You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'In this article, we calculate the mass modifications of the vector and axialvector mesons $D^*$, $B^*$, $D_1$ and $B_1$ in the nuclear matter with the QCD sum rules, and obtain the mass-shifts $\delta M_{D^*}=-71\,\rm{MeV}$, $\delta M_{B^*}=-380\,\rm{MeV}$, $\delta M_{D_1}=72\,\rm{MeV}$, $\delta M_{B_1}=264\,\rm{MeV}$, and the scattering lengths $a_{D^*}=-1.07\,\rm{fm}$, $a_{B^*}=-7.17\,\rm{fm}$, $a_{D_1}=1.15\,\rm{fm}$ and $a_{B_1}=5.03\,\rm{fm}$ for the $D^*N$, $B^*N$, $D_1N$ and $B_1N$ interactions, respectively.'
---

\
Zhi-Gang Wang [^1]\
Department of Physics, North China Electric Power University, Baoding 071003, P. R. China

PACS numbers: 12.38.Lg; 14.40.Lb; 14.40.Nd

[**[Key Words:]{}**]{} Nuclear matter, QCD sum rules

Introduction
============

The modifications of the hadron properties in the nuclear matter can affect the productions of the open-charmed mesons and the $J/\psi$ in the relativistic heavy ion collisions, the higher charmonium states, such as the $\psi'$, $\chi_{c1}$, $\chi_{c2}$, etc, are considered as the major source of the $J/\psi$ [@Jpsi-Source]. The charmed mesons can obtain mass augments or reductions in the nuclear matter, if the mass modifications are large enough, the decays of the higher charmonium states to the charmed meson pairs can be facilitated or suppressed remarkably due to the available phase-space, thus the decays to the lowest state $J/\psi$ are greatly modified [@H-to-Jspi]. For example, the higher charmonium states can decay to the $D\bar{D}$ pairs instead of decaying to the lowest state $J/\psi$, if the mass reductions of the $D$ and $\bar{D}$ mesons are large enough. On the other hand, the suppression of the $J/\psi$ production in the relativistic heavy ion collisions is considered as an important signature to identify the possible phase transition to the quark-gluon plasma [@Matsui86]. We should be careful before making definite conclusions.

The QCD sum rules is a powerful theoretical tool in studying the in-medium hadronic properties [@SVZ79], and has been applied extensively to study the light-flavor hadrons and charmonium states in the nuclear matter [@C-parameter; @Drukarev1991; @Jpsi-etac]. The works on the heavy mesons and heavy baryons are few, only the $D$, $B$, $D_0$, $B_0$, $\Lambda_c$, $\Lambda_b$, $\Sigma_c$ and $\Sigma_b$ are studied with the QCD sum rules [@Hayashigaki; @WangHuang; @DB-SR; @SumRule-D0; @SumRule-VA; @WangH]. The heavy mesons contain a heavy quark and a light quark, the existence of a light quark in the heavy mesons leads to large difference between the mass-shifts of the heavy mesons and heavy quarkonia in the nuclear matter. The former have large contributions from the light-quark condensates, while the latter are dominated by the gluon condensates [@Jpsi-etac; @Hayashigaki; @WangHuang; @DB-SR; @SumRule-D0]. In this article, we study the mass modifications of the vector mesons $D^*$, $B^*$ and axialvector mesons $D_1$, $B_1$ in the nuclear matter using the QCD sum rules. The present predictions can be confronted with the experimental data from the CBM and PANDA collaborations in the future [@CBM; @PANDA].

The article is arranged as follows: we study the mass modifications of the vector and axialvector mesons $D^*$, $B^*$, $D_1$ and $B_1$ in the nuclear matter with the QCD sum rules in Sec.2; in Sec.3, we present the numerical results and discussions; and Sec.4 is reserved for our conclusions.

Mass modifications of the $D^*$, $B^*$, $D_1$ and $B_1$ in the nuclear matter with QCD sum rules
================================================================================================

We study the mass modifications of the $D^*$ and $D_1$ mesons in nuclear matter with the two-point correlation functions $\Pi_{\mu\nu}(q)$, $$\begin{aligned}
\Pi_{\mu\nu}(q) &=& i\int d^{4}x\ e^{iq \cdot x} \langle T\left\{J_\mu(x)J_\nu^{\dag}(0)\right\} \rangle_{\rho_N} \, ,
 \end{aligned}$$ where the $J_\mu(x)$ denotes the isospin averaged currents $\eta_\mu(x)$ and $\eta_{5\mu}(x)$, $$\begin{aligned}
 \eta_\mu(x) &=&\eta_\mu^\dag(x) =\frac{\bar{c}(x)\gamma_\mu q(x)+\bar{q}(x)\gamma_\mu c(x)}{2}\, , \nonumber\\
  \eta_{5\mu}(x) &=&\eta_{5\mu}^\dag(x) =\frac{\bar{c}(x)\gamma_\mu \gamma_5q(x)+\bar{q}(x)\gamma_\mu\gamma_5 c(x)}{2}\,,\end{aligned}$$ which interpolate the vector and axialvector mesons $D^*$ and $D_1$, respectively, the $q$ denotes the $u$ or $d$ quark. The $\bar{c}q$ and $\bar{q}c$ mesons maybe obtain different mass modifications in the nuclear matter, just like the $K^+$ and $K^-$ mesons [@KaonK], for example, Hilger et al observe that there exist particle-antiparticle mass splittings for the scalar and pseudoscalar mesons [@DB-SR; @SumRule-D0]. In this article, we intend to study whether or not the decays of the higher charmonium states to the $D^*\bar{D}^*$ and $D_1\bar{D}_1$ states are facilitated in the phase-space, and prefer the average values as the particle-antiparticle mass splittings cannot modify the total mass of the particle-antiparticle pair.

