Misconceptions about Virtual Particles - Comments

[A. Neumaier]

H -> WW* -> whatever? It is one of the standard Higgs decays. The experimental papers have nice collections of references: CMS, ATLAS 1, ATLAS 2

I found 5, 1, and 0 times the word virtual in these papers. Never is the $W$ or $W^*$ labelled as virtual. Furthermore, CMD never mentions $W^*$. I found not a single occurrence of off-shell, off shell, or offshell in any of the three papers.

Figure 1 in Atlas1 (the only Feynman diagram there) clearly shows that $H\to WW$ and $H\to WW^*$ have the $W$ and $W^*$ as external legs, hence on-shell, hence resonances. Note that Figure 3 is not a Feynman diagram but a space-time diagram, and carefully displayed in a different way!

CMA and Atlas2 have no Feynamn diagram. But all three papers frequently refer to $H\to WW$ (without reference to decay products), and the two Atlas papers also frequently refer to $H\to WW^*$ (without reference to decay products), in agreement with the Feynman diagrams they draw.

I also looked at reference [4] suggested by Atlas1, which says on p.8: ''The dominant backgrounds are non-resonant $WW$, $t\bar t$, and $Wt$ production, all of which have real $W$ pairs in the final state.'' Again $W^*$ (here apparently labelled $W^{(*)}$) is not explained.

This means that based on your references it is clear that in the decay process $H\to WW$ or $H\to WW^*$, both $W$ and $W^*$ are on-shell particles (represented by external legs of Feynman diagrams), though I still have no idea what $W^*$ means. Since it seems to occur only in the combination $WW^*$, it looks as if it is perhaps just a joint notation for one of the three $W$'s and its charge conjugate?

I also looked at the particle data group (PDG) entry on the Higgs boson $H^0$. It lists $H\to W^+W^-$ in table 11.2 but not $H\to WW$ (which is however mentioned in the main text on p.19), and $H\to WW^{(*)}$ in Fig.11.16, without explaining the latter either. For the decay they talk of channels, which is standard S-matrix terminology for real particles. This PDG paper uses the word virtual a few times, each time correctly for internal lines mediating a collision process - never for the decay products $WW$ or $W^*$. It uses the word off-shell 4 times, the most relevant two on pp.16-17 in the context of simulations with off-shell bosons, which are described as simulations of the decay $H\to llll$ ''without any on-shell approximation''. This means that this inclusive decay was not simulated as a sequence of three separate decay processes ($H\to WW$ and twice $W\to ll$, which would be inaccurate in view of the other approximations made in the simulation) but as a single decay process (unresolved in space-time). A few more details are given here (in Section 9). They refer to hep-ph/0604011 for details of the simulation mechanism. They say (correctly) in the abstract ''The decay of the Standard Model Higgs boson into four leptons via a virtual W-boson or Z-boson pair is one of the most important decay modes in the Higgs-boson search at the LHC.'' There is no occurrence of $W^*$.

This means that in the simulation calculations done as part of the analysis, both $W$ particles are treated as virtual particles since these concern matrix elements of the 4-lepton decay. As always, the virtual particles appear as a calculational tool, not as a carrier of real physics.

The above shows that those publishing the experimental work were careful (and correct) in their language.

This seems not to be the case in the unpublished summary you linked to here:

Here is a Feynman diagram - it is so basic that I doubt the papers include it, but the papers I linked to are exclusively about this process. Pages: all.

That fourth paper says: ''When the mass of the Higgs boson is greater than 136 GeV the predominant mode of Higgs decay is to a pair of on/off-shell W bosons'', without giving formulas or references. (The only - global - reference given there is another summary, that neither uses the words virtual or off and doesn't mention a $W^*$.) No other usage of off-shell is resent, and the word virtual does not appear. But the Feynman diagram given shows that both $W$ and $W^*$ are virtual in the process displayed. It is one of the contributions to the decay $H\to llll$ discussed already in my analysis above. It does not describe (as claimed) the Higgs decay to a pair of W bosons, which is a different decay process with different S-matrix elements and both W-bosons on-shell.

In the light of the testimony of the three published papers you summoned, do you still want to uphold your earlier claims?

The W* in H->WW* -> ... appears as intermediate line in a Feynman diagram

The star means it is off-shell.

And could you please find a proper (published) reference where the meaning of $W^*$ and/or $W^{(*)}$ (if this is not the same) is explicitly explained?

 

[mfb]

I found 5, 1, and 0 times the word virtual in these papers. Never is the $W$ or $W^*$ labelled as virtual. Furthermore, CMD never mentions $W^*$. I found not a single occurrence of off-shell, off shell, or offshell in any of the three papers.

Those are the experimental papers. They also don't introduce what W bosons are, because they expect the reader to know that.

Figure 1 in Atlas1 (the only Feynman diagram there) clearly shows that $H\to WW$ and $H\to WW^*$ have the $W$ and $W^*$ as external legs, hence on-shell, hence resonances. Note that Figure 3 is not a Feynman diagram by a space-time diagram, and carefully displayed in a different way!

