# Large magnetocaloric effect in the kagome ferromagnet Li9Cr3(P2O7)3(PO4)2

Akshata Magar  Somesh K  Vikram Singh  School of Physics, Indian Institute of
Science Education and Research Thiruvananthapuram-695551, India  J. J. Abraham
Leibniz IFW Dresden, D-01069 Dresden, Germany  Institute for Solid State and
Materials Physics, TU Dresden, 01069 Dresden, Germany  Y. Senyk  A. Alfonsov
Leibniz IFW Dresden, D-01069 Dresden, Germany  B. Büchner  Leibniz IFW
Dresden, D-01069 Dresden, Germany  Institute for Solid State and Materials
Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, TU Dresden,
D-01062 Dresden, Germany  V. Kataev  Leibniz IFW Dresden, D-01069 Dresden,
Germany  A. A. Tsirlin  Felix Bloch Institute for Solid-State Physics, Leipzig
University, 04103 Leipzig, Germany  R. Nath<EMAIL_ADDRESS>School of
Physics, Indian Institute of Science Education and Research
Thiruvananthapuram-695551, India

###### Abstract

Single-crystal growth, magnetic properties, and magnetocaloric effect of the
$S=3/2$ kagome ferromagnet Li9Cr3(P2O7)3(PO4)2 (trigonal, space group:
$P\bar{3}c1$) are reported. Magnetization data suggest dominant ferromagnetic
intra-plane coupling with a weak anisotropy and the onset of ferromagnetic
ordering at $T_{\rm C}\simeq 2.6$ K. Microscopic analysis reveals a very small
ratio of interlayer to intralayer ferromagnetic couplings ($J_{\perp}/J\simeq
0.02$). Electron spin resonance data suggest the presence of short-range
correlations above $T_{\rm C}$ and confirms quasi-two-dimensional character of
the spin system. A large magnetocaloric effect characterized by isothermal
entropy change of $-\Delta S_{\rm m}\simeq 31$ J kg-1 K-1 and adiabatic
temperature change of $-\Delta T_{\rm ad}\simeq 9$ K upon a field sweep of 7 T
is observed around $T_{\rm C}$. This leads to a large relative cooling power
of $RCP\simeq 284$ J kg-1. The large magnetocaloric effect, together with
negligible hysteresis render Li9Cr3(P2O7)3(PO4)2 a promising material for
magnetic refrigeration at low temperatures. The magnetocrystalline anisotropy
constant $K\simeq-7.42\times 10^{4}$ erg cm-3 implies that the compound is an
easy-plane type ferromagnet with the hard axis normal to the $ab$-plane,
consistent with the magnetization data.

##  I Introduction

Kagome lattice hosts a plethora of interesting phenomena. Its frustrated
nature renders antiferromagnetic kagome insulators a natural playground for
the experimental realization of quantum spin liquid [1, *Han406,
*Carrasquilla7421]. Whereas kagome ferromagnets are not frustrated and develop
magnetic order, they are no less interesting because flat bands and Dirac
fermions expected in this setting have far-reaching implications for transport
properties. Recent work on ferromagnetic kagome metals exposed anomalous Hall
and Nernst effects, as well as chiral edge states, in several intermetallic
compounds, such as Fe3Sn2 [4], Co3Sn2S2 [5], LiMn6Sn6 [6], and UCo0.8Ru0.2Al
[7]. Concurrently, insulating kagome ferromagnets were actively studied in the
context of magnon Hall effect and other exotic properties associated with
Dirac magnons [8, 9, 10].

Ferromagnets are further interesting as materials with the large
magnetocaloric effect (MCE) that can be instrumental in cooling via adiabatic
demagnetization for reaching temperatures in the sub-Kelvin range [11, 12,
13]. This magnetic refrigeration technique is often considered as the most
energy-efficient, cost-effective (as 3He and 4He are expensive), and
environment-friendly replacement for the conventional refrigeration based on
gas compression/expansion technique. For this purpose, materials with large
magnetic moment, low magnetic anisotropy, low magnetic hysteresis, and
extremely low transition temperature are desirable [14, 15]. The nature of the
magnetic transition and the specific form of the magnetic structure are also
deciding factors for the performance of a MCE material. Ferromagnetic
insulators with second-order phase transition are proposed to be excellent MCE
materials, as only a small change in applied magnetic field is sufficient to
yield a large entropy change and adiabatic temperature change, compared to any
paramagnetic salt [12, 15]. A very few ferromagnetic insulators with low
transition temperature are reported to satisfy the above prerequisites and
qualify for low-temperature applications [16, 17, 18].

In the following, we report the magnetic properties of Li9Cr3(P2O7)3(PO4)2
(LCPP) which is a structural sibling of the recently reported $S=5/2$
Heisenberg kagome antiferromagnet Li9Fe3(P2O7)3(PO4)2 (LFPP) with a trigonal
space group $P\bar{3}c1$ [19]. LFPP shows the onset of an antiferromagnetic
(AFM) ordering below $T_{\rm N}\simeq 1.3$ K and a characteristic $1/3$
magnetization plateau below $T^{*}\simeq 5$ K. The NMR spectra along with the
NMR spin-lattice relaxation time reveal the presence of an exotic
semiclassical nematic spin liquid regime between $T_{\rm N}$ and $T^{*}$. On
the contrary, LCPP is found to be a ferromagnet and it undergoes a
ferromagnetic (FM) ordering at $T_{\rm C}\simeq 2.5$ K. LCPP exhibits a large
MCE around $T_{\rm C}$ and appear to have strong potential for cryogenic
applications such as low temperature sensors in space research, achieving sub-
kelvin temperatures for basic research, hydrogen and helium gas liquefaction
etc [11, 20].

Figure 1: (a) Corner-sharing equilateral triangles of Cr3+ form a kagome
lattice. The intraplane ($J$) and interplanar ($J_{\perp}$) coupling are
shown. The kagome lattice layers are well separated from each other with
interlayer distance of 6.9034 Å. (b) Interactions between the magnetic Cr3+
ions via PO4 tetrahedra are shown.

##  II Methods

Platelet single crystals of LCPP with the lateral size of 0.5 mm to 1 mm were
synthesized by a self-flux technique as reported in Ref. [21]. The mixture of
starting materials, Li3PO4, Cr2O3, and NH4H2PO4 in the molar ratio 15:1:9 was
kept in an alumina crucible and heated gradually to $900\degree$C. The cooling
process involves three steps. At first, the sample was cooled down to
$850\degree$C at a rate of $50\degree$C per hour and then to $600\degree$C at
a slow rate of $2\degree$C per hour. Finally, the sample was allowed to cool
naturally to room temperature. In order to dissolve the flux and separate the
crystals, the sample was treated with 1 M solution of acetic acid for five
days followed by the treatment with saturated NaCl solution and distilled
water. The final product after the treatment yields the mixture of mm-sized
single crystals and polycrystalline sample. The large-sized crystals were
hand-picked and the remaining part is grinded to get the polycrystalline
sample.

Room-temperature single-crystal x-ray diffraction (XRD) was performed on a
good quality single crystal using the Bruker KAPPA APEX-II CCD diffractometer
equipped with graphite monochromated Mo $K_{\alpha 1}$ radiation
($\lambda=0.71073$ Å). The APEX3 software was used to collect the data that
were further reduced with SAINT/XPREP followed by an empirical absorption
correction using the SADABS program [22]. The phase purity of the
polycrystalline sample was confirmed from powder XRD (PANalytical Xpert-Pro,
Cu $K_{\alpha}$ radiation with $\lambda_{av}=1.54182$ Å). The temperature-
dependent powder XRD measurement was performed in the temperature range 15 K
$\leq T\leq$ 300 K with a low-temperature (Oxford Phenix) attachment to the
diffractometer.

Magnetization ($M$) measurement was performed as a function of temperature
($T$) and magnetic field ($H$) using a superconducting quantum interference
device (SQUID) (MPMS-3, Quantum Design) magnetometer. The data were collected
in the temperature range 1.8 K $\leq T\leq$ 350 K and in the magnetic field
range 0 $\leq H\leq$ 7 T. Heat capacity ($C_{\rm p}$) as a function of $T$
(0.5 K $\leq T\leq$ 300 K) and $H$ was measured on a small piece of sintered
pellet using the relaxation technique in the physical property measurement
system (PPMS, Quantum Design). Measurements below 2 K were carried out using
an additional 3He insert in the PPMS.

High-field electron spin resonance (HF-ESR) spectroscopy was used to study the
single crystals of LCPP. For the measurements in a frequency range 75 - 330
GHz, a vector network analyzer (PNA-X from Keysight Technologies) was used and
for frequencies up to 975 GHz a modular Amplifier/Multiplier Chain (AMC from
Virginia Diodes Inc.) was used for the generation of microwaves in combination
with a hot electron InSb bolometer for detection. All measurements were
performed at a given fixed frequency in the field-sweep mode up to 16 T, using
a superconducting magnet system from Oxford Inst. The sample was mounted onto
a transmission probe head which is then inserted in a 4He variable temperature
insert (VTI) of the magnet cryostat to enable measurements in a temperature
range of $1.8-300$ K.

Density-functional (DFT) band-structure calculations were performed in the
FPLO code [23] with the Perdew-Burke-Ernzerhof flavor of the exchange-
correlation potential [24]. Correlation effects in the Cr $3d$ shell were
included on the mean-field level within DFT+$U$ using the on-site Coulomb
repulsion parameter $U_{d}=2$ eV, Hund’s coupling $J_{d}=1$ eV, and double-
counting correction in the atomic limit [25, 26]. Exchange couplings $J_{ij}$
were obtained by mapping [27] total energies of collinear magnetic
configurations onto the spin Hamiltonian,

$\mathcal{H}=\sum_{\langle ij\rangle}J_{ij}\mathbf{S}_{i}\mathbf{S}_{j}$ (1)

where the summation is over pairs, and $S=\frac{3}{2}$. Energies were
converged on a $k$ mesh with 64 points within the first Brillouin zone.
Thermodynamic properties for the model defined by Eq. (1) were obtained from
quantum Monte-Carlo simulations performed with the loop algorithm [28] of the
ALPS simulation package [29]. Finite lattices with up to 752 sites and
periodic boundary conditions were used.