At the low nuclear density, the in-medium condensates $\langle{\cal {O}}\rangle_{\rho_N}$, $$\begin{aligned}
\langle{\cal{O}}\rangle_{\rho_N} &=&\langle{\cal{O}}\rangle+\frac{\rho_N}{2M_N}\langle {\cal{O}}\rangle_N \, ,\end{aligned}$$ based on the Fermi gas model, where the $\langle{\cal{O}}\rangle$ and $\langle
{\cal{O}}\rangle_N$ denote the vacuum condensates and nuclear matter induced condensates, respectively, the $\rho_N$ is the density of the nuclear matter [@Drukarev1991]. Accordingly, the correlation functions $\Pi_{\mu\nu}(q)$ can be divided into a vacuum part $\Pi^0_{\mu\nu}(q)$ and a static one-nucleon part $T^N_{\mu\nu}(q)$ in the Fermi gas approximation for the nuclear matter [@Drukarev1991; @Hayashigaki], $$\begin{aligned}
\Pi_{\mu\nu}(q) &=&\Pi^{0}_{\mu\nu}(q)+ \frac{\rho_N}{2M_N}T^{N}_{\mu\nu}(q)\, ,
 \end{aligned}$$ where $$\begin{aligned}
T^{N}_{\mu\nu}(\omega,\mbox{\boldmath $q$}\,) &=&i\int d^{4}x e^{iq\cdot x}\langle N(p)|
T\left\{J_\mu(x)J_\mu^{\dag}(0)\right\} |N(p) \rangle\, ,\end{aligned}$$ the $|N(p)\rangle$ denotes the isospin and spin averaged static nucleon state with the four-momentum $p =
(M_N,0)$, and normalized as $\langle N(\mbox{\boldmath $p$})|N(\mbox{\boldmath
$p$}')\rangle = (2\pi)^{3} 2p_{0}\delta^{3}(\mbox{\boldmath
$p$}-\mbox{\boldmath $p$}')$ [@Hayashigaki]. The $T^{N}_{\mu\nu}(q)$ happen to be the current-nucleon forward scattering amplitudes. We can decompose the correlation functions $T^{N}_{\mu\nu}(\omega,\mbox{\boldmath $q$}\,)$ as $$\begin{aligned}
T^{N}_{\mu\nu}(\omega,\mbox{\boldmath $q$}\,) &=&T_{N}(\omega,\mbox{\boldmath $q$}\,)\left(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}\right)+\Pi^0_N(\omega,\mbox{\boldmath $q$}\,) \frac{q_\mu q_\nu}{q^2} \, ,\end{aligned}$$ according to Lorentz covariance, where the $T_{N}(\omega,\mbox{\boldmath $q$}\,)$ denotes the contributions from the vector and axialvector mesons, and the $T^0_{N}(\omega,\mbox{\boldmath $q$}\,)$ denotes the contributions from the scalar and pseudoscalar mesons. The interpolating currents $\eta_\mu(x)$ and $\eta_{5\mu}$ have non-vanishing couplings with the scalar and pseudoscalar mesons $D_0$ and $D$, respectively, i.e. $\langle 0|\eta_\mu(0)|D_0+\bar{D}_0\rangle =f_{D_0}q_\mu$ and $\langle 0|\eta_{5\mu}(0)|D+\bar{D}\rangle =if_{D}q_{\mu}$. We can exclude the contaminations by choosing the tensor structure $-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}$, and only take into account the vector and axialvector mesons through the definitions, $$\begin{aligned}
\langle 0|\eta_\mu(0)|D^*+\bar{D}^*\rangle &=&f_{D^*}M_{D^*}\epsilon_\mu\,,\nonumber\\
\langle 0|\eta_{5\mu}(0)|D_1+\bar{D}_1\rangle &=&f_{D_1}M_{D_1}\epsilon_{\mu}\,,\end{aligned}$$ with summations of the polarization vectors $\sum_\lambda \epsilon_\mu(\lambda,q)\epsilon^*_\nu(\lambda,q)=-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2}$.

In the limit of the $3$-vector $\mbox{\boldmath $q$}\rightarrow {\bf 0}$, the correlation functions $T_{N}(\omega,\mbox{\boldmath $q$}\,)$ can be related to the $D^*N$ and $D_1N$ scattering $T$-matrixes, i.e. ${{\cal T}_{D^*N}}(M_{D^*},0) =
8\pi(M_N+M_{D^*})a_{D^*}$ and ${{\cal T}_{D_1N}}(M_{D_1},0) =
8\pi(M_N+M_{D_1})a_{D_1}$, where the $a_{D^*}$ and $a_{D_1}$ are the $D^*N$ and $D_1N$ scattering lengths, respectively. Near the pole positions of the $D^*$ and $D_1$ mesons, the phenomenological spectral densities $\rho(\omega,0)$ can be parameterized with three unknown parameters $a,b$ and $c$ [@Hayashigaki], $$\begin{aligned}
\rho(\omega,0) &=& -\frac{f_{D^*/D_1}^2M_{D^*/D_1}^2}{\pi} \mbox{Im} \left[\frac{{{\cal T}_{D^*/D_1N}}(\omega,{\bf 0})}{(\omega^{2}-
M_{D^*/D_1}^2+i\varepsilon)^{2}} \right]+ \cdots \,, \\
&=& a\,\frac{d}{d\omega^2}\delta(\omega^{2}-M_{D^*/D_1}^2) +
b\,\delta(\omega^{2}-M_{D^*/D_1}^2) + c\,\delta(\omega^{2}-s_{0})\, ,\end{aligned}$$ the terms denoted by $\cdots$ represent the continuum contributions. The first term denotes the double-pole term, and corresponds to the on-shell (i.e. $\omega^2=M_{D^*/D_1}^2$) effects of the $T$-matrixes, $$\begin{aligned}
a&=&-8\pi(M_N+M_{D^*/D_1})a_{D^*/D_1}f_{D^*/D_1}^2M_{D^*/D_1}^2\, ,\end{aligned}$$ and related with the mass-shifts of the $D^*$ and $D_1$ mesons through the relation $$\begin{aligned}
\delta M_{D^*/D_1} &=&-\frac{\rho_N }{4M_Nf_{D^*/D_1}^2M_{D^*/D_1}^3}a\, ;\end{aligned}$$ the second term denotes the single-pole term, and corresponds to the off-shell (i.e. $\omega^2\neq M_{D^*/D_1}^2$) effects of the $T$-matrixes; and the third term denotes the continuum term or the remaining effects, where the $s_{0}$ is the continuum threshold.

In the limit $\omega\rightarrow 0$, the $T_{N}(\omega,{\bf 0})$ is equivalent to the Born term $T_{D^*/D_1N}^{\rm
Born}(\omega,{\bf 0})$. We take into account the Born term at the phenomenological side, $$\begin{aligned}
T_{N}(\omega^2)&=&T_{D^*/D_1N}^{\rm
Born}(\omega^2)+\frac{a}{(M_{D^*/D_1}^2-\omega^2)^2}+\frac{b}{M_{D^*/D_1}^2-\omega^2}+\frac{c}{s_0-\omega^2}\, ,\end{aligned}$$ with the constraint $$\begin{aligned}
\frac{a}{M_{D^*/D_1}^4}+\frac{b}{M_{D^*/D_1}^2}+\frac{c}{s_0}&=&0 \, .\end{aligned}$$

The contributions from the intermediate spin-$\frac{3}{2}$ charmed baryons are zero in the soft-limit $q_\mu \to 0$ [@Wangzg], where the $q_\mu$ denotes the four-momentum of the pseudoscalar mesons $P$ and vector mesons $V$ in the vertexes $B_{\frac{3}{2}}B_{\frac{1}{2}}P$ and $B_{\frac{3}{2}}B_{\frac{1}{2}}V$ for the spin-$\frac{3}{2}$ baryons $B_{\frac{3}{2}}$, spin-$\frac{1}{2}$ baryons $B_{\frac{1}{2}}$ and mesons. The contributions from the spin-$\frac{3}{2}$ charmed baryons, which are taken as the higher resonances, are included in the full $D^*N\to D^*N$ scattering amplitude, see the paragraph after Eq.(15). We take into account the intermediate spin-$\frac{1}{2}$ charmed baryons in calculating the Born terms, and parameterize the hadronic matrix elements as $$\begin{aligned}
 \langle\Lambda_c/\Sigma_c(p-q)|D^*(-q)N(p)\rangle &=&\bar{U}_{\Lambda_c/\Sigma_c}(p-q)\left[ g_{\Lambda_c/\Sigma_cD^*N}\!\not\!{\epsilon}+i\frac{g^T_{\Lambda_c/\Sigma_cD^*N}}{M_N+M_{\Lambda_c/\Sigma_c}}\sigma^{\alpha\beta}\epsilon_\alpha q_\beta\right]U_N(p)\,,\nonumber\\
 \langle\Lambda_c/\Sigma_c(p-q)|D_1(-q)N(p)\rangle &=&\bar{U}_{\Lambda_c/\Sigma_c}(p-q)\left[ g_{\Lambda_c/\Sigma_cD_1N}\!\not\!{\epsilon}+i\frac{g^T_{\Lambda_c/\Sigma_cD_1N}}{M_N+M_{\Lambda_c/\Sigma_c}}\sigma^{\alpha\beta}\epsilon_\alpha q_\beta\right]\gamma_5U_N(p)\,,\nonumber\\
 \end{aligned}$$ where the $U_N$ and $\bar{U}_{\Lambda_c/\Sigma_c}$ are the Dirac spinors of the nucleon and the charmed baryons $\Lambda_c/\Sigma_c$, respectively; the $g_{\Lambda_c/\Sigma_cD^*N}$, $g_{\Lambda_c/\Sigma_cD_1N}$, $g_{\Lambda_c/\Sigma_cD^*N}^T$ and $g_{\Lambda_c/\Sigma_cD_1N}^T$ are the strong coupling constants in the vertexes. In the limit $q_\mu \to 0$, the strong coupling constants $g_{\Lambda_c/\Sigma_cD^*N}^T$ and $g_{\Lambda_c/\Sigma_cD_1N}^T$ have no contributions.

