# Kaonic Hydrogen and Deuterium in Hamiltonian Effective Field Theory

Zhan-Wei Liu1,5<EMAIL_ADDRESS>Jia-Jun Wu4 Derek B. Leinweber2 Anthony
W. Thomas2,3 1School of Physical Science and Technology, Lanzhou University,
Lanzhou 730000, China
2Centre for the Subatomic Structure of Matter (CSSM), Department of Physics,
University of Adelaide, Adelaide SA 5005, Australia
3ARC Centre of Excellence for Particle Physics at the Terascale, Department of
Physics, University of Adelaide, Adelaide SA 5005, Australia
4School of Physical Sciences, University of Chinese Academy of Sciences
(UCAS), Beijing 100049, China
5Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China

###### Abstract

The anti-kaon nucleon scattering lengths resulting from a Hamiltonian
effective field theory analysis of experimental data and lattice QCD studies
are presented. The same Hamiltonian is then used to compute the scattering
length for the $K^{-}d$ system, taking careful account of the effects of
recoil on the energy at which the $\bar{K}N$ T-matrices are evaluated. These
results are then used to estimate the shift and width of the $1S$ levels of
anti-kaonic hydrogen and deuterium. The $K^{-}p$ result is in excellent
agreement with the SIDDHARTA measurement. In the $K^{-}d$ case the imaginary
part of the scattering length and consequently the width of the $1S$ state are
considerably larger than found in earlier work. This is a consequence of the
effect of recoil on the energy of the $\bar{K}N$ energy, which enhances the
role of the $\Lambda(1405)$ resonance.

###### pacs:

12.39.Jh, 13.30.Eg, 14.20.Lq

††preprint:  ADP-20-8/T1118

##  I Introduction

The kaon has played an important role in advancing particle physics since it
was discovered in cosmic rays. The related $\theta$-$\tau$ puzzle resulted in
the proposal of parity violation. A bound state of the kaon and nucleon was
predicted and the candidate $\Lambda(1405)$ was discovered in experiment, even
before the birth of the quark model Dalitz:1959dn (1, 2). The study of kaonic
nuclei and atoms constitutes an important source of information as we seek to
understand the nonperturbative properties of QCD and many experiments have
been designed to investigate them (see, for example, Ref. Gal:2016boi (3) for
a recent review).

In a kaonic atom an electron is replaced by a $K^{-}$, which is still bound
primarily by the attractive electromagnetic interaction. Since the kaon is
three orders of magnitude heavier than the electron, the size of the kaonic
orbit in a light atom is only $\sim$ 100 fm, while the binding energy is
$\sim$ keV. If the kaon is captured by the nucleus, it can be annihilated in
the formation of a hyperon. The behavior of these exotic atoms and nuclei
allows us to explore new aspects of hadronic interactions. For example, the
$\Lambda(1405)$ lies close below threshold in the $J^{P}=1/2^{-}$ $\bar{K}N$
system, while the lowest-lying $J^{P}=1/2^{-}$ strangeness-zero baryon, the
$N(1535)$, is far from the $\pi N$ threshold. This difference has led to a
great deal of speculation regarding the adequacy of the conventional quark
model Veit:1984jr (4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15).

Since the atomic size is beyond the range of the strong interaction, the
kaonic atom can be studied pretty well within QED. However, for the $1S$
state, the wave function $\psi_{1S}^{QED}(\vec{r}=0)$ is non-zero and the
strong interaction plays a role. Therefore, the experimental energy, $E_{1S}$,
is different from the QED prediction, $E_{1S}^{QED}$, because of the strong
interaction, $\Delta E_{1S}\equiv E_{1S}-E_{1S}^{QED}$. Moreover, the
$\bar{K}N$ system can decay to $\pi\Sigma$ and $\pi\Lambda$ and thus the $1S$
level has a width $\Gamma_{1S}$, in addition to the energy shift
$\epsilon_{1S}$, $\Delta E_{1S}\equiv
E_{1S}-E_{1S}^{QED}=\epsilon_{1S}-\frac{i}{2}\Gamma_{1S}$. The energy
distributions of X-rays emitted from the excited kaonic atom may be measured
to obtain $\Delta E_{1S}$ in experiment Bazzi:2011zj (16, 17, 18, 19, 20, 21).
The difference between the measured and QED-predicted $nP-1S$ transition
energy is essentially the same as $\Delta E_{1S}$, since the $nP$ level is
scarcely affected by the strong interaction. The SIDDHARTA experiment gave the
energy shift and width for $1S$ kaonic hydrogen as

