#  $B_{c}\to J/\psi$ helicity form factors and the $B_{c}^{+}\to
J/\psi+(P,V,\ell^{+}\nu_{\ell})$ decays

Wei Cheng  School of Science, Chongqing University of Posts and
Telecommunications, Chongqing 400065, P.R. China  State Key Laboratory of
Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing, 100190, P.R. China  Yi Zhang  Department of Physics,
Chongqing University, Chongqing 401331, P.R. China  Long Zeng  Department of
Physics, Chongqing University, Chongqing 401331, P.R. China  Hai-Bing
Fu111Corresponding author<EMAIL_ADDRESS>Department of Physics, Guizhou Minzu
University, Guiyang 550025, P.R. China  Chongqing Key Laboratory for Strongly
Coupled Physics, Chongqing 401331, P.R. China  Xing-Gang Wu<EMAIL_ADDRESS>Department of Physics, Chongqing University, Chongqing 401331, P.R. China
Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing 401331, P.R.
China

###### Abstract

In this paper, we calculate the $B_{c}\to J/\psi$ helicity form factors (HFFs)
up to twist-4 accuracy by using the light-cone sum rules (LCSR) approach.
After extrapolating those HFFs to the physically allowable $q^{2}$ region, we
investigate the $B^{+}_{c}$-meson two-body decays and semi-leptonic decays
$B_{c}^{+}\to J/\psi+(P,V,\ell^{+}\nu_{\ell})$ with $P/V$ stands for light
pseudoscalar/vector meson, respectively. The branching fractions can be
derived by using the CKM matrix element and the $B_{c}$ lifetime from the
Particle Data Group, and we obtain ${\cal B}(B_{c}^{+}\to
J/\psi\pi^{+})=(0.136^{+0.002}_{-0.002})\%$, ${\cal B}(B_{c}^{+}\to J/\psi
K^{+})=(0.010^{+0.000}_{-0.000})\%$, ${\cal B}(B_{c}^{+}\to
J/\psi\rho^{+})=(0.768^{+0.029}_{-0.033})\%$, ${\cal B}(B_{c}^{+}\to J/\psi
K^{\ast+})=(0.043^{+0.001}_{-0.001})\%$, ${\cal B}(B_{c}^{+}\to
J/\psi\mu^{+}\nu_{\mu})=(2.802^{+0.526}_{-0.675})\%$ and ${\cal
B}(B_{c}^{+}\to J/\psi\tau^{+}\nu_{\tau})=(0.559^{+0.131}_{-0.170})\%$. We
then obtain ${\cal R}_{\pi^{+}/\mu^{+}\nu_{\mu}}=0.048^{+0.009}_{-0.012}$ and
${\cal R}_{K^{+}/\pi^{+}}=0.075^{+0.005}_{-0.005}$, which agree with the LHCb
measured value within $1\sigma$-error. We also obtain ${\cal
R}_{J/\psi}=0.199^{+0.060}_{-0.077}$, which like other theoretical
predictions, is consistent with the LHCb measured value within
$2\sigma$-error. Those imply that the HFFs under the LCSR approach are also
applicable to the $B^{+}_{c}$ meson two-body decays and semi-leptonic decays
$B_{c}^{+}\to J/\psi+(P,V,\ell^{+}\nu_{\ell})$, and the HFFs obtained by using
LCSR in a new way implies that there may be new physics in the $B_{c}\to
J/\psi\ell^{+}\nu_{\ell}$ semi-leptonic decays.

##  I Introduction

The $B_{c}$-meson that consists of two different heavy quarks was first
observed by CDF collaboration CDF  in year 1998. Since then, the properties of
the $B_{c}$-meson have been?studied?extensively, cf. the reviews
Brambilla:2010cs ; Brambilla:2004wf ; Li:2018lxi . Its related decay processes
can provide good platforms not only to understand the non-perturbation
interactions by fixing the CKM-matrix elements, but also to further searches
for new physics beyond the standard model. In recent years, many ratios of
different branching fraction associated with $B_{c}\to J/\psi$ have been
measured by the LHCb collaboration, such as ${\cal
R}_{\pi^{+}/\mu^{+}\nu_{\mu}}={\cal B}(B_{c}^{+}\to J/\psi\pi^{+})/{\cal
B}(B_{c}^{+}\to J/\psi\mu^{+}\nu_{\mu})=0.0469\pm 0.0028({\rm stat.})\pm
0.0046({\rm syst.})$ Aaij:2014jxa , ${\cal R}_{K^{+}/\pi^{+}}={\cal
B}(B_{c}^{+}\to J/\psi K^{+})/{\cal B}(B_{c}^{+}\to J/\psi\pi^{+})=0.069\pm
0.019({\rm stat.})\pm 0.005({\rm syst.})$ Aaij:2013vcx , which is updated to
${\cal R}_{K^{+}/\pi^{+}}=0.079\pm 0.007({\rm stat.})\pm 0.003({\rm syst.})$
Aaij:2016tcz , and ${\cal R}_{J/\psi}={\cal B}(B_{c}^{+}\to
J/\psi\tau^{+}\nu_{\tau})/{\cal B}(B_{c}^{+}\to
J/\psi\mu^{+}\nu_{\mu})=0.71\pm 0.17(\rm stat.)\pm 0.18(\rm syst.)$
Aaij:2017tyk , where “stat.” is the statistical error and “syst.” is the
systematic error.

Many theoretical studies on those decay channels have been published in the
literature. For the branching ratio ${\cal R}_{\pi/\mu^{+}\nu_{\mu}}$,
theoretical predictions of Refs.Ke:2013yka ; AbdElHady:1999xh ;
Issadykov:2018myx  agree with the measured value of Ref. Aaij:2014jxa  within
the $2\sigma$-error and the predictions of Refs.Ebert:2003cn ; Chang:2014jca
are within $1\sigma$-error. For the branching ratio ${\cal
R}_{K^{+}/\pi^{+}}$, the predictions of Refs.Ebert:2003cn ; AbdElHady:1999xh ;
Issadykov:2018myx ; Chang:2014jca ; Wen-Fei:2013uea  agree with the latest
LHCb measurement Aaij:2016tcz  within $1\sigma$-error. For the branching ratio
${\cal R}_{J/\psi}$, the predictions of Refs.Leljak:2019eyw ;
Issadykov:2018myx ; Wen-Fei:2013uea ; Azizi:2019aaf  agree with LHCb measured
value Aaij:2017tyk  within $2\sigma$-error. The key components and main error
sources of those decays are $B_{c}\to J/\psi$ transition form factors (TFFs),
which have been studied under various approaches, such as the relativistic
model (RM) AbdElHady:1999xh , the light-front constituent quark model (LFQM)
Ke:2013yka , the covariant confined quark model (CCQM) Issadykov:2018myx , the
Bethe-Salpeter model (BSM) Chang:1992pt , the Bethe-Salpeter relativistic
quark model (BS RQM) Chang:2014jca , the relativistic quasi-potential
Schrödinger model (RQM) Ebert:2003cn , perturbation Quantum Chromodynamics
(pQCD) Wen-Fei:2013uea , the light-cone sum rule (LCSR) Leljak:2019eyw ,
Quantum Chromodynamics sum rules (QCDSR) Azizi:2019aaf , and the covariant
light-front constituent quark model (CLFQM) Huang:2018nnq , etc. There still
exist a large discrepancy between different approaches, which attract the
theorists to inquire further.

In the present paper, we shall study the $B_{c}\to J/\psi$ helicity form
factors (HFFs) instead of traditional TFFs by applying the LCSR approach. The
main difference between?the two methods is the method of decomposition. The
$\gamma$-structures of the hadronic matrix elements can be decomposed into
Lorentz-invariant structures by using covariant decomposition for the TFFs
decomposition method, while a new combination of Lorentz-invariant structures
are obtained by using the polarization vectors of the off-shell W-Boson for
the HFFs decomposition method. The use of HFFs has some advantages in
comparison to the usual treatment of TFF Bharucha:2010im ; Fu:2014pba ;
Fu:2014uea ; Fu:2016yzx ; Fu:2018vap ; Cheng:2018ouz ; Fu:2020vqd : I) The
diagonalized unitarity relations will lead to the dispersion bounds that will
be represented specifically by constraining the coefficients in the series
expansion (SE) or simplified series expansion (SSE) parameterization; II) The
polarized decay widths of a meson transition to a vector meson can be well
studied by using the HFFs; III) As the helicity amplitude has a definite spin-
parity quantum number, the helicity amplitude is convenient to consider the
contribution from the excited state of meson in the transition process. After
calculating the $B_{c}\to J/\psi$ HFFs within the LCSR, we shall extend them
into allowable physical region and then investigate the properties of
$B_{c}^{+}\to J/\psi+(P,V,\ell^{+}\nu_{\ell})$ decays.

The remaining parts of the paper are organized as follows. In Sec. II, we
present the calculation technology of the $B_{c}\to J/\psi$ HFFs and the
branching fractions for $B_{c}^{+}\to J/\psi+(P,V,\ell^{+}\nu_{\ell})$ decays.
In Sec. III, we give the numerical results and discussions. Section IV is
reserved for a summary.

##  II Theoretical Framework

The $B^{+}_{c}$-meson two-body decays $B_{c}^{+}\to J/\psi+(P,V)$ are achieved
predominantly via a $\bar{b}\to\bar{c}u\bar{q}$ transition, which can be
expressed as the corresponding hadron matrix element $\langle
J/\psi+P(V)|\mathcal{O}|B^{+}_{c}\rangle$. Hrer, the non-factoring effect is
mainly the exchange of hard gluons, so the contribution of this part can be
dealt with by perturbation theory. Considering the heavy quark limit, the
hadron matrix element under the leading order approximation of $\alpha_{s}$
can be expressed by the simple factorization formula. The corresponding
transition amplitudes can be expressed as follows:

$\displaystyle{\cal A}(B^{+}_{c}\\!\\!\to\\!J/\psi
P)\\!=\\!\frac{G_{F}\\!V_{cb}\\!V_{uq}^{\dagger}}{\sqrt{2}}a_{1}\\!\langle
P|\bar{q}\gamma^{\mu}\gamma_{5}u|0\rangle\langle
J/\psi|j_{\mu}|B^{+}_{c}\rangle,$ (1) $\displaystyle{\cal
A}(B^{+}_{c}\\!\\!\to\\!J/\psi
V)\\!=\frac{G_{F}V_{cb}V_{uq}^{\dagger}}{\sqrt{2}}a_{1}\langle
V|\bar{q}\gamma^{\mu}u|0\rangle\langle J/\psi|j_{\mu}|B^{+}_{c}\rangle,$ (2)

where $\langle P|\bar{q}\gamma^{\mu}\gamma_{5}u|0\rangle=-if_{P}q^{\mu}$ with
$P$ being a pseudoscalar meson $\pi^{+}$ or $K^{+}$, $\langle
V|\bar{q}\gamma^{\mu}u|0\rangle=m_{V}f_{V}\epsilon_{\mu}^{*}$ with $V$ bing a
vector meson $\rho^{+}$ or $K^{\ast+}$, and the hadronic matrix elements
$\langle J/\psi|j_{\mu}|B^{+}_{c}\rangle$ with
$j_{\mu}=\bar{c}\gamma_{\mu}(1-\gamma^{5})b$ can be projected as the HFFs
${\cal H}_{\sigma}^{B_{c}\to J/\psi}(q^{2})$, see Eq. (6). The
$a_{1}=C_{2}+\xi C_{1}$ with $\xi=1/N_{c}$ Buchalla:1995vs . Thus, the decay
widths of $B_{c}^{+}\to J/\psi+(P,V)$ in terms of the amplitudes ${\cal
A}(B^{+}_{c}\to J/\psi+(P,V))$ are given by

$\displaystyle\Gamma(B^{+}_{c}\to$ $\displaystyle
J/\psi\\!+\\!(P,V))\\!=\\!\frac{|{\bf p_{2}}||{\cal A}(B^{+}_{c}\to
J/\psi+(P,V))|^{2}}{8\pi m^{2}_{B_{c}}}$
$\displaystyle=\frac{G^{2}_{F}|{\mathbf{p}_{2}}|}{16\pi
m^{2}_{B_{c}}}\frac{\lambda(m_{(P,V)}^{2})}{m_{(P,V)}^{2}}\left|V_{cb}V_{uq}^{\dagger}a_{1}f_{(P,V)}m_{(P,V)}\right|^{2}$
$\displaystyle\times\sum\limits_{i}\left({\cal H}_{i}^{B_{c}\to
J/\psi}(m^{2}_{(P,V)})\right)^{2},$ (3)

where the fermi constant $G_{F}=1.166\times 10^{-5}~{}{\rm GeV}^{-2}$
Zyla:2020zbs , the momentum of the off-shell W-boson in the $B_{c}$ rest frame
is $|{\bf p_{2}}|=\lambda^{1/2}/(2m_{B_{c}}^{2})$ with the usual phase-space
factor
$\lambda=(m_{B_{c}}^{2}+m_{J/\psi}^{2}-m_{(P,V)}^{2})^{2}-4m_{B_{c}}^{2}m_{J/\psi}^{2}$.
$f_{(P,V)}$ and $m_{(P,V)}$ are the decay constants and mass. For a
pseudoscalar meson $P=\pi^{+}(K^{+})$, thus $q=d(s)$ and $i=t$, while, for a
vector meson $V=\rho^{+}(K^{\ast+})$, then $q=d(s)$ and $i=(0,1,2)$. The
corresponding mass and decay constant are listed in Table. 1.

