@Aglietti:2006tp]), the NLO QCD corrections to these LO expressions will be large.

Next-to-leading-order corrections \[sc:nlo\]
============================================

The NLO QCD corrections to Higgs-pair production via gluon fusion have been computed in the HTL, a long time ago [@Dawson:1998py]. The NLO result for the gluon-fusion cross section can be generically expressed as [@Dawson:1998py] $$\begin{aligned}
\sigma_{NLO}(pp \rightarrow H H + X) & = &
\sigma_{LO} + \Delta
\sigma_{virt} + \Delta\sigma_{gg} + \Delta\sigma_{gq} +
\Delta\sigma_{q\bar{q}} \, , \nonumber $$ $$\begin{aligned}
\sigma_{LO} & = & \int_{\tau_0}^1 d\tau~\frac{d{\cal
L}^{gg}}{d\tau}~\hat\sigma_{LO}(Q^2 = \tau s) \, , \nonumber \\ 
\Delta \sigma_{virt} & = & \frac{\alpha_s(\mu_R)}
{\pi}\int_{\tau_0}^1 d\tau~\frac{d{\cal L}^{gg}}{d\tau}~\hat
\sigma_{LO}(Q^2=\tau s)~C_{virt}(Q^2) \, , \nonumber \\ 
\Delta \sigma_{ij} & = & \frac{\alpha_{s}(\mu_R)} {\pi} \int_{\tau_0}^1
d\tau~ \frac{d{\cal L}^{ij}}{d\tau} \int_{\tau_0/\tau}^1 \frac{dz}{z}~
\hat\sigma_{LO}(Q^2 = z \tau s)\, C_{ij}(Q^2,z) \qquad (ij=gg,gq,q\bar q)
 \, ,\nonumber \\
C_{gg}(Q^2,z) & = & - z P_{gg} (z) \log \frac{\mu_F^{2}}{\tau s}
+ 6 [1+z^4+(1-z)^4] \left(\frac{\log (1-z)}{1-z} \right)_+
+ d_{gg}(Q^2,z) \, , \nonumber \\ 
C_{gq}(Q^2,z) & = & -\frac{z}{2} P_{gq}(z)
\log\frac{\mu_F^{2}}{\tau s(1-z)^2} + d_{gq}(Q^2,z) \, , \nonumber \\ 
C_{q\bar q}(Q^2,z) & = & d_{q\bar q}(Q^2,z)
\label{eq:nlocxn}\end{aligned}$$ with $\hat\sigma_{LO}(Q^2)$ denoting the partonic cross section at LO and the strong coupling $\alpha_s(\mu_R)$ is evaluated at the renormalization scale $\mu_R$. The objects $d{\cal
L}^{ij}/d\tau~(i,j=g,q,\bar 