$\displaystyle\epsilon_{1S}^{p}$ $\displaystyle=$ $\displaystyle 283\pm
36({\rm stat})\pm 6({\rm sys})~{}{\rm eV},$ $\displaystyle\Gamma_{1S}^{p}$
$\displaystyle=$ $\displaystyle 541\pm 89({\rm stat})\pm 22({\rm sys})~{}{\rm
eV}\,,$ (1)

which helps us to constrain the $K^{-}p$ interaction at low energy
Bazzi:2011zj (16, 17). The reduced mass of the anti-kaon and the deuteron is a
little larger than that of $\bar{K}N$ and thus kaonic deuterium would be
easier to form under the pure electromagnetic interaction. There has been a
proposal to search for kaonic deuterium in SIDDHARTA-2 and the J-PARC E57
experiment Curceanu:2013bxa (18, 19, 20, 21). However, technically it is far
more difficult to measure the kaonic deuteron than kaonic hydrogen.

The study of kaonic deuterium may be expected to benefit from a comparison
with earlier studies of pionic deuterium. The latter was proposed and observed
in experiment very early – see Ref. Thomas:1979xu (22) for a review. The
energy shifts and widths of the $1S$ states of pionic hydrogen and deuterium
were measured with precision at the Paul Scherrer Institute (PSI) at Villigen
Hauser:1998yd (23).

Kaonic deuterium has already been investigated by many authors Meissner:2006gx
(24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34). In recent studies, after the
resummation of contributions from kaon multiple scatterings between the two
nucleons in the deuteron, solving the full three-body Hamiltonian or Faddeev
equations, the energy shift for kaonic deuterium has been found to be of order
700$\sim$900 eV, while the width has been reported to be in the range
800$\sim$1200 eV Doring:2011xc (29, 30, 31, 32, 33, 28).

The nucleons in the deuteron have been assumed to be static in most studies of
kaonic deuterium, perhaps because this approximation has been shown to be
quite accurate for pionic deuterium. However, the situation merits further
consideration for the following reason. In the pionic case the closest
resonance, the $\Delta(1232)$, has angular momentum one and is still 150 MeV
away from the $\pi N$ threshold. Even allowing for its width, it is far away.
On the other hand, the difference between the $\Lambda(1405)$ resonance and
the $\bar{K}N$ threshold is less than 30 MeV and, moreover, its 50 MeV width
means they overlap to some extent. In such a case, even if the recoiling
nucleon shifts the center-of-mass energy of $\bar{K}N$ a relatively small
distance from the threshold, the scattering amplitude may be very different
from that at threshold, which the static approximation uses. These
considerations led us to examine the effect of recoil carefully in the
$\bar{K}d$ system.

Some earlier studies of the effect of nucleon recoil for $\bar{K}d$ scattering
was undertaken within effective field theory in Refs. Baru:2009tx (35, 36).
There non-local effects were taken into account perturbatively by using the
effective-range expansion of the $\bar{K}N$ amplitude. The recoil effect for
the double scattering contribution was found to be the order of 10-15%, as
compared to the static term Baru:2009tx (35). The effect on the single
scattering process was not discussed there, while it is a key issue in this
study.

In this work, we focus particularly on the recoil effects for the $1S$ level
shifts of anti-kaonic deuterium with Hamiltonian Effective Field Theory
(HEFT). HEFT was developed to study the low-lying resonances based on a
combined analysis of both lattice QCD data and experimental scattering data,
while preserving the constraints of chiral perturbation theory where
appropriate. It has led to important insights into the properties of the
$N(1535)$, $N(1440)$, $\Lambda(1405)$, and so on Hall:2013qba (37, 38, 12, 39,
40, 41, 42, 43, 44). In Ref. Liu:2016wxq (12), we studied both the cross
sections for $K^{-}p$ scattering and the relevant lattice QCD data and
reproduced the two-pole structure of the $\Lambda(1405)$. If not explicitly
pointed out, references of the $\Lambda(1405)$ refer to the pole close to the
experimental mass.

