EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-048 LHCb-PAPER-2016-005 June 16, 2016

Search for violations of Lorentz invariance and ${C\\!PT}$ symmetry in
${{B}^{0}_{\kern-1.45189pt{\scriptscriptstyle(}\kern-0.72594pt{s}\kern-0.43552pt{\scriptscriptstyle)}}}$
mixing

The LHCb collaboration†††Authors are listed at the end of this Letter.

Violations of ${C\\!PT}$ symmetry and Lorentz invariance are searched for by
studying interference effects in ${{B}^{0}}$ mixing and in ${{B}^{0}_{s}}$
mixing. Samples of ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays are recorded by the LHCb detector in
proton–proton collisions at centre-of-mass energies of 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
3$\mbox{\,fb}^{-1}$. No periodic variations of the particle-antiparticle mass
differences are found, consistent with Lorentz invariance and ${C\\!PT}$
symmetry. Results are expressed in terms of the Standard Model Extension
parameter $\Delta a_{\mu}$ with precisions of ${\mathcal{O}}(10^{-15})$ and
${\mathcal{O}}(10^{-14})$$\mathrm{\,Ge\kern-1.00006ptV}$ for the ${{B}^{0}}$
and ${{B}^{0}_{s}}$ systems, respectively. With no assumption on Lorentz
(non-)invariance, the ${C\\!PT}$-violating parameter $z$ in the
${{B}^{0}_{s}}$ system is measured for the first time and found to be
${{\mathcal{R}e}(z)}=-0.022\pm 0.033\pm 0.005$ and
${{\mathcal{I}m}(z)}=0.004\pm 0.011\pm 0.002$, where the first uncertainties
are statistical and the second systematic.

Published in Phys. Rev. Lett. 116 (2016) 241601

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

Lorentz invariance and the combination of charge conjugation, spatial
inversion and time reversal ($C\\!PT$) are exact symmetries in the Standard
Model (SM) of particle physics, and are deeply connected in any quantum field
theory [1]. Quantum theories that aim to describe Planck-scale physics, such
as string theory, might break these fundamental symmetries [2]. Present-day
experiments are many orders of magnitude away from the Planck energy scale of
$\sim 10^{19}\mathrm{\,Ge\kern-1.00006ptV}$, however, small effects at low
energy might still be observable. Interference effects in the mixing of
neutral mesons are sensitive to violations of $C\\!PT$ symmetry, and therefore
may provide a window to the quantum gravity scale [3]. Such effects can be
quantified in a low-energy, effective field theory, as done in the Standard
Model Extension (SME) [4, 5]. In this framework, terms that explicitly break
Lorentz and $C\\!PT$ symmetry are added to the SM Lagrangian to describe the
couplings between particles and (hypothetical) uniform tensor fields. These
fields would acquire non-zero vacuum expectation values when these symmetries
are spontaneously broken in the underlying theory. The SME couplings are
expected to be suppressed by powers of the Planck scale [6,
*Kostelecky:1994rn]. In the SME, the $C\\!PT$-violating parameters that can be
measured in neutral meson systems also break Lorentz symmetry. The amount of
$C\\!PT$ violation depends on the direction of motion and on the boost of the
particle. The SME parameters for the ${B}^{0}$ and ${B}^{0}_{s}$ systems can
be best measured with a time-dependent analysis of the decay channels
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$, using the four-velocity of the $B$ mesons [8]. The
notation $B$ refers to either ${B}^{0}$ or ${B}^{0}_{s}$ and the inclusion of
charge-conjugate processes is implied throughout this Letter. These parameters
have been measured previously, albeit with less sensitive decay modes [8], by
the BaBar collaboration for the ${B}^{0}$ system [9], and by the D0
collaboration for the ${B}^{0}_{s}$ system [10].

The LHCb detector is a single-arm forward spectrometer described in detail in
Refs. [11, 12]. Simulated events are produced using the software described in
Refs. [13, *Sjostrand:2007gs, 15, 16, 17, *Agostinelli:2002hh, 19]. The data
used in this analysis correspond to an integrated luminosity of
$3\mbox{\,fb}^{-1}$, taken at the LHC at proton–proton centre-of-mass energies
of 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$. The selection of both decay
channels is the same as used in Refs. [20] and [21]. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson is reconstructed in the
dimuon channel and the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ meson in the
${\pi}^{+}$ ${\pi}^{-}$ final state.

Interference effects from $C\\!PT$ violation can be incorporated generically
in the time evolution of a neutral $B$ meson system, described by the
Schrödinger equation $i\partial_{t}\Psi=\hat{H}\Psi$. The effective $2\times
2$ Hamiltonian is written as $\hat{H}=\hat{M}-i\hat{\Gamma}/2$ [22].
Diagonalisation gives a heavy-mass eigenstate $|B_{H}\rangle$ and a light-mass
eigenstate $|B_{L}\rangle$ with masses $m_{H,L}$ and decay widths
$\Gamma_{H,L}$. The differences between the eigenvalues are defined as
${\Delta m}\equiv m_{H}-m_{L}$ and
${\Delta\Gamma}\equiv\Gamma_{L}-\Gamma_{H}$. The differences between the
diagonal matrix elements of the effective Hamiltonian are defined as $\delta
m\equiv M_{11}-M_{22}$ and $\delta\Gamma\equiv\Gamma_{11}-\Gamma_{22}$. Any
difference between the mass or lifetime of particles and antiparticles (i.e.,
a non-zero $\delta m$ or $\delta\Gamma$) would be a sign of $C\\!PT$
violation, and is characterised by

$\displaystyle z=\frac{\delta m-i\delta\Gamma/2}{{\Delta
m}+i{\Delta\Gamma}/2}\ ,$ (1)

and the mass eigenstates are given by $|B_{H,L}\rangle=p\sqrt{1\pm
z}|{B}\rangle\mp q\sqrt{1\mp z}|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}}\rangle$. Owing to the smallness of
the $B$ mixing parameters ${\Delta m}$ and ${\Delta\Gamma}$ in the
denominator, $z$ is highly sensitive to $C\\!PT$-violating effects.

Considering only contributions to first order in $z$, the decay rate to a
$C\\!P$ eigenstate $f$ as a function of the $B$ proper decay time $t$ becomes

$\displaystyle\frac{\mathrm{d}\Gamma_{f}}{\mathrm{d}t}\propto e^{-\Gamma t}$
$\displaystyle\bigg{[}\big{[}1+{\zeta}{D_{f}}{{\mathcal{R}e}(z)}-{S_{f}}{{\mathcal{I}m}(z)}\big{]}\cosh({\Delta\Gamma}t/2)$
$\displaystyle+\big{[}{D_{f}}+{{\mathcal{R}e}(z)}({C_{f}}+{\zeta})\big{]}\sinh({\Delta\Gamma}t/2)$
$\displaystyle+{\zeta}\big{[}{C_{f}}-{D_{f}}{{\mathcal{R}e}(z)}+{\zeta}{S_{f}}{{\mathcal{I}m}(z)}\big{]}\cos({\Delta
m}t)$
$\displaystyle-{\zeta}\big{[}{S_{f}}-{{\mathcal{I}m}(z)}({C_{f}}+{\zeta})\big{]}\sin({\Delta
m}t)\bigg{]}\ ,$ (2)

where $\Gamma\equiv(\Gamma_{11}+\Gamma_{22})/2$, ${\zeta}=+1(-1)$ for an
initial $|{B}\rangle$ ($|{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}}\rangle$) state and the following
definitions are introduced:

$\displaystyle{C_{f}}\equiv\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}}\ ,\
{S_{f}}\equiv\frac{2{\mathcal{I}m}(\lambda_{f})}{1+|\lambda_{f}|^{2}}\ ,\
{D_{f}}\equiv-\frac{2{\mathcal{R}e}(\lambda_{f})}{1+|\lambda_{f}|^{2}}\ ,\
\lambda_{f}\equiv\frac{q}{p}\frac{\overline{A}_{f}}{A_{f}}\ ,$ (3)

with $A_{f}$ and $\overline{A}_{f}$ the direct decay amplitudes of a
$|{B}\rangle$ or $|{{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}}\rangle$
state to the eigenstate $f$.

For the decay ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$, the final state is $C\\!P$
odd, corresponding to $C\\!P$ eigenvalue $\eta_{f}=-1$. In the SM,
$\arg(\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}})=\pi-2\beta$, where $\beta$
is defined in terms of elements of the CKM matrix as
$\beta\equiv\arg[-({V_{{c}{d}}}{V_{{c}{b}}^{\ast}})/({V_{{t}{d}}}{V_{{t}{b}}^{\ast}})]$.
Furthermore, in the ${B}^{0}$ system, the approximation
${\Delta\Gamma_{{d}}}=0$ is made, as supported by experimental data [23].

The decay ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ is similar to
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$, but the decay width
difference $\Delta\Gamma_{{s}}$ cannot be ignored [23]. Another important
difference is that the ${{K}^{+}}{{K}^{-}}$ system mostly originates from the
$\phi(1020)$ resonance, giving the ${{K}^{+}}{{K}^{-}}$ pair an orbital
angular momentum $L=1$ (P wave). Since the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$ final state consists of
two vector mesons, its orbital angular momentum can be $L\in\\{0,1,2\\}$ for
the polarisation states $f\in\\{0,\perp,\parallel\\}$, respectively, with
corresponding $C\\!P$ eigenvalues $\eta_{f}=(-1)^{L}$. The
${{K}^{+}}{{K}^{-}}$ system has a small S-wave contribution [21], which
results in another $L=1$ component for the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}}$ final
state. These four polarisation states can be separated statistically in the
helicity formalism [24], using the three decay angles between the final-state
particles. The corresponding weak phases,
$\arg(\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}})=L\pi-{\phi_{{s}}}$, can, in the SM, be expressed
in terms of CKM matrix elements,
${\phi_{{s}}}=-2\beta_{s}\equiv-2\arg[-({V_{{t}{s}}}{V_{{t}{b}}^{\ast}})/({V_{{c}{s}}}{V_{{c}{b}}^{\ast}})]$.
The decay rate has to be modified compared to Eq. 2 to include the angular
dependence. It becomes a sum over all ten combinations of the four helicity
amplitudes,

$\displaystyle\frac{\mathrm{d}^{4}\Gamma_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}}}{\mathrm{d}t\,\mathrm{d}\vec{\Omega}}\propto\sum_{k=1}^{10}h_{k}(t)f_{k}(\vec{\Omega})\
,$ (4)

where $f_{k}(\vec{\Omega})$ are angular functions, given in Ref. [24], and
$h_{k}(t)$ are products of the amplitudes as listed in Table 1. The time
dependence of $h_{k}(t)$ is given by

$\displaystyle{A_{l}^{*}}(t){A_{m}}(t)=$
$\displaystyle\frac{{A_{l}^{*}}(0){A_{m}}(0)e^{-\Gamma_{s}t}}{1+{\zeta}{C_{f}}}$
$\displaystyle\bigg{[}a_{k}\cosh({\Delta\Gamma_{{s}}}t/2)+b_{k}\sinh({\Delta\Gamma_{{s}}}t/2)$
$\displaystyle\;\;+c_{k}\cos({\Delta m_{{s}}}t)+d_{k}\sin({\Delta
m_{{s}}}t)\bigg{]}\ ,$ (5)

with the coefficients listed in Table 2.

