It is not sufficient to merely explain that muons are more massive than electrons because they interact more strongly with the Higgs field. For example, an electron with energy of 0.511 MeV must obtain exactly the correct amount of inertia to satisfy all the physics laws. Also, there must be equivalence between inertial mass and gravitational mass. How much of this detail has been addressed?
It is possible to define asymptotic states of leptons, since the Standard Model *is* renormalizable and leptons aren't confined. Therefore the mass, m, of the electron is given by the product of the appropriate Yukawa coupling and the vev of the Higgs (taking into account the renormalization conditions and so on). Since the mass is non-zero, it is possible to define the rest frame of the electron, where the relation E=m c^2 is then satisfied.
The contributions of quarks to electroweak quantum corrections are crucial, since the electroweak theory is only then consistent as a quantum field theory when the contributions of quarks and leptons both are taken into account. This has been confirmed in a multitude of measurements at LEP, the Tevatron and the LHC.
The only subtlety concerning quarks is the fact that the pole of the quark propagator can't be given physical meaning, due to confinement. That's all. Cf., for instance, S. Coleman's lecture notes: http://inspirehep.net/record/152647
The equation, E = mc2, is an identity. You don't need to "give exactly the correct mass". Rest mass and rest energy are really just two names for the same thing. Whether you express an electron's mass as 511 keV or as 9.11e-31 kg, they mean exactly the same thing. The same way as 1 meter and 100 centimeters mean the same thing; you don't need to do anything special for a 1 meter measuring stick to also have a length of 100 cm.
The correct masses of fermions are produced by Yukawa interactions with the Higgs field, with coupling constants that give the right masses. I.e., the theory does not have the power to predict fermion masses: each unknown fermion mass is simply replaced by an unknown coupling coefficient. What the standard model provides is a sensible mechanism for fermions to acquire mass, that's all.
As gravity is not part of the standard model of particle physics, the question of equivalence of inertial and gravitational mass is not addressed by the theory.
While such an interaction could account for free quarks continuously reacquiring their characteristic rest mass from an ambient Higgs field, how could up and down quarks, for example, continuously reacquire their specific rest masses from the Higgs field while remaining confined within the structure of a nucleon - bound by the strong interaction - without releasing the nucleon's binding energy?
James: I don't know what you mean by "reacquire". In the standard model, fermions interact with the Higgs field and hence, they effectively behave as though they had rest mass. It's not about losing or reacquiring anything. Inside the nucleon, binding energies far exceed the rest masses (compare the few MeV masses of u and d to the rest mass of a proton at ~1 GeV), so the rest mass is just a small correction. One way to imagine this, I guess, would be to think of the quarks as being bound by a strong spring (the strong interaction) but also immersed in some liquid (the Higgs field) that hinders the quarks' motion slightly (this is what we interpret as their rest mass). If it were possible to cut the quarks free, only the Higgs field would remain (so the quarks would still not behave as massless particles) but this requires so much energy that new quarks are created, and end up being bound to the freshly cut ends of the spring. This would be my crude visualization of the confinement mechanism.
V. T. Toth,
Thanks for your explanation. Sorry if I've used improper terms, but isn't it considered that fundamental particles continuously interact with the Higgs field in order to 'acquire' their specific, characteristic rest masses?
As I understand, nucleon rest mass is the product of quark kinetic energy confinement - converting it to the potential energy of rest mass. This enormous, additional nucleon mass is usually attributed to the analytically derived QCD 'binding energy' of gluons, but it is the greater kinetic energy of neurons' component quarks (udd) that produce their greater rest mass of ~939.565 MeV/c^2 compared to protons' (uud) ~938.272 MeV/c^2. In any case, it is the rest mass of nucleons that is dependent on the confinement of quarks within their restricted structure - they are no longer able to move about in spacetime.
As I understand, each up and down quark would be required to interact with the Higgs field independently in order to continuously 'acquire' their specific rest masses. How can each quark individually interact with a Higgs field permeating spacetime while being continuously bound within the confined structure of the nucleon?
I would like to direct the discussion slightly. I am really talking about the inertia of leptons. This way we can avoid discussion of the binding energy of hadrons. I also define “mass” as the quantification of inertia. It is easy to talk about mass and energy being equivalent, but it is not as obvious why a lepton with a specific amount of energy has a specific amount of inertia when it is accelerated. It is often said that a photon is a massless particle. However, this is only correct for a freely propagating photon. If a photon is confined in a hypothetical reflecting box, it acquires exactly the correct amount of inertia to equal the inertia of a particle of equal energy. No Higgs field is required to give inertia to a confined photon. This is discussed in the book and paper available at http://onlyspacetime.com/ . (Chapter 1 in the book, page 12 in the paper).
John,
As merely a lay person, I cannot fully evaluate your referenced chapter, but I do find it quite interesting. However, I think you do breech upon the fundamental, inverse relation between the wave propagation state of matter and the mass of localized (confined) particles.
If it's considered that propagation energy and mass represented an identical quanta, mass might be derived as a frequency/duration of wave-particle state manifestations.
As I understand, photons only exhibit mass characteristics when their propagation energy has been absorbed - by an electron, for example. Of course, for the duration of time the photon is not propagating it does not exist as a discrete quantum particle...
Likewise, more massive particles may be distinguished by those whose propagation energy is localized (confined) for greater durations of some (sub-quantum?) time duration - I suggest that the inverse relation between inertial effects of quantum mass and wave propagation should be considered as a result rather than causation...