We first briefly review the HEFT for $\bar{K}N$ in Sec. II. Then we use it to
compute the $K^{-}p$ scattering lengths and the corresponding energy shift
$\Delta E_{1S}^{p}$, with no further adjustment of the parameters determined
in that previous work Liu:2016wxq (12). The scattering length of the $K^{-}d$
system and the energy shift $\Delta E_{1S}^{d}$ for kaonic deuterium, with and
without the recoil correction, are presented in Secs. III and IV. We conclude
with a discussion and summary in Sec. V.

Figure 1: Illustration of the Faddeev equation for $\mathcal{T}_{p}$ in kaon-
deuteron scattering. The $T_{\bar{K}N}$ can be obtained from the study of kaon
nucleon scattering, which contains the coupled-channel effect from $\bar{K}N$,
$\pi\Sigma$, $\pi\Lambda$, and $\eta\Lambda$. There are two other similar
diagrams for equations on $\mathcal{T}_{n}$ and $\mathcal{T}_{x}$ which can be
found in Ref. Kamalov:2000iy (31).

##  II Kaonic hydrogen and HEFT

The $T$ matrix for the two-body scatting process $\alpha\to\beta$ is related
to the corresponding $S$ matrix and cross section by

$\displaystyle S_{\beta\alpha}$ $\displaystyle=$
$\displaystyle\delta_{\alpha\beta}-2i\sqrt{\rho_{\alpha}\rho_{\beta}}T_{\beta\alpha},$
(2) $\displaystyle\sigma_{\beta\alpha}$ $\displaystyle=$
$\displaystyle\frac{4\pi^{3}k^{\rm cm}_{\beta}\omega^{\rm
cm}_{\alpha_{M}}\omega^{\rm cm}_{\alpha_{B}}\omega^{\rm
cm}_{\beta_{M}}\omega^{\rm cm}_{\beta_{B}}}{k^{\rm cm}_{\alpha}(E_{\rm
cm})^{2}}T_{\beta\alpha}T_{\beta\alpha}^{*}\,,$ (3)

where

$\rho_{\alpha}=\pi\frac{\omega^{\rm cm}_{\alpha_{M}}\omega^{\rm
cm}_{\alpha_{B}}}{E_{\rm cm}}k^{\rm cm}_{\alpha}\,.$ (4)

In this work, the channel label, $\alpha$ or $\beta$, refers to $\bar{K}N$,
$\pi\Sigma$, $\pi\Lambda$, and $\eta\Lambda$ but, of course, it could be
extended to more general cases. The labels $\alpha_{M}$ and $\alpha_{B}$ refer
to the meson and baryon in channel $\alpha$, respectively. $\omega^{\rm
cm}_{\alpha_{X}}$ and $k^{\rm cm}_{\alpha}$ denote the energy and spatial
momentum of the particle in the center-of-mass frame, and $E_{\rm cm}$ is the
total energy in the center-of-mass frame.

For the process $\alpha\to\alpha$, the cross section
$\sigma_{\alpha\alpha}^{\rm thr}$ at threshold can be related to the
scattering length $a_{\alpha}$

$\sigma_{\alpha\alpha}^{\rm thr}=4\pi|a_{\alpha}|^{2},$ (5)

where the scattering length is defined as

$a_{\alpha}\equiv-\lim_{k^{\rm cm}_{\alpha}\to 0}\pi\frac{\omega^{\rm
cm}_{\alpha_{M}}\omega^{\rm cm}_{\alpha_{B}}}{E_{\rm
cm}}T_{\alpha\alpha}=-\pi\mu_{\alpha}T_{\alpha}^{\rm thr}\,$ (6)

and $\mu_{\alpha}$ is the reduced mass for the channel $\alpha$.

We used HEFT to study both the cross sections of $K^{-}p\to
K^{-}p/\bar{K}^{0}n/\pi^{-}\Sigma^{+}/\pi^{0}\Sigma^{0}/\pi^{+}\Sigma^{-}/\pi^{0}\Lambda$
and the corresponding energy levels in a finite volume (for comparison with
lattice QCD simulations). The results were consistent with both the
experimental scattering data and the spectra from lattice QCD simulations
Liu:2016wxq (12). Here we want to further study the scattering length,
$a_{K^{-}p}$, within this framework. Although there are two different
scenarios which give different behavior at larger pion masses in Ref.
Liu:2016wxq (12, 45), we have checked that both yield the same threshold
observables with negligible numerical differences. Thus we can choose the
simpler scenario for this work. In our framework, the $T$ matrix can be
obtained by solving the Bethe-Salpeter equation

