∎

11institutetext: Department of Physics, Sardar Vallabhbhai National Institute
of Technology, Surat-395007, Gujarat, India. 33institutetext: Amee
Kakadiya44institutetext: 44email<EMAIL_ADDRESS>
# Spectroscopic Study of $\Omega_{cc}$, $\Omega_{cb}$ and $\Omega_{bb}$
Baryons

Amee Kakadiya  Ajay Kumar Rai

(Received: date / Accepted: date)

###### Abstract

In this paper, we presented the mass spectra of the doubly heavy $\Omega$
baryons containing one light (strange) quark and two heavy (charm and bottom)
quarks. Our predicted masses can be consider to determine the $J^{p}$ value
for the resonances detected by experimental facilities in future. The masses
of ground excited states (1S-6S, 1P-3P, 1D-2D, 1F-2F) are calculated for all
possible $J^{p}$ values, using the Hypercentral Constituent Quark Model
(hCQM), by employing screened potential as confining potential with color-
Coloumb potential. Regge trajectories are also plotted in $(J,M^{2})$ plane
for natural and unnatural parities. Doubly heavy Omega states are not declared
yet by any experimental facility. We compared our results to the predictions
gained from other theoretical approaches, and we found that our predictions
are quite close to those of them. Other properties such as magnetic moment
(for spin state $\frac{1}{2}$ and $\frac{3}{2}$) and radiative decay width are
calculated using the enuerated masses.

###### Keywords:

Doubly heavy baryon, Mass spectra, Magnetic moment, Radiative decay

##  1 Introduction

Doubly heavy $\Omega$ baryons provide an opportunity to inspect the behaviour
of strange quark in presence of two heavy quarks and also the interaction
between two heavy quarks. A serch for singly and doubly heavy baryons was held
by well-known experimental colaborations like; SELEX, BABAR, Belle, FOCUS
SELEX ; BABAR ; Belle2013 ; Belle2014 ; FOCUS . $\Xi_{cc}$ baryon is the only
member from doubly heavy family, which is detected experimentally (resonance
mass is 3621 MeV) PDG . Recently, during $pp$ collission, a bottom-charm
baryon with a single light quark is detected in mass range of 6700-7300
$MeV/c^{2}$ at LHCb experiment, via $\Lambda_{c}^{+}\pi^{-}$ and
$\Xi_{c}^{+}\pi^{-}$ decay, and they are further decaying into $pK^{-}\pi^{+}$
RAaijOmegaBC . The resonance may be either $\Xi_{cb}^{0}$ or
$\Omega_{cb}^{0}$. There is no strong evidence which confirms the isospin of
the particle. But, as $\Xi_{cc}$ and $\Xi_{cb}$/$\Omega_{cb}$ are determined
experimentally from doubly heavy family, there are major possibilities of
other doubly heavy baryons to be found in near future. Our goal of this
research is to predict the masses of ground and excited states of doubly heavy
$\Omega$ baryons, $\Omega_{cc}(scc)$, $\Omega_{cb}(scb)$ and
$\Omega_{bb}(sbb)$; having 0 isospin and -1 strangeness. Furthermore, we study
its proprties such as, magnetic moment and radiative decay. The masses are
enumerated using Hypercentral Constituent Quark Model (hCQM) Universe ; IJMPA
; FBS , incorporating the screening potential as confining potential with
color-Coulomb potential, for quantum numbers: $l=0,1,2,3$, spin-parity:
$\frac{1}{2}^{+}$, $\frac{3}{2}^{+}$ (natural parity).

Various research groups have been invastigated the doubly heavy baryons with
different theoretical approaches such as, Light-cone sum rules (LCSR)
Aliev2020 ; AlievPRD , Bathe-Salpeter equations Li2020 , Constituent quark
model ShahEPJC2017 ; ShahEPJC2016 ; Salehi , Hamiltonian model Yoshida , the
variational approach Roberts , the three-body Faddeev method Valcarce ,
relativistic approach Ebert , Salpeter model Giannuzzi , Regge phenomenology
Juhicharm ; Juhibottom , Quark model Karliner2018 , Heavy baryon chiral
perturbation theory Tong2022 , Lattice QCD Mathur ; MathurCharmBottom , NRQCD
Mohanta  etc.

This paper is as follows: The theoretical model used to predict the masses of
doubly heavy $\Omega$ baryons is described in section 2 after the
introduction. The masses of $J^{p}$ value $\frac{1}{2}$ and $\frac{3}{2}$ and
their hyperfine partners are listed in section 3 (see table 2, 3, 4), Regge
trajectories are also plotted in ($J,M^{2}$) plane for natural and unnatural
parities. The properties like; ground state magnetic moment, transition
magnetic moment and radiative decay width are discussed in section 4. And
conclusion is written in the last section of the article.

##  2 Theoretical Framework

Hypercentral Constituent Quark Model (hCQM) describes the interaction inside
the baryonic system containing three quarks. The model is deeply explianed in
our previous work Universe ; FBS ; IJMPA ; ICNFP ; triply ; ICC2019 ; DAE2019
, hence brief explaination is given here. The Jacobi coordinates are employed
to illustrate the interction between three quarks, which are Giannini ;
Giannini2015 ,

$\vec{\rho}=\frac{\vec{r_{1}}-\vec{r_{2}}}{\sqrt{2}}\hskip 14.22636ptand\hskip
14.22636pt\vec{\lambda}=\frac{\vec{r_{1}}+\vec{r_{2}}-2\vec{r_{3}}}{\sqrt{6}}.$
(1)

The hypercentral coordinates (hyperradias $x$ and hyperangle $\xi$) in terms
of Jacobi coordinates are Bijkar2000 ; Bijkar1994 ,

$x=\sqrt{\rho^{2}+\lambda^{2}}\hskip 14.22636ptand\hskip
14.22636pt\xi=arctan\left(\frac{\rho}{\lambda}\right)$ (2)

The Hamiltonian for the three quark baryonic system is expressed as,

$H=\frac{P^{2}}{2m}+V(x)$ (3)

Where, $P$ is conjugate momentum, $m$ is the reduced mass of the system and
$V(x)$ is non-relativistic interaction potential.

The expression of kinetic energy operator for the three quark system is
Giannini2015 ,

$\frac{P^{2}}{2m}=-\frac{\hbar^{2}}{2m}(\Delta_{\rho}+\Delta_{\lambda})=-\frac{\hbar^{2}}{2m}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{5}{x}\frac{\partial}{\partial
x}+\frac{L^{2}(\Omega)}{x^{2}}\right)$ (4)

Here, $L^{2}(\Omega)$ is the Grand angular operator.

The predicted mass for particular baryonic state are obtained by solving the
Schrödinger equation of it, after substituting the $\frac{P^{2}}{2m}$ term in
equation of Hamiltonian.

The non-relativistic interaction potential inside the baryonic system $V(x)$
depends on hyperradius x and falls in two terms i.e. spin dependent ($V_{SD}$)
and spin independent ($V_{SI}$) potential term Voloshin2008 ; IJMPA ; FBS .

$V(x)=V_{SD}(x)+V_{SI}(x)$ (5)

The spin dependent part of potential $V_{SD}(x)$ is Thakkar2017 ,

$V_{SD}(x)=V_{SS}(x)(\vec{S_{\rho}}\cdot\vec{S_{\lambda}})+V_{\gamma
S}(x)(\vec{\gamma}\cdot\vec{S})+V_{T}(x)\left[S^{2}-\frac{3(\vec{S}\cdot\vec{x})(\vec{S}\cdot\vec{x})}{x^{2}}\right]$
(6)

which includes spin-spin, spin-orbit and tensor term respectively.