It may also be important to consider that only perpetually propagating waves propagate linearly. Perhaps the frequency of propagation energy localization (confinement) produces the angular momentum of particle spin - thereby periodically redirecting subsequent wave state linear propagation.
What quantum process might periodically confine the propagation of particle waves - thereby producing the effect of inertial mass? Higgs field interactions appear to be the best candidate.
However, in my preceding discussion of quark confinement within the binding structure of nucleons I attempt to suggest that particles (quarks) permanently acquired their mass from the Higgs field as they were initially emitted in the early universe - that fundamental particles' rest mass is determined only as they are initially emitted (condensed). In this sense, massive particles are no long emitted from any ambient high-density Higgs field, but only from locally dense conditions, as when persistent particles decay (or are forcefully disintegrated in high-energy LHC collisions)...
Also please consider that bound (confined) particles, such as neutrons bound within atomic nuclei, can gain perhaps perpetual stability (free neutrons decay with a mean lifetime of < 15 minutes). This result is analogous to the extension of lifetime for accelerated particles resulting from time dilation...
My apologies for diverging from your intended discussion, but I think you have at least touched on a largely unconsidered fundamental aspect of quantum phenomena.
Of course, since the Standard Model is a relativistically invariant theory, so, for all physical particles, the energy-momentum 4-vector satisfies E^2-p^2c^2=m^2c^4 and, in the rest frame, where p=(0,0,0), E=mc^2. In these relations m is the Lorentz invariant mass, which is equal to the product of the Yukawa coupling and the vacuum expectation value of the Higgs. For leptons this quantity coincides with the pole of the corresponding propagator. For quarks confinement introduces subtletiies. This is standard material in a course in quantum field theory.
Just to be clear, inertial and rest mass are not the same. The inertial mass of a system is the sum of all rest masses and binding energies. The Higgs mechanism for fermion masses simply replaces rest mass with binding energy. So one way to interpret this part of the theory is to think of fermions as having zero rest mass; they couple to the Higgs field and this binding energy can be reinterpreted as an effective rest mass.
Regarding massless particles, I suggest that one relatively easy way to understand them (that is, if you understand special relativity) is to consider that all freely moving particles move along geodesics; massless particles are confined to null geodesics. When a force acts on a massless particle (i.e., when it interacts with some other field non-gravitationally) the effect of that force is that it pulls the particle away from the null geodesic. The previously lightlike trajectory of that particle now becomes a timelike trajectory. This can be viewed as the particle acquiring an effective mass.
V. T. Toth,
Regarding inertial mass, elementary particles' rest mass constitutes their inertial mass, correct?
I don't understand the statement, "The Higgs mechanism for fermion masses simply replaces rest mass with binding energy." Free elementary particles have no binding energy - correct? Compound particles do not directly interact with the Higgs field, correct?
Of course the Lorentz invariant rest mass is the inertial mass of the particle in question, since it can be shown by standard calculations to control the response of the particle to an external force. It has absolutely nothing to do with any binding energy. It *can* be computed from appropriate correlation functions of the quantum field theory.
In the standard model, even "free" fermions are not truly free, as they move in the omnipresent Higgs field. As for compound particles like, say, protons, most of their mass is the internal binding energy; the constituent quarks' so-called rest masses are effective masses that result from the Yukawa-interaction with the Higgs field.
It is possible to define asymptotic states of leptons, since the Standard Model *is* renormalizable and leptons aren't confined. Therefore the mass, m, of the electron is given by the product of the appropriate Yukawa coupling and the vev of the Higgs (taking into account the renormalization conditions and so on). Since the mass is non-zero, it is possible to define the rest frame of the electron, where the relation E=m c^2 is then satisfied.
The contributions of quarks to electroweak quantum corrections are crucial, since the electroweak theory is only then consistent as a quantum field theory when the contributions of quarks and leptons both are taken into account. This has been confirmed in a multitude of measurements at LEP, the Tevatron and the LHC.
The only subtlety concerning quarks is the fact that the pole of the quark propagator can't be given physical meaning, due to confinement. That's all. Cf., for instance, S. Coleman's lecture notes: http://inspirehep.net/record/152647
No, E = mc² even without the Higgs, it is a definition of mass.
the higs field make that the vacuum energy of the particle don't be zero this vacuum energy is the resposoble of the mass of any particle.
John, let me try to give a truly simple answer, which of course may of may not satisfy you. I am an expert in inertia generally, and equivalence, and relativity, but not Higgs, so I draw on some things I've read about the Higgs.
The Higgs field (or the Higgs bosons) have energy. Energy always has a resistance to motion (inertia) which is proportional to its gravitational mass. This mysterious fact is not explained by currently accepted theories. It is assumed because it is extremely well verified experimentally.
So if the Higgs field lends some of its energy (i.e. mass, they are the same thing) to another particle, then by axiom it is always the right proportion of inertial and gravitational mass. If the particle goes into motion, the Lorentz transforms operate identically on the original and borrowed mass, keeping everything in proportion.
In other words, there aren't details of the kind you are asking for. That would be something like a constructive theory of mass. A few have been proposed, over the last century but none has got any traction.
I just seem to be missing the main argument that since the standard model of particle physics is a relativistic (technically, Lorentz covariant) field theory E=mc^2 is part of its key ingredients: It has to be true, since we assume it to be true, as long as a mechanism (Higgs..) inside the model generates the mass! Note that this mass is the inertial (kinematic) mass!
Since gravitation is not a part of the standard model, the equivalence between gravitational and inertial mass needs to be tacked on separately, as Robert explains above.
QFT has a number of variations that claim to give the correct answers. Gravity enters into the several string theory variations. Some researchers accept it and others don't.
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