# Soft gluon symmetry/theorem The soft gluon theorem is an example of [[0009 Soft theorems|soft theorems]]. ## Leading soft gluon theorem - (at tree level) - in momentum basis $\left\langle\mathcal{O}_{1}\left(p_{1}\right) \cdots \mathcal{O}_{n}\left(p_{n}\right) \mathcal{O}^{a}(q, \varepsilon)\right\rangle_{U=1}=g_{Y M} \sum_{k=1}^{n} \frac{p_{k} \cdot \varepsilon}{p_{k} \cdot q}\left\langle\mathcal{O}_{1}\left(p_{1}\right) \cdots T_{k}^{a} \mathcal{O}_{k}\left(p_{k}\right) \cdots \mathcal{O}_{n}\left(p_{n}\right)\right\rangle_{U=1}+\mathcal{O}\left(q^{0}\right)$ - in conformal basis $\left\langle J_{z}^{a} \mathcal{O}_{1}\left(z_{1}, \bar{z}_{1}\right) \cdots \mathcal{O}_{n}\left(z_{n}, \bar{z}_{n}\right)\right\rangle_{U=1}=\sum_{k=1}^{n} \frac{1}{z-z_{k}}\left\langle\mathcal{O}_{1}\left(z_{1}, \bar{z}_{1}\right) \cdots T_{k}^{a} \mathcal{O}_{k}\left(z_{k}, \bar{z}_{k}\right) \cdots \mathcal{O}_{n}\left(z_{n}, \bar{z}_{n}\right)\right\rangle_{U=1}$ - this is just the [[0106 Ward identity]] of [[0069 Kac-Moody algebra]] - the soft pole in the momentum basis expression is absent in the conformal basis expression simply because the definition of $J^a_z$ involves the zero mode of the field strength rather than the gauge field and hence an extra factor of the soft energy - see [[2013#Strominger (Dec)]] ## Applicability - not confined - not Higgsed - otherwise no soft gluons ## $U(1)$ case - [[2019#Himwich, Strominger]] - current algebra ## Subleading soft gluon symmetry - [[2020#Banerjee, Ghosh]] - see [[0101 Gluon celestial toolkit]] for action on gluons ## Loop - unlike soft photon, the soft gluon theorem has corrections at loop level ## Refs - recent ref [[2021#Guevara, Himwich, Pate, Strominger]]