$\displaystyle
T^{I}_{\alpha\beta}(k,k^{\prime};E)=V^{I}_{\alpha\beta}(k,k^{\prime})+\sum_{\gamma}\int
q^{2}\,dq$ $\displaystyle\quad
V^{I}_{\alpha\gamma}(k,q)\,\frac{1}{E-\omega_{\gamma}(q)+i\epsilon}\,T^{I}_{\gamma\beta}(q,k^{\prime};E),$
(7)

with the Weinberg-Tomozawa potentials

$V^{I}_{\alpha\beta}(k,k^{\prime})=g_{\alpha,\beta}^{I}\frac{\left[\,\omega_{\alpha_{M}}(k)+\omega_{\beta_{M}}(k^{\prime})\,\right]\,u(k)\,u(k^{\prime})}{8\pi^{2}f^{2}\,\sqrt{2\omega_{\alpha_{M}}(k)}\,\sqrt{2\omega_{\beta_{M}}(k^{\prime})}}\,,$
(8)

where $f$ is the decay constant of the pion, the isospin $I$ can be 0 or 1.
$k$ and $k^{\prime}$ are the center-of-mass spatial momenta in the final and
initial states, respectively. The form factor
$u(k)=(1+k^{2}/\Lambda^{2})^{-2}$ is used, with $\Lambda=1$ GeV Liu:2016wxq
(12).

To fix the nonzero couplings, $g_{\alpha,\beta}^{I}$, in Ref. Liu:2016wxq (12)
we fitted the cross sections for $K^{-}p\to
K^{-}p/\bar{K}^{0}n/\pi^{-}\Sigma^{+}/\pi^{0}\Sigma^{0}/\pi^{+}\Sigma^{-}/\pi^{0}\Lambda$
with $\chi_{d.o.f}=1.5$ for laboratory energies of the kaon up to 250 MeV.
There are two poles for $\Lambda(1405)$ at $1428-23\,i$ and $1338-89\,i$ MeV
with this set of parameters. We also compared our energy levels with lattice
QCD results in Ref. Liu:2016wxq (12).

We have not fine tuned the coupling constants, $g_{\alpha,\beta}^{I}$, in this
work to exactly fit the scattering length. Instead, we use the values of
$g_{\alpha,\beta}^{I}$ found in Ref. Liu:2016wxq (12) to predict $a_{K^{-}p}$
and further check the reliability of HEFT. The scattering lengths for
$\bar{K}N$ in the isospin $I=0$ and $I=1$ channels in the HEFT, using Eq. (6),
are

$a_{I=0}=-1.77+1.08\,i~{}~{}{\rm fm},\qquad a_{I=1}=0.27+0.52\,i~{}~{}{\rm
fm}\,.$ (9)

With the isospin relation, one obtains

$a_{K^{-}p}=-0.75+0.80\,i~{}~{}{\rm fm},$ (10)

and

$\displaystyle a_{K^{-}n}=0.27+0.52\,i~{}~{}{\rm fm},\qquad
a_{\bar{K}^{0}n}=-0.75+0.80\,i~{}~{}{\rm fm},$ $\displaystyle
a_{\bar{K}^{0}n\to K^{-}p}=1.02-0.28\,i~{}~{}{\rm fm}\,.$ (11)

The relation between the energy shift $\epsilon_{1S}^{p}$ and width
$\Gamma_{1S}^{p}$ of kaonic hydrogen and $a_{K^{-}p}$ can be given by the
improved Deser formula Meissner:2004jr (46), but the logarithmic contribution
at higher orders is important for kaonic deuterium Hoshino:2017mty (32) and
thus throughout this work we use the “double-improved” Deser formula in which
the logarithmic terms are summed to all orders Baru:2009tx (35)

$\epsilon_{1S}^{p}-\frac{i}{2}\Gamma_{1S}^{p}=\frac{-2\alpha_{e}^{3}\,\mu_{K^{-}p}^{2}\,a_{K^{-}p}}{1+2\alpha_{e}\,\mu_{K^{-}p}\,(\ln\alpha_{e}-1)\,a_{K^{-}p}}\,,$
(12)

where $\alpha_{e}$ is the electromagnetic fine-structure constant. With this
resummed formula we obtain

$\epsilon_{1S}^{p}=307~{}{\rm eV},\qquad\Gamma_{1S}^{p}=533~{}{\rm eV}\,.$
(13)

These results are consistent with the result of the SIDDHARTA experiment given
in Eq. (1).

