Propagator from a space-time point to itself

[victorvmotti]

TL;DR Summary

a one-loop vacuum bubble or a double-loop vacuum bubble doesn't make an intuitive physical sense, in particular, what propagation from one point to itself would mean in space-time?

I am following a lecture note on the QFT.

But am a little confused about some parts related to the vacuum bubbles.

We define the Feynman propagator, $D_{F}(x-y)$, as giving the amplitude for a particle emitted at $x$ to propagate to $y$ (where it can be measured).

After following the LSZ reduction formalism and Wick's theorem we arrive at the Feynman Diagrams.

At first order we see disconnected diagrams. Here we deal with terms like $D_{F}(z_{1}-z_{1})D_{F}(z_{1}-z_{1})$ and also $D_{F}(z_{1}-z_{2})D_{F}(z_{2}-z_{1})$, etc.

And here is my question: what could be the physical interpretation of this term: $D_{F}(z_{1}-z_{1})$.

Based on the definition above, the term reads as the amplitude of a particle emitted at $z_{1}$ to propagate to the same point in space-time.

I see that when we write $D_{F}(z_{1}-z_{2})$, for two different points in vacuum, we can still say that such a particle is propagated. And if this was part of a fully connected Feynman diagram we interpret it physically as a virtual particle (off-shell) exchanged or propagated.

But a one-loop vacuum bubble $D_{F}(z_{1}-z_{1})$ or a double-loop vacuum bubble $D_{F}(z_{1}-z_{1})D_{F}(z_{1}-z_{1})$ doesn't make an intuitive physical sense, in particular, what propagation from $z_{1}$ to $z_{1}$ would mean here in space-time?

 

[vanhees71]

These are the socalled "tadpole diagrams" they are of course infinite, because $D_F(0)$ is undefined. You get rid of them by renormalization, though they can be important in connection with symmetries. The most simple example is the one-loop photon-polarization (photon self-energy) in scalar QED: There are two diagrams one "exchange diagram" and a tadpole diagram. Though the tadpole pole diagram is simply some divergent constant times $\eta^{\mu \nu}$ it's important for gauge invariance, because only with this diagram the total photon polarization tensor gets 4D-transverse as it must be due to the Ward Takahashi identity. So the tadpoles are finally handled by the usual renormalization procedures but can be important for Ward-Takahashi identities for theories with (gauge) symmetry.

If you don't have symmetries to respect, you can simply omit all tadpole diagrams, which can be formally seen by the fact that you can use normal ordering in the interaction part of the Hamiltonian, such that all contractions in Wick's theorem connecting two field operators at the same spacetime point vanish due to the normal ordering.

 

[PeterDonis]

I am following a lecture note on the QFT.

Which lecture notes? Please give a reference.

 

[victorvmotti]

Which lecture notes? Please give a reference.

Here is the lecture note by Timo Weigand: https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf

 

[victorvmotti]

These are the socalled "tadpole diagrams" they are of course infinite, because $D_F(0)$ is undefined. You get rid of them by renormalization

Yes, I see that there are ways to get rid of them. For example, in the lecture note itself, https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf the partition function, including the value of all such diagrams, appears both on the denominator and nominator, and therefore is canceled out. But on https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf we look at a physical interpretation of the partition function as the vacuum energy (density) of the interacting theory. I also see that we can renormalize these divergences away.

But my question is that do these vacuum double-loops, summed up in the partition function, make any physical sense individually or can we interpret them physically at all.

Are you saying that these diagrams do not contribute to the ground state energy of the theory? If they contribute then what they are describing physically?

Are they playing a any role in the non-zero vacuum expectation value of the Higgs field?

 

[dextercioby]

Here is the lecture note by Timo Weigand: https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf

This is an exellent set of notes, not as known and popular as David Tong's ones.

 

[vanhees71]

Yes, I see that there are ways to get rid of them. For example, in the lecture note itself, https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf the partition function, including the value of all such diagrams, appears both on the denominator and nominator, and therefore is canceled out. But on https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf we look at a physical interpretation of the partition function as the vacuum energy (density) of the interacting theory. I also see that we can renormalize these divergences away.

But my question is that do these vacuum double-loops, summed up in the partition function, make any physical sense individually or can we interpret them physically at all.

Are you saying that these diagrams do not contribute to the ground state energy of the theory? If they contribute then what they are describing physically?

Are they playing a any role in the non-zero vacuum expectation value of the Higgs field?

These are two different things. The first one is the issue of the socalled "disconnected diagrams" in the Feynman diagrams for the S-matrix at a given order perturbation theory. With a "disconnected diagram" one refers to a diagram which only consists of closed loops and which thus is not connected to anyone of the external points (i.e., having no external legs). It turns out that the sum of all these "vacuum bubble diagrams" add up to an undefined phase factor, which doesn't contribute to the S-matrix at all, i.e., you can factor these sum of the vacuum bubbles out and you can neglect it, because it's just the perturabative expansion of this phase factor to the considered order of the expansion, and that's why you need to consider only diagrams connected to the external legs in the depiction of your scattering process.

The ground-state energy by definition is 0 and thus the vacuum expectation value of the Hamiltonian is 0 by definition too. That means you simply cancel all these vacuum bubble diagrams.

Another place, where tadpole diagrams occur are diagrams in the $n$-point Green's functions, connected Green's functions, or proper vertex functions (i.e., the one-particle irreducible (1PI) amputated diagrams). There they usually provide constant self-energy insertions, which are anyway treated by the renormalization description. If they occur on external legs you can neglect them to begin with since you can neglect any self-energy insertions on external legs, because in calculating the S-matrix the only perturbative modification from the external legs are the wave-function renormalization factors which finally are also canceled anyway.

If tadpole self-energy insertion occur in an internal line they are also correctly treated by the renormalization procedure, because they are sub-divergences which have to be subtracted anyway as explained by the BPHZ formalism of renormalization.

As I said before, you need to keep the tadpole diagrams in the regularized theory to keep Ward-Takahashi identities of (global or local) symmetries valid at any stage of the calculation and get all the cancellations of divergences correct when renormalizing when calculating the proper vertex functions by applying the BPHZ foralism (or the Zimmermann forest formula).