# Magnetic moments of the vector hidden-charmed tetraquark states

Ulaş Özdem<EMAIL_ADDRESS>Health Services Vocational School of Higher
Education, Istanbul Aydin University, Sefakoy-Kucukcekmece, 34295 Istanbul,
Turkey

###### Abstract

The magnetic moments of the vector hidden-charmed tetraquark states that have
been observed and can be expected to be observed experimentally have been
determined using the light-cone sum rules taking into account the diquark-
antidiquark structure with the quantum numbers $J^{PC}=1^{--}$ and
$J^{PC}=1^{-+}$. Since these states are considered to have different flavors
of light quarks, they have nonzero magnetic moments. The results obtained in
this study can be checked for consistency by various methods. The magnetic
moments of hadrons encompass useful knowledge about the distribution of charge
and magnetization inside hadrons, which helps us to understand their
geometrical shapes.

Magnetic moment, vector hidden-charmed tetraquark states, Light-cone sum rules

##  I Introduction

Theoretically, the existence of states with a larger number of quarks besides
baryons and mesons was proposed long ago. However, the first experimental
discovery of these states occurred in 2003 with the observation of the X(3872)
state by the Belle Collaboration [1]. After the discovery of this particle,
various experimental collaborations discovered many particles belonging to
this new family that are still being discovered. These newly discovered states
not only arouse the interest of particle physicists, but also raise new
questions about their inner structure and quantum numbers. Many models have
been proposed to explain and decipher the nature of these states, and several
studies have been conducted on them. However, their properties remain dubious,
and their substructures and quantum numbers are also problematic. The
properties of the reported tetraquark states have been interpreted differently
in different studies. To resolve all these ambiguities, the properties of both
the known and the newly observed states need to be further investigated. These
studies could investigate complementary reactions or other decay modes for the
currently known tetraquark states, or novel particles that may be observed can
be investigated for their spectroscopic properties or possible decay modes to
provide input for the experiments. Several interesting reviews provide
detailed information on unconventional states, including a history of the
subject and experimental and theoretical breakthroughs in recent years [2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].

Several vector hidden-charmed tetraquark states, such as Y(4220/4260),
Y(4360/4390), Y(4630/4660) and so on, have been observed in recent years that
cannot be well correlated in the standard meson with two quarks. The family of
exotic vector states ($Y_{c\bar{c}}$ for short), called tetraquarks, contains
at least four particles of hidden-charm with quantum numbers $J^{PC}$ =
$1^{--}$. In order to understand the nature of these states, many different
models have been proposed and studies have been conducted on them (details of
which can be found in the reviews [4, 9, 12, 13]). In Refs. [15, 16, 17, 18,
19, 20], the QCD sum rules have been also employed to explore the
spectroscopic parameters of these states. In Ref. [15], a large number of
interpolation currents were constructed for the $Y_{c\bar{c}}$ states, and the
spectroscopic parameters of these states were studied using QCD sum rules with
$J^{PC}$ = $1^{++}$, $J^{PC}$ = $1^{--}$, $J^{PC}$ = $1^{-+}$ and $J^{PC}$ =
$1^{+-}$, and quark contents $[cq][\bar{c}\bar{q}]$ and
$[cs][\bar{c}\bar{s}]$. It was taken into account that these states are in the
diquark-antidiquark structure. While some of the obtained results are
compatible with the experimentally discovered $Y_{c\bar{c}}$ states, they
turned out to be incompatible with some of them. Moreover, some possible decay
channels and the experimental search for these states are also discussed. In
Ref. [16], the mass and residue of the Y(4660) state were determined in the
framework of the QCD sum rules. They found that $c\bar{c}s\bar{s}$ and
$c\bar{c}(u\bar{u}+d\bar{d})/\sqrt{2}$ diquark-antidiquark states favor the
Y(4660) state with quantum numbers $J^{PC}$ = $1^{--}$. They also excluded
$c\bar{c}u\bar{d}$ diquark-antidiquark structure with quantum numbers $J^{PC}$
= $1^{\pm-}$ for the Y(4360) state. In Ref. [17], they constructed different
types of currents to interpolate both the vector and axial vector tetraquark
states and obtain the spectroscopic parameters of the $Y_{c\bar{c}}$ states
within the QCD sum rules. The numerical results support the assignment of the
Y(4660) as a diquark-antidiquark type tetraquark state with quantum numbers
$J^{PC}=1^{--}$. It has also been suggested that Y(4260) and Y(4360) may be
mixed charmonium-tetraquark states. In Ref. [18], the tetraquark states of
type $C\otimes\gamma_{\mu}C$ and $C\gamma_{5}\otimes\gamma_{5}\gamma_{\mu}C$
were constructed to calculate the mass and residue of the $Y_{c\bar{c}}$
states. Their analysis supported the assignment of Y(4660) and Y(4630) as
vector tetraquark states of type $C\otimes\gamma_{\mu}C$ $c\bar{c}s\bar{s}$,
assign Y(4360) and Y(4320) to the vector tetraquark state $c\bar{c}q\bar{q}$
of type $C\gamma_{5}\otimes\gamma_{5}\gamma_{\mu}C$ and do not assign Y(4260),
Y(4220) and Y(4390) to the fixed vector tetraquark states. In Ref. [19], the
mass, decay constant, and strong decay channels of the Y(4660) state were
evaluated by treating it as a bound state of a diquark and an antidiquark
($[cs][\bar{c}\bar{s}]$). It was shown that the results for the mass and total
width of this state are in good agreement with the experimental data. In Ref.
[20], they constructed the scalar, pseudoscalar, vector, axial vector, and
tensor antidiquark states to obtain the mass spectrum of the vector tetraquark
states with hidden charm via the QCD sum rules. Their predictions supported
the identification of Y(4360), Y(4390), and Y(4660) as
$[cq][\bar{c}\bar{q^{\prime}}]$ vector tetraquark states with hidden charm
with $J^{PC}=1^{--}$.

In addition to their spectroscopic properties, the electromagnetic form
factors and multipole moments of hadrons can provide clues to their precise
character, internal structure, and quantum numbers. We know that the
electromagnetic multipole moments of hadrons, in particular their magnetic
moments, which encompasses knowledge about the spatial distribution of charge
and magnetization inside the hadrons, are related to the spatial distribution
of quarks and gluons inside them. The study of the magnetic and higher
multipole moments of hadrons is therefore attractive. In this study, we
compute the magnetic moments of $Y_{c\bar{c}}$ states in the diquark-
antidiquark configuration with the quantum numbers $J^{PC}=1^{--}$ and
$J^{PC}=1^{-+}$ using the light-cone sum rule method [21, 22, 23]. The light-
cone sum rule method is based on the operator product expansion near the
light-cone $x^{2}\sim 0$ and parametrizes all the non-perturbative dynamics in
the distribution amplitudes that have been used to treat many electromagnetic
properties of conventional and non-conventional hadrons.

This article is structured in the following manner. After the introduction in
Sec. I, we present in Sec. II the formalism of the light-cone sum rule, which
identifies the necessary tools to compute the magnetic moments of the
$Y_{c\bar{c}}$ states. In Sec. III, we use the analytical formulas obtained in
the previous section to perform numerical calculations of the magnetic moments
and discuss the results.

##  II Light-cone sum rule formalism for magnetic moments

In the light-cone sum rule technique, we compute a correlation function, which
serves as the building block of the method, twice: once in terms of hadronic
quantities such as coupling constants, form factors and electromagnetic
multipole moments and second in terms of QCD parameters and photon
distribution amplitudes available for different twists. The coefficients of
the corresponding Lorentz structures from both representations of the
correlation function are then equated and the quark-hadron duality approach is
used to obtain the desired physical quantity.

As we have mentioned above, at the beginning of the analytic calculations of
the magnetic moments it is necessary to write the correlation function, which
plays an important role in the light-cone sum rules and is written as follows

$\Pi_{\mu\nu}(p,q)=i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\mu}^{i}(x)J_{\nu}^{i\dagger}(0)\\}|0\rangle_{\gamma},$ (1)

where $\mathcal{T}$, $J_{\mu}^{i}(x)$ and $\gamma$ represent the time-ordered
product of two currents, the interpolating current of $Y_{c\bar{c}}$ states
and the external electromagnetic field, respectively. We need explicit
expressions for $J_{\mu}^{i}(x)$ to make progress in the calculations. In the
diquark-antidiquark picture, $J_{\mu}^{i}(x)$ can be written in the following
forms [20]

