Quantization Rules in Holographic QCD Models

The Schwarzschild black hole in $AdS_{5}$ is described in the Poincaré patch by the following metric:

$$ds^{2}=\frac{R^{2}}{z^{2}}\left(-f(z)dt^{2}+d\vec{x}\cdot d\vec{x}+\frac{dz^{2}}{f(z)}\right)\,,$$

(1)

where $f(z)=1-\frac{z^{4}}{z_{h}^{4}}$, $z_{h}$ denotes the position of the horizon, and $R$ represents the curvature radius of $AdS_{5}$. The black hole temperature is given by:

$$T=\frac{1}{4\pi}\left|f^{\prime}(z_{h})\right|=\frac{1}{\pi z_{h}}\,.$$

(2)

The bulk action for the massless scalar field $\phi$, which is assumed to be dual to the gauge theory operator $Tr(F^{2})$, is given by

$$S=-\int d^{5}x\sqrt{-g}e^{-\Phi}g^{MN}\partial_{M}\phi\partial_{N}\phi$$

(3)

where $\Phi=c^{2}z^{2}$ represents the dilaton field. The equation of motion derived from the action (3) and the metric (1) is

$$\partial_{\mu}\left(\sqrt{-g}e^{-\Phi}g^{\mu\nu}\partial_{\mu}\phi\right)=0.$$

(4)

Assuming a plane-wave ansatz solution of the form $\phi(\vec{x},z)=e^{-i\omega t}\phi(z)$, the equation of motion (4) reduces to:

$$\phi^{\prime\prime}+\phi^{\prime}\left(\frac{f^{\prime}}{f}-\frac{3}{z}-\Phi^{\prime}\right)+\omega^{2}\phi=0.$$

(5)

where the prime (’) denotes the derivative with respect to $z$. To simplify the calculations, we set $c=1$. Thus, the equation of motion (5) transforms into:

$$\phi^{\prime\prime}+\phi^{\prime}\left(\frac{f^{\prime}}{f}-\frac{3}{z}-2z\right)+\omega^{2}\phi=0.$$

(6)

In order to study the holographic potential in the soft wall model, we need to transform Eq. (5) into a Schrödinger-like equation. By applying the Liouville transformation:

	$\displaystyle\psi$	$\displaystyle=$	$\displaystyle e^{-B/2}\,\phi\,,$		(7)
	$\displaystyle r_{*}$	$\displaystyle=$	$\displaystyle\frac{1}{2}z_{h}\left[-\arctan\left(\frac{z}{z_{h}}\right)+\frac{1}{2}\ln\left(\frac{z_{h}-z}{z_{h}+z}\right)\right]\,,$		(8)

where $B(z)=z^{2}+3\log z$, and $r_{*}$ is the Regge–Wheeler tortoise coordinate in $AdS_{5}$, one obtains a Schrödinger-like equation:

$$\partial_{r_{*}}^{2}\psi+m_{n}^{2}\psi=V\psi\,,$$

(9)

where the potential $V$ is given by:

$$V(r_{*})=e^{B/2}\,\partial_{r_{*}}^{2}e^{-B/2}\,.$$

(10)

The above potential can be explicitly expressed as a function of the coordinate $z$ as follows:

$$V(z)=2+\frac{15}{4z^{2}}+z^{2}-\pi^{4}T^{4}\left(\frac{3}{2}z^{2}+2z^{6}\right)+\pi^{8}T^{8}\left(z^{10}-\frac{9}{4}z^{6}-2z^{8}\right),.$$

(11)

Furthermore, near the boundary and at low temperatures, the function $z(r_{)}$ can be approximated as

$$z=-r_{*}\left[1-\frac{r_{*}^{4}\pi^{4}T^{4}}{5}\right]\,.$$

(12)

Using this approximation, the effective potential (11), expanded up to fourth order in temperature, takes the form:

$$V(r_{*})=2+\frac{15}{4r_{*}^{2}}+r_{*}^{2}-\frac{12}{5}\pi^{4}r_{*}^{6}T^{4}\,.$$

(13)

It is observed that the thermal correction to the effective potential introduces a barrier in the infrared (IR) region of the tortoise coordinate. Meanwhile, the oscillator-like term proportional to $r_{*}^{2}$, the constant term 2, and the infinite barrier near the boundary, represented by the term $15/(4r_{*}^{2})$, remain unchanged from the zero-temperature case.