Table 1: Time-dependent functions $h_{k}(t)$ in Eq. 4.

$k$ | $h_{k}(t)$
---|---
$1$ | $|{A_{0}}(t)|^{2}$
$2$ | $|{A_{\parallel}}(t)|^{2}$
$3$ | $|{A_{\perp}}(t)|^{2}$
$4$ | ${\mathcal{I}m}({A_{\parallel}^{*}}(t){A_{\perp}}(t))$
$5$ | ${\mathcal{R}e}({A_{0}^{*}}(t){A_{\parallel}}(t))$

$k$ | $h_{k}(t)$
---|---
$6$ | ${\mathcal{I}m}({A_{0}^{*}}(t){A_{\perp}}(t))$
$7$ | $|{A_{\mathrm{S}}}(t)|^{2}$
$8$ | ${\mathcal{R}e}({A_{\mathrm{S}}^{*}}(t){A_{\parallel}}(t))$
$9$ | ${\mathcal{I}m}({A_{\mathrm{S}}^{*}}(t){A_{\perp}}(t))$
$10$ | ${\mathcal{R}e}({A_{\mathrm{S}}^{*}}(t){A_{0}}(t))$

Table 2: Definition of the coefficients in Eq. 5. The following definitions are used: ${\eta^{+}}\equiv(1+\eta_{l}\eta_{m})/2$, ${\eta^{-}}\equiv(1-\eta_{l}\eta_{m})/2$, ${\eta^{{\mathcal{I}m}}}\equiv i(\eta_{l}-\eta_{m})/2$, ${\eta^{{\mathcal{R}e}}}\equiv(\eta_{l}+\eta_{m})/2$. Furthermore, ${\zeta}^{+}\equiv({\zeta})^{{\eta^{+}}}$, and ${\zeta}^{-}\equiv({\zeta})^{{\eta^{-}}}$, such that ${\zeta}^{\pm}=1$ if ${\eta^{\pm}}=0$ and ${\zeta}^{\pm}={\zeta}$ otherwise.  $a_{k}=$ | $({\eta^{+}}+{\eta^{-}}{C_{f}})+{\zeta}{{\mathcal{R}e}(z)}({\eta^{{\mathcal{R}e}}}{D_{f}}+{\eta^{{\mathcal{I}m}}}{S_{f}})$
---|---
$+{{\mathcal{I}m}(z)}({\eta^{{\mathcal{I}m}}}{D_{f}}-{\eta^{{\mathcal{R}e}}}{S_{f}})\;\;$
$b_{k}=$ | $({\eta^{{\mathcal{R}e}}}{D_{f}}+{\eta^{{\mathcal{I}m}}}{S_{f}})+{{\mathcal{R}e}(z)}({\zeta}^{+}+{\zeta}^{-}{C_{f}})$
$c_{k}$= | ${\zeta}({\eta^{-}}+{\eta^{+}}{C_{f}})-{\zeta}{{\mathcal{R}e}(z)}({\eta^{{\mathcal{R}e}}}{D_{f}}+{\eta^{{\mathcal{I}m}}}{S_{f}})$
$-{{\mathcal{I}m}(z)}({\eta^{{\mathcal{I}m}}}{D_{f}}-{\eta^{{\mathcal{R}e}}}{S_{f}})$
$d_{k}$= | ${\zeta}({\eta^{{\mathcal{I}m}}}{D_{f}}-{\eta^{{\mathcal{R}e}}}{S_{f}})+{{\mathcal{I}m}(z)}({\zeta}^{+}+{\zeta}^{-}{C_{f}})$

In the SME, the dimensionless parameter $z$ is not a constant. It depends on
the four-velocity $\beta^{\mu}=(\gamma,\gamma\vec{\beta})$ of the neutral
meson as [25, 26]

$\displaystyle z=\frac{\beta^{\mu}{\Delta a_{\mu}}}{{\Delta
m}+i{\Delta\Gamma}/2}\ ,$ (6)

thereby breaking Lorentz invariance. The SME parameter $\Delta a_{\mu}$
describes the difference between the couplings of the valence quarks, within
the neutral meson, with the Lorentz-violating fields [25]. Therefore,
${B}^{0}$ and ${B}^{0}_{s}$ mesons can have different values of $\Delta
a_{\mu}$. Since $\Delta a_{\mu}$ is real [27], it follows that
${{\mathcal{R}e}(z)}{\Delta\Gamma}=-2{{\mathcal{I}m}(z)}{\Delta m}$. For $B$
mesons, ${\Delta m}\gg{\Delta\Gamma}$, and so ${\mathcal{I}m}(z)$ is two
orders of magnitude smaller than ${\mathcal{R}e}(z)$, and can be ignored in
the measurements of $\Delta a_{\mu}$. The average boost of $B$ mesons in the
acceptance of LHCb is $\left\langle\gamma\beta\right\rangle\approx 20$. It
follows from Eq. 6 that this large boost results in a high sensitivity to
$\Delta a_{\mu}$ [8].

To measure $\Delta a_{\mu}$, the meson direction needs to be determined in an
absolute reference frame. Such a frame can be defined with respect to fixed
stars [27]. In this frame, the $Z$-axis points north along the Earth’s
rotation axis, the $X$-axis points from the Sun to the vernal equinox on 1
January 2000 (J2000 epoch) and the $Y$-axis completes the right-handed
coordinate system. The latitude of the LHCb interaction point is
$46.2414^{\circ}$N, the longitude is $6.0963^{\circ}$E, and the angle of the
beam east of north is $236.296^{\circ}$. The beam axis is inclined with
respect to the geodetic plane by $3.601\mathrm{\,mrad}$, pointing slightly
upwards. The timekeeping is obtained from the LHC machine with a time stamp,
$t_{\rm LHC}$, in UTC microseconds since January $1^{\rm st}$, 1970, 00:00:00
UTC. The time, spatial coordinates and angles have negligible uncertainties
and are used to define the rotation from the coordinate system of LHCb to the
absolute reference frame. For mesons travelling along the beam axis,
${\mathcal{R}e}(z)$ can be expressed as,

$\displaystyle{{\mathcal{R}e}(z)}$ $\displaystyle=\frac{{\Delta m}}{{\Delta
m}^{2}+{\Delta\Gamma}^{2}/4}\beta^{\mu}{\Delta a_{\mu}}$
$\displaystyle\approx\frac{\gamma}{{\Delta m}}\Big{[}{\Delta
a_{0}}+\cos(\chi){\Delta a_{Z}}+\sin(\chi)\big{[}{\Delta
a_{Y}}\sin(\Omega\hat{t})+{\Delta a_{X}}\cos(\Omega\hat{t})\big{]}\Big{]}\ ,$
(7)

where $|\vec{\beta}|$ is set to unity, $\Delta a^{X,Y,Z}=-\Delta a_{X,Y,Z}$,
and $\chi=112.4^{\circ}$ is the angle between the beam axis and the rotational
axis of the Earth. The time dependence results from the Earth’s rotation,
giving a periodicity with sidereal frequency $\Omega$. The sidereal phase at
$t_{\rm LHC}=0$ is found to be $\hat{t}=(2.8126\pm 0.0014)\mathrm{\,hr}$. The
$B$ mesons are emitted at an average angle of about $5^{\circ}$ from the beam
axis. This means that the LHCb detector is mostly sensitive to the linear
combination ${\Delta a_{\parallel}}\equiv{\Delta a_{0}}+\cos(\chi){\Delta
a_{Z}}={{\Delta a_{0}}-0.38{\Delta a_{Z}}}$, while there is a much weaker
sensitivity to the orthogonal parameter, ${\Delta a_{\perp}}={0.38{\Delta
a_{0}}+{\Delta a_{Z}}}$, coming from the smaller transverse component of the
$B$ velocity. Both $\Delta a_{\parallel}$ and $\Delta a_{\perp}$ are measured
and the correlation between them is negligible.

Unbinned likelihood fits are applied to the decay-time distributions using
Eqs. 2 and 4. To obtain the SME parameters, the sidereal variation of
${\mathcal{R}e}(z)$ is taken into account by including in the fits the LHC
time and the three-momentum of the reconstructed $B$ candidate. For the
${B}^{0}_{s}$ sample, the fits are performed to the full angular distribution.

In the invariant mass distributions of the $B$ candidates, the background is
mostly combinatorial. For both decay channels, this background is
statistically subtracted using the sPlot technique [28], which allows to
project out the signal component by weighting each event depending on the mass
of the $B$ candidate. The mass models are the same as in Refs. [20, 21]. The
correlation between the shape of the invariant mass distribution and the $B$
momentum or, for the ${B}^{0}_{s}$ sample, the decay angles, leads to a small
systematic bias for both samples. This effect is included in the systematic
uncertainty. In the ${B}^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ ${K}^{+}$ ${K}^{-}$ sample,
there is a small contribution coming from misidentified ${B}^{0}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ ${K}^{+}$
${\pi}^{-}$ and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p{{K}^{-}}$ decays. This background contribution is statistically
removed by adding simulated decays with negative weights. A systematic
uncertainty is assigned to account for the uncertainty on the size and shape
of this background.

The description of the detection efficiency as a function of the decay time,
the decay-time resolution model and the flavour tagging (to distinguish
initial $B$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ mesons) are the
same as in Refs. [20] and [21] for the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ samples, respectively. This description includes
the dilution of the asymmetry due to wrong decisions of the flavour tagging
method. The decay-time resolution model and tagging calibration do not lead to
a systematic bias in the final result. A possible wrong assignment of the
primary interaction vertex (PV) to the $B$ candidate gives a small bias in the
$\Delta a_{\perp}^{{{B}^{0}}}$ parameter, which is included in the systematic
uncertainty. The inefficiency at high decay times, caused by the
reconstruction algorithms, is described by an exponential function. For the
${B}^{0}$ sample, this function is obtained from simulation and does not lead
to a systematic bias in the result. For the ${B}^{0}_{s}$ sample, the
exponential function is obtained from a data-driven method. The change in the
final result when using the correction procedure from Ref. [21] is taken as a
systematic uncertainty.

