Why Spatial String Tension and Dimensional Reduction?

This page motivates and derives the central idea behind Cheng et al., "The Spatial String Tension and Dimensional Reduction in QCD" (arXiv:0806.3264, Phys. Rev. D 78, 034506 (2008)). Two questions are answered:

See also the paper summary at The Spatial String Tension and Dimensional Reduction in QCD..

1. Why the spatial string tension?

A few distinct reasons, roughly in order of how fundamental they are:

  1. It is the cleanest probe of the magnetic sector. In the deconfined phase the electric sector is Debye-screened, so \(A_0\) correlators become perturbatively tractable at long distance. The magnetic gluons are not screened — there is no perturbative mechanism generating a magnetic mass. Spatial Wilson loops are built purely from spatial links, so they couple directly to the magnetic modes at \(p \sim g^2T\) and are essentially blind to the screened electric sector. σₛ is thus a direct handle on the scale that perturbation theory cannot reach.
  2. It demonstrates that the QGP is not a free gas. Spatial Wilson loops retain area-law behaviour, with a linearly rising pseudo-potential at large distance, at temperatures far above \(T_c\) — confinement-like behaviour persisting in the "deconfined" phase. That is a qualitative statement about high-T QCD that no amount of resummed perturbation theory reproduces.
  3. It is the natural quantitative test of dimensional reduction. This is your own use of it. Because σₛ is dominated by the soft magnetic scale, MQCD/3D Yang–Mills predicts \(\sqrt{\sigma_{s}} \propto g^2_E\) (or \(g^2_3\)) with a pure number in front — around 0.55 from independent 3D determinations. Measuring \(\frac{\sqrt{\sigma_s}}{g^2_E}\) in 4D QCD and asking at what temperature it flattens to that number is a sharp, falsifiable test of whether EQCD/MQCD actually describe QCD, and where they start to fail. Unlike most observables, the prediction is a single dimensionless constant, so there is nowhere to hide.
  4. It controls the Linde problem quantitatively. \(\sigma_s \sim (g^2T)^2\) sets the size of the non-perturbative magnetic contribution, which is exactly the term that makes the free energy non-perturbative at O(\(g^6\)). Knowing \(\sigma_s\) tells you how large that irreducible non-perturbative piece is.
  5. It has phenomenological consequences. The magnetic scale feeds into spatial screening masses of mesonic correlators at high T; the large-distance behaviour of those correlators is governed by the same non-perturbative physics, so \(\sigma_s\) propagates into observables people actually compare against thermodynamics and screening-mass data.

One caveat worth keeping in view for your write-up: in 2+1 flavour QCD the linear rise is only an intermediate-distance feature — dynamical quarks screen fundamental charge, so spatial string breaking is expected at \(RT \gg \frac{2\pi}{g^4}\), and neither EQCD nor MQCD contains fundamental charges to describe it. The string tension you extract is therefore defined in the window before breaking sets in, and that window widens with temperature.

At finite temperature the euclidean time direction is compact, \(\tau\in[0,\beta]\) with \(\beta=1/T\). A rectangular Wilson loop can lie in two kinds of plane:

defining the spatial string tension \(\sigma_s(T)\). Crucially \(\sigma_s(T)\) does not vanish above \(T_c\); it even grows with temperature. This "magnetic confinement" is a genuinely non-perturbative feature of the plasma.

So the spatial string tension is interesting precisely because it is the observable that survives deconfinement: it isolates the soft magnetic dynamics of hot QCD, which perturbation theory cannot reach (the Linde infrared problem). Its temperature dependence is a clean, quantitative test of the dimensional reduction picture.

2. Why dimensional reduction?

At high \(T\) the compact Euclidean time forces all fields into Matsubara modes,

\[ \phi(\vec x,\tau)=T\sum_{n}\ e^{i\omega_n\tau}\,\tilde\phi_n(\vec x),\qquad \omega_n=\begin{cases}2\pi n T & \text{[[boson|bosons]]}\\ (2n+1)\pi T & \text{[[fermion|fermions]]}\end{cases} \]

Every non-static mode (\(n\neq0\)) acquires an effective mass \(\ge 2\pi T\), and all fermions are non-static (their lowest frequency is \(\pi T\)). Only the static (\(n=0\)) bosonic modes — the gauge field \(A_\mu(\vec x)\) — remain light. Integrating out the heavy modes leaves a three-dimensional effective theory for the static modes: this is dimensional reduction.

