#  $T_{cc}^{+}(3875)$ relevant $DD^{*}$ scattering from $N_{f}=2$ lattice QCD

Siyang Chen  Chunjiang Shi<EMAIL_ADDRESS>Ying Chen
<EMAIL_ADDRESS>Ming Gong  Zhaofeng Liu  Wei Sun  Renqiang Zhang  Institute
of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, P.R.
China  School of Physics, University of Chinese Academy of Sciences, Beijing
100049, P.R. China  School of Physics, Nanjing University, Nanjing 210093,
P.R. China  Center for High Energy Physics, Peking University, Beijing 100871,
P. R. China

###### Abstract

The $S$-wave $DD^{*}$ scattering in the isospin $I=0,1$ channels is studied in
$N_{f}=2$ lattice QCD at $m_{\pi}\approx 350$ MeV. It is observed that the
$DD^{*}$ interaction is repulsive in the $I=1$ channel when the $DD^{*}$
energy is near the $DD^{*}$ threshold. In contrast, the $DD^{*}$ interaction
in the $I=0$ channel is definitely attractive in a wide range of the $DD^{*}$
energy. This is consistent with the isospin assignment $I=0$ for
$T_{cc}^{+}(3875)$. By analyzing the components of the $DD^{*}$ correlation
functions, it turns out that the quark diagram responsible for the different
properties of $I=0,1$ $DD^{*}$ interactions can be understood as the charged
$\rho$ meson exchange effect. This observation provides direct information on
the internal dynamics of $T_{cc}^{+}(3875)$.

##  1 Introduction

Ever since the discovery of $X(3872)$ in 2003 [1], there have been quite a lot
near-$D\bar{D}$ and $B\bar{B}$ threshold structures observed in experiments
and are generally named $XYZ$ particles [2]. They are usually assigned to be
conventional heavy quarkonia, $D\bar{D}$ ($B\bar{B}$) molecules, or
tetraquarks in phenomenological studies. Among $XYZ$ states, $Z_{c}(3900)$ may
be the most prominent candidate that has the minimal quark configuration
$c\bar{c}u\bar{d}$ and has been observed in different experiments [3, 4].
Recently, LHCb reported the first doubly-charmed narrow structure
$T_{cc}^{+}(3875)$ in the $D^{0}D^{0}\pi^{+}$ invariant mass spectrum, which
specifically has the minimal configuration of $cc\bar{u}\bar{d}$ [5]. With a
mass below the $D^{0}D^{*+}$ threshold by $-273\pm 61\pm 5^{+11}_{-14}$ keV,
its width is as small as $\Gamma=410\pm 165\pm 43_{-38}^{+18}$ keV (A
unitarised Breit-Wigner analysis gives an even smaller width $\Gamma^{U}=48\pm
2_{-14}^{0}$ keV[6]). LHCb searched other charged channels and found no
evidence for the existence of a similar structure, and therefore assigned
$T_{cc}^{+}(3875)$ to be an $I=0$ state [5, 6].

Prior to the observation of $T_{cc}^{+}(3875)$, there have been many
theoretical studies on doubly-charmed tetraquarks, whose predictions of the
mass and width of the $J^{P}=1^{+}$ isoscalar tetraquark ground state are
consistent with those of $T_{cc}^{+}(3875)$[7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41]. In the molecular picture, an early quark model
calculation predicted the existence of a $DD^{*}$ bound state below the
$DD^{*}$ threshold by $1.6\pm 1.0$ MeV [13]. Recent theoretical studies found
that light vector meson exchanges may induce an attractive interaction between
$D$ and $D^{*}$ [42, 43, 44]. One can also refer to a recent review of the
present status of theoretical studies on $T_{cc}^{+}(3875)$ in Ref.[45]. There
are also several lattice calculations performed to search exotic doubly-
charmed meson states. By using meson-meson and diquark-antidiquark
interpolators, the $N_{f}=2+1$ lattice study in Ref.[46] searched a large
range of possible exotic $J^{P}$ state spectrums with isospin-$0$ and
isospin-$\frac{1}{2}$. Ref. [30] performed series of tetraquark
$qq\bar{Q}\bar{Q}$ lattice calculation with $N_{f}=2+1+1$ and got a
$ud\bar{c}\bar{c}$ state with its energy lower than the $DD^{*}$ threshold by
$23\pm 11$ MeV after the continuum and chiral extrapolation. Another lattice
QCD calculation claimed that the ground state energy is consistent with the
$DD^{*}$ threshold [26]. The latest lattice calculation [47] explored the pole
singularity of the $DD^{*}$ scattering amplitude at the pion mass
$m_{\pi}\approx 280$ MeV and reported an $S$-wave virtual bound state pole
below the $DD^{*}$ threshold by roughly 10 MeV, which may correspond to
$T_{cc}^{+}(3875)$ when $m_{\pi}$ approaches to the physical value.

Since the LHCb experiment observed $T_{cc}^{+}(3875)$ only in the $I=0$
channel, it is conceivable that the isospin-dependent interaction plays a
vital role in its formation. The existing lattice QCD studies focus on the
$I=0$ channel from the point of view of tetraquark and $DD^{*}$ scattering and
pay little attention to the isospin-sensitive properties. Given the large
negative scattering length $a=-7.16(51)$ fm of $DD^{*}$ scattering relevant to
$T_{cc}^{+}(3875)$ determined by LHCb from theoretical analyses [6], the
characteristic size $R_{a}=|a|$ of $T_{cc}^{+}$ is too large for the present
lattice QCD to investigate it directly, which requires the lattice size $L$ to
be much larger than $R_{a}$. An alternative way is to study the relevant
$DD^{*}$ scattering in several different lattice volumes and then perform the
infinite volume extrapolation to check the existence of a bound state [48,
49]. With only one lattice on hand, we cannot explore this way yet. So we
focus on the study of the $S$-wave $DD^{*}$ scatterings in $I=0$ and $I=1$
channels, and explore if there are clear differences between them. This study
may shed light on the property of the $DD^{*}$ interaction and provide some
qualitative information for future phenomenological investigations.

This paper is organized as follows: In Section 2 we describe the lattice
setup, operator construction, and the method for studying the hadron-hadron
interaction on the lattice. The results of $DD^{*}$ scatterings in $I=0,1$
channels are presented in Section 3 and are discussed in Section 4. Section 5
is a summary of this work.

##  2 Numerical Details

###  2.1 Lattice Setup

We generate gauge configurations with $N_{f}=2$ degenerate $u,d$ quarks on an
$L^{3}\times T=16^{3}\times 128$ anisotropic lattice. We use the tadpole
improved anisotropic clover fermion action for light $u,d$ quarks [50, 51] and
tadpole improved gauge action [52, 53]. The renormalized aspect ratio is
determined to be $\xi=a_{s}/a_{t}=5.3$, and the temporal lattice spacing is
set to be $a_{t}^{-1}=6.894(51)$ GeV [54]. Using the $a_{t}$ and the $\xi$, we
get $a_{s}\approx 0.152(1)$ fm. Our bare $u,d$ quark mass parameter gives
$m_{\pi}=348.5(1.0)$ MeV, thus the value $m_{\pi}La_{s}\approx 3.9$ warrants
that the finite volume effect of this lattice setup is not important. For the
valence charm quark, we adopt the clover fermion action in Ref. [55] and the
charm quark mass parameter is tuned to give
$(m_{\eta_{c}}+3m_{J/\psi})/4=3069$ MeV. The distillation method [56] is used
to generate the perambulators for $u,d$ quarks and the valence charm quark on
our gauge ensemble. In practice, the perambulators are calculated in the
Laplacian Heaviside subspace spanned by $N_{\mathrm{vec}}=70$ eigenvectors
with the lowest eigenvalues. The parameters for the gauge ensemble are listed
in Table 1.

