EQCD is the three-dimensional effective field theory obtained from hot QCD by integrating out the hard Matsubara modes (the non-static modes with masses \(\sim 2\pi T\)). It is the first stage of dimensional reduction; a second stage of integrating out its adjoint scalar gives MQCD.

EQCD describes the static (\(n=0\)) bosonic modes of the gauge field in three spatial dimensions:

• the 3D SU(3) magnetic gauge field \(A_i(\vec x)\), \(i=1,2,3\);
• an adjoint scalar \(A_0(\vec x)\) — the static time component of the gluon field, which in 3D looks like a Higgs-type field in the adjoint representation.

\[ \mathcal{L}_{EQCD}=\tfrac12\,\mathrm{Tr}\,F_{ij}^2 +\mathrm{Tr}\,[D_i,A_0]^2 +m_E^2\,\mathrm{Tr}\,A_0^2 +\lambda_E\big(\mathrm{Tr}\,A_0^2\big)^2, \qquad F_{ij}=\partial_i A_j-\partial_j A_i - i g_E[A_i,A_j]. \]

Matching Green's functions to full QCD fixes the couplings; at leading order,

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

• \(g_E^2\) is the 3D gauge coupling (mass dimension 1); it is set by the 4D running coupling times \(T\).
• \(m_E\) is the Debye mass, the electric screening mass of \(A_0\), of order \(gT\).
• \(\lambda_E\) is the adjoint self-coupling, of order \(g^4T\), and is subleading.

EQCD captures the electric (soft, \(\sim gT\)) physics: Debye screening, the pressure to high orders, and screening masses. Its temporal component \(A_0\) is massive and, on the magnetic scale \(g^2T\), is integrated out to yield MQCD, the theory that controls the spatial string tension. See Why spatial string tension and dimensional reduction.