##  III Results and Discussion

###  III.1 X-ray Diffraction

Table 1: Crystallographic data for LCPP at room temperature, obtained from single-crystal XRD. Empirical formula | Cr3 Li9 O29 P8
---|---
Formula weight(Mr) | 930.22 g mol-1
Temperature | 296(2) K
Crystal system | Trigonal
Space group | $P\bar{3}c1$
Lattice parameters |  $a=9.668(3)$ Å $\alpha=90\degree$
|  $b=9.668(3)$ Å $\beta=90\degree$
|  $c=13.610(6)$ Å $\gamma=120\degree$
Unit cell volume | 1101.7(8) Å3
$Z$ | 2
Density (calculated) | 2.804 g cm-3
Wavelength | 0.71073 Å
Radiation type | MoK$\alpha_{1}$
Diffractometer | Bruker KAPPA APEX-II CCD
Crystal size |  $0.049\times 0.035\times 0.027$ mm3
2$\theta$ range | 2.993 to 25.997$\degree$
Index ranges | $-11\leq h\leq 11$
| $-11\leq k\leq 11$
| $-16\leq l\leq 16$
$F(000)$ | 902
Reflections collected | 6940
Independent reflections | 735 [$R_{int}$ = 0.0454]
Data / restraints / parameters | 735/0/76
Goodness-of-fit on $F^{2}$ | 1.098
Final $R$ indices [$I$ $\geq~{}$2$\sigma$($I$)] |  $R1$ = 0.0304,
|  $\omega$$R2$ = 0.0892
$R$ indices(all data) |  $R1$ = 0.0351,
|  $\omega$$R2$ = 0.0919
Largest diff. peak and hole |  $+0.445/-0.970$ e.Å-3

The crystal structure of LCPP was solved from single-crystal XRD data with
direct methods using SHELXT-2018/2 [30] and refined by the full matrix least
squares on $F^{2}$ using SHELXL-2018/3, respectively [31]. Details of the
crystal structure and the refined parameters are summarized in Table 1. LCPP
crystallizes in the trigonal space group $P\bar{3}c1$ (No. 165). The refined
atomic positions at room temperature are listed in Table III.1. These
structural parameters are in good agreement with the previous report [21].

The schematic view of the crystal structure of LCPP is presented in Fig. 1. It
illustrates the corner sharing of CrO6 octahedra and PO4 tetrahedra forming
equilateral triangles with a geometrically deformed but regular kagome
lattice. Though all of the Cr3+ – Cr3+ distances in each hexagon are equal
($\sim 4.949$ Å), the bond angles are different: three angles are about $\sim
146.3\degree$ and the remaining three angles are about $\sim 93.7\degree$. The
nearest-neighbour (NN) coupling between the Cr3+ ions in the $ab$-plane is
denoted by $J$, while the shortest interplane distance of $\sim 6.903$ Å leads
to a weak coupling ($J_{\perp}$) between the planes.

Table 2: Crystal structure of LCPP refined using single-crystal XRD data. The
atomic coordinates ($\times 10^{4}$) and the isotropic atomic displacement
parameter $U_{\rm iso}$(Å${}^{2}\times 10^{3}$), which is defined as one-third
of the trace of the orthogonalized $U_{ij}$ tensor.

Atomic |  |  |  |  |
---|---|---|---|---|---
sites |  Wyckoff |  |  |  |
|  $x$ |  $y$ |  $z$ | $U_{\rm iso}$ |
Cr(1) | $6f$ | 5676(1) | 0 | 2500 | 6(1)
Li(1) | $2b$ | 0 | 0 | 5000 | 19(3)
Li(2) | $12g$ | 3369(7) | 2367(7) | 4374(4) | 16(1)
Li(3) | $4d$ | 6667 | 3333 | 6178(7) | 15(2)
O(1) | $12g$ | 3758(2) | $-1053(2)$ | 3331(1) | 9(1)
O(2) | $6f$ | 2120(3) | 0 | 2500 | 9(1)
O(3) | $12g$ | 792(2) | $-2535(2)$ | 3440(2) | 9(1)
O(4) | $12g$ | 2299(3) | 38(2) | 4332(2) | 11(1)
O(5) | $12g$ | 6781(2) | 1893(2) | 3352(2) | 10(1)
O(6) | $4d$ | 6667 | 3333 | 4839(3) | 23(1)
P(1) | $12g$ | 2279(1) | $-894(1)$ | 3440(1) | 6(1)
P(2) | $4d$ | 6667 | 3333 | 3736(1) | 5(1)
Figure 2:  Powder XRD data measured at (a) $T=300$ K and (b) $T=15$ K. The
black solid line represents the Le-Bail fit of the data. Bragg positions are
indicated by vertical bars and the solid green line at the bottom denotes the
difference between experimental and calculated intensities. The inset of (b)
shows a representative single crystal.

In order to confirm the phase purity and to scrutinize the presence of any
structural distortions, powder XRD data were collected at various
temperatures. Le-Bail analysis of the XRD patterns was performed using
`FullProf` package [32] taking the initial structural parameters from the
single crystal data (Table 1). Figure 2(a) and (b) present the powder XRD
patterns at the highest ($T=300$ K) and lowest ($T=15$ K) measured
temperatures, respectively, along with the Le-Beil fits. All the peaks could
be indexed based on the space group $P\bar{3}c1$, suggesting phase purity of
the polycrystalline sample. The obtained lattice parameters at room
temperature are $a=b=9.6628(3)$ Å, $c=13.5769(3)$ Å, and unit-cell volume
$V_{\rm cell}\simeq 1097.85(5)$ Å3, which are consistent with the single-
crystal data. No extra peaks or features were observed in the XRD data
corroborating the absence of any structural transition or distortion down to
15 K.

Figure 3:  Variation of lattice parameters ($a$, $c$, and $V_{\rm cell}$) with
temperature. The solid line denotes the fit of $V_{\rm cell}(T)$ by Eq. (2).

The temperature variation of lattice parameters ($a$, $c$, and $V_{\rm cell}$)
is presented in Fig. 3. They are found to decrease monotonically upon cooling
down to 15 K. $V_{\rm cell}(T)$ was fitted by the equation [33]

$V_{\rm cell}(T)=\frac{\gamma U(T)}{K_{0}}+V_{0},$ (2)

where $V_{0}$ is the zero-temperature unit cell volume, $K_{0}$ is the bulk
modulus, and $\gamma$ is the Grüneisen parameter. $U(T)$ is the internal
energy and it can be expressed in terms of the Debye approximation as

$U(T)=9pk_{B}T(\frac{T}{\theta_{\rm D}})^{3}\int_{0}^{\theta_{\rm
D}/T}\frac{x^{3}}{e^{x}-1}dx.$ (3)

Here, $p$ is the number of atoms in the unit cell, $k_{\rm B}$ is the
Boltzmann constant, and the Debye temperature is represented by $\theta_{\rm
D}$. The fit (see Fig. 3) returns $\theta_{\rm D}\simeq 385$ K,
$\gamma/K_{0}\simeq 2.06\times 10^{-4}$ Pa-1, and $V_{0}\simeq 1094.2$ Å3.

###  III.2 Magnetization

Figure 4: (a) $\chi(T)$ measured in the field of $H=0.5$ T applied
perpendicular ($H\perp c$) and parallel ($H\parallel c$) to the $c$-axis. (b)
Magnetization at $T=1.8$ K as a function of applied field for both $H\perp c$
and $H\parallel c$ after correcting for the demagnetization effect. (c)
Inverse susceptibility ($1/\chi$) as a function of temperature for $H\perp c$
and solid line is the CW fit. (d) $\chi T$ vs $T$ in different fields for
$H\perp c$ in low temperatures.

Magnetic susceptibility $\chi(T)~{}[\equiv M(T)/H]$ measured in an applied
field of $H=0.5$ T perpendicular ($H\perp c$) and parallel ($H\parallel c$) to
the kagome plane is displayed in Fig. 4(a). With decreasing temperature,
$\chi(T)$ increases in a Curie-Weiss manner as expected in the high-
temperature paramagnetic (PM) regime, followed by a rapid enhancement at low
temperatures. This rapid increase suggests strong FM correlations below about
10 K. Further, the susceptibility for $H\perp c$ and $H\parallel c$ show only
a small difference even at low temperatures, which is an indication of weak
magnetic anisotropy in the compound.

For a quantitative analysis, we have plotted the inverse susceptibility
($1/\chi$) as a function of temperature in Fig. 4(c) for $H\perp c$. For
$T\geq 50~{}\rm K$, it exhibits a completely linear behaviour which was fitted
by the modified Curie-Weiss (CW) law

$\chi(T)=\chi_{0}+\frac{C}{(T-\theta_{\rm CW})}.$ (4)

Here, $\chi_{0}$ is the temperature-independent susceptibility, $C$ is the
Curie constant, and $\theta_{\rm CW}$ is the CW temperature. The fit yields
$\chi_{0}\simeq-3.6\times 10^{-4}$ cm3 mol-1, $C\simeq 1.92$ cm3 K mol-1, and
$\theta_{\rm CW}\simeq 6$ K. From the value of $C$ the effective moment is
calculated to be $\mu_{\rm eff}\simeq 3.92~{}\mu_{\rm B}$ in agreement with
the spin-only value of 3.87 $\mu_{\rm B}$ for spin-$3/2$. The positive value
of $\theta_{\rm CW}$ suggests that the dominant exchange interactions between
Cr3+ ions are FM in nature. Using the mean-field expression $J/k_{\rm
B}=-3|\theta_{\rm CW}|/zS(S+1)$ with $z=4$ neighbors on the kagome lattice, we
estimate $J/k_{\rm B}\simeq-1.2$ K. Moreover, the small peak in $\chi T$ in
low magnetic fields reveals $T_{\rm C}\simeq 2.6$ K [Fig. 4(d)]. $\chi(T)$
measured in zero-field cooled and field cooled conditions (not shown) in a
small magnetic field of $H=0.01$ T for $H\perp c$ shows no difference,
suggesting negligible hysteresis.