A Higgs cannot decay to two on-shell W bosons. It simply does not have enough mass. This is even more obvious for H->ZZ*->llll where the experiments can reconstruct all 4 leptons.

CMA and Atlas2 have no Feynamn diagram. But all three papers frequently refer to $H\to WW$ (without reference to decay products), and the two Atlas papers also frequently refer to $H\to WW^*$ (without reference to decay products), in agreement with the Feynman diagrams they draw.

They mention the W decay products everywhere, including two of the three titles, and all three abstracts.

This means that based on your references it is clear that in the decay process $H\to WW$ or $H\to WW^*$, both $W$ and $W^*$ are on-shell particles (represented by external legs of Feynman diagrams), though I still have no idea what $W^*$ means.

It is not, and I don't see how you got that impression.

Since it seems to occur only in the combination $WW^*$, it looks as if it is perhaps just a joint notation for one of the three $W$'s and its charge conjugate?

This is just nonsense. As I said, it is a notation for an off-shell W. The W* decay products have an invariant mass way below the W mass. Calculate the sum of masses for two W and compare it to the Higgs mass, it simply does not add up.
For H->ZZ*->llll this is discussed more clearly in the papers as there the decay products are all detected. As an example, this older publication refers to the Z* as "the lower-mass Z* boson" (page 7, right column) and discusses this lower mass on the next page.

This means that this inclusive decay was not simulated as a sequence of three separate decay processes ($H\to WW$ and twice $W\to ll$, which would be inaccurate in view of the other approximations made in the simulation) but as a single decay process (unresolved in space-time).

Yes of course. It is a single process. For one W you can consider production and decay as separated process if you want, for the W* you cannot as it is not on-shell. You can still calculate the total contribution of Higgs decays via WW* (W+W-, WW, ... all those notations mean the same thing, please don't overanalye who used which notation because there is no deeper meaning in it).

A few more details are given here (in Section 9). They refer to hep-ph/0604011 for details of the simulation mechanism. They say (correctly) in the abstract ''The decay of the Standard Model Higgs boson into four leptons via a virtual W-boson or Z-boson pair is one of the most important decay modes in the Higgs-boson search at the LHC.'' There is no occurrence of $W^*$.

The W* is the off-shell virtual W boson.

This means that in the simulation calculations done as part of the analysis, both $W$ particles are treated as virtual particles since these concern matrix elements of the 4-lepton decay. As always, the virtual particles appear as a calculational tool, not as a carrier of real physics.

The one W acts like a real particle. You could easily imagine a B meson there that moves a few millimeters before it decays, it would not change the situation.

It does not describe (as claimed) the Higgs decay to a pair of W bosons, which is a different decay process with different S-matrix elements and both W-bosons on-shell.

This process does not happen for a 125 GeV Higgs boson. Unless the Higgs itself is off-shell. And there we are again at the problem. Is the Higgs discussed before real? If yes, is it real if we reconstruct a mass 2 MeV away from the actual mass? 10 MeV? 1 GeV? 100 GeV? Is it never real?
How can the W or Z bosons ever be on-shell if we can observe their decay width? Where is the limit?

All those questions have an easy answer if you say "it depends on who calculates those things". But then the classification is purely arbitrary.

In the light of the testimony of the three published papers you summoned, do you still want to uphold your earlier claims?

You confirmed them, where is the problem? The W* is part of a single process H->lvlv.

And could you please find a proper (published) reference where the meaning of $W^*$ and/or $W^{(*)}$ (if this is not the same) is explicitly explained?

See "low-mass Z*" above.

 

[A. Neumaier]

This means that based on your references it is clear that in the decay process $H\to WW$ or $H\to WW^*$, both $W$ and $W^*$ are on-shell particles (represented by external legs of Feynman diagrams)

It is not, and I don't see how you got that impression.

I don't see how you got your impression. My conclusion is clearly visible from the Atlas1 paper you had cited:

Figure 1 in Atlas1 (the only Feynman diagram there) clearly shows that $H\to WW$ and $H\to WW^*$ have the $W$ and $W^*$ as external legs, hence on-shell, hence resonances. Note that Figure 3 is not a Feynman diagram but a space-time diagram, and carefully displayed in a different way!

Moreover, the paper gives numerical characteristics for these processes (in the tables) which make sense only if they are actual processes with an associated S-matrix elements (in terms of which the numbers reported are defined). There is no way in which these numbers can be associated to virtual processes.

That further decay products are mentioned in the abstracts (since they were used to deduce the numbers given) doesn't affect the fact that the result of their analysis concerns the processes stated explicitly, for which they have explicitly given the diagrams (rather than something nowhere explicitly stated in the results that you apparently read into the paper).

I'll respond to the remainder of your post after I have read more. But I still would like to see an explicit definition of the meaning of $W^*$ or $Z^*$ rather than casual phrases that assume that the meaning is already known to the reader.

Note that I have a good knowledge of QFT and of its application to QED and QCD phenomenology but I never looked into details of the standard model beyond that. I am using the present discussion to learn to understand the specific issues related to the Higgs sector - assuming that the theoretical concepts involved in its description are the same as elsewhere in QFT.

