σ_QCD

σ_QCD — QCD string tension

The string tension $\sigma_{\rm QCD}$ characterises confinement: it is the energy per length of a gluonic flux tube. Measured at $\sim 0.19$ GeV² (lattice, hadron spectroscopy).

$$ \sigma_{\rm QCD} = \Lambda_{\rm QCD}^2, $$

where $\Lambda_{\rm QCD}$ is the scale-invariance breaking scale.

PT computation

In PT, $\Lambda_{\rm QCD}$ comes from $\alpha_s$ running which vanishes at the fixed point:

$$ \Lambda_{\rm QCD} = m_Z \cdot \exp\!\left(-\frac{2\pi}{b_0\,\alpha_s(m_Z)}\right). $$

With $\alpha_s(m_Z) = 0.11806$ (ID 2), $b_0 = 11 - 2 N_f / 3 = 23/3$ for $N_f = 5$:

$$ \Lambda_{\rm QCD}^{(5)} \approx 213\ \text{MeV}, \quad \sigma = 0.1942\ \text{GeV}^2. $$

PT: 0.1942 GeV² vs PDG: 0.194 GeV². Gap: 0.10%.

Link to α'_Regge

The string tension also appears in the "Regge slope" $\alpha' = 1/(2\pi\sigma)$ controlling hadronic trajectories. See [α'_Regge](/en/observables/35).