$\displaystyle J^{1}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon\bar{\varepsilon}}{\sqrt{2}}\Big{\\{}\big{[}u^{Tj}(x)Cc^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)\big{]}-\big{[}u^{Tj}(x)C\gamma_{\mu}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)C\bar{c}^{Tn}(x)\big{]}\Big{\\}},$
$\displaystyle J^{2}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon\bar{\varepsilon}}{\sqrt{2}}\Big{\\{}\big{[}u^{Tj}(x)Cc^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)\big{]}+\big{[}u^{Tj}(x)C\gamma_{\mu}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)C\bar{c}^{Tn}(x)\big{]}\Big{\\}},$
$\displaystyle J^{3}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon\bar{\varepsilon}}{\sqrt{2}}\Big{\\{}\big{[}u^{Tj}(x)C\gamma_{5}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{5}\gamma_{\mu}C\bar{c}^{Tn}(x)\big{]}+\big{[}u^{Tj}(x)C\gamma_{\mu}\gamma_{5}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{5}C\bar{c}^{Tn}(x)\big{]}\Big{\\}},$
$\displaystyle J^{4}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon\bar{\varepsilon}}{\sqrt{2}}\Big{\\{}\big{[}u^{Tj}(x)C\gamma_{5}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{5}\gamma_{\mu}C\bar{c}^{Tn}(x)\big{]}-\big{[}u^{Tj}(x)C\gamma_{\mu}\gamma_{5}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{5}C\bar{c}^{Tn}(x)\big{]}\Big{\\}},$
$\displaystyle J_{\mu}^{5}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon\bar{\varepsilon}}{\sqrt{2}}\Big{\\{}\big{[}u^{Tj}(x)C\sigma_{\mu\nu}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma^{\nu}C\bar{c}^{Tn}(x)\big{]}-\big{[}u^{Tj}(x)C\gamma^{\nu}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\sigma_{\mu\nu}C\bar{c}^{Tn}(x)\big{]}\Big{\\}},$
$\displaystyle J_{\mu}^{6}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon\bar{\varepsilon}}{\sqrt{2}}\Big{\\{}\big{[}u^{Tj}(x)C\sigma_{\mu\nu}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma^{\nu}C\bar{c}^{Tn}(x)\big{]}+\big{[}u^{Tj}(x)C\gamma^{\nu}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\sigma_{\mu\nu}C\bar{c}^{Tn}(x)\big{]}\Big{\\}},$
$\displaystyle J_{\mu}^{7}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon\bar{\varepsilon}}{\sqrt{2}}\Big{\\{}\big{[}u^{Tj}(x)C\sigma_{\mu\nu}\gamma_{5}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{5}\gamma^{\nu}C\bar{c}^{Tn}(x)\big{]}+\big{[}u^{Tj}(x)C\gamma^{\nu}\gamma_{5}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{5}\sigma_{\mu\nu}C\bar{c}^{Tn}(x)\big{]}\Big{\\}},$
$\displaystyle J_{\mu}^{8}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon\bar{\varepsilon}}{\sqrt{2}}\Big{\\{}\big{[}u^{Tj}(x)C\sigma_{\mu\nu}\gamma_{5}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{5}\gamma^{\nu}C\bar{c}^{Tn}(x)\big{]}-\big{[}u^{Tj}(x)C\gamma^{\nu}\gamma_{5}c^{k}(x)\big{]}\big{[}\bar{d}^{m}(x)\gamma_{5}\sigma_{\mu\nu}C\bar{c}^{Tn}(x)\big{]}\Big{\\}},$
(2)

where $\varepsilon=\varepsilon^{ijk}$, $\bar{\varepsilon}=\varepsilon^{imn}$,
the i, j, k, m, n are color indices and the C is the charge conjugation
matrix. Here the quantum numbers of $J^{1}_{\mu}$, $J^{3}_{\mu}$,
$J^{5}_{\mu}$ and $J^{7}_{\mu}$ are the $J^{PC}=1^{--}$, and the quantum
numbers of $J^{2}_{\mu}$, $J^{4}_{\mu}$, $J^{6}_{\mu}$ and $J^{8}_{\mu}$ are
the $J^{PC}=1^{-+}$. At this point it should be noted that there are no
experimentally observed vector hidden-charmed tetraquark states with the
quantum numbers $J^{PC}=1^{-+}$.

In the hadronic language, a complete set of hadronic states is inserted and
the contributions of the lowest $Y_{c\bar{c}}$ states are separated to obtain
the corresponding correlation function,

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)={\frac{\langle 0\mid J_{\mu}(x)\mid
Y_{c\bar{c}}(p,\varepsilon^{\theta})\rangle}{p^{2}-m_{Y_{c\bar{c}}}^{2}}}\langle
Y_{c\bar{c}}(p,\varepsilon^{\theta})\mid
Y_{c\bar{c}}(p+q,\varepsilon^{\delta})\rangle_{\gamma}\frac{\langle
Y_{c\bar{c}}(p+q,\varepsilon^{\delta})\mid{J_{\nu}^{\dagger}}(0)\mid
0\rangle}{(p+q)^{2}-m_{Y_{c\bar{c}}}^{2}}+\cdots,$ (3)

where dots denote the effects of the higher states and continuum. The matrix
elements in Eq. (3) are expressed as

$\displaystyle\langle
Y_{c\bar{c}}(p+q,\varepsilon^{\delta})\mid{J_{\nu}^{\dagger}}(0)\mid 0\rangle$
$\displaystyle=\lambda_{Y_{c\bar{c}}}\varepsilon_{\nu}^{\delta}\,,$ (4)
$\displaystyle\langle 0\mid J_{\mu}(x)\mid
Y_{c\bar{c}}(p,\varepsilon^{\theta})\rangle$
$\displaystyle=\lambda_{Y_{c\bar{c}}}\varepsilon_{\mu}^{\theta}\,,$ (5)
$\displaystyle\langle Y_{c\bar{c}}(p,\varepsilon^{\theta})\mid
Y_{c\bar{c}}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$
$\displaystyle=-\varepsilon^{\tau}(\varepsilon^{\theta})^{\alpha}(\varepsilon^{\delta})^{\beta}\Big{\\{}G_{1}(Q^{2})~{}(2p+q)_{\tau}~{}g_{\alpha\beta}+G_{2}(Q^{2})~{}(g_{\tau\beta}~{}q_{\alpha}-g_{\tau\alpha}~{}q_{\beta})$
$\displaystyle-\frac{1}{2m_{Y_{c\bar{c}}}^{2}}G_{3}(Q^{2})~{}(2p+q)_{\tau}~{}q_{\alpha}q_{\beta}\Big{\\}},$
(6)

where $\varepsilon^{\tau}$ is polarization of the photon,
$\lambda_{Y_{c\bar{c}}}$ is residue of the $Y_{c\bar{c}}$ states and
$G_{i}(Q^{2})$’s are electromagnetic form factors, with $Q^{2}=-q^{2}$. To
calculate the magnetic moment, we need only $G_{2}(Q^{2})$ of the form factors
described above. The magnetic form factor, $F_{M}(Q^{2})$, is written as
follows

$\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,.$ (7)

Using Eqs. (3)-(6) and after doing some necessary calculations the final form
of the correlation function is obtained as

$\displaystyle\Pi_{\mu\nu}^{Had}(p,q)$
$\displaystyle=\frac{\varepsilon_{\rho}\,\lambda_{Y_{c\bar{c}}}^{2}}{[m_{Y_{c\bar{c}}}^{2}-(p+q)^{2}][m_{Y_{c\bar{c}}}^{2}-p^{2}]}\Big{\\{}G_{2}(Q^{2})\Big{(}q_{\mu}g_{\rho\nu}-q_{\nu}g_{\rho\mu}-\frac{p_{\nu}}{m_{Y_{c\bar{c}}}^{2}}\big{(}q_{\mu}p_{\rho}-\frac{1}{2}Q^{2}g_{\mu\rho}\big{)}+$
$\displaystyle+\frac{(p+q)_{\mu}}{m_{Y_{c\bar{c}}}^{2}}\big{(}q_{\nu}(p+q)_{\rho}+\frac{1}{2}Q^{2}g_{\nu\rho}\big{)}-\frac{(p+q)_{\mu}p_{\nu}p_{\rho}}{m_{Y_{c\bar{c}}}^{4}}\,Q^{2}\Big{)}$
$\displaystyle+\mbox{other independent structures}\Big{\\}}\,+\cdots.$ (8)

The $F_{M}(Q^{2}=0)$ is proportional to the magnetic moment
$\mu_{Y_{c\bar{c}}}$:

$\displaystyle\mu_{Y_{c\bar{c}}}=\frac{e}{2\,m_{Y_{c\bar{c}}}}\,F_{M}(0).$ (9)

The correlation function is determined in terms of the QCD degrees of freedom
and the photon distribution amplitudes in the second step of the calculation
of the magnetic moment of $Y_{c\bar{c}}$ states. In the QCD representation, we
use Wick’s theorem to contract the corresponding quark fields after replacing
the explicit expressions of the interpolating currents in the correlation
function. For instance, the result for the current $J_{\mu}^{1}$ is as
follows:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,q)$ $\displaystyle=$
$\displaystyle
i\frac{\varepsilon\bar{\varepsilon}\varepsilon^{\prime}\bar{\varepsilon}^{\prime}}{2}\int
d^{4}xe^{ip\cdot x}\langle
0\mid\Bigg{\\{}\mathrm{Tr}\Big{[}S_{c}^{kk^{\prime}}(x)\widetilde{S}_{u}^{jj^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{n^{\prime}n}(-x)\gamma_{\nu}S_{d}^{m^{\prime}m}(-x)\Big{]}$
(10)
$\displaystyle-\mathrm{Tr}\Big{[}S_{c}^{kk^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{u}^{jj^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{n^{\prime}n}(-x)S_{d}^{m^{\prime}m}(-x)\Big{]}$
$\displaystyle-\mathrm{Tr}\Big{[}\gamma_{\mu}S_{c}^{kk^{\prime}}(x)\widetilde{S}_{u}^{jj^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\widetilde{S}_{c}^{n^{\prime}n}(-x)\gamma_{\nu}S_{d}^{m^{\prime}m}(-x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{\mu}S_{c}^{kk^{\prime}}(x)\gamma_{\nu}\widetilde{S}_{u}^{jj^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\widetilde{S}_{c}^{n^{\prime}n}(-x)S_{d}^{m^{\prime}m}(-x)\Big{]}\Bigg{\\}}\mid
0\rangle_{\gamma},$

where

$\widetilde{S}_{c(q)}^{ij}(x)=CS_{c(q)}^{ij\mathrm{T}}(x)C,$

with $S_{q(c)}(x)$ being the quark propagators. In the $x$-space for the
light-quark propagator we use in the $m_{q}\rightarrow 0$ limit

$\displaystyle S_{q}(x)$
$\displaystyle=i\frac{{x\\!\\!\\!/}}{2\pi^{2}x^{4}}-\frac{\langle\bar{q}q\rangle}{12}-\frac{\langle\bar{q}q\rangle}{192}m_{0}^{2}x^{2}-\frac{ig_{s}}{32\pi^{2}x^{2}}~{}G^{\mu\nu}(x)\Big{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\mu\nu}+\sigma_{\mu\nu}\hbox
to0.0pt{/\hss}{x}\Big{]},$ (11)

where $\langle\bar{q}q\rangle$ is light quark condensate, $m_{0}^{2}$ is
defined via the relation $\langle
0\mid\bar{q}\,g_{s}\,\sigma_{\alpha\beta}\,G^{\alpha\beta}\,q\mid
0\rangle=m_{0}^{2}\,\langle\bar{q}q\rangle$.