Table 1: Parameters of $N_{f}=2$ gauge ensembles with degenerate $u,d$ sea quarks. $L^{3}\times T$ | $\beta$ |  $a_{t}^{-1}$(GeV) | $\xi$ | $N_{\mathrm{cfg}}$ |  $m_{\pi}$(MeV) |  $m_{J/\psi}$(MeV) | $N_{\mathrm{vec}}$
---|---|---|---|---|---|---|---
$16^{3}\times 128$ | 2.0 | $6.894(51)$ | $\sim 5.3$ | $6950$ | $348.5(1.0)$ | $3.099(1)$ | 70

###  2.2 Operators and correlation functions

In the lattice study of hadron-hadron scattering, one key task is to extract
the lattice energy levels as precisely as possible, from which the data-
sensitive scattering matrix elements can be parameterized with quantities
reflecting the scattering properties, such as the scattering phase shift and
scattering length, etc. Concerning the properties of $T_{cc}^{+}(3875)$, we
focus on the $DD^{*}$ scattering in the $J^{P}=1^{+}$ channel with the isospin
$I=0$ and $I=1$. Throughout the work, the $D(D^{*})$ operators and $DD^{*}$
operators are built in terms of smeared quark fields. We use quark bilinears
$O_{\Gamma}=\bar{q}\gamma_{5}c$ for $D$ mesons and
$O_{\Gamma}=\bar{q}\gamma_{i}c$ for $D^{*}$ mesons (here $q$ refers to $u$ for
$D^{0}$ and $d$ for $D^{+}$). Accordingly, the operators for $D(D^{*})$ mesons
moving with a spatial momentum $\vec{p}$ are obtained by the Fourier
transformation
$O_{\Gamma}(\vec{p},t)=\sum\limits_{\vec{x}}e^{-i\vec{p}\cdot\vec{x}}O_{\Gamma}(\vec{x},t)$.

Figure 1: The effective energies and dispersion relation of $D$ and $D^{*}$.
For the effective energies of $D$ (left panel) and $D^{*}$ (middle panel), the
grey bands illustrate the fittings using Eq.( 1) in the time window
$t\in[20,T-20]$. For the dispersion relations (right panel), the data points
are measured energies $E_{X}^{2}(\vec{p})$ at different momenta
$\vec{p}=\frac{2\pi}{a_{s}L}\vec{n}$ (labelled by $\vec{n}^{2}$) with $X$
referring to $D$ or $D^{*}$, and the grey bands are the fittings using Eq.
(3).

The correlation functions of $D$ and $D^{*}$ that move with spatial momentum
$\vec{p}$ are calculated precisely using the distillation method and are
parameterized as

$C_{X}(\vec{p},t)=W_{1}\cosh\left[-E_{X}(\vec{p})(\frac{T}{2}-t)\right]+W_{2}\cosh\left[-E^{\prime}_{X}(\vec{p})(\frac{T}{2}-t)\right],$
(1)

where $X$ refers to $D$ or $D^{*}$ and the second term is kept to account for
the higher state contamination. The modes $\vec{n}$ of the spatial momentum
$\vec{p}=\frac{2\pi}{La_{s}}\vec{n}$ involved in this work are
$\vec{n}=(0,0,0),(0,0,1),(0,1,1),(1,1,1),(0,0,2),(0,1,2)$. Figure 1 shows the
effective energies $E_{p}^{\mathrm{eff}}(t)$ of $D$ (left panel) and $D^{*}$
(middle panel) at different momenta $\vec{p}$, which are defined by

$E_{p}^{\mathrm{eff}}(t)=\cosh^{-1}\frac{C_{X}(\vec{p},t-1)+C_{X}(\vec{p},t+1)}{2C_{X}(\vec{p},t)},$
(2)

and the grey bands illustrate the fit results using Eq. (1) for the time
interval $t\in[20,T-20]$. The results of $E_{D}(\vec{p})$ and
$E_{D^{*}}(\vec{p})$ in the physical units are listed in Table 2 with
jackknife errors. It is seen that the hyperfine splitting $\Delta
m=E_{D^{*}}(\vec{0})-E_{D}(\vec{0})=139.70(57)$ MeV almost reproduces the
experimental values $m_{D^{*-}}-m_{D^{0}}=142.0(1)$ MeV and
$m_{D^{*+}}-m_{D^{+}}=140.6(1)$ MeV [57]. This manifests that our tuning of
charm quark mass and the scale setting scheme are reasonable. The momentum
dependence of $E_{D}(\vec{p})$ and $E_{D^{*}}(\vec{p})$ are plotted in Fig. 1
(right panel), where the shaded line is the fit results using the continuum
dispersion relation

$E_{X}^{2}(\vec{p})=m_{X}^{2}+\frac{1}{\xi^{2}}|\vec{p}|^{2}.$ (3)

The fitted $\xi$ is 5.329(12) for $D$ and 5.324(26) for $D^{*}$, both of which
are consistent with $\xi=5.3$ in Table 1.

Table 2: The energies of $D$ and $D^{*}$ at different spatial momentum modes $\vec{n}$. The energies are converted into the values in physical units with the lattice spacing $a_{t}^{-1}=6.894$ GeV.  $\vec{n}$ modes | $(0,0,0)$ | $(0,0,1)$ | $(0,1,1)$ | $(1,1,1)$ | $(0,0,2)$ | $(0,1,2)$
---|---|---|---|---|---|---
$E_{D}(\vec{p})$(GeV) | 1.88191(52) | 1.94969(63) | 2.01575(90) | 2.0793(12) | 2.1356(18) | 2.1938(33)
$E_{D^{*}}(\vec{p})$(GeV) | 2.02161(91) | 2.0841(15) | 2.1460(25) | 2.2072(30) | 2.2637(33) | 2.3107(72)

##  3 $DD^{*}$ scattering

In this work, we only focus on the $S$-wave $DD^{*}$ scattering in the isospin
$I=0$ and $I=1$ channels. The recent lattice study on $T_{cc}^{+}$ also found
that the contribution of $D$-wave scattering to the $J^{P}=1^{+}$ $DD^{*}$
system is small and can be neglected temporarily [47]. The operators for
$S$-wave $DD^{*}$ system with a relative $p=|\vec{p}|$ momentum can be built
through

$O_{DD^{*}}(p,t)=\frac{1}{N_{\vec{p}}}\sum\limits_{R\in
O}O_{D}(R\circ\vec{p},t)O_{D^{*}}(-R\circ\vec{p},t),$ (4)

where $O_{D}(\vec{p},t)$ and $O_{D^{*}}(\vec{p},t)$ are the momentum projected
single particle operators for $D$ and $D^{*}$, respectively, $R$ refers to the
rotational operations in the lattice spatial symmetry group $O$ (the
octahedral group). The operators $O_{DD^{*}}^{(I)}$ for a definite isospin $I$
is built according to the isospin combinations

$\displaystyle
I=0:~{}~{}|DD^{*}\rangle=\frac{1}{\sqrt{2}}\left(|D^{0}D^{*+}\rangle-|D^{+}D^{*0}\rangle\right)$
$\displaystyle
I=1:~{}~{}|DD^{*}\rangle=\frac{1}{\sqrt{2}}\left(|D^{0}D^{*+}\rangle+|D^{+}D^{*0}\rangle\right).$
(5)

As far as the $J^{P}=1^{+}$ $cc\bar{u}\bar{d}$ system is concerned, one may
include $O_{D^{*}D^{*}}$ operators with the same quantum numbers in the
calculation. We observed that the magnitudes of correlation functions between
$O_{DD^{*}}$ and $O_{D^{*}D^{*}}$ are very weak and the coupling effects can
be neglected (also observed in Ref. [47]). Therefore, to extract the energies
of $DD^{*}$ systems, we calculate the following correlation matrix in both
$I=1$ and $I=0$ channels in the framework of the distillation method,

$C^{(I)}(p,p^{\prime};t)=\frac{1}{T}\sum\limits_{\tau}\left\langle
O_{DD^{*}}^{(I)}(p,t+\tau)O_{DD^{*}}^{(I)}(p^{\prime},\tau)\right\rangle,$ (6)

where we average the source time slices $\tau$ to increase the statistics.
Then we solve the generalized eigenvalue problem (GEVP)
$C^{(I)}(p,p^{\prime};t)v_{p^{\prime}}^{(m)}(t,t_{0})=\lambda_{m}(t,t_{0})C^{(I)}(p,p^{\prime};t_{0})v_{p^{\prime}}^{(m)}(t,t_{0})$
to get the optimized operator
$O_{DD^{*}}^{(I)}(p_{m})=v_{p}^{(m)}(t,t_{0})O_{DD^{*}}^{(I)}(p)$ that couples
most to the $m$-th state of $DD^{*}$ system with energy
$E_{DD^{*}}^{(I)}(p_{m})$. Here $p_{m}$ is the scattering momentum of the
$m$-th state and is determined by $E_{DD^{*}}^{(I)}(p_{m})$ through the
relation