The production asymmetry between ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ mesons is included in the
modelling of the decay rates, and is taken from Refs. [29, 30]. The
corresponding uncertainties are included in the statistical uncertainty, while
a possible momentum dependence of the production asymmetry is considered as a
systematic uncertainty. The ${B}^{0}_{s}$ production asymmetry does not affect
the fit to the ${B}^{0}_{s}$ sample, since the fast ${B}^{0}_{s}$ oscillations
wash out this effect and since the decay rates for ${B}^{0}_{s}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ tags are normalised
separately.

In the fit to the ${B}^{0}$ sample, the correlation between
${\mathcal{R}e}(z^{{{B}^{0}}})$ and $C_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}$ is too large to allow
determination of ${\mathcal{R}e}(z^{{{B}^{0}}})$ without making assumptions
about the value of $C_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}$ [8]. On the other hand, to
determine $\Delta a^{{{B}^{0}}}_{\mu}$, the averages
$C_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=0.005\pm 0.020$ and
$S_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=0.676\pm 0.021$ [22] as
measured by the BaBar and Belle collaborations can be used in the fit. Since
the boost of the ${B}^{0}$ mesons is about 40 times lower in these
experiments, these values are hardly affected by possible Lorentz violation in
the SME. The value of $D_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}$ is by definition
$\sqrt{\vphantom{S^{2}}}\overline{1-S_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}}^{2}-C_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}^{2}}$. The uncertainties on
these external input values are propagated as systematic uncertainties on
$\Delta a^{{{B}^{0}}}_{\mu}$. The mass difference, ${\Delta m_{{d}}}=0.510\pm
0.003{\mathrm{\,ps^{-1}}}$ [22], is allowed to vary in the fit within its
uncertainty using a Gaussian constraint. Setting
${\Delta\Gamma_{{d}}}=0.007{\mathrm{\,ps^{-1}}}$, which corresponds to the
experimental uncertainty [23], leads to a small change in $\Delta
a_{\parallel}^{{{B}^{0}}}$, which is included in the systematic uncertainty.
The ${B}^{0}$ lifetime is allowed to vary freely in the fit.

In the fit to the ${B}^{0}_{s}$ sample, the correlation between
${\mathcal{R}e}(z^{{{B}^{0}_{s}}})$ and
$C_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}}}$ is
small owing to the additional interference terms from the helicity amplitudes,
the non-zero $\Delta\Gamma_{{s}}$ and the faster ${B}^{0}_{s}$–${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ oscillations. For this
reason, the same parameters as in Ref. [21] are varied freely in the fit, in
addition to either $\Delta a_{\mu}$ or $z$. The detection efficiency is also
described as a function of the decay angles. The shape of this angular
acceptance is obtained from simulation. The simulated events are weighted to
match the kinematic distributions in data. The uncertainty due to the limited
number of simulated events and the full effect of correcting for the kinematic
distributions in data are added to the systematic uncertainty. Systematic
effects due to the decay-angle resolution are negligibly small. The fit to the
${B}^{0}_{s}$ sample is performed simultaneously in bins of the ${K}^{+}$
${K}^{-}$ invariant mass [21]. Each bin has a different interference between
the P- and S-wave amplitudes. This effect is included in the fit and no
systematic biases are observed.

An overview of the systematic uncertainties is given in Table 3. For the
${B}^{0}$ mixing, the largest contribution comes from the uncertainty on the
external parameters $C_{f}$ and $S_{f}$. A small systematic bias is observed
in $\Delta a_{\perp}^{{{B}^{0}}}$ due to the momentum dependence of the cross-
sections of neutral kaons in the detector material. For the ${B}^{0}_{s}$
mixing, the largest contribution comes from the description of the decay-time
acceptance. Effects from the correlation between the mass and decay time and
from the accuracy of the length scale and momentum scale of the detector are
found to be negligible.

Table 3: Systematic uncertainties on $\Delta a_{\mu}$ for ${B}^{0}$ mixing and
on $\Delta a_{\mu}$ and $z$ for ${B}^{0}_{s}$ mixing. Contributions marked
with – are negligible.

${B}^{0}$ mixing | $\Delta a_{\parallel}$ | $\Delta a_{\perp}$ | $\Delta a_{X,Y}$
---|---|---|---
Source |  [$\times 10^{-15}\mathrm{\,Ge\kern-0.90005ptV}$]
Mass correlation | – | – | 0.04
Wrong PV assignment | – | 1 | –
Production asymmetry | 0.28 | 1 | 0.05
External input $C_{f}$, $S_{f}$ | 0.46 | 4 | 0.28
Decay width difference | 0.07 | – | –
Neutral kaon asymmetry | – | 1 | –
Quadratic sum | 0.54 | 4 | 0.29

${B}^{0}_{s}$ mixing | $\Delta a_{\parallel}$ | $\Delta a_{\perp}$ | $\Delta a_{X,Y}$ | ${\mathcal{R}e}(z)$ | ${\mathcal{I}m}(z)$
---|---|---|---|---|---
Source |  [$\times 10^{-14}\mathrm{\,Ge\kern-0.90005ptV}$] |  |
Mass correlation | 0.10 | 3 | 0.24 | 0.001 | 0.002
Peaking background | 0.14 | 3 | 0.15 | 0.003 | –
Decay-time acceptance | 0.30 | 7 | 0.65 | – | 0.001
Angular acceptance | 0.07 | – | – | 0.002 | 0.001
Quadratic sum | 0.36 | 8 | 0.71 | 0.003 | 0.002

The components of the SME parameter $\Delta a_{\mu}$ for ${B}^{0}$ mixing,
obtained from the fit to the sample of selected ${B}^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates, are

$\displaystyle{\Delta a_{\parallel}^{{{B}^{0}}}}$ $\displaystyle=(-0.10\pm
0.82\mathrm{\,(stat)}\pm 0.54\mathrm{\,(syst)})\times
10^{-15}\mathrm{\,Ge\kern-1.00006ptV}\ ,$ $\displaystyle{\Delta
a_{\perp}^{{{B}^{0}}}}$ $\displaystyle=(-0.20\pm 0.22\mathrm{\,(stat)}\pm
0.04\mathrm{\,(syst)})\times 10^{-13}\mathrm{\,Ge\kern-1.00006ptV}\ ,$
$\displaystyle{\Delta a_{X}^{{{B}^{0}}}}$ $\displaystyle=(+1.97\pm
1.30\mathrm{\,(stat)}\pm 0.29\mathrm{\,(syst)})\times
10^{-15}\mathrm{\,Ge\kern-1.00006ptV}\ ,$ $\displaystyle{\Delta
a_{Y}^{{{B}^{0}}}}$ $\displaystyle=(+0.44\pm 1.26\mathrm{\,(stat)}\pm
0.29\mathrm{\,(syst)})\times 10^{-15}\mathrm{\,Ge\kern-1.00006ptV}\ ,$

and the corresponding numbers for ${B}^{0}_{s}$ mixing, using ${B}^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ ${K}^{+}$
${K}^{-}$ candidates, are

$\displaystyle{\Delta a_{\parallel}^{{{B}^{0}_{s}}}}$ $\displaystyle=(-0.89\pm
1.41\mathrm{\,(stat)}\pm 0.36\mathrm{\,(syst)})\times
10^{-14}\mathrm{\,Ge\kern-1.00006ptV}\ ,$ $\displaystyle{\Delta
a_{\perp}^{{{B}^{0}_{s}}}}$ $\displaystyle=(-0.47\pm 0.39\mathrm{\,(stat)}\pm
0.08\mathrm{\,(syst)})\times 10^{-12}\mathrm{\,Ge\kern-1.00006ptV}\ ,$
$\displaystyle{\Delta a_{X}^{{{B}^{0}_{s}}}}$ $\displaystyle=(+1.01\pm
2.08\mathrm{\,(stat)}\pm 0.71\mathrm{\,(syst)})\times
10^{-14}\mathrm{\,Ge\kern-1.00006ptV}\ ,$ $\displaystyle{\Delta
a_{Y}^{{{B}^{0}_{s}}}}$ $\displaystyle=(-3.83\pm 2.09\mathrm{\,(stat)}\pm
0.71\mathrm{\,(syst)})\times 10^{-14}\mathrm{\,Ge\kern-1.00006ptV}\ .$

Figure 1 shows the result of fits of ${\mathcal{R}e}(z)$ in bins of the
sidereal phase for both samples. For the ${B}^{0}$ sample, the external
constraints on $C_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}$ and
$S_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}$ are again used. No sidereal
variation is observed. Independently of any assumption of Lorentz violation,
the complex $C\\!PT$-violating parameter $z$ in the ${B}^{0}_{s}$ system is
found to be

$\displaystyle{{\mathcal{R}e}(z^{{{B}^{0}_{s}}})}=-0.022\pm
0.033\mathrm{\,(stat)}\pm 0.003\mathrm{\,(syst)}\ ,$
$\displaystyle{{\mathcal{I}m}(z^{{{B}^{0}_{s}}})}=\phantom{+}0.004\pm
0.011\mathrm{\,(stat)}\pm 0.002\mathrm{\,(syst)}\ .$

Figure 1: Values of ${\mathcal{R}e}(z)$ obtained from fits in bins of sidereal
phase for (top) the ${B}^{0}$ sample and (bottom) the ${B}^{0}_{s}$ sample.
The solid line shows the variation of ${\mathcal{R}e}(z)$ from the $\Delta
a_{\mu}$ fits, using the average $B$ momentum.

Since the SME fits consider only one specific frequency, i.e. the sidereal
frequency, a wide range of frequencies is scanned by means of the periodogram
method. A periodogram gives the spectral power $P(\nu)$ of a frequency $\nu$
in a signal sampled at discrete, not necessarily equidistant, times. In this
analysis, the Lomb-Scargle periodogram [31] is used, as in the BaBar
measurement of $\Delta a^{{{B}^{0}}}_{\mu}$ [9].

The periodogram is determined for the term in the decay rates proportional to
$e^{-\Gamma t}{{\mathcal{R}e}(z)}$. Since negative weights cannot be used in
the periodogram, the $B$ mass windows are narrowed to
$5260<m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}}<5300{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$5350<m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}}<5390{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
compared to those used in the fits [20, 21]. In total about 5200 frequencies
are scanned in a wide range around the sidereal frequency, from 0.03 solar-
day-1 to 2.10 solar-day-1. The number of frequencies oversamples the number of
independent frequencies by roughly a factor of two, thereby avoiding any
undersampling [32]. As the data are unevenly sampled, the false-alarm
probability is determined from simulation [32], where the time stamps are
taken from data.