We draw the Feynman diagrams, calculate the Born terms and obtain the results $$\begin{aligned}
T_{D^*N}^{\rm Born}(\omega,{\bf0})&=&\frac{2f_{D^*}^2M_{D^*}^2M_N(M_H+M_N)g_{HD^*N}^2}
{\left[\omega^2-(M_H+M_N)^2\right]\left[\omega^2-M_{D^*}^2\right]^2}\,, \nonumber \\
T_{D_1N}^{\rm Born}(\omega,{\bf0})&=&\frac{2f_{D_1}^2M_{D_1}^2M_N(M_H-M_N)g_{HD_1N}^2}
{\left[\omega^2-(M_H-M_N)^2\right]\left[\omega^2-M_{D_1}^2\right]^2}\,,\end{aligned}$$ where the $H$ means either $\Lambda_c^+$, $\Sigma_c^+$, $\Sigma_c^{++}$ or $\Sigma_c^0$. The masses $M_{\Lambda_c}=2.286\,\rm{GeV}$ and $M_{\Sigma_c}=2.454\,\rm{GeV}$ from the Particle Data Group [@PDG], we can take $M_H\approx
2.4\,\rm{GeV}$ as the average value. On the other hand, there are no inelastic channels for the $\bar{D}^*  N$ and $\bar{D}_1 N$ interactions in the case of the charmed mesons $\bar{c}q$.

The scattering state $D^*N$ can translate to the scattering states $D^*N$, $\pi\Sigma_c$, $\eta\Lambda_c$, $DN$, $\pi\Lambda_c$, $\rho\Sigma_c$, $\rho\Lambda_c$, etc, we can take into account the infinite series of the intermediate baryon-meson loops with the Bethe-Salpeter equation to obtain the full $D^*N \to D^*N$ scattering amplitude, and the higher resonances, such as the $\Lambda_c(2595)$, $\Sigma_c(2800)$, etc, appear as dynamically generated baryon states [@DN-BSE]. We can saturate the full $D^*N\to D^*N$ scattering amplitude with the tree-level Feynman diagrams of the exchanges of the higher resonances, which have energy dependent widths in the Breit-Wigner formulae. There are both spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ higher resonances, neglecting the spin-$\frac{3}{2}$ contributions can lead to unknown uncertainties as the spin-$\frac{3}{2}$ contributions exist for $q_\mu\neq0$. The spin-$\frac{1}{2}$ higher resonances consist of the negative-parity charmed baryons $\Lambda_c(2595)$ and $\Sigma_c({\frac{1}{2}}^-)$ have the average mass $M_{H'}\approx 2.7\,\rm{GeV}$ [@PDG; @Wang-Negative] and other excited charmed baryons have even larger masses. The translations of the scattering state $D^*N $ to the ground states $\Lambda_c $ and $\Sigma_c$ are greatly facilitated in the phase-space. If the couplings of the $D^*N$ to the spin-$\frac{1}{2}$ ground states and spin-$\frac{1}{2}$ higher resonances are of the same magnitude, we can neglect the higher resonances without impairing the prediction ability remarkably. We admit that the imaginary parts of the inelastic scattering amplitudes therefore the imaginary parts of the scattering lengths are lost by neglecting the loop-effects (or higher resonance contributions). The real and imaginary parts of the scattering amplitudes relate with the mass and width modifications in the nuclear matter, respectively, we expect that neglecting the imaginary part of the scattering amplitude cannot impair the prediction ability remarkably (or qualitatively) for the mass-shift $\delta M_{D^*}$. In calculations, we observe that the mass-shift $\delta M_{D^*}$ does not sensitive to contributions of the ground states $\Lambda_c $ and $\Sigma_c$, see Table 1, the contributions from the spin-$\frac{1}{2}$ higher resonances maybe even smaller. If we approximate the elastic scattering amplitudes plus the real parts of the full inelastic scattering amplitudes with the elastic scattering amplitudes at the phenomenological side, where the full inelastic scattering amplitudes contain contributions from the spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ higher resonances, the mass-shifts obtained from the QCD sum rules appear as collective effects and receive contributions from the real parts (not imaginary parts) of the inelastic scattering amplitudes. In this article, we neglect the contributions from the scattering states (or continuum states) $\pi\Sigma_c$, $\eta\Lambda_c$, $DN$, $\pi\Lambda_c$, $\rho\Sigma_c$, $\rho\Lambda_c$, etc, which can lead to unknown uncertainties, but we still expect that the prediction ability survives at least qualitatively.

We carry out the operator product expansion to the condensates up to dimension-5 at the large space-like region in the nuclear matter, and obtain the analytical expressions of the correlation functions at the level of quark-gluon degree’s of freedom, $$\begin{aligned}
\Pi_{\mu\nu}(q_0,\vec{q})&=&\left(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2} \right)\sum_n C_n(q_0,\vec{q})\langle{\cal{O}}_n\rangle_{\rho_N}+\cdots\,,\end{aligned}$$ where the $C_n(q_0,\vec{q})$ are the Wilson coefficients, the in-medium condensates $\langle{\cal{O}}_n\rangle_{\rho_N}  =\langle{\cal{O}}_n\rangle+\frac{\rho_N}{2M_N}\langle
{\cal{O}}_n\rangle_N$ at the low nuclear density, the $\langle{\cal{O}}_n\rangle$ and $\langle
{\cal{O}}_n\rangle_N$ denote the vacuum condensates and nuclear matter induced condensates, respectively. One can consult Refs.[@C-parameter; @Drukarev1991] for the technical details in the operator product expansion. Then we collect the terms proportional to $\rho_N$ (or the nuclear matter induced condensates), take the quark-hadron duality, $$\begin{aligned}
T^N_{\mu\nu}(\omega,\vec{q})&=&\left(-g_{\mu\nu}+\frac{q_\mu q_\nu}{q^2} \right)\sum_n C_n(\omega,\vec{q})\langle{\cal{O}}_n\rangle_{N}+\cdots\, ,\end{aligned}$$ set $\omega^2=q^2$, and perform the Borel transform with respect to the variable $Q^2=-\omega^2$, finally obtain the following two QCD sum rules: $$\begin{aligned}
&& a \left\{\frac{1}{M^2}e^{-\frac{M_{D^*}^2}{M^2}}-\frac{s_0}{M_{D^*}^4}e^{-\frac{s_0}{M^2}}\right\}
+b \left\{e^{-\frac{M_{D^*}^2}{M^2}}-\frac{s_0}{M_{D^*}^2}e^{-\frac{s_0}{M^2}}\right\}
+ \frac{2f_{D^*}^2M_{D^*}^2M_N(M_H+M_N)g_{HD^*N}^2}{(M_H+M_N)^2-M_{D^*}^2}\nonumber\\
&&\left\{ \left[\frac{1}{(M_H+M_N)^2-M_{D^*}^2}-\frac{1}{M^2}\right]
e^{-\frac{M_{D^*}^2}{M^2}}-\frac{1}{(M_H+M_N)^2-M_{D^*}^2}e^{-\frac{(M_H+M_N)^2}{M^2}}\right\}=\left\{-\frac{m_c\langle\bar{q}q\rangle_N}{2}\right.\nonumber\\
&&\left.-\frac{2\langle q^\dag i D_0q\rangle_N}{3}+\frac{m_c^2\langle q^\dag i D_0q\rangle_N}{M^2}+\frac{m_c\langle\bar{q}g_s\sigma Gq\rangle_N}{3M^2}+\frac{8m_c\langle \bar{q} i D_0 i D_0q\rangle_N}{3M^2}-\frac{m_c^3\langle \bar{q} i D_0 i D_0q\rangle_N}{M^4}\right\}e^{-\frac{m_c^2}{M^2}}\nonumber\\
&&-\frac{1}{24}\langle\frac{\alpha_sGG}{\pi}\rangle_N\int_0^1dx \left(1+\frac{\widetilde{m}_c^2}{2M^2}\right)e^{-\frac{\widetilde{m}_c^2}{M^2}}
+\frac{1}{48M^2}\langle\frac{\alpha_sGG}{\pi}\rangle_N\int_0^1\frac{1-x}{x}\left(\widetilde{m}_c^2-\frac{\widetilde{m}_c^4}{M^2}\right)e^{-\frac{\widetilde{m}_c^2}{M^2}}\, ,\end{aligned}$$