And, spin independent part of potential is, $V_{SI}(x)=V_{conf}(x)+V_{Col}(x)$
Gandhi2018 ; GandhiIJTP2020 . Here, the screened potential is incorporated as
confining potential with the color-Coulomb potential, which cab be expressed
as,

$V_{conf}(x)=a\left(\frac{1-e^{-{\mu}x}}{\mu}\right)\hskip 14.22636ptand\hskip
14.22636ptV_{Col}(x)=-\frac{2}{3}\frac{\alpha_{s}}{x}$ (7)

where, $a$ is the string tension (different for various baryon system), the
constant $\mu$ is the the screening factor, $x$ indicates the inter-quark
separation and the parameter $\alpha_{s}$ corresponds to the strong running
coupling constant. The values of various parameters used in the calculation
are as follows (The values of $\alpha_{s}$ is given for ground state):

Table 1: Parameters $m_{s}$ | $m_{c}$ | $m_{b}$ | $\mu$ | $\alpha_{s}(\Omega_{cc})$ | $\alpha_{s}(\Omega_{cb})$ | $\alpha_{s}(\Omega_{bb})$
---|---|---|---|---|---|---
0.500 GeV | 1.275 GeV | 4.670 GeV | 0.07 | -0.1230 | -0.0748 | -0.0355

This calculation has been done in the Mathematica notebook Lucha1999 . The
obtained masses are presented in section 3.

##  3 Mass Spectra and Regge Trajectories

In this section, mass spectra of $\Omega_{cc}$, $\Omega_{cb}$ and
$\Omega_{bb}$ baryons are presented. And Regge trajectories are also plotted
for these baryons using the calculated masses to justify the results. The
masses of radial (1S-6S) and orbital (1P-3P, 1D-2D, 1F-2F) states are
enumerated using the hCQM (described in section 2) and listed in Table 2, 3
and 4.

Mass specra of $\Omega_{cc}$ baryon is shown in Table 2 and its ground state
mass is 3.736 GeV as per our calculation, which is in good accordance with the
Refs. Juhicharm , Valcarce  and Ebert  with the difference of 16 MeV, 39 MeV
and 42 MeV respectively and differ by 70-100 MeV with other comparisions given
in Table 2. Our 1P state mass is quite near to the Refs. Valcarce  and Ebert ,
differ by only 2 MeV and 9 MeV respectively. The masses of higher excited
states are lower in comapare of other predictions, because screening potential
is incorporated as confining potential in our theoretical model.

For $\Omega_{cb}$ and $\Omega_{bb}$ baryons, our results for 1S state are
7.079 GeV and 10.357 GeV respectively. Ground state mass of $\Omega_{cb}$
baryon is in good agreement with the Refs. Ebert  and Tong2022  with
difference of only 9 MeV and 1 MeV. The further excited states are also
compatible with Refs. ShahEPJC2016  and Salehi . In case of $\Omega_{bb}$
baryon, our 1S state is quite near to the results of Refs. Ebert  and
Juhibottom  (differ by 2 MeV and 7 MeV respectivel). The result of first
orbital state is matching with the results by three-body Fadeev mathod
Valcarce  and relativistic approach Ebert  with difference of 61 MeV and 41
MeV respectively. The screening effect can be seen in higher excited states.
For $\Omega_{bb}$ baryon, the effect of scrrening potential is more than in
$\Omega_{cc}$ and $\Omega_{cb}$.