The two periodograms are shown in Fig. 2. No significant peaks are found. For
the ${B}^{0}$ periodogram, the highest peak $P(\nu_{\rm max})=8.09$ is found
at a frequency of 1.5507 solar-day-1 and has a false-alarm probability of
0.57. There are 2707 (1559) sampled frequencies with a larger spectral power
than the peak at the sidereal (solar) frequency. For the ${B}^{0}_{s}$
periodogram the highest peak $P(\nu_{\rm max})=10.85$ is found at a frequency
of 1.3301 solar-day-1 and has a false-alarm probability of 0.06. There are
3386 (2356) frequencies with a larger spectral power than the sidereal (solar)
peak. The absence of any signal in the SME fits is confirmed by the absence of
significant peaks at the sidereal frequency.

Figure 2: Periodograms for (top) the ${B}^{0}$ and (bottom) the ${B}^{0}_{s}$
sample. The insets show a zoom around the solar (red dashed line) and sidereal
(blue solid line) frequencies, which have been made by highly oversampling the
frequencies in this narrow range.

The results presented here are consistent with $C\\!PT$ symmetry and Lorentz
invariance. The measurement of $\Delta a^{{{B}^{0}}}_{\mu}$ is an improvement
in precision of about three orders of magnitude compared to the one from the
BaBar collaboration [9] when the SM prediction
${\Delta\Gamma_{{d}}}=-0.0027{\mathrm{\,ps^{-1}}}$ [33] is used to scale their
result. The measurement of $\Delta a^{{{B}^{0}_{s}}}_{\mu}$ is an order of
magnitude more precise than the one from the D0 collaboration [10] (note the
different definition, ${\Delta a_{\perp}}\equiv\sqrt{{\Delta
a_{X}}^{2}+{\Delta a_{Y}}^{2}}$, in Ref. [10]). The measurement of
$z^{{{B}^{0}_{s}}}$ is the first direct measurement of this quantity.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

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LHCb collaboration

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P. Alvarez Cartelle54, A.A. Alves Jr58, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3,40, L. Anderlini18, G. Andreassi40, M. Andreotti17,g, J.E. Andrews59, R.B.
Appleby55, O. Aquines Gutierrez11, F. Archilli39, P. d’Argent12, A.
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Bachmann12, J.J. Back49, A. Badalov37, C. Baesso61, S. Baker54, W. Baldini17,
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Battista40, A. Bay40, L. Beaucourt4, J. Beddow52, F. Bedeschi24, I. Bediaga1,
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Bencivenni19, S. Benson39, J. Benton47, A. Berezhnoy33, R. Bernet41, A.
Bertolin23, F. Betti15, M.-O. Bettler39, M. van Beuzekom42, S. Bifani46, P.
Billoir8, T. Bird55, A. Birnkraut10, A. Bizzeti18,i, T. Blake49, F. Blanc40,
J. Blouw11, S. Blusk60, V. Bocci26, A. Bondar35, N. Bondar31,39, W.
Bonivento16, A. Borgheresi21,k, S. Borghi55, M. Borisyak67, M. Borsato38, M.
Boubdir9, T.J.V. Bowcock53, E. Bowen41, C. Bozzi17,39, S. Braun12, M.
Britsch12, T. Britton60, J. Brodzicka55, E. Buchanan47, C. Burr55, A.
Bursche2, J. Buytaert39, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,k, M.
Calvo Gomez37,p, P. Campana19, D. Campora Perez39, L. Capriotti55, A.
Carbone15,e, G. Carboni25,l, R. Cardinale20,j, A. Cardini16, P. Carniti21,k,
L. Carson51, K. Carvalho Akiba2, G. Casse53, L. Cassina21,k, L. Castillo
Garcia40, M. Cattaneo39, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, M.
Charles8, Ph. Charpentier39, G. Chatzikonstantinidis46, M. Chefdeville4, S.
Chen55, S.-F. Cheung56, M. Chrzaszcz41,27, X. Cid Vidal39, G. Ciezarek42,
P.E.L. Clarke51, M. Clemencic39, H.V. Cliff48, J. Closier39, V. Coco58, J.
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Craik51, A. Crocombe49, M. Cruz Torres61, S. Cunliffe54, R. Currie54, C.
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Pikies27, D. Pinci26, A. Pistone20, A. Piucci12, S. Playfer51, M. Plo
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Polycarpo2, A. Popov36, D. Popov11,39, B. Popovici30, C. Potterat2, E.
Price47, J.D. Price53, J. Prisciandaro38, A. Pritchard53, C. Prouve47, V.
Pugatch45, A. Puig Navarro40, G. Punzi24,s, W. Qian56, R. Quagliani7,47, B.
Rachwal27, J.H. Rademacker47, M. Rama24, M. Ramos Pernas38, M.S. Rangel2, I.
Raniuk44, G. Raven43, F. Redi54, S. Reichert55, A.C. dos Reis1, V. Renaudin7,
S. Ricciardi50, S. Richards47, M. Rihl39, K. Rinnert53,39, V. Rives Molina37,
P. Robbe7, A.B. Rodrigues1, E. Rodrigues55, J.A. Rodriguez Lopez64, P.
Rodriguez Perez55, A. Rogozhnikov67, S. Roiser39, V. Romanovsky36, A. Romero
Vidal38, J. W. Ronayne13, M. Rotondo23, T. Ruf39, P. Ruiz Valls68, J.J.
Saborido Silva38, N. Sagidova31, B. Saitta16,f, V. Salustino Guimaraes2, C.
Sanchez Mayordomo68, B. Sanmartin Sedes38, R. Santacesaria26, C. Santamarina
Rios38, M. Santimaria19, E. Santovetti25,l, A. Sarti19,m, C. Satriano26,n, A.
Satta25, D.M. Saunders47, D. Savrina32,33, S. Schael9, M. Schiller39, H.
Schindler39, M. Schlupp10, M. Schmelling11, T. Schmelzer10, B. Schmidt39, O.
Schneider40, A. Schopper39, M. Schubiger40, M.-H. Schune7, R. Schwemmer39, B.
Sciascia19, A. Sciubba26,m, A. Semennikov32, A. Sergi46, N. Serra41, J.
Serrano6, L. Sestini23, P. Seyfert21, M. Shapkin36, I. Shapoval17,44,g, Y.
Shcheglov31, T. Shears53, L. Shekhtman35, V. Shevchenko66, A. Shires10, B.G.
Siddi17, R. Silva Coutinho41, L. Silva de Oliveira2, G. Simi23,s, M.
Sirendi48, N. Skidmore47, T. Skwarnicki60, E. Smith54, I.T. Smith51, J.
Smith48, M. Smith55, H. Snoek42, M.D. Sokoloff58, F.J.P. Soler52, F. Soomro40,
D. Souza47, B. Souza De Paula2, B. Spaan10, P. Spradlin52, S. Sridharan39, F.
Stagni39, M. Stahl12, S. Stahl39, S. Stefkova54, O. Steinkamp41, O.
Stenyakin36, S. Stevenson56, S. Stoica30, S. Stone60, B. Storaci41, S.
Stracka24,t, M. Straticiuc30, U. Straumann41, L. Sun58, W. Sutcliffe54, K.
Swientek28, S. Swientek10, V. Syropoulos43, M. Szczekowski29, T. Szumlak28, S.
T’Jampens4, A. Tayduganov6, T. Tekampe10, G. Tellarini17,g, F. Teubert39, C.
Thomas56, E. Thomas39, J. van Tilburg42, V. Tisserand4, M. Tobin40, S. Tolk43,
L. Tomassetti17,g, D. Tonelli39, S. Topp-Joergensen56, E. Tournefier4, S.
Tourneur40, K. Trabelsi40, M. Traill52, M.T. Tran40, M. Tresch41, A.
Trisovic39, A. Tsaregorodtsev6, P. Tsopelas42, N. Tuning42,39, A. Ukleja29, A.
Ustyuzhanin67,66, U. Uwer12, C. Vacca16,39,f, V. Vagnoni15, S. Valat39, G.
Valenti15, A. Vallier7, R. Vazquez Gomez19, P. Vazquez Regueiro38, C. Vázquez
Sierra38, S. Vecchi17, M. van Veghel42, J.J. Velthuis47, M. Veltri18,h, G.
Veneziano40, M. Vesterinen12, B. Viaud7, D. Vieira2, M. Vieites Diaz38, X.
Vilasis-Cardona37,p, V. Volkov33, A. Vollhardt41, D. Voong47, A. Vorobyev31,
V. Vorobyev35, C. Voß65, J.A. de Vries42, R. Waldi65, C. Wallace49, R.
Wallace13, J. Walsh24, J. Wang60, D.R. Ward48, N.K. Watson46, D. Websdale54,
A. Weiden41, M. Whitehead39, J. Wicht49, G. Wilkinson56,39, M. Wilkinson60, M.
Williams39, M.P. Williams46, M. Williams57, T. Williams46, F.F. Wilson50, J.
Wimberley59, J. Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A.
Wotton48, K. Wraight52, S. Wright48, K. Wyllie39, Y. Xie63, Z. Xu40, Z. Yang3,
H. Yin63, J. Yu63, X. Yuan35, O. Yushchenko36, M. Zangoli15, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang3, A. Zhelezov12, Y. Zheng62, A. Zhokhov32,
L. Zhong3, V. Zhukov9, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
36Institute for High Energy Physics (IHEP), Protvino, Russia
37Universitat de Barcelona, Barcelona, Spain
38Universidad de Santiago de Compostela, Santiago de Compostela, Spain
39European Organization for Nuclear Research (CERN), Geneva, Switzerland
40Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
41Physik-Institut, Universität Zürich, Zürich, Switzerland
42Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
43Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
44NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
45Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
46University of Birmingham, Birmingham, United Kingdom
47H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
48Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
49Department of Physics, University of Warwick, Coventry, United Kingdom
50STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
51School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
52School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
53Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
54Imperial College London, London, United Kingdom
55School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
56Department of Physics, University of Oxford, Oxford, United Kingdom
57Massachusetts Institute of Technology, Cambridge, MA, United States
58University of Cincinnati, Cincinnati, OH, United States
59University of Maryland, College Park, MD, United States
60Syracuse University, Syracuse, NY, United States
61Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
62University of Chinese Academy of Sciences, Beijing, China, associated to 3
63Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
64Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
65Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
66National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
67Yandex School of Data Analysis, Moscow, Russia, associated to 32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
69Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 42

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
†Deceased


# Asymptotics of Fundamental Solution of Cauchy Problem for Parabolic Equation
with Small Parameter and Degeneration

Mark Rakhel  National Research University Higher School of Economics, Moscow,
Russia<EMAIL_ADDRESS>
###### Abstract

In this paper, the method of constructing the asymptotics of the fundamental
solution of the Cauchy problem for a degenerate linear parabolic equation with
small diffusion is considered. Based on the results obtained in [1], the study
extends them over the case of a degenerate equation. As in [1], the main
technique that allows us to switch from pseudo-differential equations to
partial differential equations is the non-oscillating WKB method. A
distinctive feature of this work is a more detailed consideration on the
characteristics of the Green’s function in terms of symplectic geometry. The
most significant intermediate result is presented as a theorem on the
properties of the fundamental solution.