$E_{DD^{*}}^{(I)}(p_{m})=\sqrt{m_{D}^{2}+p_{m}^{2}}+\sqrt{m_{D^{*}}^{2}+p_{m}^{2}}.$
(7)

In practice, the lowest four momentum modes of $\vec{p}$ are involved in the
GEVP analysis, hence the momentum modes $\vec{n}=(0,0,0)$, $(0,0,1)$,
$(0,1,1)$, $(1,1,1)$ are replaced by $m=0,1,2,3$ to present the state of the
$m$-th optimized operator. It is known that, under the periodic temporal
boundary condition, in addition to the contribution from the physical states
that all the physical degrees of freedom propagate alongside in the same time
direction, the correlation function $C^{(I)}(p,p^{\prime};t)$ also has
contribution from the so-called thermal states or wrap-around states that the
$D$ and $D^{*}$ states propagate in opposite temporal directions [58].
Therefore, the correlation function of the optimized operator
$O_{DD^{*}}^{(I)}(p_{m})$ can be parameterized as

$\displaystyle
C^{(I)}(p_{m},t)=W_{1}^{(I)}\cosh\left(E^{(I)}_{DD^{*}}(p_{m})(t-\frac{T}{2})\right)+W_{2}^{(I)}\cosh\left(\left[E_{D}(p_{m})-E_{D^{*}}(p_{m})\right](t-\frac{T}{2})\right)+W^{\prime(I)}\cosh\left(E^{\prime}(t-\frac{T}{2})\right),$
(8)

where the first term comes from the desired physical state, the second term
accounts for the contribution of the thermal state, while the third term is
introduced to account for the residual contamination from higher states. It
turns out that this function form describes $C^{(I)}(p_{m},t)$ very well in a
wide time range as shown in left panels of Figure 2 and 3.

Lüscher’s formalism provides an approach to extract the hadron-hadron
scattering properties from the energy levels of a two-meson system in a finite
box [59, 48]. When the energies $E_{DD^{*}}^{(I)}(p_{m})$ is derived
precisely, we can obtain the value of the scattering momentum $p_{m}$ using
Eq. (7). Usually, one also introduces the dimensionless quantity
$q=\frac{p_{m}La_{s}}{2\pi}$ for convenience. According to Lüscher’s
formalism, the phase shift of $S$-wave scattering can be derived from $p$ (or
$q$) by

$p\cot\delta_{0}(q)=\frac{2}{La_{s}\sqrt{\pi}}\mathcal{Z}_{00}(1,q^{2})=\frac{1}{\pi
L}\sum\limits_{\vec{n}\in Z_{3}}^{|\vec{n}|<R}\frac{1}{\vec{n}^{2}-q^{2}}-4\pi
R~{}~{}~{}(R\to\infty).$ (9)

where $\mathcal{Z}_{lm}(s,q^{2})$ is the Lüscher zeta function [59] and the
second equality above is the lattice regularized version of
$\mathcal{Z}_{lm}(s,q^{2})$ [48]. For the low-energy scattering, the effective
range expansion (ERE) up to $\mathcal{O}(p^{2})$ gives

$p\cot\delta_{0}(p)=\frac{1}{a_{0}}+\frac{1}{2}r_{0}p^{2}+\mathcal{O}(p^{4})$
(10)

where $a_{0}$ and $r_{0}$ are the $S$-wave($l=0$) scattering length and
effective range respectively. In the following, we will discuss the $DD^{*}$
scatterings in the $I=1$ and $I=0$ channels in detail.

###  3.1 The $I=0$ and $J^{P}=1^{+}$ $DD^{*}$ scattering

Figure 2: Results of the $DD^{*}(I=0)$ scattering. Left panel: Data points are
the effective energies of $DD^{*}(I=0)$ system and the grey bands are the fits
by Eq. (8) in the time window $t\in[20,T-20]$. Middle panel: Effective energy
shift $\Delta E(p_{m},t)$ defined through the ratio function $R(p_{m},t)$,
where the colored bands are from the function forms of $R(p_{m},t)$ defined
through Eq. (1) and Eq. (6). Right panel: The phase shift of $S$-wave
$DD^{*}(I=0)$ scattering, where the grey band shows the result of Eq. (10)
with best fit parameters in Eq. (12) and the red band illustrate the fitting
range

Our practical data analysis is performed through the following procedure.
First, we divide the measurements into 139 bins with each bin including 50
measurements, which is tested to saturate the statistical errors. For these
data bins, we carry out the one-eliminating jackknife analysis to the
correlation functions $C_{X}(\vec{p}_{m},t)$ ($X$ refers to $D$ and $D^{*}$)
and $C^{(I)}(p_{m},t)$ for all the momentum modes using equations Eq. (1) and
Eq. (8), respectively. In this procedure, the energies $E_{D}(\vec{p}_{m})$,
$E_{D^{*}}(\vec{p}_{m})$, $E_{DD^{*}}^{(I)}(p_{m})$ for $m=0,1,2,3$ are
obtained simultaneously along with the energy shifts $\Delta
E^{(I)}(p_{m})=E_{DD^{*}}^{(I)}(p_{m})-E_{D}(\vec{p}_{m})-E_{D^{*}}(\vec{p}_{m})$
and the squared scattering momenta $p_{m}^{2}$. As shown in the left panel of
Fig.2 as colored bands, the function form Eq. (8) describes $C^{(I)}(p_{m},t)$
very well in the time range $t\in[20,T-20]$. The dip around $t=T/2=64$ also
manifests the existence of the thermal states. The final results in the $I=0$
channel are listed in Table 3, where the energies with jackknifed errors are
converted into physical units.

Table 3: The lattice results of the $S$-wave $DD^{*}$ scattering in $I=0$ channel. Four lowest energy levels $E_{DD^{*}}^{(I)}(p_{m})$ corresponding to the four momentum modes are obtained. The energy shifts $\Delta E$ and the scattering momenta $p_{m}$ are determined accordingly. The values are in physical units converted from $a_{t}^{-1}=6.894$ GeV. The measured aspect ratio $\xi=5.33(3)$ from the dispersion relation is used to derive the dimensionless $q^{2}$. All the errors here are jackknife ones.  $\vec{p}_{m}$ modes | $m=0$ | $m=1$ | $m=2$ | $m=3$
---|---|---|---|---
$E_{D}(\vec{p}_{m})+E_{D^{*}}(\vec{p}_{m})$ | 3.9035(14) | 4.0338(19) | 4.1617(29) | 4.2864(36)
$E_{DD^{*}}^{(I=0)}(p_{m})$(GeV) | 3.8977(14) | 4.0166(15) | 4.1369(18) | 4.2682(28)
$\Delta E$(GeV) | -0.00582(22) | -0.0172(12) | -0.0248(23) | -0.0183(32)
$p_{m}^{2}$(GeV2) | -0.01134(43) | 0.22362(92) | 0.4686(20) | 0.7442(49)
$q^{2}=(p_{m}La_{s}/2\pi)^{2}$ | -0.0440(17) | 0.867(10) | 1.816(22) | 2.884(38)

It is seen that the energy shifts $\Delta
E^{(I=0)}=E_{DD^{*}}^{(I=0)}(p)-E_{D}(p)-E_{D^{*}}(p)$ are uniformly negative
for all the four momentum modes. This indicates the interaction between $D$
and $D^{*}$ in the $I=0$ channel is attractive. The energy shift $\Delta E(p)$
is also checked through the ratio function

$R(p_{m},t)\equiv\frac{C_{DD^{*}}^{(I=0)}(p_{m},t)}{C_{D}(\vec{p}_{m},t)C_{D^{*}}(\vec{p}_{m},t)}\sim
e^{-\Delta E(p_{m})t}~{}~{}(t\gg 1).$ (11)

This ratio function is used sometimes to estimate $\Delta E(p_{m})$ from the
plateau of $\Delta E(p_{m},t)\equiv\ln\frac{R(p_{m},t)}{R(p_{m},t+1)}$. The
middle panel of Fig. 2 shows $\Delta E(p_{m},t)$ for momentum modes
$m=0,1,2,3$ in $I=0$ channel, where the data points are the values from the
measured correlation functions involved in Eq. (11), and the colored bands
illustrate the results through the function forms in Eq. (1) and (8) with the
fitted parameters in Table 3 and the parameters for terms of excited states.
Obviously, $\Delta E(p_{m},t)$ does not show a plateau at all, but can be well
described by the function mentioned above. This manifests that the energy
shifts $\Delta E(p_{m})$ listed in Table 3 are derived correctly. Note that
the terms for excited states in Eq. (1) and (6) are necessary to describe the
data.