##  III results for kaonic deuterium with the static nucleon approximation

The scattering amplitude $T_{\bar{K}d}$ can be divided into two parts

$T_{\bar{K}d}=\int d^{3}\vec{q}\,\psi_{d}(\vec{q})\,\int
d^{3}\vec{q}^{\prime}\psi_{d}^{*}(\vec{q}^{\prime})\,\,\,\left(\mathcal{T}_{p}+\mathcal{T}_{n}\right),$
(14)

where $\psi_{d}(\vec{q})$ is the wave function of the nucleon in the deuteron
in momentum space and $\vec{q}$ ($\vec{q}^{\prime}$) is the relative momentum
between the two nucleons in the initial (final) state. The $\bar{K}NN$
scattering amplitude $\mathcal{T}$ can be obtained by solving the Faddeev
equations Thomas:1977aab (47) illustrated in Fig. 1

$\displaystyle\mathcal{T}_{p}$ $\displaystyle\sim$ $\displaystyle
T_{K^{-}p}\delta^{3}(\vec{q}-\vec{q}^{\prime})+\int_{\delta}T_{K^{-}p}G_{0}\mathcal{T}_{n}+\int_{\delta}T_{K^{-}p\to\bar{K}^{0}n}G_{0}\mathcal{T}_{x},$
$\displaystyle\mathcal{T}_{n}$ $\displaystyle\sim$ $\displaystyle
T_{K^{-}n}\delta^{3}(\vec{q}-\vec{q}^{\prime})+\int_{\delta}T_{K^{-}n}G_{0}\mathcal{T}_{p},$
$\displaystyle\mathcal{T}_{x}$ $\displaystyle\sim$ $\displaystyle
T_{\bar{K}^{0}n\to
K^{-}p}\delta^{3}(\vec{q}-\vec{q}^{\prime})+\int_{\delta}T_{\bar{K}^{0}n}G_{0}\mathcal{T}_{x}+\int_{\delta}T_{\bar{K}^{0}n\to
K^{-}p}G_{0}\mathcal{T}_{n}\,.$

It corresponds to the scattering length calculation, where the kaon momentum
is zero and hence the momentum of the spectator nucleon (to the kaon-nucleon
scattering) is its momentum in the deuteron center of mass. $G_{0}$ is the
propagator

$G_{0}=(\epsilon_{d}-\frac{\vec{p}_{K}^{2}}{2m_{K}}-\frac{\vec{p}_{N_{1}}^{2}}{2m_{N}}-\frac{\vec{p}_{N_{2}}^{2}}{2m_{N}}+i0^{+})^{-1},$
(16)

where $\vec{p}_{K}$, $\vec{p}_{N_{1}}$, and $\vec{p}_{N_{2}}$ are the momenta
of the intermediate particles, and $\epsilon_{d}$ is the binding energy of
deuteron.

The first term in Eq. (LABEL:eq:Faddeev) corresponds to the single scattering
(SS) of the kaon with only one nucleon, which contains the effects of the
coupling of $\bar{K}N$ to the channels $\pi\Sigma$, $\pi\Lambda$, and
$\eta\Lambda$. The remainder is generated by the multiple scattering (MS) of
the kaon between the two nucleons in the deuteron.

If one uses the static nucleon approximation, usually on the basis that the
momenta are relatively small, the momentum-dependent amplitude $T_{\bar{K}N}$
can be approximated by the value at threshold, $T_{\bar{K}N}^{\rm thr}$. With
this approximation, the multiple scattering series integration can be solved
analytically and the scattering length, $a_{K^{-}d}$, can be simplified as
Kamalov:2000iy (31)

$a_{K^{-}d}=\frac{m_{d}}{m_{K}+m_{d}}\int
d^{3}\vec{r}\,|\psi_{d}(\vec{r})|^{2}\,\,\hat{A}_{K^{-}d}(r)\,,$ (17)

where $\psi_{d}(\vec{r})$ is the wave function for nucleons in the deuteron,
and

$\hat{A}_{K^{-}d}(r)=\frac{\tilde{a}_{K^{-}p}+\tilde{a}_{K^{-}n}+(2\tilde{a}_{K^{-}p}\tilde{a}_{K^{-}n}-b_{x}^{2})/r-2b_{x}^{2}\tilde{a}_{K^{-}n}/r^{2}}{1-\tilde{a}_{K^{-}p}\tilde{a}_{K^{-}n}/r^{2}+b_{x}^{2}\tilde{a}_{K^{-}n}/r^{3}}\,,$
(18)

with $\tilde{a}_{\bar{K}N}=a_{\bar{K}N}(1+m_{K}/m_{N})$, and
$b_{x}=\tilde{a}_{\bar{K}^{0}n\to
K^{-}p}/\sqrt{1+\tilde{a}_{\bar{K}^{0}n}/r}$. This corresponds to the commonly
used fixed center approximation (e.g. Ref. Roca:2010tf (48)) in which excited
nucleon intermediate states are neglected.

