2}\ \ \ \ {\rm and}\ \ \ \
I_{2}={\pi\over 2}\epsilon^{2}\gamma R^{2}.$ (44)

Collecting terms and substituting $r=\epsilon R$ in the LHS of the flux
equation (9) we obtain

${\partial R^{2}\over\partial t}=-{\partial\over\partial z}\left(\gamma
R^{2}\right).$ (45)

While this result was obtained in a systematic way that would be useful in
cases where it is necessary, or desirable to obtain the next terms of the
expansion, the procedure is not very intuitive. In fact, the meaning and
procedence of (45) can be more easily understood by recognising that the first
order approximation corrresponds to a cylindrical vortex sheet, where
$r=r^{\prime}$ and $\gamma^{\prime}=\gamma$. Then, starting as before from (9)
we may write

${\partial r^{2}\over\partial t}=-{2\over\pi}{\partial\over\partial
z}\left(\gamma r^{2}\int_{-\infty}^{+\infty}dxM(k(x))\right)~{},$ (46)

where we have used a variant of the usual variable $x$, namely
$x=(z^{\prime}-z)/2r$. Because the value of the integral is $\pi/2$, (46)
yields the same result (45) that was obtained for the first order term by
employing the expansion method making clear its geometric interpretation.

##  VI Discussion

The present work has shown hwo the familiar axisymmetric vortex sheet dynamics
can be recast exactly in an Eulerian form which clearly displays the inherent
flux form of the motion. Previously established linear stability results are
easily recovered with this formulation. Most signficantly, the Eulerian
dynamics is well-suited to the development of a systematic reduction to a
local approximation when the aspect ratio of the system is suitably small.
This controlled approximation, which at leading order recovers well-known
empirical results EggPont , should lend itself to more rigorous studies both
of the form and validity of the governing PDEs as well as the singularities
they produce. As the underlying model includes the dynamics of an
incompressible fluid surrounding the vortex sheet, it consitutes a more
physically relevant starting point for understanding singularity formation by
moving soap films than that provided by mean curvature flows.

In light of the large body of existing work on axisymmetric vortex sheets,
there are many possible extensions of the present work, including the
introduction of swirl CaflischLiShelley  or helical symmetry CaflischLi .
Greater challenges would be to incorporate weak viscous effects Lundgren ;
Ceniceros , and to complete the Hamiltonization of the inviscid Eulerian case
in the presence of surface tension.

##  VII acknowledgments

We are indebted to Keith Moffatt for stimulating discussions at an early stage
of this project, and to John Hinch for discussions on the asymptotic analysis
and the significance of the flux form. This work was supported in part by
Established Career Fellowship EP/M017982/1 from the EPSRC (REG & AIP). REG and
AIP are grateful to the I.H.E.S., and especially Patrick Gourdon, for
hospitality during an extended visit supported by the Schlumberger Visiting
Professorship (REG).

##  VIII Appendix

Here we show some intermediate steps in the stability analysis. We begin from
the first equation of (25),

$M(k)=M(k_{R})+{dM\over dk}\bigg{|}_{k_{R}}(k-k_{R})$ (47)

where $k_{R}$ is as defined in (22), and using the relationships between the
elliptic functions and its derivatives we find

${dM\over dk}={2\over k^{2}}(E-K)-{2\over k}\left[{dE\over dk}-{dK\over
dk}\right]-K-k{dK\over dk}.$ (48)

The corresponding first-order expansions of the relevant quantities are

$k^{2}={4(R+\zeta^{\prime})(R+\zeta)\over(z-z^{\prime})^{2}+(2R+\zeta^{\prime}+\zeta))^{2}}\simeq
k_{R}^{2}\left[1+(1-k_{R}^{2})\left({\zeta^{\prime}+\zeta\over
R}\right)\right],$ (49)

which yields (26), and

${dM\over dk}\bigg{|}_{k_{R}}={2-k_{R}^{2}\over
k_{R}^{2}(1-k_{R}^{2})}E(k_{R})-{2\over k_{R}^{2}}K(k_{R}),$ (50)

where we have used the relationships $dE/dk=(E-K)/k$ and
$kdK/dk=E/(1-k^{2})-K$ GradRyz , and have substituted $k=k_{R}$. Introducing
the convenient notation

$M_{0}\equiv M(k_{R})\ \ {\rm and}\ \ M_{1}\equiv{dM\over
dk}\bigg{|}_{k_{R}}k_{R}(1-k_{R}^{2}),$ (51)

we note that, after straightforward algebra, the quantity $M_{0}+M_{1}$, which
will be needed later, reduces to $-k_{R}(E-K)$.