Table 1: The mass and decay constant of $(P,V)$-mason in the $B_{c}$-meson two body decay. | Mass | Decay constant
---|---|---
$\pi^{+}$ |  $0.140~{}\rm{GeV}$ Tanabashi:2018oca  |  $0.1304~{}\rm{GeV}$ Khodjamirian:2009ys
$K^{+}$ |  $0.494~{}\rm{GeV}$ Tanabashi:2018oca  |  $0.1562~{}\rm{GeV}$ Agashe:2014kda
$\rho^{+}$ |  $0.775~{}\rm{GeV}$ Tanabashi:2018oca  |  $0.2210~{}\rm{GeV}$ Issadykov:2018myx
$K^{*+}$ |  $0.892~{}\rm{GeV}$ Tanabashi:2018oca  |  $0.2268~{}\rm{GeV}$ Issadykov:2018myx

The amplitudes of $B^{+}_{c}$-meson semi-leptonic decays $B_{c}^{+}\to
J/\psi\ell^{+}\nu_{\ell}$ can be factored into:

${\cal
A}(B^{+}_{c}\\!\to\\!J/\psi\ell^{+}\nu_{\ell})=\frac{G_{F}V_{cb}}{\sqrt{2}}\ell^{+}\gamma^{\mu}(1-\gamma^{5})\nu_{\ell}\langle
J/\psi|j_{\mu}|B_{c}^{+}\rangle.$ (4)

After completing the square of the sime-leptonic decay amplitude Zhou:2019stx
, the corresponding decay widths in terms of the HFFs ${\cal
H}_{\sigma}^{B_{c}\to J/\psi}(q^{2})$ can be written as

$\displaystyle\Gamma(B_{c}^{+}\to$ $\displaystyle
J/\psi\ell^{+}\nu_{\ell})=\frac{G_{F}^{2}|V_{cb}|^{2}}{12(2\pi)^{3}}\int_{q^{2}_{\rm
min}}^{q^{2}_{\rm max}}dq^{2}\bigg{(}1-\frac{m_{\ell}^{2}}{q^{2}}\bigg{)}^{2}$
$\displaystyle\times\lambda\frac{|{\bf
p_{2}}|}{m_{B_{c}}^{2}}\bigg{[}\bigg{(}1+\frac{m_{\ell}^{2}}{2q^{2}}\bigg{)}\sum\limits_{i=0,1,2}({\cal
H}_{i}^{B_{c}\to J/\psi}(q^{2}))^{2}$
$\displaystyle+\frac{3m_{\ell}^{2}}{2q^{2}}({\cal H}_{t}^{B_{c}\to
J/\psi}(q^{2}))^{2}\bigg{]},$ (5)

where $\ell$ stands for the lepton $\mu$ or $\tau$, the range of integration
are from $q_{\rm min}^{2}=m_{\ell}^{2}$ to $q_{\rm
max}^{2}=(m_{B_{c}}-m_{J/\psi})^{2}$.

Then, the key component of the decay width of $B_{c}\to J/\psi$ transitions
are the HFFs ${\cal H}_{\sigma}^{B_{c}\to J/\psi}(q^{2})$ with
$\sigma=(0,1,2,t)$ defined in term of the hadronic matrix elements as follows:

$\displaystyle{\cal H}_{\sigma}^{B_{c}\to J/\psi}$
$\displaystyle(q^{2})=\sqrt{\frac{q^{2}}{\lambda}}\sum\limits_{\alpha=0,\pm,t}{\varepsilon_{\sigma}^{*\mu}(q)}$
$\displaystyle\times\langle
J/\psi(k,\varepsilon_{\alpha}(k))|\bar{c}\gamma_{\mu}(1-\gamma^{5})b|B_{c}(p)\rangle,$
(6)

where $\varepsilon_{\alpha}(k)$ with momentum $k=(k^{0},0,0,|\vec{k}|)$ are
$J/\psi$-meson longitudinal ($\alpha=0$) and transverse ($\pm$) polarization
vectors, ${\varepsilon_{\sigma}(q)}$ with transfer momentum
$q=(q^{0},0,0,-|\vec{q}|)$ are the polarization vectors of the off-shell
$W$-boson, and $\sigma=(0,1,2,t)$. These momenta satisfy the relationship
$q=(p-k)$. HFFs is a novel decomposition method that multiplies the
polarization vectors of the off-shell W-Boson (${\varepsilon_{\sigma}(q)}$) by
the hadronic matrix element. Compared with TFFs, this factor refactorizes the
hadron element by covariant decomposition instead of by the polarization
vectors of the off-shell W-Boson (${\varepsilon_{\sigma}(q)}$). It can be
proved that HFFs under this new decomposition method has Lorentzian-invariant
property as well as TFFs. A simple proof can also be made directly from the
expression of HFFs, since HFFs are composed of TFFs, and TFFs have Lorentzian-
invariant properties, HFFs also have Lorentzian invariant properties. In
conclusion, HFFs is a decomposition method as effective as the TFFs method.

Further, it can be found that the helicity amplitude consists of a single TFF
or multiple TFFs mixed in the TFFs Leljak:2019eyw , while the helicity
amplitude corresponds to HFFs in HFFs, and it can be computed directly in
LCSR.

The calculated procedure of the $B_{c}\to J/\psi$ HFFs ${\cal
H}_{\sigma}^{B_{c}\to J/\psi}(q^{2})$ under the LCSR approach is the same as
that in Ref. Cheng:2018ouz ; Fu:2020vqd . We thus will directly give the LCSR
results of $B_{c}\to J/\psi$ HFFs as follows:

$\displaystyle{\cal H}_{0}^{B_{c}\to J/\psi}(q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}due^{(m_{B_{c}}^{2}-s)/M^{2}}\frac{m_{b}f_{J/\psi}^{\bot}{\cal
Q}}{2\sqrt{\lambda}m_{J/\psi}m_{B_{c}}^{2}f_{B_{c}}}\bigg{\\{}2{\cal
S}\Theta(c(u,s_{0}))\phi_{2;J/\psi}^{\bot}(u)-\frac{\lambda~{}m_{b}~{}m_{J/\psi}~{}\tilde{f}_{J/\psi}}{u^{2}M^{2}}\hskip
5.16663pt\widetilde{\Theta}(c(u,s_{0}))$ (7) $\displaystyle\times$
$\displaystyle\int_{0}^{u}dv\phi_{2;J/\psi}^{\|}(u)+m_{J/\psi}^{2}\bigg{[}{\cal
Q}\,\Theta(c(u,s_{0}))-\frac{\lambda}{uM^{2}}\widetilde{\Theta}(c(u,s_{0}))\bigg{]}\psi_{3;J/\psi}^{\|}(u)~{}+~{}m_{b}m_{J/\psi}\tilde{f}_{J/\psi}\hskip
3.44444pt\bigg{[}\frac{\lambda}{u^{2}M^{2}}$ $\displaystyle\times$
$\displaystyle\widetilde{\Theta}(c(u,s_{0}))\int_{0}^{u}dv\phi_{3;J/\psi}^{\bot}(u)+{\cal
Q}\Theta(c(u,s_{0}))\phi_{3;J/\psi}^{\bot}(u)\bigg{]}+m_{J/\psi}^{2}{\cal
S}\bigg{[}\frac{{\cal
R}}{2u^{3}M^{4}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))\hskip
3.44444pt-\frac{3}{{2u^{2}M^{2}}}$ $\displaystyle\times$
$\displaystyle\widetilde{\Theta}(c(u,s_{0}))\bigg{]}\phi_{4;J/\psi}^{\bot}(u)-\bigg{[}\frac{\lambda{\cal
S}}{2u^{3}M^{4}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))-m_{J/\psi}^{2}\frac{{\cal
S}-4\lambda}{u^{2}M^{2}}\bigg{]}\widetilde{\Theta}(c(u,s_{0}))I_{L}(u)-m_{b}^{3}m_{J/\psi}^{3}\hskip
3.44444pt\tilde{f}_{J/\psi}$ $\displaystyle\times$
$\displaystyle\frac{\lambda}{u^{4}M^{6}}~{}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))\int_{0}^{u}dv~{}\phi_{4;J/\psi}^{\bot}(u)+m_{b}m_{J/\psi}^{3}{\tilde{f}_{J/\psi}}~{}\bigg{[}\frac{\lambda}{{u^{2}M^{4}}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))~{}+\frac{\cal
Q}{u^{2}M^{2}}~{}\hskip 3.44444pt\widetilde{\Theta}(c(u,s_{0}))\,\bigg{]}$
$\displaystyle\times$ $\displaystyle
J_{J/\psi}(u)-m_{J/\psi}^{2}\bigg{[}{\frac{3}{2}\Theta(c(u,s_{0}))+\left({\frac{{\cal
R}}{u^{2}M^{2}}-\frac{\lambda}{{2uM^{2}{\cal
Q}}}}\right)\widetilde{\Theta}(c(u,s_{0}))}\bigg{]}{H_{3}}(u)\bigg{\\}},$
$\displaystyle{\cal H}_{1}^{B_{c}\to J/\psi}(q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}due^{(m_{B_{c}}^{2}-s)/M^{2}}\frac{m_{b}f_{J/\psi}^{\bot}\sqrt{2q^{2}}}{2m_{B_{c}}^{2}f_{B_{c}}}\bigg{\\{}\Theta(c(u,s_{0}))~{}\phi_{2;J/\psi}^{\bot}(u)+m_{J/\psi}^{2}~{}\bigg{[}\frac{{\cal
R}}{u^{3}M^{4}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))~{}+\hskip
3.44444pt\frac{3}{u^{2}M^{2}}$ (8) $\displaystyle\times$
$\displaystyle\widetilde{\Theta}(c(u,s_{0}))\bigg{]}\phi_{4;J/\psi}^{\bot}(u)-\frac{{m_{J/\psi}m_{b}{\tilde{f}_{J/\psi}}}}{{2u^{2}M^{2}}}\widetilde{\Theta}(c(u,s_{0}))\psi_{3;J/\psi}^{\bot}(u)\bigg{\\}},$
$\displaystyle{\cal H}_{2}^{B_{c}\to J/\psi}(q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}~{}du~{}e^{(m_{B_{c}}^{2}-s)/M^{2}}~{}\frac{\sqrt{2q^{2}}~{}m_{b}~{}f_{J/\psi}^{\bot}}{2\sqrt{\lambda}m_{B_{c}}^{2}f_{B_{c}}}~{}\bigg{\\{}{\cal
P}~{}\Theta(c(u,s_{0}))~{}\phi_{2;J/\psi}^{\bot}(u)+2m_{J/\psi}^{2}~{}\Theta(c(u,s_{0}))~{}\psi_{3;J/\psi}^{\|}(u)$
(9) $\displaystyle+$ $\displaystyle m_{J/\psi}^{2}{\cal P}\bigg{[}\frac{\cal
R}{u^{3}M^{4}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))+\frac{3}{u^{2}M^{2}}\widetilde{\Theta}(c(u,s_{0}))\bigg{]}\phi_{4;J/\psi}^{\bot}(u)+\frac{2m_{J/\psi}^{2}}{u^{2}M^{2}}~{}{\cal
P}\widetilde{\Theta}(c(u,s_{0}))I_{L}(u)-m_{J/\psi}^{2}$ $\displaystyle\times$
$\displaystyle\bigg{[}3\Theta(c(u,s_{0}))+\frac{2{\cal
R}}{u^{2}M^{2}}\widetilde{\Theta}(c(u,s_{0}))\bigg{]}\\!H_{3}(u)-2m_{b}m_{J/\psi}\tilde{f}_{J/\psi}\Theta(c(u,s_{0}))\phi_{3;J/\psi}^{\bot}(u)-\\!2m_{b}m_{J/\psi}^{3}\tilde{f}_{J/\psi}$
$\displaystyle\times$
$\displaystyle\frac{1}{u^{2}M^{2}}\widetilde{\Theta}(c(u,s_{0}))J_{J/\psi}(u)\bigg{\\}},$
$\displaystyle{\cal H}_{t}^{B_{c}\to J/\psi}(q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}du~{}e^{(m_{B_{c}}^{2}-s)/M^{2}}\frac{m_{b}~{}m_{J/\psi}~{}f_{J/\psi}^{\bot}}{2m_{J/\psi}m_{B_{c}}^{2}f_{B_{c}}}~{}\bigg{\\{}u~{}m_{J/\psi}\Theta(c(u,s_{0}))\phi_{2;J/\psi}^{\bot}(u)-m_{b}\tilde{f}_{J/\psi}~{}\frac{\cal
Q}{u^{2}M^{2}}~{}\widetilde{\Theta}(c(u,s_{0}))$ (10) $\displaystyle\times$
$\displaystyle\int_{0}^{u}dv\phi_{2;J/\psi}^{\|}(u)-(\tilde{m}_{q}\tilde{f}_{J/\psi}-m_{J/\psi})~{}\bigg{[}\Theta(c(u,s_{0}))+\frac{{u{\cal
Q}+2{q^{2}}}}{u^{2}M^{2}}\widetilde{\Theta}(c(u,s_{0}))\bigg{]}\psi_{3;J/\psi}^{\|}(u)-m_{b}\tilde{f}_{J/\psi}$
$\displaystyle\times$
$\displaystyle\Theta(c(u,s_{0}))~{}\phi_{3;J/\psi}^{\bot}(u)+m_{b}\tilde{f}_{J/\psi}~{}\frac{\cal
Q}{u^{2}M^{2}}~{}\widetilde{\Theta}(c(u,s_{0}))\int_{0}^{u}dv~{}\phi_{3;J/\psi}^{\bot}(u)+m_{J/\psi}^{3}\bigg{[}\frac{{\cal
R}}{u^{2}M^{4}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))$ $\displaystyle+$
$\displaystyle\frac{3}{uM^{2}}\widetilde{\Theta}(c(u,s_{0}))\bigg{]}\phi_{4;J/\psi}^{\bot}(u)~{}+~{}m_{b}^{3}m_{J/\psi}^{2}\tilde{f}_{J/\psi}\frac{{\cal
Q}}{u^{4}M^{6}}\widetilde{\widetilde{\widetilde{\Theta}}}(c(u,s_{0}))\int_{0}^{u}dv\,\phi_{4;J/\psi}^{\bot}(u)-m_{J/\psi}\bigg{[}\frac{{\cal
P}}{{2u{M^{2}}}}$ $\displaystyle\times$
$\displaystyle\widetilde{\Theta}(c(u,s_{0}))~{}+~{}\frac{3}{2}~{}\Theta(c(u,s_{0}))\bigg{]}~{}H_{3}(u)~{}-~{}m_{J/\psi}\bigg{[}~{}\frac{{9{\cal
Q}\,-\,2um_{J/\psi}^{2}\,+\,15{q^{2}}}}{u^{2}M^{2}}~{}\widetilde{\Theta}(c(u,s_{0}))~{}+~{}\frac{{\cal
T}}{u^{3}M^{4}}$ $\displaystyle\times$
$\displaystyle\widetilde{\widetilde{\Theta}}(c(u,s_{0}))\bigg{]}{I_{L}}(u)+\frac{{m_{b}\tilde{f}_{J/\psi}}}{2}\bigg{[}\frac{{2m_{J/\psi}^{2}}}{u^{2}M^{2}}\widetilde{\Theta}(c(u,s_{0}))+\frac{{\cal
S}}{u^{3}M^{4}}\widetilde{\widetilde{\Theta}}(c(u,s_{0}))\bigg{]}J_{J/\psi}(u)\bigg{\\}},$

where $\tilde{f}_{J/\psi}=f_{J/\psi}^{\|}/f_{J/\psi}^{\bot}$, ${\cal
P}=m_{B_{c}}^{2}+\xi m_{J/\psi}^{2}-q^{2}$, ${\cal
Q}=m_{B_{c}}^{2}-m_{J/\psi}^{2}-q^{2}$, ${\cal
R}=um_{B_{c}}^{2}-u\bar{u}m_{J/\psi}^{2}+\bar{u}{q^{2}}$, ${\cal
S}=2m_{J/\psi}^{2}(um_{B_{c}}^{2}-um_{J/\psi}^{2}+(1-\bar{u}){q^{2}})$, ${\cal
T}=2m_{B_{c}}^{2}[-u\xi
m_{J/\psi}^{2}+{q^{2}}(1+u+u\bar{u})]+u\xi(m_{B_{c}}^{4}+m_{J/\psi}^{4})-2{q^{2}}(1+u)\bar{u}m_{J/\psi}^{2}-{q^{4}}(2+u)$
and $s=[m_{b}^{2}-\bar{u}(q^{2}-um_{J/\psi}^{2})]/u$ with $\bar{u}=1-u$,
$\xi=2u-1$. The $\Theta(c(u,s_{0}))$, $\widetilde{\Theta}[c(u,s_{0})]$,
$\widetilde{\widetilde{\Theta}}(c(u,s_{0})]$ and
$\widetilde{\widetilde{\widetilde{\Theta}}}(c(u,s_{0})]$ are the step
function. The simplified distribution functions $H_{3}(u)$, $I_{L}(u)$ and
$J_{J/\psi}(u)$ can be written as

$\displaystyle
H_{3}(u)=\int_{0}^{u}dv\Big{[}\psi_{4;J/\psi}^{\bot}(v)-\phi_{2;J/\psi}^{\bot}(v)\Big{]},$
$\displaystyle
I_{L}(u)=\int_{0}^{u}dv\int_{0}^{v}dw\Big{[}\phi_{3;J/\psi}^{\|}(w)-\frac{1}{2}\phi_{2;J/\psi}^{\bot}(w)$
$\displaystyle\hskip 24.11105pt-\frac{1}{2}\psi_{4;J/\psi}^{\bot}(w)\Big{]},$
$\displaystyle
J_{J/\psi}(u)=\int_{0}^{u}{dv}\int_{0}^{v}{dw}\Big{[}\psi_{4;J/\psi}^{\|}(w)+\phi_{2;J/\psi}^{\|}(w)$
$\displaystyle\hskip 34.44434pt-2\phi_{3;J/\psi}^{\bot}(w)\Big{]}.$ (11)

A more detailed calculation can be found in our previous work Cheng:2018ouz ;
Fu:2020vqd . Since we used a usual current in the calculation, the HFFs will
include all the Twist-2,-3,-4 light-cone distribution amplitudes (LCDAs), see
subsection III.1, which will maintain the accuracy of LCSR to a large extent.

##  III Numerical results and discussion

While performing the numerical calculations, we take the mass of ${B_{c}}$\-
and $J/\psi$-meson are $m_{B_{c}}=6.2749~{}\rm{GeV}$ and
$m_{J/\psi}=3.097~{}\rm{GeV}$ Tanabashi:2018oca , ${B_{c}}$-meson decay
constant $f_{B_{c}}=0.498\pm{0.014}~{}\rm{GeV}$ Zhong:2014fma , $J/\psi$-meson
decay constant $f_{J/\psi}^{\perp}=0.410\pm{0.014}~{}\rm{GeV}$ and
$f_{J/\psi}^{\|}=0.416\pm{0.005}~{}\rm{GeV}$ Becirevic:2013bsa ; Zeng:2021hwt
. The CKM-matrix elements will set its central values, i.e.,
$|V_{cb}|=0.0405$, $|V_{ud}|=0.974$ and $|V_{us}|=0.225$ Patrignani:2016xqp .
For the factorization scale $\mu$, we will fix it as the typical momentum
transfer of $B_{c}\to J/\psi$ Ball:2004rg , i.e.,
$\mu\simeq(m_{B_{c}}^{2}-m_{b}^{2})^{1/2}\sim 4.68~{}\rm{GeV}$.

###  III.1 $J/\psi$-meson LCDAs and $B_{c}\to J/\psi$ HFFs

An important part in the LCSR $B_{c}\to J/\psi$ HFFs is the $J/\psi$-meson
LCDAs. For the leading twist DAs $\phi_{2,J/\psi}^{\lambda}(x,\mu)$, we will
take the Wu-Huang (WH) model to carry out our analysis. Its definition is as
follows Zeng:2021hwt :

$\displaystyle\phi_{2;J/\psi}^{\lambda}(x,\mu)$
$\displaystyle=\frac{\sqrt{3}A_{J/\psi}^{\lambda}\hat{m}_{c}\beta_{J/\psi}^{\lambda}\sqrt{x\bar{x}}}{2\pi^{3/2}f_{J/\psi}^{\lambda}}\bigg{\\{}{\rm
Erf}\bigg{[}\sqrt{\frac{\hat{m}_{c}^{2}+\mu^{2}}{8(\beta_{J/\psi}^{\lambda})^{2}x\bar{x}}}\bigg{]}$
$\displaystyle-{\rm
Erf}\bigg{[}\sqrt{\frac{\hat{m}_{c}^{2}}{8(\beta_{J/\psi}^{\lambda})^{2}x\bar{x}}}\bigg{]}\bigg{\\}},$
(12)

where $\lambda=(\bot,\|)$, $\bar{x}=(1-x)$, the $c$-quark mass is taken as
$\hat{m}_{c}=1.5~{}{\rm GeV}$, and the error function ${\rm
Erf}(x)=2\int_{0}^{x}e^{-t^{2}}dt/\sqrt{\pi}$. The remaining model parameters
$A_{J/\psi}^{\lambda}$ and $\beta_{J/\psi}^{\lambda}$ can be fixed by
employing the normalization condition, i.e.,
$\int\phi_{2;J/\psi}^{\lambda}(x,\mu)dx=1$, and the second Gegenbauer moment
$a_{2;J/\psi}^{\|}(\mu_{0})=-0.379\pm{0.020}$ and
$a_{2;J/\psi}^{\bot}(\mu_{0})=-0.373_{-0.015}^{+0.026}$ Braguta:2007fh  that
are related to the leading-twist DAs, i.e.,

$\displaystyle
a_{2;J/\psi}^{\lambda}(\mu)=\dfrac{\displaystyle\int_{0}^{1}dx\phi_{2;J/\psi}^{\lambda}(x,\mu)C_{2}^{3/2}(2x-1)}{\displaystyle\int_{0}^{1}6x\bar{x}[C_{2}^{3/2}(2x-1)]^{2}}.$
(13)

The final model parameters are listed in Table. 2. Note that the model
parameters at arbitrary scale can be obtained by running the scale dependence
of the Gegenbauer moments Ball:2006nr .

