# Probing the pairing symmetry in kagome superconductors based on the single-
particle spectrum

Jinhui Liu  Tao Zhou  Corresponding author<EMAIL_ADDRESS>Guangdong Basic
Research Center of Excellence for Structure and Fundamental Interactions of
Matter, Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum
Materials, School of Physics, Guangdong-Hong Kong Joint Laboratory of Quantum
Matter, and Frontier Research Institute for Physics, South China Normal
University, Guangzhou 510006, China

###### Abstract

We investigate the single-particle spectra of recently discovered kagome
superconductors. We examine nine distinct superconducting pairing symmetries,
including the uniform $s$-wave pairing function, nearest-neighbor and next-
nearest-neighbor $s$-wave pairing, $p+ip$ pairing, $d+id$ pairing, and
$f$-wave pairing states. These pairing states can be classified into
irreducible representations of the $C_{6v}$ point group. Our findings suggest
that these pairing states can be differentiated by analyzing the energy bands,
spectral function, and local density of states. Additionally, we explore the
single impurity effect, which could potentially assist in further
distinguishing these pairing symmetries.

††preprint: APS/123-QED

##  I INTRODUCTION

The superconductivity has been discovered in several families of kagome
lattice, including the Mg2Ir3Si [1], the RT3X2 family (R=Ce, La, T=4d or 5d
transition metal, and X=Si, B or Ga) [2, 3, 4, 5, 6, 7, 8], the vanadium-based
AV3Sb5 (A=K, Rb, Cs) family [9, 10, 11], the titanium-based ATi3Bi5 (A=Rb, Cs)
family [12, 13, 14], the Ta2V3.1Si0.9 material [15] and the CsCr3Sb5 material
[16, 17]. In the past several years, the AV3Sb5 family has sparked significant
interest. Several unique properties have been exhibited, such as the double
superconducting dome feature in the temperature-pressure phase diagram [18,
19, 20, 21, 22], the time-reversal symmetry breaking charge order [18, 19, 20,
21, 22], the quantum anomalous Hall effect [23], the lack of long-range
magnetic ordering [24], a Z2 topological band structure [9], and the possible
Majorana zero modes [25]. Thus, this family presents a new platform for
investigating the interplay between the unconventional superconductivity,
competing orders, the lattice geometry, and the band topology. Furthermore,
two recently reported kagome superconductors, Ta2V3.1Si0.9 [15] and CsCr3Sb5
[16, 17], exhibit a relatively higher superconducting transition temperature
and strong correlation, respectively, providing new opportunities for studying
superconductivity within the kagome lattice.

Identifying the pairing symmetry is an essential step for understanding the
origin of the superconductivity. Theoretically, various possible pairing
symmetries have been proposed by different groups, including the $s$-wave
pairing symmetry [26, 27, 28, 29], the $p$-wave pairing symmetry [28], the
chiral $d+id$ pairing symmetry [27, 30, 31, 32, 33], and the $f$-wave pairing
symmetry [34, 33]. It was also proposed that the vortex and edge spectra can
be used to diagnosis the pairing symmetries [35]. Experimentally, a variety of
results have been proposed, primarily for the AV3Sb5 family. Signature of
nodal superconducting gap was revealed through the thermal conductivity
measurements at the zero magnetic field [19]. The existence of residual zero
energy states and the $V$-shaped superconducting gaps from the scanning
tunneling microscopy (STM) experiment also seem to propose the existing of
nodal points. However, the measurements of the spin-lattice relaxation rate,
the magnetic penetration depth, the impurity effect, and the transport
measurements indicate that the superconducting AV3Sb5 is fully gapped [36, 37,
38, 39, 40, 41]. Very recently, a nearly isotropic superconducting gap around
the Fermi surface was revealed by the angle-resolved photoemission
spectroscopy (ARPES) experiment, providing substantial evidence of a fully
gapped superconducting state [42]. It was also reminded in Ref. [42] that the
pairing symmetry still cannot be confirmed, as ARPES experiments do not
provide phase information of the superconducting gap. Moreover, in recent
years, there has been a continuous discovery of new kagome-lattice based
superconducting materials with some unusual physical properties. Therefore,
now it is of importance and timely to investigate this issue theoretically and
provide more useful information.

Theoretically, the ARPES and STM experimental results correspond to the
momentum space and real space single-particle spectra, respectively, which are
calculated from the imaginary parts of the Green’s function. A useful link for
theoretical calculations and experimental observations can be established.
Moreover, the impurity effect may be theoretically investigated based on the
single-particle spectrum and the $T$-matrix method [43]. There are also
widespread proposals of in-gap bound states induced by the impurity to resolve
pairing symmetries of various unconventional superconductors [43, 44, 45, 46,
47]. Thus, the investigation of single-particle spectra theoretically is of
great importance and could provide useful information to probe pairing
symmetry.

In this paper, we present a theoretical study of the single-particle spectra
of a kagome superconductor. We primarily concentrate on the pure
superconducting state, specifically in the phase diagram where the charge
order is absent [18, 19, 20, 21, 22]. Furthermore, we take into account all
relevant pairing possibilities based on the based on the symmetry
classification [48, 49, 50, 51, 52]. Nine possible pairing symmetries are
considered to explore single-particle spectra, and the single impurity effect
is also examined by calculating the local density of states (LDOS). It is
suggested that the in-gap resonant states are induced near the impurity for
$p+ip$ and $f$-wave pairing symmetries. We suggest that single-particle
spectra can indeed be used to distinguish between different pairing
symmetries.

The rest of the paper is organized as follows. In Sec. II, we introduce the
model and present the relevant formalism. In Sec. III, we report numerical
calculations and discuss the obtained results. Finally, we present a brief
summary in Sec. IV.

##  II MODEL AND FORMALISM

We start with a model including the normal state term ($H_{N}$) and the
superconducting pairing term ($H_{SC}$), as expressed below,

$H=H_{N}+H_{SC}.$ (1)

The kagome lattice is illustrated in Fig. 1, with one unit cell containing
three non-identical sites, depicted as red (A), blue (B) and green (C) points.
As is seen, each site has four nearest-neighbor (NN) sites. The normal state
Hamiltonian includes the NN hopping term and the chemical potential term,
expressed as

$H_{N}=-\sum_{\langle\textbf{i}\textbf{j}\rangle\sigma}(tc_{\textbf{i}\alpha\sigma}^{\dagger}c_{\textbf{j}\alpha^{\prime}\sigma}+H.c.)-\mu\sum_{\textbf{i}\sigma}c_{\textbf{i}\alpha\sigma}^{\dagger}c_{\textbf{i}\alpha\sigma},$
(2)

with $t$ and $\mu$ being the NN hopping constant and the chemical potential,
respectively. $\alpha$ and $\sigma$ are the sublattice and spin indices,
respectively. $\langle\textbf{ij}\rangle$ represents the NN bond.

$H_{SC}$ represents the superconducting pairing term. Microscopically,
superconducting pairing can be mediated by either electron-phonon interactions
or electron-electron interactions.

In the case of phonon-mediated superconductors, the superconducting coherence
length $L$ is typically quite large. Consequently, the pairing function
generally consists of a superposition of all the $n$-th pairing states in real
space, with $n$ increasing from $0$ to $L$. This usually leads to an isotropic
pairing function in momentum space. Here, we primarily focus on the electronic
structure near the Fermi energy. Assuming that the electronic structure is not
sensitive to the distortion of the gap function away from the Fermi energy, we
approximate by considering a uniform $s$-wave pairing function in momentum
space, with $\Delta({\bf k})\equiv\Delta_{0}$. This approach may provide a
qualitative description of the phonon-mediated superconductor.