Table 2: Predicted masses of radial and orbital states of $\Omega_{cc}^{+}$ baryon (in GeV) State | Present | ShahEPJC2016  | Salehi  | Yoshida  | Roberts  | Valcarce  | Ebert  | Giannuzzi  | Juhicharm  | MathurCharmBottom
---|---|---|---|---|---|---|---|---|---|---
$1^{2}S_{\frac{1}{2}}$ | 3.736 | 3.650 | 3.662 | 3.832 | 3.815 | 3.697 | 3.778 | 3.648 | 3.752 | 3.698
$2^{2}S_{\frac{1}{2}}$ | 4.078 | 4.028 | 3.964 | 4.227 | 4.180 | 4.112 | 4.075 | 4.268 |  |
$3^{2}S_{\frac{1}{2}}$ | 4.320 | 4.317 | 4.177 | 4.295 |  |  | 4.321 |  |  |
$4^{2}S_{\frac{1}{2}}$ | 4.514 | 4.570 | 4.452 |  |  |  |  |  |  |
$5^{2}S_{\frac{1}{2}}$ | 4.676 | 4.801 | 4.787 |  |  |  |  |  |  |
$6^{2}S_{\frac{1}{2}}$ | 4.813 |  |  |  |  |  |  |  |  |
$1^{4}S_{\frac{3}{2}}$ | 3.837 | 3.810 | 3.677 | 3.883 | 3.876 | 3.697 | 3.872 | 3.770 | 3.816 |
$2^{4}S_{\frac{3}{2}}$ | 4.110 | 4.085 | 3.979 | 4.263 | 4.188 | 4.160 | 4.174 | 4.334 |  |
$3^{4}S_{\frac{3}{2}}$ | 4.334 | 4.345 | 4.193 | 4.265 |  |  |  |  |  |
$4^{4}S_{\frac{3}{2}}$ | 4.521 | 4.586 | 4.467 |  |  |  |  |  |  |
$5^{4}S_{\frac{3}{2}}$ | 4.680 | 4.811 | 4.802 |  |  |  |  |  |  |
$6^{4}S_{\frac{3}{2}}$ | 4.816 |  |  |  |  |  |  |  |  |
$1^{2}P_{\frac{1}{2}}$ | 4.011 | 3.964 | 3.965 | 4.086 | 4.046 | 4.009 | 4.002 |  |  |
$1^{2}P_{\frac{3}{2}}$ | 4.004 | 3.948 | 3.966 | 4.086 | 4.052 |  | 4.102 |  | 3.975 |
$1^{4}P_{\frac{1}{2}}$ | 4.014 | 3.972 | 3.987 |  |  |  |  |  |  |
$1^{4}P_{\frac{3}{2}}$ | 4.007 | 3.981 | 3.978 |  |  |  |  |  |  |
$1^{4}P_{\frac{5}{2}}$ | 3.998 | 3.935 | 3.926 | 4.220 | 4.152 |  |  |  | 4.094 |
$2^{2}P_{\frac{1}{2}}$ | 4.253 | 4.241 | 4.194 | 4.199 | 4.135 | 4.101 | 4.251 |  |  |
$2^{2}P_{\frac{3}{2}}$ | 4.248 | 4.228 | 4.155 | 4.201 | 4.140 |  | 4.345 |  |  |
$2^{4}P_{\frac{1}{2}}$ | 4.256 | 4.248 | 4.215 |  |  |  |  |  |  |
$2^{4}P_{\frac{3}{2}}$ | 4.251 | 4.234 | 4.176 |  |  |  |  |  |  |
$2^{4}P_{\frac{5}{2}}$ | 4.244 | 4.216 | 4.156 |  |  |  |  |  |  |
$3^{2}P_{\frac{1}{2}}$ | 4.453 | 4.492 | 4.453 |  |  |  |  |  |  |
$3^{2}P_{\frac{3}{2}}$ | 4.450 | 4.479 | 4.444 |  |  |  |  |  |  |
$3^{4}P_{\frac{1}{2}}$ | 4.454 | 4.498 | 4.475 |  |  |  |  |  |  |
$3^{4}P_{\frac{3}{2}}$ | 4.451 | 4.486 | 4.466 |  |  |  |  |  |  |
$3^{4}P_{\frac{5}{2}}$ | 4.447 | 4.469 | 4.414 |  |  |  |  |  |  |
$1^{2}D_{\frac{3}{2}}$ | 4.091 | 4.133 | 4.156 |  |  |  |  |  |  |
$1^{2}D_{\frac{5}{2}}$ | 4.082 | 4.113 | 4.116 | 4.264 | 4.202 |  |  |  | 4.186 |
$1^{4}D_{\frac{1}{2}}$ | 4.101 | 4.156 | 4.215 |  |  |  |  |  |  |
$1^{4}D_{\frac{3}{2}}$ | 4.094 | 4.141 | 4.193 |  |  |  |  |  |  |
$1^{4}D_{\frac{5}{2}}$ | 4.086 | 4.121 | 4.155 |  |  |  |  |  |  |
$1^{4}D_{\frac{7}{2}}$ | 4.075 | 4.095 | 4.086 |  |  |  |  |  | 4.352 |
$2^{2}D_{\frac{3}{2}}$ | 4.318 | 4.389 | 4.429 |  |  |  |  |  |  |
$2^{2}D_{\frac{5}{2}}$ | 4.312 | 4.372 | 4.391 | 4.299 | 4.232 |  |  |  |  |
$2^{4}D_{\frac{1}{2}}$ | 4.324 | 4.407 | 4.490 |  |  |  |  |  |  |
$2^{4}D_{\frac{3}{2}}$ | 4.320 | 4.395 | 4.468 |  |  |  |  |  |  |
$2^{4}D_{\frac{5}{2}}$ | 4.314 | 4.378 | 4.430 |  |  |  |  |  |  |
$2^{4}D_{\frac{7}{2}}$ | 4.307 | 4.358 | 4.360 |  |  |  |  |  |  |
$1^{2}F_{\frac{5}{2}}$ | 4.157 | 4.287 | 4.391 |  |  |  |  |  |  |
$1^{2}F_{\frac{7}{2}}$ | 4.147 | 4.264 | 4.353 |  |  |  |  |  | 4.387 |
$1^{4}F_{\frac{3}{2}}$ | 4.167 | 4.313 | 4.467 |  |  |  |  |  |  |
$1^{4}F_{\frac{5}{2}}$ | 4.160 | 4.294 | 4.429 |  |  |  |  |  |  |
$1^{4}F_{\frac{7}{2}}$ | 4.150 | 4.271 | 4.360 |  |  |  |  |  |  |
$1^{4}F_{\frac{9}{2}}$ | 4.139 | 4.244 | 4.292 |  |  |  |  |  | 4.599 |
$2^{2}F_{\frac{5}{2}}$ | 4.372 | 4.530 | 4.726 |  |  |  |  |  |  |
$2^{2}F_{\frac{7}{2}}$ | 4.365 | 4.509 | 4.688 |  |  |  |  |  |  |
$2^{4}F_{\frac{3}{2}}$ | 4.380 | 4.552 | 4.802 |  |  |  |  |  |  |
$2^{4}F_{\frac{5}{2}}$ | 4.374 | 4.536 | 4.764 |  |  |  |  |  |  |
$2^{4}F_{\frac{7}{2}}$ | 4.367 | 4.515 | 4.695 |  |  |  |  |  |  |
$2^{4}F_{\frac{9}{2}}$ | 4.359 | 4.490 | 4.627 |  |  |  |  |  |  |
Table 3: Predicted masses of radial and orbital states of $\Omega_{cb}^{0}$ baryon (in GeV) State | Present | ShahEPJC2016  | Salehi  | Roberts  | Ebert  | Giannuzzi  | Karliner2018  | Tong2022  | Mathur  | Mohanta
---|---|---|---|---|---|---|---|---|---|---
$1^{2}S_{\frac{1}{2}}$ | 7.079 | 7.136 | 7.329 | 7.136 | 7.088 | 6.994 | 7.013 | 7.078 | 6.994 | 6.946
$2^{2}S_{\frac{1}{2}}$ | 7.435 | 7.473 | 7.475 |  |  | 7.559 |  |  |  |
$3^{2}S_{\frac{1}{2}}$ | 7.688 | 7.753 | 7.609 |  |  | 7.976 |  |  |  |
$4^{2}S_{\frac{1}{2}}$ | 7.895 | 8.004 | 7.782 |  |  |  |  |  |  |
$5^{2}S_{\frac{1}{2}}$ | 8.073 | 8.236 | 7.993 |  |  |  |  |  |  |
$6^{2}S_{\frac{1}{2}}$ | 8.227 |  |  |  |  |  |  |  |  |
$1^{4}S_{\frac{3}{2}}$ | 7.182 | 7.187 | 7.339 | 7.187 | 7.130 | 7.017 |  |  | 7.056 |
$2^{4}S_{\frac{3}{2}}$ | 7.469 | 7.490 | 7.485 |  |  | 7.571 |  |  |  |
$3^{4}S_{\frac{3}{2}}$ | 7.703 | 7.761 | 7.619 |  |  | 7.985 |  |  |  |
$4^{4}S_{\frac{3}{2}}$ | 7.903 | 8.009 | 7.792 |  |  |  |  |  |  |
$5^{4}S_{\frac{3}{2}}$ | 8.077 | 8.239 | 8.002 |  |  |  |  |  |  |
$6^{4}S_{\frac{3}{2}}$ | 8.230 |  |  |  |  |  |  |  |  |
$1^{2}P_{\frac{1}{2}}$ | 7.369 | 7.375 | 7.476 |  |  |  |  |  |  |
$1^{2}P_{\frac{3}{2}}$ | 7.364 | 7.363 | 7.470 |  |  |  |  |  |  |
$1^{4}P_{\frac{1}{2}}$ | 7.371 | 7.381 | 7.490 |  |  |  |  |  |  |
$1^{4}P_{\frac{3}{2}}$ | 7.366 | 7.369 | 7.486 |  |  |  |  |  |  |
$1^{4}P_{\frac{5}{2}}$ | 7.360 | 7.353 | 7.451 |  |  |  |  |  |  |
$2^{2}P_{\frac{1}{2}}$ | 7.620 | 7.657 | 7.619 |  |  |  |  |  |  |
$2^{2}P_{\frac{3}{2}}$ | 7.616 | 7.647 | 7.595 |  |  |  |  |  |  |
$2^{4}P_{\frac{1}{2}}$ | 7.621 | 7.662 | 7.633 |  |  |  |  |  |  |
$2^{4}P_{\frac{3}{2}}$ | 7.618 | 7.652 | 7.610 |  |  |  |  |  |  |
$2^{4}P_{\frac{5}{2}}$ | 7.614 | 7.639 | 7.594 |  |  |  |  |  |  |
$3^{2}P_{\frac{1}{2}}$ | 7.832 | 7.912 | 7.782 |  |  |  |  |  |  |
$3^{2}P_{\frac{3}{2}}$ | 7.830 | 7.903 | 7.777 |  |  |  |  |  |  |
$3^{4}P_{\frac{1}{2}}$ | 7.833 | 7.916 | 7.796 |  |  |  |  |  |  |
$3^{4}P_{\frac{3}{2}}$ | 7.831 | 7.908 | 7.793 |  |  |  |  |  |  |
