# Reference request: Regularization by differentiation with respect to external momenta

In reading Witten’s lectures on perturbative renormalization [1] (page 17) I learned of an interesting approach to renormalization/regularization in which renormalized Green functions $$\Gamma_R$$ are identified from divergent $$\Gamma$$ as the solution of a differential equation obtained by differentiating the integrand of $$\Gamma$$ with respect to external momenta to the necessary order required to render the resulting integral convergent.

This method has the appealing property that it avoids discussion of divergent bare coupling parameters, so that it might form the basis of a rigorous approach like BHPZ renormalization. Apart from one unpublished manuscript [2], I was unable to find any discussion of this procedure in textbooks or papers. Can anyone recommend a systematic account?

There is nothing mysterious about it.

Let's look at Green function $$\Gamma(x)$$ which is dependent on some parameter $$x$$. The parameter $$x$$ could be anything, such as, external momentum $$\vec{q}$$ or the mass $$m$$ of a particle. Note that $$x$$ can not be the loop integral variable, i.e. the internal momentum $$\vec{p}$$. And we also assume that $$x$$ resides in the denominator of the integrand of $$\Gamma(x)$$, so that differentiating $$\Gamma(x)$$ with $$x$$ would make it more convergent.

Let's expand $$\Gamma(x)$$ in turn of $$x$$ $$\Gamma(x) = c_0 + c_1 x + c_2x^2 + \dots$$

Now comes the punchline: for a given divergent Green function $$\Gamma(x)$$, only the first few coefficients of $$c_0, c_1, c_2, \dots$$ are divergent, while the rest are convergent.

Let's take for example the typical logarithmically divergent $$\Gamma(x)$$, where only $$c_0$$ is divergent and $$c_1, c_2, \dots$$ are convergent. The usual text book way of renormalization is to introduce some regularization to make $$c_0$$ finite. But there is actually a slick way to avoid explicit regularization: just differentiate $$\Gamma(x)$$ with respect to $$x$$: $$\frac{d\Gamma(x)}{dx}= c_1 + 2c_2x + \dots$$ Since $$\frac{d\Gamma(x)}{dx}$$ is finite ($$c_0$$ has been differentiated away!), we can proceed to calculate $$g(x) = \frac{d\Gamma(x)}{dx}$$ without explicit regularization. Once we have $$g(x)$$, we can obtain $$\Gamma(x)$$ via integration: $$\Gamma(x) = \int g(x)dx$$ The divergence of $$c_0$$ is therefore bypassed: actually $$c_0$$ is replaced by an unknown finite integration constant, which is to be determined by experimental observation.

A variation of above approach is to calculate the finite difference (rather than differentiation $$\frac{d\Gamma(x)}{dx}$$): $$\Delta\Gamma(x) = \Gamma(x) - \Gamma(x_0)$$ For more details on the finite difference approach, please see this PSE thread and references therein.

The renormalization group could be also formulated this way via differentiation against a parameter such as mass. See, for example, here.