Table 2: The determined model parameters of leading twist-2 LCDA with $\mu_{0}=1.2~{}{\rm GeV}$. $a_{2;J/\psi}^{\|}(\mu_{0})$ |  $A_{J/\psi}^{\|}$ |  $\beta_{J/\psi}^{\|}$ | $a_{2;J/\psi}^{\bot}(\mu_{0})$ |  $A_{J/\psi}^{\bot}$ |  $\beta_{J/\psi}^{\bot}$
---|---|---|---|---|---
$-0.379$ | $3818$ | $0.536$ | $-0.373$ | $3010$ | $0.548$
$-0.359$ | $1841$ | $0.579$ | $-0.353$ | $1510$ | $0.592$
$-0.400$ | $9341$ | $0.493$ | $-0.394$ | $6988$ | $0.506$

For the $J/\psi$-meson twist-3 LCDAs, we will relate it to leading-twist DAs
by employing the Wandzura-Wilczek approximation Wandzura:1977qf ; Ball:1997rj
. The specific relationships are as follows:

$\displaystyle\psi_{3;J/\psi}^{\bot}(u)$
$\displaystyle=2~{}\bigg{[}~{}\bar{u}\int_{0}^{u}dv\frac{\phi_{2;J/\psi}^{\|}(v)}{\bar{v}}+u\int_{u}^{1}dv\frac{\phi_{2;J/\psi}^{\|}(v)}{v}\hskip
0.86108pt\bigg{]},$ $\displaystyle\phi_{3;J/\psi}^{\bot}(u)$
$\displaystyle=\frac{1}{2}~{}\bigg{[}~{}\int_{0}^{u}~{}dv~{}\frac{\phi_{2;J/\psi}^{\|}(v)}{\bar{v}}\,+\int_{u}^{1}~{}dv\frac{\phi_{2;J/\psi}^{\|}(v)}{v}\bigg{]},$
$\displaystyle\psi_{3;J/\psi}^{\|}(u)$
$\displaystyle=2\,\bigg{[}\,\bar{u}\,\int_{0}^{u}dv\frac{\phi_{2;J/\psi}^{\bot}(v)}{\bar{v}}+u\,\int_{u}^{1}dv\frac{\phi_{2;J/\psi}^{\bot}(v)}{v}\hskip
0.86108pt\bigg{]},$ $\displaystyle\phi_{3;J/\psi}^{\|}(u)$
$\displaystyle=(1-2\bar{u})\\!\bigg{[}\\!\int_{0}^{u}\\!\\!dv\frac{\phi_{2;J/\psi}^{\bot}(v)}{\bar{v}}-\int_{u}^{1}\\!\\!dv\frac{\phi_{2;J/\psi}^{\bot}(v)}{v}\bigg{]},$

where $\bar{u}=(1-u)$ and $\bar{v}=(1-v)$. For the $J/\psi$-meson twist-4
LCDAs, as its contributions are usually small in comparison to that of the
twist-2,-3, we shall ignore charm-quark mass effect of the twist-4 LCDAs
Ball:1998kk  to do our analysis.

Figure 1: The extrapolated HFFs ${\cal H}_{\sigma}^{B_{c}\to J/\psi}(q^{2})$
as a function of $q^{2}$, in which the solid lines represent the center values
and the shaded bands stand for the uncertainties that are the squared of all
mentioned error sources.

As for the continuum threshold $s_{0}$ of the $B_{c}\to J/\psi$ HFFs ${{\cal
H}_{\sigma}^{B_{c}\to J/\psi}}(q^{2})$, it can usually be set near the squared
mass of $B_{c}$-meson’s first exciting state or the value between
$B_{c}$-meson ground state and higher mass contributions. We will fix it as
$s_{{\cal H}_{\sigma}}=45.0(0.5)~{}\rm{GeV}^{2}$. In contrast, to obtain the
Borel windows of the $B_{c}\to J/\psi$ HFFs, we require the continuum
contribution should be less than $65\%$ of the total LCSR. Thus, the final
Borel windows $M_{{\cal H}_{j}}^{2}(\rm{GeV}^{2})$ are $M^{2}_{{\cal
H}_{0}}=71.0(1.0)$, $M^{2}_{{\cal H}_{1}}=72.0(1.0)$, $M^{2}_{{\cal
H}_{2}}=8.5(1.0)$, and $M^{2}_{{\cal H}_{t}}=180.0(1.0)$.

Compared with the TFFs decomposition method, for a single TFFs composed of
helicity amplitude, the value of $s_{0}$ and $M^{2}$ of the two methods have
no affect to the helicity amplitude. However, for a mixture of multiple TFFs
composed of helicity amplitude, the value of $s_{0}$ and $M^{2}$ of different
TFFs is usually different, therefore, there will be a difference in the
helicity amplitude obtained by the two methods, which will affect the
following theoretical prediction.

Since the reliable region of the LCSR is in the lower and intermediate
$q^{2}$-region, we can take it as $0\leq q^{2}\leq q^{2}_{\rm LCSR,max}\approx
5~{}\rm{GeV}^{2}$ for the $B_{c}\to J/\psi$ decay, and the allowable physical
range of the momentum transfer is $0\leq q^{2}\leq q^{2}_{B_{c}\to J/\psi,{\rm
max}}$ with $q^{2}\leq q^{2}_{B_{c}\to J/\psi,{\rm
max}}=(m_{B_{c}}-m_{J/\psi})^{2}\simeq 10.10~{}\rm{GeV}^{2}$. Therefore, the
HFFs obtained by LCSR need to be extended into full allowable physical ranges,
so that the $B_{c}$-meson decays can be studied further. Here, we will adopt
the SSE method Bourrely:2008za ; Bharucha:2010im  to perform the extrapolation
due to the analyticity and unitarity consideration. The extrapolated formulas
of the $B_{c}\to J/\psi$ HFFs ${\cal H}_{\sigma}^{B_{c}\to J/\psi}(q^{2})$ can
be expressed as:

Table 3: Relations between the mass of low-lying $B_{c}$ resonances and the parametrization of HFFs ${\cal H}_{\sigma}^{\rm fit}$. Transition | $J^{P}$ |  $m_{R,i}$ (GeV) | ${\cal H}_{\sigma}^{\rm fit}(q^{2})$
---|---|---|---
| $0^{-}$ | 6.28 | ${\cal H}_{t}^{\rm fit}(q^{2})$
$b\to c$ | $1^{-}$ | 6.34 | ${\cal H}_{1}^{\rm fit}(q^{2})$
| $1^{+}$ | 6.75 | ${\cal H}_{0,2}^{\rm fit}(q^{2})$

$\displaystyle{\cal H}_{0}^{\rm fit}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{B_{0}(t)\sqrt{z(t,t_{-})}\phi_{T}^{V-A}(t)}\sum_{k=0,1,2}a_{k}^{0}z^{k},$
(15) $\displaystyle{\cal H}_{1}^{\rm fit}(t)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-z(t,0)}}{B_{1}(t)\phi_{T}^{V-A}(t)}\sum_{k=0,1,2}a_{k}^{1}z^{k},$
(16) $\displaystyle{\cal H}_{2}^{\rm fit}(t)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-z(t,0)}}{B_{2}(t)\sqrt{z(t,t_{-})}\phi_{T}^{V-A}(t)}\sum_{k=0,1,2}a_{k}^{2}z^{k},$
(17) $\displaystyle{\cal H}_{t}^{\rm fit}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{B_{t}(t)\phi_{L}^{V-A}(t)}\sum_{k=0,1,2}a_{k}^{t}z^{k},$
(18)

where $B_{i}(t)=1-q^{2}/m_{i}^{2}$ with $i=(0,1,2,t)$. The mass $m_{0,1,2,t}$
are in Table. 3, $\phi_{T,L}^{V-A}(t)=1$,
$\sqrt{-z(t,0)}=\sqrt{q^{2}}/m_{B_{c}}$,
$\sqrt{z(t,t_{-})}=\sqrt{\lambda(q^{2})}/m_{B_{c}}^{2}$, and
$z(t)=({\sqrt{t_{+}-t}-\sqrt{t_{+}-t_{0}}})/({\sqrt{t_{+}-t}+\sqrt{t_{+}-t_{0}}})$
with $t_{\pm}=(m_{B_{c}}\pm m_{J/\psi})^{2}$ and
$t_{0}=t_{+}(1-\sqrt{1-t_{-}/t_{+}})$.

Table 4: The fitted results of the parameters $a^{\sigma}_{k}$ and $\Delta$ for the HFFs ${\cal H}_{\sigma}^{B_{c}\to J/\psi}$, where we take all input parameters as their central values. | ${\cal H}_{0}^{B_{c}\to J/\psi}(q^{2})$ | ${\cal H}_{1}^{B_{c}\to J/\psi}(q^{2})$ | ${\cal H}_{2}^{B_{c}\to J/\psi}(q^{2})$ | ${\cal H}_{t}^{B_{c}\to J/\psi}(q^{2})$
---|---|---|---|---
$a_{0}^{\sigma}$ | $0.471$ | $0.507$ | $2.819$ | $0.662$
$a_{1}^{\sigma}$ | $13.409$ | $-6.746$ | $-53.780$ | $-0.428$
$\Delta$ | 0.927 | $0.917$ | $0.991$ | $0.992$
Table 5: Decay width of the two body decays $B_{c}^{+}\to J/\psi+(P,V)$ in units $a_{1}^{2}\times 10^{-15}~{}\rm{GeV}$. |  $B_{c}^{+}\to J/\psi\pi^{+}$ |  $B_{c}^{+}\to J/\psi\rho^{+}$ |  $B_{c}^{+}\to J/\psi K^{+}$ | $B_{c}^{+}\to J/\psi K^{\ast+}$
---|---|---|---|---
This Work | $1.642^{+0.082}_{-0.076}$ | $9.251^{+0.560}_{-0.604}$ | $0.123^{+0.006}_{-0.006}$ | $0.524^{+0.035}_{-0.039}$
RM AbdElHady:1999xh  | $1.22$ | $3.48$ | $0.09$ | $0.20$
RQM Ebert:2003cn  | $0.67$ | $1.8$ | $0.05$ | $0.11$
CCQM Issadykov:2018myx  | $1.22\pm 0.24$ | $2.03\pm 0.41$ | $0.09\pm 0.02$ | $0.13\pm 0.03$
BS RQM Chang:2014jca  | $1.24\pm 0.11$ | $3.59^{+0.64}_{-0.58}$ | $0.095\pm 0.008$ | $0.226\pm 0.03$
Table 6: Branching fractions (in unit of $\%$) of the two body decays $B_{c}^{+}\to J/\psi+(P,V)$ decays obtained with $a_{1}=1.038$. |  $B_{c}^{+}\to J/\psi\pi^{+}$ |  $B_{c}^{+}\to J/\psi\rho^{+}$ |  $B_{c}^{+}\to J/\psi K^{+}$ | $B_{c}^{+}\to J/\psi K^{\ast+}$
---|---|---|---|---
This Work | $0.136^{+0.002}_{-0.002}$ | $0.768^{+0.029}_{-0.033}$ | $0.010^{+0.000}_{-0.000}$ | $0.043^{+0.001}_{-0.001}$
CCQM Issadykov:2018myx  | $0.101\pm 0.02$ | $0.334\pm 0.067$ | $0.008\pm 0.002$ | $0.019\pm 0.004$

To fix the parameters $a_{k}^{\sigma}$, we will take the “quality” of fit
($\Delta$) to be less than 1%, i.e.,

$\Delta=\frac{\sum_{t}|{\cal H}_{\sigma}^{B_{c}\to J/\psi}(t)-{\cal
H}_{\sigma}^{\rm fit}(t)|}{\sum_{t}|{\cal H}_{\sigma}^{B_{c}\to
J/\psi}(t)|}\times 100\%,$ (19)

where $t\in[0,0.5,\cdots,4.5,5.0]~{}{\rm GeV}^{2}$. The determined parameters
$a_{k}^{\sigma}$ are listed in Table 4, in which all the input parameters are
set to be their central values. With the extrapolated Eq. (15)-(18) and the
fitted parameters $a^{\sigma}_{k}$, we show the extrapolated HFFs ${\cal
H}_{\sigma}^{B_{c}\to J/\psi}(q^{2})$ in whole $q^{2}$-regions in Fig. 1,
where the shaded band stands for the squared average of all the mentioned
uncertainties. Same as our previous study on pseudoscalar meson decay to
vector meson, such as $B\to\rho$ Cheng:2018ouz  and $D\to V$ Fu:2020vqd , the
transverse part of the final meson does not contribute to HFFs at the large
recoil point $q^{2}=0$. Specifically, we have ${\cal H}_{0}^{B_{c}\to
J/\psi}(0)=0.915_{-0.018}^{+0.022}$, ${\cal H}_{t}^{B_{c}\to
J/\psi}(0)=0.644_{-0.015}^{+0.018}$, and ${\cal H}_{1,2}^{B_{c}\to
J/\psi}(0)=0$, where the errors are from the $J/\psi$-meson decay constant
$f_{J/\psi}$, the $J/\psi$-meson LCDAs, the Borel parameter $M^{2}$, and the
continuum threshold $s_{0}$. All the HFFs ${\cal H}_{\sigma}^{B_{c}\to
J/\psi}(q^{2})$ are monotonically increase with the increment of $q^{2}$.

