# Why does dimensional regularization need counterterms ?

• zetafunction
In summary: Thanks, this will be helpful.In summary, dimensional regularization needs counterterms to ensure that all the infinities in the integrals are taken away. This is done by introducing a cutoff in terms of mass or energy. The pole at d=4 is the only infinity that is allowed to remain.

#### zetafunction

why does dimensional regularization need counterterms ??

if all the integrals in 'dimensional regularization' are FINITE why do we need counterterms ?? in fact all the poles of the Gamma function are simple hence the only divergent quantity is the limit as d tends to 4 of

$$1/(d-4)$$ which is ONLY a divergent quantity that could be 'absobed' or re-parametered as a divergent universal constant 'a'

Several integrals like self-energy vacuum polarization produce infinities which can be written as functions of a cutoff. The only special thing about dim. reg. is that you do not use a cutoff in terms of mass or energy but that you get a pole in 1/(d-4).

If you have an infinite quantity you can write it as (finite quantity + infinite quantity), so that means that the calculation is ambiguous. The counter term ensures that the subtraction of all infinite quantities in all Feynman integrals of the theory is done consistently.

however the pole at d=4 is SINGLE , it will give only a divergent quantity in every integral proportional to

$$1/(d-4)$$ when d-->4 perhaps it would suffice to define the divergent quantity

$$a=1/(d-4)$$ to be 'fixed' by experiments

another question if dimension of our space was for example d=3.956778899797696969695.. instead of d=4 , since there would be no poles (gamma function is perfectly defined for negative numbers except when they are negative integers) would dimensional regularization give the CORRECT finite answer, that's it if the dimension was different from an integer , there would be no poles inside the Gamma function and everything would be OK

zetafunction said:
however the pole at d=4 is SINGLE , it will give only a divergent quantity in every integral proportional to

$$1/(d-4)$$ when d-->4 perhaps it would suffice to define the divergent quantity

$$a=1/(d-4)$$ to be 'fixed' by experiments

another question if dimension of our space was for example d=3.956778899797696969695.. instead of d=4 , since there would be no poles (gamma function is perfectly defined for negative numbers except when they are negative integers) would dimensional regularization give the CORRECT finite answer, that's it if the dimension was different from an integer , there would be no poles inside the Gamma function and everything would be OK

You should have a look at the MS or MS-bar subtraction scheme (MS = minimal subtraction). Indeed only the infinities are removed, but nevertheless counter terms are required.

I would not take the approach with varying dimension too seriously; it is simply a method to parameterize the infinities w/o breaking several invariances. If you use Hamiltonian methods with mode expansion other methods e.g. heat kernel or zeta function regularization are more appropriate. In that case you do not see any deviation from D=4.

thanks Tom i wil give these methods (Zeta regularization and Heat Kernel) a look, anyway zeta function regularization has the problem of a pole at s=1 $$\zeta(1)= \infty$$

zetafunction said:
... anyway zeta function regularization has the problem of a pole at s=1

That's no problem. You introduce s just to regularize a kind of spectral- or Dirichlet series. The physical value is s=0, the pole at s=1 is unphysical. Then you continue the zeta function to s=0 avoiding s=1.

## 1. Why is dimensional regularization used in physics?

Dimensional regularization is used in physics to regulate the divergences that arise in quantum field theory calculations. These divergences occur when we try to calculate physical quantities, such as particle masses and interaction strengths, in a theory that includes interactions between particles. Dimensional regularization allows us to perform these calculations in a mathematically consistent way, avoiding the need for ad-hoc cutoffs or renormalization schemes.

## 2. What are the advantages of using dimensional regularization?

One of the main advantages of dimensional regularization is that it preserves the symmetries of the underlying theory. This is important because symmetries play a crucial role in many physical theories, and any regularization method that breaks these symmetries would lead to incorrect predictions. Additionally, dimensional regularization is a more mathematically rigorous method compared to other regularization techniques, making it a preferred choice for many physicists.

## 3. Why do we need counterterms in dimensional regularization?

In dimensional regularization, the divergent terms in a calculation are written in terms of a parameter called epsilon, which represents the deviation from the physical dimension of space-time. These divergences can be canceled out by adding counterterms, which are additional terms in the theory that depend on epsilon and are carefully chosen to cancel out the divergent terms. This allows us to obtain finite, physically meaningful results from our calculations.

## 4. Can dimensional regularization be used in all theories?

No, dimensional regularization is not applicable to all theories. It is most commonly used in quantum field theories, which describe the behavior of particles at the subatomic level. It has also been successfully applied to certain mathematical models and calculations in other areas of physics. However, it may not be suitable for all types of calculations, and other regularization methods may be more appropriate in certain cases.

## 5. Are there any limitations to dimensional regularization?

One limitation of dimensional regularization is that it can only be used in theories with a certain type of symmetry known as conformal symmetry. This means that the theory must be invariant under certain transformations that preserve angles but change distances. Additionally, dimensional regularization may not always give unique results, and in some cases, different regularization methods may produce different results. It is important for physicists to carefully consider the limitations and assumptions of any regularization method used in their calculations.