##  1 Introduction

One of the powerful methods for constructive study of partial differential
equations is the construction of asymptotic solutions. In the theory of
hyperbolic equations, methods for constructing asymptotic solutions are
distinguished, actively developed and applied, namely: the theory of the
canonical Maslov operator, the theory of Fourier integral operators. These
theories use objects and concepts of symplectic geometry: Lagrangian manifolds
and symplectic forms in phase space. This theory is less developed for
parabolic equations, a version of the theory of the Maslov canonical operator
for parabolic equations is known, but in general this area needs to be
studied. The fact is that the natural apparatus connecting hyperbolic
equations with objects of symplectic geometry is integral transforms, in
particular, the Fourier transform. For parabolic equations the use of this
apparatus leads to the appearance of complex-valued functions that are not
natural in the theory of parabolic equations. Nevertheless, another approach
can be used here. It is based on the representation of the Dirac delta
function using the action of the function of the creation and annihilation
operators on a Gaussian exponent in some abstract Hilbert space. This approach
allows one to transfer the method developed in the theory of hyperbolic
equations to parabolic equations. Similar problems in the case of non-
degenerate diffusion were solved in the classical works of S. Varadan [6], S.
Molchanov [7], V. Maslov [8] and also in the work of V. Maslov and V.
Nazaikinskii [9] on the contemporary level relatively recently. The approach
developed in these works essentially uses the non-degeneracy of the metric
corresponding to the main symbol of the parabolic equation. Since this
property is not present in particular problem, the technique mentioned above
is applied.

##  2 Delta function representations

The key point in constructing asymptotics is the representation of the Dirac
delta function via the action of the function of the creation operator on a
Gaussian exponent, which allows us to make the conversion from the Cauchy
problem for the fundamental solution to the Cauchy problem for the so-called
symbol of fundamental solution. The formula itself has been introduced and
proved in [1].

###  2.1 Delta function via Hermite polynomials

Let us consider one of the definitions of the Hermite polynomials

$H_{n}(x)=\biggm{[}\biggm{(}\frac{\partial}{\partial
z}\biggm{)}^{2}e^{xz-\frac{z^{2}}{2}}\biggm{]}\biggm{|}_{z=0}$ (1)

As it is well known, these polynomials form an orthogonal basis in $L_{2}$
with weight function $e^{-\frac{x^{2}}{2}}$. Then, in particular,
$\forall\psi(x)\in C_{0}^{\infty}:$

$\psi(x)=\frac{1}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{1}{n!}\psi_{n}H_{n}(x)e^{-\frac{x^{2}}{2}},$

where
$\psi_{n}=\langle\psi(x),H_{n}(x)e^{-\frac{x^{2}}{2}}\rangle=\int_{-\infty}^{+\infty}\psi(x)H_{n}(x)e^{-\frac{x^{2}}{2}}dx$
\- inner product. So we have the relation

$\begin{gathered}\psi(x)=\frac{1}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{1}{n!}\langle\psi(\xi),H_{n}(\xi)e^{-\frac{\xi^{2}}{2}}\rangle
H_{n}(x)e^{-\frac{x^{2}}{2}}=\\\
=\langle\psi(\xi),\frac{1}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{1}{n!}H_{n}(x)H_{n}(\xi)e^{-\frac{x^{2}+\xi^{2}}{2}}\rangle\end{gathered}$

By the definition, $\langle\delta(\xi-x),\psi(\xi)\rangle=\psi(x)$. This
implies

$\delta(\xi-x)=\frac{1}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{1}{n!}H_{n}(x)H_{n}(\xi)e^{-\frac{x^{2}+\xi^{2}}{2}}$

After considering $x=0$ and redefining variables, relation looks as follows

$\delta(x)=\frac{1}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{1}{n!}H_{n}(0)H_{n}(x)e^{-\frac{x^{2}}{2}}$
(2)

Knowing that $e^{A}=\sum_{n=0}^{\infty}\frac{1}{n!}A^{n}$, we can introduce a
small parameter $h$ and obtain

$\delta(x)=\frac{1}{\sqrt{\pi h}}e^{h\frac{\partial^{2}}{\partial y\partial
z}}e^{-\frac{(x-z)^{2}+y^{2}}{2h}}\biggm{|}_{z=y=0}\stackrel{{\scriptstyle
def}}{{=}}\frac{1}{\sqrt{\pi
h}}\mathcal{L}_{h}e^{-\frac{(x-z)^{2}+y^{2}}{2h}}\biggm{|}_{z=y=0},$ (3)

where $\mathcal{L}_{h}=e^{h\frac{\partial^{2}}{\partial y\partial z}}$.

###  2.2 Delta function via creation-annihilation operators

Creation and annihilation operators $a$ and $a^{+}$ are satisfying the
commutation relation $[a,a^{+}]=h$. For example,

$\begin{cases}a=\zeta+\frac{h}{2}\frac{\partial}{\partial\zeta}\\\
a^{+}=\zeta-\frac{h}{2}\frac{\partial}{\partial\zeta}\end{cases}$ (4)

We can define functions of these operators arranged in particular order

$\begin{cases}f(\stackrel{{\scriptstyle 2}}{{a^{+}}},\stackrel{{\scriptstyle
1}}{{a}})=\int{\tilde{f}(\alpha_{1},\alpha_{2})e^{-i\alpha_{1}a^{+}}e^{-i\alpha_{2}a}d\alpha_{1}d\alpha_{2}}\\\
g(\stackrel{{\scriptstyle 1}}{{a^{+}}},\stackrel{{\scriptstyle
2}}{{a}})=\int{\tilde{g}(\alpha_{1},\alpha_{2})e^{-i\alpha_{2}a}e^{-i\alpha_{1}a^{+}}d\alpha_{1}d\alpha_{2}}\end{cases}$

where $\tilde{f}(\alpha_{1},\alpha_{2})$ and
$\tilde{g}(\alpha_{1},\alpha_{2})$ \- are the Fourier transforms of the
functions $f(z,y)$ and $g(z,y)$
We use [4]

$F(\stackrel{{\scriptstyle 2}}{{a}},\stackrel{{\scriptstyle
1}}{{a^{+}}})=(\mathcal{L}_{h}F)(\stackrel{{\scriptstyle
1}}{{a}},\stackrel{{\scriptstyle 2}}{{a^{+}}})$

Denote $F(z,y)=e^{\frac{-(x-z)^{2}-y^{2}}{2h}}$. Then

$\mathcal{L}_{h}e^{-\frac{(x-\stackrel{{\scriptstyle
1}}{{a}})^{2}}{2h}-\frac{(\stackrel{{\scriptstyle
2}}{{a^{+}}})^{2}}{2h}}=e^{-\frac{(x-\stackrel{{\scriptstyle
2}}{{a}})^{2}}{2h}-\frac{(\stackrel{{\scriptstyle
1}}{{a^{+}}})^{2}}{2h}}=e^{-\frac{(x-a)^{2}}{2h}}e^{-\frac{(a^{+})^{2}}{2h}}$
(5)

Let us consider $\chi\in\ker(a),\quad\chi_{+}\in\ker{(a^{+})^{*}}$. Then
$a\chi=0,\quad(a^{+})^{*}\chi_{+}=0$. We found out that
$\delta(x)=\frac{1}{\sqrt{\pi h}}(\mathcal{L}_{h}F)(0,0)$. On the other hand

$\begin{gathered}(\mathcal{L}_{h}F)(0,0)=\langle\chi_{+},((\mathcal{L}_{h}F)(\stackrel{{\scriptstyle
1}}{{a}},\stackrel{{\scriptstyle
2}}{{a^{+}}})\chi\rangle\langle\chi_{+},\chi\rangle^{-1}=\\\
=\langle\chi_{+},F(\stackrel{{\scriptstyle 2}}{{a}},\stackrel{{\scriptstyle
1}}{{a^{+}}})\chi\rangle\langle\chi_{+},\chi\rangle^{-1}\end{gathered}$ (6)

The null vectors of operators (4) are
$\chi=\chi_{+}=e^{-\frac{\zeta^{2}}{h}},\quad\langle\chi_{+},\chi\rangle^{-1}=\sqrt{\frac{2}{\pi
h}}$. Substituting those expressions in (6), we get the key expression for the
Dirac delta.

$\delta(x)=\frac{\sqrt{2}}{\pi h}\langle
e^{-\frac{(x-a^{+})^{2}}{2h}}e^{-\frac{\zeta^{2}}{h}},e^{-\frac{(a^{+})^{2}}{2h}}e^{-\frac{\zeta^{2}}{h}}\rangle$
(7)

Note. Instead of using (7), we can rely on the following lemma, which was
initially presented and proven in [1].

Lemma. Any function $\phi(x,h)$ uniformly bounded for each fixed $h$ and
satisfying the Lipschitz condition $|\phi(x_{1},h)-\phi(x_{2},h)|\leq
k|x_{1}-x_{2}|$ satisfies the relation

$\phi(0,h)=\frac{1}{\sqrt{2\pi h}}\lim_{\beta\to
1-0}\int_{-\infty}^{\infty}\phi(x,h)\langle
e^{-\beta\frac{(x-a^{+})^{2}}{2h}}e^{-\frac{\zeta^{2}}{h}},\delta(\zeta)\rangle
dx$ (8)

##  3 Algorithm

The Green’s function $G(x,\xi,t,h)$ is defined as the solution to the Cauchy
problem

$\begin{cases}-h\frac{\partial G}{\partial t}+\hat{H}(\stackrel{{\scriptstyle
2}}{{x}},\stackrel{{\scriptstyle 1}}{{-h\frac{\partial}{\partial x}}})G=0\\\
G|_{t=0}=\delta(x-\xi)\end{cases}$ (9)

where $H(x,p)$ is a function analytic in $p$ and taking real values whenever
the arguments are real.

