Deriving QCD through observing deviations to QED framework

Part 1 : Experimental determination of strong coupling constant

Due to historical considerations, the examination of Quantum Chromodynamics (QCD) often involves scrutinizing deviations from the theoretical framework of Quantum Electrodynamics (QED). This approach is grounded in the fact that QED is more readily discernible to human observation compared to QCD. In this post, we explore a specific comparative analysis involving the calculation of the cross-section ratio for (e^+e^- \rightarrow e^+e^-) and (e^+e^- \rightarrow q\overline{q}).

As an experimentalist’s standpoint, if I lacked awareness of strong interactions and color charge, these two cross-sections appear remarkably similar. The distinction lies primarily in the electric charge of the produced quarks and the different flavor charges achievable at a given center-of-mass energy of the collision.

At a given energy (\sqrt{s}), the cross-section for (e^+e^- \rightarrow e^+e^-) is given by (\sigma(e^+e^- \rightarrow e^+e^-) = \frac{4\pi\alpha_{\text{em}}}{3s}). If (e^+e^- \rightarrow q\overline{q}) were a purely electromagnetic interaction, the cross-section would be (\sigma(e^+e^- \rightarrow q\overline{q}) = \frac{4\pi\alpha_{\text{em}}}{3s}e_i^2), where (e_i) is the electric charge of the quark.

For example, (\sqrt{s} = 2) GeV , there is phase space to create up, down, and strange quarks with electric charges (+\frac{2}{3}), (-\frac{1}{3}), and (-\frac{1}{3}), respectively. Consequently, the cross-section becomes (\frac{4\pi\alpha_{\text{em}}}{3s}\left(\frac{4}{9} + \frac{1}{9} + \frac{1}{9}\right) = \frac{4\pi\alpha_{\text{em}}}{3s}\frac{2}{3}) .

Therefore, the expected ratio of the cross-sections is 2/3. However, experimental measurements, conducted at DESY and later with high precision at CERN, yielded a ratio of 2. This discrepancy points to the existence of another degree of freedom in three axis that each of the quark can express itself in – color charge.

Further tests at higher energies, allowing the production of more quark flavors, confirmed the presence of color charge. Despite experimental ratios deviating from expectations, the precise reasons for these deviations need exploration.

To delve into the discrepancies, one must consider higher-order corrections beyond the leading diagrams in both QED and QCD. The next-order correction involves the radiation of gluons from quarks and anti-quarks, dependent on the strength of the strong coupling constant (\alpha_s). The resulting corrected ratio ((R)) can be expressed as (R = 3 + \frac{f_Q^2}{f_1} + \left(\frac{\alpha_s(s)}{\pi}\right) + 1.411\left(\frac{\alpha_s(s)}{\pi}\right)^2 - 12.8\left(\frac{\alpha_s(s)}{\pi}\right)^3 + \ldots). This process is experimentally observed as a three-jet event.

By measuring (R) at approximately 3.85, one can invert the equation to determine the value of (\alpha_s(s)) at a specific collision energy. In this case, solving the equation yields (\alpha_s(s) = 0.15) , highlighting that (\alpha_s(s)) is an experimentally derived quantity. This stands in contrast to the coupling constant in QED, which can be theoretically derived with great precision.

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