In Fig. 2, we plot the effective potential as a function of the tortoise coordinate $r_{*}$ for various temperatures. At zero temperature, the potential features only a well. However, at nonzero temperatures, a finite barrier emerges in the IR region. In this scenario, quantum tunneling through the barrier becomes possible. If the tunneling probability is small, the system can be interpreted as a metastable state with a finite lifetime, decaying via barrier penetration. The corresponding energy becomes complex: the real part defines the energy level and governs the usual stationary behavior over time, while the imaginary part determines the decay width and induces damping. Finally, as the temperature increases, the barrier shifts toward the UV region and gradually decreases until it ultimately vanishes.

Near the horizon, the asymptotic solutions of the equation consist of an incoming and an outgoing wave, given by:

$\displaystyle\psi(z)=e^{\mp i\omega r_{*}}\left(a_{0}+a_{1}\left(1-\frac{z}{z_{h}}\right)+a_{2}\left(1-\frac{z}{z_{h}}\right)^{2}+\cdots+a_{n}\left(1-\frac{z}{z_{h}}\right)^{n}\right)\,,$

(14)

where the coefficients $a_{0}$, $a_{1}$, $a_{2}$ and $a_{n}$ are determined by substituting this solution in the equation of motion and imposing the regularity condition at horizon. On the other hand, the Schrödinger-like equation (9) exhibit two particulars solutions near the boundary $z=0$:

	$\displaystyle\psi_{a}(z)=z^{5/2}\left(c_{0}+c_{1}z^{2}+c_{2}z^{4}+\cdots+c_{n}z^{2n}\right)$		(15)
			(16)
	$\displaystyle\psi_{b}(z)=z^{-3/2}(d_{0}+d_{1}z^{2}+d_{2}z^{4}+\cdots+d_{n}z^{2n})+b\,\psi_{a}\ln(c^{2}z^{2}).$		(17)

where the ellipses denote higher powers of $z$, and $d_{n}$, $c_{n}$, and $b$ are parameters that depend on $T$ and $\omega$. The wave function $\psi_{a}$ is normalizable due to its asymptotic behavior, $z^{5/2}$, while the wave function $\psi_{b}$ is non-normalizable, since it behaves asymptotically as $z^{-3/2}$. In the case of quasinormal modes in $AdS_{5}$ space, two boundary conditions are imposed: an ingoing wave near the horizon and a Dirichlet condition at the boundary. Thus, the non-normalizable and outgoing wave solutions are excluded to satisfy the quasinormal mode boundary conditions.

We start by analyzing the quasinormal modes with frequencies $\omega_{n}$ that are trapped near the minimum of the potential, corresponding to the very low-temperature case. As shown in Fig. 3, for a given real part of the frequency $\omega_{R}^{2}$, the horizontal line lies in the lower region of the potential and intersects it at the turning points $r_{0*}$, $r_{1*}$, and $r_{2*}$. There exist two classically allowed regions, II $=[r_{0*},r_{1*}]$, IV $=]r_{2*},r_{h*}[$ and two classically forbidden regions, I $=]0,r_{0*}[$ and III $=[r_{1*},r_{2*}]$.

According to Karnakov and Krainov (2013); Scrucca (2025), the next step involves matching the WKB wave functions across the different regions. However, special care must be taken when performing this matching due to the presence of an infinite barrier near the boundary, $r_{*}\rightarrow 0$, which behaves similarly to a centrifugal term in the potential as encountered in quantum mechanics. This infinite barrier introduces inaccuracies in the spectrum and affects the behavior of the wave function near the origin. Indeed, a closer examination reveals that the WKB approximation fails in the vicinity of the origin.

The first proposal to address the centrifugal problem in quantum mechanics was made by Langer in 1937 Langer (1937), during his WKB analysis of the radial wave equation. Langer observed inaccuracies in the behavior of the WKB wave function near the origin. To address this issue, a coordinate transformation was introduced:

$$r=e^{x},\qquad\psi(r)=\chi(e^{x}),.$$

(18)

In this way, the origin is relocated to $-\infty$, and the matching of the WKB wave function can then be performed. As a consequence, after returning to the original coordinate, the potential is modified and becomes:

$$V_{L}(r)=V(r)+\frac{1}{4r^{2}},.$$

(19)

Hence, the original potential is modified by the addition of the term $1/(4r^{2})$. This term leads to the correct spectrum and ensures the proper behavior of the WKB wave function at the origin. A more detailed discussion of the Langer transformation can be found in Refs. Koike and Silverstone (2009); Karnakov and Krainov (2013); Scrucca (2025). An alternative approach to matching the WKB solutions involves considering the exact solution in the non-quasiclassical region. In our case, for example, the solution can be obtained by analyzing the asymptotic behavior of the potential near the boundary.