Utilizing the second of the equations in (25) and collecting all the
contributions we obtain

$\gamma^{\prime}{\sqrt{rr^{\prime}}}M(k)\simeq\gamma_{0}RM_{0}+\gamma_{0}R(M_{0}+M_{1})\left({\zeta^{\prime}+\zeta\over
2R}\right)+{\tilde{\gamma}}RM_{0}.$ (52)

Using again the identity $M(k_{R})=Q_{1/2}(\chi)$, with argument
$\chi=2/k_{R}-1$, and substituting into (52) we obtain the first order result

$\gamma^{\prime}{\sqrt{rr^{\prime}}}M(k)\simeq\gamma_{0}RM(k_{R})+R{\tilde{\gamma}^{\prime}}Q_{1/2}({\chi}_{R})-\gamma_{0}Rk_{R}[E(k_{R})-K(k_{R})]\left({\zeta^{\prime}+\zeta\over
2R}\right).$ (53)

Substituting (53) into the equations of motion and retaining only first order
terms we find

$2R{\partial\zeta\over\partial t}=-{1\over\pi}{\partial\over\partial
z}\left[\int_{-\infty}^{\infty}d\alpha^{\prime}\gamma_{0}M_{0}R+R\int_{-\infty}^{\infty}d\alpha^{\prime}{\tilde{\gamma}}Q_{1/2}-\gamma_{0}\int_{-\infty}^{\infty}k_{R}[E(k_{R})-K(k_{R})]\left({\zeta^{\prime}+\zeta\over
2R}\right)~{}d\alpha^{\prime}\right].$ (54)

The first integral on the rhs does not contribute to the equation of motion,
for $\gamma_{0}=0$ identically when the fluids on both sides of the vortex
sheet are stationary, and the integral is a constant when there is a streaming
velocity in either one of the two regions. The other integrals involve the
Lagrangian parameter $\alpha$, and to be able to obtain the result in the
Eulerian frame as a function of the coordinate $z$ we note that
$ds={\sqrt{r_{\alpha}^{2}+z_{\alpha}^{2}}}~{}d{\alpha}$ and
$ds={\sqrt{1+r_{z}^{2}}}~{}dz$. Using these identities allows us to rewrite

$dz={\sqrt{1+r_{z}^{2}\over
r_{\alpha}^{2}+z_{\alpha}^{2}}}~{}d{\alpha}\simeq(1+{\cal
O}(\zeta^{2}))d{\alpha}$ (55)

making it possible to approximate $d{\alpha^{\prime}}\simeq dz^{\prime}$ to
the required order. Replacing the expressions for the perturbations (20)
into(54), rewriting all exponentials in terms of $z^{\prime}-z$ and expressing
them in terms of sines and cosines makes it clear that the only non-zero
contributions to the integrals come from the cosine terms which are the only
ones that produce even integrands. Then, defining $P=qR$, and performing the
usual change of variables $x=(z^{\prime}-z)/2R$ we find
$k_{R}(x)=1/(1+x^{2})$, and $\chi(x)=1+2x^{2}$, yielding the first one of the
two stability equations

$\beta{\hat{\zeta}}=-{1\over\pi}\left[2iP{\hat{\gamma}}\int_{0}^{\infty}~{}Q_{1/2}(\chi(x))\cos(2Px)~{}dx-i{P\gamma_{0}{\hat{\zeta}}\over
R}\int_{0}^{\infty}\left[1+\cos(2Px)\right]k_{R}[E(k_{R})-K(k_{R})]~{}dx\right].$
(56)

The integrals have been calculated with the help of Okui  and GradRyz . In
particular, after a further change of variable, $y={\sqrt{2}}\ x$, the first
integral acquires its canonical form 7.162.6 GradRyz  with $\nu=1/2$, and
$a={\sqrt{2}}\ P$, thus

$\int_{0}^{\infty}~{}Q_{1/2}(1+y^{2})\cos({\sqrt{2}}\ Py)~{}dy={\pi\over
2}K_{1}(P)I_{1}(P).$ (57)

The second integral can be divided in two contributions

$\displaystyle{\cal I}_{1}$
$\displaystyle=\int_{0}^{\infty}k_{R}[E(k_{R})-K(k_{R})]~{}dx,$ (58a)
$\displaystyle{\cal I}_{2}$
$\displaystyle=\int_{0}^{\infty}\cos(2Px)~{}k_{R}[E(k_{R})-K(k_{R})]~{}dx.$
(58b)

To calculate ${\cal I}_{1}$ we use the identity 3.2 (2) in Okui  for $p=1$ and
$a=x$

${k_{R}\over 2\pi
p^{2}}[E(k_{R})-K(k_{R})]=\int_{0}^{\infty}~{}e^{-2pt}N_{0}(at)J_{0}(at)~{}dt,$
(59)

where $N_{0}$ and $J_{0}$ are the Bessel functions. With this replacemente we
obtain ${\cal I}_{1}=-\pi/2$. The second integral can be solved using identity
6.3(5) also in Okui , for $a=x$ and $b=1$

