The spatial string tension \(\sigma_s(T)\) is the coefficient of the area law obeyed by a purely spatial Wilson loop at finite temperature. It is the order parameter for magnetic confinement and, unlike the ordinary (temporal) string tension, remains non-zero — and grows — in the deconfined phase.

For a rectangular loop \(C\) of area \(A\) lying entirely in a spatial plane,

\[ \langle W(C)\rangle \sim \exp\!\big(-\sigma_s(T)\,A\big),\qquad \sqrt{\sigma_s(T)} = \lim_{R,S\to\infty}\ \text{(area-law slope of the spatial } R\times S\text{ loop)} . \]

Equivalently \(\sigma_s\) is extracted from the large-distance slope of the spatial static potential \(V_s(R)\to \sigma_s R\).

• The temporal string tension \(\sigma(T)\) (from a loop with one edge along euclidean time) measures the electric flux tube between static quarks. It vanishes above \(T_c\): electric charges are Debye-screened, signalling deconfinement.
• The spatial string tension \(\sigma_s(T)\) measures the magnetic sector. It stays finite for all \(T\) and increases with temperature. There is no magnetic screening at leading order (the Linde infrared problem), so this is intrinsically non-perturbative.

Dimensional reduction predicts that \(\sigma_s\) is set by the 3D magnetic effective theory MQCD:

\[ \sqrt{\sigma_s(T)} = c\,g_M^2 \simeq c\,g^2(T)\,T,\qquad c=0.554(4), \]

with \(g^2(T)\) the running coupling at a thermal scale \(\mu\sim 2\pi T\). Because \(g^2(T)\) falls only logarithmically, \(\sqrt{\sigma_s}\propto g^2(T)T\) rises with \(T\). Cheng et al. (arXiv:0806.3264) verified this form down to \(T\approx1.5\,T_c\). This is the subject of Why spatial string tension and dimensional reduction.