Therefore, for our analysis, it is necessary to implement a Langer-type transformation due to the presence of an infinite barrier near the boundary in the holographic potential. We also require that the WKB wave function vanishes at the origin and consists solely of an ingoing wave at the horizon, ensuring the correct boundary conditions for quasinormal modes in $AdS_{5}$ space, as detailed in Appendix B. The WKB wave functions are given by:

$\displaystyle\left\{\begin{array}[]{llll}\Psi(r_{*})=\frac{\sqrt{i}C}{2\sqrt{p_{L}(r_{*})}}\exp\left[\frac{i}{\hbar}\int_{r_{0*}}^{r_{*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}\right]\,\,\,\,\,\,\,r_{*}>r_{0*};\\ \\ \Psi(r_{*})=\frac{C}{\sqrt{p_{L}(r_{*})}}\sin\left[\frac{1}{\hbar}\int_{r}^{r_{1*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}+\frac{\pi}{4}\right]\,\,\,\,\,\,\,r_{1*}<r_{*}<r_{0*};\\ \\ \Psi(r_{*})=\frac{C_{1}}{\sqrt{p_{L}(r)}}\exp\left[-\frac{i}{\hbar}\int_{r_{2*}}^{r_{*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}\right]+\frac{iC_{1}}{2\sqrt{p_{L}(r_{*})}}\exp\left[\frac{i}{\hbar}\int_{r_{2*}}^{r_{*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}\right]\,\,\,\,\,\,\,r_{2*}<r_{*}<r_{1*};\\ \\ \Psi(r_{*})=\frac{C_{1}}{\sqrt{p_{L}(r_{*})}}\exp\left[\frac{i}{\hbar}\int_{r_{*}}^{r_{2*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}\right]\,\,\,\,\,\,\,r_{h*}<r_{*}<r_{2*};\end{array}\right.$

(27)

where $p_{L}=\varepsilon_{n}-V_{L}(r_{*})$ is the momentum of the particle and $r_{*}$ is the tortoise coordinate and the new potential reads

$$V_{L}(r_{*})=V(r_{*})+\frac{1}{4r_{*}^{2}}\,.$$

(28)

with the potential $V(r_{*})$ determined by Eq. (10). Then, by matching the WKB solutions, we arrive at the following quantization condition:

$$\int_{r_{1*}}^{r_{0*}}\sqrt{\varepsilon_{n}-V_{L}(r_{*})}dr_{*}=\pi\left(n+\frac{1}{2}\right)-\frac{i}{4}exp\left[\frac{i}{\hbar}\int_{r_{2*}}^{r_{1*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}\right]\,.$$

(29)

Furthermore, one can expand the square root on the left-hand side of Eq. (29) in powers of the small quantity $\varepsilon_{I}$, and neglect $\varepsilon_{I}$ on the right-hand side of the equation. This reduces the problem to solving two simpler equations:

	$\displaystyle\int_{r_{1*}}^{r_{0*}}\sqrt{\varepsilon_{R}-V_{L}(r_{*})}dr_{*}=\pi\left(n+\frac{1}{2}\right)$		(30)
			(31)
	$\displaystyle\varepsilon_{I}=-\frac{1}{2}\frac{e^{{2i\int_{r_{2*}}^{r_{1*}}\sqrt{\varepsilon_{R}-V_{L}(r_{*})}dr_{*}}}}{\int_{r_{1_{*}}}^{r_{0_{*}}}\frac{1}{\sqrt{\varepsilon_{R}-V_{L}(r_{*})}}dr_{*}}\,.$		(32)

The first of the above equations corresponds to the classical Bohr-Sommerfeld quantization rule for bound states, which fully determines the real part, $\varepsilon_{R}$. The second equation is the well-known Gamow formula, where the imaginary part, $\varepsilon_{I}$, is calculated directly by substituting the previously determined value of $\varepsilon_{R}$.

