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Break1
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Hi guys! I was wondering if there is any difference choosing between d = 4 -e or d = 4 - 2e. If so, what are the impacts ?
Yes!dextercioby said:Is it e (elementary charge) or ##\epsilon##? I think the second.
Dimensional regularization is a technique used in quantum field theory to handle infinities that arise in the calculations of loop integrals due to divergences at high energies. It involves performing the calculations in a space with a non-integer number of dimensions (D), which effectively regularizes the integral by taming these infinities. After the calculations, the limit as D approaches the physical number of dimensions (usually four) is taken.
Dimensional regularization is preferred because it preserves gauge invariance and Lorentz invariance, which are fundamental symmetries in many quantum field theories, including the Standard Model of particle physics. Unlike other regularization methods, such as cutoff techniques, dimensional regularization does not introduce arbitrary parameters or break these symmetries, making the calculations more robust and theoretically consistent.
In dimensional regularization, the divergent integrals encountered in quantum field theory calculations are evaluated in a space with a dimension D that is slightly different from the physical dimension (usually 4). This change in dimensionality alters the behavior of the integrals, generally making them converge. The results are expressed in terms of D, and analytic continuation is used to handle the expressions as D approaches the physical dimension. This approach often introduces poles in 1/(D-4), which are handled separately using renormalization techniques.
Common results of dimensional regularization include the regularization of ultraviolet divergences in loop integrals. This technique often leads to expressions that contain gamma functions and poles at specific values of D (like 1/(D-4)), which indicate the presence of divergences in the original unregularized integral. These results are then used in the renormalization process to absorb the divergences into redefinitions of physical parameters (like masses and coupling constants), thereby yielding finite, physically meaningful predictions.
Dimensional regularization is primarily effective for handling ultraviolet (UV) divergences; these are associated with the behavior of integrals at high energy or short distances. However, it is not generally suitable for dealing with infrared (IR) divergences, which occur due to the behavior at low energy or long distances. For IR divergences, other techniques, such as introducing masses or using infrared cutoffs, are typically employed. Therefore, while dimensional regularization is a powerful tool for UV divergences, it is not a universal solution for all types of divergences in quantum field theories.