 

[mfb]

I don't see how you got your impression.

This is really basic experimental particle physics. So basic that the publications don't even bother explaining it in detail because every experimental particle physicist knows it. I am one of those experimental particle physicists.

Did you calculate the sum of two W or Z masses and compared it to the Higgs mass as I asked? This is a really simple check you can do to see that the W/Z cannot be both on-shell.

But I still would like to see an explicit definition of the meaning of $W^*$ or $Z^*$ rather than casual phrases that assume that the meaning is already known to the reader.

Maybe it is explained in even earlier papers, but it is so basic that I really think the authors didn't bother. The star is also irrelevant on its own - you could write WW as well, if you keep in mind that not both W can be on-shell.

(rather than something nowhere explicitly stated in the results that you apparently read into the paper)

I don't read that into the papers, I am part of the people writing papers about LHC results.

 

[A. Neumaier]

every experimental particle physicist knows it.

But they learn it from somewhere. I am sure they didn't learn it by being told (as I effectively was told by you): Here are 220 pages of experimental data discussion plus a thousand references - now try to make sense of it yourself.

From which papers can I learn it in an introductory, fully explained fashion? That was each time my question, and I never got an answer from you.

Did you calculate the sum of two W or Z masses and compared it to the Higgs mass as I asked? This is a really simple check you can do to see that the W/Z cannot be both on-shell.

Yes, I did. But I want to understand what is behind the pure assertion - the relation to the formulas of QFT. Which set of possible states is associated with $W^*$ (whatever you call it)? In which sense is it a $W$ if it hasn't its mass? What is the meaning of the branching ratio reported for a reaction $H\to WW^*$? Is it not defined through a partial width defined by a formula such as the golden section formula in which the particles are represented by external lines of Feynman diagrams - i.e., not as virtual particles? I am trying to figure out this kind of questions and your responses have given me very little help so far.

I am part of the people writing papers about LHC results.

Neither the particle data group nor the CMS and Atlas references you gave used your terminology (though they used the notation $W^*$). Do they speak a different dialect from you? Why don't they talk about virtual particles except when they refer to internal lines in a diagram?

 

[mfb]

But they learn it from somewhere. I am sure they didn't learn it by being told (as I effectively was told by you): Here are 220 pages of experimental data discussion plus a thousand references - now try to make sense of it yourself.

It usually works in a more informal way, indeed. Either you figure it out based on the masses, or you can ask: "Hey what does the star mean?" "It is off-shell".

Anyway, I used google a bit more and found Decays of the Higgs Bosons, discussing "the Higgs boson decay into WW with one off-shell W boson" on page 5. It has references to Higgs-scalar decays: $H \to W^\pm + X$ and "Rizzo Phys. Rev. D22 (1980) 389" where I didn't find a link.

It is a W because it shares its couplings.

 

[Haelfix]

I must admit that I am confused! mfb is completely correct as this is basic particle physics. Or at least if it isn't then I've been doing it wrong for the past 15 years. I also don't understand why this in anyway contradicts your insight article. If you like, you can draw one of the possible reactions as H --> Z l l, where the real Z is treated as an unstable particle that subsequently decays (in another diagram that you draw) and we reconstruct the usual peak (which we can interpret in the usual way as a pole of our S matrix process). The internal Z is a virtual particle as per your own definition!

So again, what is the problem? The whole point of this little exercise and the reason I originally brought it up is that is often convenient (experimentally) to describe things in the other way, precisely b/c there are multiple decay chains that are possible and we like to classify probabilities in a certain way (for instance does the total probability for the myriad final states of WW beat the total probability for the myriad final states of ZZ?

 

[A. Neumaier]

I also don't understand why this in anyway contradicts your insight article.

It shouldn't but on the surface it does - when I see Feynman diagrams with off-shell external legs, which shouldn't exist according to the textbook picture, and the corresponding branching ratios don't make sense in the standard way. So for me there is something more to be understood clearly enough that I can explain it in my own words. I am still reading, but gradually the confusion clears.

If you like, you can draw one of the possible reactions as H --> Z l l, where the real Z is treated as an unstable particle that subsequently decays (in another diagram that you draw) and we reconstruct the usual peak (which we can interpret in the usual way as a pole of our S matrix process). The internal Z is a virtual particle as per your own definition!

Thanks. Yesterday night I discovered the first paper http://arxiv.org/abs/hep-ph/9807536 that describes it this way. Before that I had to guess the meaning from the context, which was never clear enough. The diagram that mfb had claimed as being equivalent to $H\to WW^*$ is in fact a diagram for $H\to W^*W^*$, while I had taken the equivalence at face value. I could see that these are different only after having finally read a precise enough explanation (Figures 2 and 3 in the cited paper). I am somewhat surprised that the PDG account doesn't mention these conventions but uses the star notation without comment - at least I haven't found any explanatory comment.

