### Running quark masses

The bottom quark mass input for VBFNLO is the pole mass, $M_{b}$. This is converted to the $\overline{MS}$ mass as usual:

$$\overline{m}_{b} (M_{b}) = M_{b} \left( 1 - \frac{4}{3} \frac{\alpha_{s}(M_{b})}{\pi} + \left(1.0414N_{f} - 14.3323\right) \frac{\alpha_{s}^{2}(M_{b})}{\pi^{2}}\right).$$

As described in [1], the evolution of $m_{b}$ up to a reference scale $\mu$ can be described as

$$\setcounter{equation}{2} \overline{m}_b (\mu)= \overline{m}_b \left(m_b\right) \frac{c\big[\alpha_s(\mu) /\pi \big]}{c\big[\alpha_s(m_b) / \pi \big]} .$$

For the coefficient function $c$, the five flavor approximation [2,3] within the mass range $m_b < \mu < m_t$,

$$\setcounter{equation}{3} c(x)=\left(\frac{23}{6} \, x \right)^{\frac{12}{23}} \big[1+1.17549 \, x+1.50071 \, x^2 +0.172478 \, x^3 \big]~,$$

is used. Further evolution of $\overline{m}_b$ to a renormalization scale $\mu>m_t$ can be performed safely within the five flavor approximation, because the deviation to the six flavor scheme is less than 1% for $\mu<600$ GeV.

##### References

[1] F. Campanario, M. Kubocz, and D. Zeppenfeld, (2010), arXiv:1011.3819 [hep-ph]
[2] M. Spira, Fortsch.Phys. 46, 203 (1998), hep-ph/9705337
[3] J. Vermaseren, S. Larin, and T. van Ritbergen, Phys.Lett. B405, 327 (1997), hep-ph/9703284