Musings

Designing the 5th Dimension

I’ve just gotten back from a visit to the University of Washington which is, currently, the mecca for applications of AdS/CFT to QCD. I’ve talked a bit about some of the applications to RHIC physics. But there are lots of other interesting recent developments.

I’ll probably talk some more about them in future posts but, for today, I thought I’d summarize a paper by Erlich, Katz, Son & Stephanov.

Rather than trying to find a string background whose near-horizon geometry is holographically dual to QCD, they take a “bottom-up” approach, introducing those fields in AdS₅ which couple to the observables they are interested in. To study the low-lying mesons of QCD, they introduce 5D $SU(N_f)_L\times SU(N_f)_R$ gauge fields, $A_{L,R}^a$ which couple to the corresponding currents, $\overline{q}\gamma_\mu \frac{1}{2}(1\pm\gamma_5)t^a q$, and a 5D scalar, $X^{\overline{\alpha}\beta}$, in the $(\overline{N}_f,N_f)$ representation of $SU(N_f)_R\times SU(N_f)_L$, which couples to the quark bilinear. The geometry is taken to be that of a slice of AdS₅, $$ds^2 = \frac{1}{z^2} (dz^2 + \eta_{\mu\nu} dx^\mu dx^\nu),\qquad 0\lt z\leq z_m$$ The wall at $z=z_m$ represents the effect of confinement. Normally, this is the location where the supergravity approximation breaks down (curvatures diverge, etc.), and we need a stringy completion of the geometry, involving some D-brane. In this crude model, the 4D gauge coupling doesn’t run at all, until we hit the “IR brane,” where we simply impose some boundary condition.

The action is

where $g_5$ is the 5D gauge coupling. The (negative) mass-squared of the field $X$ is determined by the relation $m^2 = \Delta(\Delta-4)$, and the (naïve) scaling dimension, $\Delta=3$, of the quark bilinear. The UV boundary condition on $X$ is given by $$\lim_{z\to 0} \frac{2}{z} X(z) = M$$ where $M$ is the quark mass matrix. The other “branch,” as usual, gives the expectation value of the quark bilinear, $\Sigma^{\overline{\alpha}\beta}=\langle\overline{q}_R^\alpha q_L^\beta\rangle$,

Assuming no explicit flavour symmetry breaking, $M= m_q 𝟙$, the condensate $\Sigma= \sigma 𝟙$. In principle, value of $\sigma$ is determined by solving the classical equations of motion, subject to the boundary conditions for $X$ on the IR brane. Instead of specifying those, they treat $\sigma$ as another free parameter, and solve the equations of motion, using (2) as the UV boundary condition. For the gauge fields, they take the simplest possible boundary conditions, $$F_L(z_m)=F_R(z_m)=0$$ on the IR brane.

All in all, the model has 4 free parameters, $g_5, z_m, m-q,$ and $\sigma$. $g_5$ is fixed by comparing the large-$q^2$ behaviour of the 2-point function of the vector current to that of perturbative QCD, $$\frac{1}{g_5^2} = \frac{N_c}{12\pi^2}$$ The remaining three parameters are fit to data.

Assuming $N_f=2$ and isospin symmetry, some obvious observable to look at are the masses of the lowest-lying mesons, $m_\pi,m_\rho,m_{A_1}$ (respectively, the lowest mass pseudoscalar, vector, and pseudovector mesons), the pion decay constant, $f_\pi$, and the corresponding weak decay constants $F_\rho$ and $F_{A_1}$. Finally, there’s the $\rho$-$\pi$ coupling, $g_{\rho\pi\pi}$. That’s 7 observables, and 3 adjustable parameters.

The model (1) incorporates the basic structure of chiral symmetry-breaking. So, without any dynamical insight, you might expect agreement at the 10-20% level. Remarkably, the best-fit for the 7 data points has an RMS error $$\varepsilon_{RMS} = \left(\textstyle{\frac{1}{n}}\sum_i(\delta O_i/O_i)^2\right)^{1/2}$$ (here, $n=4=$ the number of observables minus the number of input parameters and $\delta O/O$ is the fractional error in each observable) of 4%.

That’s quite a bit better than we would have expected. It seems that, even in this very crude approximation, AdS/CFT captures some nontrivial features of the dynamics of QCD.