#  $C\\!P$ violation in heavy-flavour decays

Yuehong Xie
Central China Normal University
On behalf of the LHCb collaboration

(17-21 July 2023)

###### Abstract

The CKM matrix is the only source of $C\\!P$ violation in the Standard Model.
Heavy-flavour decays provide an ideal laboratory to test the CKM mechanism.
Some recent highlights in $C\\!P$ violation in heavy-flavour decays from the
LHCb and Belle II experiments are presented, including updated measurements of
the CKM angles $\beta$, $\gamma$ and $\phi_{s}$, and new results in $C\\!P$
violation in charm decays.

##  1 Introduction

The fact that charge-parity ($C\\!P$) symmetry is broken in weak interactions
has been known for more than sixty years. The discovery of large $C\\!P$
violation in $B^{0}$ decays by BaBar [1] and Belle [2] in 2001 established the
CKM mechanism [3, 4] as the leading source of $C\\!P$ violation in the
Standard Model (SM). In this theory, quark flavour transitions are fully
described by the $3\times 3$ unitary CKM matrix using four parameters,
including one irreducible complex phase responsible for $C\\!P$ violation,
which are determined from flavour physics data. Study of heavy-flavour decays
can access most CKM matrix elements, providing crucial information for
precision test of the CKM mechanism. The LHCb and Belle II experiments are
playing dominant roles in this field.

The LHCb detector [5] is a forward spectrometer at the Large Hadron Collider,
dedicated to the study of $b$\- and $c$-hadron decays. It has excellent
performance in tracking, vertexing and particle identification, and provides
an effective $b$-tagging efficiency of about 5% and a typical time resolution
of about 45 fs [6], sufficient for resolving the fast ${B}^{0}_{s}$
oscillation. LHCb has collected large samples of proton-proton collisions,
corresponding to an integrated luminosity of 3 ${\rm fb}^{-1}$ at a center-of
mass energy of 7 and 8 TeV in 2011–2012 (Run 1) and and 6 ${\rm fb}^{-1}$ at
13 TeV in 2015–2018 (Run 2). Based on these data, LHCb has performed extensive
studies of beauty and charm decays, and achieved high-precision measurements
of $C\\!P$-violating parameters [7]. This paper summarizes the recent progress
using the full LHCb Run 1 and Run 2 data samples.

The Belle II experiment [8] is a super-$B$ factory at SuperKeKB. The Belle II
physics program is complementary to that of the LHCb experiment, with unique
strengths to study $B$ decays to final states with neutrinos and neutral
particles. It provides an effective $b$-tagging efficiency of about 30% and a
typical time resolution of about 1.5 ps for study of ${B}^{0}$ mixing and
$C\\!P$ violation. Since the start of operation in 2019, Belle II has
collected $e^{+}e^{-}$ collision data at the $\Upsilon$(4S) energy
corresponding to an integrated luminosity of more than 360 ${\rm fb}^{-1}$.
This paper discusses some $C\\!P$ violation measurements based on this initial
data sample.

##  2 Measurement of $\sin 2\beta$

The CKM angle
$\beta\equiv\rm{arg}\left(-V_{\textit{cd}}^{\phantom{\ast}}V_{\textit{cb}}^{\ast}/V_{\textit{td}}^{\phantom{\ast}}V_{\textit{tb}}^{\ast}\right)$
is a key parameter to characterize mixing-induced $C\\!P$ asymmetry in
${B}^{0}$ decays, where $V_{qq^{\prime}}$ represents the relevant CKM matrix
element. The time-dependent $C\\!P$ asymmetry in ${B}^{0}$ decays via a
$b\rightarrow c\bar{c}s$ transition to a $C\\!P$ eigenstate with eigenvalue
$\eta_{f}$ follows the relation $A_{\rm CP}(t)\approx\eta_{f}\sin
2\beta\sin(\Delta m_{d}t)$, where the approximation assumes no $C\\!P$
violation in the mixing and decay. The effective value of $2\beta$ is very
sensitive to new physics in $B^{0}$-$\bar{B}^{0}$ mixing, making it a flagship
measurement for heavy-flavour experiments. Both Belle and BaBar experiments
have achieved a precision better than 0.03 combining several $b\rightarrow
c\bar{c}s$ decay modes [9, 10]. LHCb measured $\sin 2\beta=0.760\pm 0.034$ in
${{B}^{0}}\rightarrow J/\psi K^{0}_{S}$ and
${{B}^{0}}\rightarrow\psi(2S)K^{0}_{S}$ decays using Run 1 data [11],
compatible with the world average value $\sin 2\beta=0.699\pm 0.017$ [12].