${k_{R}\over
b^{2}}[E(k_{R})-K(k_{R})]=\int_{0}^{\infty}~{}K_{0}(at)J_{1}(at)~{}dt,$ (60)

to obtain

${\cal I}_{2}={\pi\over
2}\int_{0}^{\infty}{tJ_{1}(t)\over{\sqrt{4P^{2}+t^{2}}}}~{}dt.$ (61)

This can be reduded to the standard form 6.565.1 with $a=2P$ and $b=\alpha$
found in GradRyz  by noting that ${\cal I}_{2}$ can also be written as

${\cal I}_{2}={\pi\over 2}\lim_{\alpha\to
1}{\partial\over\partial{\alpha}}\left[\int_{0}^{\infty}{J_{0}(\alpha
t)\over{\sqrt{4P^{2}+t^{2}}}}~{}dt\right]={\pi\over
2}P\bigg{[}I_{1}(P)K_{0}(P)-I_{0}(P)K_{1}(P)\bigg{]}$ (62)

Finally, after collecting all the contributions and replacing them into (56)
we find the first order equation

$\beta{\hat{\zeta}}=-iP{\hat{\gamma}}K_{1}(P)I_{1}(P)-i{P\gamma_{0}{\hat{\zeta}}\over
2R}\bigg{[}1+P\left[K_{1}(P)I_{0}(P)-K_{0}(P)I_{1}(P)\right]\bigg{]}.$ (63)

Making use of standard Bessel functions identities, it is possible to further
reduce this expression to

$\left(\beta+i{P\gamma_{0}\over
R}K_{1}(P)I_{0}(P)\right){\hat{\zeta}}=-iPK_{1}(P)I_{1}(P){\hat{\gamma}}.$
(64)