Using the $\bar{K}N$ scattering lengths given in Eq. (11) and the deuteron
wave function obtained from a one-boson-exchange model Chen:2017jjn (49), the
numerical result with the static nucleon approximation is

$a_{K^{-}d}|_{\rm StaticApprox}=-1.55+1.57\,i~{}~{}{\rm fm}\,.$ (19)

With the resummed formula (12), the energy shift and width of the $1S$ level
of kaonic deuterium is

$\epsilon_{1S}^{d}|_{\rm StaticApprox}=855~{}{\rm
eV},\quad\Gamma_{1S}^{d}|_{\rm StaticApprox}=1127~{}{\rm eV}\,.$ (20)

##  IV recoil effects for kaonic deuterium

The root-mean-square radius of the deuteron is about 2 fm, which means the
momentum $q$ of the nucleons in the deuteron should be of order 1/(2 fm)
$\sim$ 100 MeV on average. The variation of momentum is not very large, but
the resonance $\Lambda(1405)$ is around the $\bar{K}N$ threshold and thus the
effect from the nucleon recoiling in the deuteron should be carefully
examined.

Firstly, we study the single scattering term of $K^{-}d$, which can be
represented as the second diagram in Fig. 1. Denoting the scattering amplitude
of a static kaon and a nucleon with momentum $\vec{q}$ in the deuteron as
$T_{\bar{K}N}(\vec{q})$, the integrated amplitude is

$\langle\,T_{\bar{K}N}^{d}\,\rangle\equiv\int
d^{3}\vec{q}\,|\psi_{d}(\vec{q})|^{2}\,\,T_{\bar{K}N}(\vec{q}).$ (21)

Obviously, the amplitude of $K^{-}d$ from the single scattering contribution
is exactly
$T_{K^{-}d}^{SS}=\langle\,T_{K^{-}p}^{d}\,\rangle+\langle\,T_{K^{-}n}^{d}\,\rangle$.

To calculate $T_{\bar{K}N}(\vec{q})$, we need to recognise that if the
spectator nucleon has momentum $\vec{q}$, the $\bar{K}N$ center of mass energy
at which the two-body T-matrix must be evaluated is effectively
$E_{\bar{K}N}=m_{d}+m_{K}-m_{N}-q^{2}/2\mu_{3}$ within Faddeev formalism where
$\mu_{3}=m_{N}(m_{N}+m_{K})/(2m_{N}+m_{K})$. We use the following relation to
obtain $T_{\bar{K}N}(\vec{q})$

$T_{\bar{K}N}(\vec{q})=T_{\bar{K}N,\bar{K}N}(k,k;E_{\bar{K}N})\,,$ (22)

where $k=qm_{K}/(m_{N}+m_{K})$.

For the multiple scattering terms, we use the approximation
$T_{\bar{K}N}\to\langle\,T_{\bar{K}N}^{d}\,\rangle$ to simplify the
calculation. Finally, we can evaluate $a_{K^{-}d}$. By replacing
$a_{\bar{K}N}$ with
$\langle\,a_{\bar{K}N}^{d}\,\rangle\equiv-\pi\mu_{\bar{K}N}\langle\,T_{\bar{K}N}^{d}\,\rangle$
in the formula given in Eq. (17), the $K^{-}d$ scattering length, including
the recoil correction, may be obtained.

##  V numerical results and discussion

More than 96% of the contribution to $\langle\,T_{\bar{K}N}^{d}\,\rangle$ in
Eq. (21) comes from $q<300$ MeV, which corresponds to $k\lesssim 110$ MeV and
$\Delta E_{\bar{K}N}\lesssim 80$ MeV, a region in which our fit with HEFT is
very reliable Liu:2016wxq (12). The wave function $\psi_{d}(\vec{q})$ is
extracted from the Bonn potential ENegele:1989pd (50). The wave function of
the deuteron is well constrained both by experiments and theory, and thus we
directly cite two different results from Refs. Chen:2017jjn (49) and
ENegele:1989pd (50) for $\psi_{d}(\vec{r})$ and $\psi_{d}(\vec{q})$ in the
coordinate and momentum spaces respectively. We do not transform between them
and simply neglect the modest differences between the models.