The magnetic isotherm ($M$ vs $H$) at $T=1.8$ K saturates in low fields of
$H_{\rm sat}{\perp c}\simeq 0.15$ T and $H_{\rm sat}{\parallel c}\simeq 0.4$ T
with the saturation magnetization of $M_{\rm sat}\sim 3.2~{}\mu_{\rm B}/{\rm
Cr}^{3+}$ and $2.8~{}\mu_{\rm B}/{\rm Cr}^{3+}$ for $H\perp c$ and $H\parallel
c$, respectively (not shown). The obtained $M_{\rm sat}$ values are close to
the calculated $M_{\rm sat}=gS\mu_{\rm B}\simeq 2.952\mu_{\rm B}$ and
$2.937\mu_{\rm B}$, taking the ESR values $g\simeq 1.968$ and 1.958 for
$H\perp c$ and $H\parallel c$, respectively. No visible hysteresis is observed
in any of the field directions. A slight difference in the saturation field
for $H\perp c$ and $H\parallel c$ may be attributed to the anisotropic
demagnetization field caused by the flat shape of the crystals [10]. The
demagnetization factor is negligible when magnetic field is parallel to the
crystal plates ($H\perp c$). However, when field is perpendicular to crystal
plate ($H\parallel c$), the demagnetization effect is considerably amplified
[34]. The data in Fig. 4(b) have been corrected for this demagnetizing field
as $H_{\rm eff}=H_{0}-4\pi NM$ where $H_{\rm eff}$ and $H_{0}$ are the
effective and applied magnetic fields, respectively, $N$ is the
demagnetization factor, and $M$ is magnetic moment in emu cm-3. To calculate
the demagnetization factor in the case of LCPP, we approximated the shape of
the sample to a rectangular strip and did the calculation following Ref. [35],
which yields $N=0.84$ for $H\parallel c$. The $H_{\rm sat}$ in both directions
after demagnetization correction is found to be almost same ($\sim 0.15$ T).

###  III.3 Heat Capacity

Figure 5: (a) Heat capacity ($C_{\rm p}$) of LCPP measured in zero applied
field. The solid line represents the simulated phonon contribution [$C_{\rm
ph}(T)$] and the dotted line represents the magnetic contribution [$C_{\rm
mag}(T)$]. Inset: Low temperature $C_{\rm p}$ measured in various applied
fields. (b) $C_{\rm mag}/\rm T$ and $S_{\rm mag}/R\ln 4$ in left and right
y-axes, respectively, are plotted as a function of temperature. Inset: $C_{\rm
mag}(T)$ vs $T$. Solid line is the power-law ($C_{\rm mag}=aT^{\alpha}$) fit
in the low-$T$ regime.

Temperature-dependent heat capacity $C_{\rm p}$ measured on the
polycrystalline sample is shown in Fig. 5. The $C_{\rm p}$ data exhibit a
sharp $\lambda$-type anomaly at $T_{\rm C}\simeq 2.5$ K demonstrating the
transition to the magnetically ordered state. Typically, in magnetic
insulators, the major contributions to $C_{\rm p}$ are from magnetic ($C_{\rm
mag}$) and phonon ($C_{\rm ph}$) parts. In high temperatures, $C_{\rm ph}$
dominates over $C_{\rm mag}$, while at low temperatures it is reverse. One can
estimate $C_{\rm mag}$ by subtracting $C_{\rm ph}$ from the total heat
capacity. First, we approximate the phonon contribution by fitting the
high-$T$ data by a linear combination of one Debye and three Einstein terms as
[36]

$C_{\rm ph}(T)=f_{\rm D}C_{\rm D}(\theta_{\rm D},T)+\sum_{i=1}^{3}g_{i}C_{{\rm
E}_{i}}(\theta_{{\rm E}_{i}},T).$ (5)

The first term in Eq. (5) is the Debye contribution to $C_{\rm ph}$, which can
be written as

$C_{\rm D}(\theta_{\rm D},T)=9nR\left(\frac{T}{\theta_{\rm
D}}\right)^{3}\int_{0}^{\frac{\theta_{\rm
D}}{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx.$ (6)

Here, $R$ is the universal gas constant, $\theta_{\rm D}$ is the
characteristic Debye temperature, and $n$ is the number of atoms in the
formula unit. The second term in Eq. (5) gives the Einstein contribution to
$C_{\rm ph}$ that has the form

$C_{\rm E}(\theta_{\rm E},T)=3nR\left(\frac{\theta_{\rm
E}}{T}\right)^{2}\frac{e^{\theta_{\rm E}/T}}{[e^{\theta_{\rm E}/T}-1]^{2}}.$
(7)

Here, $\theta_{\rm E}$ is the characteristic Einstein temperature. The
coefficients $f_{\rm D}$, $g_{1}$, $g_{2}$, and $g_{3}$ represent the fraction
of atoms that contribute to their respective parts. These values are taken in
such a way that their sum should be equal to 1 and are conditioned to satisfy
the Dulong-Petit value $\sim 3nR$ at high temperatures. The high-$T$ fit to
the $C_{\rm p}(T)$ data was then extrapolated down to low temperatures and
subtracted from $C_{\rm p}(T)$. The obtained $C_{\rm mag}/T$ is plotted as a
function of temperature in the main panel of Fig. 5(b) and the corresponding
magnetic entropy is calculated to be $S_{\rm{mag}}(T)=\int_{\rm
0\,K}^{T}\frac{C_{\rm{mag}}(T^{\prime})}{T^{\prime}}dT^{\prime}\simeq 11.6$ J
mol-1 K-1 at 12 K. This value corresponds to the expected magnetic entropy for
spin-$\frac{3}{2}$: $S_{\rm mag}=R\ln 4=11.5$ J mol-1 K-1. Unlike the
conventional magnets which release the entire entropy near the transition
temperature, LCPP releases only $\sim 40$% of the total entropy at $T_{\rm C}$
and the remaining entropy is released only above 11 K, suggesting that $T_{\rm
C}$ is partially suppressed as a result of low-dimensionality or magnetic
frustration [26].

The inset of Fig. 5(a) presents the $C_{\rm p}(T)$ data measured in different
applied fields. The influences of magnetic field is clearly reflected in the
data. The zero-field peak broadens and shifts toward high temperatures with
increasing field, which is usual for ferromagnets. At low temperatures,
$C_{\rm mag}(T)$ in zero field could be well described using power-law
($C_{\rm mag}\propto T^{\alpha}$) behavior [inset of Fig. 5(b)] with an
exponent $\alpha\sim 1.5$ that corresponds to FM spin-wave excitations [37].

###  III.4 Magnetocaloric Effect

Figure 6: (a) Isothermal magnetization ($M$ vs $H$) curves for $H\perp c$ and
(b) their corresponding Arrott plots ($M^{2}$ vs $(H/M)$) for LCPP at
different temperatures around $T_{\rm C}$. Figure 7: (a) Isothermal entropy
change ($\Delta$Sm) vs $T$ plotted upon sweeping the field to 0 T starting
from different fields from 1 T to $7$ T, calculated using $M$ vs $H$ data of
single crystals for $H\perp c$ and employing Eq. (8). (b) $\Delta S_{\rm m}$
vs $T$ plot for $5$ T and $7$ T, calculated using $C_{\rm p}(T)$ of
polycrystalline sample in Eq. (9).

Magnetocaloric effect (MCE) is an intrinsic property of magnetic materials.
Magnetic cooling is achieved by first applying magnetic field to the material
isothermally and then removing the field adiabatically. Therefore, MCE is
generally quantified by the isothermal entropy change ($\Delta S_{\rm m}$) and
adiabatic temperature change ($\Delta T_{\rm ad}$) with respect to the change
in applied field. The $\Delta S_{\rm m}$ can be calculated from either
magnetization isotherms ($M$ vs $H$) or heat capacity data measured in zero
and non-zero magnetic fields. Figure 6(a) displays the magnetic isotherms
measured in close temperature steps around $T_{\rm C}$ for $H\perp c$. The
first method utilizes Maxwell’s thermodynamic relation, $(\partial S/\partial
H)_{T}=(\partial M/\partial T)_{H}$, and $\Delta S_{\rm m}$ can be estimated
using the $M$ vs $H$ data as [38]

$\Delta S_{\rm m}(H,T)=\int_{H_{\rm i}}^{H_{\rm f}}\dfrac{dM}{dT}dH.$ (8)

Figure 7(a) presents the plot of $\Delta S_{\rm m}$ as a function of
temperature ($T$) in different values of $\Delta H=H_{\rm f}-H_{\rm i}$.
$\Delta S_{\rm m}$ vs $T$ exhibits a maximum entropy change around 4.6 K, with
a highest value of $\Delta S_{\rm m}\simeq-31$ J kg-1 K-1 for the 7 T field
change. As the magnetic anisotropy is negligibly small, no significant
difference in $\Delta S_{\rm m}$ is expected for $H\perp c$ and $H\parallel c$
[39].

Further, to cross check the large value of $\Delta S_{\rm m}$, we have also
estimated $\Delta S_{\rm m}$ from heat capacity data measured in zero field, 5
T, and 7 T. First, we calculated the total entropy at a given field as

$S(T)_{H}=\int_{T_{i}}^{T_{f}}\frac{C_{\rm p}(T)_{\rm H}}{T}dT,$ (9)

where $C_{p}(T)_{H}$ is the heat capacity at a particular field $H$ and
$T_{i}$ and $T_{f}$ are the initial and final temperatures, respectively. We
calculated $\Delta S_{\rm m}$ by taking the difference of total entropy at
non-zero and zero fields as $\Delta S_{\rm m}(T)_{\Delta\rm H}=[S(T)_{\rm
H}-S(T)_{\rm 0}]_{T}$. Here, $S(T)_{\rm H}$ and $S(T)_{\rm 0}$ are the total
entropy in the presence of $H$ and in zero field, respectively. Figure 7(b)
presents the estimated $\Delta S_{\rm m}$ as a function of temperature in 5 T
and 7 T magnetic fields. The overall shape and peak position of the $\Delta
S_{\rm m}$ curves are identical with the curves [Fig. 7(a)] obtained from the
magnetic isotherms but with a slight reduction in magnitude. This difference
in magnitude at the peak position could be related to the polycrystalline
sample used for heat capacity and single crystals for magnetic measurements
[40].

Figure 8: (a) $\Delta T_{\rm ad}$ vs $T$ plotted for different field changes
of $\Delta H=1$ T to $7$ T calculated using Eq. (10). (b) $\Delta T_{\rm ad}$
vs $T$ plotted for $\Delta H=5$ T and $7$ T calculated using Eq. (11).