$3^{4}P_{\frac{5}{2}}$ | 7.828 | 7.896 | 7.758 |  |  |  |  |  |  |
$1^{2}D_{\frac{3}{2}}$ | 7.458 | 7.545 | 7.597 |  |  |  |  |  |  |
$1^{2}D_{\frac{5}{2}}$ | 7.452 | 7.531 | 7.571 |  |  |  |  |  |  |
$1^{4}D_{\frac{1}{2}}$ | 7.464 | 7.562 | 7.634 |  |  |  |  |  |  |
$1^{4}D_{\frac{3}{2}}$ | 7.460 | 7.551 | 7.618 |  |  |  |  |  |  |
$1^{4}D_{\frac{5}{2}}$ | 7.454 | 7.536 | 7.595 |  |  |  |  |  |  |
$1^{4}D_{\frac{7}{2}}$ | 7.447 | 7.518 | 7.552 |  |  |  |  |  |  |
$2^{2}D_{\frac{3}{2}}$ | 7.695 | 7.807 | 7.768 |  |  |  |  |  |  |
$2^{2}D_{\frac{5}{2}}$ | 7.691 | 7.795 | 7.744 |  |  |  |  |  |  |
$2^{4}D_{\frac{1}{2}}$ | 7.700 | 7.821 | 7.806 |  |  |  |  |  |  |
$2^{4}D_{\frac{3}{2}}$ | 7.697 | 7.812 | 7.792 |  |  |  |  |  |  |
$2^{4}D_{\frac{5}{2}}$ | 7.692 | 7.799 | 7.767 |  |  |  |  |  |  |
$2^{4}D_{\frac{7}{2}}$ | 7.687 | 7.784 | 7.724 |  |  |  |  |  |  |
$1^{2}F_{\frac{5}{2}}$ | 7.533 | 7.702 | 7.744 |  |  |  |  |  |  |
$1^{2}F_{\frac{7}{2}}$ | 7.526 | 7.685 | 7.720 |  |  |  |  |  |  |
$1^{4}F_{\frac{3}{2}}$ | 7.541 | 7.721 | 7.792 |  |  |  |  |  |  |
$1^{4}F_{\frac{5}{2}}$ | 7.535 | 7.708 | 7.768 |  |  |  |  |  |  |
$1^{4}F_{\frac{7}{2}}$ | 7.528 | 7.690 | 7.723 |  |  |  |  |  |  |
$1^{4}F_{\frac{9}{2}}$ | 7.519 | 7.670 | 7.681 |  |  |  |  |  |  |
$2^{2}F_{\frac{5}{2}}$ | 7.758 | 7.949 | 7.956 |  |  |  |  |  |  |
$2^{2}F_{\frac{7}{2}}$ | 7.753 | 7.938 | 7.931 |  |  |  |  |  |  |
$2^{4}F_{\frac{3}{2}}$ | 7.764 | 7.965 | 8.001 |  |  |  |  |  |  |
$2^{4}F_{\frac{5}{2}}$ | 7.760 | 7.953 | 7.978 |  |  |  |  |  |  |
$2^{4}F_{\frac{7}{2}}$ | 7.755 | 7.934 | 7.935 |  |  |  |  |  |  |
$2^{4}F_{\frac{9}{2}}$ | 7.749 | 7.921 | 7.892 |  |  |  |  |  |  |
Table 4: Predicted masses of radial and orbital states of $\Omega_{bb}^{-}$ baryon (in GeV) State | Present | ShahEPJC2016  | Salehi  | Yoshida  | Roberts  | Valcarce  | Ebert  | Giannuzzi  | Juhibottom  |  |
---|---|---|---|---|---|---|---|---|---|---|---
$1^{2}S_{\frac{1}{2}}$ | 10.357 | 10.446 | 10.870 | 10.447 | 10.454 | 10.293 | 10.359 | 10.271 | 10.350 |  |
$2^{2}S_{\frac{1}{2}}$ | 10.646 | 10.730 | 10.969 | 10.707 | 10.693 | 10.604 | 10.610 | 10.830 |  |  |
$3^{2}S_{\frac{1}{2}}$ | 10.859 | 10.973 | 11.036 | 10.744 |  |  | 10.806 | 11.240 |  |  |
$4^{2}S_{\frac{1}{2}}$ | 11.039 | 11.191 | 11.123 | 10.994 |  |  |  |  |  |  |
$5^{2}S_{\frac{1}{2}}$ | 11.196 | 11.393 | 11.230 |  |  |  |  |  |  |  |
$6^{2}S_{\frac{1}{2}}$ | 11.336 |  |  |  |  |  |  |  |  |  |
$1^{4}S_{\frac{3}{2}}$ | 10.406 | 10.467 | 10.866 | 10.467 | 10.486 | 10.321 | 10.389 | 10.289 | 10.449 |  |
$2^{4}S_{\frac{3}{2}}$ | 10.662 | 10.737 | 10.964 | 10.723 | 10.721 | 10.622 | 10.645 | 10.622 |  |  |
$3^{4}S_{\frac{3}{2}}$ | 10.866 | 10.976 | 11.032 | 10.730 |  |  | 10.843 | 11.247 |  |  |
$4^{4}S_{\frac{3}{2}}$ | 11.043 | 11.193 | 11.119 | 11.031 |  |  |  |  |  |  |
$5^{4}S_{\frac{3}{2}}$ | 11.199 | 11.394 | 11.225 |  |  |  |  |  |  |  |
$6^{4}S_{\frac{3}{2}}$ | 11.337 |  |  |  |  |  |  |  |  |  |
$1^{2}P_{\frac{1}{2}}$ | 10.580 | 10.634 | 10.968 | 10.607 | 10.616 | 10.519 | 10.532 |  |  |  |
$1^{2}P_{\frac{3}{2}}$ | 10.578 | 10.629 | 10.957 | 10.608 | 10.619 | 10.520 | 10.566 |  | 10.638 |  |
$1^{4}P_{\frac{1}{2}}$ | 10.581 | 10.636 | 10.976 |  |  |  |  |  |  |  |
$1^{4}P_{\frac{3}{2}}$ | 10.579 | 10.631 | 10.963 |  |  |  |  |  |  |  |
$1^{4}P_{\frac{5}{2}}$ | 10.576 | 10.625 | 10.956 |  |  |  |  |  | 10.729 |  |
$2^{2}P_{\frac{1}{2}}$ | 10.796 | 10.881 | 11.031 | 10.796 | 10.763 | 10.683 | 10.738 |  |  |  |
$2^{2}P_{\frac{3}{2}}$ | 10.795 | 10.877 | 11.029 | 10.797 | 10.765 |  | 10.775 |  |  |  |
$2^{4}P_{\frac{1}{2}}$ | 10.797 | 10.883 | 11.039 |  |  |  |  |  |  |  |
$2^{4}P_{\frac{3}{2}}$ | 10.796 | 10.879 | 11.035 |  |  |  |  |  |  |  |
$2^{4}P_{\frac{5}{2}}$ | 10.794 | 10.874 | 11.019 |  |  |  |  |  |  |  |
$3^{2}P_{\frac{1}{2}}$ | 10.982 | 11.104 | 11.124 |  |  |  |  |  |  |  |
$3^{2}P_{\frac{3}{2}}$ | 10.981 | 11.101 | 11.111 |  |  |  |  |  |  |  |
$3^{4}P_{\frac{1}{2}}$ | 10.982 | 11.106 | 11.131 |  |  |  |  |  |  |  |
$3^{4}P_{\frac{3}{2}}$ | 10.981 | 11.103 | 11.119 |  |  |  |  |  |  |  |
$3^{4}P_{\frac{5}{2}}$ | 10.980 | 11.098 | 11.112 |  |  |  |  |  |  |  |
$1^{2}D_{\frac{3}{2}}$ | 10.660 | 10.783 | 11.032 |  |  |  |  |  |  |  |
$1^{2}D_{\frac{5}{2}}$ | 10.657 | 10.777 | 11.017 | 10.729 | 10.720 |  |  |  | 10.918 |  |
$1^{4}D_{\frac{1}{2}}$ | 10.662 | 10.789 | 11.038 |  |  |  |  |  |  |  |
$1^{4}D_{\frac{3}{2}}$ | 10.661 | 10.785 | 11.033 |  |  |  |  |  |  |  |
$1^{4}D_{\frac{5}{2}}$ | 10.658 | 10.779 | 11.019 |  |  |  |  |  |  |  |
$1^{4}D_{\frac{7}{2}}$ | 10.655 | 10.772 | 11.007 |  |  |  |  |  | 11.002 |  |
$2^{2}D_{\frac{3}{2}}$ | 10.864 | 11.012 | 11.116 |  |  |  |  |  |  |  |
$2^{2}D_{\frac{5}{2}}$ | 10.863 | 11.008 | 11.104 |  |  |  |  |  |  |  |
$2^{4}D_{\frac{1}{2}}$ | 10.866 | 11.017 | 11.126 |  |  |  |  |  |  |  |
$2^{4}D_{\frac{3}{2}}$ | 10.865 | 11.014 | 11.119 |  |  |  |  |  |  |  |
$2^{4}D_{\frac{5}{2}}$ | 10.863 | 11.009 | 11.106 |  |  |  |  |  |  |  |
$2^{4}D_{\frac{7}{2}}$ | 10.861 | 11.004 | 11.094 |  |  |  |  |  |  |  |
$1^{2}F_{\frac{5}{2}}$ | 10.727 | 10.920 | 11.105 |  |  |  |  |  |  |  |
$1^{2}F_{\frac{7}{2}}$ | 10.724 | 10.913 | 11.082 |  |  |  |  |  | 11.191 |  |
$1^{4}F_{\frac{3}{2}}$ | 10.730 | 10.927 | 11.118 |  |  |  |  |  |  |  |
$1^{4}F_{\frac{5}{2}}$ | 10.728 | 10.922 | 11.107 |  |  |  |  |  |  |  |
$1^{4}F_{\frac{7}{2}}$ | 10.725 | 10.915 | 11.094 |  |  |  |  |  |  |  |
$1^{4}F_{\frac{9}{2}}$ | 10.721 | 10.907 | 11.076 |  |  |  |  |  | 11.268 |  |
$2^{2}F_{\frac{5}{2}}$ | 10.923 | 11.136 | 11.210 |  |  |  |  |  |  |  |
$2^{2}F_{\frac{7}{2}}$ | 10.921 | 11.130 | 11.189 |  |  |  |  |  |  |  |
$2^{4}F_{\frac{3}{2}}$ | 10.925 | 11.142 | 11.226 |  |  |  |  |  |  |  |
$2^{4}F_{\frac{5}{2}}$ | 10.923 | 11.137 | 11.214 |  |  |  |  |  |  |  |
$2^{4}F_{\frac{7}{2}}$ | 10.921 | 11.132 | 11.202 |  |  |  |  |  |  |  |
$2^{4}F_{\frac{9}{2}}$ | 10.919 | 11.125 | 11.179 |  |  |  |  |  |  |  |

