Dimensional regularization

Dimensional regularization analytically continues loop integrals from an integer spacetime dimension to

d = d_{0} - ϵ

. Ultraviolet logarithms then appear as poles in

1/ ϵ

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Euclidean massive loop integral ( $I_{d, n} (m)$ )

 0  0

For

m^{2} > 0

and

n > d /2

, Schwinger parameterization and a Gaussian integral give

I_{d, n} (m) = \int d^{d} k (2 π)^{- d} (k^{2} + m^{2})^{- n} = Γ (n - d /2) (m^{2})^{d /2 - n} / [(4 π)^{d /2} Γ (n)]

. Its meromorphic continuation supplies dimensional regularization poles.

Massive scalar bubble pole in four dimensions ( $\int \frac{d ^{4 - ϵ} k}{( 2 π ) ^{4 - ϵ}} \frac{1}{( k ^{2} + m ^{2} ) (( p - k ) ^{2} + m ^{2} )} = \frac{1}{8 π ^{2} ϵ} + O (1)$ )

 0  0

For positive

m^{2}

, a Feynman parameter and a momentum shift turn the integral into

\int_{0}^{1} I_{d, 2} (m^{2} + x (1 - x) p^{2}) d x

. The Euclidean massive loop integral has factor

Γ (ϵ /2) = 2/ ϵ + O (1)

, while its other factors tend to

(4 π)^{- 2}

. The pole is independent of the external momentum and the mass, so it is removable by a local counterterm.

Minimal subtraction scheme ( $MS$ scheme)

 1  0

The minimal subtraction scheme chooses counterterms that remove only the poles in the dimensional regulator

ϵ

, without additional finite terms.

Simple-pole formulas for minimal-subtraction renormalization group functions ( $β (λ) = (λ \partial_{λ} - 1) f_{1} (λ), γ_{m^{2}} (λ) = λ b_{1}^{'} (λ)$ )

 0  0

In the minimal subtraction scheme, let

λ_{B} = μ^{ϵ} F

F = λ + \sum_{k \geq 1} f_{k} (λ) / ϵ^{k}

, and

m_{B}^{2} = m^{2} H

H = 1 + \sum_{k \geq 1} b_{k} (λ) / ϵ^{k}

. Define the renormalization-group beta function at fixed bare parameters by

β = μ d λ / d μ = - ϵ λ + β (λ)

, and the running mass function by

γ_{m^{2}} = μ d lo g m^{2} / d μ

The bare coupling is scale independent, so

0 = ϵ F + β F^{'}

. Its finite coefficient is

f_{1} + β - λ f_{1}^{'}

, giving the first formula. Similarly

0 = γ_{m^{2}} H + β H^{'}

has finite coefficient

γ_{m^{2}} - λ b_{1}^{'}

. The remaining poles cancel recursively:

(λ \partial_{λ} - 1) f_{k + 1} = β f_{k}^{'}

and

λ b_{k + 1}^{'} = γ_{m^{2}} b_{k} + β b_{k}^{'}

. Thus the simple poles determine the finite renormalization group functions, while higher poles obey consistency conditions.

Modified minimal subtraction scheme ( $\overline{MS}$ scheme)

 0  0

The modified minimal subtraction scheme removes the dimensional pole together with the universal constants conventionally grouped as

- γ + lo g 4 π

. This simplifies renormalization-group formulas while differing from minimal subtraction only by finite parameter redefinitions.

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Dimensional regularization is a mathematical technique used in quantum field theory to handle ultraviolet divergences (infinities) that arise in loop integrals during the calculation of Feynman diagrams. The method involves extending the number of spacetime dimensions from the usual integer values (like 4 in our physical universe) to a complex or arbitrary value, typically denoted as $d$.

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