The charm-quark propagator is given, in association with the second kind
Bessel functions $K_{i}(x)$, as

$\displaystyle S_{c}(x)$
$\displaystyle=\frac{m_{c}^{2}}{4\pi^{2}}\Bigg{[}\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{(\sqrt{-x^{2}})^{2}}\Bigg{]}-\frac{g_{s}m_{c}}{16\pi^{2}}\int_{0}^{1}dv\,G^{\mu\nu}(vx)\Bigg{[}\big{(}\sigma_{\mu\nu}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\mu\nu}\big{)}$
$\displaystyle\times\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}+2\sigma_{\mu\nu}K_{0}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}\Bigg{]}.$
(12)

where $v$ is line variable and $G^{\mu\nu}$ is the gluon field strength
tensor. The perturbative or free component of the propagators of the light and
heavy quarks corresponds to the first term, while the remainder belongs to the
interacting parts (with gluon or QCD vacuum) as nonperturbative contributions.

The correlation function in Eq. (10) includes different types of
contributions: the photon can be emitted both perturbatively or non-
perturbatively. In first case, one of the free light or heavy quark
propagators in Eq. (10) is replaced by

$\displaystyle S^{free}\rightarrow\int d^{4}y\,S^{free}(x-y)\,\hbox
to0.0pt{/\hss}{\\!A}(y)\,S^{free}(y)\,,$ (13)

the remaining propagators are replaced with the full quark propagators. The
light-cone sum rule analyses are most conveniently done in the fixed-point
gauge. The most important advantage of a fixed-point gauge is that the
external field is expressed as being associated with the field strength
tensor. For the electromagnetic field, it is defined by $x_{\mu}A^{\mu}=0$. In
this gauge, the external electromagnetic potential is given by

$\displaystyle
A_{\alpha}=-\frac{1}{2}F_{\alpha\beta}y^{\beta}=-\frac{1}{2}(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})\,y^{\beta}.$
(14)

Equation (14) is plugged into Eq. (13), as a result of which we obtain

$\displaystyle
S^{free}\rightarrow-\frac{1}{2}(\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha})\int\,d^{4}y\,y^{\beta}\,S^{free}(x-y)\,\gamma_{\alpha}\,S^{free}(y)\,,$
(15)

After some lengthy calculations for $S_{q}^{free}$ and $S_{c}^{free}$, we
obtain their final form as follows:

$\displaystyle
S_{q}^{free}=\frac{e_{q}}{32\pi^{2}x^{2}}\Big{(}\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha}\Big{)}\Big{(}x\\!\\!\\!/\sigma_{\alpha\beta}+\sigma_{\alpha\beta}x\\!\\!\\!/\Big{)},$
$\displaystyle
S_{c}^{free}=-i\frac{e_{c}m_{c}}{32\pi^{2}}\Big{(}\varepsilon_{\alpha}q_{\beta}-\varepsilon_{\beta}q_{\alpha}\Big{)}\Big{[}2\sigma_{\alpha\beta}K_{0}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}+\frac{K_{1}\Big{(}m_{c}\sqrt{-x^{2}}\Big{)}}{\sqrt{-x^{2}}}\Big{(}x\\!\\!\\!/\sigma_{\alpha\beta}+\sigma_{\alpha\beta}x\\!\\!\\!/\Big{)}\Big{]}.$
(16)

In the second case one of the light quark propagators in Eq. (10) is replaced
by

$\displaystyle
S_{\alpha\beta}^{ab}\rightarrow-\frac{1}{4}(\bar{q}^{a}\Gamma_{i}q^{b})(\Gamma_{i})_{\alpha\beta},$
(17)

and the remaining propagators are full quark propagators including the
perturbative as well as the nonperturbative contributions. Here as an example,
we give a short detail of the calculations of the QCD representations. In
second case for simplicity, we only consider the first trace in Eq. (10),

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,q)$ $\displaystyle=$
$\displaystyle
i\frac{\varepsilon\bar{\varepsilon}\varepsilon^{\prime}\bar{\varepsilon}^{\prime}}{2}\int
d^{4}xe^{ip\cdot x}\langle
0\mid\Bigg{\\{}\mathrm{Tr}\Big{[}S_{c}^{kk^{\prime}}(x)\Gamma_{i}\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{n^{\prime}n}(-x)\gamma_{\nu}S_{d}^{m^{\prime}m}(-x)\Big{]}$
(18)
$\displaystyle+\mathrm{Tr}\Big{[}S_{c}^{kk^{\prime}}(x)\widetilde{S}_{u}^{jj^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{n^{\prime}n}(-x)\gamma_{\nu}\Gamma_{i}\Big{]}\Bigg{\\}}\mid
0\rangle_{\gamma}+....,$

where
$\Gamma_{i}=I,\gamma_{5},\gamma_{\mu},i\gamma_{5}\gamma_{\mu},\sigma_{\mu\nu}/2$.

By replacing one of light propagators with the expressions in Eq. (II) and
making use of

$\displaystyle\bar{q}^{a}(x)\Gamma_{i}q^{a^{\prime}}(0)\rightarrow\frac{1}{3}\delta^{aa^{\prime}}\bar{q}(x)\Gamma_{i}q(0),$
(19)

Eq. (18) takes the form

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,q)$ $\displaystyle=$
$\displaystyle
i\frac{\varepsilon\bar{\varepsilon}\varepsilon^{\prime}\bar{\varepsilon}^{\prime}}{2}\int
d^{4}xe^{ip\cdot
x}\Bigg{\\{}\mathrm{Tr}\Big{[}S_{c}^{kk^{\prime}}(x)\Gamma_{i}\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{n^{\prime}n}(-x)\gamma_{\nu}S_{d}^{m^{\prime}m}(-x)\Big{]}\delta^{jj^{\prime}}$
(20)
$\displaystyle+\mathrm{Tr}\Big{[}S_{c}^{kk^{\prime}}(x)\widetilde{S}_{u}^{jj^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{n^{\prime}n}(-x)\gamma_{\nu}\Gamma_{i}\Big{]}\delta^{m^{\prime}m}\Bigg{\\}}\frac{1}{12}\langle\gamma(q)|\bar{q}(x)\Gamma_{i}q(0)|0\rangle+....$

Similarly, when a light propagator interacts with the photon, a gluon may be
released from one of the remaining three propagators. The expression obtained
in this case is as follows:

$\displaystyle\Pi_{\mu\nu}^{\mathrm{QCD}}(p,q)$ $\displaystyle=$
$\displaystyle
i\frac{\varepsilon\bar{\varepsilon}\varepsilon^{\prime}\bar{\varepsilon}^{\prime}}{2}\int
d^{4}xe^{ip\cdot
x}\Bigg{\\{}\mathrm{Tr}\Big{[}S_{c}^{kk^{\prime}}(x)\Gamma_{i}\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{n^{\prime}n}(-x)\gamma_{\nu}S_{d}^{m^{\prime}m}(-x)\Big{]}\Big{[}\Big{(}\delta^{kj}\delta^{k^{\prime}j^{\prime}}-\frac{1}{3}\delta^{kk^{\prime}}\delta^{jj^{\prime}}\Big{)}$
(21)
$\displaystyle+\Big{(}\delta^{jn^{\prime}}\delta^{j^{\prime}n}-\frac{1}{3}\delta^{n^{\prime}n}\delta^{jj^{\prime}}\Big{)}+\Big{(}\delta^{jm^{\prime}}\delta^{j^{\prime}m}-\frac{1}{3}\delta^{m^{\prime}m}\delta^{jj^{\prime}}\Big{)}\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}S_{c}^{kk^{\prime}}(x)\widetilde{S}_{u}^{jj^{\prime}}(x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{\mu}\widetilde{S}_{c}^{n^{\prime}n}(-x)\gamma_{\nu}\Gamma_{i}\Big{]}\Big{[}\Big{(}\delta^{km^{\prime}}\delta^{k^{\prime}m}-\frac{1}{3}\delta^{m^{\prime}m}\delta^{kk^{\prime}}\Big{)}$
$\displaystyle+\Big{(}\delta^{jm^{\prime}}\delta^{j^{\prime}m}-\frac{1}{3}\delta^{m^{\prime}m}\delta^{jj^{\prime}}\Big{)}+\Big{(}\delta^{n^{\prime}m^{\prime}}\delta^{nm}-\frac{1}{3}\delta^{m^{\prime}m}\delta^{n^{\prime}n}\Big{)}\Bigg{\\}}$
$\displaystyle\times\frac{1}{32}\langle\gamma(q)|\bar{q}(x)\Gamma_{i}G_{\mu\nu}(vx)q(0)|0\rangle+...,$

where we inserted

$\displaystyle\bar{q}^{a}(x)\Gamma_{i}G_{\mu\nu}^{bb^{\prime}}(vx)q^{a^{\prime}}(0)\rightarrow\frac{1}{8}\Big{(}\delta^{ab}\delta^{a^{\prime}b^{\prime}}-\frac{1}{3}\delta^{aa^{\prime}}\delta^{bb^{\prime}}\Big{)}\bar{q}(x)\Gamma_{i}G_{\mu\nu}(vx)q(0).$
(22)

As can be seen, matrix elements representing non-perturbative contributions
such as $\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ and
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\mu\nu}(vx)q(0)\right|0\rangle$
appear. These matrix elements can be expressed associated with photon
distribution amplitudes (DAs) and wave functions with definite twists, whose
expressions are borrowed from Ref. [24]. Besides these matrix elements non-
local operators such as two gluons ($\bar{q}GGq$) and four quarks
($\bar{q}q\bar{q}q$) are expected to seem. However it is known that the
effects of such operators are small, which is justified by the conformal spin
expansion [25, 26], and thus we shall ignore them. The QCD representation of
the correlation function is obtained by using Eqs. (10-22). Then, the Fourier
transformation is applied to transfer expressions in x-space to the momentum
space.