 

[vanhees71]

It usually works in a more informal way, indeed. Either you figure it out based on the masses, or you can ask: "Hey what does the star mean?" "It is off-shell".

Yep, this notation you find everywhere. E.g., sometimes you confuse your experimental colleagues by talking about "virtual photons", writing $\gamma^*$, when you in fact mean dileptons, but that's just slang again. I'd not say that there's any problem with it in the scientific community. The same is true for $W^*$. I guess it just means a $W$ line that is far off-shell (in the sense of the pole mass or peak mass of the $W$ meson, where only the former is of course a well-defined gauge-invariant object). What's of course observed is not such a far-off-shell W boson but the stable decay products showing up as signals from the detectors. Puristically speaking any resonance is never on-shell, because it has not a well-defined mass but a mass distribution given by its spectral function (i.e., the imaginary part of its (retarded!) propagator). So with some right you should put a star on every resonance, when you write a short-hand reaction like $H \rightarrow WW$. The two W's decay further anyway, and you detect the stable endproducts.

The same thing you have in the heavy-ion dilepton context with $\rho$ mesons way below the "mass shell", where it appears in Dalitz-decay diagrams as a vectorm-meson resonance model for the electromagnetic transition form factor, e.g., in the decay of baryon resonances or other mesons (like the $\omega$ or $\eta$). The $\rho$ meson has a low-mass tail down to $2m_{\text{e}}$, and thus it has even spectral strength there, but what goes into the rates is anyway the squared $\rho$ propagator in the em. current-current correlation function.

I think, you fight against windmills, if you try to get rid of the usual slang in the HEP/HI community. Sometimes you have confusion, and then you can clarify it by reminding the confused colleagues about the meaning of resonances vs. particles, which are defined as asyptomptic free states (external lines of Feynman diagrams) and that a resonance with finite width around its pole mass can only occur as external lines in a very specific sense when calculating decay rates in an approximate way.

Where you really have still a lot of unnecessary confusion is when it comes to oscillations (particularly neutrino oscillations). There you must explicitly take care of the production and detection mechanism to make proper sense of the hand-wavingly derived "plane-wave oscillation formula", but that's another topic.

 

[A. Neumaier]

Puristically speaking any resonance is never on-shell, because it has not a well-defined mass but a mass distribution given by its spectral function (i.e., the imaginary part of its (retarded!) propagator).

It is on-shell in the analytic continuation with a well-defined complex mass. Otherwise it would be impossible to compute cross section for processes associated with them, without taking into account all their decay products. It would not even be possible to give it an unambiguous meaning! And it has a state with a well-defined dynamics in time since it is (almost, ignoring very tiny tails at negative masses) equivalent to a state in the undeformed, physical Hilbert space involving contributions from the scattering spectrum with a mass distribution given by its spectral function. None of these can be said of a virtual particle.

I think, you fight against windmills, if you try to get rid of the usual slang in the HEP/HI community.

I never tried to do that. My goal is to fully understand what they say and do, in clear and unambiguous terms, and pointing out where and how sloppy terminolgy is used and what the latter really means. I want to have for myself a clear mental picture and a precise way of talking about it.

 

[vanhees71]

Sure, it's also a question of time scales. On a time scale of a typical nuclear reaction a neutron or a pion is a very stable particle and thus can be treated almost always as such.

 

[A. Neumaier]

"Rizzo Phys. Rev. D22 (1980) 389"

I think the intended reference was p.722, http://journals.aps.org/prd/abstract/10.1103/PhysRevD.22.722

 

[A. Neumaier]

"Hey what does the star mean?" "It is off-shell".

If you like, you can draw one of the possible reactions as H --> Z l l, where the real Z is treated as an unstable particle that subsequently decays (in another diagram that you draw) and we reconstruct the usual peak (which we can interpret in the usual way as a pole of our S matrix process). The internal Z is a virtual particle as per your own definition!

So for me there is something more to be understood clearly enough that I can explain it in my own words.

I think I understand now in my own words what happens in the $H\to WW^*$ decay. I added the following two sections to the collection of definitions in the Insight article The Physics of Virtual Particles.

Branching fractions. If an unstable particle can decay in several different ways, the branching fraction of each single decay (or group of decays) is the relative frequency of this decay (or group of decays) compared to all decays. It can be computed from the S-matrix elements of all individual processes.

External lines and off-shell particles.
As a consequence of the description of S-matrix elements, the external lines usually correspond to on-shell particles. and then describe real particles before and after a collision or decay. However, there is the custom of using (generalized) Feynman diagrams also in certain cases where one or more out-particles are off-shell (typically denoted by a *). An example (see Figure 2 in http://arxiv.org/abs/hep-ph/9807536) is the Higgs decay $H\to WW^*$ in which one of the $W$ produced is off-shell, hence not a real particle but an unobservable label. Such a Feynman diagram is short-hand for a family of Feynman diagrams obtained by attaching to each off-shell particle another vertex and its admissble interaction partners, in case of the $W^*$ two leptons. Thus the single Feynman diagram visualizing the decay $H\to WW^*$ stands in fact for $H\to WX$, where $X$ are two external lepton lines attached to a vertex that turns $W^*$ into an internal line, as it should be for off-shell particles. The branching fractions for decays involving off-shell particles must be interpreted in the same way. For example, the branching factor for the decay $H\to WW^*$ is defined as the inclusive branching factor for all $H\to WX$, where $X$ are two observable leptons consistent with the standard model interactions.