## References

  * (1) Robinson, N. D. and Steen, P. H. 2001 Observations of Singularity Formation during the Capillary Collapse and Bubble Pinch-off of a Soap Film Bridge _J. Coll. and Interface Science_ 241 448-458.
  * (2) Maxwell, J. C. Capillary Action: The stability of the catenoid _The Collected paper of James Clerk Maxwell_ 2 583-585.
  * (3) Day, R. F., Hinch, E. J. and Lister, J. R. 1998 Self-Similar Capillary Pinchoff of an Inviscid Fluid _Phys. Rev. Lett._ 80 704-707
  * (4) Chen, Y. -J. and Steen, P. H. 1997 Dynamics of inviscid capillary breakup: collapse and pinchoff of a film bridge _J. Fluid Mech._ 341 245-267.
  * (5) Colding, T. H., Minicozzi, W. P. and Pedersen, E. K. 2015 Mean Curvature Flow _Bull. Am. Math. Soc._ 52, 297-333.
  * (6) Eggers, J 1997 Nonlinear dynamics and breakup of free-surface flows _Rev. Mod. Phys._ 69 865-929
  * (7) Goldstein, R.E., Pesci, A.I. and Shelley, M. J. 1993 Topology Transitions and Singularities in Viscous Flows _Phys. Rev. Lett._ 70 3043-3046
  * (8) Pugh, M. C. and Shelley, M. J. 1998 Singularity Formation in Thin Jets with Surface Tension. _Com. in Pure and Applied Mathematics_ 51 733-795
  * (9) Goldstein, R.E., Pesci, A.I. and Shelley, M. J. 1995 Attracting Manifold for a Viscous Topology Transition _Phys. Rev. Lett._ 75 3665-3668
  * (10) Goldstein, R.E., Pesci, A.I. and Shelley, M. J. 1998 Instabilities and Singularities in Hele-Shaw Flow _Physics of Fluids_ 10 2701-2723
  * (11) Constantin, P., Dupont, T.F., Goldstein, R.E., Kadanoff, L.P., Shelley, M.J. and Zhou, S.-M. 1993 Droplet Breakup in a Model of the Hele-Shaw Cell _Phys. Rev. E_ 47, 4169-4181
  * (12) Dupont, T.F., Goldstein, R.E., Kadanoff, L.P. and Zhou, S.-M. 1993 Finite-Time Singularity Formation in Hele-Shaw Systems _Phys. Rev. E_ 47, 4182-4196
  * (13) Eggers, J. and Dupont, T.F. 1994 Drop formation in a one-dimensional approximation of the Navier-Stokes equation _J. Fluid Mech._ 262 205-221
  * (14) de Bernardinis B., Moore D.W. 1987 A Ring-Vortex Representation of an Axi-Symmetric Vortex Sheet In: Hussaini M.Y., Salas M.D. (eds) _Studies of Vortex Dominated Flows. ICASE NASA LaRC Series_ Springer, New York, NY 33-43
  * (15) Caflisch, R. E. and Li X.-F 1992 Lagrangian Theory for 3D Vortex Sheets with Axial or Helical Symmetry _Transport Theory and Statistical Physics_ 21 559-578
  * (16) Baker, G. R., Meiron, D. I. and Orzag S. A. 1982 Generalized Vortex Methods for Free-Surface Flow Problems _J. Fluid Mech._ 123 477-501
  * (17) Nie, Q and Baker, G. R. 1998 Application of Adaptive Quadrature to Axi-symmetric Vortex Sheet Motion _J. Computational Physics_ 143 49-69
  * (18) Nie, Q 2001 The Nonlinear Evolution of Vortex Sheets with Surface Tension in Axisymmetric Flows _J. Computational Physics_ 174 438-459
  * (19) Hou, T. Y., Lowengrub, J. S. and Shelley, M. J. 1997 The long-time motion of vortex sheets with surface tension _Physics of Fluids_ 9 1033-1954
  * (20) Helmholtz, H. 1858 Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen _Angew. math._ 55 25-55
  * (21) Maxwell, J. C. 1891 Circular Currents: Two Parallel Circles. _Electricity and Magnetism_ Dover-Edition 1954 701 339-340
  * (22) Eggers, J. 2000 Singularities in Droplet Pinching with Vanishing Viscosity _SIAM J. Appl. Math._ 60 1997-2008
  * (23) Peng, D., Merriman, B., Osher, S., Zhao, H. and Kang, M. 1999 A PDE-based fast local level set method. _J. Comput. Phys._ 155, 410-438.
  * (24) Herrmann, M. 2005 A Eulerian level set/vortex sheet method for two-phase interface dynamics _J. Comp. Phys._ 203 539-571
  * (25) Meleshko, V. V. 2010 Coaxial axisymmetric vortex rings: 150 years after Helmholtz _Theor. Comput. Fluid Dyn._ 24 403-431
  * (26) Dyson, F. W. 1893 The potential of an anchor ring. Part II _Phil. Trans. R. Soc. Lond. A_ 184, 1041-1106
  * (27) Novikov, E. A. 1985 Hamiltonian description of axisymmetric vortex flows and the system of vortex rings _Phys. Fluids_ 28, 2921-2922
  * (28) Fernandez, O. E. 2009 The Hamiltonization of Nonholonomic Systems and its Applications Ph.D. Thesis, The University of Michigan
  * (29) Alterman, Z 1961 Kelvin-Helmholtz Instability in Media of Variable Density _Physics of Fluids_ 4 1177-1179
  * (30) Sterling, A. M. and Sleicher, C. A. 1975 The Instability of Capillary Jets _J of Fluid Mech._ 68 477-495
  * (31) Rayleigh, Lord 1878 On the instability of jets _Proc. Lond. Math. Soc._ 10, 4-13
  * (32) Okui, S. 1974 Complete Elliptic Integrals Resulting from Infinite Integrals of Bessel Functions _J. Research Nat. Bureau of Standards-B Math. Sci._ 78B 113-135
  * (33) Gradshteyn, I. S. and Ryzhik, I.M. 1980 _Tables of Integrals, Series and Products_ (London, Academic Press)
  * (34) Caflisch, R. E., Li, X. and Shelley, M. J. 1993 The collapse of an axi-symmetric, swirling vortex sheet _Nonlinearity_ 6, 843-867
  * (35) Lundgren, T. S. 1989 A free surface vortex method with weak viscous effects _Mathematical Aspects of Vortex Dynamics_ ed by R. E. Caflisch (SIAM,Philadelpha, PA), 68-79
  * (36) Ceniceros, H.D. and Roma, A. M. 2004 Study of the long-time dynamics of a viscous vortex sheet with a fully adaptive nonstiff method _Phys. Fluids_ 16, 4285-4318


# Mixing and ${C\\!P}$ violation in beauty and charm at LHCb

Jordy Butter  on behalf of the LHCb Collaboration  Department of Physics,
Cavendish Laboratory, JJ Thomson Avenue,
Cambridge, CB3 0HE, UK

###### Abstract

These proceedings discuss recent measurements by the LHCb experiment on mixing
and ${C\\!P}$ violation with beauty and charm mesons, as presented at the
Moriond QCD 2024 conference. All discussed measurements show agreement with
the Standard Model.

##  1 Beauty measurements

The study of ${B}$ hadrons provides a great laboratory to study the CKM sector
of the Standard Model (SM) and look for manifestations of physics beyond the
SM. First, direct determinations of the weak angle $\gamma$ using
${{B}^{\pm}}\rightarrow D^{*}h^{\pm}$ and ${{B}^{0}}\rightarrow
DK^{*}(892)^{0}$ decays are discussed. Afterwards, mixing-induced ${C\\!P}$
violation is studied using $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$,
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}{{K}^{-}}$
and
${{B}^{0}}\rightarrow\psi(\rightarrow{\ell^{+}}{\ell^{-}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}(\rightarrow{{\pi}^{+}}{{\pi}^{-}})$ decays, probing the CKM angles
$\gamma$, $\beta_{s}$ and $\beta$, respectively. Finally, a measurement of
${\Delta\Gamma_{{s}}}$ is shown.

