Holographic nuclear matter in presence of chemical potential at finite temperature

Calculations in quantum chromodynamics (QCD) for the matter at low temperature and considerable chemical potential are complicated. Two important examples of matter in this regime are nuclear matter inside compact stars (including neutrons and protons in the simplest combinations with possible Cooper pairing), and deconfined matter (including three-flavor quarks). An important property of compact stars is their rich phase diagram where differs them from ordinary matter in the condensed matter systems. They include both nuclear and deconfined matters. However, studying both phases in one field theory model is very difficult. Also, these systems are strongly coupled, and study of them with conventional lattice QCD models is challenging, see [1] for recent progress and the other models like Nambu-Jona-Lasinio (NJL) [2] and quark-meson [3] have problems, they can be beneficial to get some insight into quark matter. Still, they usually do not include nuclear matter.
The original version of gauge/string correspondence is $N=4$ supersymmetric Yang-Mills theory for dual field theory and type IIB string theory for gravity. This exciting method is helps study strongly coupled gauge theory, especially for zero or small baryon chemical potential. Important studies include application to the heavy-ion collision.
In this paper, we use Sakai-Sugimoto (SS) model and study QCD properties. Already, the full spectrum of the top-down $D4$-$D8$ brane system of Sakai-Sugimoto model calculated by Ref. [4]. We can account for a single model for nuclear and quark matter, where the parameters of the model are the ’t Hooft coupling $\lambda$, the Kaluza-Klein mass $M_{KK}$, and the asymptotic separations of $D8$ and $\bar{D}8$ branes which denoted by $L$. In this model, fundamental chiral fermions have been introduced through $N_{f}$ $D8$ and $\bar{D}8$ branes. These are in the background of $N_{c}$ $D4$ branes [5], $N_{c}$ and $N_{f}$ are respectively numbers of color and flavor. Although, these are another works where Sakai-Sugimoto considered, so effect of chemical potential on the nuclear force by using the string theory already studied by Ref. [6].
We now work in the $N_{c}>>N_{f}$ limit, that this limit caused to neglect effects of flavor branes on the background geometry, and also we consider the chiral limit when $D8$ and $\bar{D}8$ branes are not connected. however, you can see Ref. [7] for discussing non-zero quark mass. Baryons are studied as $D4$-branes wrapped around the 4-sphere of the background geometry in the Sakai-Sugimoto model [8], here, this is the same to instanton with the non-zero topological charge on the connected flavor branes of the model [9], baryonic matter at first, non-zero density, and temperature was considered in a point-like approximation of the instantons on the flavor branes [10]. The instanton has been further developed including finite-size effects [11]. It has been studied by Ref. [12], but here we extend [13] and consider a new profile for instanton which depends on the temperature. Indeed, the temperature dependence was already studied in Ref. [12] where the numerical analysis was only carried out in the zero temperature regime. Now, we would like to do numerical analysis of the finite temperature.
We are working in the decomapctified limit of the SS model[14, 15, 16] and this is the limit where the asymptotic separation of D8 branes ($L$) is small compared to the compactified radius of the extra dimension. In another word, this limit have been provided by fixing $L$ and choosing a very large radius. Since a large radius corresponds to a very small Kaluza-Klein mass, the critical temperature for deconfinement goes to zero in this limit then this limit is suitable for studying the phase transitions, which now depends on the baryon chemical potential and temperature.
The holographic description of quark matter in this setup is giving by disconnected flavor branes. In Ref. [13], it was shown that nuclear and quark matter could be continuously transformed into each other when we have considered the interaction between instantons at zero temperature. Our work is an extension of Ref. [17], by accounting for the interactions of the instantons in bulk. It has been done by using the exact two-instanton solution in flat space, which is a special case of the general Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction [10], and which has been discussed previously in the context of the SS model [18] to study the nucleon-nucleon interaction [19, 20]. As we shall see, the instanton interactions are crucial for this observation. There are various other, in some aspects complementary, approximations to many-baryon phases in the Sakai-Sugimoto model, see for instance Refs. [20, 21]. One of them is based on a homogeneous ansatz for the gauge fields in the bulk [22], which is expected to yield a better approximation for large densities, which is less transparent from a physical point of view because it is not built from single instantons.
This paper is organized as follows: In section 2, we introduce DBI and CS actions, section 3 including deformation parameters and depending on the temperature. In section 4, we explained results like a phase diagram, connecting two phases and the speed of sound. Finally, in section 5, we give a conclusion and summary of results.