$$\begin{aligned}
&& a \left\{\frac{1}{M^2}e^{-\frac{M_{D_1}^2}{M^2}}-\frac{s_0}{M_{D_1}^4}e^{-\frac{s_0}{M^2}}\right\}
+b \left\{e^{-\frac{M_{D_1}^2}{M^2}}-\frac{s_0}{M_{D_1}^2}e^{-\frac{s_0}{M^2}}\right\}
+ \frac{2f_{D_1}^2M_{D_1}^2M_N(M_H-M_N)g_{HD_1N}^2}{(M_H-M_N)^2-M_{D_1}^2}\nonumber\\
&&\left\{ \left[\frac{1}{(M_H-M_N)^2-M_{D_1}^2}-\frac{1}{M^2}\right]
e^{-\frac{M_{D_1}^2}{M^2}}-\frac{1}{(M_H-M_N)^2-M_{D_1}^2}e^{-\frac{(M_H-M_N)^2}{M^2}}\right\}=\left\{\frac{m_c\langle\bar{q}q\rangle_N}{2}\right.\nonumber\\
&&\left.-\frac{2\langle q^\dag i D_0q\rangle_N}{3}+\frac{m_c^2\langle q^\dag i D_0q\rangle_N}{M^2}-\frac{m_c\langle\bar{q}g_s\sigma Gq\rangle_N}{3M^2}-\frac{8m_c\langle \bar{q} i D_0 i D_0q\rangle_N}{3M^2}+\frac{m_c^3\langle \bar{q} i D_0 i D_0q\rangle_N}{M^4}\right\}e^{-\frac{m_c^2}{M^2}}\nonumber\\
&&-\frac{1}{24}\langle\frac{\alpha_sGG}{\pi}\rangle_N\int_0^1dx \left(1+\frac{\widetilde{m}_c^2}{2M^2}\right)e^{-\frac{\widetilde{m}_c^2}{M^2}}
+\frac{1}{48M^2}\langle\frac{\alpha_sGG}{\pi}\rangle_N\int_0^1\frac{1-x}{x}\left(\widetilde{m}_c^2-\frac{\widetilde{m}_c^4}{M^2}\right)e^{-\frac{\widetilde{m}_c^2}{M^2}}\, ,\end{aligned}$$

where $\widetilde{m}_c^2=\frac{m_c^2}{x}$.

Differentiate above equation with respect to $\frac{1}{M^2}$, then eliminate the parameter $b$, we can obtain the QCD sum rules for the parameter $a$. With the simple replacements $m_c \to m_b$, $M_{D^*} \to M_{B^*}$, $M_{D_1} \to M_{B_1}$, $\Lambda_c \to \Lambda_b$ and $\Sigma_c \to \Sigma_b$, we can obtain the corresponding QCD sum rules for the mass-shifts of the $B^*$ and $B_1$ mesons in the nuclear matter, where we take the approximation $M_H=\frac{M_{\Sigma_b}+M_{\Lambda_b}}{2}\approx 5.7\,\rm{GeV}$ [@PDG].

The present approach was introduced by Koike and Hayashigaki to study the spin-isospin averaged meson-nucleon scattering lengths and the relevant mass-shifts for the $\rho$, $\omega$, $\phi$ mesons in the nuclear matter [@Koike1996]. The heavy mesons contain a heavy-quark and a light-quark. The existence of a heavy quark in the heavy mesons results in much difference between the in-medium properties of the heavy mesons and light mesons. The heavy quark interacts with the nuclear matter through the exchange of the intermediate gluons and the modifications of the gluon condensates in the nuclear matter are mild, while the modifications of the quark condensates in the nuclear matter are rather large. We expect that the convergent behaviors of the heavy-light type interpolating currents are better than that of the light-light type interpolating currents, if the correlation functions $\Pi_{\mu\nu}(q)$ are expanded in terms of the external parameter $\rho_N$. The approach developed for the light mesons still works for the heavy mesons in the operator product expansion side. In the phenomenological side, we take the lowest order Born terms plus the elastic scattering amplitudes to approximate the phenomenological spectrum, the present article shares both the advantages and shortcomings of the approach developed in Ref.[@Koike1996].

In Ref.[@Klingl-Refe], Klingl, Kaiser and Weise use an effective Lagrangian which combines chiral $SU(3)$ dynamics with vector meson dominance to calculate the forward vector-meson-nucleon scattering amplitudes, and take them as input parameters in the hadronic side of the QCD sum rules in nuclear matter, and observe a remarkable degree of consistency with the operator product expansion at the quark level. In Ref.[@Leupold-Refe], Leupold and Mosel study the electromagnetic current-current correlation functions in the nuclear matter, and expand the QCD sum rules in terms of the finite squared three-momentum $\vec{q}^2$, and observe that the QCD sum rule can provide an interesting and non-trivial consistency check for the hadronic models, but cannot rule out the hadronic models which predict a different behavior of the vector mesons with different $\vec{q}^2$. In this article, the hadronic side of the QCD sum rules of the order ${\cal O }(\rho_N)$ consists of the elastic scattering amplitudes and the lowest Born terms, and can lead to stable QCD sum rules with the suitable Borel parameters in a finite range, so the hadronic model is consistent with the operator product expansion side with $\vec{q}^2=0$ [@Leupold-Refe]. According to Ref.[@Klingl-Refe], there are both real and imaginary parts in the scattering amplitudes, neglecting the imaginary parts of the scattering amplitudes therefore the imaginary parts of the scattering lengths miss the inelastic contributions such as the transitions $D^*N \to$, $\pi\Sigma_c$, $\eta\Lambda_c$, $DN$, $\pi\Lambda_c$, $\rho\Sigma_c$, $\rho\Lambda_c$, etc, the approximation results in unwanted uncertainties, it is the shortcoming of the present method. We can calculate the scattering amplitudes with the effective field theory based on the heavy quark symmetry and chiral symmetry, and keep the imaginary parts of the scattering amplitudes explicitly, and study the loop effects, the tedious calculation is beyond the present work.