###  III.2 The two body decays $B_{c}^{+}\to J/\psi+(P,V)$

Table 7: The branching ratios ${\cal R}_{\pi^{+}/\mu^{+}\nu_{\mu}}$, ${\cal R}_{K^{+}/\pi^{+}}$ and ${\cal R}_{J/\psi}$ with the errors are squared average from various input parameters. | $\mathcal{R_{\pi^{+}/\mu^{+}\nu_{\mu}}}$ | ${\cal R}_{K^{+}/\pi^{+}}$ | ${\cal R}_{J/\psi}$
---|---|---|---
This work | $0.048^{+0.009}_{-0.012}$ | $0.075^{+0.005}_{-0.005}$ | $0.199^{+0.060}_{-0.077}$
LHCb Aaij:2014jxa  | $0.047\pm 0.005$ |  |
LHCb Aaij:2013vcx  |  | $0.069\pm 0.019$ |
LHCb Aaij:2016tcz  |  | $0.079\pm 0.008$ |
LHCb Aaij:2017tyk  |  |  | $0.71\pm 0.25$
RM AbdElHady:1999xh  | $0.0525$ | $0.074$ |
LFQM Ke:2013yka  | $0.058$ | $0.075$ |
CCQM Issadykov:2018myx  | $0.061\pm 0.012$ | $0.076\pm 0.015$ | $0.24\pm 0.05$
BS RQM Chang:2014jca  | $0.064^{+0.007}_{-0.008}$ | $0.072^{+0.019}_{-0.008}$ |
RQM Ebert:2003cn  | $0.050$ | $0.077$ |
pQCD Wen-Fei:2013uea  |  |  | $0.29$
LCSR Leljak:2019eyw  |  |  | $0.23\pm 0.01$
QCD-SR Azizi:2019aaf  |  |  | $0.25\pm 0.01$
CLFQM Huang:2018nnq  |  |  | $0.248\pm 0.006$
pQCD Shen:2014msa  |  |  | $0.26$
NRQCD Qiao:2012hp  |  | $0.075$ |
Figure 2: LCSR prediction for the branching ratio ${\cal R}_{K^{+}/\pi^{+}}$. Table 8: Branching fractions (in unit of $\%$) of the $B_{c}^{+}$-meson semileptonic decays within uncertainties. Other theoretical predictions are also listed here to make a comparison. | $B_{c}^{+}\to J/\psi\mu^{+}\nu_{\mu}$ | $B_{c}^{+}\to J/\psi\tau^{+}\nu_{\tau}$
---|---|---
This Work | $2.802^{+0.526}_{-0.675}$ | $0.559^{+0.131}_{-0.170}$
RM AbdElHady:1999xh  | $2.01$ |
CCQM Issadykov:2018myx  | $1.67\pm 0.33$ | $0.40\pm 0.08$
BSM Chang:1992pt  | $2.33$ |
BS RQM Chang:2014jca  | $1.73$ |
RQM Ebert:2003cn  | $1.67\pm 0.33$ | $1.23$
pQCD Wen-Fei:2013uea  | $1.01$ | $0.29$
LCSR Leljak:2019eyw  | $2.24^{+0.57}_{-0.49}$ | $0.53^{+0.16}_{-0.14}$
QCD-SR Azizi:2019aaf  | $1.93^{+0.50}_{-0.60}$ | $0.49^{+0.10}_{-0.14}$

After setting all the input parameters and getting the extrapolated $B_{c}\to
J/\psi$ HFFs, we can now analyze the $B_{c}$-meson decays numerically. For the
$B_{c}$-meson two-body decay $B_{c}^{+}\to J/\psi+(P,V)$, its decay width can
be calculated by employing the Eq. (3). We list our results in Table. 5. As a
comparison, RM AbdElHady:1999xh , CCQM Issadykov:2018myx , BS RQM
Chang:2014jca  and RQM Ebert:2003cn  predictions are also shown. After further
taking into account the $B_{c}$-meson lifetime $\tau_{B_{c}}^{\rm
exp}=(0.507\pm 0.009)~{}{\rm ps}$ from Particle Data Group Zyla:2020zbs , we
can get the corresponding branching fractions that are listed in Table. 6. In
order to compare with the experiments, we calculate the branching ratio ${\cal
R}_{K^{+}/\pi^{+}}$, which is given by

${\cal R}_{K^{+}/\pi^{+}}=\frac{{\cal B}(B_{c}^{+}\to J/\psi K^{+})}{{\cal
B}(B_{c}^{+}\to J/\psi\pi^{+})}.$ (20)

The numerical results are listed in Table. 7. Although our results related to
the two-body decay width in Table. 5 and branching fractions in Table. 6 are
larger than that of other theories, our results ${\cal R}_{K^{+}/\pi^{+}}$ are
consistent with the latest LHCb experiment results within the
$1\sigma$-errors. For a more intuitive comparison, we show ${\cal
R}_{K^{+}/\pi^{+}}$ in Fig. 2.

###  III.3 The semi-leptonic decays $B_{c}^{+}\to J/\psi\ell^{+}\nu_{\ell}$

Figure 3: The differential decay width of the semileptonic $B_{c}^{+}\to
J/\psi\ell^{+}\nu_{\ell}$ decays with $\ell=\mu$ for left and $\ell=\tau$ for
right, where the solid and shade bands correspond to their central values and
the uncertainties respectively.

Figure 4: The LCSR predictions on branching ratios ${\cal R}_{J/\psi}$ and
${\cal R}_{\pi/\mu^{+}\nu_{\mu}}$ that can be calculated by Eq. (22) and Eq.
(21) respectively.

With the $B_{c}^{+}\to J/\psi\ell^{+}\nu_{\ell}$ semi-leptonic decays Eq. (5)
and $B_{c}$ lifetime, we can calculate the branching fractions of the semi-
leptonic decays $B_{c}^{+}\to J/\psi\ell^{+}\nu_{\ell}$ that are listed in
Table. 8. For comparison, we also present other theoretical predictions, i.e.
RM AbdElHady:1999xh , CCQM Issadykov:2018myx , BS RQM Chang:2014jca , RQM
Ebert:2003cn , pQCD Wen-Fei:2013uea , LCSR Leljak:2019eyw  and QCD-SR
Azizi:2019aaf . We find that all the theoretical branching fraction
predictions of the $\mu$-lepton decay channel are greater than that of the
$\tau$-lepton decay channel. That may be caused by the small mass of
$\mu$-lepton. To illustrate this effect more clearly, we have shown the
differential decay width in Fig. 3, where the solid and shade bands correspond
to their central values and the uncertainties respectively. In addition, we
can see that even though they were drawn by applying the same formula,
however, for $q^{2}\sim q^{2}_{\rm min}$, $\mu$\- and $\tau$-lepton decay
channel are significantly different, and there is an obvious sharp increase
for $\mu$-lepton decay channel, which results in the branching fraction of
$\mu$ decay channel being significantly larger than that of the $\tau$ decay
channel.

Considering that $R_{J/\psi}$ and $R_{\pi/\mu^{+}\nu_{\mu}}$ were measured by
LHCb experiment, their definitions are as follows:

$\displaystyle{\cal R}_{J/\psi}=\frac{{\cal B}(B_{c}^{+}\to
J/\psi\tau^{+}\nu_{\tau})}{{\cal B}(B_{c}^{+}\to J/\psi\mu^{+}\nu_{\mu})},$
(21) $\displaystyle{\cal R}_{\pi^{+}/\mu^{+}\nu_{\mu}}=\frac{{\cal
B}(B_{c}^{+}\to J/\psi\pi^{+})}{{\cal B}(B_{c}^{+}\to
J/\psi\mu^{+}\nu_{\mu})}.$ (22)

We collect our results in Table. 7 by using those formula. For comparison, the
predictions of LHCb Aaij:2014jxa , LHCb Aaij:2016tcz , LHCb Aaij:2017tyk , RM
AbdElHady:1999xh , LFQM Ke:2013yka , CCQM Issadykov:2018myx , BSM Chang:1992pt
, BS RQM Chang:2014jca , RQM Ebert:2003cn , pQCD Wen-Fei:2013uea , LCSR
Leljak:2019eyw , QCD-SR Azizi:2019aaf , CLFQM Huang:2018nnq  are also shown.
For convenience, we have shown it in Fig. 4. For ${\cal
R}_{\pi^{+}/\mu^{+}\nu_{\mu}}$, all theoretical predictions are consistent
with the LHCb experimental value Aaij:2014jxa  within the $2\sigma$-errors.
For $R_{J/\psi}$, all theoretical predictions are less than the measured value
of the LHCb experiment Aaij:2017tyk . In addition, our predictions of ${\cal
R}_{\pi^{+}/\mu^{+}\nu_{\mu}}$ and ${\cal R}_{K^{+}/\pi^{+}}$ are also
consistent with other theories, which also shows that the HFFs method is an
effective method from the predictions. Therefore, this is a new, complete and
effective method for calculating the decay of Bc mesons. Using this method to
calculate the ${\cal R}_{J/\psi}$ ratio in SM is also useful.

##  IV summary

In this study, we have investigated the $B_{c}\to J/\psi$ HFFs and keep up to
twist-4 accuracy within the LCSR approach. As shown in Fig. 1, similar to
other pseudoscalar meson decays to vector meson, the $J/\psi$-meson transverse
component will not have any contribution to the HFFs of the $B_{c}\to J/\psi$
transition at the point $q^{2}=0$. With the extrapolated $B_{c}\to J/\psi$
HFFs, we study the semi-leptonic decays $B_{c}\to J/\psi\ell^{+}\nu_{\ell}$
with $\ell=(\mu,\tau)$ and the two-body decays $B_{c}\to J/\psi+(P,V)$ with
$P=(\pi^{+},K^{+})$ and $V=(\rho^{+},K^{*+})$. The corresponding decay widths
and branch fractions predictions are listed in Table. 7. We observe that our
results are larger than that of the other theories.

To compare with the experiments, we use these predictions to further study the
three kinds of branching ratios, i.e., ${\cal R}_{\pi^{+}/\mu^{+}\nu_{\mu}}$ ,
${\cal R}_{K^{+}/\pi^{+}}$, and ${\cal R}_{J/\psi}$. The results are listed in
Tables 5, 6, 8. Meanwhile, we also provided two images for comparsion as shown
in Figs. 2 and 4 to give our results more clarity. For
$\mathcal{R_{\pi^{+}/\mu^{+}\nu_{\mu}}}$ and ${\cal R}_{K^{+}/\pi^{+}}$, our
results are consistent with other theorical predictions and the LHCb
experimental results within the $1\sigma$-errors. For ${\cal R}_{J/\psi}$, our
predictions are close to the those obtained using other theories, but all of
the theoretical predictions were smaller than that of the LHCb experimental
predictions. Therefore, we believe that the HFFs obtained by the LCSR approach
are also applicable to the $B^{+}_{c}$ meson two-body decays and semi-leptonic
decays $B_{c}^{+}\to J/\psi+(P,V,\ell^{+}\nu_{\ell})$. According to the ${\cal
R}_{J/\psi}$, calculating the HFFs in LCSR in a new way shows that there may
be new physics in the $B_{c}\to J/\psi\ell^{+}\nu_{\ell}$ semi-leptonic
decays.