For electron-interaction mediated pairings, the superconducting pairing may
occur over short ranges in real space and exhibit directionality within the
lattice model. The superconducting pairing term in real space can be
represented as follows:

$H_{SC}=\sum_{\textbf{i}\textbf{j}}(\Delta_{\textbf{ij}}c_{\textbf{i}\alpha\uparrow}^{\dagger}c_{\textbf{j}\alpha^{\prime}\downarrow}^{\dagger}+H.c.),$
(3)

where ${\bf i}$ and ${\bf j}$ are the lattice sites of the two paired
electrons. The phases of the superconducting order parameter might depend on
the direction of the two pairing electrons, giving rise to distinct pairing
symmetries. When transforming the pairing function into momentum space, an
additional momentum-dependent factor is generally introduced.

Typically, for unconventional superconductors, the pairing states between
nearest-neighbor (NN) sites or next-nearest-neighbor (NNN) sites are
considered. Pairings involving longer distances between electrons result in
higher-order pairing channels, which in turn lead to an increased number of
nodal lines [48]. However, the presence of more nodal lines usually
corresponds to higher free energy, making them less favorable. Consequently,
longer-distance pairings are not considered.

We examine nine types of pairing symmetries in the context of
superconductivity. These include the uniform $s$-wave pairing, as well as the
pairing between NN sites and NNN sites. For both the NN-site pairing and the
NNN-site pairing, four different pairing symmetries, namely, the $s$-wave
pairing symmetry, the chiral $p+ip$-wave pairing symmetry, the chiral
$d+id$-wave pairing symmetry, and the $f$-wave pairing symmetry, are examined.

Figure 1: Schematic illustration of the kagome lattice. The red and blue
vectors symbolize the NN and NNN bonds, respectively. The dashed lines
indicate two non-equivalent mirror planes, $\sigma_{d}$ and $\sigma_{v}$,
respectively.

For the NN-site pairing, $\Delta_{\textbf{ij}}$ represents the superconducting
pairing between the two NN sites, with
$\Delta_{\textbf{ij}}=\Delta_{\textbf{i},\textbf{i}+\textbf{e}_{j}}=\Delta_{j}=\Delta_{0}e^{i\phi_{j}}$.
The superconducting pairing symmetry is determined by the parity and the phase
$\phi_{j}$. Given that a single unit cell encompasses three lattice sites, and
each site possesses four NN sites, there is a total of twelve non-equivalent
NN vectors. These comprise the six red vectors seen in Fig. 1 and their
respective inverse vectors. Then the order parameters of pairing, denoted as
$\Delta_{\textbf{ij}}$, along these six vectors for different pairing
symmetries are presented in Table 1.

Table 1: The superconducting order parameters, corresponding to various red bond vectors within the NN pairing. $\Delta_{\textbf{ij}}$ | $s$ | $p+ip$ | $d+id$ | $f$
---|---|---|---|---
$\Delta_{\rm{III_{A},I_{B}}}$ | $\Delta_{0}$ | $\Delta_{0}$ | $\Delta_{0}$ | $\Delta_{0}$
$\Delta_{\rm{I_{B},I_{C}}}$ | $\Delta_{0}$ | $e^{i\frac{\pi}{3}}\Delta_{0}$ | $e^{i\frac{2\pi}{3}}\Delta_{0}$ | $-\Delta_{0}$
$\Delta_{\rm{I_{C},II_{A}}}$ | $\Delta_{0}$ | $e^{i\frac{2\pi}{3}}\Delta_{0}$ | $e^{i\frac{4\pi}{3}}\Delta_{0}$ | $\Delta_{0}$
$\Delta_{\rm{II_{A},II_{B}}}$ | $\Delta_{0}$ | $-\Delta_{0}$ | $\Delta_{0}$ | $-\Delta_{0}$
$\Delta_{\rm{II_{B},III_{C}}}$ | $\Delta_{0}$ | $e^{i\frac{4\pi}{3}}\Delta_{0}$ | $e^{i\frac{8\pi}{3}}\Delta_{0}$ | $\Delta_{0}$
$\Delta_{\rm{III_{C},III_{A}}}$ | $\Delta_{0}$ | $e^{i\frac{5\pi}{3}}\Delta_{0}$ | $e^{i\frac{10\pi}{3}}\Delta_{0}$ | $-\Delta_{0}$

Similarly, for the NNN-site pairing term, twelve non-equivalent NNN bond
vectors exist. The corresponding superconducting order parameters
$\Delta_{\textbf{ij}}$ for various pairing symmetries along the six blue
vectors are illustrated in Table 2. It’s important to note that for both NN
and NNN pairing, an additional constraint condition must be satisfied, with
$\Delta_{\textbf{ij}}=\xi\Delta_{\textbf{ji}}$, where $\xi$ is $+1$ for
$s$-wave and $d+id$ pairing symmetries, and $\xi$ is $-1$ for $p+ip$ and
$f$-wave pairing symmetries.

Table 2: The superconducting order parameters, corresponding to various blue bond vectors within the NNN pairing. $\Delta_{\textbf{ij}}$ | $s$ | $p+ip$ | $d+id$ | $f$
---|---|---|---|---
$\Delta_{\rm{i_{B},iii_{C}}}$ | $\Delta_{0}$ | $\Delta_{0}$ | $\Delta_{0}$ | $\Delta_{0}$
$\Delta_{\rm{i_{C},iii_{A}}}$ | $\Delta_{0}$ | $e^{i\frac{\pi}{3}}\Delta_{0}$ | $e^{i\frac{2\pi}{3}}\Delta_{0}$ | $-\Delta_{0}$
$\Delta_{\rm{ii_{A},i_{B}}}$ | $\Delta_{0}$ | $e^{i\frac{2\pi}{3}}\Delta_{0}$ | $e^{i\frac{4\pi}{3}}\Delta_{0}$ | $\Delta_{0}$
$\Delta_{\rm{iii_{B},i_{C}}}$ | $\Delta_{0}$ | $-\Delta_{0}$ | $\Delta_{0}$ | $-\Delta_{0}$
$\Delta_{\rm{iii_{C},ii_{A}}}$ | $\Delta_{0}$ | $e^{i\frac{4\pi}{3}}\Delta_{0}$ | $e^{i\frac{8\pi}{3}}\Delta_{0}$ | $\Delta_{0}$
$\Delta_{\rm{iii_{A},ii_{B}}}$ | $\Delta_{0}$ | $e^{i\frac{5\pi}{3}}\Delta_{0}$ | $e^{i\frac{10\pi}{3}}\Delta_{0}$ | $-\Delta_{0}$

We take the distance between the NN sites as the length unit, then we can
transform the whole Hamiltonian into the momentum space, with,

$H=\sum_{\bf
k}\Psi_{\textbf{k}}^{\dagger}\mathcal{H}_{\textbf{k}}\Psi_{\textbf{k}},$ (4)

where
$\Psi_{\textbf{k}}^{\dagger}=(c_{\textbf{k}A\uparrow}^{\dagger},c_{\textbf{k}B\uparrow}^{\dagger},c_{\textbf{k}C\uparrow}^{\dagger},c_{-\textbf{k}A\downarrow},c_{-\textbf{k}B\downarrow},c_{-\textbf{k}C\downarrow})$,
is the basis vector. $\mathcal{H}_{\textbf{k}}$ is a $6\times 6$ matrix,
expressed as,

$\mathcal{H}_{\textbf{k}}=\begin{bmatrix}\mathcal{H}_{0}(\textbf{k})&\Delta(\textbf{k})\\\
\Delta^{\dagger}(\textbf{k})&-\mathcal{H}_{0}(\textbf{k}).\end{bmatrix}$ (5)

$\mathcal{H}_{0}(\textbf{k})$ is a $3\times 3$ matrix and is expressed as

$\mathcal{H}_{0}(\textbf{k})=\begin{pmatrix}-\mu&-2t\cos k_{1}&-2t\cos
k_{2}\\\ -2t\cos k_{1}&-\mu&-2t\cos k_{3}\\\ -2t\cos k_{2}&-2t\cos
k_{3}&-\mu\\\ \end{pmatrix},$ (6)

where $k_{1}=\frac{\sqrt{3}}{2}k_{x}+\frac{1}{2}k_{y}$,
$k_{2}=\frac{\sqrt{3}}{2}k_{x}-\frac{1}{2}k_{y}$ and $k_{3}=-k_{y}$.