From the lemma follows that if $V_{\beta}(x,\xi,y,t,h)$ is a solution to the
Cauchy problem

$\begin{cases}-h\frac{\partial V_{\beta}}{\partial
t}+H(\stackrel{{\scriptstyle 2}}{{x}},\stackrel{{\scriptstyle
1}}{{-h\frac{\partial}{\partial x}}})V_{\beta}=0\\\
V_{\beta}|_{t=0}=e^{-\beta\frac{(x-\xi-y)^{2}}{2h}}\end{cases}$ (10)

then the general form of the fundamental solution can be represented by the
relation

$G(x,\xi,t,h)=\frac{1}{\sqrt{2\pi h}}\lim_{\beta\to
1-0}V_{\beta}(x,\xi,a^{+},t,h)e^{-\frac{\zeta^{2}}{h}}\biggm{|}_{\zeta=0}$
(11)

Function $V(x,t,y,h)$ is called by a symbol of the fundamental solution and
its argument $y$ corresponds to the creation operator $a^{+}$.

It is proposed to search for the symbol using the non-oscillating WKB method,
that is, to look for asymptotic approximation $V_{\beta,N}(x,\xi,y,t,h)$

$V_{\beta,N}=e^{-\frac{\Phi_{\beta}}{h}}(\varphi_{0}+h\varphi_{1}+...+O(h^{N}))$

In this paper, we restrict ourselves to the first term

$V_{\beta}=e^{-\frac{\Phi_{\beta}}{h}}(\phi_{0}+O(h))$ (12)

Hence, for function $\Phi_{\beta}(x,\xi,y,t)$ we get the Hamilton-Jacobi
equation

$\begin{cases}\frac{\partial\Phi_{\beta}}{\partial
t}+H(x,\frac{\partial\Phi_{\beta}}{\partial x})=0\\\
\Phi_{\beta}|_{t=0}=\beta\frac{(x-\xi-y)^{2}}{2}\end{cases}$ (13)

and for $\varphi_{0}(x,\xi,y,t)$ \- Transport equation

$\begin{cases}\Pi\varphi=0\\\ \varphi_{0}|_{t=0}=P(x)\end{cases}$ (14)

where $\Pi=\frac{\partial}{\partial t}+H_{p}\frac{\partial}{\partial
x}+\frac{1}{2}H_{pp}(\Phi_{\beta})_{xx}$, and function $P(x)\in
C_{0}^{\infty}$ looks as following

$P(x)=\begin{cases}1,|x|<\delta_{1}\\\ 0,|x|\geq\delta\end{cases}$

if $\delta<\delta_{1}$. This means that we restrict ourselves to considering
solutions in the neighborhood of $x=0$ and initial conditions with support
from the same neighborhood. The characteristic system of (13) is the Hamilton
system

$\begin{cases}\dot{x}=H_{p},\quad x|_{t=0}=x_{0}\\\ \dot{p}=-H_{x},\quad
p|_{t=0}=\beta(x_{0}-\xi-y)\end{cases}$

and the solution to the Hamilton-Jacobi equation can be found with the
integration of form $\omega=pdx$ and projection $x_{0}(x,\xi,y,t)$ as long as
$J_{0}=|\frac{\partial x}{\partial x_{0}}(x_{0},\xi,y,t)|\neq 0$

The expression for the solution to the Cauchy problem (14) is well known

$\varphi_{0}=\frac{1}{\sqrt{J_{0}}}e^{\frac{1}{2}\int_{0}^{t}{H_{xp}d\tau}}$
(15)

The action of the function of the creation operator on the exponent before
taking the limit, namely $V(x,\xi,a^{+},t,h)e^{-\frac{\zeta^{2}}{h}}$ can be
also figured out by dint of the non-oscillating WKB method. Eventually, the
explicit formula for the fundamental solution is as follows

$G(x,\xi,t,h)=\frac{1}{\sqrt{2\pi h}}\lim_{\beta\to
1-0}{\frac{e^{\frac{1}{h}(\frac{\hat{y}^{2}}{2}-\Phi_{\beta}(x,\xi,\hat{y},t))}}{\sqrt{J_{0}}\sqrt{1-\frac{\partial^{2}\Phi_{\beta}}{\partial
y^{2}}(x,\xi,\hat{y},t)}}}e^{\frac{1}{2}\int_{0}^{t}{H_{xp}d\tau}}(1+O(h))$
(16)

where $\hat{y}=\hat{y}(x,t,\xi)$ is the implicit function defined as a
solution to the equation $y=\frac{\partial\Phi}{\partial y}(x,t,\xi)$.

##  4 New geometric method

A more detailed consideration of the construction described above indicates to
us the presence of a new symplectic structure, an analogue of which arises in
the theory of Fourier integral operators, namely:

1) Extended four-dimensional phase space with variables $(x,y,p_{x},p_{y})$
and Hamilton system

$\begin{cases}\dot{x}=H_{p}(x,p_{x}),\quad x|_{t=0}=x_{0}\\\
\dot{p_{x}}=-H_{x}(x,p_{x}),\quad p_{x}|_{t=0}=\beta(x_{0}-y-\xi)\\\
\dot{y}=0,\quad y|_{t=0}=y\\\ \dot{p_{y}}=0,\quad
p_{y}|_{t=0}=-\beta(x_{0}-y-\xi)\end{cases}$ (17)

wherein $y$ and $p_{y}$ correspond to the creation operator.
2) New fundamental form

$\omega_{y}\in T^{*}(\mathbb{R}_{x}^{n})\times
T^{*}(\mathbb{R}_{y}^{n}),\quad\omega_{y}=p_{x}dx-p_{y}dy$ (18)

3) Function $V$ appears in the variables $(x,t)$ after the projection

$\pi:\Lambda_{t}^{2n}\to(\mathbb{R}_{x}^{n},y=p_{y})$ (19)

which comes to be the product of the configuration space and the diagonal of
the complement $\mathbb{R}_{y,p_{y}}^{n}$.
As mentioned above, phase $\Phi$ is found by integrating the form and applying
the inverse mapping

$\Phi_{\beta}=(\pi^{-1})^{*}\int\omega_{y},$ (20)

where $(\pi^{-1})^{*}$ is the mapping induced by $\pi^{-1}$.
It is easy to see that

$\begin{gathered}\int{\omega}_{y}=\Phi_{\beta}|_{t=0}-\int_{0}^{y}{p_{y}dy}+\int_{0}^{t}{{(pH_{p}-H)d\tau}}=\\\
=\frac{\beta}{2}(x_{0}-\xi)^{2}+\int_{0}^{t}{(pH_{p}-H)d\tau}\end{gathered}$

If $(pH_{p}-H)\geq 0$, then for each $x_{0}$ and $y$

$\Phi_{\beta}\geq 0$

So, according to the new notation the expression for the fundamental solution
takes the form

$G(x,t,\xi,h)=\frac{1}{\sqrt{2\pi h}}\lim_{\beta\to
1-0}{\frac{e^{-\frac{1}{h}(\pi^{-1})^{*}\int\omega_{y}}}{\sqrt{J}}}e^{\frac{1}{2}\int_{0}^{t}{H_{xp}d\tau}}(1+O(h))$
(21)

where $J=|\frac{\partial(x,y-p_{y})}{\partial(x_{0},y)}|$

##  5 Example: Heat equation

Let us consider as an example the one-dimensional heat equation with constant
coefficients and a small parameter $h$

$\frac{\partial u}{\partial t}-h\frac{\partial^{2}u}{\partial x^{2}}=0$ (22)

the fundamental solution of which is well known. Let us find it first using a
sequential algorithm and then with the geometric method.
The Green’s function for (22) is the function $G(x,\xi,t,h)$, satisfying

$\begin{cases}\frac{\partial G}{\partial t}-h\frac{\partial^{2}G}{\partial
x^{2}}=0\\\ G|_{t=0}=\delta(x-\xi)\end{cases}$ (23)

The problem for the symbol $V(x,\xi,y,t,h)$

$\begin{cases}\frac{\partial V}{\partial t}-h\frac{\partial^{2}V}{\partial
x^{2}}=0\\\ V|_{t=0}=e^{-\frac{(x-\xi-y)^{2}}{2h}}\end{cases}$ (24)

Non-oscillating WKB method leads to the Hamilton-Jacobi and the Transport
equations for functions $\Phi(x,t,y)$ and $\varphi(x,t,y)$

$\begin{cases}\frac{\partial\Phi}{\partial t}+(\frac{\partial\Phi}{\partial
x})^{2}=0\\\ \Phi|_{t=0}=\frac{(x-\xi-y)^{2}}{2}\end{cases}$ (25)

The Hamilton system is

$\begin{cases}\dot{x}=2p,\quad x|_{t=0}=x_{0}\\\ \dot{p}=0,\quad
p|_{t=0}=x_{0}-y-\xi\end{cases}$

and the solution to the problem (25) is as follows

$\Phi=\int{pdx}\big{|}_{x_{0}=x_{0}(x,\xi,y,t)}=\frac{(x-\xi-y)^{2}}{2(1+2t)}$

A solution to the Transport equation $\varphi_{0}(x,\xi,y,t)$ in this case is

$\varphi_{0}=\frac{1}{\sqrt{\frac{\partial x}{\partial
x_{0}}}}=\frac{1}{\sqrt{1+2t}}$

Now we can write down the explicit formula for $V$

$V=\frac{1}{\sqrt{1+2t}}e^{-\frac{(x-\xi-y)^{2}}{2(1+2t)h}}$ (26)

Then in the expression for the symbol (26), instead of $y$, you need to
substitute the creation operator
$a^{+}=\zeta-\frac{h}{2}\frac{\partial}{\partial\zeta}$ and calculate the
action of the function from the operator on the Gaussian exponent
$V(x,\xi,a^{+},t,h)e^{-\frac{\zeta^{2}}{h}}|_{\zeta=0}$. This action is
calculated through finding the function $W(x,\ xi,t,h,\alpha,\zeta)$, which is
a solution to the problem

$\begin{cases}\frac{\partial W}{\partial\alpha}-V(x,\xi,a^{+},t,h)W=0\\\
W|_{\alpha=0}=e^{-\frac{\zeta^{2}}{h}}\end{cases}$ (27)

Then $W(x,\xi,t,h,1,0)=V(x,\xi,a^{+},t,h)e^{-\frac{\zeta^{2}}{h}}|_{\zeta=0}$.
The problem (27) is again solved using the non-oscillating WKB method. So,
repeating the steps described above we obtain the required expression for the
Green’s function of the heat equation

$G(x,\xi,t,h)=\frac{1}{\sqrt{4\pi ht}}e^{-\frac{(x-\xi)^{2}}{4th}}$ (28)

Now let’s find a fundamental solution using the ”geometric” method. In this
case, instead of the system (5), we write the extended system of
characteristics

$\begin{cases}\dot{x}=2p,\quad x|_{t=0}=x_{0}\\\ \dot{p_{x}}=0,\quad
p_{x}|_{t=0}=\beta(x_{0}-y-\xi)\\\ \dot{y}=0,\quad y|_{t=0}=y\\\
\dot{p_{y}}=0,\quad p_{y}|_{t=0}=-\beta(x_{0}-y-\xi)\end{cases}$

From this system, we now need to express $x_{0}(x,\xi,t)$ and $y(x,\xi,t)$.
The expressions for them are as follows

$\begin{gathered}x_{0}=\frac{-x(1-\beta)-2\beta\xi t}{1-\beta+2\beta t}\\\
y=\frac{-\beta(x-\xi)}{1-\beta+2\beta t}\end{gathered}$

and the Jacobian of the extended system is

$J=\frac{\partial(x,y-p_{y})}{\partial(x_{0},y)}=1-\beta+2\beta t$

Finally, by integrating the form $\omega_{y}$ and substituting the required
values into the formula (21), we obtain an expression for the Green’s function
of the heat equation (28).