Our calculation has been started with the following Dirac-Born-Infeld and Chern-Simons action,

$$S=S_{DBI}+S_{CS},$$

(2.1)

where $S_{DBI}$ is given as

$$S_{\rm DBI}=2T_{8}V_{4}\int_{0}^{1/T}d\tau\int d^{3}X\int_{u_{c}}^{\infty}du\,e^{-\Phi}\sqrt{{\rm det}(g+2\pi\alpha^{\prime}{\cal F})},$$

(2.2)

where $\alpha^{\prime}$ is related to string length $\ell_{s}$ via $\alpha^{\prime}=\ell_{s}^{2}$, $T_{8}=\frac{1}{(2\pi)^{8}\ell_{s}^{9}}$ is $D8$-brane tension, $V_{4}=\frac{8\pi^{2}}{3}$ is the volume of the 4-sphere. One should notice that $u_{c}$ is the location of the tip of the connected flavor branes $D8$ and $\bar{D}8$. The dilaton field is given via $e^{\Phi}=g_{s}\left(\frac{U}{R}\right)^{\frac{3}{4}}$, where $R$ is the curvature radius and $g_{s}$ is the string coupling. Also,

$$u=\frac{U}{M_{KK}R^{3}},$$

(2.3)

is holographic coordinate which is dimensionless quantity. The integral is taken over the position space, imaginary time $\tau$ with the temperature $T$ and over the holographic coordinate $u$, also $g$ is the induced metric on the flavor branes. The detailed form of its induced metric $g$ on the $D8$ brane, abelian and non-abelian field strengths can be found in Ref. [13]. Also,

$${\cal F}_{\mu\nu}=\hat{F}_{\mu\nu}+F_{\mu\nu}\ \,,\qquad F_{\mu\nu}=F_{\mu\nu}^{a}\sigma_{a},$$

(2.4)

with the Pauli matrices $\sigma_{a}$ and $\mu,\nu=0,1,2,3,u$. In our ansatz, the only non-zero abelian field strength is $\hat{F}_{0u}$ and non-abelian field strengths are $F_{iu}$, and $F_{ij}$. All other field strengths are set to zero.
By using the mentioned ansatz, the CS contribution is given by,

$$S_{\rm CS}=\frac{N_{c}}{4\pi^{2}}\int_{0}^{1/T}d\tau\int d^{3}X\int_{U_{c}}^{\infty}du\hat{A}_{0}Tr[F_{ij}F_{kU}]\epsilon_{ijk}.$$

(2.5)

Therefore, we can obtain

$${\cal L}=u^{5/2}\sqrt{(1+u^{3}f_{T}x_{4}^{\prime 2}-\hat{a}_{0}^{\prime 2}+g_{1})(1+g_{2})}-\hat{a}_{0}n_{I}q(u),$$

(2.6)

where the prime denotes derivative with respect to $u$ and $n_{I}$ represent the instanton density per flavor . Also, $f_{T}=1-u_{T}^{3}/u^{3}$ with $u_{T}=(\frac{4\pi T}{3M_{KK}})^{2}$. The instanton density is given by $n_{I}$, and $\hat{a}_{0}$ is denoted the abelian gauge field with boundary condition so $\hat{a}_{0}(\infty)=\mu$. Also the embedding function of the flavor branes has been shown by $x_{4}(u)$ so the boundary conditions are $x_{4}(u_{c})=0$ and $x_{4}(\infty)=LM_{KK}/2$. The functions $g_{1}$, $g_{2}$ and $q$ in Ref. [13] are defined by the following relations

$$g_{1}\equiv\frac{f_{T}n_{I}}{3\gamma}\frac{\partial z}{\partial u}q(u),\ \ \ \ \ \ \ \ \ \ g_{2}\equiv\frac{\gamma n_{I}}{3u^{3}}\frac{\partial u}{\partial z}q(u),$$