Figure 2: (a) The normal state Fermi surface. (b) The normal state energy
bands along the highly symmetric lines.

The quasiparticle energy bands can be derived by diagonalizing the
aforementioned matrix. With the chosen values of $t=1$ and $\mu=0.1$, the
normal state Fermi surface, along with the energy bands along the highly
symmetric lines, are depicted in Figs. 2(a) and 2(b), respectively. The normal
state Fermi surface constitutes a large pocket surrounding the $\Gamma$ point,
with the upper Van Hove singularity positioned slightly below the Fermi
energy. These characteristics are in line with the previously reported
experimental findings and theoretical computations for kagome superconductors
[9, 29, 35, 15].

$\Delta(\textbf{k})$ is the $3\times 3$ superconducting pairing matrix. For
the uniform $s$-wave pairing, $\Delta(\textbf{k})$ is expressed as,

$\Delta^{\mathrm{uni}}_{s}(\textbf{k})=\begin{pmatrix}\Delta_{0}&0&0\\\
0&\Delta_{0}&0\\\ 0&0&\Delta_{0}\end{pmatrix}.$ (7)

For the NN pairing with four different pairing symmetries, the corresponding
matrices $\Delta(\textbf{k})$ are obtained based on Table 1, expressed as,

$\Delta^{\mathrm{NN}}_{s}(\textbf{k})=\Delta_{0}\begin{pmatrix}0&2\cos
k_{1}&2\cos k_{2}\\\ 2\cos k_{1}&0&2\cos k_{3}\\\ 2\cos k_{2}&2\cos
k_{3}&0\end{pmatrix},$ (8) $\begin{aligned}
&\Delta^{\mathrm{NN}}_{p+ip}(\textbf{k})=\Delta_{0}\\\
&\begin{pmatrix}0&-2i\sin k_{1}&-(\sqrt{3}+i)\sin k_{2}\\\ -2i\sin
k_{1}&0&-(\sqrt{3}-i)\sin k_{3}\\\ -(\sqrt{3}+i)\sin k_{2}&-(\sqrt{3}-i)\sin
k_{3}&0\end{pmatrix}\end{aligned},$ (9) $\begin{aligned}
&\Delta^{\mathrm{NN}}_{d+id}(\textbf{k})=\Delta_{0}\\\ &\begin{pmatrix}0&2\cos
k_{1}&-(1+\sqrt{3}i)\cos k_{2}\\\ 2\cos k_{1}&0&-(1-\sqrt{3}i)\cos k_{3}\\\
-(1+\sqrt{3}i)\cos k_{2}&-(1-\sqrt{3}i)\cos
k_{3}&0\end{pmatrix}\end{aligned},$ (10)

and

$\Delta^{\mathrm{NN}}_{f}(\textbf{k})=\Delta_{0}\begin{pmatrix}0&-2i\sin
k_{1}&2i\sin k_{2}\\\ -2i\sin k_{1}&0&-2i\sin k_{3}\\\ 2i\sin k_{2}&-2i\sin
k_{3}&0\end{pmatrix}.$ (11)

For the NNN pairing, according to Table 2, the corresponding
$\Delta(\textbf{k})$ matrices are expressed as,

$\Delta^{\mathrm{NNN}}_{s}(\textbf{k})=\Delta_{0}\begin{pmatrix}0&2\cos
k_{4}&2\cos k_{5}\\\ 2\cos k_{4}&0&2\cos k_{6}\\\ 2\cos k_{5}&2\cos
k_{6}&0\end{pmatrix},$ (12)
$\displaystyle\Delta^{\mathrm{NNN}}_{p+ip}(\textbf{k})=\Delta_{0}$ (13)
$\displaystyle\begin{pmatrix}0&(\sqrt{3}+i)\sin k_{4}&(\sqrt{3}-i)\sin
k_{5}\\\ (\sqrt{3}+i)\sin k_{4}&0&2i\sin k_{6}\\\ (\sqrt{3}-i)\sin
k_{5}&2i\sin k_{6}&0\end{pmatrix},$ $\begin{aligned}
&\Delta^{\mathrm{NNN}}_{d+id}(\textbf{k})=\Delta_{0}\\\
&\begin{pmatrix}0&-(1+\sqrt{3}i)\cos k_{4}&-(1-\sqrt{3}i)\cos k_{5}\\\
-(1+\sqrt{3}i)\cos k_{4}&0&2\cos k_{6}\\\ -(1-\sqrt{3}i)\cos k_{5}&2\cos
k_{6}&0\end{pmatrix}\end{aligned},$ (14)

and

$\Delta^{\mathrm{NNN}}_{f}(\textbf{k})=\Delta_{0}\begin{pmatrix}0&-2i\sin
k_{4}&2i\sin k_{5}\\\ -2i\sin k_{4}&0&2i\sin k_{6}\\\ 2i\sin k_{5}&2i\sin
k_{6}&0\end{pmatrix},$ (15)

with $k_{4}=\frac{\sqrt{3}}{2}k_{x}-\frac{3}{2}k_{y}$,
$k_{5}=-\frac{\sqrt{3}}{2}k_{x}-\frac{3}{2}k_{y}$ and $k_{6}=\sqrt{3}k_{x}$.

The bare Green’s function matrix [$\hat{G}_{0}(\textbf{k},\omega)$] in the
momentum space can be obtained through diagonalizing the above Hamiltonian. It
is defined as $\hat{G}_{0}(\textbf{k},\omega)$, with the elements being
expressed as,

$G_{0ij}(\textbf{k},\omega)=\sum_{n}\frac{u_{i,n}(\textbf{k})u_{j,n}^{*}(\textbf{k})}{\omega-
E_{n}(\textbf{k})+i\Gamma},$ (16)

where $u_{in}(\textbf{k})$ and $E_{n}({\bf k})$ are the eigenvectors and the
eigenvalue of the Hamiltonian matrix, respectively.

Quasiparticle excitations can, in principle, be described by the imaginary
parts of the Green’s function. In particular, the spectral function can be
obtained from the diagonal elements of the Green’s function and is expressed
as:

$A(\textbf{k},\omega)=-\frac{1}{\pi}{\rm
Im}\sum^{3}_{i=1}[G_{0ii}(\textbf{k},\omega)+[G_{0i+3,i+3}(\textbf{k},-\omega)].$
(17)

The information about the superconducting pairing can be obtained from the
off-diagonal elements of the Green’s function (also called the anomalous
Green’s function). We define a momentum-dependent pairing function $F({\bf
k})$ by integrating all the occupied quasiparticle states as follows:

$F({\bf k})=\int_{-\infty}^{0}\mathrm{Im}\sum^{3}_{i=1}G_{0i,i+3}({\bf
k},\omega)d\omega.$ (18)

We consider an additional point impurity to study the impurity effect. The
additional Hamiltonian contributed by the impurity is expressed as,

$H_{imp}=V_{s}\sum_{\sigma}c_{\textbf{r}_{0}\alpha\sigma}^{\dagger}c_{\textbf{r}_{0}\alpha\sigma},$
(19)

with $V_{s}$ being the impurity strength. $\textbf{r}_{0}$ is the site where
the impurity locates.