Regge trajectories of $\Omega_{cc}$, $\Omega_{cb}$ and $\Omega_{bb}$ for
natural and unnatural parities are shown in Figure 2 \- 6. They are plotted in
$(J,M^{2})$ plane, using calculated masses presented in Table 2, 3 and 4. Our
plots are shown in F The nature of Regge trajectories for baryons is suppose
to be linear. Here, the Regge lines are linear at first sight, but for higher
excited states beyond the $J^{p}$ value $\frac{7}{2}^{-}$, they are going to
intersect each other. This is caused by the confining potential (screening
potential) used in our theoretical model, called screening effect.

Figure 1: The $M^{2}\rightarrow J$ Regge trajectory of $\Omega_{cc}^{+}$
baryon with natural parity

Figure 2: The $M^{2}\rightarrow J$ Regge trajectory of $\Omega_{cc}^{+}$
baryon with unnatural parity

Figure 3: The $M^{2}\rightarrow J$ Regge trajectory of $\Omega_{cb}^{0}$
baryon with natural parity

Figure 4: The $M^{2}\rightarrow J$ Regge trajectory of $\Omega_{cb}^{0}$
baryon with unnatural parity

Figure 5: The $M^{2}\rightarrow J$ Regge trajectory of $\Omega_{bb}^{-}$
baryon with natural parity

Figure 6: The $M^{2}\rightarrow J$ Regge trajectory of $\Omega_{bb}^{-}$
baryon with unnatural parity

##  4 Magnetic moment and Radiative decay

###  4.1 Magnetic Moment

Magnetic moment is an intrinsic property of a baryon, which can be determine
by using spin, charge and effective mass of its constituent quarks. The
expression of magnetic moment for a baryon can be obtain by operating the
spin-flavour wave function to the $z$-component of the magnetic moment
operator GandhiDecay ; BPatel ,

$\mu_{B}=\sum_{q}\langle\Phi_{sf}\|\hat{\mu}_{qz}\|\Phi_{sf}\rangle\quad\quad
q=s,c,b$ (8)

Here, $\Phi_{sf}$ represents the spin-flavour wave-function of the baryon and
$\hat{\mu}_{qz}$ is the magnetic moment operator.
The magnetic moment of individual quark is given by,

$\mu_{q}=\frac{e_{q}}{2m_{q}^{eff}}\cdot\sigma_{q}$ (9)

where, $e_{q}$ and $\sigma_{q}$ are charge and spin of the individual
constituent quark of the baryonic system respectively and $m_{q}^{eff}$ is the
effective mass of constituent quark which is expressed as given below
GandhiDecay :

$m_{q}^{eff}=m_{q}\left(1+\frac{\langle H\rangle}{\sum_{q}m_{q}}\right)$ (10)

Here, $\langle H\rangle=M-\sum_{q}m_{q}$; where, $M$ is predicted mass of the
particular baryonic state. And $m_{q}^{eff}$ is the mass of bounded quark
inside the baryon with consideration of the interaction with other two quarks.
The magnetic moment of spin states $\frac{1}{2}$ and $\frac{3}{2}$ for
$\Omega_{cc}$, $\Omega_{cb}$ and $\Omega_{bb}$ baryons are calculated using
above equations and masses presented in Section 3. Our results are shown in
the Table 5 and compared with the other magnetic moment results obtained by
different approaches. Our calculated magnetic moment results are differ by
0.02-0.52 $\mu_{N}$ from Ref. ShahEPJC2016 , by 0.007-0.4 $\mu_{N}$ from Ref.
BernotasPRD , by 0.004-0.1 $\mu_{N}$ from Ref. BPatel , by 0.01-0.1 from Ref.
Albertus  and by 0.01-1 $\mu_{N}$ from Ref. DhirPRD2021 .

Table 5: Magnetic moment of ground state of doubly heavy $\Omega$ baryons (in $\mu_{N}$) Baryon | $J^{p}$ | Expression | Present | ShahEPJC2016  | BernotasPRD  | BPatel  | Albertus  | DhirPRD2021  |
---|---|---|---|---|---|---|---|---|---
$\Omega_{cc}^{+}$ | $\frac{1}{2}^{+}$ | $\frac{4}{3}\mu_{c}-\frac{1}{3}\mu_{s}$ | 0.704 | 0.692 | 0.668 | 0.785 | 0.635 | 0.7109 |
|  |  |  |  |  |  |  | $\pm 0.0017$ |
$\Omega_{cb}^{0}$ | $\frac{1}{2}^{+}$ | $\frac{2}{3}\mu_{c}+\frac{2}{3}\mu_{b}-\frac{1}{3}\mu_{s}$ | 0.447 | 0.439 | 0.034 | 0.397 | 0.368 | -0.6202 |
|  |  |  |  |  |  |  | $\pm 0.00001$ |
$\Omega_{bb}^{-}$ | $\frac{1}{2}^{+}$ | $\frac{4}{3}\mu_{b}-\frac{1}{3}\mu_{s}$ | 0.113 | 0.108 | 0.120 | 0.109 | 0.101 | 0.1135 |
|  |  |  |  |  |  |  | $\pm 0.0008$ |
$\Omega_{cc}^{*+}$ | $\frac{3}{2}^{+}$ | $2\mu_{c}+\mu_{s}$ | 0.283 | 0.285 | 0.332 | 0.121 | 0.139 | 0.1871 |
|  |  |  |  |  |  |  | $\pm 0.0026$ |
$\Omega_{cb}^{*0}$ | $\frac{3}{2}^{+}$ | $\mu_{c}+\mu_{b}+\mu_{s}$ | -0.181 | -0.181 | -0.111 | -0.317 | -0.261 | -0.2552 |
|  |  |  |  |  |  |  | $\pm 0.0016$ |
$\Omega_{bb}^{*-}$ | $\frac{3}{2}^{+}$ | $2\mu_{b}+\mu_{s}$ | -0.718 | -1.239 | -0.730 | -0.711 | -0.662 | -0.6999 |
|  |  |  |  |  |  |  | $\pm 0.0017$ |

###  4.2 Radiative Decay

The $\gamma$ emission while changing the spin state $\frac{3}{2}$ to
$\frac{1}{2}$ is known as ‘Radiative decay’. The radiative decay width is
expressed as GandhiDecay ; Shah2016cpc ,

$\Gamma=\frac{k^{3}}{4\pi}\frac{2}{2J+1}\frac{e^{2}}{2m_{p}^{2}}\mu_{B\rightarrow
B^{\prime}}^{2}$ (11)

where, $k$ is photon energy, $J$ is total angular momentum of the initial
baryonic state, $m_{p}$ is the mass of proton (in MeV) and $\mu_{b}$
transition magnetic moment for the particular radiative decay.