 

[DarMM]

The same applies to unstable particles. Strictly speaking they are never particles but resonances and as such appear in the description of scattering matrix elements as internal lines.

A difference between resonances and virtual particles is that resonances are actual states in the Hilbert Space, i.e. they are present even non-perturbatively.

Also the debate about the "inner structure of protons" is funny. I think, the whole debate is much ado about nothing or say about sloppy language in the QFT community. Any practitioner of QFT,...

I agree that this debate is in a sense "about nothing", but many people, even professionals do not think as you seem to believe they do. They think virtual particles are really physical states, not just perturbative labels and that the proton is really a "sea" of quarks and gluons, rather than this being an approximation model.

That is why it is still worth pointing these things out, sloppy language can lead to sloppy thinking.

 

[DarMM]

I think the much more interesting issue is the series is asymptotically divergent. How can we get answers from a divergent series?

We can get answers because the series is an asymptotic series, i.e. a specific kind of divergent series where the early terms do give a good approximation. In such a series the series approaches the full result up until the point where the error is of order $e^{-\frac{1}{\lambda}}$, with $\lambda$ the expansion parameter. At this point it begins to diverge.

No surprise in QFT as $e^{-\frac{1}{\lambda}}$ is basically an instanton contribtuion.

 

[atyy]

A difference between resonances and virtual particles is that resonances are actual states in the Hilbert Space, i.e. they are present even non-perturbatively.

So this Physics FAQ is wrong?
http://www.desy.de/user/projects/Physics/Quantum/virtual_particles.html
"Then, the use of virtual particles as a communication channel is completely consistent with quantum mechanics and relativity. That's fortunate: since all particle interactions occur over a finite time interval, in a sense all particles are virtual to some extent."

Presumably the answer should be that virtual particles do not exist, whereas resonances are short lived states in Hilbert space which can be seen by a suitable measurement?

 

[RockyMarciano]

So this Physics FAQ is wrong?
http://www.desy.de/user/projects/Physics/Quantum/virtual_particles.html
"Then, the use of virtual particles as a communication channel is completely consistent with quantum mechanics and relativity. That's fortunate: since all particle interactions occur over a finite time interval, in a sense all particles are virtual to some extent."

Presumably the answer should be that virtual particles do not exist, whereas resonances are short lived states in Hilbert space which can be seen by a suitable measurement?

IMO this whole discussion is pointless. All that is ever measured are interactions(that's why the meaningful quantities come from the vertices in the diagrams), never "real particles" in the idealized infinite time of the free field theory nor "virtual particles". The detections, whether one refers to them as clicks, dots or tracks or scattering cross sections all refer to interactions and it is nonsensical to debate whether the particles exist as it will only depend on the kind of book-keeping one previously decides to use based on their particular prejudices.
Resonances are just inelastic scattering cross sections and again measure short lived interactions. If one decides to call them "particles" it is totally irrelevant if one classifies them as virtual or as idealized states in Hilbert space in the free theory. It doesn't affect the physics at all since the physics deals with the interacting fields.

 

[A. Neumaier]

So this Physics FAQ is wrong?

It tells (just as wikipedia) the ''popular science'' story based on the silent (but meaningless) identifications mentioned in post #58, not the ''research level'' one which has precise conventions. In addition, the Physics FAQ makes some other meaningless statements such as ''A virtual particle with momentum p corresponds to a plane wave filling all of space''. In fact it treats virtual particles as if they had wave functions (which would make them on-shell).

 

[mfb]

It tells (just as wikipedia) the ''popular science'' story based on the silent (but meaningless) identifications mentioned in post #58, not the ''research level'' one which has precise conventions. In addition, the Physics FAQ makes some other meaningless statements such as ''A virtual particle with momentum p corresponds to a plane wave filling all of space''. In fact it treats virtual particles as if they had wave functions (which would make them on-shell).

As discussed before, there is no clear line between a W* that is clearly off-shell and a muon that is treated as real particle, despite its finite lifetime and corresponding uncertainty in the invariant mass of its decay products. Yes, the muon is a much better approximation to a real particle with a proper mass, but it is still just an approximation.

 

[A. Neumaier]

there is no clear line between a W* that is clearly off-shell and a muon that is treated as real particle

What is regarded as existent (i.e., modeled by a state rather than symbolically by an internal line) in a particular experimental situation is in borderline cases a matter of modeling. But since it changes the predictions (factorized or unfactorized computations in the simulation!), one can in principle distinguish the two situation - though (due to limited data and/or simulation accuracy) perhaps not in practice.

In any case, it makes no sense (except in sloppy talk) to model a particle as off-shell and then claim it has a lifetime or another spatio-temporal property since the latter are properties of on-shell particles only. If one wants something to have a lifetime one must model it as on-shell (even when the sloppy talk says otherwise).

