Error estimation of $\alpha_s$

Question

I have calculated the strong coupling constant $\alpha_s$ using and approximate solution of the Renormalization Group Equation $$\mu_R^2\frac{d\alpha_s}{d\mu_R^2}=-(b_0 \alpha_s^2 + b_1\alpha_s^3 + b_2\alpha_s^4 + ...)$$ given by $$\alpha(\mu_R^2)=\frac{1}{b_0t}(1-\frac{b_1}{b_0^2}\frac{\ln t}{t}+\frac{b_1^2(\ln^2 t-\ln t -1)+b_0b_2}{b_0^4t^2}-...)$$ with $t\equiv\ln\frac{\mu_R^2}{\Lambda^2}$ and using as $\mu_R$ as the center of mass energy of the $c \bar c$ ($\mu_R=\frac{m_c}{2}c_F\alpha_s$) meson, which also depends on $\alpha_s$ like $M_{c\bar c}=2m_c-(c_F\alpha_s)^2\frac{m_c}{2} $ (and taking $M_{c\bar c}$ as some experimental value). You can find the explanation here).

My question is: how can I estimate the error commited taking just the first term or the first two terms in the $\alpha(\mu_R^2)$ equation?

Answer 1

I don't know if there is a way of getting from first principles an error estimación for field theory predictions. I thought that knowing that in puré yang-mills was almost the thing that you needed for getting the millenium prize. I heard that in the QCD-pheno comunity they have some euristics, but they aren't first principle thing. I'll like to know the real answer to this question, so I will make a little noise with this almost-fake answer.

Answer 2

The terms in the equation for $\alpha_s(\mu_R^2)$ as cited gives schematically the one loop, two loop etc running couplings in terms of $\Lambda_{\text{QCD}}$ and RGE coefficients $b_i$.

$\Lambda_{\text{QCD}}$ is a beast carrying many 'indices' in the sense that a numerical value attached to it depends on the order of the running, the number of active flavours and a reference scale $\mu_o^2$, commonly taken to be the $Z$ mass. In Schwartz's 'Quantum Field Theory and the Standard Model', he calculates values for this parameter at various loop orders assuming a constant flavour running with $n_f=5$ using the well known world average value for $\alpha_s(M_z^2) = 0.1181 \pm 0.0013.$

This approximate solution for the runnings assumes a constant number of active flavours throughout the evolution which introduces an error since no flavour decoupling mechanisms are in play. The package RunDec has in built functionality which will do this decoupling for you automatically. Comparing its output for $\alpha_s$ at e.g two loop with various values of $\mu_R^2$ against that obtained using the naive 'poor mans' approach formula yields percentage differences starting off in the range of $\sim 11$% and growing as $\mu_R^2$ is increased.

Clearly for precision calculations needed today for descriptions of LHC phenomena, this is a noticeable error and so such a formula is not to be used in practice.