Baryons from Supergravity

TL;DR

This work provides a concrete holographic realization of baryons in the gauge/gravity duality by modeling a baryon vertex as a D5-brane wrapped on $S^5$ with $N$ fundamental strings. It delivers a quantitative analysis of baryon energies across ${\cal N}=4$ SYM, non-SUSY YM in three and four dimensions, and finite-temperature settings, revealing a strong-coupling $E_B \propto 1/L$ behavior for small separations and a temperature-driven screening at large separations, along with a linear confinement-like energy in non-conformal backgrounds. A key new result is a bound on the number of quarks $k$ in a stable baryon for ${\cal N}=4$ theories, namely $\frac{5N}{8} \le k \le N$, and the finding that, in non-SUSY theories, finite-energy $k<N$ baryons do not exist. Together, these findings illuminate how baryon structure and confinement vs. conformal behavior emerge in strongly coupled gauge theories from their stringy holographic duals, with concrete predictions for baryon spectra and temperature effects.

Abstract

We study the construction of baryons via supergravity along the line suggested recently by Witten and by Gross and Ooguri. We calculate the energy of the baryon as a function of its size. As expected the energy is linear with N. For the non-supersymmetric theories (in three and four dimensions) we find a linear relation which is an indication of confinement. For the {\cal N} = 4 theory we obtain the result (E L= - {const.}) which is compatible with conformal invariance. Surprisingly, our calculation suggests that there is a bound state of k quarks if N\geq k\geq 5N/8. We study the {\cal N} = 4 theory also at finite temperature and find the zero temperature behavior for small size of the baryon, and screening behaviour for baryon, whose size is large compared to the thermal wavelength.

Figures (4)

• Figure 1: The Baryon Vertex
• Figure 2: The two ways to obtains $N$ fermionic quarks at the boundary. At small distances (compared to the wavelength of the temperature) the lowest energy configuration is the one with the D5-brane above the horizon. In this configuration there is a potential between the quarks and the vertex. At large distances the lowest energy configuration is the one with the D5-brane located at the horizon. Since $g_{tt}$ vanishes at the horizon the contribution to the energy from the horizontal parts of the strings (as well as from the D5-brane) vanishes. Hence, the energy does not depend on the size of the baryon. In other words the energy of this configuration is simply $N$ times the energy of a string stretched between $U=\infty$ and the horizon which explains the subtraction made at eq.(\ref{['ui']}).
• Figure 3: The baryon in non supersymmetric theories
• Figure 4: The $k < N$ "baryon" vs. $k$ free quarks. Since the longitudinal metric vanishes at $U=0$ the the surface $U=0$ is in fact a point and hence the vertex is smeared along this "surface" $U=0$. As a result the string can move freely at the boundary.