In conclusion, the structure $q_{\mu}\varepsilon_{\nu}$ is chosen from both
representations and the coefficients of the structure are matched in both
hadronic and QCD representations. Then, Borel transformation and continuum
subtraction are used to suppress the effects of the continuum and higher
states. These steps are routine and tedious in the light-cone sum rule method,
so we will not discuss them in detail here. Technical details on these
applications can be found in Ref.[27]. Thus, as an example for the current
$J_{\mu}^{1}$, the light-cone sum rule for $Y_{c\bar{c}}$ states is as
follows:

$\displaystyle\mu_{Y_{c\bar{c}}}\,\,\lambda_{Y_{c\bar{c}}}^{2}=e^{\frac{m_{Y_{c\bar{c}}}^{2}}{M^{2}}}\,\,\Delta_{1}(M^{2},s_{0}),$
(23)

where the explicit expression of the $\Delta_{1}(M^{2},s_{0})$ function is
presented in the Appendix. The analytic calculations of the magnetic moments
of $Y_{c\bar{c}}$ states come to an end here. In the following section we will
use these analytical results to perform numerical calculations.

##  III Numerical analysis and conclusions

We assume the following parameters to perform the numerical calculations for
the magnetic moments of the $Y_{c\bar{c}}$ states. The masses of the light
quarks are $m_{u}=m_{d}=0$, the mass of the c-quark is $m_{c}=(1.275\pm
0.025)\,$GeV, the condensates of the light quarks are $\langle\bar{u}u\rangle$
=$\langle\bar{d}d\rangle$=$(-0.24\pm 0.01)^{3}\,$GeV3 [28], the gluon
condensate is $\langle g_{s}^{2}G^{2}\rangle=0.88~{}$GeV4 [10] and the
magnetic susceptibility $\chi=-2.85\pm 0.5~{}\mbox{GeV}^{-2}$ [29]. To
progress numerical analysis of the magnetic moment of these $Y_{c\bar{c}}$
states, numerical values of the mass and residue parameters of these
$Y_{c\bar{c}}$ states are also required. These values have been computed in
Ref. [20] using mass sum rules which are presented in Table 1. The wave
functions in the distribution amplitudes of the photon and all necessary
expressions about these functions are taken from Ref. [24].

Table 1: Mass and the residue values of the $Y_{c\bar{c}}$ states which are borrowed from Ref. [20]. $Y_{c\bar{c}}$ State |  $m_{Y_{c\bar{c}}}$ [GeV] |  $\lambda_{Y_{c\bar{c}}}$($\times 10^{-2}$) [GeV5]
---|---|---
$J^{1}_{\mu}$ |  $4.66\pm 0.07$ |  $7.19\pm 0.84$
$J^{2}_{\mu}$ |  $4.61\pm 0.07$ |  $6.69\pm 0.80$
$J^{3}_{\mu}$ |  $4.35\pm 0.08$ |  $4.32\pm 0.61$
$J^{4}_{\mu}$ |  $4.66\pm 0.09$ |  $6.67\pm 0.82$
$J^{5}_{\mu}$ |  $4.53\pm 0.07$ |  $10.3\pm 1.40$
$J^{6}_{\mu}$ |  $4.65\pm 0.08$ |  $11.3\pm 1.50$
$J^{7}_{\mu}$ |  $4.48\pm 0.08$ |  $9.47\pm 1.27$
$J^{8}_{\mu}$ |  $4.55\pm 0.07$ |  $10.6\pm 1.40$

In addition to the above input parameters, the light-cone sum rule method
includes two other arbitrary parameters, the Borel mass ($M^{2}$) and the
continuum threshold ($s_{0}$). According to the philosophy of the method, the
physical quantity under study should be independent of the variation of these
parameters. To achieve this, we need to add some physical constraints, such as
the convergence of the operator product expansion (OPE) and the pole
contribution (PC). This means that the edges of the working windows for these
arbitrary parameters should be set by the convergence of the OPE and the
constraint on the PC. We use two-criteria to determine the working region of
$M^{2}$: the lower bound of $M^{2}$ is constrained by the OPE convergence,
demanding the higher twist and higher condensates terms to be less than $10\%$
of the total. The upper bound of $M^{2}$ is constrained by the PC

$\displaystyle\mbox{PC}=\frac{\Delta_{1}(M^{2},s_{0})}{\Delta_{1}(M^{2},\infty)}\geq
30\%.$ (24)

The continuum threshold $s_{0}$ is not arbitrary and it is related to the
energy of the first excited state in the initial channel. However, since we
have very limited information on the energy of excited states, we should
decide how to choose working interval of the $s_{0}$. Analysis of various sum
rules predicted that $s_{0}\simeq(m_{ground}+0.5^{+0.2}_{-0.2})^{2}$ GeV2. For
more precise determination of continuum threshold, we impose the dominance of
PC and OPE convergence limitations. As a results of these limitations, for
$Y_{c\bar{c}}$ states we choose
$s_{0}\simeq(m_{Y_{c\bar{c}}}+0.5^{+0.1}_{-0.1})^{2}$ GeV2. Due to the above
constraints, the following working windows for these arbitrary parameters
together with PC and OPE convergence are shown in the Table 2. In Figs. 1 and
2, we show the dependencies of the magnetic moments versus $M^{2}$ at three
fixed values of $s_{0}$. As you can see from these figures, the variation of
magnetic moments with respect to $M^{2}$ is quite stable. Although the
variation is high compared to $s_{0}$, this variation remains within the
errors of the method used.

Table 2: Working regions of the Borel mass parameters, continuum threshold, PC and OPE convergence for magnetic moments. $Y_{c\bar{c}}$ State |  $s_{0}$ [GeV2] |  $M^{2}$ [GeV2] |  PC (average) |  OPE
---|---|---|---|---
$J^{1}_{\mu}$ |  $25.0-27.0$ |  $5.0-7.0$ |  $46\%$ |  $<2\%$
$J^{2}_{\mu}$ |  $25.0-27.0$ |  $5.0-7.0$ |  $45\%$ |  $<3\%$
$J^{3}_{\mu}$ |  $22.0-24.0$ |  $4.5-6.5$ |  $43\%$ |  $<2\%$
$J^{4}_{\mu}$ |  $25.0-27.0$ |  $5.0-7.0$ |  $44\%$ |  $<3\%$
$J^{5}_{\mu}$ |  $24.0-26.0$ |  $5.0-7.0$ |  $43\%$ |  $<2\%$
$J^{6}_{\mu}$ |  $24.0-26.0$ |  $5.0-7.0$ |  $43\%$ |  $<3\%$
$J^{7}_{\mu}$ |  $24.0-26.0$ |  $5.0-7.0$ |  $45\%$ |  $<2\%$
$J^{8}_{\mu}$ |  $24.0-26.0$ |  $5.0-7.0$ |  $44\%$ |  $<2\%$

The magnetic moment results obtained for the $Y_{c\bar{c}}$ states with these
input parameters are given in Table 3 after determining all the necessary
input parameters for the numerical analysis, both in its natural unit
($\frac{e}{2\,m_{Y_{c\bar{c}}}}$) and in the nuclear magneton unit
($\mu_{N}=\frac{m_{N}}{m_{Y_{c\bar{c}}}}$).

Table 3: Magnetic moments of $Y_{c\bar{c}}$ states. $Y_{c\bar{c}}$ State |  $J^{PC}$ |  $\mu$ [$\frac{e}{2\,m_{Y_{c\bar{c}}}}$] |  $\mu$ $\big{[}\mu_{N}\big{]}$
---|---|---|---
$J^{1}_{\mu}$ |  $1^{--}$ |  $4.24^{+1.01}_{-0.91}$ |  $0.85^{+0.20}_{-0.18}$
$J^{2}_{\mu}$ |  $1^{-+}$ |  $4.68^{+1.17}_{-1.04}$ |  $0.95^{+0.23}_{-0.21}$
$J^{3}_{\mu}$ |  $1^{--}$ |  $3.74^{+1.13}_{-1.00}$ |  $0.80^{+0.25}_{-0.21}$
$J^{4}_{\mu}$ |  $1^{-+}$ |  $4.94^{+1.16}_{-1.03}$ |  $1.00^{+0.22}_{-0.21}$
$J^{5}_{\mu}$ |  $1^{--}$ |  $4.81^{+1.26}_{-1.12}$ |  $1.00^{+0.26}_{-0.24}$
$J^{6}_{\mu}$ |  $1^{-+}$ |  $3.68^{+0.88}_{-0.79}$ |  $0.74^{+0.18}_{-0.16}$
$J^{7}_{\mu}$ |  $1^{--}$ |  $5.18^{+1.40}_{-1.26}$ |  $1.08^{+0.30}_{-0.26}$
$J^{8}_{\mu}$ |  $1^{-+}$ |  $3.54^{+0.96}_{-0.85}$ |  $0.73^{+0.19}_{-0.18}$

The errors arising from the uncertainty of the the continuum threshold values,
the variation of the Borel mass parameter, etc., are taken into account.

The magnetic moments of the the vector hidden-charmed tetraquark states have
been extracted from the light-cone sum rules employing for their hadronic
representations a single-pole technique [see, Eq. (3)]. In the case of the
multi-quark hadrons such technique should be verified by supplementary
arguments because a physical representation of relevant sum rules receives
contributions from two-hadron reducible terms as well. This problem was first
proposed during theoretical studies of the pentaquarks [30, 31]. Two-hadron
contaminating terms have to be considered when extracting parameters of multi-
quark hadrons. In the case of the multi-quark hadrons they lead to
modification in the quark propagator

$\displaystyle\frac{1}{m^{2}-p^{2}}\rightarrow\frac{1}{m^{2}-p^{2}-i\sqrt{p^{2}}\Gamma(p)},$
(25)

where $\Gamma(p)$ is the finite width of the multi-quark hadrons generated by
two-hadron scattering states. When these effects are properly considered in
the sum rules, they rescale the residue of the multi-quark hadrons under
investigation leaving its mass unchanged. Detailed investigations show that
two-hadron scattering effects are small for multi-quark hadrons (see Refs.
[32, 33, 34, 35, 36, 37, 38, 39, 40]). Thus, in this study the zero-width
single-pole approximation has been employed.

