Anomalous and linear holographic hard wall models for glueballs and the pomeron

In this work we propose improved holographic hard wall (HW) models by the inclusion of anomalous dimensions in the dual operators that describe glueballs inspired by the AdS/CFT correspondence. The anomalous dimensions come from well known semiclassical gauge/string duality analysis showing a dependence with the logarithm of spin $S$ of the boundary states. We show that these logarithm anomalous dimensions of the high spin operators combined with the usual HW model allow us to match the pomeron trajectory and give glueball masses that are better than that of the original HW and soft wall models in comparison with lattice data. We also build up other anomalous HW models considering that the logarithm anomalous dimensions can be approximated by a truncated series of odd powers of the difference $\sqrt{S}-1/\sqrt{S}$. These models also fit the pomeron trajectory and produce good glueball masses. Then, we consider an anomalous dimension that is proportional to $\sqrt{S}$, providing reasonable results. Finally, we propose an asymptotic linear anomalous HW model that effective dimensions for high spins operators are of the form $\mathrm{\Delta }=a\sqrt{S}+b$, where $a$ and $b$ are constants to be fixed by comparison with the soft pomeron trajectory. In this last model, the Regge trajectory is asymptotically linear even for very high spins ($J\sim 100$) matching the soft pomeron trajectory accurately and generates glueball masses with deviations with respect to the lattice data better than the original HW and soft wall models.

Figure 1

Plot of $J\times M^2$ with masses expressed in GeV and $J=2$ to 10 in the AHWlog model with anomalous dimensions given by Eq. (46). The dots represent the glueball masses shown in the second column of Table 3 with the corresponding error bars. The straight line corresponds to a linear fit of the soft pomeron trajectory $J=1.08\pm 0.21+(0.25\pm 0.01)M^2$ with ${\chi }^2/\mathrm{ndf}=3.76/3=1.25$.

Figure 2

Plot of $J\times M^2$ with masses expressed in GeV and $J=2$ to 10 in the anomalous model ${\mathrm{HW}}_{\log 2}$, Eq. (47). The dots represent the glueball masses shown in the fifth column of Table 3 with the corresponding error bars. The straight line corresponds to a linear fit matching the soft pomeron trajectory $J=1.08\pm 0.36+(0.25\pm 0.02)M^2$ with ${\chi }^2/\mathrm{ndf}=15.2/3=5.1$.

Figure 3

Plot of $J\times M^2$ with masses $M$ expressed in GeV and $J=2$ to 10, using the anomalous HW model (54) with conformal dimension Eq. (53), truncated at $N=0$ with coefficients $a=1.38$ and $b=2.67$. The dots with error bars represent the glueball masses shown in Table 4 for $N=0$, and the straight line corresponds to a linear fit given by $J=1.08\pm 0.28+(0.25\pm 0.01)M^2$ with ${\chi }^2/\mathrm{ndf}=5.21/3=1.74$.

Figure 4

Plot of $J\times M^2$ with $M$ expressed in GeV and $J=2$ to 10, using the approximate anomalous dimension Eq. (53), truncated at $N=1$ with $a=2.86$, $b=1.00$. The dots with error bars represent the glueball masses shown in Table 4 for $N=1$, and the straight line corresponds to a linear fit given by $J=1.08\pm 0.04+(0.250\pm 0.002)M^2$ with ${\chi }^2/\mathrm{ndf}=0.312/3=0.104$.

Figure 5

Regge trajectory $J\times M^2$ for the anomalous linear HW model defined by Eq. (58), with $S_0=0$ and $0\le S\le 100$. The dots with error bars represent the glueball masses obtained from this equation and their values corresponding to states from $J=2$ to $J=10$ are shown in Table 6. The straight line is the corresponding linear fit in the range from $J=2$ to $J=100$, matching the soft pomeron trajectory with a very precise angular coefficient, $J=1.08\pm 0.01+(0.25144\pm 0.00006)M^2$ with ${\chi }^2/\mathrm{ndf}=2.58/48=0.054$.