Recently, LHCb updated the $\sin 2\beta$ measurement using the Run 2 data
sample. The measured time-dependent $C\\!P$ asymmetry and the fit projection
are shown on the left of Figure 1. The direct $C\\!P$ violation parameter,
$C$, is found to be consistent with zero, and mixing-induce $C\\!P$ violation
is measured to be $S=\sin 2\beta=0.717\pm 0.013(\rm stat)\pm 0.008(\rm syst)$
[13]. A combination of the LHCb Run 1 and Run 2 results yields $\sin
2\beta=0.724\pm 0.014$. With Run 2 data, LHCb overtakes the $B$ factories in
the precision of $\sin 2\beta$ measurement (Figure 1 right), and improves the
precision of the world average by 35%. The new world average, $\sin
2\beta=0.708\pm 0.011$, is consistent with the indirect determinations by the
CKMfitter group [14] and UTfit group [15]. LHCb plans to accumulate 300 ${\rm
fb}^{-1}$ of proton-proton collisions after a phase-II upgrade, and improve
the precision of $\sin 2\beta$ to 0.003 [16].

Using the $\Upsilon$(4S) data collected in 2019–2021 and corresponding to 190
${\rm fb}^{-1}$, Belle II has measured the ${B}^{0}$ mass difference $\Delta
m_{d}=0.516\pm 0.008(\rm stat)\pm 0.004(\rm syst)\;{\rm ps}^{-1}$ in
${{B}^{0}}\rightarrow D^{(*)-}\pi^{+}$ decays [17], and the $C\\!P$ violation
parameter $\sin 2\beta=0.720\pm 0.062(\rm stat)\pm 0.016(\rm syst)$ in the
decay ${{B}^{0}}\rightarrow J/\psi K^{0}_{S}$ [18]. Belle II has also reported
measurements of effective $\sin 2\beta$ value in penguin-dominated
$b\rightarrow s$ transitions, such as ${{B}^{0}}\rightarrow\phi K^{0}_{S}$
[19], ${{B}^{0}}\rightarrow K^{0}_{S}\pi^{0}$ [20] and ${{B}^{0}}\rightarrow
K^{0}_{S}K^{0}_{S}K^{0}_{S}$ [21]. While being statistically limited, these
early measurements demonstrate Belle II’s ability to study ${B}^{0}$ mixing
and $C\\!P$ violation. It plans to collect 50 ${\rm ab}^{-1}$ at the
$\Upsilon$(4S) energy, which will enable the key parameter $\sin 2\beta$ to be
measured at a precision of 0.005 [8].

Figure 1: Left: asymmetry between $B^{0}$\- and $\bar{B}^{0}$-tagged decays as
a function of the decay time obtained using LHCb Run 2 data, overlaid with the
fit projection; right: comparison of the two-dimensional contours in the $C$
versus $S$ plane from LHCb, BaBar, Belle and Belle II.

##  3 Measurement of ${B}^{0}_{s}$ mixing phase $\phi_{s}$

The ${B}^{0}_{s}$ mixing phase $\phi_{s}$ is the counterpart of the ${B}^{0}$
mixing phase $2\beta$. Its value is precisely predicted to be very small in
the SM, $\phi_{s}^{\rm SM}=-2\beta_{s}=-0.0368^{+0.0009}_{-0.0006}$ rad [14],
where
$\beta_{s}\equiv-\rm{arg}\left(-V_{\textit{cb}}^{\phantom{\ast}}V_{\textit{cs}}^{\ast}/V_{\textit{tb}}^{\phantom{\ast}}V_{\textit{ts}}^{\ast}\right)$,
and sensitive to new physics in ${B}^{0}_{s}$-$\bar{B}^{0}_{s}$ mixing. The
$\phi_{s}$ angle can be accessed in time-dependent $C\\!P$ asymmetries in
${B}^{0}_{s}$ decays to $C\\!P$ eigenstates via $b\rightarrow c\bar{c}s$
transitions. The decay ${{B}^{0}_{s}}\rightarrow J/\psi\phi$ is the golden
mode to measure $\phi_{s}$. A flavour-tagged time-dependent analysis is
required to determine the mixing-induced $C\\!P$ asymmetry
$S=\eta_{f}\sin\phi_{s}$. In addition, an angular analysis is needed to
statistically separate the $C\\!P$-even and odd components of the $J/\psi\phi$
system. The LHCb, ATLAS and CMS experiments have all measured the $\phi_{s}$
parameter and the ${B}^{0}_{s}$ decay width parameter, $\Delta\Gamma_{s}$, in
${{B}^{0}_{s}}\rightarrow J/\psi\phi$ decays using Run 1 and part of Run 2
data [22, 23, 24]. While the $\phi_{s}$ values agree well, there is some
tension between the three measurements of $\Delta\Gamma_{s}$. Combining these
results with LHCb measurements in other $b\rightarrow c\bar{c}s$ modes leads
to the world average $\phi_{s}=-0.049\pm 0.019$ rad [12].