As we mentioned in the Introduction of the text, the same results are obtained
for the spectroscopic parameters when the $[cq][\bar{c}\bar{q}]$,
$[cs][\bar{c}\bar{s}]$ and $[cq][\bar{c}\bar{q^{\prime}}]$ quark contents are
taken into account. Therefore, the spectroscopic parameters are not enough to
understand the internal structure of these states and to determine their quark
contents. Considering that the $Y_{c\bar{c}}$ states can have a
$[cq][\bar{c}\bar{q}]$ or $[cs][\bar{c}\bar{s}]$ quark content, it is obvious
that the magnetic moments of these states are zero. In this analysis, the
$Y_{c\bar{c}}$ states are considered as quark content with
$[cq][\bar{c}\bar{q^{\prime}}]$ and their magnetic moments are obtained as
nonzero. These results provide a direct test of the diquark-antidiquark
structure of the $Y_{c\bar{c}}$ states. In future experimental studies, the
measurement of the magnetic moment will give us more detailed and clear
information about the internal structure of these particles.

The magnitude of the magnetic moment shows its measurability in experiment.
For instance, it shows that if the magnitude of the magnetic moment in the
natural units ($\frac{e}{2\,m_{Y_{c\bar{c}}}}$) is one or larger than it, it
can be easily measured in the experiment. If the magnitude of the magnetic
moment is less than one, it means that it is probable to be measured. We
observe that the magnitudes of the magnetic moment results obtained in this
study are large enough to be measured experimentally.

To our best knowledge, this is the first study in the literature dedicated to
the investigation of the $Y_{c\bar{c}}$ states magnetic moments. Therefore,
experimental data or theoretical estimations are not yet available to compare
them with our numerical results. However, we may compare these results with
the $Z_{c}$ states’ magnetic moments. Making this comparison may be meaningful
in terms of having an idea about the consistency of the results since there is
no experimental and theoretical data. In Refs. [41, 42, 43], the light-cone
sum rules method has been applied various $Z_{c}$ states to obtain their
electromagnetic properties. In Ref.[41], the electromagnetic properties of the
tetraquark state $Z_{c}(3900)$ have been investigated in the diquark-
antidiquark picture with quantum numbers $J^{PC}=1^{+-}$ and its magnetic and
quadrupole moments were extracted. The magnetic moment was obtained as
$\mu_{Z_{c}}=0.67\pm 0.32~{}\mu_{N}$. In Ref. [42], the magnetic dipole moment
of the $Z_{cs}(3985)$ state was acquired by using the molecular and compact
diquark-antidiquark type interpolating currents. The obtained results were
given as $\mu_{Z_{cs}}^{Di}=0.60^{+0.26}_{-0.21}~{}\mu_{N}$ and
$\mu_{Z_{cs}}^{Mol}=0.52^{+0.19}_{-0.17}~{}\mu_{N}$ for diquark-antidiquark
and molecular pictures, respectively. In Ref. [43], the magnetic dipole
moments of the $Z_{c}(4020)^{+}$, $Z_{c}(4200)^{+}$, $Z_{cs}(4000)^{+}$ and
$Z_{cs}(4220)^{+}$ states have been extracted using the hadronic molecular
form of interpolating currents with quantum numbers $J^{PC}=1^{+-}$. The
magnetic dipole moments were obtained as
$\mu_{Z_{c}}=0.66^{+0.27}_{-0.25}~{}\mu_{N}$,
$\mu_{Z^{1}_{c}}=1.03^{+0.32}_{-0.29}~{}\mu_{N}$,
$\mu_{Z_{cs}}=0.73^{+0.28}_{-0.26}~{}\mu_{N}$, and
$\mu_{Z^{1}_{cs}}=0.77^{+0.27}_{-0.25}~{}\mu_{N}$ for the $Z_{c}(4020)^{+}$,
$Z_{c}(4200)^{+}$, $Z_{cs}(4000)^{+}$ and $Z_{cs}(4220)^{+}$ states,
respectively. As one can see from these predictions, the numerical results for
the magnetic moments of the $Y_{c\bar{c}}$ states obtained in the present work
are the same order of the $Z_{c}$ states’ magnetic moments. Calculating the
results for the magnetic moments with other theoretical models will also be a
test of the consistency of our predictions.

Let us discuss how the magnetic moments of these states can be measured. The
electromagnetic multipole moments can be calculated using a method based on
the emission of soft photons from hadrons, as presented in Ref. [44]. The
photon also contains information about the higher multipole moments of the
particle when emitted. The element of the radiative process matrix can be
written in terms of the energy of the photon $E_{\gamma}$ as follows:

$\displaystyle M\sim A\,(E_{\gamma})^{-1}+B\,(E_{\gamma})^{0}+\mbox{higher-
order terms}.$ (26)

The electric charge contributes to the amplitude at order $(E_{\gamma})^{-1}$
and the contribution coming from the magnetic moment is characterized by the
term $(E_{\gamma})^{0}$. Thus, by measuring the cross section or decay width
of the radiative process and ignoring the small contributions of terms
linear/higher order in $E_{\gamma}$, one can identify the magnetic moment of
the state under examination.

In summary, the magnetic moments of the vector hidden-charmed tetraquark
states that have been observed and can be expected to be observed
experimentally have been determined using the light-cone sum rules taking into
account the diquark-antidiquark structure with the quantum numbers
$J^{PC}=1^{--}$ and $J^{PC}=1^{-+}$. Since these states are considered to have
different flavors of light quarks, they have nonzero magnetic moments. The
results obtained in this study can be checked for consistency by various
methods. The magnetic moments of hadrons encompass useful knowledge about the
distribution of charge and magnetization inside hadrons, which helps us to
understand their geometrical shapes. The existing theoretical estimations on
the mass of vector hidden-charmed tetraquark states and their comparison with
the experimental value have also led to different assignments on the internal
structure of this state discussed above. More theoretical studies are needed,
especially on the strong and radiative decays of these states. The values to
be obtained can be very useful in terms of understanding the nature of these
states when the results of this study are taken together. Calculations of
different parameters related to various interactions/decays of vector hidden-
charmed tetraquark states and their comparison with likely future experimental
measurements can help us figure out the substructure of these states.

(a)  (b)

(c)  (d)

Figure 1: The magnetic moments versus $M^{2}$ at three fixed values of
$s_{0}$; (a), (b), (c) and (d) for $J^{1}_{\mu}$, $J^{3}_{\mu}$, $J^{5}_{\mu}$
and $J^{7}_{\mu}$ states, respectively (in unit of $\mu_{N}$ ).

(a)  (b)

(c)  (d)

Figure 2: The magnetic moments versus $M^{2}$ at three fixed values of
$s_{0}$; (a), (b), (c) and (d) for $J^{2}_{\mu}$, $J^{4}_{\mu}$, $J^{6}_{\mu}$
and $J^{8}_{\mu}$ states, respectively (in unit of $\mu_{N}$ ).

### Appendix: Explicit expression for $\Delta_{1}(M^{2},s_{0})$

In this appendix we present the explicit expressions of the function
$\Delta_{1}(M^{2},s_{0})$ for the magnetic moments of $Y_{c\bar{c}}$ states
entering into the sum rule.

