Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

August 7, 2006

Chiral Symmetry Breaking


Because of bug 70132, the text in the SVG figure below (and in most of the other SVG figures on this blog) is spooged in Mozilla’s native SVG renderer. The only current workaround is to disable the native renderer in about:config and use the Adobe plugin instead.


Bah. I give up. Fonts converted to outlines.

There was a beautiful talk on Friday by David Sahakyan about his work with Parnachev on the Sakai-Sugimoto model (and the contemporaneous, but somewhat more extensive work by Aharony, Sonnenschein and Yankielowicz). This is the most QCD-like of all the AdS/CFT backgrounds (in a certain limit, it is QCD with massless quarks) and it exhibits both confinement and U(N f)×U(N f)U(N f) diagU(N_f)\times U(N_f)\to U(N_f)_{\text{diag}} chiral symmetry breaking.So I thought it would be nice to give a little review.

Let’s start with a stack of N c1N_c \gg 1 D4-branes, wrapped on a circle (x 4x 4+2πRx_4\sim x_4+2\pi R). The theory on the worldvolume of the D4 branes is a 5-dimensional supersymmetric U(N c)U(N_c) gauge theory, whose 't Hooft coupling is λ 5=g 5 2N c=(2π) 2g s sN c \lambda_5 = g_5^2 N_c = (2\pi)^2 g_s \ell_s N_c The UV completion of this theory is not 5-dimensional, but 6-dimensional, the A N c1A_{N_c-1} (2,0) superconformal theory in D=6D=6. At length scales much longer than RR, the theory is effectively 4-dimensional. With supersymmetry-preserving boundary condition on the S 1S^1, it would reduce to N=4N=4 SYM. Instead, we will choose the bounding spin structure for the fermions on the S 1S^1, which breaks supersymmetry. The 4D fermions pick up masses of order 1/R1/R, and the scalars in the adjoint of SU(N c)SU(N_c) pick up 1-loop mass-squareds of order λ 416π 21R 2 \frac{\lambda_4}{16\pi^2} \frac{1}{R^2} where λ 4=λ 52πR\lambda_4=\tfrac{\lambda_5}{2\pi R} is the 4D 't Hooft coupling. The six neutral scalars (corresponding to the center-of-mass postion of the D4-branes in x 5,x 9x_5,\dots x_9 and the U(1)U(1) Wilson line on the circle) remain massless, but decouple.

This theory reduces to pure Yang-Mills in 4D. In the limit where λ 41\lambda_4 \ll 1 the coupling is weak at the cutoff scale, 1/R1/R, and the theory confines at a scale exponentially smaller than the cutoff. The supergravity approximation is valid in the opposite limit, where λ 5/R1\lambda_5/R \gg 1.

The near-horizon geometry is

ds 2=(uu 0) 3/2(dt 2+dxdx+f(u)dx 4 2)+(u 0u) 3/2(du 2f(u)+u 2dΩ 4 2) {\mathrm{d} s}^2 = \left(\tfrac{u}{u_0}\right)^{3/2}(-{\mathrm{d} t}^2+ \mathrm{d}\vec{x}\cdot\mathrm{d}\vec{x} +f(u) {\mathrm{d} x_4}^2 ) +\left(\tfrac{u_0}{u}\right)^{3/2}\left(\tfrac{{\mathrm{d} u}^2}{f(u)}+ u^2 {\mathrm{d} \Omega_4}^2\right)

where u 0 =(πg sN c) 1/3 s, u Λ=(23R) 2u 0 3, f(u) =1(u Λu) 3, G (4) =2πN cd 4Ω 4V 4, e ϕ=g s(uu 0) 3/4 \begin{aligned} u_0 &= (\pi g_s N_c)^{1/3} \ell_s,\quad &u_\Lambda = \left(\tfrac{2}{3 R}\right)^2 u_0^3, \\ f(u) &= 1-\left(\tfrac{u_\Lambda}{u}\right)^3,\\ G^{(4)} &= 2\pi N_c \tfrac{\mathrm{d}^4\Omega_4}{V_4},\quad & e^\phi = g_s \left(\tfrac{u}{u_0}\right)^{3/4} \end{aligned} which is weakly-curved1 at u=u Λu=u_\Lambda for λ 5/R1\lambda_5/R \gg 1.

For large λ 5/R\lambda_5/R, the gravity model undergoes a Hawking-Page transition, which is the gravitational dual of the confinement/deconfinement transition, at a temperature T=12πRT= \tfrac{1}{2\pi R}. Wick rotate tiτt\to i\tau, and periodically identify ττ+1/T\tau\sim \tau+ 1/T. In the low-temperature phase, the geometry (1), in which the x 4x_4-circle shrinks to zero size at u=u Λu=u_\Lambda, dominates. In the high temperature phase, it’s the geometry where the τ\tau-circle shrinks to zero size at u T=(4πT3) 2u 0 3 u_T = \left(\tfrac{4\pi T}{3}\right)^2 u_0^3 which dominates. When T=12πRT=\tfrac{1}{2\pi R}, the two circles have the same radius, and there’s an obvious symmetry under x 4τx_4\leftrightarrow \tau. But the two geometries are not continuously connected, and the transition is 1st order.