Equations (30) and (32) provide all the information required to determine the quasinormal modes trapped in the potential minimum region. The procedure begins by substituting the potential expression from Eq. (28) into Eq. (30) and solving it numerically to determine the real part of the frequency. Once the value of $\varepsilon_{R}$ is obtained, it is inserted into the Gamow formula (32), where the imaginary part is calculated by numerically evaluating the integral. Finally, the quasinormal frequency is determined by taking the square root: $\omega_{n}=\sqrt{\varepsilon_{n}}=\sqrt{\varepsilon_{R}-i\varepsilon_{I}}$.

As the temperature increases, the finite barrier in the effective potential begins to decrease. Consequently, the quasinormal modes with frequency $\omega_{n}$, which were previously trapped at the minimum of the effective potential, gradually approach the top of the potential barrier, as illustrated in Fig. 4. The extension of the Bohr–Sommerfeld quantization rule to quasi-stationary states near the top of the potential barrier was developed in Refs. Popov et al. (1968, 1991b, 1991a). A more detailed discussion of this approach can be found in Ref. Karnakov and Krainov (2013).

The WKB wave solutions considered in this regime are given by:

$\displaystyle\left\{\begin{array}[]{llll}\Psi(r_{*})=\frac{\sqrt{i}C}{2\sqrt{p_{L}(r_{*})}}\exp\left[i\int_{r_{1*}}^{r_{0*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}\right]\,\,\,\,\,\,\,r_{*}>r_{0*};\\ \\ \Psi(r_{*})=\frac{C}{\sqrt{p_{L}(r_{*})}}\sin\left[\int_{r_{*}}^{r_{1*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}+\frac{\pi}{4}\right]\,\,\,\,\,\,\,r_{1*}<r_{*}<r_{0*};\\ \\ \Psi(r_{*})=\frac{C_{1}}{\sqrt{p_{L}(r_{*})}}\exp\left[\frac{i}{\hbar}\int_{r_{*}}^{r_{2*}}p_{L}(r_{*}^{\prime})dr_{*}^{\prime}\right]\,\,\,\,\,\,\,r_{h*}<r_{*}<r_{2*};\end{array}\right.$

(44)

These solutions take the same form as those for quasi-stationary states in the under-barrier region. However, since the WKB approximation fails near the top of the potential barrier, the authors of Refs. Popov et al. (1968, 1991b, 1991a) employed the parabolic approximation to solve the Schrödinger equation in the region $r_{2*}<r_{*}<r_{1*}$. Following the same procedure, we obtain the generalized Bohr–Sommerfeld quantization rule Popov et al. (1968, 1991b, 1991a):

$$\int_{r_{0*}}^{r_{1*}}\sqrt{\varepsilon_{n}-V_{L}(r_{*})}dr_{*}=\pi\left(n+\frac{1}{2}\right)-\frac{1}{2}\Xi(\lambda),$$

(45)

where

$$\Xi(\lambda)=\lambda(1-\ln\lambda)+\frac{1}{2i}\ln\left[\frac{\Gamma\left(\frac{1}{2}+i\lambda\right)}{\Gamma\left(\frac{1}{2}-i\lambda\right)\left(1+e^{-2\pi\lambda}\right)}\right],$$

(46)

and

$$\lambda=\frac{1}{\pi}\int_{r_{2*}}^{r_{1*}}\sqrt{V_{L}(r_{*})-\varepsilon_{n}}dr_{*}.$$

(47)

Here, $r_{0*}$, $r_{1*}$, and $r_{2*}$ are the turning points, and $\Gamma$ denotes the gamma function. The computation of quasinormal frequencies involves evaluating these integrals, which, in our case, must be performed numerically. Accordingly, the frequencies are obtained via numerical methods.

In Table 5, we present the results for the ground state computed using both the shooting method and the WKB approximation. Additionally, Table 6 shows the corresponding results for the first three excited states, $n=1$, $n=2$, and $n=3$. As observed, the WKB approximation provides reasonable agreement with the shooting method for the real part of the frequency. However, the imaginary part of the frequency exhibits a lower level of accuracy compared to the results obtained using the shooting method.