The scattering phase shift $p\cot\delta_{0}(q^{2})$ is obtained by using Eq.
(9) at each $q^{2}$ and is plotted as data points in the right panel of Fig.
2, where dashed lines illustrate the function form in Eq. (9). The fit to the
four data points of lower $q^{2}$ using Eq. (10) gives

$a_{0}^{(I=0)}=0.538(33)~{}\mathrm{fm},~{}~{}~{}r_{0}^{(I=0)}=0.99(11)~{}\mathrm{fm}.$
(12)

Our results are in line with $a_{0}\sim 1$ fm and $r_{0}\sim 1.0$ fm
determined in Ref [47] at a lighter pion mass $m_{\pi}=280$ MeV. Both results
indicate the attractive interaction of $DD^{*}$ in the $I=0$ channel. Since we
have only one lattice volume, we cannot make a proper discussion on the
existence of a bound state yet.

Table 4: The lattice results of the $S$-wave $DD^{*}$ scattering in $I=1$ channel (similar to Table 3).  $\vec{p}_{m}$ modes | $m=0$ | $m=1$ | $m=2$ | $m=3$
---|---|---|---|---
$E_{D}(\vec{p}_{m})+E_{D^{*}}(\vec{p}_{m})$(GeV) | 3.9035(14) | 4.0338(19) | 4.1617(29) | 4.2864(36)
$E_{DD^{*}}^{(I=1)}(p_{m})$(GeV) | 3.9120(13) | 4.0405(14) | 4.1628(16) | 4.2836(22)
$\Delta E^{(I=1)}$(GeV) | 0.00851(23) | 0.0067(12) | 0.0011(23) | -0.0028(33)
$p_{m}$(GeV) | 0.1289(17) | 0.52131(73) | 0.7226(11) | 0.8815(18)
$q^{2}=(p_{m}La_{s}/2\pi)^{2}$ | 0.0644(19) | 1.053(12) | 2.024(24) | 3.012(36)

###  3.2 The $I=1$ and $J^{P}=1^{+}$ $DD^{*}$ scattering

Figure 3: Results of the $DD^{*}(I=I)$ scattering. The three panels are
similar to those of Fig. 2.

The data analysis of the $I=1$ $DD^{*}$ scattering takes the same procedure as
that for the $I=0$ case. The results of $E_{DD^{*}}^{(I=1)}(p_{m})$ are listed
in Table 4 along with the values of corresponding energy shift $\Delta
E^{(I=1)}$, the scattering momentum $p_{m}$ etc.. The major results of $I=1$
$DD^{*}$ scattering are illustrated in Fig.3 similar to Fig. 2 for the $I=0$
case: The left panel shows the the effective energies of $C^{(I=1)}(p,t)$ and
the related fits using Eq. (8). The middle panel shows the check of the energy
shift $\Delta E^{(I=1)}$ for different momentum $\vec{p}_{m}$. The right panel
is for the $S$-wave phase shift of the $DD^{*}(I=1)$ scattering, which is
obtained from the scattering momentum $p_{m}$. It is seen that
$E_{DD^{*}}^{(I=1)}(p)$ is higher than $E_{D}(\vec{p})+E_{D^{*}}(\vec{p})$
when it is not far from the $DD^{*}$ threshold (the lowest two energy levels
of $E_{DD^{*}}^{(I=1)}(p)$). This reflects a repulsive interaction for the
low-energy $D$ and $D^{*}$ scattering in the $I=1$ channel. When the
scattering momentum $p$ is larger, the energy shift gets smaller and smaller,
and is finally consistent with zero within the error. This is in striking
contrast to the case of $I=0$ where the energy shift is uniformly negative in
a large range of the scattering momentum. Accordingly, the corresponding
$q^{2}$ for the two higher energies are consistent with integers, such that
when the phase shift is determined through Eq. (9), its error blows up, as
shown in the right panel of Fig. 3. The fit to these phase shifts using Eq.
(10) gives the scattering length and the effective range as

$a_{0}^{(I=1)}=-0.433(43)~{}\mathrm{fm},~{}~{}~{}r_{0}^{(I=1)}=-3.6(1.0)~{}\mathrm{fm}.$
(13)

##  4 Discussion

Figure 4: The components of the correlation function $C^{(I)}(p,t)$. Left
panel: The schematic quark diagrams of the four terms $D$, $C_{1}(\pi/\rho)$,
$D^{\prime}$ and $C_{2}(\rho)$ that contribute to $C^{(I)}(p,t)$. Right panel:
The relative magnitudes of the four terms for the case of $\vec{p}=0$, which
are scaled by the $C_{D}(\vec{p}=0,t)C_{D^{*}}(\vec{p}=0,t)$.

In the previous section, we present the numerical results of the $S$-wave
$DD^{*}$ scattering in the $I=0$ and $I=1$ channels. The major observation is
that $DD^{*}$ interaction is attractive for $I=0$ in a wide momentum range and
repulsive for $I=1$ when the energy of $DD^{*}$ is near the $DD^{*}$ mass
threshold. This is conceptually in agreement with the observation of LHCb [5]
that the $T_{cc}^{+}$ state is found only in the $D^{0}D^{*+}$ system.

In order to understand the isospin-dependent interaction of $DD^{*}$, let us
take a closer look at the quark diagrams (after the Wick contraction) that
contribute to the correlation functions $C^{(I)}(p,t)$. There are four
distinct terms whose schematic quark diagrams are shown in the left part of
Fig. 4: The diagram on the upper left side is called $D$ (direct) term which
comes from the direct contractions between $O_{D}(O_{D^{*}})$ in the sink and
source operators. The diagram on the upper right side is called the
$C_{1}(\pi/\rho)$ (crossing) term which involves either the $u,d$ quark
exchange effects (as illustrated in the figure) or charm quark exchange (if
flipping upside down the positions of $D^{0}$ and $D_{j}^{*+}$ on the right-
hand side). In the lower-left diagram, $D^{\prime}$ is the direct contraction
between $D$ and $D^{*}$. The lower right diagram $C_{2}(\rho)$ is also a $u,d$
quark exchange one. As such $C^{(I)}(p,t)$ can be abbreviated as

$C^{(I)}(p,t)=D-C_{1}(\pi/\rho)+(-)^{I+1}\left(D^{\prime}-C_{2}(\rho)\right),$
(14)

where the minus signs of $C$ terms come from the single quark loops after Wick
contraction.