The construction is organized by a hierarchy of scales, valid because \(g\ll1\) at high \(T\) (asymptotic freedom):

\[ \underbrace{2\pi T}_{\text{hard}}\ \gg\ \underbrace{gT}_{\text{soft (electric)}}\ \gg\ \underbrace{g^2 T}_{\text{ultrasoft (magnetic)}} . \]

Each scale is removed in turn, producing a tower of effective theories: full QCD \(\to\) EQCD \(\to\) MQCD.

3. EQCD: integrating out the hard scale \(2\pi T\)

Removing the non-static modes (mass \(\sim 2\pi T\)) gives Electrostatic QCD — a 3D SU(3) gauge theory for the static magnetic field \(A_i\) plus the static time component \(A_0\), which becomes an adjoint scalar:

\[ \mathcal{L}_{EQCD}=\tfrac12\,\mathrm{Tr}\,F_{ij}^2 +\mathrm{Tr}\,[D_i,A_0]^2 +m_E^2\,\mathrm{Tr}\,A_0^2 +\lambda_E\big(\mathrm{Tr}\,A_0^2\big)^2 . \]

The matched parameters, at leading order, are

\[ g_E^2=g^2(T)\,T,\qquad m_E^2=\Big(\tfrac{N_c}{3}+\tfrac{N_f}{6}\Big)g^2(T)\,T^2,\qquad \lambda_E\sim g^4 T . \]

Here \(g_E^2\) (dimension of mass in 3D) is the 3D gauge coupling, and \(m_E\sim gT\) is the Debye mass — the screening mass of the electrostatic gluon \(A_0\). See EQCD.

4. MQCD: integrating out the soft scale \(gT\)

The adjoint scalar \(A_0\) is heavy on the magnetic scale, \(m_E\sim gT\gg g^2 T\), so it too is integrated out. What remains is pure 3D SU(3) Yang–Mills — Magnetostatic QCD:

\[ \mathcal{L}_{MQCD}=\tfrac12\,\mathrm{Tr}\,F_{ij}^2,\qquad g_M^2=g_E^2\big[1+\mathcal{O}(g)\big]=g^2(T)\,T\big[1+\mathcal{O}(g)\big]. \]

MQCD contains only the ultrasoft magnetic gluons; it is a confining 3D gauge theory with a single dimensionful parameter \(g_M^2\). See MQCD.

5. The prediction for \(\sigma_s(T)\)

A spatial Wilson loop involves only the magnetic field \(A_i\), so it is computed entirely within MQCD. Because MQCD is a confining 3D theory, the loop obeys an area law and defines a string tension \(\sigma_M\). In three dimensions \(g_M^2\) carries dimension of mass, so on dimensional grounds

\[ \sqrt{\sigma_M}=c\,g_M^2,\qquad c=\text{pure number}. \]

3D lattice simulations of SU(3) Yang–Mills fix the constant (Karsch–Laermann–Lütgemeier; Teper):

\[ \frac{\sqrt{\sigma_M}}{g_M^2}=c=0.554(4). \]

Identifying \(\sigma_s(T)=\sigma_M\) and using \(g_M^2\simeq g^2(T)\,T\) yields the dimensional-reduction prediction

\[ \boxed{\ \frac{\sqrt{\sigma_s(T)}}{T}=c\,g^2(T)\,\big[1+\mathcal{O}(g)\big]\ },\qquad c=0.554(4). \]

Since the running coupling \(g^2(T)\) decreases only logarithmically (evaluated at a thermal scale \(\mu\sim 2\pi T\)), the combination \(\sqrt{\sigma_s}\sim g^2(T)\,T\) grows with temperature — consistent with the persistence of magnetic confinement above \(T_c\).

6. What Cheng et al. found

Measuring \(\sigma_s(T)\) on the lattice in (2+1)-flavour QCD (physical strange mass, near-physical light masses; \(N_\tau=4,6,8\)) and comparing to the boxed formula with a properly 2-loop-defined \(g^2(T)\), they found that dimensional reduction reproduces the data remarkably well down to \(T\approx 1.5\,T_c\) — much lower than one might expect for a weak-coupling expansion, and even in the presence of light dynamical quarks. The spatial string tension is therefore a striking confirmation that the soft/ultrasoft magnetic sector of the quark-gluon plasma is governed by the 3D effective theories EQCD and MQCD.

References