LSZ reduction and Renormalization constants

[earth2]

Hey folks,

i have a question about LSZ and how to take into account the renormalization constants of the theory in question.

In the derivation, only the field strength renormalization enters as a factor of Z (or square root thereof) but some mates said that also the vertex renormalization constants should be taken into account for LSZ. I've looked about a trillion times into Peskin and Schroeder (e.g. chapter 7) but i don't see how the vertex renormalization constant has anything to do with LSZ...

Where am I going wrong?

Thanks,
earth2

 

[ParticleGrl]

The LSZ reduction formula uses the pole mass, and the measured charge. This implicitly makes use of vertex and mass corrections.

 

[Lapidus]

The LSZ reduction formula uses the pole mass, and the measured charge. This implicitly makes use of vertex and mass corrections.

Would you mind to explain that a bit further?

As I understand that LSZ formula connects the correlation function (transition amplitude/ Green function) with the S-matrix.

A correlation function with four scalar fields can be represented by all kinds of Feynman diagrams with four external legs. Problem, all external legs are associated with off-shell propagators, and also disconnected graphs appear.

The LSZ formula gets rid of the external propagators and all the disconnected graphs by turning the correlation function into a S-matrix.

The in and out states must be far in the past and far in the future, respectively. That's because so that all interactions turn off, since by deriving the LSZ formula we had to assume that the fields behave like free fields, i.e. fields that do not interact.

Even if particles evolved far in time and all interactions are almost shut off, the particles/ fields still interact with themselves. That's why we need to rescale/ renormalize them. Again, it is because to make the LSZ formula work, that we derived under the premise that the fields act like free fields.

OK, is that complete nonsense that I just wrote? What is wrong, what less wrong?

I would really, really appreciated if someone could help shedding some light on what's going on (deep down!) with the LSZ formula.

 

[Polyrhythmic]

The in and out states must be far in the past and far in the future, respectively. That's because so that all interactions turn off, since by deriving the LSZ formula we had to assume that the fields behave like free fields, i.e. fields that do not interact.

That's not quite correct. The whole point of the LSZ formula is to describe an interaction by allowing us to calculate scattering amplitudes. The LSZ formula describes an interacting theory, as long as we make sure that certain normalizations hold, one of them being that the vacuum expectation value of the field is equal to zero. These normalizations enter the Lagrangian of the theory in the form of renormalization constants.

 

[Lapidus]

thanks for replying!

That's not quite correct. The whole point of the LSZ formula is to describe an interaction by allowing us to calculate scattering amplitudes. The LSZ formula describes an interacting theory, as long as we make sure that certain normalizations hold, one of them being that the vacuum expectation value of the field is equal to zero. These normalizations enter the Lagrangian of the theory in the form of renormalization constants.

Ok, so a more correct wording would be 'behave like free fields in the sense that have the same normalizations as free fields'.

But still (I have started another https://www.physicsforums.com/showthread.php?t=490516") I have not fully understood what the LSZ does good to us. You say it describes interactions by allowing us to calculate scattering amplitudes. But we, or rather many Professors in their textbooks and notes can do that also without them.

I suspect it has something to do we 'amputated' and 'disconnected' Feynamn diagrams, and that we get rid off them by using the LSZ formula when going from correlation functions to S-matrices. Can that be?

 

[Polyrhythmic]

Well, to my knowledge, without the LSZ formula, one cannot calculate physically measurable quantities like scattering cross sections (which require the S-matrix of a process). Of course, you can introduce and discuss perturbation theory and Feynman diagrams without talking about cross sections, but the LSZ formula is what gives the connection.
I'm not sure about the professors and textbooks you are talking about, but the way I learned it, you need the LSZ formula. I can recommend the book by Srednicki, which is available for free at: http://www.physics.ucsb.edu/~mark/qft.html .
It's my main source for studying QFT (I'm still a student, so correct me if I'm wrong ;) ), and I can recommend it to you. It even contains a chapter devoted to the LSZ formula alone.

 

[Lapidus]

Zee, Mandl, Griffiths, many others calculate cross sections without LSZ formula. Even Peskin and Schroeder introduces the LSZ formula (chapter 7) after they have carried out some cross section and decay rates calculations without LSZ (chapter 5).

But take Srednicki, chapter ten, equation 10.5. What happens here? First, we note that disconnect diagrams do not contribute. Second, in the diagrams (representing the correlation function), the propagators of the external lines get canceled by the Klein-Gordon operators in the LSZ formula.

I assume that in the diagrams the momentum of the external lines are off-shell. The LSZ formula canceles them now and gives us Feynman graphs with 'external lines amputated'.

But again, I'm not sure why that is.

What does the corellation function physically mean? (Vaccum transition amplitude of interacting fields which gives Feynman graphs with off-shell external lines??)

