My question relates to the very origin of the elementary particles. I understand that the Higgs field breaks the mass symmetry of the elementary particles and that the Higgs field-based mass value for each particle is dependent upon the strength of interaction between each elementary particle and a Higgs field. This leads to the mass differences between elementary particles that we observe today. My question is what determines the difference in the strength of coupling to the Higgs field for different elementary particles? Does each kind of elementary particle differ in some way before each of them couple to a Higgs field? Or are they fully symmetrical in every way, i.e. of the same quantum state), and it is only later that they take on their distinct properties as elementary particles through after interaction with the Higgs field(s)? Does some fraction of these identical particles (or their fields) interact with a different Higgs field, each kind of Higgs field(s) interaction creating a different elementary particle? That is, is the set of elementary particles created at the instant of their creation or only after interacting with the appropriate Higgs field? To put is more succinctly, were there a set of distinguishable elementary particles before their future interaction with Higgs fields?
The gauge boson masses are a consequence of how the symmetry is broken. The fermions on the other hand have their masses picked out of a pick and mix bag. They are what we measure them to be, I'm not aware of an explanation for the specific observed hierarchy of fermion masses. Anyone else?
People of course try to explain it, and there are many models around, but I don't think any of them stand out of the crowd as "obviously" the right idea. Usually these are associated with ideas about unification, i.e. GUT models.
What is puzzling (at least to me) is that all couplings to other fields are quantized. I.e. we see only 1/3e, 2/3e, 1e electric charges, weak isospin and weak hypercharge are quantized as well, color charge of all quarks is also quantized - it is either 0 for non-quarks and gluons, and the same magnitude for all quarks and gluons (I guess we can say it's equals 1 in some units for them). But for Higgs field, no such thing is seen. Rest masses of all particles seem pretty arbitrary and they are not simple multiples of an "elementery mass". Is there a reason why it's so?
The fermion masses are due to the (Yukawa) couplings of each field to the Higgs; these are more analogous to the electromagnetic and strong coupling constants than they are to the quantum numbers you quote, since those come from the group theory (which representations of the gauge groups the fields transform under). It is still not very nice that there are so many of these couplings floating around the Higgs sector of course, and I think everyone agrees with you on this. It is legit to guess that they should all have some common origin. But in the Standard Model they are just free parameters, fitted to reproduce the observed fermion masses, and that's all there is to it.
Even worse, for the quarks, the up-like mass states and the down-like mass states are not quite orthogonal, and this generation mixing makes possible cross-generation decays. This mixing matrix, the Cabibbo-Kobayashi-Maskawa matrix, has 3 CP-preserving angles and 1 CP-violating phase angle -- even more parameters. There's been a lot of speculation about "horizontal symmetries" between the quark generations, but without any clear solution. Adding neutrino masses makes things worse: 3 masses and 4 or more angles, with the masses being small and the 3 CP-preserving angles being large.
The gauge couplings have to be quantized because of gauge symmetry. Also, if they were not quantized, triangular anomalies would be generated destroying the consistency of the theory. There is no requirement for a similar quantization of the Yukawa couplings (Not at the standard model level anyways. There might be hidden relation s that only become obvious at some higher energy. This hidden relations are common features of GUT theories
I'll now derive the masses and mixing angles from the Yukawa interaction terms. L = left-handed quarks, R = right-handed quarks (either up-like or down-like) D = differential operator, m = mass matrix from (Yukawa couplings) * (vacuum Higgs value) m^{+} is the Hermitian conjugate of m Simplified Lagrangian: L^{+}.D(L) + R^{+}.D(R) + L^{+}.m.R + R^{+}.m^{+}.L Equations of motion: D(L) = m.R D(R) = m^{+}.L D^{2}(L) = M.L where M = m.m^{+} Diagonalize M: M = V.m_{obs}^{2}.V^{-1} where m_{obs} is a diagonal matrix of observed masses. This yields a relationship between the weak-interaction states and the mass states: L_{weak} = V.L_{mass} Since the weak states are related for up-like (U) and down-like (D) quarks, we get L_{weak} = V_{U}.L_{U,mass} = V_{D}.L_{D,mass} Thus, L_{U,mass} = V_{UD}.L_{D,mass} where V_{UD} = V_{U}^{-1}.V_{D} - Let's now count parameters. For n generations, the original matrices have 4n^{2} parameters, because they can be complex and asymmetric. However, the observable quantities, the masses and cross-generation mixings, have only n^{2} + 1 parameters: 2n masses, n(n-1)/2 CP-preserving angles, and (n-1)(n-2)/2 CP-violating phases. - Since the generations mix, M_{U} and M_{D} must have different eigenvector matrices. Since we have CP violation, at least one of M_{U} and M_{D} must be complex, since if they were both real, then they can both have real eigenvectors, thus making no CP violation. So we are stuck with direct-from-Higgs mass matrices m_{U} and m_{D} being complex and completely general, without any clear hints as to how to simplify them.
