3D Integral to 4D integral in Width Calculation

[Hepth]

While this is more a mathematics problem, it really belongs here for those theorists who have experience with QFT calculations.
This is sort of a generic question, but can I turn a 1->3 decay width integral into a single 4-D integral over one of the momenta.
Such as:

$$\Gamma = \frac{1}{2 M_X} (2 \pi)^{4-9} \int \frac{d^3 p_1}{2 E_1}\frac{d^3 p_2}{2 E_2}\frac{d^3 k}{2 E_k} \delta^4 (P-p_1-p_2-k) |M|^2$$

into the form:

$$\Gamma = (...) \int \frac{d^4 k}{(2 \pi)^4} |M|^2$$

without any delta functions in the k integral.

I know I can replace the p2 integral times the delta with a 1-dimensional delta

$$\int \frac{d^3 p_2}{2 E_2} \delta^4(P-p_1-p_2-k) = \delta((P-p_1-k)^2-m_2^2)$$

using

$$d^4 k \theta(E_k) \delta(k^2-\mu^2) = \int \frac{d^3 k}{2 E_k}$$

but still I'm left with a delta function for at least the k_0 piece. I know how to ACTUALLY calculate the integrals, but I'm trying to get them into this form so I can plug them into FeynCalc/PHi to get the integrals in terms of the Passarino-Veltman integrals.

Or is there another, BETTER way of getting tree level calculations that diverge into the Passarino-Veltman integral form to cancel divergences from 1-loop corrections?

Again, I know there are many other ways of doing this (DREG/cutoffs/PV/OpticalThm) but I'm trying to automate some divergence cancellation with Mathematica specifically using the Pass-Velt functions (A0,B0,C0,D0).

Thanks!

 

[AndyW]

Thank you for your question. It is possible to turn a 1->3 decay width integral into a single 4-D integral over one of the momenta, as shown in your proposed form. However, it is important to note that this is not a straightforward process and requires careful consideration and understanding of the underlying physics.

In general, the Passarino-Veltman integrals are used to handle loop corrections in quantum field theory calculations. These integrals are typically used in dimensional regularization (DREG) to regulate ultraviolet divergences. The Passarino-Veltman integrals can also be used to handle tree level calculations, as you have mentioned, but this is not a standard practice as it can lead to complications and inconsistencies.

In order to properly automate divergence cancellation using Mathematica and the Passarino-Veltman integrals, it is important to have a strong understanding of the underlying physics and the limitations of using these integrals. It may be beneficial to consult with a colleague or mentor who has experience with this specific approach, or to seek out additional resources and literature on the topic.

In summary, while it is possible to turn a 1->3 decay width integral into a single 4-D integral over one of the momenta, it is important to carefully consider the underlying physics and potential limitations of using this approach. It may be beneficial to explore other methods for handling divergences in your calculations, such as DREG or cutoffs, to ensure accurate and consistent results. Thank you for your question and best of luck with your research.