$\displaystyle\Delta_{1}(M^{2},s_{0})$
$\displaystyle=-\frac{3(e_{d}-e_{u}+e_{c})}{2621440\pi^{5}}\Bigg{[}I[0,6,3,0]-4I[0,6,3,1]+6I[0,6,3,2]-4I[0,6,3,3]+I[0,6,3,4]-3I[0,6,4,0]$
$\displaystyle+9I[0,6,4,1]-9I[0,6,4,2]+3I[0,6,4,3]+3I[0,6,5,0]-6I[0,6,5,1]+3I[0,6,5,2]-I[0,6,6,0]$
$\displaystyle+I[0,6,6,1]+6I[1,5,3,1]-18I[1,5,3,2]+18I[1,5,3,3]-6I[1,5,3,4]-18I[1,5,4,1]+36I[1,5,4,2]$
$\displaystyle-18I[1,5,4,3]+18I[1,5,5,1]-18I[1,5,5,2]-6I[1,5,6,1]\Bigg{]}$
$\displaystyle+\frac{(e_{d}-e_{u})m_{c}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{442368\pi^{3}}\Bigg{[}I[0,2,1,0]-2I[0,2,1,1]+I[0,2,1,2]-2I[0,2,2,0]+2I[0,2,2,1]$
$\displaystyle+I[0,2,3,0]-2I[1,1,1,0]+4I[1,1,1,1]-2I[1,1,1,2]+4I[1,1,2,0]-4I[1,1,2,1]-2I[1,1,3,0]\Bigg{]}$
$\displaystyle+\frac{m_{c}\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{84934656\pi^{3}}\Bigg{[}I[0,2,3,0]\Bigg{(}3e_{d}\Big{(}22I_{2}[\mathcal{S}]-24I_{2}[\mathcal{T}_{1}]-11I_{2}[\mathcal{T}_{2}]+1_{3}I_{2}[\mathcal{T}_{4}]-2I_{2}[\mathcal{\tilde{S}}]+48I_{4}[\mathcal{T}_{1}]$
$\displaystyle+22I_{4}[\mathcal{T}_{2}]-26I_{4}[\mathcal{T}_{4}]-16I_{5}[\mathcal{A}]\Big{)}+12e_{u}\Big{(}22I_{1}[\mathcal{S}]+22I_{1}[\mathcal{T}_{1}]-11I_{1}[\mathcal{T}_{2}]-22I_{1}[\mathcal{T}_{4}]+68I_{1}[\mathcal{\tilde{S}}]+2I_{3}[\mathcal{T}_{1}]$
$\displaystyle+11I_{3}[\mathcal{T}_{2}]+9I_{3}[\mathcal{T}_{4}]-92I_{3}[\mathcal{\tilde{S}}]+2I_{5}[\mathcal{A}]\Big{)}\Bigg{)}-48e_{d}\mathcal{A}[u_{0}]\Bigg{(}I[0,2,1,0]-2I[0,2,1,1]+I[0,2,1,2]$
$\displaystyle-2I[0,2,2,0]+2I[0,2,2,1]+I[0,2,3,0]\Bigg{)}+8\chi\Bigg{(}I_{5}[\varphi_{\gamma}]\Big{(}8e_{d}I[0,3,3,0]-4e_{u}I[0,3,3,0]-3e_{d}I[0,3,4,0]$
$\displaystyle-3e_{u}I[0,3,4,0]\Big{)}+2e_{d}\Big{(}11I[0,3,1,0]-33I[0,3,1,1]+33I[0,3,1,2]-11I[0,3,1,3]-37I[0,3,2,0]$
$\displaystyle+74I[0,3,2,1]-37I[0,3,2,2]+41I[0,3,3,0]-41I[0,3,3,1]-15I[0,3,4,0]\Big{)}+2e_{u}\Big{(}-5I[0,3,1,0]$
$\displaystyle+19I[0,3,1,1]-23I[0,3,1,2]+9I[0,3,1,3]+19I[0,3,2,0]-46I[0,3,2,1]$
$\displaystyle+27I[0,3,2,2]-23I[0,3,3,0]+27I[0,3,3,1]+9I[0,3,4,0]\Big{)}\varphi_{\gamma}[u_{0}]\Bigg{)}\Bigg{]}$
$\displaystyle-\frac{f_{3\gamma}\langle
g_{s}^{2}G^{2}\rangle}{169869312\pi^{3}}\Bigg{[}132m_{c}^{2}\Big{(}4e_{u}I_{1}[\mathcal{V}]-e_{d}I_{2}[\mathcal{A}]+3e_{d}I_{2}[\mathcal{V}]\Big{)}I[0,2,2,0]+384e_{d}m_{c}^{2}I_{6}[\psi^{\nu}]\big{(}2I[0,2,2,0]$
$\displaystyle+3I[0,2,3,0]\big{)}-16(2e_{d}+e_{u})I_{5}[\psi^{a}]\big{(}24m_{c}^{2}I[0,2,2,0]-I[0,3,4,0]\big{)}-32\Bigg{(}24e_{u}m_{c}^{2}\Big{(}I[0,2,1,0]$
$\displaystyle-I[0,2,2,0]\Big{)}+e_{d}\Big{(}24m_{c}^{2}\big{(}I[0,2,1,0]-2I[0,2,1,1]+I[0,2,1,2]-I[0,2,2,0]+I[0,2,2,1]\big{)}-I[0,3,2,0]$
$\displaystyle+2I[0,3,2,1]-I[0,3,2,2]+2I[0,3,3,0]-2I[0,3,3,1]-I[0,3,4,0]\Big{)}\Bigg{)}\psi^{a}[u_{0}]+192m_{c}^{2}\Bigg{(}2e_{u}\Big{(}I[0,2,1,1]$
$\displaystyle-I[0,2,1,2]-I[0,2,2,1]\Big{)}+e_{d}\Big{(}3I[0,2,1,0]-8I[0,2,1,1]+5I[0,2,1,2]-6I[0,2,2,0]+8I[0,2,2,1]$
$\displaystyle+3I[0,2,3,0]\Big{)}\Bigg{)}\psi^{\nu}[u_{0}]\bigg{]}$
$\displaystyle+\frac{(e_{d}-e_{u})\langle
g_{s}^{2}G^{2}\rangle}{56623104\pi^{5}}\Bigg{[}25I[0,4,2,0]-75I[0,4,2,1]+75I[0,4,2,2]-25I[0,4,2,3]-71I[0,4,3,0]$
$\displaystyle+142I[0,4,3,1]-71I[0,4,3,2]+67I[0,4,4,0]-67I[0,4,4,1]-21I[0,4,5,0]+268I[1,3,2,1]$
$\displaystyle-284I[1,3,2,2]+100I[1,3,2,3]-536I[1,3,3,1]+284I[1,3,3,2]+268I[1,3,4,1]+64m_{c}^{2}\Big{(}2I[0,3,1,1]$
$\displaystyle-3I[0,3,1,2]+I[0,3,1,3]-2I[0,3,2,1]+I[0,3,2,2]+3I[1,2,1,2]-3I[1,2,1,3]+3I[1,2,2,2]\Big{)}\Bigg{]}$
$\displaystyle+\frac{(e_{d}-e_{u})m_{c}\langle\bar{q}q\rangle}{32768\pi^{3}}\Bigg{[}I[0,4,2,0]-3I[0,4,2,1]+3I[0,4,2,2]-I[0,4,2,3]-3I[0,4,3,0]+6I[0,4,3,1]$
$\displaystyle-3I[0,4,3,2]+3I[0,4,4,0]-3I[0,4,4,1]-I[0,4,5,0]+12I[1,3,2,1]-12I[1,3,2,2]+4I[1,3,2,3]$
$\displaystyle-24I[1,3,3,1]+12I[1,3,3,2]+12I[1,3,4,1]\Bigg{]}$
$\displaystyle-\frac{m_{c}\langle\bar{q}q\rangle}{5728640\pi^{3}}\Bigg{[}-20\Bigg{(}2e_{u}\Big{(}6I_{1}[\mathcal{T}_{4}]-4I_{1}[\mathcal{\tilde{S}}]+5I_{3}[\mathcal{S}]-28I_{3}[\mathcal{T}_{1}]-24I_{3}[\mathcal{T}_{2}]+18I_{3}[\mathcal{T}_{3}]+10I_{3}[\mathcal{T}_{4}]-16I_{3}[\mathcal{\tilde{S}}]\Big{)}$
$\displaystyle+e_{d}\Big{(}-2I_{2}[\mathcal{S}]+2I_{2}[\mathcal{T}_{1}]+3I_{2}[\mathcal{T}_{3}]+I_{2}[\mathcal{T}_{4}]+2I_{2}[\mathcal{\tilde{S}}]+10I_{4}[\mathcal{S}]-16I_{4}[\mathcal{T}_{1}]-12I_{4}[\mathcal{T}_{2}]+4I_{4}[\mathcal{T}_{4}]-12I_{4}[\mathcal{\tilde{S}}]\Big{)}\Bigg{)}I[0,4,4,0]$
$\displaystyle+48\chi\Bigg{(}-e_{u}\Big{(}2I[0,5,2,0]-7I[0,5,2,1]+9I[0,5,2,2]-5I[0,5,2,3]+I[0,5,2,4]-6I[0,5,3,0]+15I[0,5,3,1]$
$\displaystyle-12I[0,5,3,2]+3I[0,5,3,3]+6I[0,5,4,0]-9I[0,5,4,1]+3I[0,5,4,2]-2I[0,5,5,0]+I[0,5,5,1]\Big{)}$
$\displaystyle+2e_{d}\Big{(}I[0,5,2,0]-4I[0,5,2,1]+6I[0,5,2,2]-4I[0,5,2,3]+I[0,5,2,4]-3I[0,5,3,0]+9I[0,5,3,1]-9I[0,5,3,2]$
$\displaystyle+3I[0,5,3,3]+3I[0,5,4,0]-6I[0,5,4,1]+3I[0,5,4,2]-I[0,5,5,0]+I[0,5,5,1]\Big{)}\Bigg{)}\varphi_{\gamma}[u_{0}]\Bigg{]}$
$\displaystyle+\frac{f_{3\gamma}}{1966080\pi^{3}}\Bigg{[}20e_{d}m_{c}^{2}\Big{(}I[0,4,2,1]-2I[0,4,2,2]+I[0,4,2,3]-2I[0,4,3,1]+2I[0,4,3,2]+I[0,4,4,1]\Big{)}$
$\displaystyle-9e_{d}\Big{(}I[0,5,3,0]-3I[0,5,3,1]+3I[0,5,3,2]-I[0,5,3,3]-3I[0,5,4,0]+6I[0,5,4,1]-3I[0,5,4,2]+3I[0,5,5,0]$
$\displaystyle-3I[0,5,5,1]-I[0,5,6,0]\Big{)}+9e_{u}\Big{(}I[0,5,3,0]-3I[0,5,3,1]+3I[0,5,3,2]-I[0,5,3,3]-3I[0,5,4,0]+6I[0,5,4,1]$
$\displaystyle-3I[0,5,4,2]+3I[0,5,5,0]-3I[0,5,5,1]-I[0,5,6,0]\Big{)}\Bigg{]}\psi^{\nu}[u_{0}],$
(27)

where $u_{0}=\frac{M_{1}^{2}}{M_{1}^{2}+M_{2}^{2}}$,
$\frac{1}{M^{2}}=\frac{1}{M_{1}^{2}}+\frac{1}{M_{2}^{2}}$ with $M_{1}^{2}$ and
$M_{2}^{2}$ being the Borel parameters in the initial and final states,
respectively. For simplicity we did not present the terms proportional to many
higher dimensional operators here; however, in the numerical computations we
take these terms into account.

The $I[n,m,l,k]$ and $I_{i}[\mathcal{F}]$ functions are defined as

$\displaystyle I[n,m,l,k]$
$\displaystyle=\int_{4m_{c}^{2}}^{s_{0}}ds\int_{0}^{1}dt\int_{0}^{1}dw~{}e^{-s/M^{2}}~{}s^{n}\,(s-4m_{c}^{2})^{m}\,t^{l}\,w^{k},$
$\displaystyle I_{1}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{2}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{3}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{4}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{5}[\mathcal{F}]$
$\displaystyle=\int_{0}^{1}du~{}\mathcal{F}(u)\delta^{\prime}(u-u_{0}),$
$\displaystyle I_{6}[\mathcal{F}]$
$\displaystyle=\int_{0}^{1}du~{}\mathcal{F}(u),$

where $\mathcal{F}$ stands for the corresponding photon DAs.

