Divergence of the tree level scattering amplitude in quantum field theory

Question

I'm studying a toy theory in quantum field theory. There are two free fields: a real massive scalar field $\phi$ with mass $M$ and a complex massive scalar field $\Psi$ with mass $m$.

They are coupled by $$ \mathcal{L} \subset g \Psi \Psi^\dagger \phi $$ I'm well aware that this interaction term results in a Lagrangian which is unbounded below, but it's just a toy model I'm using to try to get a grasp on the basics of quantum field theory.

Now, when I go to compute the tree-level scattering amplitude for $\Psi\Psi \to \Psi\Psi$ scattering I end up with a $T$-channel diagram and a $U$-channel diagram. (In both diagrams the two $\Psi$ particles come in, exchange an off-shell $\phi$ particle, and scatter into their final states).

The sum of these two diagrams gives me the total matrix element, which goes like $$ \frac1{(t-M^2)} + \frac1{(u-M^2)} $$ ignoring the factor's of $i$ and the factor of $g^2$. In computing this amplitude I worked in the center of mass/momentum frame, with the initial four momentum of the two particles being $(E(p),0,0,\pm p)$, with the final four momentum of the two particles being $(E(p),0, \pm \cos(θ)p, \pm \sin(θ)p)$. This ensures that four-momentum is conserved.

With this parametrization, $t$ becomes $-4(p\sin(θ/2))^2$, and $u$ becomes $-4(p\cos(θ/2))^2$. Things get interesting in the limit as $M \to 0$, i.e., the mass of the phi particle approaches zero. In this case the scattering amplitude becomes unbounded as $θ$ approaches zero or $\pi$. This is, to my mind, a highly unphysical result.

Is this divergence a result of a fundamental issue with this toy-model? How can we make sense of this result? What is the physical interpretation of a divergent scattering amplitude? Worse still is the fact that the integrated/total cross section also appears to diverge in all of these cases.

Answer 1

What you found is a very simple example of an infra-red divergence, which plague all physical theories with massless particles.

This kind of divergences are already present in the classical case (see for example ref.1), and they usually signal that you are asking an unphysical question, not that the theory itself is unphysical. For example, if you have massless particles it becomes meaningless to ask about the total number of them in a certain physical configuration, while asking about the total energy is a well-posed question. This is reflected in the mathematics of the theory through divergences: the first question results in a divergent expression, while the second does not.

These divergences are inherited to the quantum mechanical case. For example, IR divergences are ubiquitous in QED, but it can be proven that these divergences always cancel out for "meaningful cross-sections", i.e., for predictions that are well-posed in the sense of the previous paragraph. This is sometimes known as the Bloch-Nordsieck cancellation, or in the more general case of the Standard model, as the Kinoshita-Lee-Nauenberg theorem. See for example refs.2,3,4.

In your particular case, as you are dealing with a scalar theory as opposed to a gauge boson, the analysis is slightly simpler (although one must in general abandon on-shell renormalisation schemes if we are to have massless particles). This is discussed in ref.5.

For a further discussion of infra-red divergences, see refs. 6-8.

References

1. Itzykson C., Zuber J.-B. Quantum field theory, section 4-1-2.

2. Peskin, Schroesder. An introduction To Quantum Field Theory, section 6.5.

3. Itzykson C., Zuber J.-B. Quantum field theory, section 8-3-1.

4. Schwartz M.D. Quantum Field Theory and the Standard Model, chapter 20.

5. Srednicki M. Quantum Field Theory, chapters 26 and 27.

6. Ticciati R. Quantum Field Theory for Mathematicians, section 19.9.

7. Pokorski S. Gauge Field Theories, sections 5.5 and 8.7.

8. Weinberg S. Quantum theory of fields, Vol.1. Foundations, chapter 13.

Answer 2

This is a very crude answer. But this will help in your queries. You see the problem with the above method you implied is that at tree level $\psi \psi - > \psi \psi$ the internal propagator is $\phi$ particle. Now the propagator for massive and massless particles are very different. For massless particle the equation of motion in momentum space is not trivially invertible. This you cannot set M = 0 in the above equation. For massless $\phi$ you will need to find the propagator first.