###  1.1 Direct determinations of the CKM angle $\gamma$

The experimental sensitivity to the CKM angle $\gamma$ is constrained most
notably by ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ decays, where $D$ is an
admixture of ${{D}^{0}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mesons and the interference
between $b\rightarrow c$ and $b\rightarrow u$ amplitudes is used to measure
$\gamma$. In these decays, the differences in the $D$-decay Dalitz-plot
distributions between ${{{B}^{+}}}$ and ${{{B}^{-}}}$ decays are due to the
${C\\!P}$-violating phase $\gamma$, which is probed via time-integrated
analyses.

These proceedings show two $\gamma$ measurements using ${{B}^{\pm}}\rightarrow
D^{*}h^{\pm}$ decays, where $D^{*}\rightarrow D\gamma/{{\pi}^{0}}$ and
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$, using
proton-proton data collected between 2011 and 2012 (Run 1) and between 2015
and 2018 (Run 2), corresponding to an integrated luminosity of 3 and
$6\mbox{\,fb}^{-1}$, respectively. The rich interference structure in
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$ decays, with
$h={\pi,K}$, is explored through a model-independent approach, not relying on
a $D$ decay amplitude model. The interference effects are studied by measuring
the yields of the ${{{B}^{+}}}$ and ${{{B}^{-}}}$ decays in bins of $D$ decay
Dalitz phase space. The key difference between these two measurements is that
one fully reconstructs the final-state particles [1], whereas the other does
not reconstruct the neutral $\gamma$ or ${{\pi}^{0}}$ coming from the $D^{*}$
decay [2]. The values of $\gamma=(69^{+13}_{-14})^{\circ}$ and
$\gamma=(92^{+21}_{-17})^{\circ}$ are found for the fully and the partially
reconstructed analysis, respectively.

Figure 1: Confidence-level contours projected in the
$\gamma$–$r_{{{B}^{0}}}^{DK^{*}}$ plane, obtained from ${{B}^{0}}\rightarrow
DK^{*}(892)^{0}$ decays with (left) $D\rightarrow K\pi(\pi\pi)$,
$\pi\pi(\pi\pi)$ and $KK$ only and (right) combined with
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$. The
contours contain $68.3\%$, $95.4\%$ and $99.7\%$ of the distribution.

Additionally, two measurements using ${{B}^{0}}\rightarrow DK^{*}(892)^{0}$
decays are presented, using Run 1 and Run 2 data. Despite its relatively small
branching fraction, the study of the ${{B}^{0}}\rightarrow DK^{*}(892)^{0}$
decay provides a precise determination of $\gamma$ due to the relatively
similar magnitude of the interfering amplitudes. The first analysis studies
interference effects using $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-}$ decays by measuring the ${{B}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ decay yields across the $D$
phase space [3]. These yields are used to determine the weak phase
$\gamma=(49^{+22}_{-19})^{\circ}$, the amplitude ratio
$r_{{B}^{0}}=0.271^{+0.065}_{-0.066}$ and the strong phase
$\delta_{{B}^{0}}=(236^{+19}_{-21})^{\circ}$. The other ${{B}^{0}}\rightarrow
DK^{*0}$ measurement studies final states with $D\rightarrow K\pi(\pi\pi)$,
$\pi\pi(\pi\pi)$ and $KK$ [4]. Invariant mass fits are used to measure
asymmetries and ratios of the ${{B}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}$ decay rates, which are
functions of $\gamma$, $r_{{B}^{0}}$ and $\delta_{{B}^{0}}$. Four solutions
are found, but a combination with the previously described analysis favours
the values $\gamma=(63.3\pm 7.2)^{\circ}$, $r_{{B}^{0}}=0.233\pm 0.016$ and
$\delta_{{B}^{0}}=(191.9\pm 6.0)^{\circ}$. Figure 1 shows the confidence-level
contours of the corresponding solutions projected in the
$\gamma$–$r_{{{B}^{0}}}^{DK^{*}}$ plane.

###  1.2 Mixing-induced ${C\\!P}$ violation

In certain ${B}$ hadron decays ${C\\!P}$ violation manifests itself in the
interference between the mixed and the unmixed decay amplitudes. This creates
a time-dependent ${C\\!P}$ asymmetry, dependent on the complex phases of the
CKM matrix.