Recently, LHCb reported an improved measurement of $\phi_{s}$ obtained using
the full Run 2 data sample in the decay ${{B}^{0}_{s}}\rightarrow J/\psi\phi$
[25]. The left plot in Figure 2 shows the measured time-dependent $C\\!P$
asymmetry. No evidence for $C\\!P$ violation is observed, and the parameters
are measured to be $\phi_{s}=-0.039\pm 0.022(\rm stat)\pm 0.006(\rm syst)$ rad
and $\Delta\Gamma_{s}=0.0845\pm 0.0044(\rm stat)\pm 0.0024(\rm syst)\;{\rm
ps}^{-1}$. A combination with previous independent LHCb measurements in
$b\rightarrow c\bar{c}s$ decays yields the LHCb average value,
$\phi_{s}=-0.031\pm 0.018$ rad. This new result improves the precision of the
world average by 15%. As seen in the right plot of Figure 2, the new world
average, $\phi_{s}=-0.039\pm 0.016$ rad, is dominated by the LHCb measurement
and consistent with the indirect determinations [14, 15]. Looking to the
future, LHCb aims to achieve a precision of 0.003 rad for $\phi_{s}$ after the
phase II upgrade using a data sample of 300 ${\rm fb}^{-1}$ [16].

LHCb has also studied time-dependent $C\\!P$ violation in the penguin-
dominated decay ${{B}^{0}_{s}}\rightarrow\phi\phi$, which proceeds via a
$b\rightarrow s\bar{s}s$ transition and can probe quantum effects of new
particles in the loop decay process. Previously, LHCb measured the
$C\\!P$-violating phase, $\phi_{s}^{s\bar{s}s}=-0.073\pm 0.115(\rm stat)\pm
0.027(\rm syst)$ rad [26], using data taken in 2011–2017. A recent update of
this analysis using the full Run 1 and Run 2 data samples leads to the
improved measurement, $\phi_{s}^{s\bar{s}s}=-0.074\pm 0.069$ rad [27]. This is
the most precise measurement of mixing-induced $C\\!P$ asymmetry in any
penguin-dominated $B$ decay to date, and agrees with the SM prediction of tiny
$C\\!P$ violation [28, 29, 30]. The $C\\!P$-violating parameters are also
measured separately for different polarization states of the $\phi\phi$ pair,
and no sign of polarization dependence is observed.

Figure 2: Left: asymmetry between $B_{s}^{0}$\- and $\bar{B}_{s}^{0}$-tagged
decays as a function of the decay time obtained using LHCb Run 2 data,
overlaid with the fit projection; right: comparison and combination of
measurements from different experiments.

##  4 Measurement of CKM angle $\gamma$

The CKM angle $\gamma\equiv\textrm{arg}(-V_{ud}V^{*}_{ub}/V_{cd}V_{cb}^{*})$
can be measured through interference of $b\rightarrow u$ and $b\rightarrow c$
quark transitions in purely tree-level decays. These measurements are free of
new physics effect and provide a SM benchmark, which can be compared to
indirect determinations inferred from measurements of other observables
sensitive to new physics. Any significant discrepancy will be a clear sign of
physics beyond the SM. The golden modes to measure $\gamma$ are $B\rightarrow
Dh$ decays, where $h$ is a charged kaon or pion meson, and $D$ is a
superposition of $D^{0}$ and $\bar{D}^{0}$ mesons, which can be reconstructed
in many different modes. Other $B$ decay modes, such as $B\rightarrow
D^{*0}h$, are also used to measure $\gamma$.

