An effective field theory (EFT) describes physics at a chosen energy/length scale by keeping only the relevant low-energy fields and integrating out ("matching away") the heavier degrees of freedom. The heavy physics survives only through the values of a finite set of couplings in the EFT. Dimensional reduction of hot QCD into EQCD and MQCD is a textbook example.

If a theory has a large scale \(M\) and one is interested in energies \(E\ll M\), the effects of the heavy modes can be expanded in powers of \(E/M\):

\[ \mathcal{L}_{\text{eff}} = \sum_i c_i(M)\,\mathcal{O}_i, \]

a sum of local operators \(\mathcal{O}_i\) built from the light fields, with coefficients \(c_i\) fixed by matching the two theories' predictions for the same low-energy observable. Higher-dimension operators are suppressed by powers of \(1/M\), so a few terms suffice to a given accuracy.

• Matching: compute an observable in both the full and effective theories at the boundary scale and equate them to fix the \(c_i\).
• Running: renormalization-group evolution resums large logarithms between the scales.

The thermal scales separate as \(2\pi T\gg gT\gg g^2T\), giving a tower of EFTs:

\[ \text{QCD}_{4D}\ \xrightarrow{\ \text{integrate out }2\pi T\ }\ \text{[[EQCD]]}\ \xrightarrow{\ \text{integrate out }gT\ }\ \text{[[MQCD]]} . \]

Matching fixes \(g_E^2=g^2(T)T\), the Debye mass \(m_E\), and finally \(g_M^2\simeq g^2(T)T\). The spatial string tension is then a pure prediction of the last EFT, MQCD. This EFT logic is what makes dimensional reduction quantitative and testable — see Why spatial string tension and dimensional reduction.