## Acknowledgment

W. Cheng and H. B. Fu would like to thank the Institute of Theoretical Physics
in Chongqing University (CQUITP) for kind hospitality. This work was supported
in part by the National Natural Science Foundation of China under Grant
No.11765007, No.11625520, No.11947406 and No.12047564, the Project of Guizhou
Provincial Department of Science and Technology under Grant No.KY[2019]1171,
the Project of Guizhou Provincial Department of Education under Grant
No.KY[2021]030, the China Postdoctoral Science Foundation under Grant
Nos.2019TQ0329, 2020M670476, the Chongqing Graduate Research and Innovation
Foundation under Grant No.ydstd1912, the Fundamental Research Funds for the
Central Universities under Grant No.2020CQJQY-Z003, and the Project of Guizhou
Minzu University under Grant No. GZMU[2019]YB19.

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# Exploring the Efficacy of Large Language Models (GPT-4) in Binary Reverse
Engineering

Saman Pordanesh Dr. Benjamin Tan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This study investigates the capabilities of Large Language Models (LLMs),
specifically GPT-4 [1], in the context of Binary Reverse Engineering (RE).
Employing a structured experimental approach, we analyzed the LLM’s
performance in interpreting and explaining human-written and decompiled codes.
The research encompassed two phases: the first on basic code interpretation
and the second on more complex malware analysis. Key findings indicate LLMs’
proficiency in general code understanding, with varying effectiveness in
detailed technical and security analyses. The study underscores the potential
and current limitations of LLMs in reverse engineering, revealing crucial
insights for future applications and improvements. Also, we examined our
experimental methodologies, such as methods of evaluation and data
constraints, which provided us with a technical vision for any future research
activity in this field.

###### Index Terms:

Large Language Models (LLMs), GPT-4, Binary Reverse Engineering, Code
Interpretation, Malware Analysis, Decompiled Code Analysis, AI-Assisted
Reverse Engineering

##  I Introduction

The advent of Large Language Models (LLMs), a groundbreaking AI technology
employing deep learning, has revolutionized the way we interact with machines.
These advanced models can understand human language and generate relevant,
context-specific content for various Natural Language Processing (NLP) [2]
tasks. This advancement has sparked considerable interest across diverse
professional fields as people seek to leverage these tools to enhance the
efficiency and effectiveness of their work.

In computer science, particularly in reverse engineering, challenges in code
comprehension and pattern recognition are common. LLMs, however, are showing
promising capabilities in various computer science-related tasks, including
interpreting and explaining code using these sophisticated deep learning
models [3].

Reverse engineering of binary files, a critical aspect of computer science,
involves deciphering the underlying code and design logic of binary files.
This process is naturally complex and time-consuming [4], as it requires
retracing the steps of code compilation. The potential application of LLMs in
Binary Reverse Engineering sparked our curiosity, raising questions about
their performance, current capabilities, and areas for further improvement in
this challenging field. A key area of interest is the application of LLMs in
understanding decompiled binary code, a task markedly different from
interpreting standard, human-written code.

To address these questions, our research seeks to establish a meaningful
connection between reverse engineering tasks and LLMs’ capabilities. We aim to
explore these two domains’ intersections, enabling us to conduct more in-depth
investigations and address more specific, real-world challenges.

This report presents the initial stage of our research journey. We delve into
various infrastructures, application perspectives, and the scope of coverage
in this field. Additionally, we survey the technical aspects of our
experimental methods, including evaluation, implementation, and the
constraints of the tools. Our goal is to provide a comprehensive understanding
of the role and potential of LLMs in building up Binary Reverse Engineering
processes.

###  I-A Overview of the Literature

We did a detailed literature survey of at least ten related papers and
provided a strong background of what we had done in the experimental phase.
The landscape of AI-assisted reverse engineering and code analysis has seen
significant advancements, as evidenced by recent research. Studies like those
by Pearce et al. [5] and Wong et al. [6] have demonstrated the potential of
large language models like OpenAI’s Codex [7] to enhance the understanding and
recompilability of decompiled code. These advancements suggest significant
matches between AI tools and traditional reverse engineering techniques,
suggesting a good option for addressing complex software analysis and
cybersecurity challenges. Furthermore, the efforts by Davis et al. [8], and
Al-Kaswan et al. [9] in binary code summarization underscore the burgeoning
role of AI in generating concise, understandable summaries of complex binary
code. This is crucial in a landscape where the volume and complexity of
malware and software rapidly expand.

Moreover, the application of AI in educational contexts, as explored by Sarsa
et al. [3], and in malware analysis, as developed by Novkovicé et al. [10],
and Hajipour et al. [11], highlight the versatility of AI tools. Creating
specialized datasets and models, such as BinT5 [12] and MLARES, demonstrate
the expanding capabilities of AI in providing insightful analysis and support
in diverse areas of programming and cybersecurity. This body of work
collectively indicates a significant shift towards integrating advanced AI
techniques in software engineering, offering innovative solutions to
longstanding challenges in code analysis, malware classification, and
programming education.

###  I-B Our Research Direction

Based on our literature survey, which we talked about briefly in the previous
section, experiments were discovering LLM’s capability in some areas like code
explanations [3, 6, 9], recompiling a decompiled code [6, 9], and training
LLMs for security tasks [13, 10, 8].

Generally, these publications focus on machine learning and neural network
capability to ease reverse engineering tasks like malware analysis and
malicious code or file recognition. We wrapped our thoughts around these
different fields and decided to be more focused on code explanation, which
keeps this research close to the code understanding aspect of the LLMs. The
reason is that firstly, we found these areas have more capacity for
improvement and make some RE jobs easier. Also, some RE tasks like malware
detection are already being researched and developed by well-known
cybersecurity companies for commercial purposes. However, without going deep
into malware detection, papers around this field provided us with helpful
materials on research jobs like data collection, evaluation methods, and
analysis approaches.

##  II Setups & Tools

This section outlines our study’s specific tools and technologies, detailing
the range of tools and technological frameworks utilized. It also illuminates
our data collection strategies throughout the experiment.

###  II-A Model Selection - GPT4

Central to this experiment is exploring Large Language Model (LLM)
performance. With various models available, including OpenAI’s Codex [7],
GPT-3.5 [14], GPT-4 [1], and Facebook’s LlaMa [15], our choice fell on GPT-4.
Selected based on benchmarks Trustbit LLM Benchmark [16, 17, 18], GPT-4 stands
out as one of the most advanced LLMs in NLP, positioning us to evaluate the
forefront of this technology and later form robust recommendations for
improvement.

###  II-B Tools

####  II-B1 Python Scripting and Data Storage

The experiment interfaces with GPT-4 via its API using a unique API key. All
operations, including data transformation, prompt creation, communication with
the model, and result collection, are conducted through Python scripting.
Data, encompassing examination details, outcomes, and evaluation results, are
stored in .csv format for simplicity. While adequate for this study’s scale,
more extensive future studies may necessitate more advanced data storage
solutions. Our GitHub repository [19] consolidates all scripts and data,
ensuring effective version control.

####  II-B2 Decompiling Tools - Ghidra and RetDec

We utilized two distinct open-source decompiling methods: Ghidra [20] and
RetDec [21]. Ghidra, an NSA-developed tool, offers decompiling, disassembling,
and code language detection capabilities. RetDec, on the other hand, supports
various programming languages and processor architectures. Both tools were
crucial in our experiment for different scenarios.

###  II-C Data

We discuss our data collection strategies and methods here, though subsequent
sections will provide more detailed descriptions of the data specific to each
phase.

####  II-C1 Dataset 1 - Simple C Programming Problems

This dataset comprises 70 simple C programming problems sourced from Programiz
[22]. These examples range from 20 to 60 lines and are stored in a .csv file
with columns for problem name, explanatory notes, original code, and a
comment-free version of the code. This dataset primarily supports the first
phase of our experiment, with additional data points added for each scenario
as needed.

####  II-C2 Dataset 2 - Malware Source Codes in C

The second dataset features 15 malware source codes in C, all obtained from
open-source GitHub [23] repositories, complete with comments and
documentation. These codes underwent a manual compile-decompile process,
yielding two versions of decompiled code using Ghidra [20] and RetDec [21]
decompilers. Stored in a .csv file, the data includes columns for the source,
platform applicability, original code, and both versions of decompiled code.
This dataset was integral to the latter phases of our study.

##  III Experiment Design

In this section, we detail the various phases and scenarios that represent the
experimental journey. Each phase is carefully designed to explore distinct
aspects of Large Language Models (LLMs) in Binary Reverse Engineering. We will
explain the objectives of each phase, the methodologies employed, and the
intended outcomes of these investigative exercises.

This experiment is structured into two distinct phases, each containing
various scenarios tailored to diverse evaluation perspectives. This diversity
ensures a comprehensive exploration of the subject matter, focusing on
specific objectives and methodologies in each phase.

More details like scripts, data, prompts, and rubrics can be found on the
project GitHub repository [19] and the documentation there.

###  III-A Phase 1: Basic Code Interpretation

Figure 1: Workflow Diagram of Phase 1 - Basic Code Interpretation and Analysis
in Large Language Model Experimentation

Phase 1 lays the foundational groundwork for our experiment, focusing on
establishing the infrastructure, refining data-gathering methodologies, and
assessing the basic capabilities of GPT-4 in explaining code. This phase is
divided into three distinct scenarios designed to evaluate the model’s
performance. Please see Figure 1 for a visualized workflow in this phase.

Data Source: We selected 70 examples of C programming exercises from Programiz
[22], ranging from essential ”Hello World” programs to intermediate-level
geomatics calculations. These examples, typically used in C programming
tutorials, perfectly align with our objective of simplicity for this initial
phase.

Objective: The primary goal in this phase is to observe and analyze how LLMs,
GPT-4 [1], respond to straightforward requests for code explanations. We aim
to gain insights into the model’s interpretative skills and foundational
understanding of programming concepts when it faces human written code and
something more ambiguous like uncommented and decompiled code. This phase
serves as a stepping stone for more complex analyses in the next phase.

####  III-A1 Scenario 1: Original Code Explanation

In Scenario 1, our approach involved feeding the Large Language Model (LLM)
the original code from tutorials, complete with comments and in-code
documentation. The prompt was carefully tailored to include the application
name and programming language, guiding the model to provide concise, precise,
and descriptive explanations. The objective of this scenario was to observe
the LLM’s performance in explaining simple code that includes all the typical
aspects of human-written code.

####  III-A2 Scenario 2: Stripped Code Explanation

Moving to Scenario 2, the approach differed slightly. All comments and
documentation were removed from the code while retaining the original function
and variable names and their structural integrity. The prompt was, on purpose,
made more ambiguous by omitting the function application name and the
reference to ”C programming.” The aim was to assess the LLM’s capability to
interpret code with reduced contextual hints, thereby evaluating its
adaptability and understanding in somewhat ambiguous settings.

####  III-A3 Scenario 3: Decompiled Code Explanation

In Scenario 3, we took a different route. All code samples were compiled and
then decompiled using Python automation along with Ghidra [20] as the
decompilation tool. The decompiled functions then served as the basis for
generating prompts for each data entry, maintaining a descriptiveness similar
to that of Scenario 2. The primary focus here was to test the LLM’s ability to
explain decompiled code, which lacks meaningful names, comments,
documentation, or structure, posing a more significant challenge in code
explanation.