(2.7)

where $u=(u_{c}^{3}+u_{c}z^{2})^{1/3}$ shows the relation between $z$ and $u$, also

$$q(u)=2\frac{\partial z}{\partial u}D(z),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D(z)=(1-p)D_{0}(z)+\frac{p}{2}D_{int}(d,z),$$

(2.8)

where the function $D(z)$ is instanton profile on the flavor branes which is depends on the lattice parameter $p$. So, the solution for the single instanton is given by

$$D_{0}(z)=\frac{3\rho^{4}}{4(\rho^{2}+z^{2})^{5/2}},$$

(2.9)

where $\rho$ is the width in the holographic direction. We can control the interaction between the instantons of this model by introducing the instanton interaction as

$$D_{int}(z)=\frac{3\sqrt{2}\rho^{8}}{4}\frac{h_{1}S_{1}+h_{2}S_{2}}{(a^{2}+b)^{5/2}b^{2}}.$$

(2.10)

This result has been derived by Ref. [13] from ADHM approach, all parameters, $h_{1},S_{1},h_{2},S_{2},a$ and $b$ are defined in the appendix A. We have also introduced $d$ as the overlap parameter

$$d=\frac{\gamma}{\rho}(\frac{6\pi^{4}r}{\lambda^{2}n_{I}}),$$

where $\gamma$ is a deformation parameter. It is interpreted as a distance between two instantons that normalized by twice their spatial width; when $d\rightarrow 0$, then instantons will overlap.
In the above equations, we have parameters $p$ and $r$ that all information about lattice structures are carried by them. For example, cubic, a body-centered cubic and a face-centered cubic crystal we have $(p,r)=(6,1),(8,3\sqrt{3}/4)$ and $(12,\sqrt{2})$ respectively. For non-interacting approximation, we consider $p=0$.
We have checked for other lattice configurations and saw that the results do not depend significantly on the lattice configurations, then we present our results only for cubic lattice $p=12,r=\sqrt{2}$.

The main novelty of this paper is to study the new temperature dependency of the instantons shape. We introduced instantons in our configuration setup. In the shape of instanton, $\rho$ is the width in the holographic direction while $\rho/\gamma$ is the spatial width; as a result $\gamma$ is a deformation parameter that characterizes the deviation of the instanton from SO(4) symmetry. For the single instanton at large t’Hooft coupling, the leading-order expression has been considered for deconfined geometry [12]:

$$(F_{iz})^{2}_{(1)}=\frac{12(\rho/\gamma)^{4}}{\gamma^{2}[x^{2}+(z/\gamma)^{2}+(\rho/\gamma)^{2}]^{4}},$$

(3.1)

where the subscript ”(1)” indicated the single-instanton solution, also $x^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ and $z$ indicates the holographic coordinate where $z\epsilon[-\infty,\infty]$. Moreover, parameters $\rho$ and $\gamma$ that described the shape of the instanton are given by the following expressions,

	$\displaystyle\rho$	$\displaystyle=$	$\displaystyle\frac{\rho_{0}u_{c}^{3/4}}{\sqrt{\lambda}}\left[\frac{f_{T}(u_{c})}{\beta_{T}(u_{c})}\right]^{1/4}\sqrt{\alpha_{T}(u_{c})},$
	$\displaystyle\gamma$	$\displaystyle=$	$\displaystyle\frac{3\gamma_{0}u_{c}^{3/2}}{2}\sqrt{\alpha_{T}(u_{c})}.$		(3.2)