##  6 Construction of the asymptotics of the fundamental solution for a linear
degenerate parabolic equation

Let us consider in details the case $H(x,p)=x^{2}p^{2}$, i.e., we seek a
fundamental solution to the problem

$\begin{cases}\frac{\partial u}{\partial
t}-hx^{2}\frac{\partial^{2}u}{\partial x^{2}}=0\\\
u|_{t=0}=u_{0}(x)\end{cases}$ (29)

namely, the function $G(x,t,\xi,h)$, which is a solution to the Cauchy problem

$\begin{cases}\frac{\partial G}{\partial
t}-hx^{2}\frac{\partial^{2}G}{\partial x^{2}}=0\\\
G|_{t=0}=\delta(x-\xi)\end{cases}$ (30)

The solution to the original problem (29) is the function $u(x,t)$, which is
defined as the convolution of the Green’s function with the initial data

$u(x,t,h)=\int{G(x,\xi,t,h)u_{0}(\xi)d\xi}$

Note. Let $\xi=0$. Then in the problem (30) you can separate the variables,
namely, look for a solution in the form $G=f(t)\delta(x)$. Solving the
ordinary differential equation, we get

$G=e^{2ht}\delta(x)=\delta(x)(1+O(h))$

This fact shows the absence of smoothing, in contrast to ordinary parabolic
equations (for example, the heat equation).

The Cauchy problem for the symbol has the form

$\begin{cases}\frac{\partial V_{\beta}}{\partial
t}-hx^{2}\frac{\partial^{2}V_{\beta}}{\partial x^{2}}=0\\\
V_{\beta}|_{t=0}=e^{-\beta\frac{(x-\xi-y)^{2}}{2h}}\end{cases}$ (31)

and after applying the WKB method, we obtain the problem for finding the phase
$\Phi_{\beta}(x,\xi,y,t)$

$\begin{cases}\frac{\partial\Phi_{\beta}}{\partial
t}+x^{2}(\frac{\partial\Phi_{\beta}}{\partial x})^{2}=0\\\
\Phi_{\beta}|_{t=0}=\beta\frac{(x-\xi-y)^{2}}{2}\end{cases}$ (32)

and the amplitude $\phi_{0}(x,\xi,y,t)$

$\begin{cases}\frac{\partial\phi_{0}}{\partial
t}+2(\Phi_{\beta})_{x}\frac{\partial\phi_{0}}{\partial
x}+(\Phi_{\beta})_{xx}\phi_{0}=0\\\ \phi_{0}|_{t=0}=P(x)\end{cases}$ (33)

Hamilton characteristic system for (32)

$\begin{cases}\dot{x}=H_{p}=2x^{2}p,\quad x|_{t=0}=x_{0}\\\
\dot{p}=-H_{x}=-2xp^{2},\quad p|_{t=0}=\beta(x_{0}-\xi-y)\end{cases}$ (34)

From such a system we find $x=x(x_{0},\xi,y,t)$ and $p=p(x_{0},\xi,y,t)$:

$\begin{cases}x=x_{0}e^{2\beta x_{0}(x_{0}-\xi-y)t}\\\
p=\beta(x_{0}-\xi-y)e^{-2\beta x_{0}(x_{0}-\xi-y)t}\end{cases}$

The Jacobian of the mapping $J_{0}=\frac{\partial x}{\partial
x_{0}}(x_{0},\xi,y,t)$ is here

$J_{0}=e^{2\beta x_{0}(x_{0}-\xi-y)t}(1+2\beta tx_{0}(2x_{0}-\xi-y))$

If we do not introduce the parameter $\beta$ (or immediately set it to one),
such a Jacobian will take zero values for $y=const$, and the Lagrangian
manifold is S-shaped at some points. This is clearly seen at $t\approx 2/3$.

In terms of the new geometric method, a similar picture is the graph of the
projection of the section of the Lagrangian manifold
$\lambda\in\mathbb{R}^{4}$ by the plane $y=const$ onto the space $(x,p_{x})$.
But in the very four-dimensional space $(x,y,p_{x},p_{y})$ for sufficiently
small $t$ such pictures do not appear.

Lemma. The function $G_{\beta}$ satisfies the following relations (in the case
of distributions)

$\lim_{\beta\to 1-0}G_{\beta}(0,\xi,t,h)=\delta(\xi)$ (35)

and

$\lim_{\beta\to 1-0}(\psi(x),G_{\beta}(x,0,t,h))=\psi(0)(1+O(h)),$ (36)

$\psi$ \- test function.

###### Proof.

To prove the first relation we use the fact that if $x=0$, then $x_{0}=0$. The
function $G_{\beta}(0,\xi,t,h)$ is then written out explicitly

$G_{\beta}=\frac{e^{-\frac{\beta\xi^{2}}{2(1-\beta)h}}}{\sqrt{2\pi
h}{\sqrt{1-\beta}}}$

and it equals $\delta(\xi)$.
The second relation can be proved using the following fact

$\int_{-\infty}^{\infty}G_{\beta}(x_{0}(x,\xi,t),\xi,t,h)dx=\int_{-\infty}^{\infty}G_{\beta}(x_{0},\xi,t,h)\frac{dx}{dx_{0}}dx_{0}$

The convergence of the integral is then easily verified, and again we get a
delta function. ∎

###  6.1 Existence and properties of the solution

As noted in the relevant section, the improved algorithm introduces a special
symplectic structure, including the projection $\ pi$, which is the
composition of the projection onto the $x$ axis with the projection onto the
diagonal $y=p_{y}$. Thus, in the course of solving, it becomes necessary to
express $x_{0}(x,\xi,\beta,t)$ and $\hat{y}(x,\xi,\beta,t)$ from the system

$\begin{cases}x(x_{0},\xi,y,\beta,t)=x\\\
y=p_{y}(x_{0},\xi,y,\beta,t)\end{cases}$

For the case $H(x,p)=x^{2}p^{2}$ this system looks as follows

$\begin{cases}x=x_{0}e^{2\beta x_{0}(x_{0}-\xi-y)t}\\\
y=-\beta(x_{0}-\xi-y)\end{cases}$ (37)

Expressing $y(x_{0},\xi,\beta,t)=-\frac{\beta(x_{0}-\xi)}{1-\beta}$ from the
second equation of the system (37) and substituting into the first, we get an
expression for $x(x_{0},\xi,\beta,t)$

$x=x_{0}e^{\frac{2\beta x_{0}(x_{0}-\xi)t}{1-\beta}}$ (38)

$x_{0}(x,\xi,\beta,t)$, as noted above, cannot be expressed explicitly. But we
can consider the Jacobian of the map by substituting $y=y(x0,\xi,\beta,t)$

$J_{0}(x_{0},\xi,\beta,t)=\frac{\partial x(x_{0},\xi,\beta,t)}{\partial
x_{0}}=e^{\frac{2\beta x_{0}(x_{0}-\xi)t}{1-\beta}}\big{(}1+\frac{2\beta
t}{1-\beta}(2x_{0}^{2}-x_{0}\xi)\big{)}$ (39)

For $0<\beta<1$ for any bounded $x$ and $\xi$ this Jacobian is nowhere equal
to zero for sufficiently small $t$, which means that the solution will be a
smooth function.

For $\beta\to 1-0$, the expression (38) is defined only for two values of
$x_{0}$: $x_{0}=0$ and $x_{0}=\xi$, and for each of these values the Jacobian
is defined and distinct from zero, which means that the solution also exists
as a smooth function.

It is worth noting that the formula for the fundamental solution contains the
Jacobian of the extended system
$J=\frac{\partial(x,y-p_{y})}{\partial(x_{0},y)}$, that is, (39) is multiplied
by $\frac{\partial(y-p_{y})}{\partial y}=1-\beta$, and for $\beta\to 1-0$, the
denominator no longer has an infinitely large factor.

Since $x=0\Leftrightarrow x_{0}=0$, this point is the support of the delta
function, which remains at the point $x=0$ for any $t>0$. If $x\not=0$, then
any solution $x^{*}$ of the equation of characteristics lies on the trajectory
outgoing from the point $x_{0}=\xi$, which is the support of the initial data.

###  6.2 asymptotics of the logarithmic limit of the fundamental solution as
$t\to+0$

To construct the asymptotics of the limit of the fundamental solution as
$t\to+0$, we need to introduce the following lemma [3]

Lemma. Suppose that there exists a smooth solution to the problem

$\begin{cases}\frac{\partial\Phi}{\partial t}+H(x,\frac{\partial\Phi}{\partial
x})=0\\\ \Phi|_{t=0}=\frac{(x-\xi-y)^{2}}{2}\end{cases}$ (40)

for $0\leq t\leq T$, i.e. $J=\frac{\partial x(x_{0},\xi,y,t)}{\partial
x_{0}}\neq 0$. Then for $t>0$

$G(x,t,\xi,h)=\frac{1}{\sqrt{2\pi
h}}{\frac{e^{-\frac{S(x,t,\xi)}{h}}}{\sqrt{J_{0}}}}e^{\frac{1}{2}\int_{0}^{t}{H_{xp}d\tau}}(1+O(h))$
(41)

where the function $S(x,t,\xi)$ is determined by the expression

$S(x,t,\xi)=\int_{0}^{t}{(pH_{p}(x,p,\tau)-H(x,p,\tau))d\tau}$ (42)

and $x=x(\xi,p_{0},t)$, $p=p(\xi,p_{0},t)$ is the solution of the Hamilton
system

$\begin{cases}\frac{dx}{d\tau}=H_{p}(x,p,\tau),\quad x|_{\tau=0}=\xi\\\
\frac{dp}{d\tau}=-H_{x}(x,p,\tau),\quad
p|_{\tau=0}=p_{o}\in\mathbb{R}\end{cases}$ (43)

and $J_{0}=\frac{\partial x(x_{0},\xi,p_{o})}{\partial p_{0}}\neq 0$.