Then the T-matrix and the full Green’s function are expressed as [43],

$\hat{T}(\omega)=\frac{\hat{U}}{\hat{I}-\hat{U}\hat{G}_{0}(\omega)}$ (20)

and

$\hat{G}(\textbf{r},\textbf{r}^{\prime},\omega)=\hat{G}_{0}(\textbf{r},\textbf{r}^{\prime},\omega)+\hat{G}_{0}(\textbf{r},0,\omega)\hat{T}(\omega)\hat{G}_{0}(0,\textbf{r}^{\prime},\omega).$
(21)

Here
$\hat{G}_{0}(\textbf{r},\textbf{r}^{\prime},\omega)=\frac{1}{N}\sum\limits_{k}\hat{G}_{0}(\textbf{k},\omega)e^{i\textbf{k}\cdot(\textbf{r}-\textbf{r}^{\prime})}$
is the Fourier transformation of $\hat{G}_{0}(\textbf{k},\omega)$. $\hat{I}$
is the identity matrix. We put a point impurity on the sublattice A. $\hat{U}$
is a $6\times 6$ matrix with $U_{11}=-U_{44}=V_{s}$. The LDOS near the
impurity can be obtained as

$\rho(\textbf{r},\omega)=-\frac{1}{\pi}{\rm
Im}[G_{22}(\textbf{r},\textbf{r},\omega)+G_{55}(\textbf{r},\textbf{r},-\omega)].$
(22)

##  III RESULTS AND DISCUSSION

###  III.1 Group classification of superconducting states

The superconducting pairing symmetries can be classified based on the general
symmetry classification [48, 49, 50, 51, 52]. The superconducting phase
transition from the normal state to the superconducting state is associated
with symmetry breaking, which can be described using the superconducting order
parameter. This order parameter can be classified according to the basis
functions of the irreducible representations for the symmetry group of the
crystal lattice.

In the case of a two-dimensional kagome lattice, it belongs to the $C_{6v}$
point group which has twelve elements in six conjugacy classes. This includes
a six-fold rational symmetry and two non-equivalent mirror planes,
$\sigma_{d}$ and $\sigma_{v}$, as shown in Fig. 1. There are six non-
equivalent irreducible representations for the $C_{6v}$ point group, and their
character table is presented in Table III.

Table 3: The character table of the $C_{6v}$ point group. $C_{6v}$ | $E$ | $2C^{\pm}_{6}$ | $2C^{\pm}_{3}$ | $C_{2}$ | $3\sigma_{v}$ | $3\sigma_{d}$
---|---|---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 | 1 | 1 | 1
$A_{2}$ | 1 | 1 | 1 | 1 | -1 | -1
$B_{1}$ | 1 | -1 | 1 | -1 | 1 | -1
$B_{2}$ | 1 | -1 | 1 | -1 | -1 | 1
$E_{1}$ | 2 | 1 | -1 | -2 | 0 | 0
$E_{2}$ | 2 | -1 | -1 | 2 | 0 | 0

$E_{1}$ and $E_{2}$ are the two-dimensional irreducible representations for
this point group. Each element in these representations is represented by a
two-dimensional matrix, which can be found in Table IV.

Table 4: The $E_{1}$ and $E_{2}$ irreducible representations. $C_{6v}$ | $E$ | $2C^{\pm}_{6}$ | $2C^{\pm}_{3}$ | $C_{2}$ | $3\sigma_{v}$ | $3\sigma_{d}$
---|---|---|---|---|---|---
$E_{1}$ | $\left[{\begin{array}[]{cc}1&0\\\ 0&1\end{array}}\right]$ | $\left[{\begin{array}[]{cc}\frac{1}{2}&\mp\frac{\sqrt{3}}{2}\\\ \pm\frac{\sqrt{3}}{2}&\frac{1}{2}\end{array}}\right]$ | $\left[{\begin{array}[]{cc}-\frac{1}{2}&\mp\frac{\sqrt{3}}{2}\\\ \pm\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{array}}\right]$ | $\left[{\begin{array}[]{cc}-1&0\\\ 0&-1\end{array}}\right]$ | $\left[{\begin{array}[]{cc}-1&0\\\ 0&1\end{array}}\right]$ | $\left[{\begin{array}[]{cc}1&0\\\ 0&-1\end{array}}\right]$
$E_{2}$ | $\left[{\begin{array}[]{cc}1&0\\\ 0&1\end{array}}\right]$ | $\left[{\begin{array}[]{cc}-\frac{1}{2}&\pm\frac{\sqrt{3}}{2}\\\ \mp\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{array}}\right]$ | $\left[{\begin{array}[]{cc}-\frac{1}{2}&\mp\frac{\sqrt{3}}{2}\\\ \pm\frac{\sqrt{3}}{2}&-\frac{1}{2}\end{array}}\right]$ | $\left[{\begin{array}[]{cc}1&0\\\ 0&1\end{array}}\right]$ | $\left[{\begin{array}[]{cc}1&0\\\ 0&-1\end{array}}\right]$ | $\left[{\begin{array}[]{cc}1&0\\\ 0&-1\end{array}}\right]$

The simplest basis functions of the irreducible representations can be
obtained through Tables III and IV, and they are listed in the second column
of Table V. The corresponding order parameter phase illustrations in the first
Brillouin zone based on these simplest basis functions are given in the third
column. Among them, the simplest basis function of the $A_{2}$ irreducible
representation has multiple nodal lines, leading to a higher free energy for
the $A_{2}$ superconducting state. This state can only be obtained with long-
range pairing interactions and is unlikely to be realized. Therefore, it is
not discussed in this work. For the $E_{1}$ and $E_{2}$ irreducible
representations, the basis functions have two components, and the actual
pairing order parameter should be a linear combination of them. Generally,
chiral pairing states with a $\pi/2$ phase shift between these two components
are favorable [50].

Table 5: The simplest basis functions of the irreducible representations and symmetry schematic in the first Brillouin zone from the basis functions. $C_{6v}$ | Simplest basis functions | Symmetry schematic
---|---|---
$A_{1}$ | 1 |
$A_{2}$ | $k_{x}k_{y}(k_{x}^{2}-3k_{y}^{2})(k_{y}^{2}-3k_{x}^{2})$ |
$B_{1}$ | $k_{x}(k_{x}^{2}-3k_{y}^{2})$ |
$B_{2}$ | $k_{y}(k_{y}^{2}-3k_{x}^{2})$ |
$E_{1}$ | $(k_{x},k_{y})$ |
$E_{2}$ | $(2k_{x}k_{y},k_{x}^{2}-k_{y}^{2})$ |
Figure 3: The intraband pairing functions for the eight unconventional pairing
channels. For the $p+ip$ and $d+id$ pairing symmetries, the pairing functions
are complex including the real parts and the imaginary parts, which are
presented separately.

We here obtain the pairing functions by defining the pairing terms in the real
space of the kagome lattice. This method ensures that the obtained pairing
functions in the momentum space are naturally associated with the basis
functions of the irreducible representations [51, 52]. The uniform $s$-wave
pairing symmetry generates a uniform superconducting pairing and does not
break any lattice symmetry, thus belonging to the $A_{1}$ irreducible
representation. For the other eight unconventional pairing channels, possible
anisotropic energy gaps may exist. The anisotropic behaviors can be observed
more clearly by exploring the pairing functions in the momentum space.