Numerical results and discussions
=================================

In calculations, we have assumed that the linear density approximation is valid at the low nuclear density, $\langle{\cal{O}}\rangle_{\rho_N}=\langle0|{\cal{O}}|0\rangle+\frac{\rho_N}{2M_N}\langle
N|{\cal{O}}|N\rangle=\langle{\cal{O}}\rangle+\frac{\rho_N}{2M_N}\langle
{\cal{O}}\rangle_N$ for a general condensate $\langle{\cal{O}}\rangle_{\rho_N}$ in the nuclear matter. The input parameters are taken as $\langle\bar{q} q\rangle_N={\sigma_N \over m_u+m_d } (2M_N)$, $\langle\frac{\alpha_sGG}{\pi}\rangle_N= (-0.65\pm 0.15) \,{\rm {GeV}} (2M_N)$, $\langle q^\dagger iD_0 q\rangle_N=(0.18\pm0.01) \,{\rm{GeV}}(2M_N)$, $\langle\bar{q}g_s\sigma G q\rangle=3.0\,{\rm GeV}^2(2M_N) $, $\langle \bar{q} iD_0iD_0
q\rangle_N+{1\over8}\langle\bar{q}g_s\sigma G
q\rangle_N=0.3\,{\rm{GeV}}^2(2M_N)$, $m_u+m_d=12\,\rm{MeV}$, $\sigma_N=(45\pm 10)\,\rm{MeV}$, $M_N=0.94\,\rm{GeV}$, $\rho_N=(0.11\,\rm{GeV})^3$ [@C-parameter], $m_c=(1.35\pm 0.1)\,\rm{GeV}$ and $m_b=(4.7\pm 0.1)\,\rm{GeV}$ at the energy scale $\mu=1\,\rm{GeV}$.

The value of the strong coupling constant $g_{DN\Lambda_c}$ is $g_{\Lambda_c DN}=6.74$ from the QCD sum rules [@Nielsen98], while the average value of the strong coupling constants $g_{\Lambda_cDN}$ and $g_{\Sigma_cDN}$ from the light-cone QCD sum rules is $\frac{g_{\Lambda_cDN}+g_{\Sigma_cDN}}{2}=6.775$ [@Khodjamirian1108], those values are consistent with each other. The average value of the strong coupling constants $g_{\Lambda_c D^*N}$ and $g_{\Sigma_c D^*N}$ from the light-cone QCD sum rules is $\frac{g_{\Lambda_c D^*N}+g_{\Sigma_cD^*N}}{2}=3.86$ [@Khodjamirian1108]. In this article, we take the approximation $g_{\Lambda_cD^*N}\approx g_{\Sigma_cD^*N}
\approx g_{\Lambda_cD_1N}\approx g_{\Sigma_cD_1N}\approx g_{\Lambda_bB^*N}\approx g_{\Sigma_bB^*N}
\approx g_{\Lambda_bB_1N}\approx g_{\Sigma_bB_1N}\approx3.86$.

For the well established vector mesons $D^*$ and $B^*$, we take the values from the Particle Data Group, $M_{D^*}=2.01\,\rm{GeV}$ and $M_{B^*}=5.325\,\rm{GeV}$ [@PDG], the decay constants $f_{D^*}$ and $f_{B^*}$ are determined by the QCD sum rules, $f_{D^*}=0.270\,\rm{GeV}$ and $f_{B^*}=0.195\,\rm{GeV}$ [@SRreview], where the threshold parameters are taken as $s^0_{D^*}=(5-7)\,\rm{GeV}^2$ and $s^0_{B^*}=(33-37)\,\rm{GeV}^2$ [@SRreview; @Kho-Decay], here we have neglected the uncertainties of the decay constants. We can take the threshold parameters as $s^0_{D^*}=6.5\,\rm{GeV}^2$, $s^0_{B^*}=35\,\rm{GeV}^2$ and the Borel parameters as $T^2_{D^*}=(1.6-2.6)\,\rm{GeV}^2$, $T^2_{B^*}=(4.0-6.0)\,\rm{GeV}^2$ to reproduce the values $M_{D^*}=2.01\,\rm{GeV}$, $M_{B^*}=5.325\,\rm{GeV}$, $f_{D^*}=0.270\,\rm{GeV}$ and $f_{B^*}=0.195\,\rm{GeV}$ approximately for the QCD sum rules in the vacuum [^2]. The mass of the axialvector meson $D^{0}_1(2430)$ is $M_{D_1}=(2427\pm26\pm25)\,\rm{MeV}$ from the Particle Data Group [@PDG], and the axialvector meson $B_1$ has not been observed yet. We calculate the hadronic parameters of the axialvector mesons $D_1$ and $B_1$ using the QCD sum rules in the vacuum, and obtain the values $M_{D_1}=2.42\,\rm{GeV}$, $M_{B_1}=5.75\,\rm{}GeV$, $f_{D_1}=0.305\,\rm{GeV}$ and $f_{B_1}=0.255\,\rm{GeV}$ with the threshold parameters $s^0_{D_1}=8.5\,\rm{GeV}^2$, $s^0_{B_1}=39\,\rm{GeV}^2$ and the Borel parameters $T^2_{D_1}=(2.0-3.0)\,\rm{GeV}^2$, $T^2_{B_1}=(5.0-7.0)\,\rm{GeV}^2$. The value $M_{D_1}=2.42\,\rm{GeV}$ reproduces the experimental data $M_{D_1}=(2427\pm26\pm25)\,\rm{MeV}$ well [@PDG]. The prediction of the mass $M_{B_1}$ satisfies the relation $M_{B_1}-M_{B^*}\approx M_{D_1}-M_{D^*}$. For the explicit expressions of the QCD sum rules in the vacuum derived from the correlation functions $\Pi^{0}_{\mu\nu}(q)$, one can consult Ref.[@Wang-CPL] and the references therein.

In the QCD sum rules, the phenomenological hadronic spectrum does not depend on the Borel parameters, the continuum threshold parameters $\sqrt{s_0}$ are usually taken as $\sqrt{s_0}=M_{gr}\sim M_{ra}$, where the $gr$ and $ra$ denote the ground states and first radial excited states respectively, so there are uncertainties come from the continuum threshold parameters. We choose suitable Borel parameters to satisfy the two criteria (pole dominance and convergence of the operator product expansion) of the QCD sum rules. The optimal Borel parameters result in rather stable QCD sum rules which are not sensitive to the threshold parameters and have Borel platforms. The phenomenological hadronic spectrum in a definite channel (in other words, for a definite interpolating current) survives in different QCD sum rules, we can choose the same threshold parameters but different Borel parameters to satisfy the two criteria of the QCD sum rules in studying the correlation functions $\Pi^0_{\mu\nu}$ and $T_{\mu\nu}^N$. In this article, the threshold parameters are taken as $s^0_{D^*}=(6.5\pm0.5)\,\rm{GeV}^2$, $s^0_{D_1}=(8.5\pm0.5)\,\rm{GeV}^2$, $s^0_{B^*}=(35\pm1)\,\rm{GeV}^2$ and $s^0_{B_1}=(39\pm1)\,\rm{GeV}^2$, respectively, which satisfy the relations $s^0_{D^*,D_1,B_1}=(M_{D^*,D_1,B_1}+0.4\sim 0.6\, \rm{GeV})^2$ and $s^0_{B^*}=(M_{B^*}+0.5\sim 0.7\, \rm{GeV})^2$. In general, the energy gap between the ground state and the first radial excited state is about $0.5\,\rm{GeV}$.