Also

	$\displaystyle\alpha_{T}(u_{c})$	$\displaystyle\equiv$	$\displaystyle 1-\frac{5u_{T}^{3}}{8u_{c}^{3}},$
	$\displaystyle\beta_{T}(u_{c})$	$\displaystyle\equiv$	$\displaystyle 1-\frac{u_{T}^{3}}{8u_{c}^{3}}-\frac{5u_{T}^{6}}{16u_{c}^{6}}.$		(3.3)

At zero temperature we have $f_{T}(u_{c})=\alpha_{T}(u_{c})=\beta_{T}(u_{c})=1$. Also $\rho_{0}$ and $\gamma_{0}$ are free parameters which have been introduced in Refs. [12] and [17]. Our purpose is to investigate how important thermodynamic properties of the given system, such as the phase diagram, change. In Table 1, as well as Ref. [23], we have obtained the physical dimensionful quantities from their dimensionless counterparts $t,\mu,n_{I},M$. We have abbreviated $\lambda_{0}=\frac{\lambda}{4\pi}$ and $M_{KK}$ is Kaluza-Klein mass.

Here, we present the main equations and formulas for computing thermodynamic quantities of the system. In next section we will use these equations to find these quantities by numeric techniques.

• •

  Free energy
  The main quantity is the free energy of the system, one finds it as

  $$\Omega=\int_{u_{c}}^{\infty}du{\cal L},$$

  (3.4)

  which is yields the following expression

  $$\Omega=\int_{u_{c}}^{\infty}du\left(u^{5/2}\eta(u)+\frac{\ell}{2}k-\mu n_{I}\right),$$

  (3.5)

  where

  $$\eta(u)=\sqrt{1+g_{1}}\sqrt{1+g_{2}-\frac{k^{2}}{u^{8}f_{T}}+\frac{(n_{I}Q)^{2}}{u^{5}}},$$

  (3.6)

  where $k$ is an integration constant.
  

  To minimize the free energy, $\Omega$ with respect to free parameters, at given $T$ and $\mu$, we calculate $\frac{\partial\Omega}{\partial k}=\frac{\partial\Omega}{\partial n_{I}}=\frac{\partial\Omega}{\partial u_{c}}=0$. Therefore, one finds the three main stationarity equations as
  

  	$\displaystyle\frac{\ell}{2}$	$\displaystyle=$	$\displaystyle\int_{u_{c}}^{\infty}du\,x_{4}^{\prime}\,,$		(3.7a)
  	$\displaystyle\mu n_{I}$	$\displaystyle=$	$\displaystyle\int_{u_{c}}^{\infty}du\,u^{5/2}\left[\frac{g_{1}\zeta^{-1}+g_{2}\zeta}{2q}\left(q-\frac{d}{3}\frac{\partial q}{\partial d}\right)+\frac{\zeta n_{I}^{2}Q}{u^{5}}\left(Q-\frac{d}{3}\frac{\partial Q}{\partial d}\right)\right]\,,$		(3.7b)
  	$\displaystyle st$	$\displaystyle=$	$\displaystyle 2u_{c}^{7/2}+\int_{u_{c}}^{\infty}du\,u^{5/2}\left\{7-\zeta\left[7(1+g_{2})+2\frac{(n_{I}Q)^{2}}{u^{5}}+\frac{k^{2}}{u^{8}f_{T}}\right]+\right.$		(3.7c)
  			$\displaystyle\left.\frac{g_{1}\zeta^{-1}+g_{2}\zeta}{2q}\left(5q\!-\!\frac{3d}{2}\frac{\partial q}{\partial d}+\frac{\rho}{2}\frac{\partial q}{\partial\rho}\right)-\frac{\zeta n_{I}^{2}Q}{u^{5}}\left(\frac{3d}{2}\frac{\partial Q}{\partial d}-\frac{\rho}{2}\frac{\partial Q}{\partial\rho}\right)\right\}$

  and

  $$Q(u)=\int_{uc}^{u}dvq(v)$$

  (3.8)