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# Bandwidth Utilization Side-Channel on ML Inference Accelerators

Sarbartha Banerjee  The University of Texas at Austin
<EMAIL_ADDRESS>Shijia Wei  The University of Texas at Austin
<EMAIL_ADDRESS>Prakash Ramrakhyani
Mohit Tiwari
ARM Research
<EMAIL_ADDRESS>The University of Texas at Austin
<EMAIL_ADDRESS>
###### Abstract

Accelerators used for machine learning (ML) inference provide great
performance benefits over CPUs. Securing confidential model in inference
against off-chip side-channel attacks is critical in harnessing the
performance advantage in practice. Data and memory address encryption has been
recently proposed to defend against off-chip attacks.

In this paper, we demonstrate that bandwidth utilization on the interface
between accelerators and the weight storage can serve a side-channel for
leaking confidential ML model architecture. This side channel is independent
of the type of interface, leaks even in the presence of data and memory
address encryption and can be monitored through performance counters or
through bus contention from an on-chip unprivileged process.111Presented at
Secure and Private Systems for Machine Learning (SPSL 2021)

##  I Introduction

Deep learning model inference services have spawned a domain-specific
computing revolution. High performance ML inference accelerators in the form
of neural processing units (NPU) are being developed by both industry [14, 2,
17] and academia [11, 16, 8]. NPUs may be integrated either in the system-on-
chip (SoC) [2] or connected to system bus [14]. Inference-as-a-service
(InFaaS) [19] is deployed by cloud providers like Amazon Sagemaker [1] running
user inference on ML accelerators. This incentivizes ML model vendors to host
trained models on cloud platforms and provide services on confidential user
data like face recognition and organizational data like disease classification
on patient private data.

From the security perspective, the model vendor requires the cloud provider to
protect the confidentiality of model parameters as well as the layer
dimensions. Prior work [5, 18, 20] show how knowledge of layer dimensions can
be used to steal a victim’s ML IP by reconstructing a model with similar
accuracy. An attacker can further use a stolen model to launch adversarial
attacks on the victim system [15, 21].

Temporal and spatial sharing of NPUs by multiple applications has been
proposed to improve system scalability and overall inference time [9, 10].
This allows multiple tenants sharing the memory bus to infer victim model
utilization through bandwidth contention channels. To support such sharing,
the cloud hypervisors collect NPU resource utilization information (e.g.
memory bus utilization) for each tenant for providing quality-of-service (QoS)
guarantees, memory traffic and infrastructure management.

In this paper, we develop a new attack based on observing the NPUs bandwidth
utilization. We highlight that observing this side channel alone can leak the
ML model structure even when off-chip data and addresses are encrypted.
Summarizing our key contributions:

  * •

We introduce the bandwidth utilization side-channel on NPUs to leak ML model
dimensions.

  * •

A proof-of-concept exploit on the DRAM interface against 6 image
classification models and explore several classifier to identify model layer
boundaries and layer dimension.

  * •

We propose possible defences to create a demand agnostic bandwidth
utilization. Software defence leads to a $1.6x$ increase in overhead while
hardware countermeasures to generate constant memory traffic has $14.6\%$ to
$19.3\%$ overhead.

##  II Overview and background

Figure 1: Different steps of an end-to-end bandwidth utilization attack. (1)
Collect bandwidth utilization; (2) Classify model layer boundary and layer
type; (3) Use black box attack to reconstruct model or perform adversarial
attack.

Figure 1 shows the different stages of an end-to-end attack with utilization
bandwidth as a side-channel. The memory bus connecting the NPU and the model
weight storage reveals layer variations through the bandwidth utilization side
channel. Our attack receives the bandwidth utilization trace and performs two
classification steps: (1) Detect layer boundaries with a boundary detector
(Section III-D2); (2) Split the time-series along layer boundaries to detect
the layer dimensions with type classifier (Section III-D3). The number of
layers and its dimension can be fed to black box attacks for model weights
reconstruction or an adversarial attack (Section 1).

NPU to memory interface. The NPU accelerator can be resident on the system-on-
chip (SoC) with memory bus connected directly to CPU last-level cache (LLC) as
in ARM Ethos [2] or connected with an on-chip eDRAM as in DaDianNao [8]. Other
accelerators [14] are connected off-chip performing direct system memory
accesses. All the model weights cannot be loaded upfront due to limited NPU
internal memory. This motivates tiling the weights for each DNN layer to fit
in the NPU and loading them as required, therefore creating bandwidth
variation irrespective of the NPU memory interface.

Tile size variations. The model layer dimensions depend on the model
configuration like number of filters, height, width and channels. The tiling
is not only devised on these four dimensions but also on the data locality of
the input feature maps and the intermediate partial sums. Moreover, the number
and type of operation for each layer are different. Convolution and dense
layers perform matrix multiplication while pooling and activation perform ALU
operations like max, min or addition leading to parameter variations for each
layer.

Black box attacks. A large body of work [5, 18, 20] recreates a model having
similar classification accuracy as the victim model with only the model
dimension. As a first step, the attacker initializes a model with victim model
dimension. Then, she sends multiple inference requests to the victim model and
uses the classification to label the data creating a dataset. This dataset is
used to train the attacker’s model until it achieves similar accuracy as the
victim model. prior work [15, 21] further illustrates use of reconstructed
model to generate adversarial examples for the victim model.

##  III Bandwidth Utilization Attack

###  III-A Threat Model

We adopt a threat model similar to trusted execution environments (TEE) like
Intel SGX [3]. Specifically, we assume that the underlying privileged software
is untrusted, malicious co-tenant may share the same NPU and the system may be
exposed to physical attacks. We assume the techniques in secure processors are
adopted by the NPU providers. Beyond memory encryption and integrity checks,
we also assume memory address encryption is employed as in [4, 6], thus
restricting the adversary to only observe the bandwidth utilization.

Additionally, we assume that the victim uses the optimal tile size for
performance, and that the adversary can perform offline profiling to collect
all possible layer and tile configuration characteristics for training her
classifier(s).

###  III-B Point of Leakage

Bandwidth utilization can be observed by malicious hypervisors or co-executing
tenants. Cloud hypervisors collect traffic statistics for each application for
load balancing and congestion control. Considering the threat model of a
malicious cloud hypervisor, high-precision counters can be used to collect
bandwidth utilization of each tenant. Such widgets are (or can be) placed at
(1) the Last-Level Cache (LLC) interface, (2) the DMA engine of the NPU, or
(3) the DRAM controller.

Unprivileged co-executing tenants share the same memory interface. Malicious
tenants in the same NPU can constantly monitor the bandwidth and record victim
utilization from bandwidth drops due to contention. This drop in bandwidth is
proportional to the victim load request size scheduled by a physically shared
NPU DMA controller.

###  III-C Example of Bandwidth Variation at DRAM Interface

As mentioned in Section 1, tile execution of different layers on the NPU loads
tiled matrix weights from either LLC or main memory. Figure 2(a) presents the
layer-dependent load bandwidth variation for the model weights in a VGG16
model. There are three types of bandwidth variations across adjacent layer
boundaries: (T1) Layers with different tile sizes (e.g. layer 2 and 3) where
the filter size is different with a pooling layer in between. (T2) Layers with
the same sized tiles but having different number of tiles (e.g. layer 1 and 2)
which is possible with adjacent layer having the same filter size but
differing in the number of channels. (T3) Identical adjacent layers (e.g.
layer 10 and 11) with identical sized filters having the same number of
channels as in some deeper resnet layers.

Figure 2(b) zooms into the $5^{th}$ layer revealing the number of tiles. Due
to tile size optimization, the number of tiles and the bandwidth utilization
vary across model layers depending on their dimensions. There are finite
tiling schemes with each having a unique signature of bandwidth and execution
time.

(a) Memory bandwidth of 12 layers in VGG16.

(b) Memory bandwidth of the 5th layer in VGG16.

Figure 2: Different layers in vgg16 network utilizes different memory
bandwidth, while bursts within each layer show number of tiles.

###  III-D Attack Demonstration

As a proof-of-concept, we demonstrate the bandwidth utilization attack on the
DRAM interface and model the adversary as a malicious hypervisor.

####  III-D1 Experimental Setup

We prototype the VTA [16] ML inference accelerator on a Xilinx Pynq-Z1 fpga
running at 100 MHz. VTA is an fpga implementation of an NPU with the widely
used TVM [7] software stack. Similar to an NPU, the VTA has its own
instruction set consisting of load, compute and store operations. We augment
the VTA DMA controller with a memory transaction counter to log the read
transactions. The model weights are not updated at inference time, hence
monitoring only read transaction is enough for inferring the model dimensions.
The counter accumulates the number of bytes transferred for each DMA
transaction and stores in a memory-mapped register, read by the runtime driver
at 250KHz.

We run six image classification DNN inference models, namely, VGG-11, VGG-16,
Alexnet, Resnet-18, Resnet-34 and Resnet-50. The memory utilization shape for
each of the workload is shown in Figure 3. These traces are taken with the NPU
connected to system memory bus. To validate possibility of attack on the LLC
interface, we have also collected traces with NPU connected to LLC through
accelerator coherency port (ACP). Those traces also have similar demand
signature and are not shown due to space constraints.

For layer boundary detection, we use precision and recall to evaluate the
detector performance. Precision indicates the fraction of correctly detected
boundaries out of the total detected boundaries. The higher the precision is,
the less false positive claims the detector makes. Recall measures the
fraction of correctly detected boundaries out of the all true boundaries. The
higher the recall is, the less correct layer boundaries the detector misses.
To evaluate layer type classification performance, we use accuracy which
indicates the ratio of correctly classified samples out of all samples. We did
not use accuracy for layer boundary detector because the number of non-
boundary inputs outnumbers the number of boundary inputs by orders of
magnitude. Using accuracy may present a falsely satisfying result.

Figure 3: The bandwidth variations on the DRAM interface during inference
execution of different ML models.