In Fig.1, we plot the mass-shifts $\delta M$ versus the Borel parameter $M^2$ at large intervals. From the figure, we can see that the values of the mass-shifts are rather stable with variations of the Borel parameter at the intervals $M^2=(4.5-5.4)\,\rm{GeV}^2$, $(6.5-7.6)\,\rm{GeV}^2$, $(22-24)\,\rm{GeV}^2$ and $(34-37)\,\rm{GeV}^2$ for the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; in other words, the uncertainties originate from the Borel parameter $M^2$ are less than $1\%$. The main contributions come from the terms $\pm m_c\langle\bar{q}q\rangle_N$ and $\pm m_b\langle\bar{q}q\rangle_N$, see Eqs.(18-19) and Fig.2, the operator product expansion is well convergent. The spectral densities at the level of the quark-gluon degrees of freedom consist of the medium-induced condensates, and have the form $A_1 \delta(s-m_{c/b}^2)+A_2\delta(s-\widetilde{m}_{c/b}^2)$, where the $A_1$ and $A_2$ denote the coefficients, we carry out the integrals, $$\begin{aligned}
  \int_{m_{c/b}^2}^{s_0}ds\left[ A_1 \delta(s-m_{c/b}^2)+A_2\delta(s-\widetilde{m}_{c/b}^2)\right]e^{-\frac{s}{M^2}}\, ,
  \end{aligned}$$ to obtain the right side of the QCD sum rules in Eqs.(18-19), there are no perturbative terms to approximate the continuum contributions at the regions $s>s_0$. At the phenomenological side, the exponential factors $$\begin{aligned}
 e^{-\frac{s_0}{M^2}}&=&e^{-(1.20-1.44)}, \, e^{-(1.12-1.31)}, \,e^{-(1.46-1.59)},  \,e^{-(1.05-1.15)},
 \end{aligned}$$ at the intervals $M^2=(4.5-5.4)\,\rm{GeV}^2$, $(6.5-7.6)\,\rm{GeV}^2$, $(22-24)\,\rm{GeV}^2$ and $(34-37)\,\rm{GeV}^2$ for the $D^*$, $D_1$, $B^*$, $B_1$ mesons respectively, where we take the central values of the threshold parameters, the corresponding exponential factors of the ground states are $$\begin{aligned}
  e^{-\frac{M^2_m}{M^2}}&=&e^{-(0.75-0.90)}, \,e^{-(0.77-0.90)}, \,e^{-(1.18-1.29)}, \,e^{-(0.89-0.97)},
 \end{aligned}$$ where the $m$ stands for the $D^*$, $D_1$, $B^*$, $B_1$ mesons respectively; the continuum contributions are suppressed more efficiently. Furthermore, we expect that the couplings of an special interpolating current to the excited states are more weak than that to the ground state mesons. For example, the decay constants of the pseudoscalar mesons $\pi(140)$ and $\pi(1800)$ have the hierarchy: $f_{\pi(1300)}\ll f_{\pi(140)}$ from the Dyson-Schwinger equation [@CDRoberts], the lattice QCD [@Latt-pion], the QCD sum rules [@QCDSR-pion], etc, or from the experimental data [@pion-exp].

We can take the Borel windows as $M^2=(4.5-5.4)\,\rm{GeV}^2$, $(6.5-7.6)\,\rm{GeV}^2$, $(22-24)\,\rm{GeV}^2$ and $(34-37)\,\rm{GeV}^2$ for the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively, and obtain mass-shifts $\delta M_{D^*}=-71^{+20}_{-23}\,\rm{MeV}$, $\delta M_{B^*}=-380^{+82}_{-91}\,\rm{MeV}$, $\delta M_{D_1}=72^{+22}_{-20}\,\rm{MeV}$, $\delta M_{B
_1}=264^{+76}_{-69}\,\rm{MeV}$, respectively; and the scattering lengths $a_{D^*}=-1.07^{+0.30}_{-0.34}\,\rm{fm}$, $a_{B^*}=-7.17^{+1.53}_{-1.71}\,\rm{fm}$, $a_{D_1}=1.15^{+0.35}_{-0.32}\,\rm{fm}$ and $a_{B_1}=5.03^{+1.46}_{-1.31}\,\rm{fm}$ for the $D^*N$, $B^*N$, $D_1N$ and $B_1N$ interactions, respectively. For the technical details in analyzing the uncertainties, one can consult Ref.[@Baxi].

In Fig.3, we plot the mass-shifts $\delta M$ versus the Borel parameter $M^2$ and the strong coupling constants $g^2$. From the figure, we can see that the mass-shifts decrease (increase) monotonously with increase of the squared strong coupling constants $g^2$ for the vector mesons $D^*$ and $B^*$ (axialvector mesons $D_1$ and $B_1$) in the Borel windows. The precise values of the mass-shifts and scattering lengths are presented in Table 1.

Although the present QCD sum rules are not stable with variations of the Borel parameters, the uncertainties originate from the Borel parameters are less than $1\%$ in the Borel windows, i.e. we choose suitable platforms to avoid large uncertainties. On the other hand, we can take moments of the correlation functions and derive QCD sum rules to study the mass-shifts as in Ref.[@DB-SR], the present QCD sum rules are equal to that derived from the first two moments of the correlation functions, i.e. we take the weight functions to be $1$ and $s$.

In the present work and Refs.[@Hayashigaki; @WangHuang], the correlation functions are divided into a vacuum part and a static one-nucleon part, and the nuclear matter induced effects are extracted explicitly; while in Refs.[@DB-SR; @SumRule-D0], the pole terms of the hadronic spectral densities are parameterized as $\frac{\rm{Im}\Pi(\omega,0)}{\pi}=F_{+}\delta(\omega-M_{+})-F_{-}\delta(\omega+M_{-})$, where $M_{\pm}=M\pm\Delta M$ and $F_{\pm}=F\pm\Delta F$, and QCD sum rules for the mass center $M$ and the mass splitting $\Delta M$ are obtained. For the pseudoscalar $D$, $\bar{D}$ mesons, Hayashigaki obtains the mass-shift $\delta M_{D}=-50\,\rm{MeV}$ [@Hayashigaki], while Hilger, Thomas and Kampfer obtain the mass-shift $\delta M_{D}=+45\,\rm{MeV}$ [@DB-SR]. For scalar $D_0$, $\bar{D}_0$ mesons, the mass-shift $\delta M_{D_0}=M-M_{D_0}<0$ obtained by Hilger and Kampfer [@SumRule-D0] differs from the result $\delta M_{D0} = +69\,\rm{MeV}$ obtained by Wang and Huang [@WangHuang]. In Ref.[@SumRule-VA], Hilger, Kampfer and Leupold study the chiral partners of charmed mesons in the nuclear matter, and focus on the differences between the pseudoscalar and scalar as well as vector and axialvector D mesons and derive the corresponding Weinberg type sum rules, while the mass-shifts are not presented. In the present work and Refs.[@Hayashigaki; @WangHuang], the inelastic scattering amplitudes (or the loop-effects) are neglected, which can lead to unknown uncertainties; while in Refs.[@DB-SR; @SumRule-D0], the simple hadronic spectral densities miss the complex energy dependence. The two approaches both have shortcomings, and the corresponding predictions can be confronted with the experimental data in the future.