Similarly, the adiabatic temperature change $\Delta T_{\rm ad}$ can be
estimated from either the combination of zero-field heat capacity and the
magnetic entropy change obtained from magnetic isotherms or from the heat
capacity alone measured in different magnetic fields. Using the heat capacity
in zero field and magnetization isotherm data, the estimation of $\Delta
T_{\rm ad}$ can be done as [41]

$\Delta T_{\rm ad}=\int_{H_{\rm i}}^{H_{\rm f}}\dfrac{T}{C_{\rm
p}}\dfrac{dM}{dT}dH.$ (10)

The dependence of $\Delta T_{\rm ad}$ on $T$ for different magnetic fields is
shown in Fig. 8(a). The maximum value of $\Delta T_{\rm ad}$ is obtained to be
$\sim 40$ K for $\Delta H=7$ T. However, as explained in Ref. [42] the above
expression overestimates $\Delta T_{\rm ad}$ since $T/C_{p}$ is not constant
over the range of applied fields as it was assumed. It is evident from the
inset of Fig. 5(a) that $C_{\rm p}$ at low temperatures is changing
drastically as we apply magnetic field and this change should be taken into
account while calculating the entropy for that particular field. Therefore, we
tried to estimate $\Delta T_{\rm ad}$ by taking the difference in temperatures
corresponding to two different fields with constant (same) entropy as [42]

$\Delta T_{\rm ad}(T)_{\Delta H}=[T(S)_{H_{f}}-T(S)_{H_{i}}].$ (11)

$\Delta T_{\rm ad}$ vs $T$ for $\Delta H=5$ T and $7$ T calculated by this
method is shown in Fig. 8(b). The maximum value of $\Delta T_{\rm ad}$ at 7 T
is around $\sim 9$ K which is significantly smaller than the value obtained
using the former method [Eq. (10)]. A similar difference has been reported
earlier for ErAl2 [42]. The latter method is considered to be more reliable.
It is expected to provide accurate results as the effect of magnetic field on
$C_{\rm p}$ is accounted for.

Figure 9: (a) Relative cooling power ($RCP$ = $\Delta S^{\rm
peak}_{m}\times\delta T_{FWHM}$) as a function of magnetic field and the inset
shows the value of entropy change at the peak position $\Delta S^{\rm
peak}_{m}$. Solid lines are the fits as described in the text. (b) The
exponent $n$ plotted as a function of temperature which is obtained from
fitting power law to $\Delta S_{m}$ vs field isotherms. Inset shows the
$\Delta$Sm isotherms for temperatures near and well above $T_{\rm C}$.

Note that the large values of $\Delta S_{\rm m}$ and $\Delta T_{\rm ad}$ are
not sufficient to characterize the potential of a material for the magnetic
refrigeration applications. Another important parameter is the relative
cooling power ($RCP$) which is a measure of the amount of heat transferred
between the cold and hot reservoirs in a refrigeration cycle. Mathematically,
it can be expressed as

${RCP}=\int_{T_{\rm{cold}}}^{T_{\rm hot}}\Delta S_{\rm m}(T,H)~{}dT,$ (12)

where, $T_{\rm cold}$ and $T_{\rm hot}$ correspond to temperatures of cold and
hot reservoirs, respectively. The formula for $RCP$ can be approximated as

$|{RCP}|_{\rm approx}=\Delta S_{\rm m}^{\rm peak}\times\delta T_{\rm FWHM},$
(13)

where $\Delta S_{m}^{\rm peak}$ and $\delta T_{\rm FWHM}$ are the maximum
value of entropy change (or the peak value) and full width at half maximum of
the $\Delta S_{m}$ curve, respectively. $RCP$ as a function of $H$ calculated
using the $\Delta S_{m}$ data from Fig. 7(a) is plotted in Fig. 9(a). The
maximum value of $RCP$ is calculated to about $\sim 284$ J kg-1 at 7 T.

The application of a MCE material is also decided by the nature of its
magnetic phase transition. In materials with a first-order phase transition,
though the peak height of the $\Delta S_{\rm m}$ and $\Delta T_{\rm ad}$ vs
$T$ curves is large but the curve width is not very broad, which limits the
relevance of these materials in a cyclic operation. The second problem with
the first-order transitions is the energy loss due to magnetic and thermal
hysteresis [43]. Further, relatively large magnetic fields are required to
perturb the first-order magneto-structural transitions and induce large MCE
which is another drawback of these materials. On the other hand, materials
with second-order phase transition do not show very large peaks, but their
$RCP$ values are large due to the increased curve width and the absence of
thermal hysteresis, both effects making them promising for practical
applications. In order to analyze the nature of the phase transition, we
construct the Arrott plot [44] in Fig. 6(b) by using the isothermal
magnetization data presented in Fig. 6(a). Clearly, the slope of $M^{2}$ vs
$H/M$ curves is positive in the entire measured temperature range, well below
and above $T_{\rm C}$. According to the Banerjee criterion [45], positive
slope implies the second-order phase transition. This confirms the continuous
second-order nature of the PM to FM phase transition in LCPP.

Furthermore, MCE is also utilized to characterize the nature of a phase
transition [41, 46, 47]. According to the scaling hypothesis [43], the $\Delta
S_{m}(T)$ curves for different values of $\Delta H$ should collapse on a
single universal curve when the $\Delta S_{m}(T)$ is normalized to its peak
value $\Delta S_{m}^{\rm peak}$. However, due to the low transition
temperature, the universal curve construction is implausible with the present
data. Therefore, we performed only the power-law analysis of $\Delta S_{m}$
and $RCP$. In Fig. 9(a) main panel and inset, we fitted the $RCP$ and $\Delta
S_{m}^{\rm peak}$ data by power laws of the form $RCP\propto H^{N}$ and
$|\Delta S_{m}^{\rm peak}|\propto H^{n}$, respectively. The exponents $N$ and
$n$, which are related to the critical exponents ($\beta$, $\gamma$, and
$\delta$), are estimated to be $N\simeq 0.7$ and $n\simeq 0.5$. In order to
perceive the temperature dependence of $n$, we fitted the field-dependent
isothermal magnetic entropy change $\Delta S_{m}(H)$ at various temperatures
across the transition using the power law $\Delta S_{m}\propto H^{n}$ [see
inset of Fig. 9(b)] [46]. The obtained $n$ vs $T$ data are plotted in Fig.
9(b) and provide information concerning the nature of the transition. For
instance, for a second-order magnetic transition the exponent should have the
value $n\simeq 2$ in the paramagnetic region ($T>>T_{\rm C}$) and $n(T)$
typically exhibits a minimum near $T_{\rm C}$ [46]. Indeed, our $n(T)$
demonstrates the expected behavior, further confirming the second-order
magnetic transition in LCPP.

Table 3: Comparison of the adiabatic temperature change ($\Delta T_{\rm ad}$), maximum entropy change ($\Delta S_{m}^{\rm peak}$), and relative cooling power ($RCP$) of LCPP with some known magnets having low transition temperatures ($T_{\rm C}$ or $T_{\rm N}$) and large MCE in a field change of $\Delta H=5$ to 8 T. More compounds with the similar behaviour are listed in Refs. [48, 40, 49]. System | $T_{\rm C}/T_{\rm N}$ | $|\Delta T_{\rm ad}|$ | $|\Delta S_{\rm m}^{\rm peak}|$ | $RCP$ | $\Delta H$ | Ref.
---|---|---|---|---|---|---
| (K) | (K) | (J kg-1 K-1) | (J kg-1) | (T) |
LCPP | 2.6 | 9 | 31 | 284 | 7 | This work
HoMnO3 | 5 | 6.5 | 13.1 | 320 | 7 | [50]
ErMn2Si2 | 4.5 | 12.9 | 25.2 | 365 | 5 | [18]
EuTi0.9V0.3O3 | 4.5 | 17.4 | 41.4 | 577 | 7 | [17]
GdCrTiO5 | 0.9 | 15.5 | 36 | – | 7 | [51]
EdDy2O4 | 5 | 16 | 25 | 415 | 8 | [52]
EuHo2O4 | 5 | 12.7 | 30 | 540 | 8 | [52]
Mn32 | 0.32 | 6.7 | 18.2 | – | 7 | [53]
HoB2 | 15 | 12 | 40.1 | - | 5 | [54]
EuTiO3 | 5.6 | 21 | 49 | 500 | 7 | [48]

In Table 3, we compare the main parameters of LCPP with those of well-studied
magnets having low transition temperatures and large MCE. Though the $\Delta
S_{\rm m}^{\rm peak}$ value of LCPP is comparable to the values for most of
the potential low-temperature magnetic refrigerant materials, the width of
$\Delta S_{\rm m}$ vs $T$ curves is not very broad. Due to which LCPP has a
slightly reduced value of $RCP$ compared to others. Nevertheless, the obtained
value of $RCP\simeq 284$ J kg-1 is still significantly large and LCPP may have
strong prerequisites for cryogenic applications in sub-Kelvin temperatures
[11, 20].

###  III.5 Electron Spin Resonance

####  III.5.1 Frequency Dependence at $T=1.8$ K

The ferromagnetic resonance (FMR) measurements on LCPP were performed at
$T=1.8$ K ($<T_{\rm C}$) for the two orientations of the applied magnetic
field, $H\parallel c$ and $H\perp c$. The frequency ($\nu$) vs resonance field
($H_{\rm res}$) dependence of the FMR signal is shown in Fig. 10 together with
few selected spectra (right axis). The spectra measured at fields
perpendicular to the $c$-axis are relatively broader and more distorted when
compared to the $H\parallel c$ measurements. The $\nu$ vs $H_{\rm res}$ data
were fitted with a spin-wave model for FMR, $\nu=h^{-1}g_{\parallel}\mu_{\rm
B}(H_{\rm res}-H_{\rm a})$ for $H\parallel c$ orientation and
$\nu=h^{-1}g_{\perp}\mu_{\rm B}[H_{\rm res}(H_{\rm res}+H_{\rm a})]^{1/2}$ for
$H\perp c$ orientation to obtain the $g$-factors and the anisotropy field
$H_{\rm a}$. It was found that the $g$-factors in both orientations are
somewhat different amounting to $g_{\perp}=1.968\pm 0.003$ and
$g_{\parallel}=1.958\pm 0.003$. Such a slightly anisotropic $g$-tensor is
typical for a Cr3+ ion ($3d^{3}$, $S=3/2$, $L=3$) in a distorted octahedral
ligand coordination [55].