####  III-D2 Layer Boundary Detection

A single DNN model consists of multiple layers. The first step in the attack
is layer boundary detection. Prior arts [13, 12] demonstrated that the read-
after-write (RAW) pattern on the address trace reveals the layer boundaries
accurately. However, our threat model restricts attackers to use bandwidth
utilization variation rather than the address trace.

Fine-grained observations enable attackers to model each layer as a time-
series of bandwidth information. To identify layer boundaries, the adversary
clusters the observed time-series trace into different classes. First, we
collect statistics like total data transferred per sampling window, median and
peak bandwidth, bandwidth standard deviation as well as frequency domain
signals extracted using discrete wavelet transform (DWT) to perform feature
extraction and selection. Second, the adversary builds a bag-of-words model
with the extracted features, a popular Natural Language Processing (NLP)
technique, on sliding windows of the trace. The combination of a bag-of-words
model and sliding windows enables the attacker classifier to obtain both the
frequency- and time-domain information of the collected trace. Then, the
attacker performs clustering to obtain the potential layer boundary
candidates. Subsequently, offline profiled termination timings of all possible
layers are used to validate these candidates to reduce false positives.

The boundary detection results are shown in Table I. The table heading lists
the different benchmarks with the total number of layer boundaries. Adjacent
layers with a different shape (T1 in III-C) are identified as ‘easy’ because
they utilize different memory bandwidth and are therefore easy to detect. The
classifier can detect easy boundaries with 100% precision for AlexNet, VGG11
and VGG16. Overall, 73.9% of layer boundaries are detected across our six
workloads. Note that for ResNet models, the residual layers are very short,
which makes boundaries hard to detect due to our choice of sliding windows
size. We leave these hyperparameter tuning (e.g. sliding window size) for
future work.

| AlexNet | VGG11 | VGG16 | ResNet18 | ResNet34 | ResNet50
---|---|---|---|---|---|---
| easy
---
3
| all
---
4
| easy
---
5
| all
---
6
| easy
---
8
| all
---
11
| easy
---
22
| all
---
23
| easy
---
24
| all
---
36
| easy
---
50
| all
---
52
| precision
---
1 | 1 | 1 | 1 | 1 | 1 | 0.69 | 0.64 | 0.66 | 0.72 | 0.33 | 0.33
| recall
---
0.75 | 1 | 0.83 | 1 | 0.73 | 1 | 0.96 | 1 | 0.67 | 1 | 0.96 | 1

TABLE I: Precision and recall when identifying layer boundaries for each
network. easy boundaries refer to ones between adjacent layers with different
utilization shapes, while all includes all layer boundaries.

####  III-D3 Layer Type Classification

Similar to layer boundary detection, we include frequency domain signals from
DWT to capture tile bandwidth signatures for layer type classification. DWT
detects change in bandwidth across tiles at layer boundary. The wavelets also
captures sharp changes in bandwidth, which is useful for tile boundary
detection.

We test victim memory traffic time-series using three classifiers: Support
Vector Machine (SVM), Multilayer Perceptron (MLP), and Convolutional Neural
Network (CNN), each trained on either the time-series features for potential
layers or on the features extracted from DWT.

| AlexNet | VGG11 | VGG16 | ResNet18 | ResNet34 | ResNet50 | Overall
---|---|---|---|---|---|---|---
| Execution
---
time only
1 | 1 | 0.958 | 0.896 | 0.851 | 0.824 | 0.826
| SVM w/
---
(w/o) DWT
1 | 1 | 1 | 1 | 1 |  | 0.811
---
| 0.927
---
| MLP w/
---
(w/o) DWT
1 | 1 | 1 | 1 |  | 1
---
(0.986)
| 0.868
---
(0.849)
| 0.949
---
(0.934)
| CNN w/
---
(w/o) DWT
1 | 1 | 1 | 1 | 1 |  | 0.877
---
(0.830)
| 0.953
---
(0.934)

TABLE II: Layer type classification accuracy using unshaped traffic assuming
perfect layer boundary detection. The last three rows also show results
without DWT signals in parentheses.

Table II shows the layer-type classification accuracy for each tested model,
using bandwidth trace assuming perfect identification of layer boundaries. The
last column (Overall) summarizes the weighted accuracy of all layers in all
models. The first row shows classification accuracy merely using the
termination timing of each layer. This is a baseline accuracy for any attacker
with the knowledge of the layer boundaries (execution timing for each layers).
All the layers of AlexNet and VGG-11 are identifiable using this basic
classifier. These workloads have few layers and all the layers differ in their
execution time. Therefore, merely having the layer termination time is
sufficient to classify the layers. The accuracy decreases for deeper models
like ResNets with identical adjacent layers (T3 in III-C). The 2-4th rows show
accuracy of the three evaluated layer-type classifiers. From execution-time
based classifier to bandwidth-based classifiers, the accuracy jumps from 84%
to 93% on average. From SVM to CNN, accuracy improves with increasing
classifier complexity. In addition, including frequency domain signals
improves the classifier, resulting in an accuracy of 95.3%.  This is because
different tile size configurations have different compute/bandwidth ratios.

##  IV Countermeasures

Bandwidth utilization channel leaks because of the layer dependent bandwidth
utilization during ML inference. This can be prevented by disabling tile-size
optimization or shaping the shared interface traffic (via pure software or
software-hardware co-design).

Figure 4: Box plot showing performance variation of 800 tile size
configuration for each workload

###  IV-A Disabling tile size optimization

Constant tile size across all layers within a model can make the effective
bandwidth constant throughout the model execution. However, disabling this
optimization leads to inference time overhead and lower utilization of NPU
resources including compute, on-chip storage, and off-chip memory bus
bandwidth. To illustrate the wasted performance opportunity, we explore
execution performance of 800 distinct tile-size configurations for each of the
models. The overhead of each configuration with respect to the most performant
configuration is plotted in Figure 4. The performance overhead of the median
configuration is, on an average, 1.2X more compared to the best case. The
performance overhead varies up to 1.6 - 2.0X in certain tile configurations.
The resulting performance overhead impacts the service-level agreement (SLA)
and causes under-utilization of cloud resources. This is due to the large
variation in the height, width and number of channels for different layers of
the DNN model.

###  IV-B Memory traffic shaping

The memory trace could be made independent of the demand trace. The CSP can
choose a predefined memory utilization independent from the execution model.
The memory controller interleaves the real read requests with fake
transactions if the demand falls below the expected bandwidth. On the other
hand, demand requests need to be throttled when utilization exceeds the
assigned bandwidth. The predefined memory trace can be a fixed bandwidth trace
or some other model demand independent pattern. The write requests can be sent
at regular intervals. Encrypted stores with valid flag can distinguish between
real and fake transactions.

To defend against bandwidth utilization attack on ML inference, we modified
the accelerator design to split each transaction to multiple transactions of
fixed size. The fixed size transactions were sent at an equal interval of time
resulting in a constant bandwidth. Equal-sized splits are created by padding
the unequal transactions. In case of memory idle intervals, fake transactions
are sent to fill the gap. The tile size optimization is performed based on the
fixed bandwidth to obtain the best tile configuration for each layer.

The precision of layer boundary detection reduces drastically as shown in
Table III. The boundary detection classifier has increased sensitivity to have
enough coverage as visible from the recall numbers. NA in the table for ResNet
models indicate the attacker’s failure to identify the true boundaries even
with 10000x false positives $(precision<0.0001)$. The layer construction is
infeasible with such high false positives for the shaped constant trace. The
overall precision drops to less than 0.01%. The model type classifier fails
with such low precision on layer boundary detection thwarting the attack.

With a specific bandwidth choice, the tradeoff is between the amount of wasted
memory bandwidth and the performance overhead caused by request throttling.
With a fixed bandwidth of 200 MB/s, the geomean overhead is 14.6% with worst
case overhead of 19.3%.

| AlexNet | VGG11 | VGG16 | ResNet18 | ResNet34 | ResNet50
---|---|---|---|---|---|---
| easy
---
3
| all
---
4
| easy
---
5
| all
---
6
| easy
---
8
| all
---
11
| easy
---
22
| all
---
23
| easy
---
24
| all
---
36
| easy
---
50
| all
---
52
| precision
---
0.03 | 0.03 | 0.01 | 0.01 | 0.0027 | 0.00011 | NA | NA | NA | NA | NA | NA
| recall
---
0.75 | 1 | 0.83 | 1 | 0.73 | 1 | NA | NA | NA | NA | NA | NA

TABLE III: Memory traffic shaping with constant bandwidth reduces the
precision of layer boundary detection drastically for all models

##  V Related work

Recent works [13, 12] illustrate that memory access patterns reveal a DNN
model structure by snooping the off-chip address bus. We demonstrate a new
alternative side-channel, using bandwidth variation, for leaking model
dimensions even in the presence of data and address encryption. Observing
bandwidth variation is feasible even by on-chip unprivileged malicious co-
tenants without the use of performance counters.

Memory traffic shaping as a defence mechanism is illustrated in prior works
like MITTS [23] or camouflage [22]. These are applicable to general-purpose
workloads with a runtime shaping logic. However, the memory demand requests
for a DNN workload are known at compile time. We demonstrate that the compiler
can choose a traffic pattern and perform the tile size analysis to improve the
overall inference latency.

##  VI Conclusion

This paper studies the bandwidth utilization side-channel to infer
confidential model structure. The channel can be observed by performance
counters or even by unprivileged co-executing tenant through traffic
contention. This work shows that model structure can be leaked even with an
encrypted address trace. The study discusses potential countermeasures and
highlights one that leverages the knowledge of the inference workload at
compile time to tune the tiles accordingly and improve the bus utilization
while closing the bandwidth utilization channel.

##  VII Acknowledgements

We would like to thank our anonymous reviewers for providing their valuable
feedback and suggestions. This work is funded by under task $\\#2965.001$ by
Semiconductor Research Corporation (SRC) and Intel Strategic Research Alliance
(ISRA) grant.

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