In the limit $m_q \to 0$, the quark condensate $\langle\bar{q}q\rangle$ serves as the order parameter and indicates that the chiral symmetry is broken. The quark condensate undergoes reduction in the nuclear matter, $\langle\bar{q}q\rangle_{\rho_N}=\langle\bar{q}q\rangle+\frac{\rho_N}{2M_N}\langle\bar{q}q\rangle_N$, the chiral symmetry is partially restored, for example, the in-medium nucleon mass $ M_N^*$ can be approximated as $M_N^*=-\frac{8\pi^2}{M^2}\langle\bar{q}q\rangle_{\rho_N}$ and the mass reduction is rather large; on the other hand, there appear new medium-induced condensates, for example, the $\langle \bar{q} iD_0iD_0q\rangle_N$, $\langle\bar{q}g_s\sigma G q\rangle_N$, etc, which also break the chiral symmetry. In the present case, the medium-induced condensates are associated with the large heavy quark masses $m_Q$, $m_Q^2$, $m_Q^3$ or $m_Q^4$, the net effects do not always warrant that the chiral symmetry is monotonously restored with the increase of the density of the nuclear matter. The light vector current $\bar{q}\gamma_\mu q$ and axialvector current $\bar{q}\gamma_\mu\gamma_5 q$ are invariant under the chiral transformation $q\to e^{i\alpha\gamma_5}q$, however, the heavy vector current $\bar{Q}\gamma_\mu q$ and axialvector current $\bar{Q}\gamma_\mu\gamma_5 q$ are mixed with each other under the transformation, the heavy quark currents $\bar{Q}\gamma_\mu q$ and $\bar{Q}\gamma_\mu\gamma_5 q$ are not conserved in the limit $m_q \to 0$, it is better to take the doublets $(D^*,D_1)$, $(B^*,B_1)$ as the parity-doublets rather than the chiral-doublets. If we take into account the flavor $SU(3)$ symmetry of the light quarks, the chiral $SU(3)_L \times SU(3)_R$ transformations require that the ground states $(\bar{D}^{*0}, D^{*-}, D^{*-}_s)$ and $(B^{*+}, B^{*0}, B^{*0}_s)$ have their chiral partners $(\bar{D}^{0}_1, D^{-}_1, D^{-}_{s1})$ and $(B^{+}_1, B^{0}_1, B^{0}_{s1})$, respectively, those parity-doublets are chiral-doublets. When the density of the nuclear matter is large enough, the order parameter $\langle\bar{q}q\rangle_{\rho_N} \to 0$, the chiral symmetry is restored, the Fermi gas approximation for the nuclear matter does not survive, there are free of the non-perturbative contributions from the condensates, and the parity-doublets (or chiral-doublets) maybe have degenerated masses approximately. In the present case, we study the parity-doublets (or chiral-doublets) in the low nuclear density, the mass breaking effects of the parity-doublets (or chiral-doublets) maybe even larger.

The axialvector current $\bar{u}(x)\gamma_\mu \gamma_5 c(x)$ interpolates the axialvector meson $D_1(2430)$ has non-vanishing coupling with the scattering state $D^*\pi$, in the soft $\pi$ limit, the coupling constant $\lambda_{D^*\pi}$ can be estimated as $$\begin{aligned}
\langle 0|\bar{u}(x)\gamma_\mu \gamma_5 c(x)|D^*\pi\rangle&=&-\frac{i}{f_\pi}\langle 0|\left[Q_5,\bar{u}(x)\gamma_\mu \gamma_5 c(x)\right]|D^*\rangle \nonumber \\
&=&-\frac{i}{f_\pi}\langle 0|\bar{d}(x)\gamma_\mu  c(x)|D^*\rangle =-\frac{if_{D^*}M_{D^*}\epsilon_\mu}{f_\pi}=i\lambda_{D^*\pi}\epsilon_\mu \, ,\\
&=&f_{D_1}M_{D_1}\varepsilon_\mu \frac{i}{p^2-M_{D_1}^2} \langle D_1|D^*\pi\rangle \, ,\end{aligned}$$ where the axial-charge $Q_5=\int d^3y d^{\dagger}(y) \gamma_5 u(y)$, and the $\epsilon_\mu$ and $\varepsilon_\mu$ are the polarization vectors of the vector and axialvector mesons $D^*$ and $D_1$, respectively [@LeeDa], the formula in Eq.(24) survives beyond the soft $\pi$ limit. The coupling constant $\lambda_{D^*\pi}$ is a large quantity and cannot be neglected. The rescatterings $$\begin{aligned}
 D^*\pi &\to& D^*\pi \, , \nonumber \\
 D^*\pi &\to& D^*\pi, D_s^*\bar{K}, D^* \eta \to D^*\pi\,,  \nonumber \\
 D^*\pi &\to& D^*\pi, D_s^*\bar{K}, D^* \eta \to D^*\pi, D_s^*\bar{K}, D^* \eta\to D^*\pi \, ,  \nonumber \\
 D^*\pi &\to& D^*\pi, D_s^*\bar{K}, D^* \eta \to D^*\pi, D_s^*\bar{K}, D^* \eta\to D^*\pi, D_s^*\bar{K}, D^* \eta\to D^*\pi \, ,
 \end{aligned}$$ $\cdots$ also have contributions to the hadronic spectral densities. In the heavy meson chiral unitary approach, we can use the Bethe-Salpeter equation to perform the summation of the infinite series of the intermediate meson-loops (such as the $D^*\pi$, $D_s^*\bar{K}$, $D^* \eta$) to obtain the full $D^*\pi \to D^*\pi$ scattering amplitude, and generate the axialvector meson $D_1(2430)$ dynamically [@PANDA; @GuoFK]. If we saturate the full $D^*\pi \to D^*\pi$ scattering amplitude with the exchanges of the intermediate axialvector meson $D_1(2430)$, the $D^*\pi$ rescattering effects lead to the renormalization $\frac{1}{p^2-M_{D_1}^2+i\epsilon} \to \frac{1}{p^2-M_{D_1}^2-\widehat{\Pi}(p)+i\epsilon}$ in the hadronic representation of the correlation functions, where the $\widehat{\Pi}(p)$ denotes the renormalized self-energy of the intermediate $D^*\pi$ loops, and contributes a finite imaginary part to modify the dispersion relation. In fact, the contributions of the intermediate meson-loops are very large, we have to take the mass $M_{D_1}$ as the bare mass $\stackrel{ \rm \circ} M_{D_1}$ to absorb the real part of the un-renormalized self-energy to reproduce the physical mass, the net effects are embodied in the finite imaginary part. We can take into account those meson-loops effectively by taking the following replacement for the hadronic spectral density, $$\begin{aligned}
\delta\left(s-M_{D_1}^2\right) &\to& \frac{1}{\pi}\frac{\sqrt{s}\Gamma_{D_1}(s)}{\left(s-M_{D_1}^2\right)^2+s\Gamma_{D_1}^2(s)} \, ,\end{aligned}$$ here we neglect the complicated renormalization procedure for simplicity [@WangIJTP]. Furthermore, we neglect the energy dependence of the width, and approximate it with the experimental value from the Particle Data Group. In Ref.[@WangIJTP], we observe that a width about (or less than) $400\,\rm{MeV}$ cannot change the prediction significantly, the $\delta$ function approximation for the spectral densities still survives. In the present case, $\Gamma_{D_1}=384^{+107}_{-75}\pm 75\,\rm{MeV}$ from the Particle Data Group [@PDG], the contaminations from the intermediate state $D^*\pi$ are expected to be small. Analogical discussions can be applied to the contaminations from the intermediate state $B^*\pi$.

The negative scattering lengths $a_{D^*}=-1.07\,\rm{fm}$ and $a_{B^*}=-7.17\,\rm{fm}$ indicate that the $D^*N$ and $B^*N$ interactions are attractive, it is possible to form the $D^*N$ and $B^*N$ bound states; while the positive scattering lengths $a_{D_1}=1.15\,\rm{fm}$ and $a_{B_1}=5.03\,\rm{fm}$ indicate that the $D_1N$ and $B_1N$ interactions are repulsive, it is difficult to form the $D_1N$ and $B_1N$ bound states. According to the observations of the present work and Refs.[@Hayashigaki; @WangHuang], we can draw the conclusion tentatively that the negative parity heavy mesons decrease their masses in the nuclear matter while the positive parity heavy mesons increase their masses. The decays of the high charmonium states to the negative parity charmed meson pairs $D\bar{D}$ and $D^*\bar{D}^*$ are facilitated, while the decays to the positive parity charmed mesons pairs $D_0\bar{D}_0$ and $D_1\bar{D}_1$ are suppressed. The $J/\psi$ production can obtain additional suppressions due to mass modifications of the negative parity charmed mesons $D$ and $D^*$ in the nuclear matter.

