Correlation function of flavored fermion in holographic QCD

By using the gauge-gravity duality, we investigate the correlation function of flavored fermion in the ${\mathrm{D}}_p/{\mathrm{D}}_{p+4}$ model as top-down approaches of holographic QCD for $p=4,3$. The bulk spinor, as the source of the flavored fermion in QCD, is identified to the worldvolume fermion on the flavor ${\mathrm{D}}_{p+4}$-branes and the standard form of its action can be therefore obtained by the T-duality rules in string theory. Keeping this in hand, we afterwards generalize the prescription for two-point correlation function in the $\mathrm{AdS}/\mathrm{CFT}$ dictionary into general D-brane backgrounds and apply it to the case of $p=4,3$, i.e., the $\mathrm{D}4/\mathrm{D}8$ and $\mathrm{D}3/\mathrm{D}7$ approaches, respectively. Resultantly, our numerical calculation with the bubble background always displays discrete peaks in the correlation functions which imply the bound states created by the flavored fermions as the confinement in QCD. With the black brane background, the on-shell condition illustrated by the correlation function covers basically the dispersion curves of fermion obtained by the hard thermal loop approximation in the hot medium. Finally, we interpret the flavored fermions in the bubble background as baryons by taking into account a baryon vertex, then find the two-point correlation function is able to fit the lowest baryon spectrum. In this sense, we conclude remarkably that our top-down approach in this work could reveal the fundamental properties of QCD both in the confined and deconfined phases.

Figure 1

The correspondence of the bulk field $\psi$ and boundary operator $\chi$ in the ${\mathrm{D}}_p/{\mathrm{D}}_{p+4}$ system. Here $r$ refers to the radial direction as the holographic direction. $\chi$ is a hadronic field as a gauge-invariant operator produced by multiple fundamental quark fields, or equivalently by multiple $(p,p+4)$ strings.

Figure 2

The configuration of the $\mathrm{D}4/\mathrm{D}8$ model in the bubble (confined) background (a) and black brane (deconfined) background (b) on $r-x^4$ plane. Red line refers to the D8-branes ($\mathrm{D}8$) and anti-D8-branes ($\overline{\mathrm{D}8}$).

Figure 3

The correspondence of the bulk fields ${\psi }_{R,L}$ and flavored fermions ${\chi }_{R,L}$ consisting of chiral quarks in the $\mathrm{D}4/\mathrm{D}8$ model. The $N_f$ pairs of D8- and anti-D8-branes are located at $Z\to \pm \infty$ at boundary providing $U{(N_f)}_R\times U{(N_f)}_L$ chiral symmetry in the dual theory. The bulk field ${\psi }_{R,L}$ as source couples to the chiral quarks in ${\chi }_{R,L}$ respectively as it is in QCD.

Figure 4

Density plot of the confined retarded Green function $G_R^{(1,2)}$ as the spectral function of $1+3$ dimensional QCD from the $\mathrm{D}4/\mathrm{D}8$ model. The white regions refer to the peaks in the Green function and the dashed lines refer to $\omega =\mathrm{k}$ as the light cones. The parameters are chosen as ${\mathrm{\Lambda }}_l=2,l=0,{\mathcal{T}}_c=1$ in the unit of $M_{KK}=1$.

Figure 5

The confined retarded Green function $G_R^{(1,2)}$ as the spectral function of $1+3$ dimensional QCD from the $\mathrm{D}4/\mathrm{D}8$ model. The parameters are chosen as ${\mathrm{\Lambda }}_l=2,\mathrm{k}=0,{\mathcal{T}}_c=1$ in the unit of $M_{KK}=1$.

Figure 6

The imaginary part of confined retarded Green function $G_R^{(1)}$ as the spectral function from the $\mathrm{D}4/\mathrm{D}8$ model with various ${\mathrm{\Lambda }}_l$ for $\mathrm{k}=0$.

Figure 7

The 3d plot of the imaginary part of the Green function $G_R^{(1,2)}$ from the black D4-brane background. The parameters are chosen as ${\mathrm{\Lambda }}_l=2,l=0,{\mathcal{T}}_d=1$ in the unit of $2\pi T=1$.

Figure 8

Density plot of the imaginary part of the Green function $G_R^{(1,2)}$ from the black D4-brane background. The white regions represent the protrusion or pit, i.e., the peaks, in the Green functions which are fitted by green lines. The dashed lines refer to $\omega =\pm \mathrm{k}$ as the light cone and the parameters are chosen as ${\mathrm{\Lambda }}_l=2,l=0,{\mathcal{T}}_d=1$ in the unit of $2\pi T=1$.

Figure 9

Density plot of the confined retarded Green function $G_R^{(1,2)}$ as the spectral function of the $1+2$ dimensional QCD from the $\mathrm{D}3/\mathrm{D}7$ model. The parameters are chosen as $M_{KK}=1,{\mathrm{\Lambda }}_l=2,L=0$. The white regions refer to the peaks in the Green function and dashed lines refer to the light cones.

Figure 10

The confined retarded Green function $G_R^{(1,2)}$ as the spectral function of the $1+2$ dimensional QCD from the $\mathrm{D}3/\mathrm{D}7$ approach. The parameters are set as $M_{KK}=1,{\mathrm{\Lambda }}_l=2,L=0$.

Figure 11

Imaginary part of the confined retarded Green function $G_R^{(1,2)}$ with $L=0$ (solid blue line) and $L=1$ (dashed red line).

Figure 12

The 3d plot of the imaginary part of the Green function $G_R^{(1,2)}$ from the black D3-brane background. The parameters are chosen as ${\mathrm{\Lambda }}_l=2,l=0,\mathcal{T}=1$ in the unit of $2\pi T=1$.

Figure 13

Density plot of the imaginary part of the Green function $G_R^{(1,2)}$ from the black D3-brane background. The white regions represent the protrusion or pit, i.e., the peaks, in the Green functions which is fitted by green lines. The dashed lines refer to $\omega =\pm \mathrm{k}$ as the light cone and the parameters are chosen as ${\mathrm{\Lambda }}_l=2,l=0,\mathcal{T}=1$ in the unit of $2\pi T=1$.

Figure 14

The D-brane configuration of ${\mathrm{D}}_p/{\mathrm{D}}_{p+4}$ system with a baryon vertex. The bulk field $\psi$ is created by $N_c$ $(p,p+4)$ strings as quarks and the dual field $\chi$ at boundary is produced by $N_c$ quark points as a baryon field.