• •

  Entropy density
  By switching on the temperature in the system, the dimensionless entropy density is denoted by, $s$, which is entropy of the system. First, we give the derivation of it from the free energy, one knows that $\Omega=\Omega(u_{c},k,n_{I},\rho,\gamma,d)$ then,

  $$s=-\frac{\partial\Omega}{\partial t}=-\frac{\partial\Omega}{\partial u_{c}}\frac{\partial u_{c}}{\partial t}-\frac{\partial\Omega}{\partial k}\frac{\partial k}{\partial t}-\frac{\partial\Omega}{\partial n_{I}}\frac{\partial n_{I}}{\partial t}-\frac{\partial\Omega}{\partial\rho}\frac{\partial\rho}{\partial t}-\frac{\partial\Omega}{\partial\gamma}\frac{\partial\gamma}{\partial t}-\frac{\partial\Omega}{\partial d}\frac{\partial d}{\partial t}-\frac{\partial\Omega}{\partial t}.$$

  (3.9)

  The first three terms are zero because of stationary point, also we have,

  $$\frac{\partial\Omega}{\partial\rho}=\int_{u_{c}}^{\infty}du\,u^{5/2}\left(\frac{1}{2q}(g_{1}\zeta^{-1}+g_{2}\zeta)\frac{\partial q}{\partial\rho}+\frac{n_{I}^{2}}{u^{5}}Q\frac{\partial Q}{\partial\rho}\zeta\right),$$

  (3.10)

  $$\frac{\partial\Omega}{\partial\gamma}=\int_{u_{c}}^{\infty}du\,u^{5/2}\left(\frac{1}{2q}(g_{1}\xi^{-1}+g_{2}\zeta)\frac{\partial q}{\partial\gamma}+\frac{n_{I}^{2}}{u^{5}}Q\frac{\partial Q}{\partial\gamma}\zeta+\frac{1}{2}(\frac{g_{1}\zeta^{-1}+g_{2}\zeta}{\gamma})\right),$$

  (3.11)

  and

  $$\frac{\partial\Omega}{\partial d}=\int_{u_{c}}^{\infty}du\,u^{5/2}\left(\frac{1}{2q}(g_{1}\zeta^{-1}+g_{2}\zeta)\frac{\partial q}{\partial d}+\frac{n_{I}^{2}}{u^{5}}Q\frac{\partial Q}{\partial d}\zeta\right).$$

  (3.12)

  Moreover,

  $$\frac{\partial\Omega}{\partial t}=\frac{3u_{T}^{3}}{t}\int_{u_{c}}^{\infty}\frac{du}{\sqrt{u}f_{T}}\left(g_{1}\zeta^{-1}+\frac{k^{2}}{u^{8}f_{T}}\zeta\right).$$

  (3.13)

  Finally, one can obtain,

  $\displaystyle s$

  $\displaystyle=\int_{u_{c}}^{\infty}du\left[\frac{3u_{T}^{3}}{t}\frac{1}{\sqrt{u}f_{T}}(g_{1}\zeta^{-1}+\frac{k^{2}}{u^{8}f_{T}}\zeta)+u^{5/2}\mathcal{X}\right],$

  (3.14)

  where we defined

  	$\displaystyle\mathcal{X}$	$\displaystyle\equiv$	$\displaystyle\left(\frac{1}{2q}(g_{1}\zeta^{-1}+g_{2}\zeta)\frac{dq}{d\rho}+\frac{n_{I}^{2}}{u^{5}}Q\frac{dQ}{d\rho}\zeta\right)\frac{d\rho}{dt}$		(3.15)
  		$\displaystyle+$	$\displaystyle\frac{1}{2}(\frac{g_{1}\xi^{-1}+g_{2}\zeta}{\gamma})\frac{d\gamma}{dt}+(\frac{g_{1}\zeta^{-1}+g_{2}\zeta}{2q}\frac{dq}{dd}+\frac{n_{I}^{2}}{u^{5}}Q\frac{dQ}{dd}\xi)\frac{dd}{dt},$

  and

  $$\zeta=\frac{\sqrt{1+g_{1}}}{\sqrt{1+g_{2}-\frac{k^{2}}{u^{8}f_{T}}+\frac{(n_{I}Q)^{2}}{u^{5}}}}.$$

  (3.16)

  

  We use (3.16) for deriving our results with or without the interaction between the instantons, except for calculating the thermal baryon mass.

