# Does The Use Of The Zeta Function Bypass Renormalization

• A
• bhobba
In summary, the use of the Zeta function does not allow for bypassing renormalization. While it can be used as a regularization method, it still requires the subtraction of divergent terms to obtain finite results. Additionally, it is necessary for renormalization schemes to agree on certain properties, such as the flow of beta functions and anomalous dimensions, indicating that not all divergences can be eliminated through the use of the Zeta function.
bhobba said:
I am trying to figure out if the use of the Zeta function allows renormalization to be bypassed.
No, it cannot be bypassed in this way. Ones still needs the subtractions and determine their value such that the divergences cancel. Zeta-regularization is only a way to do the bookkeeping.

bhobba and vanhees71
You have to disinguish between regularization and renormalization: Regularization is some way to make sense of the divergent integrals you get from evaluating the Feynman rules for diagrams with loops. There are many possibilities around in the literature. The most simple is a cut-off, but that's uncomfortable, because it destroys Lorentz and gauge invariance. The most used regularization nowadays is dimensional regularization, where you analytically continue the space-time integrals over 4-dimensional spacetime momenta to an arbitrary d-dimensional momenta with ##d## not necessarily integer. Another one is ##\zeta##-function regularization. These have the advantage of making the calculations Lorentz and gauge invariant at any step.

At the end you want the result for the physical values of the parameters introduced in the regularization prescription (i.e., cut-off to ##\infty## or dimensions to ##4##). The divergent integrals are of course divergent when taking this limit. So you have to subtract the divergent part, and for renormalizable theories this can be done in a way that is equivalent to adding contributions to the Lagrangian which are of the same form as the terms already there, which lumps the infinities into the unobservable "bare normalization and coupling constants as well as masses", and everything is expressed in terms of the finite coupling constants that are observed.

A way to renormalize without an explicit regularization step is to read the Feynman rules as Feynman rules for the integrands of the loop integrals and you subtract the counterterms from them before you do the then finite integrals. This is known as BPHZ renormalization (named after Boguliubov and Parasiuk, Hepp, and Zimmermann). For details, see my QFT script:

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf

bhobba and DarMM
Some - but not all! - divergences are "automatically" renormalized by zeta/dimensional regularization (both methods lie within the general purview of analytic continuation methods of regularization). For example, consider the integral
$$I_{d,s} = \int \frac{d^d p}{(2 \pi)^d} \frac{1}{(p^2 + m^2)^s}.$$
This is clearly divergent for any ##d>2s##, but analytic continuation gives you the answer
$$I_{d,s} = \frac{\Gamma(s - d/2)}{\Gamma(s) (4 \pi)^{d/2}}m^{d - 2s}.$$
This expression has poles if ##s - d/2## is a negative integer, but is perfectly finite everywhere else. So then loop integrals of this form are finite for all odd ##d## provided ##s## is an integer.

But certain properties of renormalization need to agree between all regularization schemes. For example, the flow of the beta functions, the analytic properties of Callan-Symanzik equations, anomalous dimensions in scale invariant theories, etc. So clearly zeta reg can't get rid of all divergences, since it also needs to be able to contain this important physical information.

bhobba

## 1. What is the Zeta function and how is it related to renormalization?

The Zeta function is a mathematical function used in number theory and other branches of mathematics. In physics, it is used to calculate the sum of infinite series, which is a common occurrence in renormalization. The Zeta function is used to regulate the divergent sums that arise in renormalization calculations.

## 2. Can the Zeta function completely bypass the need for renormalization?

No, the Zeta function cannot completely bypass renormalization. While it can regulate divergent sums, it does not address the underlying issue of infinities in quantum field theory. Renormalization is still necessary to remove these infinities and make meaningful predictions.

## 3. How does the use of the Zeta function affect the renormalization process?

The use of the Zeta function simplifies the renormalization process by providing a method to regulate divergent sums. This allows for more efficient and accurate calculations in quantum field theory.

## 4. Are there any limitations to using the Zeta function in renormalization?

Yes, there are limitations to using the Zeta function in renormalization. It can only be used for certain types of divergent sums and may not work for all renormalization calculations. Other regularization methods may need to be used in conjunction with the Zeta function.

## 5. Is the use of the Zeta function widely accepted in the scientific community?

Yes, the use of the Zeta function in renormalization is widely accepted in the scientific community. It has been successfully used in various fields of physics, including quantum electrodynamics and quantum chromodynamics. However, it is still an active area of research and other regularization methods are also being explored.

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