The contribution of these terms to $C^{(I)}(p,t)$ are checked to have the
hierarchy $D\gg C_{2}(\rho)\gtrsim C_{1}(\pi/\rho)\gg D^{\prime}$ with each
level being smaller by roughly two orders of magnitude, as shown in the right
panel of Fig. 4, in which the magnitudes of $D$, $C_{1}(\pi/\rho)$,
$C_{2}(\rho)$ and $D^{\prime}$ at $\vec{p}=0$ are scaled by the product of
single meson correlation functions $C_{D}(\vec{p}=0,t)$ and
$C_{D^{*}}(\vec{p}=0,t)$. The contribution of $D^{\prime}$ term is very small
and negligible in the following discussion. The $C_{1}(\pi/\rho)$ term
contributes equally to $C^{(I=1)}(p,t)$ and $C^{(I=0)}(p,t)$, while the
contributions of $C_{2}(\rho)$ have opposite signs for $I=0$ and $I=1$ and are
necessarily responsible for the energy difference of $E_{DD^{*}}^{(I=0)}(p)$
and $E_{DD^{*}}^{(I=1)}(p)$. This is understood as follows. As shown in the
right panel of Fig. 4, the linear behaviors of
$C_{1}(\pi/\rho)/(C_{D}C_{D^{*}})$ and $C_{2}(\rho)/(C_{D}C_{D^{*}})$ with a
positive slope in the intermediate time range imply that, if the time
dependence of $D$ term is approximately $A_{0}e^{-E_{0}t}$, then the time
dependence of $C_{1}(\pi/\rho)$ and $C_{2}(\rho)$ is also approximately
exponential, and can be expressed qualitatively as
$A_{0}\epsilon_{i}e^{-E_{i}t}$ with $i=1,2$ referring to $C_{1}(\pi/\rho)$ and
$C_{2}(\rho)$, respectively. Accordingly, the curves in the figure indicate
$\epsilon_{i}\sim\mathcal{O}(10^{-2})\ll 1$ (and numerically positive) and
$E_{i}-E_{0}=-\delta E_{i}<0$. In the mean time, the curve for $C_{2}(\rho)$
is uniformly higher than that for $C_{1}(\pi/\rho)$ and thereby implies
$\epsilon_{1}<\epsilon_{2}$. Thus one has

$E_{DD^{*}}^{(I)}\approx\ln\frac{C^{(I)}(p,t)}{C^{(I)}(p,t+1)}\approx
E_{0}+\epsilon_{1}\delta E_{1}e^{\delta E_{1}t}+(-)^{I+1}\epsilon_{2}\delta
E_{2}e^{\delta E_{2}t}$ (15)

in the time range $t\in[20,50]$ where $\delta E_{i}t\ll 1$.

The physical meaning can be clearly understood in Eq. (15). As mentioned
previously, $D$ term is very close to the product of the single-particle
correlation functions $C_{D}(p,t)$ and $C_{D^{*}}(p,t)$, such that $E_{0}$ can
be taken as an approximation to $E_{D}(p)+E_{D^{*}}(p)$. The second term on
the right-hand side of Eq. (15) comes from the $C_{1}(\pi/\rho)$ contribution
and is positive for both $I=0,1$ channels. This means the $C_{1}(\pi/\rho)$
term reflects a repulsive interaction. In contrast, the third term, which is
contributed from $C_{2}(\rho)$, is positive for $I=1$ and negative for $I=0$
and manifests a repulsive interaction for $I=1$ and an attractive interaction
for $I=0$. On the other hand, since $\epsilon_{2}$ and $\delta E_{2}$ are
larger than $\epsilon_{1}$ and $\delta E_{1}$, respectively, such that
$\epsilon_{1}\delta E_{1}e^{\delta E_{1}t}<\epsilon_{2}\delta E_{2}e^{\delta
E_{2}t}$. In other words, the combined effects of the $C_{1}(\pi/\rho)$ and
$C_{2}(\rho)$ contribution result in a negative energy shift from the non-
interacting $DD^{*}$ energy $E_{D}(p)+E_{D^{*}}(p)$, which reflects the
totally attractive interaction between $D$ and $D^{*}$ in the $S=0$ and $I=0$
channel.

On the hadron level, the four terms depicted in Fig. 4 can be interpreted as
follows:

  * 1.

$D$ term: It involves two separately closed quark diagrams, each of which is
the propagator of $D(D^{*})$ meson. After the gauge averaging, the two parts
can have an interaction mediated by at least two gluons that are necessarily
in a color singlet. Intuitively, the light quark lines can zigzag, such that
the crossing of the zigzag $u,d$ quark lines can induce complicated meson
exchange effects, such as $\sigma$, $\omega$, etc., on the hadron level.
Either gluon exchanges on the quark level or meson exchanges on the hadron
level, the resultant effects are very tiny since the contribution of this term
is very close in magnitude (after the subtraction of the contribution from the
wrap-around states) to the product of the correlation functions of single $D$
and $D^{*}$ mesons.

  * 2.

$D^{\prime}$ term: This also involves two closed quark diagrams, however, each
one connects two different mesons $D$ and $D^{*}$. This diagram contributes to
$C_{DD^{*}}(p,t)$ only when color singlet gluon exchanges (at least two gluons
also) take place between the two parts after the gauge average. On the hadron
level, the interaction can be mediated by $\eta,\omega$, etc. However,
empirically in our study, it is found these effects are very weak, and the
contribution from the $D^{\prime}$ term is negligible in comparison with the
other terms.

  * 3.

$C_{1}(\pi/\rho)$ term: As shown in the right upper part of Fig. 4, there are
explicit $u,d$ quark exchanges between $D$ and $D^{*}$ during their temporal
propagation. This exchange effect can be viewed as that of the charged meson
($\pi^{\pm}$,$\rho^{\pm}$, etc.) on the hadron level. If we flip the positions
of $D^{0}$ and $D^{*+}$ on the right-hand side, the figure implies a
$c\bar{c}$ exchange process, and accordingly charmonium $V_{c}$ ($J/\psi$,
$\psi^{\prime}$, etc.) exchange process on the hadron level. Since
$C_{1}(\pi/\rho)$ contributes equally to $C_{DD^{*}}^{(I=0)}(p,t)$ and
$C_{DD^{*}}^{(I=1)}(p,t)$, according to our discussion above, these
intermediate meson exchanges on the hadron level result in a repulsive
interaction to the $DD^{*}$ system. Note that vector meson exchange models
[43, 44] also obtain a repulsive interaction for the $J/\psi$ exchange.

  * 4.

$C_{2}(\rho)$: This term also comes from the $u,d$ quark exchanges. On the
hadron level, since the $\mathcal{P}$-parity conservation prohibits the
$DD\pi$ interaction, the effect of light quark exchange can be reflected
mainly by the charged $\rho$ exchange, which provides an attractive
interaction for the $S$-wave $I=0$ $DD^{*}$ system and a repulsive interaction
for the $S$-wave $I=1$ $DD^{*}$ system. Furthermore, the observation
$E_{DD^{*}}^{(I=0)}(p)<E_{D}(p)+E_{D^{*}}(p)$ indicates that this attractive
$\rho$-exchange effect overcomes the repulsive interaction reflected by
$C_{1}(\pi/\rho)$ term and results in a total attraction interaction. This
result is in qualitative agreement with those in Refs. [42, 43, 44].

##  5 Summary

The $S$-wave $DD^{*}$ scattering are investigated from $N_{f}=2$ lattice QCD
calculations on a lattice with $m_{\pi}\approx 350$ MeV and
$m_{\pi}La_{s}\approx 3.9$. Benefited from the large statistics, several
lowest energy levels of the $DD^{*}$s of isospin $I=0$ and $I=1$ are
determined precisely through the distillation method and by solving the
relevant generalized eigenvalue problems. In the $I=1$ case, the $DD^{*}$
energy $E_{DD^{*}}^{(I=1)}(p)$ is higher than the corresponding non-
interacting $DD^{*}$ energy $E_{D}(\vec{p})+E_{D^{*}}(\vec{p})$ when closing
to the $DD^{*}$ threshold, and manifests a repulsive interaction between $D$
and $D^{*}$. But when the scattering momentum $p$ is large, the difference of
$E_{DD^{*}}^{(I=1)}(p)$ and $E_{D}(\vec{p})+E_{D^{*}}(\vec{p})$ becomes
smaller and even indiscernible. In the $I=0$ case, the $DD^{*}$ energy
$E_{DD^{*}}^{(I=0)}(p)$ is uniformly lower than
$E_{D}(\vec{p})+E_{D^{*}}(\vec{p})$ when $p$ goes up to around $800$ MeV, and
reflects definitely an attractive interaction between $D$ and $D^{*}$ in the
$I=0$ state. This is consistent with the experimental assignment $I=0$ for
$T_{cc}^{+}(3875)$ given a $DD^{*}$ bound state. Based on these energy levels,
the $S$-wave phase shifts of $DD^{*}$ scattering in $I=0,1$ channels are
derived using Lüscher’s finite volume formalism. The effective range
expansions give the following scattering lengths $a_{0}^{(I=0,1)}$ and the
effective ranges $r_{0}^{(I=0,1)}$