The transition magnetic moment for the $\gamma$ emission, can be calculated by
the sandwiching spin-flavour wave functions of initial baryon state($B$) and
final baryon state($B^{\prime}$) with $z$ component of magnetic moment
operator, which is expressed as below:

$\mu_{B\rightarrow B^{\prime}}=\langle\Phi_{B}\|\mu_{B\rightarrow
B^{\prime}}\|\Phi_{B^{\prime}}\rangle$ (12)

The spin-flavour wave function of initial baryon($\Phi_{B}$) state and final
baryon state($\Phi_{B}^{\prime}$) can be determined as described in
Majethiya2009 .

The transition magnetic moment and radiative decay widths for doubly heavy
$\Omega$ baryons are shown in Table 6 and compared with the results of Ref. Li
and DhirPRD2021 .

Table 6: Transition magnetic moment (in $\mu_{N}$) and radiative decay width (in keV) of doubly heavy $\Omega$ baryons Transition | Expression | Transition magnetic moment | Radiative decay width |
---|---|---|---|---
|  | Present | Li  | DhirPRD2021  | Present | DhirPRD2021  |
$\Omega_{cc}^{*+}\rightarrow\Omega_{cc}^{+}\gamma$ | $\frac{\sqrt{2}}{3}(\mu_{s}-\mu_{b})$ | 0.847 | 0.96 | 0.9111 | 2.947 | 1.973 |
|  |  |  | $\pm 0.0013$ |  | $\pm 0.029$ |
$\Omega_{cb}^{*0}\rightarrow\Omega_{cb}^{0}\gamma$ | $\frac{\sqrt{2}}{3}(\mu_{s}-\mu_{b})$ | 0.713 | 0.69 | -0.7896 | 2.255 | 0.579 |
|  |  |  | $\pm 0.0010$ |  | $\pm 0.014$ |
$\Omega_{bb}^{*-}\rightarrow\Omega_{bb}^{-}\gamma$ | $\frac{\sqrt{2}}{3}(\mu_{s}-\mu_{b})$ | 0.499 | 0.48 | 0.4906 | 0.121 | 0.0426 |
|  |  |  | $\pm 0.0014$ |  | $\pm 0.0018$ |

##  5 Conclusion

The mass spectra of doubly heavy omega baryons with strangeness -1 for all
possible hyperfine states are determined in this work. To enumereate the
masses of radial and orbital states, Hypercentral Constituent Quark Model
(hCQM) has been utilized. In model, screening potential has been incorporated
as a confining part with the color-Coulomb potential. Mass spectra of
$\Omega_{cc}$, $\Omega_{cb}$ and $\Omega_{bb}$ baryons have been enumerated by
solving six-dimentional Schrödinger equation in Mathematica notebook. Our
results are in good agreement with the results from various theoretical
approaches ShahEPJC2016 ; Salehi ; Yoshida ; Roberts ; Valcarce ; Ebert ;
Giannuzzi ; Juhicharm ; Juhibottom ; Karliner2018 ; Tong2022 ; Mathur ;
MathurCharmBottom ; Mohanta . For $\Omega_{cc}$ baryon, the recent result
obtained by Lattice QCD calculation MathurCharmBottom  is quite matching with
our ground state with difference of 38 MeV. The first orbital state is only
differ by 47 MeV from Ref. ShahEPJC2016 . After 3P state, the screening effect
is visible in $\Omega_{cc}$ baryon. Looking into the results of $\Omega_{cb}$
baryon, our ground state is only 34 MeV differ from Refs. Mathur  and Roberts
, which are the results obtained using lattice QCD and variational approach
respectively. Our 1P states is only 6 MeV less than Ref. ShahEPJC2016 . From
3P state onwards, masses are going to suppress due to screened potential. The
ground state of $\Omega_{bb}$ baryon is only 7 MeV lower than Ref. Juhibottom
. Here, screening is visible after 2P state. Though doubly heavy baryon states
are not experimentally established, we have no experimental proof to justify
our results. Our goal is to offer a solid theoretical evidence to the
resonances that will be discovered experimentally in the near future. The
results calculated in this paper will be helpful to determine $J^{p}$ value of
newly detected experimental resonances.

After the mass spectra calculation, Regge trajectories are plotted in
$(J,M^{2})$ plane using the enumerated masses for $\Omega_{cc}$, $\Omega_{cb}$
and $\Omega_{bb}$ baryons. The nature of Regge trajectories validate the
obtained masses. Regge lines follow the linearity, but effect Regge lines
slightly seems contracting rather than being parallel, due to screening
effect.

The properties like ground state magnetic moment, transition magnetic moment
and radiative decay width for $\Omega_{cc}$, $\Omega_{cb}$ and $\Omega_{bb}$
baryons are determined using the calculated mass spectra. The results of
ground state magnetic moment are compared to Refs. ShahEPJC2016 ; BernotasPRD
; BPatel ; Albertus ; DhirPRD2021  and found quite near to them. Transition
magnetic moment and radiative dacay width are also calculated and compared to
Ref. DhirPRD2021  and Li . Our results are in accordance with the compared
results.

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11institutetext: Department of Diagnostic and Interventional Radiology,
University Hospital Aachen, Aachen, Germany 22institutetext: Else Kroener
Fresenius Center for Digital Health, Medical Faculty Carl Gustav Carus,
Technische Universität Dresden, Dresden, Germany 33institutetext: Department
of Medicine I, University Hospital Dresden, Dresden, Germany 44institutetext:
Medical Oncology, National Center for Tumor Diseases (NCT), University
Hospital Heidelberg, Heidelberg, Germany

# On Instabilities of Unsupervised Denoising Diffusion Models in Magnetic
Resonance Imaging Reconstruction

Tianyu Han 11 Sven Nebelung 11 Firas Khader 11 Jakob Nikolas Kather 223344
Daniel Truhn 11

###### Abstract

Denoising diffusion models offer a promising approach to accelerating magnetic
resonance imaging (MRI) and producing diagnostic-level images in an
unsupervised manner. However, our study demonstrates that even tiny worst-case
potential perturbations transferred from a surrogate model can cause these
models to generate fake tissue structures that may mislead clinicians. The
transferability of such worst-case perturbations indicates that the robustness
of image reconstruction may be compromised due to MR system imperfections or
other sources of noise. Moreover, at larger perturbation strengths, diffusion
models exhibit Gaussian noise-like artifacts that are distinct from those
observed in supervised models and are more challenging to detect. Our results
highlight the vulnerability of current state-of-the-art diffusion-based
reconstruction models to possible worst-case perturbations and underscore the
need for further research to improve their robustness and reliability in
clinical settings.

###### Keywords:

Magnetic Resonance Imaging Image Reconstruction Denoising Diffusion Models.

##  1 Introduction

Magnetic Resonance Imaging (MRI) is essential for medical diagnostics,
especially for brain diseases, due to its detailed, non-invasive imaging
capabilities. However, MRI faces challenges like long acquisition times and
high sensitivity to motion. Recent advancements, particularly denoising
diffusion models, promise to accelerate MRI by reconstructing high-quality
images from undersampled data. Unlike traditional methods, these models can
operate without paired training data. However, our study reveals a critical
vulnerability: susceptibility to minimal worst-case perturbations, leading to
significant inaccuracies in reconstructed images. Our research explores the
robustness of diffusion models in MRI reconstruction, investigating
adversarial perturbations and proposing strategies to enhance resilience. We
aim to advance reliable diffusion models in clinical settings.