A previous combination of all LHCb measurements sensitive to the angle
$\gamma$ and charm mixing parameters led to the value
$\gamma=(65.4^{+3.8}_{-4.2})^{\circ}$ [31], which dominates the world average
of direct measurements, $\gamma=(67\pm 4)^{\circ}$ [12]. Since then, some new
measurements of $\gamma$ have emerged. A binned analysis of
$B^{\pm}\rightarrow D[K^{\mp}\pi^{\pm}\pi^{\pm}\pi^{\mp}]h^{\pm}$ decays using
the full Run 1 and Run 2 data samples and hadronic parameters of $D$ decays
from CLEO-c and BESIII [32] has yielded the result
$\gamma=(54.8^{+3.8\;+0.6\;+6.7}_{-5.8\;-0.6\;-4.3})^{\circ}$ [26], where the
first uncertainty is statistical, the second systematic and the third due to
the external inputs. This is one of the most precise single measurements to
date. A weak constraint on $\gamma$ is also obtained in $B^{\pm}\rightarrow
D[h^{\pm}h^{\prime\mp}\pi^{0}]h^{\pm}$ decays [33]. A new LHCb combination
including these two results gives $\gamma=(63.8^{+3.5}_{-3.7})^{\circ}$ [34],
in good agreement with the indirectly determined value
$\gamma=(65.7^{+1.3}_{-1.2})^{\circ}$ [14]. Figure 3 (left) compares the
constraints on $\gamma$ from decays of different $B$ species, showing a
tension at $2.2\sigma$ level between ${B}^{0}$ and $B^{+}$ mesons. Figure 3
(right) shows the continuous improvement of the LHCb $\gamma$ combination
result in the past decade. LHCb aims to improve the $\gamma$ precision to
better than $1^{\circ}$ with 50 ${\rm fb}^{-1}$ from phase-I upgrade and
$0.4^{\circ}$ with 300 ${\rm fb}^{-1}$ from phase-II upgrade. To achieve these
goals, the hadronic parameters of $D$ decays need be significantly improved,
using more quantum-correlated $D\bar{D}$ data from BESIII and the planned
Super $\tau$-charm factory [35].

Belle II is starting to contribute to the $\gamma$ measurement. Through
combined analysis of Belle and Belle II data, it has measured $\gamma=(78.4\pm
11.4\pm 0.5\pm 1.0)^{\circ}$ [36], and obtained $C\\!P$ asymmetries in
$B\rightarrow Dh$ decays with $D\rightarrow K^{0}_{S}K^{\pm}\pi^{\mp}$ that
can be used to constrain $\gamma$ [37]. Belle II is expected to measure
$\gamma$ with a precision of $1$-$2$∘ using 50 ${\rm ab}^{-1}$ of data [8].

Figure 3:  Left: the 1$-$CL value as a function of $\gamma$ from the
combination using inputs from $B_{s}^{0}$, $B^{0}$ and $B^{+}$ mesons and all
species together; right: evolution of the LHCb $\gamma$ combination result.

##  5 Measurement of direct $C\\!P$ violation in charm decays

In 2019, LHCb observed significant $C\\!P$ violation in the difference between
the time-integrated $C\\!P$ asymmetries in $D^{0}\rightarrow K^{+}K^{-}$ and
$D^{0}\rightarrow\pi^{+}\pi^{-}$ decays, $\Delta A_{{C\\!P}}=(-15.4\pm
2.9)\times 10^{-4}$ [38], using the Run 2 data sample. The quantity $\Delta
A_{{C\\!P}}$ could receive contributions from the decay amplitudes, the mixing
and the interference of decay and mixing, but the exact breakdown is unknown.
Recently, LHCb measured the time-integrated $C\\!P$ asymmetry in the decay
$D^{0}\rightarrow K^{+}K^{-}$ using the same data sample [39]. Combining this
result with the $\Delta A_{{C\\!P}}$ measurement determines the direct $C\\!P$
asymmetries in the two decay processes. The obtained value for
$D^{0}\rightarrow\pi^{+}\pi^{-}$ is $a_{d}(\pi^{+}\pi^{-})=(23.2\pm 6.1)\times
10^{-4}$, which is $3.8\sigma$ from zero (Figure 4). This is the first
evidence for direct $C\\!P$ violation in $D^{0}$ decays. LHCb also searched
for direct $C\\!P$ violation in the decays
$D^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ [40], $D_{(s)}^{+}\rightarrow
K^{-}K^{+}K^{+}$ [41] and $D^{0}\rightarrow h^{+}h^{-}\mu^{+}\mu^{-}$ [42],
with no evidence found.

Figure 4:  Central values and confidence regions in the
$a^{d}_{\pi^{-}\pi^{+}}$ vs $a^{d}_{K^{-}K^{+}}$ plane for the LHCb results
obtained with data taken during 2010–2018 and 2010–2012.

##  6 Summary

Using proton-proton data collected in the first two operation periods, the
LHCb experiment has obtained high-precision measurements of $C\\!P$ violation
in beauty and charm decays, which all agree with the SM predictions. The Belle
II experiment is ramping up and producing interesting results. A deeper
investigation of $C\\!P$ violation to fully understand its nature and origin
is a long-term effort that requires synergies of LHCb, Belle II and charm
experiments as well as theoretical physicists.