Now, following Sakai and Sugimoto, we add N fN cN_f\ll N_c D8-branes at x 4=0x_4=0 and N fN_f D8¯\overline{\text{D8}}-branes at x 4=Lx_4=L, for 0<LπR0\lt L\leq \pi R. This configuration is, of course, unstable (because of the mutual attraction of the D8- and D8¯\overline{\text{D8}}-branes). But that instability can be systematically suppressed by taking g s0g_s\to 0 and g sN f0g_s N_f\to 0 while holding g sN cg_s N_c fixed. In this “probe” limit, the introduction of the D8 branes does not affect the geometry (1). Nor does it affect the “bulk” thermodynamics (the number of degrees of freedom of the gauge theory scales like N c 2N cN fN_c^2 \gg N_c N_f.

Quantizing the 4-8 strings, the ground state of the NS sector is massive, while the ground state of the R sector is comprised of massless fermions in the (N f,N c)(N_f,N_c) representation of U(N f)×U(N c)U(N_f)\times U(N_c), “quarks”. In the probe limit, the gauge coupling on the D8-branes vanishes, and the theory has a U(N f)×U(N f)U(N_f)\times U(N_f) global symmetry.

Now let’s look at the near-horizon geometry, dual to the low-energy gauge theory. In the low-temperature phase (1), the x 4x_4 circle shrinks to zero size at u=u Λu=u_\Lambda. At that point, the D8- and D8¯\overline{\text{D8}}-branes merge smoothly. Rather than separate stacks of D8- and D8¯\overline{\text{D8}}-branes, there is only a single stack which starts out, asymptotically on the cigar, as a stack of D8-branes located at x 4=0x_4=0. It curves around at the tip of the cigar, and emerges asymptotically as a stack of D8¯\overline{\text{D8}}-branes located at x 4=Lx_4=L. The would-be U(N f)×U(N f)U(N_f)\times U(N_f) symmetry is broken to U(N f) diagU(N_f)_{\text{diag}}.

Correspondingly, among the mesons of the theory (strings stretched between the D8- and D8¯\overline{\text{D8}}-branes) are a set of massless Goldstone bosons, localized near the tip of the cigar. The pion decay constant, as found by Sakai and Sugimoto, is f π 2=127π 4λ 4N cR 2 f_\pi^2 = \frac{1}{27\pi^4}\frac{\lambda_4 N_c}{R^2}

In the high temperature (deconfined) phase, the x 4x_4 circle is not capped off. Instead it’s the τ\tau circle that is capped off at u=u Tu=u_T. So there exists a solution in which the one has separate stacks of D8- and D8¯\overline{\text{D8}}-branes at x 4=0,Lx_4=0,L. But there’s also a solution where the two stacks bend around and rejoin at some u=u min(T)u=u_{\text{min}}(T). In the former solution, chiral symmetry is unbroken; in the latter, it is broken to U(N f)U(N_f), just as in the confining phase.

Which solution is favoured is an energetic calculation, comparing the DBI action for the two solutions. The result is that the transition temperature is T χ=0.154/LT_\chi = 0.154/L. This, of course, hold only so long as T χ>T dT_\chi \gt T_d. T d =12πR T χ ={0.154/L T χ>T d(or, equivalently,L/R<0.97) T d otherwise \begin{aligned} T_d &= \tfrac{1}{2\pi R}\\ T_\chi &= \begin{cases}0.154/L& T_\chi\gt T_d\, (\text{or, equivalently,}\, L/R\lt 0.97)\\ T_d& \text{otherwise}\end{cases} \end{aligned}

Phase diagram of Aharony et al
Phase diagram of the Sakai-Sugimoto model.

So the phase diagram found by Aharony et al looks like the figure at left. All of the phase transitions are first order. In the pale blue region below TR=1/2πTR=1/2\pi, the theory confines and chiral symmetry is broken. In the pink region, the theory is deconfined but, nonetheless, chiral symmetry is broken. Above all the lines, chiral symmetry is unbroken and the theory is deconfined.

Physically, the existence of a phase with chiral symmetry-breaking but without confinement seems rather exotic. It’s a feature of the large λ 4\lambda_4 limit that deserves to be better-understood. Also, the x 4τx_4\leftrightarrow\tau duality at T=12πRT=\tfrac{1}{2\pi R} persists as we lower λ 4\lambda_4 (though the supergravity approximation breaks down). However, for small λ 4\lambda_4, we expect the deconfinement transition at a temperature T d12πRT_d \ll \tfrac{1}{2\pi R}. What happens at T=12πRT=\tfrac{1}{2\pi R}? Is there a phase transition? Finally: before adding the probe D8-branes, the theory had a UV completion as a 6 dimensional quantum field theory. What about after adding the D8-branes? Is there a UV completion, besides the full 10 string theory?

1 Note that, in the limit we are taking (N cN_c\to \infty, λ 5\lambda_5 fixed), the position, at which the dilaton becomes large, scales like uN c 4/3u\sim N_c^{4/3}\to\infty.

Posted by distler at August 7, 2006 1:43 AM

0 Comments & 0 Trackbacks

Post a New Comment