Related, what does the generating functional Z(J) physically mean?

thanks

 

[Polyrhythmic]

I just took a quick look at Peskin and Schroeder, and noticed that in section 7, they note that they had actually cheated in section 5 and that a correct derivation actually requires the LSZ-formula.
Eq. 10.5 is the LSZ formula for two ingoing and two outgoing particles, it describes a scattering process. We have the correlation function which can be expressed through the generating functional Z(J) in terms of Feynman diagrams. They further show that only connected contribute. I'm not sure how you come to the conclusion that the external lines are off-shell, I don't see any obvious reason for this. Also, the propagators for those external lines do not get cancelled, they simply disappear after evaluating the LSZ formula, but that doesn't manipulate the Feynman diagrams in any way. To my knowledge, the scattering matrix (which is what the LSZ formula gives) can't be visualized by Feynman diagrams. What you do is:

1.) Express the correlation function in terms of the generating functional.
2.) Take a look at the LSZ-formula, check which diagrams actually contribute.
3.) Calculate the scattering amplitudes by inserting the remaining contributions into the LSZ-formula
4.) Calculate cross-sections.

The correlation function gives you the vacuum expectation value of a product of operators. In the special case where you have products of two field operators, the correlation function gives you the Green function (propagator).
The generating functional Z(J) is just the path integral expression of <0|0> with 0 being the vacuum state, so basically it's a "correlation function without any operators". Expanding it into Taylor-series gives us Feynman diagrams.

 

[Lapidus]

Thanks for the reply!

I'm not sure how you come to the conclusion that the external lines are off-shell, I don't see any obvious reason for this. Also, the propagators for those external lines do not get cancelled, they simply disappear after evaluating the LSZ formula, but that doesn't manipulate the Feynman diagrams in any way.

The reason is that after taking the functional derivatives, the sources are removed. The particle propagated from the vertex to the source. Propagators represent 'virtual' particles, thus represent per definition off-shell particles. The external propagators have to get canceled, since they are 'unphysical'. The Klein-Gordon operators in the LSZ formula do that, or as you say they simply disappear after evaluating the LSZ formula.

To my knowledge, the scattering matrix (which is what the LSZ formula gives) can't be visualized by Feynman diagrams.

Again, many textbook do it, for example Peskin and Schroeder or Zee. They get the right results, though they cheat somewhere since the right way to is to do is using the LSZ formula. But where and how do they cheat? That was my question.

The correlation function gives you the vacuum expectation value of a product of operators. ...The generating functional Z(J) is just the path integral expression of <0|0> with 0 being the vacuum state, so basically it's a "correlation function without any operators".

So are you suggesting, that when I square them I get probabilities for measurable processes?

How can that be, when the correlation function represents 'off shell' processes and the generating functional Z(J) is dependent on some source J, which is just a mathematical crutch, that must be set to zero in the end of the calculation for a real physical process?

 

[Polyrhythmic]

Thanks for the reply!

The reason is that after taking the functional derivatives, the sources are removed. The particle propagated from the vertex to the source. Propagators represent 'virtual' particles, thus represent per definition off-shell particles. The external propagators have to get canceled, since they are 'unphysical'. The Klein-Gordon operators in the LSZ formula do that, or as you say they simply disappear after evaluating the LSZ formula.

This is not true. The sources are removed, but what remains are propagators of on-shell particles. If this weren't the case, the whole scenario for example of two particle scattering would be treated in a nonsensical way: We take two incoming particles into a scattering process which for some reason would be virtual particles (?) and then we remove their propagators (??), so we would basically end up with no propagators for nothing at all.

On-shell particles are represented by external propagators, whereas virtual particles are internal propagators. (See for example Srednicki Figure 10.1.)

Again, many textbook do it, for example Peskin and Schroeder or Zee. They get the right results, though they cheat somewhere since the right way to is to do is using the LSZ formula. But where and how do they cheat? That was my question.

I must admit that I didn't check Peskin/Schroeder's derivation in detail, I will do it when I have time.

So are you suggesting, that when I square them I get probabilities for measurable processes?

Measurable quantities like scattering cross sections depend on the square of the scattering matrix elements.

How can that be, when the correlation function represents 'off shell' processes and the generating functional Z(J) is dependent on some source J, which is just a mathematical crutch, that must be set to zero in the end of the calculation for a real physical process?

The correlation function doesn't represent off-shell processes. It consists of all contributing diagrams that in turn consist of on-shell and off-shell contributions.

 

[Lapidus]

Polyrhytmic, do you see the product of four Klein-Gordon operators in the LSZ formula, equation 10.5 in Srednicki? Then look at the equation of the correlation functions 10.9, do you see the products of the four external propagators? What do you conclude?

Fourier transform 10.5 and you get a product of the four external propagators. Why should there be a term with the product of the four external propagators in the LSZ formula (which links the S-matrix with the correlation function)?

what remains are propagators of on-shell particles.

There is no such thing as propagators for on-shell particles.

the whole scenario for example of two particle scattering would be treated in a nonsensical way: We take two incoming particles into a scattering process which for some reason would be virtual particles (?) and then we remove their propagators (??), so we would basically end up with no propagators for nothing at all.

Here you confuse S-matrix with correlation function. When you want to visualize S-matrices with Feynman diagrams , then all external lines correspond to physical particles. Some texts do that, exactly those that do not use the LSZ approach!