In the momentum space, there are three normal state energy bands in the normal
state, as seen in Fig. 2. The pairing functions are expressed as the $3\times
3$ matrices [Eqs.(7-15)]. To demonstrate the symmetries of the pairing
functions, the pairing function matrices are transformed into the band basis
with $\tilde{\Delta_{\bf k}}=V_{\bf k}^{\dagger}\Delta_{\bf k}V_{\bf k}$,
where $V_{\bf k}$ is the eigenvector matrix of the $3\times 3$ normal state
matrix $\mathcal{H}_{0}(\textbf{k})$. The diagonal components of the matrix
$\mathcal{H}_{0}(\textbf{k})$ reflect the intra-band pairing. For the kagome
superconductor, band $2$ is important due to the existence of the Van Hove
singularity points near the Fermi energy in this band, as shown in Fig. 2.

The numerical results of intraband pairing functions for the eight
unconventional pairing channels in the momentum space for band $2$ are
presented in Fig. 3. The symmetries of the different pairing channels are
clearly reflected, and the obtained pairing symmetries are indeed associated
with irreducible representations. They are consistent with the basis functions
of irreducible representations. The $s$-wave pairing symmetries belong to the
$A_{1}$ irreducible representation. The $f_{NN}$ pairing symmetry and the
$f_{NNN}$ one belong to the $B_{2}$ and $B_{1}$ irreducible representations,
respectively. The $p+ip$ pairing symmetries and $d+id$ pairing symmetries
belong to the $E_{1}$ and $E_{2}$ irreducible representations, respectively.

For the NNN $p+ip$ and $d+id$ pairing symmetries, the overall symmetric
behavior is consistent with the basis functions of the $E_{1}$ and $E_{2}$
irreducible representations. However, additional nodal lines are induced,
indicating the existence of a higher-order pairing term. In principle, the
higher-order pairing term may be induced by the long-range pairing interaction
and then more nodal lines may exist [48]. Numerically, it has been checked
that when farther pairing terms are considered, more nodal lines will be
induced.

Figure 4: Similar to Fig.3 but the pairing functions are obtained from the
anomalous Green’s function [Eq.(18)].

Exploring the pairing symmetry based on the momentum-dependent anomalous
Green’s function is also useful. The intensity plots of the pairing function
obtained by the anomalous Green’s function [defined by Eq.(18)] in the
momentum space for all of the eight unconventional pairing channels are
presented in Fig. 4. The symmetric features of the anomalous Green’s functions
are generally consistent with the pairing functions shown in Fig. 3. However,
the anomalous Green’s functions are renormalized by the quasiparticle energy
bands, causing them to decrease significantly when moving away from the normal
state Fermi surface. They can reflect the pairing order parameters around the
Fermi surface.

###  III.2 Single particle spectra

The pairing symmetry for the superconductor cannot be directly obtained
through experiments. However, useful information can be acquired by studying
the electronic structure in the superconducting state. In this context, let’s
consider the single particle spectra in various superconducting states that
exhibit different pairing symmetries. After the diagonalization of the
$6\times 6$ superconducting Hamiltonian [Eq.(5)] with $\Delta_{0}=0.1$, the
energy bands along the symmetric lines are displayed in Fig. 5. It is
important to note that, in the normal state, the energy bands along the
$\Gamma-K$ and $M-\Gamma$ lines intersect the Fermi energy at the Fermi
momentum coordinates $k^{A}_{F}$ and $k^{B}_{F}$, respectively [Fig. 2]. In
the superconducting state, an energy gap may appear at moments $k^{A}_{F}$ or
$k^{B}_{F}$. As shown in Fig. 5, the energy bands of the $s$-wave pairing
symmetry, the NN $p+ip$ pairing symmetry, and the NNN $s$-wave pairing
symmetry are fully gapped. The energy gap for the uniform $s$-wave pairing
symmetry is isotropic, with $\Delta(k^{A}_{F})=\Delta(k^{B}_{F})=\Delta_{0}$.
However, for the NN $p+ip$ pairing symmetry and the NNN $s$-wave pairing
symmetry, the energy gap is anisotropic, where
$\Delta(k^{A}_{F})>\Delta(k^{B}_{F})$.

Figure 5: Superconducting energy bands along highly symmetric lines for the
uniform $s$-wave pairing state ($s_{uni}$) and eight unconventional pairing
states.

The numerical findings also reveal that, for the NN $s$-wave pairing symmetry
and NN $d+id$ pairing symmetry, the system is fully gapped with a slight
energy gap. In the NN $s$-wave pairing symmetry, a gap of about
$0.1\Delta_{0}$ occurs at both Fermi momentum points. For the NN $d+id$
pairing symmetry, the energy gap is approximately $0.1\Delta_{0}$ at the
momentum $k^{A}_{F}$, and about $0.18\Delta_{0}$ at the momentum $k^{B}_{F}$.

The energy bands for the NN $f$-wave, NNN $p+ip$ pairing symmetry, NNN $d+id$
pairing symmetry, and NNN $f$-wave pairing symmetry display nodal points. For
the NN $f$-wave pairing symmetry and the NNN $d+id$ pairing symmetry, a
sizable energy gap exists at the momentum $k^{A}_{F}$, while the gap closes at
the momentum $k^{B}_{F}$. The energy gaps of the NNN $p+ip$ pairing symmetry
and the NNN $f$-wave pairing symmetry manifest at the momentum $k^{B}_{F}$,
and disappear at the momentum $k^{A}_{F}$. These quantitative results strongly
suggest that the quasiparticle energy bands are heavily dependent on the
pairing symmetries and can be used to distinguish different pairing
symmetries.

Figure 6: The intensity plots of the spectral function as functions of the
energy and the momentum along the highly symmetric lines for different
superconducting pairing states.

The numerical results of energy bands qualitatively agree with the pairing
functions depicted in Figs. 3 and 4. In experimental terms, quasiparticle
energy bands can be directly explored using ARPES experiments. On theoretical
grounds, the ARPES spectra can be defined utilizing the spectral function. The
intensity plots of the spectral functions, in relation to the momentum and the
energy, along the highly symmetric lines are shown in Fig. 6. The
quasiparticle dispersions for various pairing symmetries are vividly depicted.
It is noteworthy that, the energy bands in the superconducting state have the
particle-hole symmetry, however, the spectral weight for each band is distinct
and hence, only fragments of energy bands can be clearly discerned in the
spectral function spectra. The potential energy gaps at the Fermi level for
typical pairing symmetries can also be identified through the spectral
functions. Substantial consistency is observed in the corresponding results
between the energy bands and the spectral functions shown in Figs. 5 and 6.

It is also informative to evaluate the spectral functions at a constant energy
to investigate the momentum dependence of the quasiparticle excitation in the
whole Brillouin zone. We are interested in the low energy quasiparticle
excitation. When the energy bands display nodal points, we exhibit the zero
energy spectral functions. For fully gapped superconducting states, energies
are chosen as the minimum energy of the quasiparticle. Subsequently, we
illustrate the intensity plots of the spectral functions as a function of
momentum at a constant energy in Fig. 7. Three qualitatively different spectra
are revealed. For the uniform $s$-wave and the NN $s$-wave pairing symmetries.
The low energy quasiparticle excitation emerges along the entire Fermi
surface, indicating that the superconducting gap is isotropic along the entire
Fermi surface. For the NN $p+ip$ pairing, the NN $f$-wave pairing, the NNN
$s$-wave pairing, and the NNN $d+id$ pairing, the quasiparticle excitations
emerge near the $M$ point (at the momentum $k^{B}_{F}$ and its symmetric
points), designating the superconducting gap at momentum $k^{B}_{F}$ to be
miniscule or nonexistent. For the other three pairing symmetries (the NN
$d+id$ pairing, the NNN $p+ip$ pairing, and the NNN $f$-wave pairing), the
quasiparticle excitations emerge around the momentum $k^{A}_{F}$ and its
symmetric points, implying that for these pairing symmetries, the
superconducting gap at the momentum $k^{A}_{F}$ is miniscule or zero. These
findings are qualitatively consistent with the numerical results along the
highly symmetric lines presented in Figs. 5 and 6.