Figure 10: Left vertical scale: Frequency as a function of resonant field
measured for both orientations at $T=1.8$ K. The hollow red and solid black
squares represent the measured resonance fields for orientations $H\parallel
c$ and $H\perp c$, respectively. Right vertical scale: Selected HF-ESR
spectra.

Further, the anisotropy field $H_{\rm a}$ was obtained as $\sim 2864$ Oe from
the fit. Generally, $H_{\rm a}$ consists of two contributions [56]:

$H_{\rm a}=4{\pi}M-2K/M.$ (14)

The first term accounts for the shape anisotropy and the second term is the
intrinsic magnetocrystalline anisotropy. In Eq. (14), the first term assumes
the shape anisotropy of a thin plate with $N=1$.

Using the saturation magnetization value, $M_{\rm
s}=g_{\parallel}S=2.94~{}\mu_{\rm B}/{\rm Cr}^{3+}$, the magnetocrystalline
anisotropy constant $K=-7.42\times 10^{4}$ erg cm-3 was obtained. The negative
sign of $K$ implies that LCPP is an easy-plane ferromagnet with the hard
magnetic axis normal to the $ab$-plane. We note that no other resonance
excitations could be found in a frequency range up to 975 GHz (4 meV).

Figure 11: Temperature dependence of the HF-ESR spectra at an excitation
frequency of 141.310 GHz for (a) $H\perp c$ and (b) $H\parallel c$. Line shape
distortions at $T<20$ K in the $H\perp c$ field geometry are instrumental
artifacts arising due to the strong magnetization of the sample. Figure 12:
Temperature dependence of (a) the resonance field $H_{\rm res}$ and (b) the
resonance shift $\delta H=H_{\rm res}(T)-H_{\rm res}(300~{}{\rm K})$ (left
$y$-axis) and the magnetization $M$ in an applied field of $H=0.5$ T (right
$y$-axis), for both $H\parallel c$ and $H\perp c$.

####  III.5.2 Temperature Dependence

HF-ESR spectra of LCPP at various temperatures were measured at a fixed
excitation frequency of $\nu=141.310$ GHz for both field directions [see Figs.
11(a) and (b)]. The resonance fields $H_{\rm res}$ were obtained from the
absorption minima of each spectrum and are plotted against temperature in Fig.
12(a). Interestingly, the $H_{\rm res}$ vs $T$ dependence for both
orientations does not converge to a paramagnetic line immediately at $T_{\rm
C}=2.8$ K. Only above 10 K, the resonance fields for $H\parallel c$ and
$H\perp c$ orientations rapidly start to decrease and increase, respectively,
toward the expected paramagnetic position and almost merge around 100 K at a
field corresponding to the $g$-factor, $g=1.97$. The lineshape distortions in
measurements at low temperatures for $H\perp c$ are accounted for in the
enlarged error bars.

The shift of the resonance position $\delta H(T)$ from the paramagnetic one is
shown in Fig. 12(b) (left $y$-axis). $\delta H(T)$ is positive and larger when
the external field is parallel to the magnetic hard axis, as compared to the
smaller negative shift for the in-plane field geometry, as expected for an
easy-plane ferromagnet. Such a shift cannot be ascribed entirely to the shape
anisotropy, which should play a role also in the paramagnetic state if the
sample’s magnetization $M$ is large. The $M(T)$ curve plotted in Fig. 12(b)
(right $y$-axis) for comparison decreases right above $T_{\rm C}$ more rapidly
than the $\delta H(T)$ dependence. Therefore, the line shift observed for both
orientations well above the Curie temperature may be indicative of short-range
FM spin correlations on the fast ESR time scale, typical for low-dimensional
magnets such as, e.g., the quasi-two-dimensional van der Waals compound
Cr2Ge2Te6 [57]. This is also the case for LCPP owing to its layered crystal
structure.

###  III.6 Microscopic Analysis

Our DFT calculations return nearest-neighbor exchange coupling $J/k_{\rm
B}\simeq-0.5$ K within the kagome planes. This value is somewhat dependent on
the choice of the DFT+$U$ parameters, but the negative sign is robust and
suggests ferromagnetic nature of the kagome network in LCPP. The origin of
this ferromagnetic coupling deserves some attention, as the sibling Fe3+
compound is clearly antiferromagnetic [19]. Superexchange theory stipulates
that antiferromagnetic couplings are mediated by hoppings between half-filled
orbitals, whereas ferromagnetic couplings arise from hoppings between the
half-filled and empty orbitals. In LCPP with the $3d^{3}$ Cr3+ magnetic ion,
these orbitals have the $t_{2g}$ and $e_{g}^{\sigma}$ characters,
respectively. Trigonal symmetry of the crystal structure further splits the
$t_{2g}$ levels into $a_{1g}$ and $e_{g}^{\pi}$.

Figure 13:  (a) PBE density of states for LCPP with the Fermi level placed at
zero energy. (b) Nearest-neighbor Cr–Cr hopping amplitudes within the kagome
plane. Figure 14:  Magnetic susceptibility of LCPP measured in the applied
field of 0.01 T upon field cooling and its fit using the model of
ferromagnetic kagome planes (dashed line) and coupled ferromagnetic kagome
planes ($J_{\perp}/J=0.01$, solid line). The inset shows calculated specific
heat for the coupled kagome planes, with $T_{C}\simeq 2.8$ K.

DFT band structure of LCPP calculated on the PBE level features narrow
$t_{2g}$ ($a_{1g}+e_{g}^{\pi}$) bands around the Fermi level and almost
equally narrow $e_{g}^{\sigma}$ bands centered at around 2.0 eV (Fig. 13).
This band structure is metallic because neither magnetism nor correlation
effects have been taken into account. The small band width of 0.25 eV for the
$t_{2g}$ bands indicates that antiferromagnetic contribution to the exchange
couplings should be minor. The $t_{2g}-e_{g}^{\sigma}$ hoppings are small too,
but somewhat larger than the $t_{2g}-t_{2g}$ hoppings, as shown in Fig. 13(b).
Therefore, ferromagnetic contribution to the exchange becomes predominant, and
the overall coupling is ferromagnetic. Magnetic susceptibility of LCPP is well
described by the model of nearest-neighbor ferromagnetic kagome planes with
$J/k_{\rm B}=-1.2$ K ($g=1.995$). A minute interlayer coupling
$J_{\perp}/J=0.01$ improves the fit below 3.5 K and leads to the Curie
temperature $T_{C}=2.8$ K in a perfect agreement with the experimental $T_{\rm
C}\simeq 2.6$ K (Fig. 14). Our DFT calculations corroborate this result and
reveal a weakly ferromagnetic $J_{\perp}$ with $J_{\perp}/J\simeq 0.02$.

##  IV Summary

We synthesized single crystals of LCPP and confirmed trigonal $P\bar{3}c1$
symmetry of this compound with the lattice constants $a=b=9.668(3)$ Å and
$c=13.610(6)$ Å at room temperature. Green-colored LCPP is a rare example of
an insulating kagome ferromagnet. Ferromagnetic order below $T_{C}\simeq 2.6$
K is driven by the in-plane FM coupling $J/k_{\rm B}\simeq-1.2$ K supplied
with a minute inter-plane coupling $J_{\perp}/J=0.02$, which is also FM in
nature. The incomplete release of the magnetic entropy at $T_{\rm C}$ and the
increased width of the ESR line above $T_{\rm C}$ both suggest quasi-2D
magnetic behavior caused by the strong spatial anisotropy of FM couplings. The
overall magnetic behavior of LCPP has striking resemblance with that of other
insulating kagome ferromagnets, such as $\alpha$-MgCu3(OD)6Cl2 and Cu[1,3-bdc]
[58, 10]. Magnetization and ESR measurements on single crystals indicate a
weak easy-plane anisotropy. The critical scaling of magnetization suggests a
non-mean-field type second-order nature of the phase transition at $T_{\rm
C}$. The low $T_{\rm C}$ combined with the large values of $\Delta S_{\rm m}$,
$\Delta T_{\rm{ad}}$, and $RCP$ render LCPP a promising magnetocaloric
material for low-temperature applications.

###### Acknowledgements.

AM, SK, VS, and RN would like to acknowledge SERB, India for financial support
bearing sanction Grant No. CRG/2019/000960.

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# Coup de grace to the charged Higgs solution of $P_{5}^{\prime}$ and
$R_{D^{(*)}}$ discrepancies

Syuhei Iguro<EMAIL_ADDRESS>Institute for Theoretical Particle Physics
(TTP), Karlsruhe Institute of Technology (KIT), Engesserstraße 7, 76131
Karlsruhe, Germany  Institute for Astroparticle Physics (IAP), Karlsruhe
Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, 76344
Eggenstein-Leopoldshafen, Germany

###### Abstract

We consider a general two Higgs doublet model which can simultaneously solve
discrepancies in neutral B meson decay ($b\to s\ell\overline{\ell}$
distribution) and charged B meson decay ($b\to c\tau\overline{\nu}$) with a
charged Higgs. The model contains two additional neutral scalars at the same
mass scale and predicts distinctive signals at the LHC. Based on the recent
same-sign top search by the ATLAS collaboration, we found the constraint on
the scalar mass spectrum. To probe the remaining mass window, we propose a
novel $cg\to t\tau\overline{\tau}$ process at the LHC.
——————————————————————————————————————————–
Keywords: Two Higgs Doublet Model, $b\to s\ell\overline{\ell}$, $b\to
c\tau\nu$, Top-Associated Scalar Production