A well-known example of this type of ${C\\!P}$ violation is the study of
$B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ decays, where the interfering
amplitudes have a comparable magnitude and a weak-phase difference of
$\gamma-2\beta_{s}$. This analysis has recently been performed using Run 2
data [5] and makes use of the measurement of the ${{B}^{0}_{s}}$–${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ mixing frequency $\Delta
m_{s}$ with $B^{0}_{s}\\!\rightarrow D_{s}^{-}\pi^{+}$ decays to calibrate
detector effects and for the value of $\Delta m_{s}$ [6]. First, a likelihood
fit is performed to the invariant mass of the ${{B}^{0}_{s}}$ and
${{D}^{-}_{s}}$ mesons to separate the signal from the background. Afterwards,
a decay-time fit is performed to extract a set of five ${C\\!P}$ observables.
The corresponding ${C\\!P}$ asymmetry plot is provided in Fig. 3, where the
shift between the two lines is a manifestation of ${C\\!P}$ violation. The
decay-time fit requires a good understanding of the decay-time acceptance, the
resolution of the decay-time measurement and the flavour tagging, the main
detector effects diluting the sensitivity to ${C\\!P}$ violation. Finally,
assuming $\phi_{s}=-2\beta_{s}$ and using the LHCb combination of $\phi_{s}$
[7], the measured ${C\\!P}$ observables are used to find the CKM angle
$\gamma=(74\pm 11)^{\circ}$, the strong-phase difference $\delta=(346.9\pm
6.6)^{\circ}$ and the amplitude ratio $r_{D_{s}K}=0.327\pm 0.038$. This is the
most precise determination of $\gamma$ with ${{B}^{0}_{s}}$ mesons and agrees
with the LHCb combination of $\gamma$ [8].

The most important constraint for the CKM angle $\beta_{s}$ is the study of
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$ decays, where
the measurable ${C\\!P}$-violating phase $\phi_{s}$ is equal to $-2\beta_{s}$
in the SM, neglecting subleading loop contributions. The corresponding Run 2
measurement is performed using
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}{{K}^{-}}$
decays near the $\phi(2010)$ resonance [7]. The measurement requires a
flavour-tagged, time-dependent angular analysis, leading to the most precise
determination of ${C\\!P}$ violating phase $\phi_{s}$, the ${{B}^{0}_{s}}$
decay-width difference ${\Delta\Gamma_{{s}}}$ and the difference in the
average ${{B}^{0}_{s}}$ and ${{B}^{0}}$ decay width,
${\Gamma_{{s}}}-{\Gamma_{{d}}}$, to date. This measurement finds
$\phi_{s}=-0.039\pm 0.022\pm 0.006\mathrm{\,rad}$,
${\Delta\Gamma_{{s}}}=0.0845\pm 0.0044\pm 0.0024{\mathrm{\,ps}}^{-1}$ and
${\Gamma_{{s}}}-{\Gamma_{{d}}}=-0.0056^{+0.0013}_{-0.0015}\pm
0.0014{\mathrm{\,ps}}^{-1}$, where the uncertainties are statistical and
systematic. The result is compatible with ${C\\!P}$ symmetry, other $\phi_{s}$
measurements and the SM expectation.

Also, the most precise single determination of the CKM angle $\beta$ is
presented, using
${{B}^{0}}\rightarrow\psi(\rightarrow{\ell^{+}}{\ell^{-}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}(\rightarrow{{\pi}^{+}}{{\pi}^{-}})$ decays collected in Run 2 [9]. The
considered final states are
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{-}}{\mu^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$,
$\phi(2S)(\rightarrow{\mu^{-}}{\mu^{+}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ and ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow
e^{+}e^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$. Due to the small
${{B}^{0}}$ mass-width difference, the ${C\\!P}$ asymmetry is approximately
$\mathcal{A}^{{C\\!P}}(t)\approx S\sin(\Delta m_{d}t)-C\cos(\Delta m_{d}t)$,
where $S$ is equal to $\sin(2\beta)$ in the SM, besides CKM-suppressed
contributions from penguin topologies. The determination of the ${C\\!P}$
observables requires a flavour-tagged and time-dependent analysis, resulting
in $S_{\psi{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=0.717\pm 0.013\pm 0.008$
and $C_{\psi{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=0.008\pm 0.012\pm
0.003$. Figure 3 shows the measured ${C\\!P}$ asymmetry as a function of
${{B}^{0}}$ decay time.

Figure 2:  $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ mixing asymmetries of
the (blue) ${{D}^{-}_{s}}{{K}^{+}}$ (red) ${{D}^{+}_{s}}{{K}^{-}}$. The decay
time is folded into a single oscillation period.