Figure 7: The intensity plots of the spectral function as a functions of the
momentum at the constant energy for different superconducting pairing states.
The black hexagons are the first Brillouin zone.

We now focus on studying the numerical results of the LDOS. The LDOS spectra
at the NN site to the impurity for different pairing states with varying
impurity strengths ($V_{s}$) are shown in Fig .8. Without the impurity
($V_{s}=0$), the LDOS spectra can be used to ascertain whether the system is
fully gapped. If the system is indeed fully gapped, the LDOS at low energies
equates to zero and the spectrum manifests in a ’U’ shape. If nodal points
exist, the LDOS will not completely disappear at any non-zero energy and the
spectrum adopts a ’V’ shape. Our numerical results suggest that for the NN
$f$-wave state, NNN $p+ip$ state, NNN $d+id$ state, and NNN $f$-wave state,
the LDOS spectrum creates a ’V’ shape, implying that with these pairing
symmetries, the system is not fully gapped, congruent with the numerical
results of the quasiparticle energy bands.

Let us examine the influence of impurity effects. As shown in Fig .8, the
effects of impurity indeed heavily depend on the pairing symmetries. For all
the three $s$-wave states discerned, no in-gap states exist. Two subdued in-
gap peaks near the gap edge are induced by the impurity for the NN $d+id$
state. For the NNN $d+id$ pairing state, some additional in-gap states are
brought forth by the impurity, specifically, a sizeable bump exists at a
positive energy. For the two spin triplet pairing states ($p+ip$ state or
$f$-wave state), there are sharp resonant peaks inside the energy gap. The
existing sharp resonant peaks might serve as a distinctive characteristic for
the spin triplet pairing in the kagome superconductor.

Figure 8: The LDOS spectra at the NN site of the impurity with different
impurity strengths $V_{s}$ and different pairing symmetries. From the bottom
to the top, $V_{s}$ increases from 0 to 60.

Theoretically, the impurity effects are responsive to the phase of the
superconducting order parameter. The resonant states generally emerge when the
sign of the superconducting gap changes based on the Andreev reflection
picture [44, 45, 46]. Here, as seen in Tables 1 and 2, for the three $s$-wave
states, no sign changes occur, whereas, for the $p+ip$ pairing and $f$-wave
pairing, the sign change generally arises due to the properties of the spin
triplet pairing thereby creating sharp in-gap resonant states. For the the
$d+id$ pairing, the phase of the superconducting gap changes, generating the
in-gap states, while in this case there are no sign changes occurring thus no
strong resonant states exist.

An alternative method to explain the existence of these in-gap states can be
derived from an analysis of the denominator of the T-matrix $D(\omega)$, where
$D(\omega)=|\hat{I}-\hat{U}\hat{G}_{0}(\omega)|$. As the energy resides inside
the superconducting gap, the imaginary part of $D(\omega)$ is generally small.
A resonant state emerges when its real part approaches zero at a particular
low energy. The real and imaginary parts of $D(\omega)$ with $V_{s}=60$ for
different pairing symmetries are shown in Fig. 9. The fundamental features of
the impurity effect illustrated in Fig. 8 can be fully accounted for. For both
the $p+ip$ pairing states and the $f$-wave pairing states, the real and
imaginary parts of $D(\omega)$ tend to zero near the Fermi energy, resulting
in resonant states. For the NN $d+id$ pairing state, the real part of
$D(\omega)$ intersects the zero axis at an energy near the gap edge, whereas
at the gap edge, the absolute value of the imaginary part, $\mid$
Im$D(\omega)$ $\mid$, generally increases in a steplike manner due to the
existence of the superconducting coherent peak. As a result, the impurity-
induced in-gap peak is quite weak. For the NN $d+id$ pairing state, the real
part of $D(\omega)$ does not cross the zero axis, while it is minimal at low
energies, leading to the in-gap bump structure. For the three $s$-wave pairing
states, the real part of $D(\omega)$ is consistently large, thereby resulting
in no in-gap states for these pairing states.

Figure 9: The real and imaginary parts of $D(\omega)$ as a function of the
energy $\omega$ with the impurity strength $V_{s}=60$.

Our numerical results indicate that single-particle spectra are heavily
influenced by pairing symmetries, making them effective tools for
distinguishing between different pairing states. Comparing our theoretical
findings with recent experiments on kagome superconductors provides valuable
insights.

The AV3Sb5 family has attracted considerable experimental interest. Impurity
effects in the AV3Sb5 family have been investigated using STM experiments. In
these experiments, no in-gap states were detected in the superconducting state
[41], initially ruling out the possibilities of $p+ip$ and $f$-wave pairing
for this family. A recent ARPES experiment [42] reported an isotropic
superconducting gap, with the energy gap around the Fermi surface ($\Delta$)
and the superconducting critical temperature $T_{c}$ yielding a ratio of
$2\Delta/K_{B}T_{c}\approx 3.44$. This ratio closely aligns with that of a
conventional superconductor, suggesting that the AV3Sb5 family may be phonon-
mediated superconductors. As previously mentioned, the electronic structure is
assumed to be insensitive to the distortion of the gap function and the
uniform $s$-wave pairing state may describe qualitatively phonon-mediated
superconductors. Our numerical findings reveal that the uniform $s$-wave
pairing symmetry indeed produces an isotropic superconducting gap around the
Fermi surface [Figs. 6 and 7]. In addition to the uniform $s$-wave pairing,
the NN-$s$-wave pairing also generates an isotropic pairing around the Fermi
surface. However, for NN $s$-wave pairing, the energy gap around the Fermi
surface is minimal, with $\Delta\approx 0.1\Delta_{0}$. Moreover, the
superconducting critical temperature is generally determined by the gap
magnitude $\Delta_{0}$. As such, for the NN $s$-wave pairing symmetry, we
expect this ratio to be significantly smaller than the experimentally observed
value. The other seven unconventional pairing states generate anisotropic
energy gaps around the Fermi surface, as seen from Figs. 6 and 7, which
deviate from the experimental results [42].

Recently, several different kagome-lattice-based superconductors have been
reported, such as the ATi3Bi5 family [12, 13, 14], the Ta2V3.1Si0.9 material
[15], and the CsCr3Sb5 material [16, 17]. Some physical properties of these
new kagome superconductors differ from those of AV3Sb5. The superconducting
transition temperature of the ATi3Bi5 family is approximately 4.8 K, higher
than the AV3Sb5 compounds (around 2.5 K). Additionally, this family of
materials exhibits electronic nematicity and no charge-density-wave states
[12, 13, 14]. The Ta2V3.1Si0.9 material has a much higher superconducting
critical temperature, with Tc at about 7.5 K [15]. Furthermore, it has been
proposed that the CsCr3Sb5 material exhibits a strong correlation effect [16,
17]. It is worth noting that no STM and ARPES experiments have been reported
for these new kagome-based superconducting materials to date. Further studies
may be required to explore the corresponding pairing symmetries and single-
particle spectra.