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velocity matrix is diagonal, the current is diagonalized by the wave-function
components $\psi_{\pm}(x)$ themselves (labeled according to the sign $\pm$ of
their contribution to the current), which have the character of the right- and
left-moving waves ($\mathcal{N}_{+}=\mathcal{N}_{-}=1$), respectively. The
scalar projections (30) read

$\bar{\Psi}_{\pm}[\hat{\psi}(x)]=\sqrt{v_{z}}\psi_{\pm}(x)$ (94)

and the general BC (72), spelled out in terms of the wave-function components,
reads

$\sqrt{v_{z}}\psi_{+}(0)={\text{e}}^{-{\text{i}}\nu}\sqrt{v_{z}}\psi_{-}(0).$
(95)

As in Sec. XII.1, all possible forms of the BC are parameterized by the phase-
shift angle $\nu\in[0,2\pi)\sim S^{1}$; each value of $\nu$ on the unit circle
corresponds to a distinct BC.

###  XIII.2 Bound state for the general boundary condition

Figure 6:  (a) The bulk spectrum and (b) the bound state of the linear-in-
momentum two-component model with the Hamiltonian $\hat{H}_{1}(\hat{p})$ [Eq.
(92)] and general boundary condition (95). A single bound state with the
energy $\mathcal{E}_{1}(\nu)$ [Eq. (97), red] exists in the half $-\pi<\nu<0$
of the ${\text{U}}(1)\sim S^{1}$ parameter space of the BC. The energy
$d_{x}+\mathcal{E}_{2}(L_{2}(\nu))$ of the bound state of the quadratic-in-
momentum one-component model [Eqs. (74) and (78)], obtained as a low-energy
expansion around the minimum of the upper band, is also shown in dashed red.
Their quantitative asymptotic agreement [Eqs. (108) and (109)] at small $\nu$
illustrates the general connection (Sec. XIV) between the continuum models
with the same total number $\mathcal{N}$ of degrees of freedom, but different
numbers of components $M$ and order of momentum $N$, $(M,N)=(1,2)$ and $(2,1)$
here. The box schematically shows the validity range of the quadratic-in-
momentum model, where the agreement is observed.

The bulk spectrum of the Hamiltonian (92) consists of two bands

$\varepsilon_{1,\pm_{b}}(p)=\pm_{b}\sqrt{v_{z}^{2}p^{2}+d_{x}^{2}}.$ (96)

Bound states can exist only at energies in the gap region
$\epsilon\in(-d_{x},d_{x})$ between the bands. Using the analytical method of
Sec. VIII, we find that, for the system occupying the half-line $x\geq 0$,
there exists one bound state, with the wave function

$\hat{\psi}_{\text{bs}}(x)=\sqrt{\frac{|p_{\text{bs}}|}{2}}\left(\begin{array}[]{c}{\text{e}}^{-{\text{i}}\frac{\nu}{2}}\\\
{\text{e}}^{+{\text{i}}\frac{\nu}{2}}\end{array}\right){\text{e}}^{{\text{i}}p_{\text{bs}}x},\mbox{
}p_{\text{bs}}=-\frac{d_{x}}{v_{z}}{\text{i}}\sin\nu,$

and energy

$\mathcal{E}_{1}(\nu)=d_{x}\cos\nu,$ (97)

shown in Fig. 6. The bound state exists only in the range $-\pi<\nu<0$ of the
phase-shift angle parameterizing the general BC (95), i.e., in the half of its
parameter space ${\text{U}}(1)\sim S^{1}$. The explanation and multiple
general implications of this result have been provided in Ref. 9.

##  XIV Connection between the number of components and order of momentum

The same universal form (52) of the general BCs for Hamiltonians with
different numbers of wave-function components $M$ and momentum orders $N_{m}$,
but the same total number $\mathcal{N}=\sum_{m=1}^{M}N_{m}$ [Eq. (24)] of
degrees of freedom, strongly suggests a connection between these two types of
degrees of freedom. Such connection can indeed be established and is quite
insightful. We demonstrate this connection for the two models considered above
in Secs. XII and XIII, with the same minimal total number $\mathcal{N}=2$, but
different composition in terms of $M$ and $N_{m}=N$.

The bulk bands (96) of the linear-in-momentum ($N=1$) Hamiltonian (92) for the
two-component ($M=2$) wave function (Sec. XIII) are asymptotically quadratic
in the vicinity of their extrema at $p=0$ (Fig. 6):

$\varepsilon_{1,\pm_{b}}(p)=\pm_{b}\left[d_{x}+\frac{v_{z}^{2}}{2d_{x}}p^{2}+\mathcal{O}(p^{4})\right].$

For each of the bands, the system can asymptotically be described by a
quadratic-in-momentum ($N=2$) Hamiltonian (74) for a one-component ($M=1$)
wave function (Sec. XII).