Saying (as the Physics FAQ does) ''A virtual particle with momentum p corresponds to a plane wave filling all of space'' is scientifically empty since this sentence cannot be translated to something more definite. The plane wave can formally be associated to a state only, in this case probably a bare state in a simplified model - but to associate on the formal level (not in one's fantasy) a plane wave with an internal line in a Feynman diagram has no function at all - it helps neither do nor to interpret calculations. One could as well associate plane waves to the momentum of a car, rocket or planet...

Thus the status of this FAQ entry is not better than a lie-to-children.

 

[mfb]

What is regarded as existent (i.e., modeled by a state rather than symbolically by an internal line) in a particular experimental situation is in borderline cases a matter of modeling. But since it changes the predictions
(factorized or unfactorized computations in the simulation!), one can in principle distinguish the two situation - though (due to limited data and/or simulation accuracy) perhaps not in practice.

Your model never influences physics. It would be like claiming 2 meters is a fundamental concept of nature if you use one model above 2 meters and a different one below that just because they are better approximations.

In any case, it makes no sense (except in sloppy talk) to model a particle as off-shell and then claim it has a lifetime or another spatio-temporal property since the latter are properties of on-shell particles only. If one wants something to have a lifetime one must model it as on-shell (even when the sloppy talk says otherwise).

The decay width is an observable - it does not care about the model you use. Every unstable particle has a decay width, if you measure the invariant mass of its decay products precise enough you will always get different values for different decays.

 

[A. Neumaier]

The decay width is an observable - it does not care about the model you use.

An observed decay width always needs for its interpretation a model involving a process observed in time, hence a state, hence a real particle (resonance). A virtual particle never has a decay width since it does not make formal sense to talk about its temporal properites. Thus one cannot model a decay width by a virtual particle.

But where you may have a choice (and hence a subjective aspect depending on the model used) is in the interpretation of a particular composite decay process for which an experimental decay width is available. You may model it as a single process with unresolved time, expressed by intermediate off-shell virtual particles. Or you may model it as a partially time-resolved process with an intermediate on-shell resonance and a factorizing computation. In borderline cases, the two models will produce approximately the same decay width. At a given finite resolution the models may agree, in which case one cannot decides whether the intermediate particle is on- or off-shell. But with sufficient computing time and experimental accuracy it should in principle be possible to distingush the two situations. This is what I was referring to in my post.

 

[mfb]

An observed decay width always needs for its interpretation a model involving a process observed in time, hence a state, hence a real particle (resonance). A virtual particle never has a decay width since it does not make formal sense to talk about its temporal properites. Thus one cannot model a decay width by a virtual particle.

I'm not talking about virtual particles now, I'm discussing the "on-shell"/"off-shell" difference which does not have a clear dividing line. The W* in a Higgs decay is clearly off-shell, while muons in decays are considered as on-shell. And there is no sharp line dividing those two cases. You can treat one as off-shell and one as on-shell, fine, but that is just your model used, it is not the physics behind it. You can also treat the muon as off-shell particle (with way too much computing effort) or the W* as "not so far away from on-shell particles" (with a huge error).

 

[A. Neumaier]

You can treat one as off-shell and one as on-shell, fine, but that is just your model used, it is not the physics behind it.

So what is the same physics behind the two different treatments?

 

[vanhees71]

I'm not talking about virtual particles now, I'm discussing the "on-shell"/"off-shell" difference which does not have a clear dividing line. The W* in a Higgs decay is clearly off-shell, while muons in decays are considered as on-shell. And there is no sharp line dividing those two cases. You can treat one as off-shell and one as on-shell, fine, but that is just your model used, it is not the physics behind it. You can also treat the muon as off-shell particle (with way too much computing effort) or the W* as "not so far away from on-shell particles" (with a huge error).

Well, and the good thing is that it doesn't matter, how you call the diagrammatic elements of a Feynman diagram symbolizing a mathematical expression to calculate an S-matrix element. You just calculate it, square it, multiply it with the appropriate kinematical factors, and you get (within a model, of course, because without a model, you'd have no Feynman diagram to begin with) a lifetime/decay width or cross section.

There's, of course, a well defined meaning when calculating, say at tree level, the life time or decay width of a muon, which you treat as an "on-shell" external line in the corresponding diagram, using the Feynman rules from QFD, as there is a well-defined meaning for the diagrams describing a Higgs-boson decay via W mesons, where you finally of course measure not W mesons directly but their decay products. How call them, on shell or off shell is completely irrelevant. What counts is the well-defined meaning of an (approximately calculated) S-matrix element, and this you can compare with counts in the detectors at the LHC and be excited about the discovery of the Higgs boson of the Standard Model checking all the details about its nature, maybe hoping for a deviation from the Standard Model to find new physics.

 

[mfb]

So what is the same physics behind the two different treatments?

What is different? After a long time, you get stable particles flying away. W* or muon are "just" our description how those particles got created.