##  IV SUMMARY

In summary, we analyzed the single-particle spectra of kagome superconductors
considering nine different pairing symmetries. The superconducting states with
different pairing symmetries can be classified into irreducible
representations of the $C_{6v}$ point group. Based on the numerical results of
the energy bands and spectral function, three distinct pairing states were
found, notably, the isotropic fully gapped states (uniform $s$-wave and NN
$s$-wave), the anisotropic fully gapped states (NN $p+ip$ pairing, NN $d+id$
pairing, and NNN $s$-wave), and the nodal states (NN $f$-wave, NNN $p+ip$
pairing, NNN $d+id$, and NNN $f$-wave). The anisotropic fully gapped pairing
states and the nodal pairing states can be further discerned by investigating
the momentum-dependent superconducting gap and nodal points. The role of the
impurity effect was also studied. For the $p+ip$ pairing and the $f$-wave
pairing, sharp resonant peaks were revealed near an impurity. Whereas some in-
gap features were observed for the $d+id$ symmetry, no in-gap features were
observed for the three $s$-wave pairing states. These elements can be employed
to probe the pairing symmetries of kagome superconductors.

To conclude, our work holds substantial significance in this field of study
for three primary reasons. Firstly, there is no clear consensus presently
among researchers regarding the pairing mechanism and symmetry of kagome
superconductors; our work offers pivotal insights and knowledge that can be of
value to experts working in this particular domain. Secondly, we have
considered every relevant pairing symmetry, based on the spatial point group
symmetry of the kagome lattice. Recently, new kagome superconductor compounds,
including the ATi3Bi5 family [12, 13, 14], Ta2V3.1Si0.9 material [15], and
CsCr3Sb5 material [16, 17], have been reported. Our theoretical findings are
not merely applicable to the early existing kagome superconductors but may
extend to these novel compounds, underlining the timeliness and relevancy of
our research. Lastly, for the first time, our study provides a systematic
analysis of single-particle spectra, including the spectral function and the
single impurity effect of kagome superconductors. Our findings highlight the
potential use of single-particle spectra as an effective tool in determining
the pairing symmetries of these type of superconductors, promising to
facilitate a more comprehensive understanding of this rapidly evolving
research area.

###### Acknowledgements.

This work was supported by the NSFC (Grant No.12074130) and the Natural
Science Foundation of Guangdong Province (Grant No. 2021A1515012340).

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# Measurement of $C\\!P$ asymmetries in $D^{0}\rightarrow hh$ decays

Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
E-mail on behalf of the LHCb collaboration

###### Abstract:

The latest measurements of the individual time-integrated $C\\!P$ asymmetry in
the Cabibbo-suppressed decays ${{D}^{0}}\rightarrow hh$ decays are presented.
The results are based on $pp$ collision data, corresponding to an integrated
luminosity of 3$\mbox{\,fb}^{-1}$, collected with the LHCb detector at centre-
of-mass energies of 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$. A combination of
the asymmetries using both prompt and secondary charm decays is presented

$\displaystyle A_{{C\\!P}}({{K}^{-}}{{K}^{+}})=(0.04\pm
0.12\mathrm{\,(stat)}\pm 0.10\mathrm{\,(syst)})\%,$ $\displaystyle
A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})=(0.07\pm 0.14\mathrm{\,(stat)}\pm
0.11\mathrm{\,(syst)})\%.$

These are the most precise measurements from a single experiment. The result
for $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ is the most precise determination of a
time-integrated $C\\!P$ asymmetry in the charm sector to date, and neither
measurement shows evidence of $C\\!P$ asymmetry.

Charge-parity ($C\\!P$) violation in weak decays is described by the Standard
Model (SM) and is governed by a single complex phase in the Cabibbo-Kobayashi-
Maskawa (CKM) matrix. The charmed $D$ meson system is suitable for probing the
$C\\!P$ violation in the up-type quark sector. Recent studies of $C\\!P$
violation in weak decays of $D$ mesons have not found evidence of $C\\!P$
violation, while the phenomenon is well established in decays of $K$\- and
$B$-meson systems [1, 2, 3, 4, 5].

To distinguish the two ${C\\!P}$-conjugate decays, the flavour of the
${{D}^{0}}$ at production must be known. At LHCb, two different techniques are
employed. The flavour of the ${{D}^{0}}$ can be tagged by the charge of the
soft pion, $\pi^{+}_{s}$, in the strong decay
${{D}^{*+}}\rightarrow{{D}^{0}}\pi^{+}_{s}$ for prompt ${{D}^{0}}$ decays [6],
or by the charge of the muon in the semileptonic decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}}\rightarrow{{D}^{0}}\mu^{-}{{\overline{\nu}}_{\mu}}X$
for secondary charm decays [7]. The time-integrated $C\\!P$ asymmetry in the
decay rates of the singly Cabibbo-supressed process ${{D}^{0}}\rightarrow
K^{-}K^{+}$

$A_{{C\\!P}}({{D}^{0}}\rightarrow
K^{-}K^{+})\equiv\frac{\Gamma(D^{0}\rightarrow K^{-}K^{+})-\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow
K^{-}K^{+})}{\Gamma(D^{0}\rightarrow K^{-}K^{+})+\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow K^{-}K^{+})},$ (1)

was measured recently at LHCb [8]. This observable cannot be accessed directly
due to the presence of production asymmetry of the ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mesons at $pp$ collider, and
due to the presence of detection asymmetries. The latter can origin from an
asymmetric interaction of the charged particles with the material of the
detector. One example for this is the different cross-section for interaction
with matter of ${K}^{+}$ and ${K}^{-}$. Any asymmetry in the detector e.g.
some inefficient regions, faulty electronics, etc could also contribute to the
detection asymmetry. The effect of such asymmetries can cancel to first order
with the regular swap of the magnetic field of the dipole magnet at LHCb.

The measured raw asymmetry in the number of background-subtracted number of
signal decays

$\displaystyle A_{\rm
raw}\equiv\frac{N({{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}})-N({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{-}}{{K}^{+}})}{N({{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}})+N({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{-}}{{K}^{+}})},$
(2)

is related to the observable of interest, $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$,
via

$\displaystyle A_{{C\\!P}}(D^{0}\rightarrow K^{-}K^{+})\approx A_{\rm
raw}(D^{0}\rightarrow K^{-}K^{+})-A_{P}(D^{*+})-A_{D}(\pi^{+}_{s}).$ (3)

This approximation is valid only for small asymmetries, this is why a
particular care has been taken to exclude regions with large asymmetries,
related to detector acceptance effects [6, 8].

To cancel the effect of the nuisance detection and production asymmetry, a set
of Cabibbo-favoured decays, $D^{0}\rightarrow K^{-}\pi^{+},D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+},D^{+}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}\pi^{+}$, in which no $C\\!P$
violation is expected, is used. For example, by exploiting the prompt
$D^{0}\rightarrow K^{-}\pi^{+}$ channel, one can cancel the ${{D}^{*+}}$
production asymmetry, and the soft pion detection asymmetry. However, a new
detection asymmetry is present due to the asymmetric final state
$K^{-}\pi^{+}$. The effect of this asymmetry is mitigated using the other two
calibration channels leading to

$\displaystyle A_{{C\\!P}}(D^{0}\rightarrow K^{-}K^{+})$ $\displaystyle=$
$\displaystyle A_{\rm raw}(D^{0}\rightarrow K^{-}K^{+})-A_{\rm
raw}(D^{0}\rightarrow K^{-}\pi^{+})$ $\displaystyle+$ $\displaystyle A_{\rm
raw}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{+})-A_{\rm raw}(D^{+}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}\pi^{+})$ $\displaystyle+$
$\displaystyle A_{D}({{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}).$

$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ now only depends on measurable raw
asymmetries and the calculable ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}$ detection asymmetry [9]. The
method to determine $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ has been first used in
Ref. [7].