As discussed in more detail in Sec. IX, the corresponding effective low-energy
model can be derived using a systematic low-energy-expansion procedure.
Crucially, not only the bulk Hamiltonian (for which the procedure is well-
known as part of the $k\cdot p$ method [34]), but also the associated BC can
always be derived within the same procedure. We now present this procedure,
which establishes the connection between the two models.

We do so for the minimum of the upper band $\varepsilon_{1,+_{b}}(p)$. We
consider the wave-function vector

$\hat{\psi}(x)=\hat{\chi}_{+_{b}}\psi_{2}(x)+\hat{\chi}_{-_{b}}\delta\psi(x)$
(98)

with components $\psi_{2}(x)$ and $\delta\psi(x)$ in the basis of the
eigenstates

$\hat{\chi}_{\pm_{b}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
\pm_{b}1\end{array}\right)$ (99)

at $p=0$, with the energies $\varepsilon_{1,\pm_{b}}(0)=\pm_{b}d_{x}$,
respectively. The stationary Schrödinger equation in this basis reads

$\left[\left(\begin{array}[]{cc}d_{x}&v_{z}\hat{p}\\\
v_{z}\hat{p}&-d_{x}\end{array}\right)-\epsilon\hat{1}\right]\left(\begin{array}[]{c}\psi_{2}(z)\\\
\delta\psi(z)\end{array}\right)=\hat{0},$

from which we express

$\delta\psi(x)=\frac{1}{\epsilon+d_{x}}v_{z}\hat{p}\psi_{2}(x)$ (100)

in terms of $\psi_{2}(x)$. Inserting this form into the other equation, we
obtain the closed equation

$d_{x}\psi_{2}+v_{z}\hat{p}\frac{1}{\epsilon+d_{x}}v_{z}\hat{p}\psi_{2}=\epsilon\psi_{2}$
(101)

for $\psi_{2}(x)$. At energies

$\epsilon=d_{x}+\delta\epsilon$ (102)

close to the upper-band minimum $\varepsilon_{1,+}(0)=d_{x}$, such that
$|\delta\epsilon|\ll d_{x}$, it is justified, to leading order, to substitute
$\epsilon\rightarrow d_{x}$ in the denominators in Eqs. (100) and (101) to
obtain

$\delta\psi(x)=\frac{v_{z}}{2d_{x}}\hat{p}\psi_{2},$ (103)
$\frac{v_{z}^{2}}{2d_{x}}\hat{p}^{2}\psi_{2}(x)=\delta\epsilon\psi_{2}(x),$
(104)

respectively. Equation (104) has the form of the stationary Schrödinger
equation

$\hat{H}_{2}(\hat{p})\psi_{2}(x)=\delta\epsilon\psi_{2}(x)$

for the wave function $\psi_{2}(x)$ with the Hamiltonian

$\hat{H}_{2}(\hat{p})=h_{2}\hat{p}^{2},\mbox{
}h_{2}=\frac{v_{z}^{2}}{2d_{x}},$ (105)

and energy $\delta\epsilon$. Equation (103) provides the expression for the
“remainder” $\delta\psi(x)$. For energies $\epsilon$ in the vicinity of
$d_{x}$ and momentum $p$ in the vicinity of $0$, this remainder is
parametrically small, $\delta\psi(x)\ll\psi_{2}(x)$, and the wave function
(98) has the dominant character of the $\hat{\chi}_{+_{b}}$ eigenstate at
$p=0$. Nonetheless, this remainder is essential for the correct derivation of
both the Hamiltonian (105) above and the BC for it, as we now show.

To this accuracy, the two-component wave function (98) of the linear
Hamiltonian (92) has the form [Eqs. (99) and (103)]

$\hat{\psi}(x)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\psi_{2}(x)+l_{2}\hat{p}\psi_{2}(x)\\\
\psi_{2}(x)-l_{2}\hat{p}\hat{\psi}_{2}(x)\end{array}\right),$

with

$l_{2}=\frac{v_{z}}{2d_{x}}.$ (106)

Inserting it into the BC (95), we obtain the BC for the wave function
$\psi_{2}(x)$ in the form

$\psi_{2}(0)+l_{2}\hat{p}\psi_{2}(0)={\text{e}}^{-{\text{i}}\nu}[\psi_{2}(0)-l_{2}\hat{p}\psi_{2}(0)].$
(107)