Before providing the average $\langle\,T_{\bar{K}N}^{d}\,\rangle$, we show the
scattering amplitude $T_{\bar{K}N}(\vec{q})$ in Fig. 2. By comparing the two
x-axes, one can see that even if the momentum $q$ increases from 0 to 200 MeV
where the wave function $\psi_{d}$ is extremely close to 0, the center-of-mass
energy $E_{\bar{K}N}$ of the $\bar{K}N$ system varies over a relatively small
range, roughly 1430 to 1400 MeV. Usually such a small change of $E_{\bar{K}N}$
would not be expected to lead to large variations in related amplitudes. For
example, the blue dashed lines for the amplitudes $T_{\bar{K}N}(I=1)$ are
relatively flat for isospin 1. However, the issue is very different when there
is a resonance or bound state in a channel below and close to the threshold.

Figure 2: The scattering amplitude $T_{\bar{K}N}(\vec{q})$ for a static kaon and a nucleon of momentum $q$ in the deuteron. The lower x-axis is the spatial momentum $q$ of a nucleon, while the upper x-axis is the center-of-mass energy $E_{\bar{K}N}$ of $\bar{K}N$, and they are related by $E_{\bar{K}N}=(m_{d}/2+m_{K})-q^{2}/2\mu_{3}$ with $\mu_{3}=m_{N}(m_{N}+m_{K})/(2m_{N}+m_{K})$. The deuteron wave function is also shown as the green dotted lines. Table 1: Comparison of scattering lengths $a_{K^{-}d}$ in units of fm. The first four columns are from Refs. Kamalov:2000iy (31, 29, 30, 32). The last four columns are the results in this work with (✓) and without ($\times$) the multiple scattering contributions or the recoil effects, respectively. $a_{K^{-}d}$ | Kamalov Kamalov:2000iy (31) | Doring Doring:2011xc (29) | Mizutani Mizutani:2012gy (30) | Hoshino Hoshino:2017mty (32) | (MS ✓, recoil $\times$) | (MS $\times$, recoil ✓) | (MS $\times$, recoil $\times$) | (MS ✓, recoil ✓)
---|---|---|---|---|---|---|---|---
Re | $-1.62$ | $-1.46$ | $-1.58$ | $-1.42$ | $-1.55$ | -0.06 | -0.58 | $-0.59$
Im | 1.91 | 1.08 | 1.37 | 1.60 | 1.57 | 2.55 | 1.59 | 2.70

In our previous work Liu:2016wxq (12) we searched for two poles for the
$\Lambda(1405)$ in the $I=0$ channel within HEFT, finding one close to the
$\bar{K}N$ threshold with width about 50 MeV. As $E_{\bar{K}N}$ passes by the
pole, the scattering amplitude $T_{\bar{K}N}(I=0)$ changes rapidly, as shown
by the red dashed and dot-dashed lines in Fig. 2. The phase shift usually
crosses $90^{\tiny o}$ when a resonance appears, which is equivalently
exhibited by the real part of the amplitude changing its sign for
$E_{\bar{K}N}$ around 1410 MeV (red dashed). The imaginary part of
$T_{\bar{K}N}(I=0)$ is related to the experimental $\pi\Sigma$ invariant mass
distribution and thus the red dot-dashed line shows a bump Hemingway:1984pz
(51).

From this analysis, we would expect that the average value of
$\langle\,T_{\bar{K}N}^{d}\,\rangle$ may deviate significantly from the
threshold value in the $I=0$ channel. Equivalently speaking,
$\langle\,a_{I=0}^{d}\,\rangle$ is very different from the scattering length
$a_{I=0}$, while $\langle\,a_{I=1}^{d}\,\rangle$ should be still close to
$a_{I=1}$. We find

$\langle\,a_{I=0}^{d}\,\rangle=-1.07+3.01\,i~{}~{}{\rm
fm},\qquad\langle\,a_{I=1}^{d}\,\rangle=0.32+0.40\,i~{}~{}{\rm fm}\,,$ (23)

which in comparison with the threshold scattering lengths in Eq. (9) shows
that the average imaginary part in the channel with $I=0$ increases by more
than 1 fm. This will, of course, increase the width of the $1S$ $K^{-}d$
energy level significantly.