Figure 3:
${{B}^{0}}\rightarrow\psi(\rightarrow{\ell^{+}}{\ell^{-}}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}(\rightarrow{{\pi}^{+}}{{\pi}^{-}})$ time-dependent ${C\\!P}$-asymmetry

###  1.3 Measurement of $\Delta\Gamma_{s}$

The measurements of the ${{B}^{0}_{s}}$ decay-width difference
$\Delta\Gamma_{s}$ using
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$ decays show a
tension between the LHC experiments [10]. Consequently, performing
measurements using other decay modes is important. As such, $\Delta\Gamma_{s}$
is determined as the decay-with difference between ${C\\!P}$-even
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\eta^{\prime}$ and
mostly ${C\\!P}$-odd
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\pi}^{-}}{{\pi}^{+}}$
decays, collected between 2011 and 2018 [11]. The yield ratio of these decays
in bins of decay time depends on ${\Delta\Gamma_{{s}}}$, allowing for a
$\chi^{2}$ fit to obtain ${\Delta\Gamma_{{s}}}$. Finally, the value of
$\Delta\Gamma_{s}=0.087\pm 0.012\pm 0.009{\mathrm{\,ps}}^{-1}$ is reported,
which is compatible with the HFLAV average.

##  2 Charm measurements

${C\\!P}$ violation in $D$ mesons has been observed for the first time in 2019
[12]. The compatibility of this result with SM is debated, therefore new
measurements of ${C\\!P}$ violation in charm meson decays are crucial to
clarify the picture [13]. These proceedings discuss the time-dependent study
of charm mixing and ${C\\!P}$ violation in
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays and the usage of the energy-
test method to study ${C\\!P}$ violation in the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{0}}$ decays. The neutral
$D$ meson flavour at production in these measurements is determined by
requiring it to originate from ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$
decays.

###  2.1 Charm mixing and ${C\\!P}$ violation with
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$

In this measurement, the Run 2 dataset is exploited to probe mixing and
${C\\!P}$-violating parameters by measuring the decay-time dependency of the
ratio of wrong sign (WS) ${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ and right
sign (RS) ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays, $R_{{K}{\pi}}^{+}(t)$, and the ratio of ${C\\!P}$-conjugated
processes, $R_{{K}{\pi}}^{-}(t)$ [14]. The ratios are expanded as
$R_{{K}{\pi}}^{\pm}(t)\approx R_{{K}{\pi}}(1\pm
A_{{K}{\pi}})+\sqrt{R_{{K}{\pi}}(1\pm A_{{K}{\pi}})}(c_{{K}{\pi}}\pm\Delta
c_{{K}{\pi}})(t/\tau_{{{D}^{0}}})+(c_{{K}{\pi}}^{\prime}\pm\Delta
c_{{K}{\pi}}^{\prime})(t/\tau_{{{D}^{0}}})^{2}$, where $c_{{K}{\pi}}$ and
$c_{{K}{\pi}}^{\prime}$ are sensitive to mixing parameters, while
$A_{{K}{\pi}}$, $\Delta c_{{K}{\pi}}$ and $\Delta c_{{K}{\pi}}^{\prime}$ are
sensitive ${C\\!P}$ violation in the decay, the mixing and their interference,
respectively. The WS-to-RS yield ratio in 18 bins of ${{D}^{0}}$ decay time is
used to extract $R_{K\pi}=(343.1\pm 2.0)\times 10^{-5}$, $c_{K\pi}=(51.4\pm
3.5)\times 10^{-4}$ and $c^{\prime}_{K\pi}=(13\pm 4)\times 10^{-6}$. No
evidence for ${C\\!P}$ violation is found.

###  2.2 Search for $C\\!P$ violation using the energy-test method

A search for ${C\\!P}$ violation is conducted in the phase space of
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{\pm}}{{\pi}^{\mp}}$ decays [15] and
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{0}}$ decays [16] using the
energy-test method. This method studies local ${C\\!P}$ violation by checking
if the phase spaces of ${{D}^{0}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decays follow the same
underlying distribution. A test statistic $T$ is measured using the data which
is compared with the $T$ distribution under the null hypothesis. The resulting
p-values correspond to $70\%$ for
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{-}}{{\pi}^{+}}$, $66\%$ for
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{+}}{{\pi}^{-}}$ and $62\%$ for
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{0}}$; showing no evidence
for ${C\\!P}$ violation.

##  3 Conclusions

Great progress is made in both beauty and charm measurements in LHCb. The
experimental uncertainty on $\gamma$ has dropped 3 degrees, with the latest
HFLAV average providing $\gamma=(66.5^{+2.8}_{-2.9})^{\circ}$ [10], whereas
the indirect determination provides $\gamma=(66.3^{+0.7}_{-1.9})^{\circ}$
[17]. The sensitivity gap between direct and indirect determinations of
$\gamma$ is narrowing, and closing this gap is one of the physics goals of
LHCb. Additionally, the most precise determinations of the CKM angles $\beta$
and $\beta_{s}$ are discussed. Finally, mixing and ${C\\!P}$ violation in
charm mesons is probed, further constraining the charm-mixing parameters and
finding agreement with ${C\\!P}$ symmetry and the SM.