How call them, on shell or off shell is completely irrelevant.

Someone in this thread argued that there was a difference between those categories I think.

 

[vanhees71]

Of course there is a difference, but it's totally irrelevant. I rarely use the words "on-shell" and "off-shell" in my daily conversation although we also use QFT and Feynman diagrams all the time. Sometimes you talk about "on-shell" and "off-shell" form factors or the like. There was never ever any problem understanding each other.

What's sometimes a problem is, however, the notion of resonances, and that often resonances are also a model-dependent notion, particularly if you have them involved (again in our case often in electromagnetic transition form factors of hadrons in vector-meson-dominance approaches) in "far-off-shell situations", i.e., where you probe the "far-off-shell tails" of broad spectral functions of such resonances as the $\rho$ meson. Then there's often confusion about "what's a $\rho$ meson" or "what's the $\rho$ meson's spectral shape" etc. Then it's good to have the arguments given in the Insights article at hand.

What I wanted to say is only, that now this discussion goes a bit out of sense, because it's now becoming a purely semantical fight about the meaning of a $W^*$ vs. a $W$ although the meaning is very clear in terms of (perturbative) Feynman diagrams and the underlying QFT (the GSW model of the electroweak interaction within the Standard Model in this case).

 

[A. Neumaier]

After a long time, you get stable particles flying away. W* or muon are "just" our description how those particles got created.

After a long time, we are all dead, and physics no longer matters.
Physics happens on every time scale, not only at arbitrarily long time.

W* or muon are "just" our description how those particles got created.

No. A $W^*$ is never responsible for the creation of these particles, while a $W$ can be. For getting created is a process in time, and this requires for its description a state - honce (on the level of the particle language) an on-shell particle.

 

[mfb]

No. A $W^*$ is never responsible for the creation of these particles, while a $W$ can be.

But we get leptons from both in a Higgs decay.

For getting created is a process in time, and this requires for its description a state - honce (on the level of the particle language) an on-shell particle.

And there is the "on-shell" again. No unstable particle is "exactly on-shell". Not the W, and not even the muon.

 

[A. Neumaier]

But we get leptons from both in a Higgs decay.

No. We get leptons from Higgs, and we calculate it either in partially resolved time as $H\to Wll$ and $W\to ll$ or unresolved in time as $H\to llll$. The $W^*$ is only a formal book-keeping device, not something existing in time, hence nothing from which we can gat anything - except figuratively.

No unstable particle is "exactly on-shell". Not the W, and not even the muon.

Of course they are on-shell, on a mass shell with a complex rest mass, as explained in the companion Insight article. That's what defines an unstable particle or resonance. They have an associated eigenstate for the analytically continued Hamiltonian in the second sheet of the Riemannian surface defined by the resolvent. In terms of the original Hilbert space they have a state formed by a superposition of states in the continuous spectrum of $H$ (the system Hamiltonain in the rest frame of the unstable particle), with energies corresponding to a continuous range of masses - which may be visible as a resonance peak. Thus their real effective mass (defined by $m=H/c^2$) is unsharp, in the same sense as the position of a particle in a beam is unsharp. But this doesn't make unstable particles off-shell.

 

[mfb]

Fine, then we use different definitions for "on-shell". Forget about the word. You can include the muon in the Higgs decay for $H \to e^-e^+ (e^- \nu_\mu \bar\nu_e) (e^+ \bar\nu_\mu \nu_e)$, brackets for clarity. Is the muon gone?

 

[A. Neumaier]

Fine, then we use different definitions for "on-shell". Forget about the word. You can include the muon in the Higgs decay for $H \to e^-e^+ (e^- \nu_\mu \bar\nu_e) (e^+ \bar\nu_\mu \nu_e)$, brackets for clarity. Is the muon gone?

In this process as written, time is not resolved at all, and the S-matrix elements (that only describe the in-out behavior, not the behavior at finite time) are computed in terms of Feynman integrals described by virtual particles in which no muon is present. Thus the muon is gone from the description - there is no occasion to talk about it. If you want to have a more detailed picture in time you either need to factor the S-matrix in terms of on-shell but unstable intermediate particle states such as a muon, or you need to work in terms of a hydrodynamic or kinetic description where states are abstracted from the CTP (Schwinger-Keldysh) formalism. In the latter case, the muon appears as an on-shell resonance state.

So the unstable muon exists once you sufficiently resolve the time, but it is invisible in the S-matrix elements of the total process when the latter are computed in unfactored form.

 

[vanhees71]

Be, however warned that "off-shell transport" is a very tricky business, much trickier than S-matrix theory, were you deal with asymptotic states and don't care about transient states, let alone the very ambigous if not impossible interpretation of Heisenberg field operators and their expectation values at finite times. In defining the S-matrix properly, by the way a lot is hidden by the ingenious use of $\mathrm{i} \epsilon$ prescriptions (aka. the Gell-Mann-Low description). In short, it is very difficult if not impossible to interpret Heisenberg-picture field operators in terms of particles. The particle interpretation is only unambiguously working for asymptotic free states. In any case a transport theory for broad resonances is very delicate and not fully solved (after more then 20 years of research, including practical implementations of some ideas concerning off-shell transport). Also the full quantum equations (Kadanoff-Baym equations)are not fully developed to the extent that you can use it for simulations of real-world problems (in heavy-ion collision physics), because they are simply to difficult and "CPU demanding" to bring them to the computer. There are, however some studies, with simpler toy models like $\phi^4$ in (1+2) dimensions using a $\Phi$-derivable approximation up to the two-point level (look for papers by Juchem & Cassing and also for Berges).

