How can a pion have a mass, given it's a “field mediator” and created/destroyed continuously?

Maybe some of my assumptions here are basically wrong, but isn't it true that

• pion is the "mediator" for the strong force field.
• the quantum field theory basically says that there are no fields, instead all forces are caused by interchanging of mediator particles all the time.

So to me this look like two particles in a nucleus, which are hold together by the strong force, are "sending" these pion particles all the time. But all references say that these particles have mass! Why doesn't this mean that mass is created/destroyed all the time? Is mass of the whole nuclei affected in some way by the constant stream of pions? If the question is basically irrelevant, what are the most important parts that I am missing here?

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whyever would you think that a theory called Quantum Field Theory says that there are no fields? In fact, the fields are the fundamental objects, whereas particles are 'just' excitations; more so, virtual particles acting as force mediator aren't really particles, but (arguably) artifacts of perturbation theory –  Christoph Jul 5 '12 at 20:33
I'll obviously have to study this more, but aren't for example pions real, detected, observable particles? I've read we can even "observe" effects of free pions from cosmic rays? –  Cray Jul 5 '12 at 20:43
pions are, but virtual pions are not –  Christoph Jul 5 '12 at 20:46
Learn some QM, they said; it'll be fun, they said. :P –  Cray Jul 5 '12 at 20:59

The fundamental force carriers are gauge bosons and the gauge-symmetries forbid them to have a mass. The only way they can gain a mass is through the Higgs mechanism, as in the case of the W and the Z, i.e. the gauge symmetry has to be spontaneously broken.

So in a sense you are right, the fundamental force mediators should be massless but not for the reason you have given and there is a get out clause if the Higgs mechanism is at play.

But the pions are not fundamental force mediators and are not gauge bosons so they are not required to be massless for this reason. However there is an interesting twist to all this: One can in fact understand the pions as the Nambu-Goldstone bosons of the spontaneously broken chiral symmetry (broken by the non zero vacuum expectation value of the quark condensate). If the chiral symmetry were exact then the pions would indeed be required to be massless (by Goldstone's theorem). However the chiral symmetry is only an approximate symmetry as it is explicitly broken by the quark masses. For this reason the pions are allowed to have a (small) mass and are thus called 'pseudo'-Nambu–Goldstone bosons.

I hope the above helps to explain the situation in relation to the masses of the force mediators and the pions. However I sense that your real confusion is in the fact that you cannot understand how a force mediator can have a mass as you think this will in some way effect the interacting particles in a way that is not observed. You seem to be worried about the interacting particles having to create and destroy mass. But why be so concerned about mass? Even with a massless force mediator the interacting particles still need to create and destroy the energy of the transferred mediator and mass is simply one form of energy, so by your reasoning you should be equally well concerned about both massive and massless force carriers.

But your reasoning should just not concern you, why are you worried about this constant exchange of energy? After all, the particles are interacting and how else could they do this? A very simple analogy of how exchange of a massive particle could create a force would be if two people sat in boats and kept throwing a heavy ball between each other, they would move apart and the exchange of the massive ball would appear to be producing a repulsive force.

The above analogy does not get too close to the real situation and would for example fail to explain attractive forces, but it at least shows that you should not be concerned about the transfer of mass or energy in such a situation. To really understand how virtual particles can transmit a force then you have to do the calculations and understand that virtual particles are really not particles at all but are non-particle like disturbances in their respective fields. You should also remember that any interacting particle is constantly creating and destroying virtual particles in its vicinity - the creation and annihilation of virtual particles are simply part and parcel of the definition of the particle in the first place.

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You are right, the creation/destruction of energy is equally confusing to me, I just took mass as an example, it seemed easier to construct a question about. But is the bottomline this: that all force mediator particles are in fact "virtual", and could never be detected, even if one could somehow, say, get in between two protons in a nucleus and try to detect what is going on there? (surely impossible proposition, but if it was possible) ? Also, then what does the explanation of a field by quantified mediators really achieve? What does it explain if they are all virtual? –  Cray Jul 6 '12 at 6:43
Yes, all force mediators are virtual particles. A particle is defined as an asymptotic state infinitely far from other particles. So virtual particles are not these and can not be detected as such. They should simply be thought of as disturbances in the field and the name particle is a bit of a misnomer. I would say the use in thinking of particle mediators is in the Feynman diagram approach where we represent the interaction of fields by exchange of one or more virtual particles to represent a perturbative calculation. But this is more a mathematical tool than a true picture of reality. –  Mistake Ink Jul 6 '12 at 11:43

Two points:

• Pion exchange (or more generally light meson exchange) is a reasonable approximation (AKA an "effective theory") for the strong force as it acts on nucleons in the nucleus, but it is not the real strong force which is mediated by gluons.

In other words, pions are not really force mediators at all.

• The isn't actually a problem with massive mediators. The weak force is carried by the W and Z bosons with masses of 80 and 90 GeV respectively.

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I thought that force particles could be massive (because they're only virtual anyway) but only under the condition that the force is limited in range. Does that make any sense? –  AlanSE Jul 5 '12 at 20:25
@AlanSE Yeah, heavy carriers introduce range limits, and the pion mass is more or less responsible for the range of the inter-nucleon force. As I say the pion exchange model is a good effective theory. –  dmckee Jul 5 '12 at 20:28
So widely used approximation is one that assumes creation/destruction of mass/energy? Despite the energy conservation principle? That's the part I don't get, how can we say that any field mediators (ie, particles that are created all the time I guess?) can have mass, if that contradicts the energy conservation? –  Cray Jul 5 '12 at 20:38
@Cray: the hand-wavy argument is that because of uncertainty, anything goes as long as the action doesn't exceed $\hbar$ –  Christoph Jul 5 '12 at 20:43