## References

## References

  * [1] LHCb Collaboration, R. Aaij, et al.  Measurement of the CKM angle $\gamma$ using the ${{B}^{\pm}}\rightarrow{{D}^{*}}{h}^{\pm}$ channels.  JHEP 12 (2023) 013.  arXiv:2310.04277.
  * [2] LHCb Collaboration, R. Aaij, et al.  A model-independent measurement of the CKM angle $\gamma$ in partially reconstructed ${{B}^{\pm}}\rightarrow{{D}^{*}}h^{\pm}$ decays with ${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$ ($h=\pi,K$).  JHEP 02 (2024) 118.  arXiv:2311.10434.
  * [3] LHCb Collaboration, R. Aaij, et al.  Measurement of the CKM angle $\gamma$ in the ${{B}^{0}}\rightarrow{{D}^{0}}{{K}^{*0}}$ channel using self-conjugate ${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{h}^{+}{h}^{-}$ decays.  Eur. Phys. J. C84 (2024) 206.  arXiv:2309.05514.
  * [4] LHCb Collaboration, R. Aaij, et al.  Study of $C\\!P$ violation in ${{B}_{({s})}^{0}}\rightarrow{D}K^{*}(892)^{0}$ decays with ${D}\rightarrow{K}{\pi}({\pi}{\pi})$, ${\pi}{\pi}({\pi}{\pi})$, and ${K}{K}$ final states.  JHEP 05 (2024) 025.  arXiv:2401.17934.
  * [5] LHCb Collaboration, R. Aaij, et al.  Measurement of $C\\!P$ asymmetry in ${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ decays.  (LHCb-CONF-2023-004), 2023.
  * [6] LHCb Collaboration, R. Aaij, et al.  Precise determination of the ${{B}^{0}_{s}}$–${{\kern 1.25995pt\overline{\kern-1.25995ptB}{}}{}^{0}_{s}}$ oscillation frequency.  Nature Physics 18 (2022) 1.  arXiv:2104.04421.
  * [7] LHCb Collaboration, R. Aaij, et al.  Improved measurement of ${C\\!P}$ violation parameters in ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}{{K}^{-}}$ decays in the vicinity of the $\phi(1020)$ resonance.  Phys. Rev. Lett. 132 (2024) 051802.  arXiv:2308.01468.
  * [8] LHCb Collaboration, R. Aaij, et al.  Simultaneous determination of the CKM angle $\gamma$ and parameters related to mixing and ${C\\!P}$ violation in the charm sector.  (LHCb-CONF-2022-003), 2022.
  * [9] LHCb Collaboration, R. Aaij, et al.  Measurement of ${C\\!P}$ violation in ${{B}^{0}}\rightarrow{\psi}(\rightarrow\ell^{+}\ell^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}(\rightarrow{{\pi}^{+}}{{\pi}^{-}})$ decays.  Phys. Rev. Lett. 132 (2024) 021801.  arXiv:2309.09728.
  * [10] Y. Amhis et al.  Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of 2021.  Phys. Rev. D107 (2023) 052008.  arXiv:2206.07501.
  * [11] LHCb Collaboration, R. Aaij, et al.  A measurement of $\Delta\Gamma_{s}$.  arXiv:2310.12649.
  * [12] LHCb Collaboration, R. Aaij, et al.  Observation of ${C\\!P}$ violation in charm decays.  Phys. Rev. Lett. 122 (2019) 211803.  arXiv:1903.08726.
  * [13] LHCb Collaboration, R. Aaij, et al.  $U$-spin-${C\\!P}$ anomaly in charm.  Phys. Rev. D108 (2023) 035005.  arXiv:2210.16330.
  * [14] LHCb Collaboration, R. Aaij, et al.  Measurement of ${D}^{0}$–${\kern 1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}$ mixing and search for $C\\!P$ violation with ${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays.  in preparation.
  * [15] LHCb Collaboration, R. Aaij, et al.  Search for $C\\!P$ violation in the phase space of ${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{\pm}}{{\pi}^{\mp}}$ decays with the energy test.  JHEP 03 (2024) 107.  arXiv:2310.19397.
  * [16] LHCb Collaboration, R. Aaij, et al.  Search for CP violation in the phase space of ${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ decays with the energy test.  JHEP 09 (2024) 129.  arXiv:2306.12746.
  * [17] J. Charles et al.  Current status of the standard model CKM fit and constraints on $\Delta F=2$ new physics.  Phys. Rev. D91 (2015) 073007.  arXiv:1501.05013. updated results and plots available at http://ckmfitter.in2p3.fr/.