The results are obtained using a data sample of proton-proton ($pp$)
collisions at centre-of-mass energies of 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$ collected by the LHCb detector in 2011 and
2012, corresponding to approximately 3${\mbox{\,fb}^{-1}}$ of integrated
luminosity.

The LHCb detector [10, 11] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing b or c quarks. The detector elements that are particularly relevant
to this analysis are: a silicon-strip vertex detector surrounding the pp
interaction region that allows c- and b-hadrons to be identified from their
characteristically long flight distance; a tracking system that provides a
measurement of momentum, $p$, of charged particles; and two ring-imaging
Cherenkov detectors that are able to discriminate between different species of
charged hadrons.

The production and detection asymmetries depend on the kinematics of the
particles involved. If the kinematic distributions are very different, this
may lead to an imperfect cancellation of the nuisance asymmetries in
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$. This effect is alleviated by equalising the
corresponding kinematical distributions by assigning weights to each candidate
[12].

The raw asymmetries and the signal yields are determined from binned
likelihood fits to the $\delta m=m({{D}^{*+}})-m({{D}^{0}})$, distributions in
the ${D}^{0}$ decay modes, and to the invariant mass distributions
$m({{D}^{+}})$ in the ${D}^{+}$ channels (see Fig. 1). The yields of the decay
modes, before and after the weighting procedure, are listed in Table 1.

Table 1: Signal yields of the four channels before and after the kinematic weighting. In the case of the weighted samples, effective yields are given. Channel | Before weighting | After weighting
---|---|---
${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ | $5.56\,\text{M}$ | $~{}~{}1.63\,\text{M}$
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ | $32.4\,\text{M}$ | $~{}~{}2.61\,\text{M}$
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ | $37.5\,\text{M}$ | $13.67\,\text{M}$
${{D}^{+}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ | $1.06\,\text{M}$ | $~{}~{}1.06\,\text{M}$

Figure 1: Fits to the ${\delta m}$ and to the $m({{D}^{+}})$ distributions
corresponding to the whole data sample and both flavours. Data samples after
the kinematic weighting described in the text are used.

The non-adequateness of the fit model, and the presence of peaking background
can cause biases in the determination of individual raw asymmetries, and their
effects are studied using pseudoexperiments, and simulated events. Possible
presence of large local asymmetries can spoil the
$A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ measurement since Eq. 3 is valid only for
small asymmetries. This systematic uncertainty is accounted for by excluding
additional regions of the detector acceptance, and comparing the result to the
nominal one.

The non-cancellation of detection and production asymmetries is accounted for
by modifying the configuration of the procedure. Another way to asses this
systematic contribution is by modelling the kinematic dependancies of the raw
asymmetries, and combining them with per-event weights from data. Presence of
secondary charm decays can bias the measurement due to the different
asymmetries related to the production of secondary ${D}^{0}$ decays. The
uncertainty was calculated by measuring the fraction of secondary decays, and
using the measured production asymmetries at LHCb as an input. A systematic
uncertainty on the calculation of the neutral kaon detection asymmetry is also
taken into account. All other investigated sources are found to be negligible.

A summary of the systematic uncertainties considered can be found in Table 2
and added in quadrature to obtain the overall systematic uncertainty.

Table 2: Systematic uncertainties from the different categories. The quadratic sum is used to compute the total systematic uncertainty.  Category |  Systematic uncertainty[$\%$]
---|---
Determination of raw asymmetries: |
Fit model | $0.025$
Peaking background | $0.015$
Cancellation of nuisance asymmetries: |
Additional fiducial cuts | $0.040$
Weighting configuration | $0.062$
Weighting simulation | $0.054$
Secondary charm meson | $0.039$
Neutral kaon asymmetry | $0.014$
Total | $0.10$

The time-integrated $C\\!P$ asymmetry in
${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ decays measured using prompt charm
decays is determined to be

$\displaystyle A_{{C\\!P}}^{\mathrm{prompt}}({{K}^{-}}{{K}^{+}})=(0.14\pm
0.15\mathrm{\,(stat)}\pm 0.10\mathrm{\,(syst)})\%.$ (5)

This result can be combined with previous LHCb measurements of the same and
related observables. A combination with $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$
measured using secondary charm decays [7] yields

$\displaystyle A_{{C\\!P}}^{\rm comb}({{K}^{-}}{{K}^{+}})=(0.04\pm
0.12\mathrm{\,(stat)}\pm 0.10\mathrm{\,(syst)})\%.$ (6)

Since the same ${D}^{+}$ decay channels are employed for the cancellation of
detection asymmetries, the result is partially correlated with
$A_{{C\\!P}}^{\mathrm{prompt}}({{K}^{-}}{{K}^{+}})$. The statistical
correlation coefficient is $\rho_{\rm stat}=0.36$. The difference in $C\\!P$
asymmetries between ${D}^{0}$ $\rightarrow$ ${K}^{-}$ ${K}^{+}$ and ${D}^{0}$
$\rightarrow$ ${\pi}^{-}$ ${\pi}^{+}$ decays, $\Delta A_{{C\\!P}}^{prompt}$,
is measured at LHCb using prompt charm decays [6]. A combination of the
measurement of $A_{{C\\!P}}^{\mathrm{prompt}}({{K}^{-}}{{K}^{+}})$ with
$\Delta A_{{C\\!P}}$ yields a value for

$\displaystyle
A_{{C\\!P}}^{\mathrm{prompt}}({{\pi}^{+}}{{\pi}^{-}})=A_{{C\\!P}}^{\mathrm{prompt}}({{K}^{+}}{{K}^{-}})-\Delta
A_{{C\\!P}}^{prompt}=(0.24\pm 0.15\mathrm{\,(stat)}\pm
0.11\mathrm{\,(syst)})\%.$ (7)

The statistical correlation coefficient of the two measurements is $\rho_{\rm
stat}=0.24$.

A combination with the measurement using secondary charm results in

$\displaystyle A_{{C\\!P}}^{\rm comb}({{\pi}^{-}}{{\pi}^{+}})=(0.07\pm
0.14\mathrm{\,(stat)}\pm 0.11\mathrm{\,(syst)})\%.$

Figure 2: Measurements of $C\\!P$ violation asymmetries in ${D}^{0}$
$\rightarrow$ ${K}^{-}$ ${K}^{+}$ and ${D}^{0}$ $\rightarrow$ ${\pi}^{-}$
${\pi}^{+}$ decays. Alongside the two LHCb measurements, presented in this
Letter (green ellipse) and in Ref.[7] (blue ellipse), and their combination
(red ellipse), the latest value of the Heavy Flavour Averaging Group [13] is
shown (black ellipse). The latter already includes the measurement of $\Delta
A_{{C\\!P}}$ with muon(pion)-tagged ${D}^{0}$ decays, using
$3(1)\,$$\mbox{\,fb}^{-1}$ $pp$ collision data collected with the LHCb
detector [7, 14]. The $68\%$ confidence level contours are displayed where the
statistical and systematic uncertainties are added in quadrature.

Fig. 2 shows the LHCb measurements of $C\\!P$ asymmetry using both prompt and
secondary ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays. Additionally, the latest
combined values of the Heavy Flavour Averaging Group [13] for these quantities
are presented.

In conclusion, no evidence of $C\\!P$ violation is found in the Cabibbo-
suppressed decays ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$. These results are obtained
assuming that there is no $C\\!P$ violation in ${D}^{0}$–${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing and no direct $C\\!P$
violation in the Cabibbo-favoured ${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$,
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}}{{\pi}^{+}}$ decay modes. The
combined LHCb results are the most precise measurements of the individual
time-integrated $C\\!P$ asymmetries $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ and
$A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})$ from a single experiment to date.

## References

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