$\displaystyle a_{0}^{(I=0)}$ $\displaystyle=$
$\displaystyle~{}0.538(33)~{}\mathrm{fm},~{}~{}~{}r_{0}^{(I=0)}=0.99(11)~{}\mathrm{fm},$
$\displaystyle a_{0}^{(I=1)}$ $\displaystyle=$
$\displaystyle-0.433(43)~{}\mathrm{fm},~{}~{}~{}r_{0}^{(I=1)}=-3.6(1.0)~{}\mathrm{fm}.$
(16)

To understand the isospin dependence of the $DD^{*}$ interaction, further
analysis is performed on the components of $DD^{*}$ correlation functions. It
is found that the difference of the $I=0$ and $I=1$ $DD^{*}$ correlation
functions comes mainly from the $C_{2}(\rho)$ term that $D$ and $D^{*}$
exchange $u,d$ quarks when propagating in time. This term can be viewed as the
charged vector $\rho$ meson exchange in the hadron level and contributes to
the $I=0$ and $I=1$ $DD^{*}$ correlation functions with opposite signs. As a
result, it raises the $DD^{*}$ energy in the $I=1$ channel, and pulls it down
in the $I=0$ channel. This provides a shred of strong evidence that the
$DD^{*}$ interaction induced by the charged $\rho$ meson exchange may play a
crucial role in the formation of $T_{cc}^{+}(3875)$. This is in qualitative
agreement with the results of recent phenomenological studies [42, 43, 44].

## Acknowledgements

We thank Prof. Q. Zhao of IHEP for valuable discussions. This work is
supported by the Strategic Priority Research Program of Chinese Academy of
Sciences (No. XDB34030302), the National Key Research and Development Program
of China (No. 2020YFA0406400) and the National Natural Science Foundation of
China (NNSFC) under Grants No.11935017, No.11775229, No.12075253,
No.12070131001 (CRC 110 by DFG and NNSFC), No.12175063. The Chroma software
system [60] and QUDA library [61, 62] are acknowledged. The computations were
performed on the HPC clusters at the Institute of High Energy Physics
(Beijing) and China Spallation Neutron Source (Dongguan), and the CAS
Sunrise-1 computing environment.

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# Effect of nonmagnetic Ti substitution on the structural, magnetic and
transport properties in pyrochlore iridate Eu2(Ir1-xTix)2O7

Sampad<EMAIL_ADDRESS>B. Majid, M. Modakb, Swapan K.
<EMAIL_ADDRESS>and S.
<EMAIL_ADDRESS>a Department of Physics, Visva-
Bharati, Santiniketan 731235, India
b Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India
cRamsaday College, Amta, Howrah 711401, India
d Acharya Jagadish Chandra Bose College, 1/1B, A. J. C. Bose Road, Kolkata
700020, India

###### Abstract

We have studied the effect of nonmagnetic Ti substitution Eu2(Ir1-xTix)2O7
with the help of electrical transport and magnetic measurement. The minor
structural modification enhances the orbital overlapping and favours its
electrical transport properties with Ti doping though the tuning of SOC and U
with site dilution opposes it. As a result, metal insulator transition (MIT)
is disappeared and resistivity of the system throughout the temperature
increases with Ti doping. The nature of the conduction mechanism at low
temperature follows power law like variation. As the Ti4+ is nonmagnetic, the
introduction of Ti at Ir site dilutes the magnetic interaction at Ir
octahedral network, which in turn decreases the magnetic moment and magnetic
frustration in the system though the magnetic irreversibility temperature is
hardly affected by Ti.

Introduction

Recently, in condensed matter physics 5d transition metal mainly iridium oxide
have created much attention due to their exotic magnetic and electronic
properties originating from interplay between spin orbit coupling (SOC) and
electronic correlation (U)Rau ; Cao ; Wan . Pyrochlores offer a perfect
illustration for exploring the interplay of SOC and U in 5d transition metal
oxide compounds due to their interpenetrating corner sharing tetrahedral
structure tend to form a narrow flat band at Fermi level that enhance the
effect of SOC and electron correlation energy, both the energy scale are
comparable to the band width of the transition metal. Within the realm of 5d
transition metal based pyrochlore compounds, particular attention is drawn to
pyrochlore iridate R2Ir2O7 (where R represents a rare earth element) owing to
its captivating transport and magnetic characteristics, including phenomena
such as spin liquid, spin glass, Weyl semimetal behavior, and metal insulator
transitions.Zhang ; Yanagishima ; Matsuhira ; Matsuhira1 ; Nakatsuji  In these
compounds system goes from a metallic state for R= Pr to insulating state for
R=Gd-Lu depending on the ionic radius of the rare earth atom and in the
intermediate state shows a metal insulator transition for R=Nd-Sm.Matsuhira ;
Matsuhira1  As it is believed that there exist a topological phase near the
MIT phase boundary, the family of pyrochlore iridate compound with R= Nd, Eu
and Sm are the promising candidate to give exotic topological phase.Wan

To analyse the physical property mostly for Ir lattice among the pyrochlore
iridates our special interest on Eu2Ir2O7 (EIO) as we can avoid f-d exchange
interactionChen  for nonmagnetic Eu. So this is the most preferable compound
to study the role of SOC and electronic correlation associated with the
system. The analysis of resistivity data reveals that EIO undergoes from
paramagnetic metal to Weyl semimetal (in an intermediate temperature range)
and, finally, to an antiferromagnetic insulator at the lowest
temperature.Tafti  Besides Terahertz optical conductivity measurement confirm
the Weyl semimetal state in this compound.Sushkov  The ground state of the
pyrochlore iridate system can be tuned by band filling of Ir 5d state via
chemical substitution. Previous study have shown that different band filling
in EIO compound gives various interesting magnetic and transport properties.
The doping of Sr at Eu site in EIO gives non-Fermi liquid behavior above
TMI.Banerjee  Recently the Weyl semimetal like behavior is observed in the
polycrystalline EIO compound by doping Bi at Eu site by Telang et al.Telang
The doping of Ca at Eu site in EIO gives doping induced metallic phase,Kaneko
although the doping of Cu at Ir site in EIO does not influence the MIT.Mondal

Our current study is focused on exploring the interplay between SOC and U
within the pyrochlore iridate system and we have examined the impact of
substituting Ti4+ ions at the Ir4+ site. The minor structural change with
doping enhances the transport properties but the tuning of SOC and U with site
dilution, leads to contrary outcome, resulting in an overall increase in
resistivity. The substitution of Ti4+ (3d0) at Ir4+ (5d5) site not only
changes the SOC and U of the compound but also dilutes the magnetic structure
of the system.

Experimental details All the polycrystalline samples Eu2(Ir1-xTix)2O7, where
$x$=0, 0.05, 0.1, 0.15 were prepared by solid state reaction method. High
purity ingredient powder Eu2O3, IrO2 and TiO2 were mixed in stoichiometric
ratio and grounded well. After pressing the mixture powder in pellet form,
heated at 1273 K for 3 days with several intermediate grinding. All the
samples were characterized by powder X-ray diffraction (XRD). The room
temperature XRD measurement was taken by X-ray diffractometer with Cu Kα
radiation. Structural parameters were determined using standard Rietveld
refinement technique with Fullprof software package. XPS measurements at room
temperature were carried out by using an Omicron Multiprobe Electron
Microscopy System equipped with a monochromatic Al Kα X-ray source ($h\nu$ =
1486.7 eV). All the XPS spectra have been analyzed by using PeakFit software,
where Shirley method was used for background subtraction. Magnetic measurement
was taken by Superconducting Quantum Interference Device Magnetometer (SQUID-
VSM) of Quantum Design in the temperature range 3 K - 300 K. Electrical and
magnetic transport measurement were carried out by four probe method with
temperature range 2 K-300 K using Physical Properties Measurement System
(PPMS).

Result

Figure 1: (a) Room temperature XRD patterns of the compounds Eu2(Ir1-xTix)2O7,
where $x=0,0.05,0.1,0.15$ and red and black stars indicate the non reacting
IrO2 and Eu2O3 oxides respectively. (b) Rieteveld of the $x$=0.1 compound,
where scattered data are observed data and red line is fit to the data. Figure
2: (a) Variation of lattice parameter a (b) Positional parameter of O1 atom
(X) (c) Ir-O-Ir bond angle and (d) Ir-O bond length with Ti concentration
($x$).