Of course, a muon is a good example for an "almost stable" particle, i.e., you can take it as a stable particle in QED and put the weak interaction perturbatively on top. The muon becomes a very narrow resonance, and it makes some sense to use the (tree-level) diagram to calculate its decay widht/lifetime although "strictly speaking" the muon is not a asymptotic free state anymore when taking into account its finite lifetime due to the weak interaction.

Of course, you must not take the formal $t \rightarrow \pm \infty$ limits in the Gell-Mann-Low prescription to define S-matrix elements as transition rates between asymptotic free states too literally. It's rather a matter of physical (finite) time scales, i.e., a separation between the distances involving the (approximately) free moving particles in the initial and final states compared to the usually very short duration of the collision itself. Usually you don't resolve the collision ("transient state") dynamics in time and are content with S-matrix elements as transition rates between asymptotic free states.

Of course, there's some tension between mathematical rigor and the physicists' intuition about the scattering process, having to do with Gell-Mann-Low and Lehmann-Symanzik-Zimmermann (LSZ) reduction and taking the limits. As far as I know, there's no mathematically rigorous definition of these limits and thus the so overwhelmingly successful S-matrix elements used to evaluate the Standard Model and comare it to the measured cross sections.

 

[Buzz Bloom]

Nothing virtual happens. The dry facts are that two real particles are created from gravitational energy (from two gravitons or from an external gravitational field), not from the vacuum. One particle escapes, the other is absorbed. A valid description is given on p.645 of the book
B.W. Carroll and D.A. Ostlie, An Introduction to Modern Astrophysics, 2nd. ed., Addison Wesley 2007.

Hi @A. Neumaier:

It has taken a month, but my local library has finally gotten for me a copy of the Carroll & Ostlie book you recommended. I think I now understand the phenomenon of Hawking Radiation more clearly than I had before, but some aspects I still find confusing.

This is what I now think I understand:

The total matter-energy M producing a black hole with an event horizon at radius R_s=2GM/c² includes the energy of the gravitational field that exists outside the event horizon. When this energy creates a pair of real particles, which is a common occurrence, and the tidal force moves one of the pair towards the event horizon while the other escapes towards infinity, which happens only a relatively small fraction of the time, the escaping particle is Hawking Radiation, and it's mass-energy is subtracted from the total mass energy M of the black hole.​

What are the particles that make up the energy of the static gravitational field? Are they virtual gravitons?

I have been thinking about the similar phenomenon about a sphere with a net charge distributed uniformly on its surface. What are the particles that make up the contents of the static electric field about this sphere? Are they virtual photons? If the field is strong enough, won't pairs of real particles also occasionally be formed, from the static electric field, (almost ?) all of which will then very soon afterwards annihilate each other? However, is it not possible for some of these pairs of particles to not annihilate each other? If these pairs are charged, then would this not also result in a weakening of the electric field? But, perhaps this phenomenon only creates pairs of uncharged particles. What would the effect be if a pair of non-charged particles did not annihilate each other? Would there then be some form of radiation emitted from the sphere?

Regards,
Buzz

 

[A. Neumaier]

the particles that make up the contents of the static electric field about this sphere? Are they virtual photons?

No. A static field is not made up of photons. It is made up of the fields of the particles in the sphere, which in turn is due to the fact that every charged physical particle carries with it an electromagnetic field - in a good approximation a Coulomb field.

In formal terms, physical electrons and nuclei are so-called infraparticles. If one describes these particles in perturbation theory based on the bare theory with a cutoff, they look like bare particles accompanied by a cloud of infinitely many arbitrarily soft photons. But this picture does not survive infrared renormalization via coherent states, hence has no physical meaning beyond pure analogy to condensed matter physics.

In a very strong classical electromagnetic field, real (not virtual) electron-positron pairs are created spontaneously. For example, http://arxiv.org/abs/hep-th/0005078 treats the corresponding case with scalar particles.

What are the particles that make up the energy of the static gravitational field? Are they virtual gravitons?

No. In a quantum field theory of gravity and matter, all massive physical particles also carry a gravitational field; everything is completely analogous. A static gravitational field behaves quantum mechanically in the same way, except that it can also produce two photons, because photons are their own antiparticles. Or two neutrinos, etc.. There is no energy barrier for photon production since photons are massless, but to achieve a noticeable effect, the field has to be extremely strong. (To date, Hawking radiation has not been experimentally demonstrated, primarily for this reason. The black holes with very strong fields are fortunately too far away to affect us significantly.)