# Phi^4 theory two-loop contributions

• gobbles
In summary, the two-loop contribution to the ##\phi^4## propagator is significant and can be canceled by a counterterm diagram.
gobbles
Wherever I see calculations of two-loop contributions to the ##\phi^4## propagator (such as Peskin, page 328, on the bottom), only the sunset diagram (aka the Saturn diagram) is considered, but not, say, the two-loop diagram involving a loop on top of a loop (looks like this: _8_). Does it not contribute? As far as I can tell, it does and the expression for it is
##\int\frac{d^4k}{k^2-m^2}\frac{d^4q}{q^2-m^2}##
with a high degree of divergence (##\Lambda^4##). Am I correct?

Depending on the renormalization scheme, this diagram might be completely canceled by a counterterm diagram. Recall that there is a counterterm introduced to cancel the one-loop quadratic divergence in the propagator. When computing the two-loop propagator you have to include a diagram like yours but with the upper loop replaced by this counterterm. This might completely cancel your diagram.

Yes, but it's a nice very simple example for renormalization. Of course, it's just the tadpole diagram squared. So you can calculate the tadpole diagram (say in dimensional regularization). Then you'll see that there are divergences going like ##A/\epsilon^2##, where ##A## is some constant. This you can subtract. It's the overall divergence. But then there are terms like ##B/\epsilon \ln(m^2/\mu^2)##, where ##m## is the mass of the particle and ##\mu## the renormalization scale (entering the dim.-reg. regularization scheme to keep the coupling constant dimensionless at all space-time dimensions). These you get rid of by using the one-loop counter terms for the two sub-divergences, where one counterterm is the tadpole loop on top of the "8" and the other the coupling-constant counter term from the lower loop of the "8". After this you should get rid of all negative powers of eps, and thus you are at a finite result. For the on-shell scheme, where you consider ##m## as the renormalized mass of the particle, you get 0 for the "double tadpole" as for the "single tadpole" diagram.

## 1. What is Phi^4 theory and what does it represent?

Phi^4 theory is a quantum field theory that describes the interactions of a scalar field (represented by the symbol Phi) with itself. It is used to study the behavior of particles at the quantum level and has applications in particle physics, condensed matter physics, and statistical mechanics.

## 2. What are two-loop contributions in Phi^4 theory?

Two-loop contributions in Phi^4 theory refer to the calculations of the scattering amplitudes between particles at the second-order level. This involves analyzing the interactions between particles and their virtual counterparts, which play a significant role in understanding the behavior of particles at the quantum level.

## 3. Why is studying two-loop contributions important in Phi^4 theory?

Studying two-loop contributions in Phi^4 theory is important because it allows us to make more accurate predictions about the behavior of particles. At the two-loop level, we are able to account for more complex interactions between particles, which can significantly impact the results of our calculations.

## 4. What are some current research topics related to Phi^4 theory two-loop contributions?

Some current research topics related to Phi^4 theory two-loop contributions include studying the effects of higher-order corrections on the behavior of particles, exploring new methods for calculating two-loop contributions, and investigating the connections between Phi^4 theory and other quantum field theories.

## 5. How do two-loop contributions in Phi^4 theory affect our understanding of the universe?

Two-loop contributions in Phi^4 theory play a crucial role in our understanding of the universe at the quantum level. They allow us to make more accurate predictions about the behavior of particles, which can then be applied to various fields of physics and contribute to our overall understanding of the fundamental laws of nature.

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