Room temperature XRD patterns for all the compounds Eu2(Ir1-xTix)2O7, where
$x$=0, 0.05, 0.1, 0.15 are shown in fig. 1(a). There is no significant change
in the XRD pattern after doping, indicates that crystallographic structure and
symmetry are retained with Ti doping. All the observed data are refined on the
basis of cubic structure with space group Fd-3m. Fig. 1(b) shows the Rietveld
refinement of the compound $x$=0.1. All the compounds are nearly in pure phase
with the existence of some impurity phases. These phases occur due to
nonreacting oxides IrO2, Eu2O3, which are marked by red and black star
respectively. The value of goodness of fit (defined as R${}_{wp}/R_{exp}$) is
found to be around 1.20, 1.06, 1.0, 1.54 for $x$=0, 0.05, 0.10 and 0.15
respectively, which indicates good fitting. Fig. 2(a) shows lattice parameter
’a’ decreases with Ti doping systematically due to substitution of smaller
ionic radius Ti4+ (0.605 $\AA$) compare to Ir4+ (0.625$\AA$). In the
pyrochlore structure two sets of oxygen atom present. Therefore the compound
can be written as Eu2Ir2(O16)(O2). The crystallographic position of the four
nonequivalent atoms are Eu at the 16d site ($\frac{1}{2}$, $\frac{1}{2}$,
$\frac{1}{2}$), Ir at the 16c site (0, 0, 0), O1 at the 48f site (X,
$\frac{1}{8}$, $\frac{1}{8}$) and O2 at the 8b site ($\frac{3}{8}$,
$\frac{3}{8}$, $\frac{3}{8}$). Ir cation are coordinated with six O1 atoms and
form corner shared IrO6 octahedra, where each O1 atom is equal distance from
Ir atom. Though position of the O1 atoms delivers distortion in the IrO6
octahedra. For a perfect and undistorted octahedra, the value of the X is
0.3125 and Ir ions reside under the cubic field.Gardner  Deviation from the
ideal value of X generates a distortion in the octahedra, which lower the
cubic symmetry in the crystal structure. The value of X for parent compound is
0.352, which is larger than the ideal vale, suggesting a compressed and
distorted IrO6 octahedra. Fig. 2(b) shows the value of the X is decreasing
with doping, suggesting the system headed toward perfect octahedra and gives
rise to reduced crystal field. The bond angle Ir-O-Ir and bond length Ir-O
related to IrO6 octahedra have an impact role in the orbital overlapping and
charge transfer mechanism, which contribute to the electronic properties of
the system. Fig. 2(c-d) shows the doping dependents bond angle Ir-O-Ir and
bond length Ir-O. Here bond angle is increasing with doping. When bond angle
stretching toward 180∘, maximum electrons hopping between two Ir atom via O
atom. Thus increasing of bond angle Ir-O-Ir delivers a increase of electron
hopping. The decrease of bond length indicates the enhance of overlapping
between Ir(5d) and O(2p) orbitals, which promotes larger hoping of Ir
electrons. So increases of bond angle and decreases of bond length, both the
structural modification enhance the orbital overlapping and favors its
electrical transport properties.

Figure 3: (a-b) show Ir 4f core level X-ray photoelectrons spectra for $x$=0
and 0.1.

In this system magnetic and transport properties are dominated by transition
metal Ir. Therefore, it is important to know the charge state of the
transition metal. We have performed X-ray photoelectrons spectroscopy (XPS)
measurement for the samples where fig. 3(a,b) depict the Ir 4f core level
spectra for $x$=0, 0.1 respectively. Ir 4f core level spectra is split into
4f7/2 and 4f5/2 electronic states around the binding energy 62 and 65 eV
respectively (as shown by dashed blue line) due to spin orbit coupling (SOC).
All the spectra can be fitted by taking the peaks due to Ir4+ with some weak
additional peaks arising at the higher binding energy 64 and 67.7 eV (as shown
by pink dotted line) which are correspond to 4f7/2 and 4f5/2 states of Ir5+
ion. We observe the existence of the Ir5+ peak for both the compounds. The
calculation of area under the curve for XPS spectrum reveals that for $x$=0,
0.1 compounds, the amount of Ir4+ and Ir5+ are 95.9% and 4.1%, 96% and 4%
respectively. Both the compounds contain near about equally amount of Ir5+
state and majority of Ir is in Ir4+ charge state. The small amount of Ir5+ in
both the compounds could be due to unavoidable nonstoichiometry of the
sample.Ishikawa ; Zhu

Figure 4: Temperature dependent resistivity of the compounds Eu2(Ir1-xTix)2O7,
where $x$=0, 0.05, 0.1, 0.15. Inset: Temperature derivative of the resistivity
as a function of temperature.

Fig. 4 demonstrates the temperature dependent electrical transport phenomena
for all the samples. Parent compound shows metal insulator transition near
about 120 K, below which the temperature dependent resistivity ($\rho$(T))
sharply increases with decrease in temperature. Such type of resistivity
behaviour is previously reported.Ishikawa  The temperature derivative
resistivity (d$\rho$/dT) changes sign from positive to negative upon cooling
from room temperature for $x$=0 but does not change sign with Ti doping, shown
in inset fig. 4. This indicates MIT is disappearing with Ti and tends the
system into insulating state. $\rho$(T) for all the compounds increases
systematically with Ti and at low temperature for the highest doped compound
($x$=0.15) increases four order of magnitude. This behaviour was also reported
for a Ti doped layered Sr2Ir1-xTixO4, where $\rho$(T) increases systematically
with Ti and end member Sr2TiO4 shows highly insulating nature.Gatimu ; Ge

Figure 5: ln-ln plot of the $\rho$(T) data for all the compounds where red line is fit to the data. Table 1: Fitting parameters $\alpha$ obtained from Power law fitting for the samples Eu2(Ir1-xTix)2O7 system. Sample |  Temperature range (K) |  $\alpha$
---|---|---
$x$=0.0 | 3-23 | 0.53
$x$=0.05 | 3-55 | 0.86
$x$=0.10 | 3-55 | 1.32
$x$=0.15 | 3-20 | 2.64

To further investigate in more details, we have tried to fit the temperature
dependent resistivity data at low temperature region with power law

$\rho=\rho_{0}T^{-\alpha}$ (1)

where $\alpha$ is the exponent, shown in fig. 5. The fitting temperature
region and $\alpha$ value are reported in Table-1. The parent compound fit
within the temperature region 3-20 K and the value is 0.53, which was reported
by Telang et al.Telang  For the doped samples range is 3-50 K except $x$=0.15
and $\alpha$ increases with Ti doping systematically. In a 3D disordered
system Mott variable range hoping can be expected.

$\rho=\rho_{0}\hskip 5.69046ptexp((T_{0}/T)^{1/4})$ (2)

where, T0 = $\frac{21.2}{N(E_{F})\xi^{3}}$, $N(E_{F})$ and $\xi$ are density
of states at Fermi level and localization length. Ishikawa et al. described
the transport behaviour by using VRH model for single crystal sample.Ishikawa
We have also tried to fit the resistivity data with VRH model. But data fits
only a limited temperature region (3-10 K). Polycrystalline sample have
extrinsic scattering due to grain boundaries, which plays an important role in
transport mechanism. Attempt to fit the data with Arhenious plot

$\rho=\rho_{0}\hskip 5.69046ptexp((E_{g}/KT))$ (3)

also fails, does not give the satisfactory result.

Figure 6: Temperature dependent magnetization measured at 1 KOe in ZFC FC
protocol for Eu2(Ir1-xTix)2O7, where $x$=0, 0.05, 0.1, 0.15.