Next, since we want to investigate the behavior of the function $S(x,t,\xi)$
as $t\to+0$, we represent $t$ in the form $t=\nu t^{\prime}$, where
$t^{\prime}$ is a fixed number and $\nu\to+0$. Changing the variable
$t=\nu\sigma$, from the lemma we receive

$S(x,t,\xi)=\nu\int_{0}^{t}{(pH_{p}(X,p,\tau)-H(X,p,\tau))d\tau}$ (44)

where $X$ and $p$ satisfy

$\begin{cases}\frac{dX}{d\sigma}=\nu H_{p}(X,p),\quad X|_{\sigma=0}=\xi\\\
\frac{dp}{d\sigma}=-\nu H_{x}(X,p),\quad
p|_{\sigma=0}=p_{o}\in\mathbb{R}\end{cases}$ (45)

Solving (45), we have

$\begin{cases}X=\xi+\nu\sigma H_{p}(\xi,p_{0})+O(\nu^{2})\\\ p=p_{0}-\nu\sigma
H_{x}(\xi,p_{0})+O(\nu^{2})\end{cases}$ (46)

Substituting the expressions for $X$ and $p$ into (44) with $\nu\to+0$,
eventually we obtain

$\begin{gathered}S=\nu
t^{\prime}(pH_{p}(\xi,p_{0})-H(\xi,p_{0}))+O(\nu^{2})=\\\
=t(pH_{p}(\xi,p_{0})-H(\xi,p_{0}))+O(t^{2})\end{gathered}$ (47)

Let $X|_{\sigma=t^{\prime}}=x\in\mathbb{R}$. Denote $Y=\frac{x-\xi}{\nu
t^{\prime}}$. Expanding $p_{0}$ into a power series with respect to $\nu$, we
have $p_{0}=P_{0}+\nu P_{1}+...+\nu^{n}P_{n}$. Then $P_{0}=P_{0}(Y,\xi)$ can
be found from the relation

$\frac{x-\xi}{\nu t^{\prime}}=H_{p}(\xi,P_{0})$ (48)

In our case $H=x^{2}p^{2}$ and

$S=\nu\int_{0}^{t^{\prime}}X^{2}p^{2}d\sigma$ (49)

Relation (48) looks as

$\frac{x-\xi}{\nu t^{\prime}}=2\xi^{2}P_{0}$ (50)

If $|x-\xi|\leq c\nu$, then $P_{0}$ is a smooth function as long as $\xi\neq
0$. As was mentioned in (36), given $\xi=0$, the fundamental solution is the
Dirac delta function and function $S$ has the form

$S(x,\beta)=\frac{x^{2}}{2(1-\beta)},\quad\beta\to 1-0$ (51)

If $\xi\neq 0$, we have

$P_{0}=\frac{Y}{2\xi^{2}}$ (52)

Thus, $P_{0}\sim\frac{1}{\nu}$ for each fixed $\xi$ and $\nu\to+0$. This
implies that we need to set $P=\nu p$ in (45) in order to obtain $X$ and $P$
by using regular perturbation methods. The system (45) now takes the form

$\begin{cases}\frac{dX}{d\sigma}=2X^{2}P\\\
\frac{dP}{d\sigma}=-2XP^{2}\end{cases}$ (53)

with boundary conditions

$X|_{\sigma=0}=\xi,\quad X|_{\sigma=t^{\prime}}=x$ (54)

According the regular perturbation theory, we seek the solution of such system
in the form

$X=X_{0}+\nu X_{1}+...+\nu^{N}X_{N},\quad P=P_{0}+\nu P_{1}+...+\nu^{N}P_{N}$
(55)

After substituting the expansions for $X$ and $P$ in (53), we obtain systems
for $X_{0}$ and $P_{0}$

$\begin{cases}\frac{dX_{0}}{d\sigma}=2X_{0}^{2}P_{0}\quad
X_{0}|_{\sigma=0}=\xi\\\ \frac{dP_{0}}{d\sigma}=-2X_{0}P_{0}^{2}\quad
X_{0}|_{\sigma=t^{\prime}}=x\end{cases}$ (56)

for $X_{1}$ and $P_{1}$

$\begin{cases}\frac{dX_{1}}{d\sigma}=4X_{0}X_{1}P_{0}+2X_{0}^{2}P_{1}\quad
X_{1}|_{\sigma=0}=0\\\
\frac{dP_{1}}{d\sigma}=-2X_{1}P_{0}^{2}-4X_{0}P_{0}P_{1}\quad
X_{1}|_{\sigma=t^{\prime}}=0\end{cases}$ (57)

and so on. The function $S$ in this case has the form

$S=\nu^{-1}S_{0}+S_{1}+\nu S_{2}+O(\nu^{2})$ (58)

where $S_{j}$ can be found from (49).

##  7 Conclusion. Investigation of the Hamilton system in the general one-
dimensional case

The achieved results can be transferred to the case of more general single-
dimensional Hamiltonians

$H(x,p)=x^{2}a^{2}(x)p^{2},\quad a\neq 0$ (59)

Let us denote $A(x)=xa(x)$. The characteristic system for the corresponding
Hamilton-Jacobi equation thus appears the following way

$\begin{cases}\dot{x}=H_{p}=2A^{2}p,\quad x|_{t=0}=x_{0}\\\
\dot{p}=-H_{x}=-2AA^{\prime}p^{2},\quad
p|_{t=0}=p_{0}=\beta(x_{0}-\xi-y)\end{cases}$ (60)

When the second equation of the system (60) is divided by the first one, the
ordinary differential equation is obtained

$\frac{dp}{dx}=-\frac{A^{\prime}}{A}p$

or the equivalent to it

$\frac{dp}{d}+\frac{dA}{A}=0$

Integrating the both parts, we get

$p(x_{0},t)A(x(x_{0},t))=C_{0}$ (61)

The constant $C_{0}$ is defined by the initial conditions and after the
insertion of $y=-\frac{\beta(x_{0}-\xi)}{1-\beta}$ has the form

$C_{0}=p_{0}A(x_{0})=\frac{\beta x_{0}a(x_{0})(x_{0}-\xi)}{1-\beta}$ (62)

Expressing $p=\frac{C_{0}}{A}$ from (61) and inserting it in the first
equation of the system (60), the ordinary differential equation is obtained

$\frac{dx}{dt}=2AC_{0}$

Separating the variables and integrating, taking into account the initial
data, we have

$\int_{x_{0}}^{x}\frac{dy}{ya(y)}=\frac{2\beta
tx_{0}a(x_{0})(x_{0}-\xi)}{1-\beta}$ (63)

Let us indicate a couple of factors, that could be find out from the
expression (63):

1) $x=0\Leftrightarrow x_{0}=0$. Inasmuch as with the fixed $t$ and
$0<\beta<1$ there is a bounded expression on the right, and the integral on
the left diverges at zero value of one of the integration limits, the equality
itself (63) is possible only in the case when the other integration limit is
equal to zero.

2) With $\beta\to 1-0$, on the contrary, the expression on the right will
increase infinitely, and since the integral on the left is bounded, equality
is possible only for $x_{0}=0$ or $x_{0}=\xi$.

Now let us estimate the properties of the Jacobian $\frac{\partial x}{\partial
x_{0}}$. The integral in the expression (63) is a function of $x$ and $x_{0}$.
We denote

$g(x,x_{0})=\int_{x_{0}}^{x}\frac{dx}{xa(x)}$

Differentiate both sides of the expression (63) by $x_{0}$

$\frac{\partial g}{\partial x}\frac{\partial x}{\partial x_{0}}+\frac{\partial
g}{\partial x_{0}}=2t\frac{\partial C_{0}}{\partial x_{0}}$

Thus, the expression for Jacobian gets the following form

$\frac{\partial x}{\partial x_{0}}=\frac{1}{\frac{\partial x}{\partial
x_{0}}}\biggm{(}2t\frac{\partial C_{0}}{\partial x_{0}}-\frac{\partial
g}{\partial x_{0}}\biggm{)}$ (64)

According to the rule of differentiation of the integral with the variable
limits of integration, we have

$\frac{\partial g}{\partial x}=\frac{1}{xa(x)},\quad\frac{\partial g}{\partial
x_{0}}=-\frac{1}{x_{0}a(x_{0})}$

The derivative of a constant is equal to

$\frac{\partial C_{0}}{\partial
x_{0}}=\frac{\beta}{1-\beta}\big{(}a(x_{0})(2x_{0}-\xi)+x_{0}a^{\prime}(x_{0})(x_{0}-\xi)\big{)}$

Substituting the obtained values of partial derivatives in (64), we have the
following

$\frac{\partial x}{\partial
x_{0}}=\frac{xa(x)}{x_{0}a(x_{0})}\biggm{(}1+\frac{2\beta
tx_{0}a(x_{0})}{1-\beta}\big{(}a(x_{0})(2x_{0}-\xi)+x_{0}a^{\prime}(x_{0})(x_{0}-\xi)\big{)}\biggm{)}$
(65)

As it was mentioned before, if $x=0$, than $x_{0}=0$ and backwards. Moreover,
according to the condition of the problem $a\neq 0$, which means that the term
in front of the parentheses does not affect the equality of the Jacobian to
zero. Then it remains to examine the expression in parentheses

$1+\frac{2\beta
tx_{0}a(x_{0})}{1-\beta}\big{(}a(x_{0})(2x_{0}-\xi)+x_{0}a^{\prime}(x_{0})(x_{0}-\xi)\big{)}$
(66)

The considerations here are similar to the ones that were mentioned before the
case $H=x^{2}p^{2}$ described above

With $0<\beta<1$ the expression (66) also never turns into zero, if we
restrict $x$ and $\xi$ and consider sufficiently small $t$.

When $\beta\to 1-0$, then $x_{0}=0$ or $x_{0}=\xi$, and for these values the
expression (66) will also be different from zero everywhere.

So, the solution at all points will be a smooth function, generally speaking,
for sufficiently small $t$ and in the case of a local consideration of the
initial data and the solution, as explained in Section 3.

An obstacle to constructing a solution for finite times is the caustics - the
supports of the singularities of the projection of the Lagrangian manifold
$\Lambda$ onto the space $(x,t)$. A standard way to overcome this obstacle is
to use the construction of Maslov tunneling canonical operator [9]. You can
also use the construction of a generalized asymptotic solution to the Cauchy
problem [2]. In this paper, such issues are not considered.

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