Subtleties of non-Abelian D-brane actions and their effect on holographic heavy-light meson spectra

TL;DR

The paper revisits holographic heavy–light mesons in the D3–D7 framework at zero temperature, focusing on non-Abelian flavor dynamics via the Myers non-Abelian DBI action. By enforcing Hermiticity of the induced metric and performing a careful mixed-determinant expansion, the authors derive modified fluctuation equations that yield a scalar–vector mass splitting: scalar HL modes become heavier while transverse vector HL modes become lighter, removing the degeneracy found in previous analyses. At finite 't Hooft coupling, the vector HL spectrum exhibits a qualitatively different dependence on the heavy-to-light quark mass ratio, and the heavy-light vector ground state can even decrease with increasing heavy mass, with large-λ scaling M^2_meson ∝ m_light^4 / m_heavy^2. The work provides a robust, broadly applicable framework for incorporating non-Abelian flavor dynamics into holographic models and establishes criteria for connecting to or extending beyond the vacuum, finite-temperature, and deformed backgrounds.

Abstract

We revisit the holographic description of heavy light mesons in the D3-D7 system at zero temperature, analyzing the dynamics of the coupled probe D7 branes through the non-Abelian Dirac-Born-Infeld action. Distinct quark masses are realized by separating the flavor branes, producing holographic flavor hierarchies. We refine the calculation made in previous works: we impose Hermiticity on the induced metric and fix the expansion of the determinant for matrix valued fields. Implementing these improvements yields modified fluctuation equations and quantitatively different meson spectra: the scalar modes become heavier while the vector modes become lighter, removing the degeneracy reported in the literature. At finite 't Hooft coupling, we also observe a qualitatively different dependence of the vector modes on the quark masses. The resulting prescription provides a consistent, broadly applicable framework for incorporating non-Abelian flavor dynamics into holographic models and can be readily extended to situations away from the vacuum.

Figures (9)

• Figure 1: D7-branes' embedding in the $(\rho,X^8)$-subspace. They are placed at $X^9=0$ and their position along the $X^8$ direction, $m_\text{heavy}$ and $m_\text{light}$, is constant in $\rho$. Strings stretching between the same D7-branes correspond to light-light (LL) and heavy-heavy (HH) mesonic excitations. String stretching between flavor branes placed at different positions in the $X^8$ corresponds to the heavy-light (HL) mesons.
• Figure 2: The plots show the scalar (left panel) and transverse vector (right panel) meson spectrum squared (times $\lambda/\pi$), normalized by the average quark mass $\mathbf{\overline m_q}$, as a function of $1/\lambda$. The solid line corresponds to the analytic result \ref{['eq:mesonspectrumlambda']} for $\widetilde{r} = 1$, and $n = 0$ in both plots. The red dots are the numerical results for the meson spectrum.
• Figure 3: The plot shows the $n=0$ (blue) and $n=1$ (red) scalar meson mass squared (times $\lambda/(\mathbf{m}^2_\mathbf{{q,\text{light}}}\pi)$) as a function of the mass of the heavy quark $\mathbf{m_{q,\text{heavy}}}$ (rescaled by $\mathbf{m_{q,\text{light}}}$), for fixed $\lambda=2^4$ (left panel) and $\lambda=3^4$ (right panel). The blue and red dots correspond respectively to the $n=0$ and $n=1$ heavy-light meson mass computed from \ref{['eq:eqzzz']}, while the blue and red crosses correspond respectively to the $n=0$ and $n=1$ masses obtained in Erdmenger:2007vj.
• Figure 4: The plot shows the mass squared of the lightest transverse vector mode (times $\lambda/( \mathbf{m}^2_\mathbf{{q,\text{light}}}\pi)$) as a function of $\mathbf{m_{q,\text{heavy}}}/\mathbf{m_{q,\text{light}}}$ and several values of the 't Hooft coupling $\lambda = \{2^4,3^4,3^4,4^4,5^4,6^4,7^4\}$. The dots are the numerical results for the meson mass, while the solid curves are interpolations of the data points.
• Figure 5: The plot shows the transverse vector meson spectrum squared (times $\lambda/(\mathbf{m}^2_\mathbf{{q,\text{light}}}\pi)$), for fixed $\lambda=2^4$, as a function of the ratio $\mathbf{m_{q,\text{heavy}}}/\mathbf{m_{q,\text{light}}}$ in the regime where it is very large. The dots correspond to the numerical solutions for $\mathbf{m_{q,\text{heavy}}}/\mathbf{m_{q,\text{light}}}=\{50,75,100,150,200,250,300,400,500\}$, while the blue solid curve is the numerical fit \ref{['eq:massratiofit']}.