Fig. 6 represent temperature dependent magnetization (M) for all the sample
Eu2(Ir1-xTix)2O7, where $x$=0, 0.05, 0.1, 0.15 under zero field cooled (ZFC)
field cooled (FC) protocol with applied field 1 KOe. $x$=0 compound shows a
magnetic irreversibility around Tirr = 120 K, below which there is a
bifurcation between ZFC and FC magnetization. The nature of the magnetic state
for Eu2Ir2O7 compound was well described in literature.Ishikawa  In this
compound below Tirr, Ir moments order in an antiferromagnetic all-in/all-out
magnetic order, revealed by Resonant x-ray diffraction (RXD) and muon spin
rotation and relaxation ($\mu$SR) measurements.Zhao ; Sagayama  All the
compounds show magnetic irreversibility and irreversibility temperature Tirr
for $x$=0, 0.05, 0.1, 0.15 compounds are 120, 118, 117, 115 K respectively.
Tirr shifting towards lower temperature systematically with Ti doping and
below Tirr the difference between ZFC-FC magnetization decreases with doping.
We also observe that ZFC-FC curve shifting gradually towards the lower
magnetization value with increasing of Ti concentration due to dilution of the
magnetic lattice by nonmagnetic Ti4+. The change of Tirr with doping is not
substantial where Tirr decreases only by $\approx$ 5 K up to 15% Ti doping.
Such type of magnetic behavior was also observed for Cu and Ru doping at Ir
site in Eu2Ir2O7.Mondal ; Wu  However, the hole doping at rare earth site in
EIO significantly changes the irreversibility temperature with doping.Banerjee
; Telang  Thus in these system the tuning of SOC and U by 3d/4d transition
metal can play a role to minor change in irreversibility temperature. Besides,
thermomagnetic irreversibility in a system depends on magnetic interaction,
disorder and magnetic anisotropy. To check whether such effects are
responsible for the minor change in irreversibility temperature further
investigation is needed.

Figure 7: Temperature dependent inverse susceptibility with Curie fit for
Eu2(Ir1-xTix)2O7, where $x$=0, 0.05, 0.1, 0.15

Temperature dependent inverse susceptibility in the entire temperature range
for all the compounds are shown in fig. 7. In order to study the magnetic
interaction, we have fitted the data in the temperature region 175-300 K with
the Curie Weiss law $\chi=\frac{C}{T-\theta{p}}$, $\theta{p}$ is Curie Weiss
temperature, and Curie constant $C=\frac{N_{A}\mu_{eff}^{2}}{3K_{B}}$, and
$\mu_{eff}$ is the effective paramagnetic moment. All the fitted parameters
$\mu_{eff}$, $\theta_{p}$ and frustration parameter defined as
$f=|\theta_{p}|/T_{irr}$ are shown in table-2. We observe negative
$\theta_{p}$ for all the compounds, indicates the nature of the magnetic
interaction in these compounds is antiferromagnetic, where strength of the
interaction decreases with the introduction of Ti. Pyrochlore structure is
inherently frustrated due to their geometrical structure, promotes the exotic
ground states like spin glass,Gingras  spin liquid Nakatsuji ; Gardner1  etc.
The reported $f$ value for a highly frustrated pyrochlore oxide Y2Mo2O7 is
8.8, gives a spin glass like behaviour.Gardner2  But in our system for the
parent compound $f$ value is lower than the highly frustrated system, which
decreases with Ti. The effective paramagnetic moment $\mu_{eff}$ decreases
with Ti due to incorporation of nonmagnetic Ti4+ (3d0) at Ir4+ (5d5) site.

Table 2: $\mu_{eff}$, $\theta_{p}$ and $f$ for Eu2(Ir1-xTix)2O7 system. Sample |  $\mu_{eff}$ ($\mu_{B}$/f.u.) |  $\theta_{p}$ (K) |  $f$
---|---|---|---
x=0.0 | 8.33 | -306 | 2.55
x=0.05 | 7.45 | -270 | 2.29
x=0.10 | 7.25 | -260 | 2.22
x=0.15 | 7.02 | -257 | 2.23
Figure 8: Isothermal magnetization at 3 K for Eu2(Ir1-xTix)2O7, where $x$=0,
0.05, 0.1, 0.15

Isothermal magnetization curve (M-H)for all the sample measured at 3 K up to
field 50 kOe shown in fig. 8. Both the parent and Ti doped samples show linear
behavior with field without any hysteresis. Magnetic moment up to highest
applied magnetic field (50 kOe) decreases with Ti doping due to site dilution
with nonmagnetic Ti4+.

Discussion In Eu2Ir2O7, most of the Ir is in the Ir4+ (5d5) charge state with
small amount of Ir5+ which is not changing with Ti doping. In this system
structural arrangement of IrO6 octahedra plays an important role in the
transport and magnetic properties. The structural evaluation takes place with
Ti doping. The doping of smaller ionic radius Ti at Ir site reduces the volume
of the unit cell and also bond angle increases and bond length decreases. All
the magnetic and transport properties of the system is governed by Ir 5d
electrons, which have a strong spin orbit coupling and electronic correlation,
both are comparable to the energy scale of Ir. In presence of IrO6 octahedral
environment, 5d orbital splits into t2g and eg energy level. It is believed
that in presence of strong SOC further split the lower energy level t2g into
spin Jeff=3/2 quadruplet with fully filled and spin Jeff=1/2 doublet with
partially filled. Now if there exists on site electron correlation then a Mott
like gap can open in the half filled Jeff=1/2 state, which makes the system
Jeff=1/2 Mott insulator. As Ti being a 3d transition metal, it has higher
electronic correlation (U) effect compare to 5d transition metal Ir. Thus the
substitution of nonmagnetic Ti4+ (3d0) at Ir4+ (5d5) site not only dilute the
magnetic network of the system but also tune its SOC and U. The analysis of
the resistivity data, dilution of the Ir-O-Ir connectivity, tuning of the SOC
and U and structural modification need to be considered. As the increase of
bond angle Ir-O-Ir and decrease of bond length Ir-O enhances the orbital
overlapping between Ir 5d and O 2p orbitals. This phenomena is likely to
favour the electron transport mechanism. As a result resistivity was supposed
to decrease with Ti doping. However, the site dilution and tuning of SOC and U
by Ti doping may affect the electrical transport of the system. As a result
the resistivity of the system increases with Ti doping as well as MIT is
disappearing. $\rho$(T) also follows the power law like behaviour where power
coefficient increases with Ti doping. The reason for the magnetic
irreversibility is due to the field induced moment.Mondal  There is an
unavoidable mixing of nonstoichiometry which produce nonmagnetic Ir5+ state,
deviates the all-in-all-out spin structure. With the application of field each
octahedra picks up a moment, resulting as a field induced moment. As Ti4+
(3d0) is nonmagnetic, introduction of Ti at Ir site, Ti4+ reduce the magnetic
interaction in the octahedral network. As a result, magnetization of the
system decreases with the incorporation of Ti at Ir site. Tirr decreases very
marginally with Ti doping. The tuning of SOC and U by Ti doping can play a
role in minor change in irreversibility temperature. Besides various factors
like disorder, magnetic interaction and anisotropy may affect the
irreversibility temperature, further investigation needed. In this system the
value of magnetic frustration parameter $f$ is lower than the highly
frustrated pyrochlore oxide, which decreases with Ti due to magnetic site
dilution by Ti4+.

Conclusion From the analysis of electrical transport and magnetic property
measurement in pyrochlore iridate Eu2(Ir1-xTix)2O7, we conclude that with the
introduction of nonmagnetic Ti influences the electrical transport and
magnetic property of the system. Minor structural changes due to Ti doping
enhances the orbital overlapping between Ir(5d) and O(2p) orbitals and favours
the transport mechanism. But, as the $3d$ transition metal Ti has higher
electron correlation than $5d$ transition metal Ir, the introduction of Ti at
Ir site tune the SOC and U of the system and site dilution occurs. As a result
resistivity of the system overall increases systematically with the
incorporation of Ti. The nonmagnetic Ti4+ sits at the vertices of the Ir
octahedra, dilute the magnetic interaction between Ir4+. We find that magnetic
moment and magnetic frustration decreases with Ti though magnetic
irrevesibility temperature is hardly influenced by Ti.

Acknowledgements S. M. acknowledges Prof. S. Banerjee and Dr. S. K. Mandal
(Supervisor) for their valuable contribution. S. M. would like to thank Prof.
S. Hazra, Mr. G. Sarkar, SPMS Division, SINP for XPS measurement and Dr. M. K.
Mukhopadhyay, Dr. R. Dev Das, SPMS Division, SINP for XRD measurement. This
work is partially supported by SERB, DST, GOI under TARE project (Grant No.
TAR$/2018/000546$).

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