Let us point out that the generalized Bohr–Sommerfeld rule (45) reduces to the quantization condition for modes trapped near the potential minimum (29) in the limit $\lambda\rightarrow\infty$. The parameter $\lambda$ serves as an indicator of the quasinormal modes location: large values of $\lambda$ correspond to modes near the potential minimum, while small values indicate proximity to the top of the potential barrier Karnakov and Krainov (2013).

Now, let us examine the case in which the modes are localized above the potential barrier, see Fig. 5. There is a connection between the transition of the modes to the above-barrier region and the behavior of the turning points. At very low temperatures, the turning points $r_{0*}$, $r_{1*}$, and $r_{2*}$ are real. As the temperature increases, the turning point $r_{2*}$ gradually approaches $r_{1*}$ until they collide. The temperature at which this collision occurs corresponds to the point where the quasinormal modes reach the top of the potential barrier. Above this temperature, the modes lie entirely in the above-barrier region. Furthermore, the turning points $r_{1*}$ and $r_{2*}$, which were previously real, become complex.

The localization of the modes above the barrier can be interpreted as a signal of the dissociation of the glueball particle on the gauge theory side. Within the framework of quantum mechanics and atomic physics Karnakov and Krainov (2013), it has been emphasized that above-barrier resonance states exhibit large widths, often exceeding the spacing between neighboring states. Consequently, these states lack distinct individual properties. Hence, it is reasonable to interpret this behavior as an indication of particle dissociation in the dual field theory.

To illustrate this behavior, we plot the spectral function at different temperatures in Fig. 6. The figure shows that the peak of the spectral function¹¹1The spectral function of the scalar field is derived in Appendix C. decreases significantly as the modes move into the above-barrier region. Specifically, the temperatures at which the modes reach the top of the potential barrier are approximately 76.9 MeV for the ground state, 57.7 MeV for the first excitation, 48.5 MeV for the second excitation, and 42.7 MeV for the third excitation.

Although the turning points $r_{1*}$ and $r_{2*}$ shift into the complex plane, the WKB approximation remains applicable as $\lambda$ increases, indicating that the quasi-stationary states move away from the vicinity of the top of the barrier. This extension has been thoroughly studied in Refs. Mur et al. (1993a, b). The generalized Bohr–Sommerfeld quantization rule (45) for quasinormal modes above the potential barrier can be obtained through analytical continuation, implemented via the formal substitution Mur et al. (1993b); Karnakov and Krainov (2013):

$$\lambda\rightarrow\lambda e^{-2\pi i},$$

(48)

applied to the quantization rule (45). The limit $\left|\lambda\right|\gg 1$ is considered to ensure the validity of the WKB approximation. Consequently, we do not consider solutions located very close to the top of the barrier. Under this assumption, Eq. (45) takes the following form Mur et al. (1993a); Karnakov and Krainov (2013):

$$\oint_{C´}\sqrt{\varepsilon_{n}-V_{L}(r_{*})}dr_{*}=\int_{r_{2*}}^{r_{0*}}\sqrt{\varepsilon_{n}-V_{L}(r_{*})}dr_{*}=\pi\left(n+\frac{1}{2}\right)\,.$$

(49)

where the integration contour $C´$ encloses the turning points $r_{0*}$ and $r_{2*}$. In the context of quantum mechanics, this formula was applied in Ref. Alvarez (1988) to compute the resonances of the anharmonic oscillator.

It can be shown that WKB corrections can still be incorporated to achieve more accurate frequency calculations Mur et al. (1993a, b). Here, we take into account the first-order correction. Thus, Eq. (49) can be rewritten as:

$$\int_{r_{2*}}^{r_{0*}}\sqrt{\varepsilon_{n}-V_{L}(r_{*})}dr_{*}=\pi\left(n+\frac{1}{2}\right)+I_{1}\,,$$

(50)

where

$\displaystyle I_{1}=\frac{d^{2}}{d\varepsilon_{n}^{2}}\int_{r_{2*}}^{r_{0*}}\frac{4\left(\varepsilon_{n}-V_{L}\right)^{2}+\left(2V_{T}-2V_{L}+r_{*}V_{T}^{\prime}\right)\left[12\left(\varepsilon_{n}-V_{L}\right)+\left(2V_{T}-2V_{L}+r_{*}V_{T}^{\prime}\right)\right]}{24r^{2}_{*}\sqrt{\varepsilon_{n}-V_{L}}}dr_{*}\,.$

(51)