††preprint: P3H–23–010, TTP23–05

##  I Introduction

The current flavor anomalies in B meson decays $\it{e.g.}$ deviations in
angular distribution in $b\to s\mu\overline{\mu}$ processes, so-called
$P_{5}^{\prime}$ Aaij  _et al._ (2013a, b, c); Chatrchyan  _et al._ (2013);
Aaij  _et al._ (2014, 2015a, 2016, 2020); Sirunyan  _et al._ (2021a); Aaij
_et al._ (2021a, b)#1#1#1Different from lepton flavor universality ratio
$R_{K^{(*)}}={\rm{BR}}(B\to K^{(*)}\mu\overline{\mu})/{\rm{BR}}(B\to
K^{(*)}e\overline{e})$, there is sizable hadronic parameter dependence. For
instance sizable charm hadronic contributions would also explain the
deviation, see Refs, Ciuchini  _et al._ (2016, 2022) for instance. On the
other hand, the tension between measured BR($B_{s}\to\phi\mu\overline{\mu}$)
Aaij  _et al._ (2021c), BR($\Lambda_{b}\to\Lambda\mu\overline{\mu}$) Aaij  _et
al._ (2015b) and BR($B\to K^{(*)}\mu\overline{\mu}$) Aaij  _et al._ (2014) and
the SM predictions Ball and Zwicky (2005); Horgan  _et al._ (2014, 2015);
Bharucha  _et al._ (2016); Detmold  _et al._ (2013); Bobeth  _et al._ (2011,
2012) can be relaxed with the vector contribution., and lepton flavor
violation of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}\to
D^{(*)}\tau\overline{\nu}$ Lees  _et al._ (2012, 2013); Huschle  _et al._
(2015); Hirose  _et al._ (2017, 2018); Abdesselam  _et al._ (2019); Caria  _et
al._ (2020); Aaij  _et al._ (2015c, 2018a, 2018b); Bordone  _et al._ (2020);
Iguro and Watanabe (2020); Bernlochner  _et al._ (2022); Amhis  _et al._
(2022) can be solved with a light charged scalar ($H^{+}$) from a generic two
Higgs doublet model (G2HDM) Iguro and Omura (2018); Kumar (2022)#2#2#2The
possibility was originally pointed out in Ref. Iguro and Omura (2018), and
recently revisited in Ref. Kumar (2022). It is noted that thanks to the
relaxed constraint from $B_{c}\to\tau\overline{\nu}$ Alonso  _et al._ (2017);
Celis  _et al._ (2017); Blanke  _et al._ (2019); Aebischer and Grinstein
(2021) and the experimental shift, $H^{+}$ can now explain
$R_{D^{(*)}}={\rm{BR}}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}\to
D^{(*)}\tau\overline{\nu})/{\rm{BR}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}\to D^{(*)}\ell\overline{\nu})$ within
$1\,\sigma$ Iguro . For the individual explanation, see, Refs. Hu  _et al._
(2017); Arnan  _et al._ (2017); Arhrib  _et al._ (2017); Li  _et al._ (2018);
Crivellin  _et al._ (2019) for $b\to s\ell\overline{\ell}$ and Refs. Crivellin
_et al._ (2012, 2013); Cline (2016); Crivellin  _et al._ (2016); Lee (2017);
Iguro and Tobe (2017); Martinez  _et al._ (2018); Fraser  _et al._ (2018);
Athron  _et al._ (2022); Iguro (2022); Blanke  _et al._ (2022); Fedele  _et
al._ (2022) for $R_{D^{(*)}}$. . Although a significant deviation in the
lepton flavor universality test in $b\to s\ell\overline{\ell}$ transition
where $\ell=e,\,\mu$ has disappeared in the recent LHCb measurement LHC
(2022a, b) thanks to the improved electron tagging method. Furthermore, the
deviation in $B_{s}\to\mu\overline{\mu}$ has gone CMS  and, consequently the
explicit priority of the vector$-$axial vector (V$-$A) like interaction no
longer exists Ciuchini  _et al._ (2022). Those recent changes brought charged
Higgs solution back into the game and makes it more appealing. There days, due
to the disappearance of $R_{K^{(*)}}$ puzzle, there is a psychological tone
down for the B anomalies, though, it is a fact that there are still about
$3\sim 4\,\sigma$ discrepancies in $b\to s\ell\overline{\ell}$ and $b\to
c\tau\overline{\nu}$ processes.

Interestingly a successful charm penguin contribution to the flavor universal
vector operator of $b\to s\ell\overline{\ell}$ and tree level $b\to
c\tau\overline{\nu}$ transition are both controlled by the common
$\overline{b}_{L}c_{R}H^{-}$ interaction where the corresponding Yukawa
coupling is denoted as $\rho_{u}^{tc}$. In the G2HDM, the coupling
$\rho_{u}^{tc}$ induces $\overline{t}_{L}c_{R}\phi$ interaction where
$\phi=H,\,A$ denotes additional neutral scalars which are SU(2)L partners of
the charged Higgs. It is noted that the additional doublet with sizable
$\rho_{u}^{tc}$ is discussed with the spontaneous CP violating scenario Lee
(1973); Nierste  _et al._ (2020) and the electroweak baryogenesis Fuyuto  _et
al._ (2018).#3#3#3They used the closed time path formalism Riotto (1998) to
evaluate the produced baryon number.

The available mass range of the charged scalar for the simultaneous
explanation is bounded from the above based on the $\tau\overline{\nu}$
resonance searches at the LHC Sirunyan  _et al._ (2019a) as $m_{H^{+}}\leq
400\,{\rm GeV}$ Iguro  _et al._ (2019). Although in Ref. Blanke  _et al._
(2022) we theoretically showed that the $b+\tau\overline{\nu}$ resonance
search is a powerful tool to probe the remaining parameters, the corresponding
experimental search has not been performed.

Different from recent studies which mainly focus on the charged scalar
collider phenomenology in light of deviations in B meson decays Iguro (2022);
Blanke  _et al._ (2022); Desai  _et al._ (2022), we consider the collider
signal of additional neutral scalars. Although connection between sizable
$\rho_{u}^{tc}$ and neutral scalars mediated multi-top final states at the LHC
has been discussed in Refs. Iguro and Tobe (2017); Gori  _et al._ (2018);
Kohda  _et al._ (2018); Iguro  _et al._ (2019); Hou  _et al._ (2018, 2021);
Hou and Modak (2021),#4#4#4See, also Refs. Hall and Wise (1981); Hou  _et al._
(1997); Atwood  _et al._ (1997); Altunkaynak  _et al._ (2015) for the earlier
works to probe $\rho_{u}^{tc}$ in a flavor changing top decay. last summer,
the ATLAS collaboration reported the game changing result ATL (a). They
searched for the G2HDM in top-associated processes and directly set the upper
limit on $\rho_{u}^{tc}$. In this letter, we reinterpret the constraint in
light of the simultaneous explanation and propose an additional process to
cover the remaining parameter space thorough the neutral scalars. Thanks to
the electroweak precision data even after the controversial CDF result
Aaltonen  _et al._ (2022), the mass of those additional scalars ($m_{\phi}$)
should be similar to $m_{H^{+}}$ up to $\mathcal{O}(v)$ where $v=246\,{\rm
GeV}$ denotes the vacuum expectation value. Therefore it would be natural to
consider the LHC phenomenology to fully probe the interesting parameter space.

The outline of the letter is given as follows. In Sec. II we introduce the
model setup and explain the relevant parameters. The favored region and upper
limit on the additional scalars are summarized in Sec. III. In Sec. IV we
investigate the model prediction of top-associated processes. Summary and
discussion will be given in Sec. V.

##  II Model setup

We consider a two Higgs doublet model (2HDM) where an additional scalar
doublet is introduced to the SM. The general scalar potential of the model is
given as

$\displaystyle\rm{V}(H_{1},\,$ $\displaystyle
H_{2})=M_{11}^{2}H_{1}^{\dagger}H_{1}+M_{22}^{2}H_{2}^{\dagger}H_{2}-\left(M_{12}^{2}H_{1}^{\dagger}H_{2}+{\rm
h.c.}\right)$
$\displaystyle+\frac{\lambda_{1}}{2}(H_{1}^{\dagger}H_{1})^{2}+\frac{\lambda_{2}}{2}(H_{2}^{\dagger}H_{2})^{2}+\lambda_{3}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})$
$\displaystyle+\lambda_{4}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})+\frac{\lambda_{5}}{2}(H_{1}^{\dagger}H_{2})^{2}$
$\displaystyle+\left\\{\lambda_{6}(H_{1}^{\dagger}H_{1})+\lambda_{7}(H_{2}^{\dagger}H_{2})\right\\}(H_{1}^{\dagger}H_{2})+{\rm
h.c.}.$ (1)

Here, we work in the Higgs basis where only one doublet takes the VEV Georgi
and Nanopoulos (1979); Donoghue and Li (1979):

$\displaystyle H_{1}=\begin{pmatrix}G^{+}\\\
\frac{1}{\sqrt{2}}(v+h+iG^{0})\end{pmatrix},\,H_{2}=\begin{pmatrix}H^{+}\\\
\frac{1}{\sqrt{2}}(H+iA)\end{pmatrix},$ (2)

where $G^{+}$ and $G^{0}$ denotes the NG bosons. It is noted that alignment
where the SM h lives in $H_{1}$ is considered to avoid the constraint from
$t\to ch$ Tumasyan  _et al._ (2022a, b); ATL (2023). For simplicity, we
further assume the CP-conserving scalar potential and then one can define the
CP-even and -odd scalar mass eigenstates. The SM-like Higgs is $h$ and $H$ and
$A$ correspond to additional the CP-even and -odd neutral scalars. Masses
differences among additional scalars are given as,

$\displaystyle m_{H}^{2}$
$\displaystyle=m_{A}^{2}+\lambda_{5}v^{2},\,\,\,m_{H^{+}}^{2}=m_{A}^{2}-\frac{\lambda_{4}-\lambda_{5}}{2}v^{2}.$
(3)

It is noted that other potential couplings does not affect the following
discussion.

When the both doublets couple to all fermions, the Higgs bosons have flavor
violating interactions in general. In this letter we take the bottom-up
approach and introduce the interaction Lagrangian of the heavy scalars
relevant to $b\to s\ell\overline{\ell}$ and $b\to c\tau\overline{\nu}$,

$\displaystyle{\cal L}_{int}$
$\displaystyle=\,\rho_{u}^{tc}\frac{H+iA}{\sqrt{2}}(\overline{t}P_{R}c)+\rho_{e}^{\tau\tau}\frac{H-iA}{\sqrt{2}}(\overline{\tau}P_{R}\tau)$
$\displaystyle+V^{*}_{td_{i}}\rho_{u}^{tc}H^{-}(\overline{d_{i}}P_{R}c)-\rho_{e}^{\tau\tau}H^{-}(\overline{\tau}P_{L}\nu_{\tau})+{\rm{h.c.}},$
(4)

where $P_{L/R}=(1\mp\gamma_{5})/2$ and $V$ are a chirality projection operator
and Cabbibo-Kobayashi-Maskawa matrix Cabibbo (1963); Kobayashi and Maskawa
(1973), respectively. The neutral scalar interaction and the charged scalar
interaction are related by the SU(2)L rotation. We assume that other Yukawa
coupling to be small ($\ll\mathcal{O}(10^{-2})$) for simplicity. For the more
detailed phenomenological analysis with other Yukawa couplings, see Refs.
Crivellin  _et al._ (2013); Iguro and Tobe (2017); Iguro and Omura (2019). We
will also discuss this point in Sec. V.