We recognize that this BC does have the form (77) of the general BC for the
quadratic-in-momentum Hamiltonian $\hat{H}_{2}(\hat{p})$ [Eq. (74)]. However,
importantly, its parameters $\nu$ and $l_{2}$ are not arbitrary, but are
determined by those of the initial linear-in-momentum two-component model (92)
and (95), and so is the parameter $h_{2}$ of the bulk Hamiltonian (105): the
angle $\nu$ in Eq. (107) is just that of the BC (95) and the length scale
$l_{2}$ has a specific value (106) arising from the parameters of the
Hamiltonian (92). This is in contrast to the situation with the general BC
(78) for the quadratic model, where the angle $\nu$ and length scale $l$ are
arbitrary. This difference constitutes two conceptually different “sides” of
the formalism of general BCs and the relation between the two models is
another example of the realization of the general BCs within a microscopic
model, both general points discussed in Sec. IX.

Further, we recognize that the scale $l_{2}$ is the “microscopic” scale
determining the range of validity ($pl_{2}\ll 1$, $|\delta\epsilon|\ll d_{x}$)
of the low-energy model (105) and (107): its wave function $\psi_{2}(x)$ must
vary at scales much larger than $l_{2}$.

According to Secs. XII and XIII, for $-\pi<\nu<0$, both the initial linear-in-
momentum model and its quadratic-in-momentum low-energy model have a bound
state, with the respective energies [Eqs. (81) and (102)]

$\mathcal{E}_{1}(\nu)=d_{x}\cos\nu=d_{x}[1-\tfrac{1}{2}\nu^{2}+\mathcal{O}(\nu^{4})],$
(108)
$d_{x}+\mathcal{E}_{2}(L_{2}(\nu))=d_{x}-\frac{h_{2}}{L_{2}^{2}(\nu)}=d_{x}[1-\tfrac{1}{2}\nu^{2}+\mathcal{O}(\nu^{4})],\mbox{
}L_{2}(\nu)=l_{2}\cot\tfrac{\nu}{2}.$ (109)

We do observe quantitative asymptotic agreement between these energies of the
bound state to the leading order in $|\nu|\ll 1$, as shown in Fig. 6. For
larger $|\nu|\sim 1$, outside of the validity range $|\delta\epsilon|\ll
d_{x}$ of the quadratic-in-momentum low-energy model, the energies differ.

This establishes the relation between both the Hamiltonians and BCs of the
models with same total number $\mathcal{N}=2$ of degrees of freedom, but
different numbers of the wave-function components $M$ and orders of momentum
$N$. We see that this relation is not a one-to-one correspondence. Rather, the
system with less components and higher momentum order ($N=2$, $M=1$) is the
low-energy limit of the system with more components and lower momentum order
($N=1$, $M=2$) and therefore, the validity range of the former is smaller.
Analogous relations can be established between models with more degrees of
freedom, $\mathcal{N}\geq 4$.

##  XV Generalizations and extensions of the formalism to higher dimensions,
junctions, and interfaces

The formalism of general BCs, presented in this work for a half-infinite 1D
system, allows for a number of natural generalizations and extensions to
systems with other geometries. The fundamental principle of the wave-function-
norm conservation always holds, and to derive general BCs, a similar procedure
of finding all possible BCs that satisfy the current conservation principle
has to be carried out.

First, the formalism could be extended to systems of higher dimensions. One
can immediately distinguish between the cases of preserved and broken
translation symmetry along the surface. For preserved translation symmetry,
the momentum along the surface is conserved and becomes a parameter for the
family of effective 1D Hamiltonian for the motion perpendicular to the
surface. Hence, one could expect the problem to reduce to the 1D problem
(possibly, with some necessary adaptations). When translation symmetry along
the surface is broken (due to, e.g., surface roughness, disorder), e.g., as
considered in Ref. 24, the problem becomes more complicated and may not
readily reduce to an effective 1D problem.

Next, one could adapt this formalism to junctions and interfaces [15] between
the systems described by different CMs, which could have completely different
band structures. The current continuity at the junction or interface is still
the guiding principle that leads to the general BCs. Equivalently, one can
reduce the geometry to that of one half-infinite system by mapping all half-
infinite subsystems onto one region $x>0$. For example, in the case of two
coupled 1D systems, one at $x>0$ and the other at $x<0$, with the respective
currents $j_{\gtrless}(x)$, one can map the $x<0$ system onto the $x>0$ region
by the change of coordinate $x\rightarrow-x$, in which case the net current of
the new system will be $j(x)=j_{>}(x)-j_{<}(-x)$. The current continuity
requirement $j_{<}(0)=j_{>}(0)$ is then equivalent to the current
nullification requirement $j(0)=0$.

