Editing Renormalization (section)

=== Running couplings ===
To minimize the contribution of loop diagrams to a given calculation (and therefore make it easier to extract results), one chooses a renormalization point close to the energies and momenta exchanged in the interaction. However, the renormalization point is not itself a physical quantity: the physical predictions of the theory, calculated to all orders, should in principle be ''independent'' of the choice of renormalization point, as long as it is within the domain of application of the theory. Changes in renormalization scale will simply affect how much of a result comes from Feynman diagrams without loops, and how much comes from the remaining finite parts of loop diagrams. One can exploit this fact to calculate the effective variation of [[Coupling constant|physical constants]] with changes in scale. This variation is encoded by [[Beta function (physics)|beta-function]]s, and the general theory of this kind of scale-dependence is known as the [[renormalization group]].

Colloquially, particle physicists often speak of certain physical "constants" as varying with the energy of interaction, though in fact, it is the renormalization scale that is the independent quantity.  This [[Coupling constant#Running coupling|''running'']] does, however, provide a convenient means of describing changes in the behavior of a field theory under changes in the energies involved in an interaction.  For example, since the coupling in [[quantum chromodynamics]] becomes small at large energy scales, the theory behaves more like a free theory as the energy exchanged in an interaction becomes large – a phenomenon  known as [[asymptotic freedom]].  Choosing an increasing energy scale and using the renormalization group makes this clear from simple Feynman diagrams; were this not done, the prediction would be the same, but would arise from complicated high-order cancellations.

For example,
<math display="block">I=\int_0^a \frac{1}{z}\,dz-\int_0^b \frac{1}{z}\,dz=\ln a-\ln b-\ln 0 +\ln 0</math>
is ill-defined.

To eliminate the divergence, simply change lower limit of integral into {{mvar|ε<sub>a</sub>}} and {{mvar|ε<sub>b</sub>}}:
<math display="block">I=\ln a-\ln b-\ln{\varepsilon_a}+\ln{\varepsilon_b} = \ln \tfrac{a}{b} - \ln \tfrac{\varepsilon_a}{\varepsilon_b}</math>

Making sure {{math|{{sfrac|''ε<sub>b</sub>''|''ε<sub>a</sub>''}} → 1}}, then {{math|''I'' {{=}} ln {{sfrac|''a''|''b''}}.}}