For the later convenience we show the approximate formulae for the partial
decay width,

$\displaystyle\Gamma(\phi\to\tau\overline{\tau})\simeq\frac{|\rho_{e}^{\tau\tau}|^{2}}{16\pi}m_{\phi},\,\Gamma(\phi\to
tc)\simeq\frac{3|\rho_{u}^{tc}|^{2}m_{\phi}}{16\pi}\beta^{2}(m_{\phi}),$ (5)

where $\Gamma(\phi\to tc)=\Gamma(\phi\to
t\overline{c})+\Gamma(\phi\to\overline{t}c)$ and
$\beta(m_{\phi})=\left(1-\frac{m_{t}^{2}}{m_{\phi}^{2}}\right)$ are
defined.#5#5#5In this letter we neglect light fermion masses, though, one can
trivially include the effect.

##  III Summary of the available parameter region

First we consider the charged Higgs contribution to flavor universal $b\to
s\ell\overline{\ell}$. Since the coupling dependence is different among $b\to
s\ell\overline{\ell}$ (induced by the charm penguin
$\propto|\rho_{u}^{tc}|^{2}$) and the most constraining flavor process,
$B_{s}-\overline{B_{s}}$ mixing (charged Higgs box
$\propto|\rho_{u}^{tc}|^{4}$), we can set an upper limit on the charged Higgs
mass Iguro and Omura (2018); Kumar (2022). The relevant Hamiltonian for $b\to
s\ell\overline{\ell}$ in our model is given as

$\displaystyle{\mathcal{H}}_{\rm{eff}}$ $\displaystyle=-\frac{\alpha
G_{F}}{\sqrt{2}\pi}V_{tb}V_{ts}^{*}C_{9}(\overline{s}\gamma^{\mu}P_{L}b)(\overline{l}\gamma_{\mu}l)+\rm{h.c.},$
(6)

where $l=e,\,\mu$ and $\tau$. We note that contribution from $Z$ penguin is
small enough to neglect. We follow the prescription in Ref. Kumar (2022) and
use the following numerical formula,

$\displaystyle
C_{9}^{l}(\mu_{b})\simeq-0.95\left(\frac{|\rho_{u}^{tc}|}{0.7}\right)^{2}\left(\frac{200\,{\rm
GeV}}{m_{H^{+}}}\right)^{2}.$ (7)

This should be compared with the recent global fit to $b\to
s\ell\overline{\ell}$ data of $C_{9}^{l}(\mu_{b})=-0.95\pm 0.13$ Hurth  _et
al._ .#6#6#6This fit does not include $B_{s}\to\mu\overline{\mu}$ and lepton
flavor universality observables e.g. $R_{K^{(*)}}$. Since $C_{9}$ operator
does not contribute to $B_{s}\to\mu\overline{\mu}$, the result will be
unchanged, though. The similar result is also reported in Ref. Ciuchini  _et
al._ (2022). In Fig. 1, we show $1\,(2)\,\sigma$ favored region in green
(yellow) on the $m_{H^{+}}$ vs. $\rho_{u}^{tc}$ plane. Since we also has the
upper limit on the mass as $m_{H^{+}}\leq 400\,{\rm GeV}$ and the lower limit
form the LEP experiment Abbiendi  _et al._ (2013), we focus on $100\,{\rm
GeV}\leq m_{H^{+}}\leq 400\,{\rm GeV}$. As mentioned above $B_{s}$ meson
mixing puts the most stringent flavor constraint Di Luzio  _et al._ (2019)
which is shown in magenta.

Figure 1:  The favored region of $C_{9}^{\ell}$ is shown in green
($1\,\sigma$) and yellow ($2\,\sigma$) on the $\rho_{u}^{tc}$ vs. $m_{H^{+}}$
plane. $B_{s}-\overline{B_{s}}$ mixing constraint excludes the magenta region.
Cyan, purple, blue regions are excluded by low mass di-jet resonance searches.
The orange dashed line corresponds to the upper limit from the same-sign top
search adopted from Ref. ATL (a) assuming $m_{H^{+}}=m_{H}$. See the main text
for further detail.

In this mass region, di-jet resonance searches at the LHC are able to set the
upper limit on $\rho_{u}^{tc}$ Iguro (2022). We overlay the constraint from
the (bottom flavored) di-jet searches in blue Sirunyan  _et al._ (2018a),
purple Aaboud  _et al._ (2019) and cyan Sirunyan  _et al._ (2018b) where
BR($H^{+}\to\overline{b}c$)=1 is assumed. It is noted that as we will see soon
later, we need a hierarchy of $|\rho_{u}^{tc}|\gg|\rho_{e}^{\tau\tau}|$ for
the simultaneous explanation. As a result $H^{+}\to\overline{b}c$ is the
dominant decay mode in the minimal set up of Eq. (4) and hence the exclusion
discussed above is unaffected.#7#7#7The stau search constraint Aad  _et al._
(2020a); CMS (2022) on the charged Higgs is very weak due to
BR($H^{+}\to\overline{b}c)\simeq 1$. See, Fig. 4 of Ref. Iguro  _et al._
(2022a). We see that di-jet constraints touch the interesting parameter
region. Run 2 full data would be possible to improve the constraint further.

Figure 2:  The representative diagrams for the same-sign top final state at
the LHC. In the numerical evaluation we include the charge conjugated
processes also. The dominant contribution comes from the right diagram.

We move onto the explanation of the $R_{D^{(*)}}$ discrepancy. The relevant
interaction Hamiltonian is given as

$\displaystyle{\mathcal{H}}_{\rm{eff}}=2\sqrt{2}G_{F}V_{cb}C_{S_{L}}^{\tau}(\overline{c}P_{L}b)(\overline{\tau}P_{L}\nu_{\tau}).$
(8)

The charged Higgs contribution including renormalization group running
corrections Alonso  _et al._ (2014); Jenkins  _et al._ (2014); González-Alonso
_et al._ (2017); Aebischer  _et al._ (2017), is approximately given as

$\displaystyle|C_{S_{L}}^{\tau}(\mu_{b})|\simeq
0.83\left(\frac{|\rho_{u}^{tc*}\rho_{e}^{\tau\tau}|}{0.03}\right)\left(\frac{200\,{\rm
GeV}}{m_{H^{+}}}\right)^{2}.$ (9)

Adopting the analytic formulae of $R_{D^{(*)}}$ in Ref. Iguro
(2022)#8#8#8Those analytic formulae used in Ref. Iguro (2022) are consistent
with the recent result Iguro  _et al._ (2022b) within the uncertainty. latest
$1\,\sigma$ explanation is realized with
$0.68\lesssim|C_{S_{L}}^{\tau}(\mu_{b})|\lesssim 1.13$.#9#9#9To fit the
$R_{D^{(*)}}$ data $\rho_{u}^{tc*}\rho_{e}^{\tau\tau}$ needs to have a complex
phase, however, this does not change the following discussion. By combining
Eqs. (7, 9), one can see that the simultaneous explanation requires the large
magnitude difference in $\rho_{u}^{tc}$ and $\rho_{e}^{\tau\tau}$.

So far we focused on the charged Higgs phenomenology, however, neutral scalar
mass spectrum is constrained with the LHC data and electroweak precision
observables. The last summer the ATLAS collaboration reported the result of
the G2HDM search in top-associated processes ATL (a) for
$m_{\phi}\geq\,200\,{\rm GeV}$.#10#10#10To adopt the experimental data and
extend the constraint down to $m_{\phi}\simeq m_{t}$, detailed distribution
data is necessary. Although this data is not available in Ref. ATL (a) and
thus beyond scope of this letter. The relevant signal events include the same-
sign top quarks. In Fig. 1, the constraint directly taken from Ref. ATL (a) is
shown in the orange dashed line assuming $m_{H}=m_{H^{+}}$.#11#11#11In this
analysis they only considered $H$ to be present and ignore $A$ for simplicity.
If there is a mild mass difference of $\mathcal{O}(10)\,{\rm GeV}$, the
constraint will be more stringent by a factor of $\sqrt{2}$.  It is observed
that this same-sign top search would exclude the $b\to s\ell\overline{\ell}$
explanation for $m_{\phi}\geq\,200{\rm GeV}$. Although there is a loophole in
this same-sign top bound. There are two-types of the contributing Feynman
diagrams, namely t-channel (left) and s-channel (right) as shown in Fig. 2. In
both diagrams, due to the different CP nature of $H$ and $A$, the amplitude
cancels in the mass degenerate limit. The destructive interference for the
dominant s-channel approximately happens up to the width difference Hou  _et
al._ (2018). For the simultaneous explanation, $\rho_{u}^{tc}$ needs to be as
large as 0.7 (0.8) for $m_{H^{+}}=200\,(250)\,{\rm GeV}$ and hence the total
width of $\Gamma_{\phi}=0.8\,(3.5)\,{\rm GeV}$ is predicted. This indicates
that $|\lambda_{5}|\leq{\mathcal{O}}(10^{-2})$ is necessary for the
simultaneous explanation with $m_{\phi}\geq 200\,{\rm GeV}$. To simplify the
analysis and evade the constraint we set $m_{A}=m_{H}$ in the following.

Figure 3:  The $\chi^{2}$ based on $S$ and $T$ parameters before (dashed) and
after (solid) the recent CDF result is shown as a function of $m_{\phi}$. For
blue, orange and green lines, $m_{H^{+}}=150,\,200,\,250\,$GeV are fixed. The
gray vertical line corresponds to $m_{\phi}=m_{t}$.