The application schemes outlined above in Sec. XI should readily apply to
these other geometries as well, once the formalism has been generalized.

##  XVI Conclusion and outlook

In this work, we deliberately focused only on the derivation, substantiation,
elucidation, and expansion of the formalism of general BCs, while leaving its
numerous possible applications to future works. Already the existing
applications [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 7, 8, 23,
24, 25, 29, 26, 27, 28, 9] demonstrate the advantages of the formalism,
summarized in Sec. X. In all application scenarios, the formalism provides an
exhaustive characterization of the bound-state structure. Therefore, perhaps
most importantly, the formalism of general BCs allows one to not only check
and confirm the expected properties of bounds states (such as topological
properties), but also to discover new effects and features, which may turn out
to be quite unusual and unanticipated.

###### Acknowledgements.

The author is thankful to B. Trauzettel, E. M. Hankiewicz, V. N. Golovach, and
F. S. Bergeret for useful discussions and to Deutsche Forschungsgemeinschaft
(DFG) for financial support through the Grant No. KH 461/1-1.

##  Appendix A Possible redundancies of the parametrization of general
boundary conditions

One of the key values of the universal form (52) of general BCs is that they
are parameterized by unitary matrices in a nonredundant one-to-one (bijective)
way: each unitary matrix specifies one set of BCs corresponding to one
admissible Hilbert space.

Since one can derive the general BCs by resolving the current-nullification
constraint in many different ways, it is worthwhile to point out the
redundancies that may arise when the procedure is performed arbitrarily. Two
types of redundancies can be pointed out.

(i) Equivalence transformations. One can perform equivalence transformations
upon the BCs. These do not change the Hilbert space and are therefore
redundant. It is important to make sure that the parameters of the equivalence
transformations are not regarded as meaningful parameters of the BCs.

(ii) Additional linearly dependent relations, i.e., when there are more than
$\Lambda=\mathcal{N}/2$ relations of which only $\Lambda$ are linearly
independent. As an example, for linear-in-momentum Hamiltonians ($N=1$) for
the multicomponent ($M$) wave function $\hat{\psi}(x)$, the BCs are often
presented in the form [16, 17, 18, 25, 20, 21]

$\hat{M}\hat{\psi}(0)=\hat{0},$

where $\hat{M}$ is a square matrix of dimensions $M\times M$. Since in the
correct BCs there can only be $\Lambda=\mathcal{N}/2=M/2$ linearly independent
relations, the rank of the matrix $\hat{M}$ is only $M/2$ and half of the
relations in this form are redundant: one could keep only $M/2$ linearly
independent relations, which can always be presented in the form (52).

##  Appendix B $\hat{J}$ for $N=4$

As an explicit illustration of the general property of the eigenvalues of the
current matrix presented in Sec. VII, consider the fourth-order-in-momentum
($N=4$) Hamiltonian $\hat{H}(\hat{p})=h_{2}\hat{p}^{2}+h_{4}\hat{p}^{4}$ for a
one-component ($M=1$) wave function. Its current matrix (59) reads

$\hat{J}=\left(\begin{array}[]{cccc}0&J_{2}&0&J_{4}\\\ J_{2}&0&J_{4}&0\\\
0&J_{4}&0&0\\\ J_{4}&0&0&0\\\ \end{array}\right),\mbox{
}J_{2}=\frac{h_{2}}{l},\mbox{ }J_{4}=\frac{h_{4}}{l^{3}}.$

The eigenvalues of $\hat{J}$ are

$\mathcal{J}_{\pm_{a}\pm_{b}}(h_{2},h_{4};l)=\frac{1}{2}\left(\pm_{a}\sqrt{4J_{4}^{2}+J_{2}^{2}}\pm_{b}J_{2}\right)=\frac{1}{2}\left[\pm_{a}\sqrt{4\left(\frac{h_{4}}{l^{3}}\right)^{2}+\left(\frac{h_{2}}{l}\right)^{2}}\pm_{b}\left(\frac{h_{2}}{l}\right)^{2}\right].$

We see that, indeed, there are two positive $\mathcal{J}_{+_{a}\pm_{b}}>0$ and
two negative $\mathcal{J}_{-_{a}\pm_{b}}<0$ eigenvalues and their signs remain
the same regardless of the relation $h_{4}/(h_{2}l^{2})$ between the
parameters; in particular, they are the same as in the limit of dominant
$h_{4}$, when
$\mathcal{J}_{\pm_{a}\pm_{b}}(|h_{4}|\gg|h_{2}|l^{2})=\pm_{a}|h_{4}|/l^{3}$.

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