####  III-A4 Evaluation: Unified Approach Across Scenarios

Finally, a consistent evaluation methodology was applied across all scenarios.
Utilizing Python scripting for automation, we compared the explanations
generated by the LLM against the original explanations sourced from the
tutorials, employing the BLEU score [24] for this purpose. The BLEU score,
commonly used in translation quality assessment, enabled us to quantify the
similarity between the LLM’s output and the reference explanations. This
unified approach not only streamlined the comparison process but also
critically evaluated the effectiveness of BLEU scoring in the context of this
experimental study.

###  III-B Phase 2: Malware Analysis

Figure 2:  Workflow Diagram of Phase 2 - Advanced Analysis of Malware-Reversed
Engineered Applications Using Large Language Models

In Phase 2, our investigation explores the LLM’s proficiency in explicating
real-world, malware-reversed engineered applications, mainly written in C.
This phase was carefully structured with specific scenarios to comprehensively
evaluate the LLM’s capabilities in code explanation under varying conditions.

Dataset Composition: We collected a set of 15 malware source codes from
publicly accessible GitHub [23] repositories. The selection criteria were
centered around the extent of in-code comments and documentation, ensuring a
rich dataset for analysis. This selection aims to mirror real-world
complexities experienced in reverse engineering scenarios. Please see Figure 2
for a visualized workflow in this phase

Decompilation Process: All sourced codes were compiled in a secure, isolated
Linux environment and subsequently decompiled using RetDec [21]. The choice of
RetDec over Ghidra [20] for this phase was driven by RetDec’s user-
friendliness and ability to generate cleaner decompiled code, making future
manual assessments easier.

Scenario Design: To thoroughly assess the LLM’s performance, we developed
various scenarios differing in dataset size, quality, and complexity. These
scenarios were strategically designed to understand the different faces of the
LLM’s code explanation abilities. The following sections provide more
information about the objectives and methodologies for each scenario.

####  III-B1 Scenarios 1&2: Variable and Function Name Detection

In these initial scenarios, we evaluated the LLM’s proficiency in renaming
decompiled functions and variables to more human-readable, meaningful, and
contextually relevant names. For each decompiled malware application, we
crafted a prompt that included the code and requested the LLM to suggest new
names for functions and variables, ensuring they are meaningful and align with
the overall application context. The results were compiled in JSON format,
documenting original and suggested names.

The responses were assessed using a manual scoring method based on a rubric
that considered the meaningfulness and effectiveness of the newly suggested
names.

####  III-B2 Scenario 3: Enhanced Clarity and Structure

In Scenario 3, our approach involved presenting the LLM with eight binary
(Yes/No) questions applied to each decompiled code sample. This experiment has
replicated the original code, which has a more precise and descriptive code
structure, including in-code comments and original function/variable names.
The purpose was to assess the LLM’s ability to provide binary responses that
accurately reflect the code’s attributes and functionalities.

The questions were carefully chosen to cover a range of analyses, including
the program’s primary functions, network interactions, scalability, and
potential security vulnerabilities. Specific areas of focus were the detection
of suspicious API usage, unauthorized network activities, and unnecessary
encryption practices. Furthermore, the questions delved into whether the
program showed any signs of being stealthy or having persistence mechanisms,
such as running in the background or auto-starting.

The selection of these particular questions was grounded in the fundamental
inquiries a reverse engineer would challenge during the initial stages of code
analysis. This choice aims to test the LLM’s proficiency in basic reverse
engineering tasks.

To evaluate the LLM’s performance, we compared its responses from the
decompiled code against those obtained from the original code. This comparison
operated assuming that the LLM could accurately interpret and respond to the
questions when passing the original code. This assumption establishes a
benchmark for assessing the LLM’s effectiveness in understanding and analyzing
the more challenging decompiled code form.

####  III-B3 Scenario 4: Enhanced Analysis Approach

Scenario 4 focuses on the LLM’s ability to analyze decompiled code, requiring
it to generate a short and general analysis from three angles: Functionality
Overview, Key Observations, and Security Analysis. The LLM’s task was to
produce an analysis within a 200-word limit, mirroring the approach a reverse
engineer might take in quickly assessing a decompiled code’s features.

The selection of these three perspectives was intentional, aimed at examining
the depth and accuracy of the LLM’s analysis in critical areas of reverse
engineering. To evaluate the LLM’s performance, we compared its analyses of
the original and decompiled codes, using the original code as a benchmark for
accuracy.

####  III-B4 Scenario 5: Comprehensive Code Analysis via Questionnaire

Scenario 5 builds on the previous scenarios to delve deeper into code analysis
and explanation tasks. This scenario introduces a seven-question survey
designed to examine the structure and functionality of code. The questions aim
to uncover the primary function of the code, the roles of key functions and
variables, error handling mechanisms, execution flow, dependencies on external
libraries, and potential security issues. This comprehensive set of questions
mirrors the analytical process a reverse engineer undertakes when dissecting a
decompiled code.

The rationale behind these questions is to capture the core of reverse
engineering analysis, focusing on crucial aspects that determine the code’s
functionality and potential security vulnerabilities. The answers to these
questions, obtained from the original and decompiled code, provide a basis for
comparison. By assuming the original code responses are accurate, we can
effectively measure the LLM’s proficiency in understanding and explaining
complex code structures and behaviors, an essential skill in reverse
engineering.

####  III-B5 Evaluation

In the evaluation phase of these experiments, we employed a careful manual
process with a set of detailed rubrics tailored to each scenario. These
rubrics serve as a scoring guide, allowing us to objectively assess the LLM’s
responses against a set standard – the original code’s explanations. Each
scenario’s unique nature involves distinct scoring methodologies and ranges,
ensuring an appropriate and fair evaluation of the LLM’s performance in
diverse contexts. The results of this evaluation, along with the rubrics and
prompts used, will be elaborated in the Results section of the report. For a
deeper dive into our methods, readers can refer to the project documentation
available on our GitHub repository [19].

##  IV Results and Analysis

In this section, we delve into the findings from our evaluations across
different scenarios, dissecting the outcomes based on the respective
methodologies and rubrics employed. Each subsection will interpret the
numerical results and explore their implications in the context of our
experimental framework.

###  IV-A Phase 1

The analysis of BLEU score results (Figures 3-5), computed for each of the 70
code samples across three distinct scenarios, the scores that were revealed
were significantly lower than the threshold considered adequate for making
meaningful comparisons. According to BLEU score guidelines [24], a score more
excellent 0.2 is indicative of a reliable comparison, yet our findings fall
short of this benchmark. The accompanying scatter diagrams (Figures 3-5)
demonstrate a majority of scores below 0.2, with only a handful approaching
this value. This pattern underscores the limitations of relying on automated
evaluation methods in their current state for assessing code explanations.

Figure 3: BLEU score results for Scenario 1. Figure 4: BLEU score results for
Scenario 2. Figure 5: BLEU score results for Scenario 3.

Consequently, we recognize the necessity of abstaining from automated
evaluation methods for the time being. While helpful in saving time, these
methods require extensive exploration to be effective in the context of code
explanation evaluation. Hence, our research path shifts towards Phase 2,
prioritizing manual evaluation techniques for a more meaningful and in-depth
analysis. This change in strategy indicates a temporary break in analyzing the
results of Phase 1, redirecting our focus towards more hands-on evaluation
methods in the following phases.

###  IV-B Phase 2

This section will discuss our findings from the manual evaluation of Phase 2,
which covered all five scenarios. Each scenario had its unique rubric and
scoring system tailored to its complexity and specific aspects.

####  IV-B1 Scenario 1&2

In these scenarios, we focused on how well GPT-4 [1] could rename functions
and variables in 15 decompiled malware codes. We evaluated the effectiveness
and meaningfulness of each renamed entity. The meaningfulness score reflected
how understandable the new names were for users, while the effectiveness score
measured their relevance to other parts of the code.

TABLE I: 42 Functions Renaming Task Average Evaluated Scores Aspect | Average Score (out of 3)
---|---
Meaningfulness | 2.83
Effectiveness | 2.29
TABLE II: 69 Variables Renaming Task Average Evaluated Scores Aspect | Average Score (out of 3)
---|---
Meaningfulness | 2.49
Effectiveness | 2.09

In these scenarios, we focused on how well GPT-4 could rename functions (Table
I) and variables (Table II) in 15 decompiled malware codes. We evaluated the
effectiveness and meaningfulness of each renamed entity. The meaningfulness
score reflected how understandable the new names were for users, while the
effectiveness score measured their relevance to other parts of the code.

The higher scores for function names compared to variable names suggest that
GPT-4 might be more aware of the code’s general context, aiding it in choosing
more appropriate function names. In contrast, variable names seem to require a
deeper analysis of the code’s complexities, which might be a current
limitation of the model.

Furthermore, the lower effectiveness scores, compared to the meaningfulness
scores for both functions and variables, indicate a challenge for the model in
establishing logical connections between different parts of the code. This
highlights a potential area for improvement in the model’s ability to
understand and reflect code elements’ specific roles and interactions.

For a detailed breakdown of the evaluation rubrics, please refer to the
project documentation [19].

####  IV-B2 Scenario 3

In this experiment phase, as previously outlined in our experimental design,
we subjected both the original and decompiled code to a series of eight Yes/No
questions. These questions were sent to the LLM on each code sample, and the
responses derived from the original code were used as benchmarks for
comparison against the responses obtained from the decompiled codes.

TABLE III: Analysis of Question-Based Responses Question number | Number of wrong answer | % wrong answer
---|---|---
1 | 0/15 | 0.0%
2 | 4/15 | 26.7%
3 | 3/15 | 20.0%
4 | 5/15 | 33.3%
5 | 5/15 | 33.3%
6 | 3/15 | 20.0%
7 | 7/15 | 46.7%
8 | 4/15 | 26.7%
TOTAL | 31/120 | 25.62%
TABLE IV: Analysis of Application-Based Responses Application id | # Wrong Answers | % Wrong Answers
---|---|---
1 | 2 | 25.0%
2 | 2 | 25.0%
3 | 2 | 25.0%
4 | 0 | 0.0%
5 | 2 | 25.0%
6 | 5 | 62.5%
7 | 2 | 25.0%
8 | 4 | 50.0%
9 | 0 | 0.0%
10 | 3 | 37.5%
11 | 0 | 0.0%
12 | 0 | 0.0%
13 | 4 | 50.0%
14 | 5 | 62.5%
15 | 1 | 12.5%

Two key tables of results support our analysis. Table III details the
frequency of incorrect responses for each question across the 15 code samples.
This provides insights into the consistency and reliability of the LLM’s
analysis capabilities. Table IV focuses on the number of incorrect responses
per individual code sample. This aspect of the analysis potentially reflects
each code sample’s varying complexity and clarity.

Our primary focus will be on the first table, as it offers a more controlled
view of the LLM’s response patterns to our specific questions. The second
table’s insights show how code complexity impacts the LLM’s performance, which
gives us less flexibility for modification, as any code can be passed to the
model by the user.

Question-Based Analysis

  * •

Function Identification (Q1): GPT-4 performed flawlessly in identifying the
main functions in the code across all malware samples. This indicates a strong
capability to recognize structural elements of code.

  * •

Network-Related Program (Q2) & Scalability (Q3): Errors in these areas were
relatively low (26.7% and 20.0%, respectively), suggesting some capability in
understanding network-related aspects and scalability but with room for
improvement.

  * •

Suspicious API Calls (Q4) & Network Activity (Q5): Higher error rates of 33.3%
in both questions highlight a challenge in accurately identifying potentially
malicious API usage and unauthorized network activities.

  * •

Encryption Routines (Q6): With a 20.0% error rate, GPT-4 can detect unrelated
encryption routines, although it is not entirely foolproof.

  * •

Stealth Techniques (Q7): The highest error rate (46.7%) was observed here.
This suggests that GPT-4 struggles more significantly recognizing
cybersecurity manipulation code implementation.

  * •

Persistence Mechanisms (Q8): An error rate of 26.7% indicates a moderate
challenge in identifying code ensuring auto-start or background activity.

Application-Based Analysis

  * •

Consistent Performance Across Some Applications: Some applications (4, 9, 11,
12) showed no errors, highlighting instances where GPT-4 could accurately
respond to all questions.