We show the $K^{-}d$ scattering length after taking the recoiling effect into
consideration

$a_{K^{-}d}=-0.59+2.70\,i~{}~{}{\rm fm}\,.$ (24)

With the resummed formula of Eq. (12), the energy shift and width of $1S$ kaon
deuterium are

$\epsilon_{1S}^{d}=803~{}{\rm eV},\qquad\Gamma_{1S}^{d}=2280~{}{\rm eV}.$ (25)

By comparing these results with those given in Eq. (20) of Sec. III, we can
clearly see that the recoil effect plays a very important role for the
$K^{-}d$ system.

Figure 3: Comparison of the energy shifts and widths for the $1S$ state of
kaonic deuterium. The data with green triangles are from Refs. Doring:2011xc
(29, 30, 31, 32) and $\Delta E_{1S}^{d}$ are extracted from the scattering
lengths of $K^{-}d$ using the resummed formula (12). The data with blue
circles are from Refs. Revai:2016muw (33, 28, 32) where the dynamical
equations directly provide the energy level of kaonic deuterium without the
help of $K^{-}d$ scattering length. For example, there are two points labeled
with “Hoshino2017” which are from the same paper Hoshino:2017mty (32), but the
left blue circle is directly obtained by solving the Schrödinger equation
while the right green triangle is translated with the double-improved Deser
formula (12) from the $K^{-}d$ scattering length. Our result (labeled with
hollow star) without the recoiling effect is close to other works, but the
result after considering the recoiling effect (labeled with a filled star) has
a much larger width.

We compare the scattering lengths $a_{K^{-}d}$ in this work with some other
studies in Table 1 and Fig. 3. Without including the effect of recoil our
results in the fifth column are close to those of earlier work. That is, our
result would be close to others if the momentum of the nucleon in the deuteron
is neglected. However, our corrected result has a much larger width.

The contribution to $a_{K^{-}d}$ from the single scattering term is exactly

$a_{K^{-}d}^{SS}=\frac{\mu_{\bar{K}d}}{\mu_{\bar{K}N}}\left(\,\frac{1}{2}\langle\,a_{I=0}^{d}\,\rangle+\frac{3}{2}\langle\,a_{I=1}^{d}\,\rangle\,\right).$
(26)

We divide $a_{K^{-}d}$ into $a_{K^{-}d}^{SS}$ and $a_{K^{-}d}^{MS}$, which are
from the single and multiple scattering contributions, respectively

$a_{K^{-}d}=a_{K^{-}d}^{SS}+a_{K^{-}d}^{MS}=(-0.06+2.55i)+(-0.53+0.15i)~{}{\rm
fm}\,.$ (27)

By comparing this with

$a_{K^{-}d}|_{\rm StaticApprox}=(-0.58+1.59i)+(-0.97-0.02i)~{}{\rm fm}\,,$
(28)

the main difference in the imaginary part clearly arises from the single
scattering term. This gives us confidence in the conclusion that the ground
state of kaonic deuterium is broad. Our final result for the $K^{-}d$
scattering length is

$a_{K^{-}d}=\,-0.59\,+\,2.70\,i~{}~{}{\rm fm}\,.$ (29)

In summary, we first established that our earlier Hamiltonian effective field
theory study of the $\bar{K}p$ system did indeed reproduce the empirical
$K^{-}p$ scattering length, as well as producing an energy shift and width for
the ground state of kaonic hydrogen which is consistent with the SIDDHARTA
experiment. The calculation was then extended to the the $K^{-}d$ system,
where it was found that the effect of recoil in the energy argument of the
$\bar{K}p$ T-matrix within the Faddeev formalism is very significant, making
the $1S$ level of kaonic deuterium considerably more broad and short lived.

## Acknowledgement

Z.W.L thanks Kan Chen for checking the program of kaon nucleon scattering
length. This project is supported by the National Natural Science Foundation
of China under Grants Nos. 11705072 and 11965016, CAS Interdisciplinary
Innovation Team (Z.W.L), the Thousand Talents Plan for Young Professionals
(J.J.W), and the Australian Research Council through ARC Discovery Project
Grants Nos. DP150103101 and DP180100497 (A.W.T) and DP150103164 and
DP190102215 (D.B.L).

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