           $g^2$               0        10        20        30        40        50     
  ------------------------ --------- --------- --------- --------- --------- --------- --
   $\delta M_{D^*}$ (MeV)    $-75$     $-72$     $-70$     $-67$     $-65$     $-62$   
   $\delta M_{B^*}$ (MeV)   $-382$    $-381$    $-380$    $-380$    $-379$    $-378$   
   $\delta M_{D_1}$ (MeV)    $70$      $71$      $73$      $74$      $76$      $78$    
   $\delta M_{B_1}$ (MeV)    $262$     $263$     $264$     $265$     $266$     $267$   
       $a_{D^*}$ (fm)       $-1.13$   $-1.09$   $-1.05$   $-1.02$   $-0.98$   $-0.94$  
       $a_{B^*}$ (fm)       $-7.20$   $-7.18$   $-7.17$   $-7.15$   $-7.14$   $-7.13$  
       $a_{D_1}$ (fm)       $1.11$    $1.14$    $1.16$    $1.19$    $1.21$    $1.24$   
       $a_{B_1}$ (fm)       $5.00$    $5.02$    $5.04$    $5.05$    $5.07$    $5.09$   

  :  The mass-shifts $\delta M$ and the scattering lengths $a$ versus the strong coupling constants $g^2$.

![The mass-shifts $\delta M$ versus the Borel parameter $M^2$, the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively. ](mass-DV.EPS "fig:"){width="7cm"} ![The mass-shifts $\delta M$ versus the Borel parameter $M^2$, the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively. ](mass-DA.EPS "fig:"){width="7cm"} ![The mass-shifts $\delta M$ versus the Borel parameter $M^2$, the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively. ](mass-BV.EPS "fig:"){width="7cm"} ![The mass-shifts $\delta M$ versus the Borel parameter $M^2$, the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively. ](mass-BA.EPS "fig:"){width="7cm"}

![The contributions from different terms versus the Borel parameter $M^2$ in the operator product expansion, where the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; the $\alpha$, $\beta$, $\lambda$ and $\tau$ denote the $\langle\bar{q} q\rangle_N$, $\langle q^\dagger iD_0 q\rangle_N$, $\langle \bar{q} iD_0iD_0
q\rangle_N+ \langle\bar{q}g_s\sigma Gq\rangle_N $, and $\langle\frac{\alpha_sGG}{\pi}\rangle_N$ terms, respectively. ](Fraction-DV.EPS "fig:"){width="7cm"} ![The contributions from different terms versus the Borel parameter $M^2$ in the operator product expansion, where the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; the $\alpha$, $\beta$, $\lambda$ and $\tau$ denote the $\langle\bar{q} q\rangle_N$, $\langle q^\dagger iD_0 q\rangle_N$, $\langle \bar{q} iD_0iD_0
q\rangle_N+ \langle\bar{q}g_s\sigma Gq\rangle_N $, and $\langle\frac{\alpha_sGG}{\pi}\rangle_N$ terms, respectively. ](Fraction-DA.EPS "fig:"){width="7cm"} ![The contributions from different terms versus the Borel parameter $M^2$ in the operator product expansion, where the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; the $\alpha$, $\beta$, $\lambda$ and $\tau$ denote the $\langle\bar{q} q\rangle_N$, $\langle q^\dagger iD_0 q\rangle_N$, $\langle \bar{q} iD_0iD_0
q\rangle_N+ \langle\bar{q}g_s\sigma Gq\rangle_N $, and $\langle\frac{\alpha_sGG}{\pi}\rangle_N$ terms, respectively. ](Fraction-BV.EPS "fig:"){width="7cm"} ![The contributions from different terms versus the Borel parameter $M^2$ in the operator product expansion, where the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; the $\alpha$, $\beta$, $\lambda$ and $\tau$ denote the $\langle\bar{q} q\rangle_N$, $\langle q^\dagger iD_0 q\rangle_N$, $\langle \bar{q} iD_0iD_0
q\rangle_N+ \langle\bar{q}g_s\sigma Gq\rangle_N $, and $\langle\frac{\alpha_sGG}{\pi}\rangle_N$ terms, respectively. ](Fraction-BA.EPS "fig:"){width="7cm"}

![The mass-shifts $\delta M$ versus the Borel parameter $M^2$, the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; the $\alpha$, $\beta$, $\gamma$, $\lambda$, $\tau$ and $\rho$ correspond the strong coupling constants $g^2=0$, $10$, $20$, $30$, $40$ and $50$, respectively. ](Coupling-DV.EPS "fig:"){width="7cm"} ![The mass-shifts $\delta M$ versus the Borel parameter $M^2$, the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; the $\alpha$, $\beta$, $\gamma$, $\lambda$, $\tau$ and $\rho$ correspond the strong coupling constants $g^2=0$, $10$, $20$, $30$, $40$ and $50$, respectively. ](Coupling-DA.EPS "fig:"){width="7cm"} ![The mass-shifts $\delta M$ versus the Borel parameter $M^2$, the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; the $\alpha$, $\beta$, $\gamma$, $\lambda$, $\tau$ and $\rho$ correspond the strong coupling constants $g^2=0$, $10$, $20$, $30$, $40$ and $50$, respectively. ](Coupling-BV.EPS "fig:"){width="7cm"} ![The mass-shifts $\delta M$ versus the Borel parameter $M^2$, the $A$, $B$, $C$ and $D$ denote the $D^*$, $D_1$, $B^*$ and $B_1$ mesons, respectively; the $\alpha$, $\beta$, $\gamma$, $\lambda$, $\tau$ and $\rho$ correspond the strong coupling constants $g^2=0$, $10$, $20$, $30$, $40$ and $50$, respectively. ](Coupling-BA.EPS "fig:"){width="7cm"}

Conclusion
==========

In this article, we calculate the mass-shifts of the vector and axialvector mesons $D^*$, $B^*$, $D_1$ and $B_1$ in the nuclear matter using the QCD sum rules. We take the linear approximation at the low density of the nuclear matter, and extract the mass-shifts and scattering lengths explicitly, $\delta M_{D^*}=-71\,\rm{MeV}$, $\delta M_{B^*}=-380\,\rm{MeV}$, $\delta M_{D_1}=72\,\rm{MeV}$, $\delta M_{B
_1}=264\,\rm{MeV}$, $a_{D^*}=-1.07\,\rm{fm}$, $a_{B^*}=-7.17\,\rm{fm}$, $a_{D_1}=1.15\,\rm{fm}$ and $a_{B_1}=5.03\,\rm{fm}$. Our numerical results indicate that the $D^*N$ and $B^*N$ interactions are attractive while the $D_1N$ and $B_1N$ interactions are repulsive; it is possible (difficult) to form the $D^*N$ and $B^*N$ ($D_1N$ and $B_1N$) bound states. The $J/\psi$ production can obtain additional (no additional) suppression due to mass modification of the vector meson $D^*$ (axialvector meson $D_1$) in the nuclear matter. The present predictions can be confronted with experimental data in the future.

Acknowledgements {#acknowledgements .unnumbered}
================

This work is supported by National Natural Science Foundation, Grant Number 11075053, and the Fundamental Research Funds for the Central Universities.

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[^1]: E-mail,<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>.

[^2]: We use the $T^2$ to denote the Borel parameters in the vacuum.