##  2 Related Works

###  2.1 DL-based end-to-end solution

Model-based image reconstruction methods consider imaging systems as a linear
operator A that maps anatomical ground truth to the signal domain.
Specifically, a noisy observation y given by $m$ sparse measurements can be
defined as

$\textbf{y}\in\mathbb{R}^{m}=\textbf{Ax}+\epsilon,$ (1)

where $\textbf{x}\in\mathbb{R}^{n}$ is the unknown,
$\textbf{A}\in\mathbb{R}^{m\times n}$ ($n>m$) denotes an operator randomly
samples k-space data, and $\epsilon\in\mathbb{R}^{m}$ is the measurement
noise. When using standard Cartesian acquisition, we can factorize operator A
into an operator $\mathcal{P}(\Lambda)$ and an invertible matrix
$T\in\mathbb{R}^{n\times n}$ that corresponds to Fourier transform:
$\textbf{A}=\mathcal{P}(\Lambda)T$. K-space lines are selected during
acquisition if $\Lambda_{ii}=1$ in $\mathcal{P}(\Lambda)$, where
$\Lambda\in\\{0,1\\}^{n\times n}$ is a diagonal matrix with $tr(\Lambda)=m$.

Given the rise of DL, it’s a natural flow to approximate the inverse model
using convolutional neural networks (CNNs). In the CNN formulation, one force
x to be well-approximated by the CNN reconstruction by using the following
objective:

$\underset{\theta}{\min.}\;\lambda\left\|\textbf{x}-f_{\text{cnn}}(\textbf{A}^{H}\textbf{y}\,|\,\theta)\right\|_{2}^{2}+\left\|\textbf{Ax}-\textbf{y}\,\vphantom{f_{\text{cnn}}(\textbf{A}^{H}\textbf{y}\,|\,\theta)}\right\|_{2}^{2}.$
(2)

Here, we denote $f_{\text{cnn}}(\cdot\,|\,\theta)$ as a CNN parameterized by
$\theta$. The CNN reconstruction can be considered as resolving a de-aliasing
issue in the spatial domain because $\textbf{A}^{H}\textbf{y}$ is severely
affected by aliasing from sub-Nyquist sampling. However, the performance of
directly optimizing Equation 2 is inadequate since the CNN reconstruction and
the data fidelity are optimized independently. The CNN is specifically trained
to reconstruct the sequence without knowing the prior details of the obtained
data in k-space because it works purely in the image domain.

###  2.2 Bayesian image reconstruction

In the Bayesian picture, the MRI measurement y and tissue signal x are coupled
by a measurement distribution in this probabilistic formulation:
$p(\textbf{y}\,|\,\textbf{x})=q_{\epsilon}(\textbf{y}-\textbf{Ax})$, where
$q_{\epsilon}(\cdot)$ stands for the noise distribution. The conditional
distribution $p(\textbf{y}\,|\,\textbf{x})$ represents a forward process of
measuring y from x, which is also described by the linear forward model
(Equation 1). The reconstruction problem is then viewed as drawing samples
from the posterior distribution $p(\textbf{x}\,|\,\textbf{y})$. In general, we
can obtain such a posterior using Bayes’ theorem:
$p(\textbf{x}\,|\,\textbf{y})=p(\textbf{y}\,|\,\textbf{x})p(\textbf{x})/p(\textbf{y})$.
The following Bayes’ rule for score functions results from taking gradients
concerning $x$ on both sides of this expression: $\nabla_{\textbf{x}}\log
p(\textbf{x}\,|\,\textbf{y})=\nabla_{\textbf{x}}\log
p(\textbf{y}\,|\,\textbf{x})+\nabla_{\textbf{x}}\log p(\textbf{x})$. Note, the
data prior term $\nabla_{\textbf{x}}\log p(\textbf{x})$ can be efficiently
estimated by a denoising diffusion model [4, 13]. Incorporating measured
observation $\nabla_{\textbf{x}}\log p(\textbf{y}\,|\,\textbf{x})$ into our
system is an essential step that transforms an unguided diffusion into a
conditional one. In general, one can add such a correction term to
unconditional diffusion steps either via approximating the likelihood gradient
or directly performing closed-form data consistency.

###  2.3 Reconstructing MRI using diffusion prior and posterior sampling

Given a time-dependent diffusion-based model
$s_{\theta^{\ast}}(\textbf{x}_{t},t)$ that has been trained to approximate the
data score function via a diffusion process $\\{\textbf{x}_{t}\\}_{t=0}^{T}$
that was produced by perturbing x with an SDE. The procedure of unconditional
sampling chooses a series of time steps and iterates by
$\hat{x}_{t_{i-1}}=h(\hat{x}_{t_{i}},\,z_{i},\,s_{\theta^{\ast}}(\hat{\textbf{x}}_{t},t_{i}))$,
where function $h$ represents an SDE solver and
$z_{i}\sim\mathcal{N}(0,\textit{I})$. In MRI reconstruction, function $h$
needs an additional step $k$ prepended to itself to impose the constraint
implied by measurements, resulting in

$\displaystyle\hat{x}_{t_{i}}^{\prime}$
$\displaystyle=k(\hat{x}_{t_{i}},\,\hat{y}_{t_{i}},\,\lambda)$ (3)
$\displaystyle\hat{x}_{t_{i-1}}$
$\displaystyle=h(\hat{x}_{t_{i}}^{\prime},\,z_{i},\,s_{\theta^{\ast}}(\hat{\textbf{x}}_{t},t_{i})).$

The above $k$ function interacts with measured k-space entries and thus should
follow an image fidelity objective and a k-space consistency objective:

$\hat{x}_{t}^{\prime}=T^{-1}\left[\lambda\Lambda
P^{-1}(\Lambda)\hat{y}_{t}+(1-\lambda)\Lambda
T\hat{x}_{t}+(\textit{I}-\Lambda)T\hat{x}_{t}\right].$ (4)

##  3 Worst-Case Instabilities in MRI reconstruction

As demonstrated in Fig. 1, our worst-case noise is designed to be small in
k-space but can induce a significant mismatch in the slice recovered by the
undersampled version of the perturbed ground truth. Let $f$:
$\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ be a trained neural network $f$
mapping an undersampled measurement to an image. Finding an adversarial
direction $\delta\in\mathbb{R}^{m}$ in the measurement domain can be viewed as
solving an optimization with the following objective:

$\displaystyle\underset{\delta:\left\|\delta\right\|\leq\epsilon}{\max}\mathbb{E}_{\mathbf{y}}\left[\left\|f(\mathbf{y};\theta)-f(\mathbf{y}+\delta;\theta)\right\|_{2}^{2}\right].$
(5)

Here, we confine allowed perturbation sets to be a hypersphere $l_{2}$ ball
around any $\mathbf{y}$ with a norm $\epsilon$, i.e.,
$\left\|\delta\right\|_{2}\leq\epsilon\left\|\mathbf{y}\right\|_{2}$.
Following [1], a projected gradient descent (PGD) method [7] was used to
maximize the objective in Equation 5.

Figure 1:  MRI reconstruction can be vulnerable to worst-case perturbations,
which add noise to the original k-space signal and manipulate the
reconstruction process of undersampled data. The resulting reconstructions can
show false gray matter structures that are difficult for humans to detect (see
zoomed-in plots).

Both white- and black-box perturbations were evaluated in this study. In a
white-box scenario, we have full access to the target reconstruction model
$\theta$ and can perform gradient-based PGD attacks on its reconstruction.
Second, in the black-box setting, we tested the success rate of transferring
the above perturbations to the remaining models, e.g., diffusion-based
reconstructions.