On the other hand, additional neutral scalars dominantly decay to
$\tau\overline{\tau}$ for $m_{\phi}\leq m_{t}$. In that case, the electroweak
pair production of neutral scalars results in multiple $\tau$ final state.
Such a region is studied in Ref. Chun  _et al._ (2015) and even only with Run
1 data Aad  _et al._ (2014) we can exclude our scenario of $m_{\phi}\leq
m_{t}$. Furthermore do not have an explicit new physics signal with the Run 2
full data Tumasyan  _et al._ (2022c); ATL (b) and hence the exclusion is
robust.

Besides, electroweak precision observables are helpful to further constrain
the mass spectrum. We consider $S$ and $T$ parameter constraint#12#12#12Since
the deviation in $U$ parameter is suppressed in this model and the uncertainty
in $S$ and $T$ parameters will be reduced considerably, we set $U=0$. Peskin
and Takeuchi (1990, 1992) both excluding and including recent controversial
CDF result Aaltonen  _et al._ (2022). More concretely we use

$\displaystyle S=0.00\pm 0.07,~{}~{}T=0.05\pm 0.06,$ (10)

with the correlation of $\rho=0.92$ Zyla  _et al._ (2020) (denoted as 2021
fit) and

$\displaystyle S=0.086\pm 0.077,~{}~{}T=0.177\pm 0.070,$ (11)

with the correlation of $\rho=0.89$ based on the global fit de Blas  _et al._
(2022) (denoted as 2023 fit). Fig. 3 shows $\chi^{2}$ of $S$ and $T$
parameters as a function of $m_{\phi}$ where $m_{H^{+}}=$150 GeV (blue), 200
GeV (orange) and 250 GeV (green) is fixed. Dashed and solid lines are drawn
based on 2021 fit and and 2023 fit. We see that the favored $m_{\phi}$ is
different depending on the fit data. For $m_{H^{+}}=150$ GeV, 2023 fit
disfavors $m_{t}\leq m_{\phi}\leq 200\,{\rm GeV}$ more than $2\,\sigma$, while
2021 fit allows the mass window.

In short section summary, for the simultaneous explanation we need to set
$m_{t}\leq m_{\phi}\leq 200\,{\rm GeV}$ or $\mathcal{O}$(1) GeV level mass
degeneracy among neutral scalars.

##  IV Exotic top processes

In order to fully probe the remaining mass window of $m_{\phi}$ we propose
another top-associated process, namely $gc\to c\to t\phi\to
t\tau\overline{\tau}$ where the relevant diagram is shown in Fig.
4.#13#13#13It would be worthwhile to mention that $t\overline{t}$ inclusive
cross section measurement still has an uncertainty of 70 pb Aad  _et al._
(2020b) and does not exclude the scenario with $gc\to c\to t\phi\to
t\overline{t}c$ channel. In the mass window, even with the hierarchical
coupling structure, BR($\phi\to\tau\overline{\tau}$) could be sizable due to
the phase space suppression in $\phi\to tc$ decay. The production cross
section is calculated using MadGraph5_aMC@NLO Alwall  _et al._ (2014) using
NNPDF2.3 Ball  _et al._ (2013) at the leading order in the five flavor scheme
with $\sqrt{s}=13$ TeV. Fig. 5 shows the cross section in pb as a function of
$m_{\phi}$. The prediction of the $1\,\sigma$ simultaneous explanation was
obtained by fixing the charged Higgs mass $m_{H^{+}}=150$ GeV (blue), 200 GeV
(orange), 250 GeV (green) and $m_{\phi}$ (black). It is observed that bands
are overlapping and the cross section is as large as 30 fb$\sim$10 pb for the
mass window.#14#14#14For the numerical analysis we include $\phi\to
H^{\pm}W^{\mp}$ if the phase space is available. A heavier charged scalar
predicts the larger signal rate since it requires larger couplings.

Figure 4:  Representative diagram for $gc\to c\to t\phi\to
t\tau\overline{\tau}$.

Estimating the size of the electroweak SM back ground (BG) is not difficult
even for our mass range. For instance, $tZq$ and $thq$ production contribute
to $t+\tau\overline{\tau}+q$ final state with cross section of $\simeq 50$ fb
Sirunyan  _et al._ (2019b) and $\simeq 5$ fb de Florian  _et al._ (2016) where
$\tau\overline{\tau}$ comes from $Z$ and $h$ decay for each. Therefore the
contribution from those processes are expected to be moderate. On the other
hand, it is not easy to estimate the precise amount of the miss-tag associated
BG e.g. from $tW^{-}q\to t\tau\overline{\nu}+\not{j}$ and $t\overline{t}\to
tW^{-}j\to t\tau\overline{\nu}+\not{j}$ where slashed final state will be
miss-tagged as a hadronically decaying $\tau$ ($\tau_{h}$). For the precise
determination we need a considerable help from the experimental side and thus
investigating the sensitivity of this channel is beyond the scope of this
letter.#15#15#15The charge asymmetry of the top quark would help to improve
the sensitivity since the SM single top has the production asymmetry, while
our signal does not have this feature. Actually Ref. Sirunyan  _et al._
(2021b) searched for the $thq$ production with $h\to\tau\overline{\tau}$ with
Run 2 full data. They set the upper limit of $\mu=8.1^{+8.2}_{-7.5}$ where
$\mu$ denotes a signal strength. This approximately leads to the upper limit
on $\sigma(thq\to t\tau\overline{\tau}q)\lesssim 100\,$fb for
$m_{\tau\tau}=125\,{\rm GeV}$. Since the invariant mass of our signal is
larger, the corresponding SMBG would be smaller and thus we can expect the
better sensitivity.

Figure 5:  Prediction of $\sigma(pp\to t\phi\to t\tau\overline{\tau})$ [pb] as
a function of $m_{\phi}$ for the simultaneous explanation of deviations in
$b\to s\ell\overline{\ell}$ and $b\to c\tau\overline{\nu}$.

##  V Summary and discussion

Recently the charged Higgs solution to B anomalies became more interesting
than ever. The charged Higgs need to interact with left-handed bottom quark
and thus can be a part of an additional doublet. Hence a two Higgs doublet
model is a minimal model and there are also two additional neutral scalars.
The Yukawa interaction of those scalars are related by SU(2)L rotation and the
simultaneous explanation predicts distinctive signal at the LHC. The
theoretical proposals to probe the solution via charged Higgs mediated
processes was made last year, however, the crucial process has not been tested
experimentally yet. Although, in the meantime, the ATLAS experiment reported
the game changing constraint on the neutral scalars. In this letter we
reinterpret the ATLAS constraint and obtained the condition for the mass
spectrum of the additional neutral scalars: ${\mathcal{O}}(1)\,{\rm GeV}$ mass
degeneracy among $H$ and $A$ or $m_{t}\leq m_{\phi}\leq 200\,{\rm GeV}$ where
$\phi$ denotes $H$ and $A$. We also pointed out that the signal cross section
of $gc\to t\phi\to t\tau\overline{\tau}$ could be as large as 10 fb$\sim$10 pb
for the mass window.

Imposing a U(1) Peccei-Quinn symmetry Peccei and Quinn (1977),
$\\{H_{1},\,H_{2}\\}\to\\{H_{1},\,H_{2}e^{i\alpha}\\}$ can prohibit
$\lambda_{5}$ and realize the mass degeneracy of additional neutral scalars
Branco  _et al._ (2012). Although this symmetry should be broken since we also
need Yukawa couplings, $\rho_{u}^{tc}$ and $\rho_{e}^{\tau\tau}$ and therefore
the more complicated setup is necessary Ferreira and Silva (2011); Serôdio
(2013); Chiang  _et al._ (2015); Björkeroth  _et al._ (2019).

In general, other couplings $\it{e.g.}$ di-bottom quark coupling, namely
$\rho_{d}^{bb}$ would be non-negligible. For instance, one would think that
$\mathcal{O}(10^{-2})$ of $\rho_{d}^{bb}$ could reduce the branching ratio of
$\phi\to\tau\overline{\tau}$ thanks to the color factor and revive the
scenario with $m_{\phi}\leq m_{t}$.#16#16#16It is noted that even in that case
$C_{S_{R}}^{\tau}$, where the chirality of quarks are flipped in Eq. (8), can
not be large to affect $R_{D^{(*)}}$ due to the $V_{cb}$ suppression Iguro and
Tobe (2017). Although this is difficult since the ATLAS collaboration searched
additional particles in flavor changing top decays set $\mathcal{O}(10^{-4})$
upper bound on BR$(t\to qX)\times$BR$(X\to b\overline{b})$ very recently ATL
(2023). Therefore an additional coupling to bottom quarks, does not save the
scenario. Since $c\to b$ miss tagging rate, $\epsilon_{c\to b}$ is about
$15\sim 20\,\%$ ATL (2022), even if neutral scalars decay into charm quarks,
the scenario is difficult to survive the constraint. On the other hand,
$\rho_{d}^{bb}$ would be able to reduce signal rate of $gc\to
t\tau\overline{\tau}$ process.

It would to worthwhile to emphasize that the ATLAS bound ATL (a) does not
necessarily kill the solo $R_{D^{(*)}}$ solution even without mass degeneracy.
This is because that the contribution to $C_{S_{L}}$ is proportional to the
coupling product of $\rho_{u}^{tc*}\rho_{e}^{\tau\tau}$ (see, Eq. (9)) and
hence the larger $\rho_{e}^{\tau\tau}$ allows the smaller $\rho_{u}^{tc}$. If
we want to avoid the ATLAS bound on $\rho_{u}^{tc}$ by setting
$m_{A},\,m_{H}\leq 200\,{\rm GeV}$ instead, electroweak precision parameters
at $2\,\sigma$ give the upper limit on the charged Higgs mass as
$m_{H^{+}}\leq 270\,{\rm GeV}$ ($290\,{\rm GeV}$) for Eq. (10) (Eq. (11)). In
this case, $t+\tau\overline{\tau}$ would provide a key test since
BR$(\phi\to\tau\overline{\tau})$ will be amplified compared to the scenario
for the simultaneous explanation.

## Acknowledgements

I would like to thank Ulrich Nierste, Marco Fedele, Hiroyasu Yonaha and Teppei
Kitahara for the useful discussion and great encouragement. I appreciate
Masaya Kohda for the exchange on the same-sign top signal in 2018. I also
appreciate Javier Montejo Berlingen, Tamara Vazquez Schroeder and Shigeki
Hirose for the detailed information of Ref. ATL (a). The work is supported by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
grant 396021762-TRR 257.

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