• •

  Baryon density
  Because of numeric approach for solving the main equations, we should check our results carefully. For this reason we calculate the number density of instantons or baryon density, $n_{I}$ from the following equation

  $$n_{I}=-\frac{\partial\Omega_{baryon}}{\partial\mu},$$

  (3.17)

  where $\Omega$ is the baryonic free energy and $\mu$ is chemical potential. We do this calculation one time to make sure that our results match together, in Fig. 1, we show results for small temperatures $t\simeq 0.05,0.07,0.08,0.1$ and large temperature $t\simeq 0.141,0.143,0.145,0.147$ and choice of parameters $\rho_{0}=4.3497$ and $\gamma_{0}=3.7569$,$\lambda=15.061$. Of course, this diagram can be drawn for other parameters, but, if we change the parameters $\rho$ and$\gamma$ the type of transition will be different, in some regions we will have a second-order phase transition. For example, at the temperature of $t=0.1$ and $\rho=1.5$ and $\gamma=2$ the phase transition from the meson to the quark is second order, generally for $\rho=1.5$ in the area of $0.8\leq\gamma\leq 3$ at $t=0.1$ we have a second-order phase transition.
  In the lower temperature at large chemical potential, we have a second turning point and then, we have unstable phase. The features that our selected parameters have, by considering $M_{KK}=\frac{3185MeV}{\pi}$ and $\ell=1$, is that with properties of nuclear mater have been fitted such as baryon mass, binding energy, saturation density and incompressibility. As it is clear in the left panel, there is a turning point where shows the first order baryon onset for all temperatures, the calculation that leads to this Fig. 1 was first presented in Ref. [23] for $t=0$ and we recapitulate it for $t\neq 0$, as the figure shows, the geometry of the plot does not change when the temperature is taken into account. Only small amounts are changed. For the small chemical potential for temperatures $t\leq 0.1$, baryonic phase is preferred. But in the right panel, we plotted $n_{I}$ with respect to $\mu$ for large temperatures, for largest temperature, there is not turning point and probably we have second order phase transition. But these solutions are not stable and chirally symmetry phase is preferred.
  

In this paper, we use the Sakai-Sugimoto holographic model. In this model, baryons are described by the instantons in bulk. Shape of instantons are defined by parameters $\rho$ and $\frac{\rho}{\gamma}$, $\rho$ is the width in the holographic directions and $\frac{\rho}{\gamma}$ is spatial width. These parameters are expressed to upset the symmetry $SO(4)$. In this paper, what has been done is that the parameters $\rho$ and $\gamma$ are considered temperature-dependent. The temperature dependence is determined by the relations (3). This dependence causes let sentences be added to the entropy quantity, and let the principal equations give us other solutions under the influence of this dependence. According to these solutions, we study the thermodynamics of some properties such as the phase diagram, the quantity of the connection between the baryon and quark phase, and the speed of sound. It is observed that the effect of temperature on the parameters causes a change in the critical chemical potential from baryonic to quark state. It slightly changes the speed of sound in small chemical potentials. At some point, the speed of sound is not dependent on the chemical potential. Also, consider the temperature, there is still a connection between the baryon phase and the quark. In the references section, we also mentioned that the mass of baryons at different temperatures has different values. Indeed, we don’t check the chiral phase transition in the small ’t Hooft coupling, so it will be interesting issue for the future work.

We would like to thank Andreas Schmitt and K. Bitaghsir Fadafan for valuable comments and discussion.