Algorithm 1 Worst-case perturbation in k-space

A model with its current parameter $\theta$: $f_{\theta}$; Loss function:
$L=-L_{2}$; Partially measured k-space: $ksp$; Acquisition mask: $M$

Adam Optimizer $Opt$; Perturbation strength: $\epsilon$; Number of iterations:
$T$; Learning rate: $\alpha$; Constant $c=1e4$

Initialize perturbation: $\delta\leftarrow\delta_{r}+i\cdot\delta_{i}$, where
$\delta_{r},\delta_{i}\sim\mathcal{N}(0,\textit{I})$

$\delta\leftarrow\delta\times\frac{\|ksp\|}{\|\delta\|\times c}$

Initialize Optimizer:
$Opt\leftarrow\texttt{Adam}(\texttt{param}=[\delta],\texttt{lr}=\alpha)$

Get standard reconstruction: $x\leftarrow f_{\theta}(ksp\odot M,M)$

for i in range(T) do

$\hat{x}\leftarrow f_{\theta}((ksp+\delta)\odot M,M)$

$Opt.\texttt{zero\\_grad()}$

$loss\leftarrow L(x,\hat{x})$

$loss.\texttt{backward()}$

if $\|\delta\|>\epsilon$ then

$\delta\leftarrow\delta\times\frac{\|ksp|\|}{\|\delta\|}\cdot\epsilon$

end if

end for

return $\delta$

##  4 Experiments and Results

###  4.1 Data

All neuroimaging data were employed from the Alzheimer’s Disease Neuroimaging
Initiative (ADNI), a multicenter, longitudinal study of 2,463 participants
between the ages of 55 and 90 who had or were at high risk of developing
dementia and Alzheimer’s disease. Each participant has gone through $T_{1}$
weighted structure MRI measured by Magnetization Prepared RApid Gradient Echo
(MP-RAGE) [8]. For our study, we utilized 80% of the dataset, corresponding to
1,970 participants and 13,651 scans, for training, and randomly selected 108
scans from the remaining 20% of ADNI participants (493 individuals) for
testing. Using 80% of the dataset for training and validation follows a
standard 80-20 train-validation split, ensuring robust model evaluation.

###  4.2 Supervised baselines

CNNs have established a new state-of-the-art MRI reconstruction, vastly beyond
the traditional baselines. One typical approach utilizes auto-encoder
architectures, such as U-Net [12, 15], which solves the medical inverse
problem in an end-to-end fashion. We selected a Unet-based baseline
(ResUnet++) as it is the most widely used CNN backbone in MRI image
reconstruction. In experiments, we trained a ResUnet++ [5] model with a batch
size of 16 and a learning rate of 0.001, using 50 epochs on ADNI training set.
Another branch of models such as ADMM [14] and i-RIM [9, 10] generalizes the
idea of iterative compressed sensing reconstruction that unrolls the data-flow
graph via a cascade of neural networks. We selected i-RIM model due to its
superior performance in vairous MRI reconstruction challenges, especially, the
FastMRI challenge [9, 6]. In our experiments, the i-RIM model was trained
using a batch size of 4 and a learning rate of 0.001 with 50 epochs. We
trained both supervised models with an acceleration factor of 8.

###  4.3 Denoising diffusion reconstruction

Our training method is similar to Song et al [13]. Since the Predictor-
Corrector (PC) sampler has generally higher performance for VE-SDEs, we employ
it here in place of the numerical SDE solver to generate samples. The
corrector in this PC sampler follows Langevin dynamics that solely rely on the
scores, whereas the predictor refers to a numerical solver for the reverse-
time SDE. We perform an additional data consistency [11, 13] step to adapt the
PC sampler for solving inverse problems. 1,000 noise scales and 1 step of the
Langevin correction for each noise scale were selected, resulting in a total
of 2,000 steps of score model evaluation in the PC sampler. Besides, the
signal-to-noise ratio (SNR) $\eta$ controls the step size $\epsilon$ in
Langevin dynamics. In our setting, $\eta$ was set to 0.517 and the $\lambda$
in the data consistency operation was set to 1.0.

Figure 2:  We designed experiments to evaluate the susceptibility of trained
i-RIM and ResUnet++ models to white- and black-box attacks (a and b).

###  4.4 Experimental design

First, we trained supervised and unsupervised models on reconstruction tasks
in the training cohort and evaluated the performance in the test cohort with
three acceleration schemes. We demonstrated diffusion models deliver
comparable performance to state-of-the-art supervised models while showing
significantly better generalization to unknown acquisition processes. Then, as
demonstrated in Fig. 2, we assessed the robustness of the trained models
against white- and black-box adversarial perturbations. Both scenarios are
important as a white-box attack corresponds to malfunctions of an internal MRI
system, while black-box perturbations reveal vulnerabilities of current deep-
learning reconstruction towards possible external adversarial interferences.

###  4.5 Quantitative and visual evaluation

Figure 3:  We visualized the impact of perturbation amplitude on model
performance, measured by the $\Delta$SSIM metric. Subplot (a) shows that all
models experienced a drastic drop in SSIM as the perturbation amplitude
increased using worst-case perturbations generated by i-RIM. Similar findings
were observed with adversarial perturbations via the ResUnet model, in (b).

####  4.5.1 Worst-case instabilities of supervised models.

The results of our experiments are presented in Fig. 3, which shows the SSIM
loss as a function of perturbation strength. Subplots c and d of Fig. 3 show
the white-box perturbations [3] obtained by attacking i-RIM and ResUnet++
models, respectively. We found that both i-RIM (the blue line in subplot (a))
and ResUnet++ (the green line in subplot (b)) are unstable and can be easily
biased by tiny adversarial perturbations.

####  4.5.2 Worst-case transferability to diffusion models.

Figure 4:  To demonstrate, we crafted worst-case black-box perturbations [2]
using an independent ResUnet++ model and applied them to unsupervised
diffusion reconstruction and supervised i-RIM. The application of worst-case
inference to unsupervised reconstruction can create misleading artifacts in
brain tissue, which can be seen as red arrows in the subplot below.

Empirical evidence for the transferability of adversarial examples has been
investigated in classification applications, but rarely demonstrated in
regression tasks such as image reconstruction. In Fig. 2a, we explore a black-
box scenario where adversarial perturbations are generated against a surrogate
model, such as an i-RIM, rather than the actual models used for
reconstruction, such as diffusion. Similar to supervised models, as shown in
Figures 3, the unsupervised diffusion model is also susceptible to worst-case
distribution shifts in the form of adversarial perturbations. Even with a
minimal adversarial perturbation of $\epsilon=0.01$ crafted from ResUnet++
parameters, the diffusion model distorts the gray matter structure in its
reconstruction (Fig. 4). This result is significant because diffusion models
are trained similarly to denoisers, as evidenced by the appearance of a
Gaussian-noise-like artifact when we adversarially perturb them (Fig. 4).

##  5 Conclusion

In summary, our study highlights the vulnerability of both state-of-the-art
supervised models and unsupervised diffusion models to adversarial
perturbations from the MRI signal domain. We found that worst-case
perturbations can effectively transfer between independently trained
regression models, similar to the transferability observed in classification
tasks. While diffusion models are generally robust against anatomical and
test-time distribution shifts, our findings indicate that even tiny
adversarial perturbations can cause these models to generate fake tissue
structures that may mislead clinicians. Furthermore, at larger perturbation
amplitudes, diffusion models exhibit noise-like artifacts that are distinct
from those observed in supervised models and may be more difficult for
clinicians to detect.

We hypothesize that the main reason for this vulnerability is due to the
perturbed K-space misleads the reverse iterative diffusion process, creating
nonphysical artifacts. Classical regularization techniques like total variance
regularization might offer better robustness in such scenarios.

####  5.0.1

The authors have no competing interests to declare that are relevant to the
content of this article.

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## Supplementary Information

Figure S1:  Comparisons of the pSNR change of all models against
$l_{2}$-bounded adversaries with increasing perturbation amplitudes in the
frequency space. left, A worst-case perturbation was generated by using the
trained i-RIM model. Comparisons in the left subplot correspond to our
experiments outlined in Fig. 2a. Right, A worst-case perturbation was
generated by using the ResUnet++ model. Comparisons in the right subplot
correspond to our experiments outlined in Fig. 2b.  Figure S2:  Top,
Reconstructions of both supervised and unsupervised methods under the mid-
level perturbation generated from ResUnet++. Bottom, Mean absolute error map
of all reconstructions. The diffusion model, trained like a denoiser, exhibits
an unstable reconstruction with a distinct noisy pattern that is difficult to
distinguish from the clean reconstruction.