  * •

Variable Performance: Other applications (6, 13, 14) had high error
percentages (above 50%), showing significant challenges in those specific
cases. This variance suggests that GPT -4’s performance heavily depends on the
specific nature of the code it analyzes.

  * •

Overall Error Rate: An overall error rate of 25.62% across all questions and
applications suggests that while GPT-4 is generally reliable, there are
significant areas where its analysis does not align with the benchmarked
correct responses.

GPT-4 shows strong capabilities in understanding essential structural elements
of code, such as function identification. The model struggles with some
delicate aspects of code analysis, particularly in detecting stealth
techniques and certain malicious activities. Also, the second table of results
shows variable performance across different applications, suggesting that the
complexity of the code may influence GPT -4’s effectiveness.

####  IV-B3 Scenario 4

In this section, we delve into the model’s capability to comprehensively
explain decompiled code, which is crucial for initial reverse engineering
analysis. We assessed the model’s performance across four essential
perspectives (Table V), using the explanations from the original code as
benchmarks. Each aspect was scored on a scale of 0 to 5 to quantify the
model’s performance in presenting primary information related to reverse
engineering tasks.

TABLE V: Performance of GPT-4 in code explanation including 4 different perspectives Aspect | Average Score (out of 5)
---|---
1\. Functionality Overview | 2.4
2\. Key Observations | 1.73
3\. Security Analysis | 2.13
4\. Clarity and Structure | 2.33

In Functionality Overview, the GPT-4 displayed a moderate understanding of the
code’s primary functions, with an average score of 2.4. This indicates a
reasonable capability to grasp basic functionalities but reveals
inconsistencies, especially in complex codes.

Moving forward on Key Observations, scoring 1.73 on average, the model
struggled in the key observations category. This reflects challenges in
providing in-depth technical insights, which are crucial in reverse
engineering.

With an average score of 2.13 in security analysis, the LLM showed a mixed
ability to identify security risks. It could detect obvious vulnerabilities in
some instances but lacked consistency across various code samples.

Also, in Clarity and Structure, the model’s average score of 2.33 in clarity
and structure suggests that while explanations were generally logical, they
sometimes lacked organization, especially in complex scenarios.

Overall, GPT -4’s performance varied across different aspects of code
explanation. It was relatively better at explaining basic functionalities but
less effective at detailed technical and security analyses. This suggests that
while GPT-4 can assist in code explanation, it currently requires human
expertise for more complex, nuanced tasks, particularly in security-critical
scenarios.

####  IV-B4 Scenario 5

In this scenario, we presented the GPT-4 with original and decompiled code,
asking it to answer seven targeted short-answer questions (Table VI). These
questions were designed to probe the model’s depth of understanding and
ability to extract and articulate key aspects of the code. We used the answers
derived from the original code as a benchmark to evaluate the GPT -4’s
responses to the decompiled code.

TABLE VI: Performance on answering 7 reverse engineering-related short answer questions Aspect | Average Score (out of 5)
---|---
1\. Primary Functionality | 3.04
2\. Key Functions Description | 3.43
3\. Role of Selected Variable | 2.21
4\. Error Handling Mechanism | 2.93
5\. Flow of Execution | 2.71
6\. External Libraries/Dependencies | 2.68
7\. Evident Security Concerns | 2.64

Here is a breakdown of the results analysis of answering 7 short answer
questions, which were scored manually based on a rubric:

  1. 1.

Primary Functionality (Avg. Score: 3.04): We are supposed to have a score
higher than other sections here, as in previous scenarios, we witnessed that
the model’s ability to provide a general overview of the code application is
better than analyzing details.

  2. 2.

Key Functions Description (Avg. Score: 3.43): In this section, we have got the
best score, which is mainly about how the model can identify the main
functions of the code. This is another task that focuses on a more
comprehensive view of the application and fewer details.

  3. 3.

Role of Selected Variable (Avg. Score: 2.21): The lower scores in this
category indicate challenges in understanding the nuanced roles of individual
variables within the code. However, variables in the original and decompiled
codes vary, and we will talk more about it in the limitation and discussion
sections.

  4. 4.

Error Handling Mechanism (Avg. Score: 2.93): The LLM demonstrated an average
understanding of error handling mechanisms in original and decompiled codes.

  5. 5.

Flow of Execution (Avg. Score: 2.71): The model’s ability to trace and explain
the code’s execution flow was average, highlighting potential difficulties in
understanding complex control structures. The flow of execution can be a good
parameter for comparison, as the final purpose of the application in both the
original and decompiled codes is pretty similar.

  6. 6.

External Libraries/Dependencies (Avg. Score: 2.68): The scores here reflect a
fair ability to identify external dependencies, suggesting an understanding of
the code’s broader context. We are supposed to get a better score here as the
main libraries for both code versions need to see some similarities, so we
identified this task more accessible than others.

  7. 7.

Evident Security Concerns (Avg. Score: 2.64): The model was moderately
effective in identifying security issues, indicating a need for enhanced
capabilities in security analysis. However, the question is ambiguous by
itself, which makes answering this question hard for LLM. It can be
interpreted as overall security issues, which the code may conduct as an
application after compilation, or just security flaws on the code itself,
making it vulnerable to manipulations.

The results from this scenario suggest that while the LLM is average at
extracting and explaining high-level functionalities and key functions, it
struggles with more nuanced aspects, such as variable roles and detailed
execution flows. Its performance in recognizing external libraries and
pinpointing security concerns was adequate but indicated potential areas for
further development.

We can discuss the comparison and questionnaire themselves, which may have
caused this scenario to be a bit nonstandard for giving us a sound vision of
GPT -4’s performance on such a detailed analysis, which will be discussed
later.

##  V Discussion and Limitations

###  V-A Experiment Results

In this discussion, we reflect on the experimental methods, evaluation
techniques, and the implications of the results for understanding the
performance of large language models (LLMs), in this case GPT-4, in reverse
engineering tasks. However, the following sections will discuss doubts about
our assumptions about experimental methods and evaluation techniques.

The first phase’s results were not within a reasonable range for meaningful
discussion, highlighting evaluation limitations we will explore further.
However, this phase was instrumental in shaping our approach to subsequent
phases, particularly in data collection and experiment implementation. We
discovered the complexity of data preparation, focusing on easily
understandable samples with in-code comments and pre-written explanations to
manage complexity effectively.

In the second phase, our findings offer more substantial insights. The first
scenario demonstrated the GPT -4’s proficiency in renaming variables and
functions in a contextually relevant manner. While the scores for
meaningfulness were higher, the effectiveness ratings also indicated a degree
of reliability. This suggests potential areas for improvement in LLMs,
particularly in establishing a reasonable channel of relations throughout the
code components before choosing a new variable or function’s name.

Scenario 3 revealed limitations in the GPT -4’s accuracy, especially in
addressing network-related and security-specific queries. This reflects the
GPT -4’s training on general code datasets and underscores the need for
specialized, reverse engineering-focused datasets to enhance understanding in
these areas.

The analysis of the fourth scenario indicated a shortfall in providing general
explanations from reverse engineering viewpoints. Despite achieving less than
50% accuracy across all subsections, there were instances of better
performance in sections requiring a broad overview of the code. This suggests
a potential reliance on the LLMs for identifying general application purposes,
though it struggled with a more detailed network (cybersecurity perspectives)
and structural questions.

Finally, scenario 5, which delved into detailed explanations, mirrored the
earlier patterns: stronger performance in broader functionality queries and a
weaker grasp of intricate code relationships. Interestingly, responses
involving ”functions” proceed better than those about ”variables,” and the
model was more adept in identifying error handling mechanisms, a common
feature in network applications, rather than manipulation mechanism detection
tasks, which are more related to reverse engineering. This again points to the
need for more reverse engineering-specific training to enhance the LLM’s
ability to distinguish and interpret complex coding patterns.

###  V-B Methods of Experiment

In analyzing our research methods and their applicability to evaluating LLM
performance in reverse engineering tasks, we confront various challenges and
insights that will shape future research directions. Any discussion here will
help us to find a better systematic way of experimenting in the next stage of
our research journey.

During the first phase, the limitations of the BLEU score [24] as an
evaluation tool became evident. Originally designed for translation tasks, the
BLEU score’s effectiveness diminishes in assessing technical, computer
science-related content. Its algorithmic focus on word-for-word comparisons
fails to capture the nuanced understanding necessary for interpreting complex
technical explanations. Thus, while BLEU scores may have a place in this
research domain, their utility is restricted to scenarios with highly specific
and limited explanatory expectations, which was not the case in our study.

The second phase’s initial scenarios (1 and 2) showcased a more reliable
evaluation method. These scenarios benefitted from direct human judgment with
scoring based on a well-defined rubric. This approach underscored the value of
human evaluators in contexts where nuanced understanding and contextual
interpretation are crucial.

However, scenarios 3, 4, and 5 faced challenges due to their foundational
assumption of the LLM’s accuracy in interpreting original code. This
assumption overlooks the inherent differences between human-written code and
its decompiled equivalent. The distinct structural and methodological
variations between these code forms, although having the same result in
execution after compilation, mean that equivalent questions can yield
differing yet valid responses for both codes, which makes these responses less
comparable. Furthermore, our question selection could have been more
effective. A focus on broader, general code aspects would likely yield more
meaningful comparisons between original and decompiled code. This is
especially relevant when our questions do not fully align with this standard.

Data collection presented another significant challenge, particularly in the
second phase, where our focus on C code malware with existing comments and
documentation restricted our data sources. Relying primarily on GitHub [23]
repositories limited the diversity and reliability of our dataset, thus
impacting the generalizability of our findings. Additionally, the inconsistent
nature of the dataset contributed to an uneven experimental landscape.

In addition, budget constraints related to the use of the GPT-4 API further
limited our research scope, restricting the number of iterations we could
afford.

All in all, our experience highlights the superiority of manual assessments in
this context. The absence of specialized, automated evaluation tools for such
complex tasks requires a more indicative approach. Ideally, a comparative
analysis should involve two versions of decompiled code: one raw and one
reviewed by expert reverse engineers. This benchmarked code, complete with
expert annotations, could serve as a reference point for evaluating LLM
responses. Developing a tailored questionnaire derived from the annotations on
the benchmarked decompiled code would enable a more precise assessment of the
LLM’s capabilities in this delicate field. This data set would solve both
valid benchmark and dataset constraints.

##  VI Conclusion and Future Works

The findings from our comprehensive investigation into the use of LLMs,
particularly GPT-4 [1], in Binary Reverse Engineering tasks reveal both the
potential and the limitations of these advanced models. The initial phase of
the study highlighted the challenges in automated evaluation methods,
prompting a shift towards manual evaluation in subsequent phases. This change
in approach facilitated a more in-depth understanding of LLMs’ capabilities in
code explanation and interpretation.

In the second phase, GPT-4 demonstrated a noteworthy ability to provide
contextually relevant names for functions and variables and a general
understanding of code functionalities. However, the model’s performance varied
significantly when delving into more complex aspects such as detailed
technical insights, security analysis, and understanding intricate code
relationships. Notably, the model showed stronger capabilities in explaining
broader functionalities and identifying key functions compared to its
performance in nuanced aspects like variable roles and specific execution
flows.

The research also brought to light the limitations of existing evaluation
methodologies in this domain. The BLEU score [24], while useful in certain
contexts, proved inadequate for assessing complex technical explanations.
Furthermore, the reliance on human-written code as a benchmark for comparison
with decompiled code introduced inherent challenges due to structural
differences between these code forms.

In future research, our focus will shift toward developing more sophisticated
evaluation tools specifically designed for technical language analysis,
particularly in the context of reverse engineering. Recognizing the need for
broader and more varied datasets, efforts will be directed toward expanding
the range of code types and sources. This will help create a comprehensive
platform to test and understand the full potential of large language models
(LLMs). Additionally, we plan to undertake comparative studies using expert-
reviewed decompiled code as benchmarks. Addressing the constraints of
resources, particularly in terms of budget and access to LLMs, will also
direct us to use other open-source models for our examination.

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