Elementary fermions have the oddity that neutrino masses are *very* tiny to the charged ones. The most massive neutrino flavor likely has a mass of around 0.05 eV, and the less massive ones 0.01 eV or less. A common solution to this conundrum is the "seesaw model", where there exist right-handed neutrino modes with huge Majorana masses. These masses mix with the Higgs-generated ones to make the tiny ones that we observe. Simplified Lagrangian: L^{+}.D(L) + R^{+}.D(R) + L^{+}.m.R + R^{+}.m^{+}.L + R^{+}.M.R^{+} + R.M^{*}.R Kinetic terms, Higgs-Dirac terms, right-handed Majorana terms M is the right-handed neutrinos' Majorana mass matrix. It is symmetric, but it may be complex. The observed masses one finds from M.M^{*}. Equations of motion: D(L) = m.R D(L^{+}) = R^{+}.m^{+} D(R) = m^{+}.L + M.R^{+} D(R^{+}) = L^{+}.m + R.M^{*} Create mixtures of the left-handed and right-handed parts: R' = R + L^{+}.m.M^{*-1} R'^{+} = R^{+} + M^{-1}.m^{+}.L L' = L - R^{+}.M^{*-1}.m^{T} L'^{+} = L^{+} - m^{*}.M^{-1}.R One gets equations of motion D(R') = M'.R'^{+} D(R'^{+}) = R'.M'^{+} D(L') = m'.L' D(L'^{+}) = L'.m'^{+} where M' = M + m^{*}.M^{-1}.m^{+} m' = - m.M^{*-1}.m^{T} One gets two sets of Majorana modes, mostly right-handed modes with masses close to the original right-handed mass, and mostly left-handed modes with masses much smaller than the Higgs-Dirac masses: O(m_{Higgs-Dirac})^{2}/O(M_{right-handed Majorana}) For m(observed) = 0.05 eV = 0.5*10^{-10} GeV m(Higgs-Dirac estimate) = 30 GeV we get M(right-handed) = 2*10^{13} GeV approaching GUT energies.
Now for what GUT's would have to say about elementary-fermion-Higgs interactions. The mass-related terms are the Yukawa interactions that give masses by the Higgs mechanism, and also the right-handed-neutrino mass terms. (Minimal Supersymmetric) Standard Model mass-related terms: y_{U}.Q.U.H_{u} + y_{D}.Q.D.H_{d} + y_{N}.L.N.H_{u} + y_{E}.L.E.H_{d} + M_{RN}.N.N Left-handed doublets: Q = quarks, L = leptons Right-handed singlets: U = up, D = down, N = neutrino, E = electron Higgs particles: H_{u}, H_{d} (MSSM), H_{u} = ε.H_{d}^{*} (Standard Model) GUT's: SU(5) from (MS)SM: F = elementary fermions, H = Higgs particles F(1) = N F(10) = Q + U + E F(5*) = L + D H(5) = H_{u} + down (s)quarklike "Higgs triplet" H(5*) = H_{d} + down (s)quarklike "Higgs triplet" Mass-related terms: y_{U}.F(10).F(10).H(5) + y_{D}.F(10).F(5*).H(5*) + y_{N}.F(5*).F(1).F(5) + M_{RN}.F(1).F(1) y_{E} = y_{D}^{T} One gets the unification of the masses of the down-like quarks and the charged leptons. The next step is SO(10), and I'll express it as a superset of SU(5): F(16) = F(1) + F(10*) + F(5) -- gauge-group spinor F(10) = F(5) + F(5*) -- gauge-group vector Mass-related terms: y.F(16).F(16).H(10) y = y_{U} = y_{D} = y_{N} = y_{E} -- it is symmetric, though it may be complex Mass unification is too successful! No cross-generation decays, no CP violation, and no right-handed-neutrino masses. So breaking of SO(10) must somehow produce all of these effects.
That's not right (I don't think). You can have cross-generation Yukawa interactions and CP violation even before Symmetry breaking.
How would that work? Here's the problem for SO(10). The interaction that gives the Standard-Model EF-Higgs interactions is y_{ij}*(F_{i}.F_{j}.H) y = Yukawa couplings, symmetric but may be complex y can be diagonalized: y = V.Y.V^{T}, where V^{-1} = V^{T} (inverse = transpose), and Y is a diagonal matrix of eigenvalues. Let F_{i}V_{ij} = F'_{j} Then the term becomes Y_{i}*(F'_{i}.F'_{i}.H) thus making the F' generations independent of each other.
Electromagnetic coupling constant is the same for all particles, so the forces they experience are proportional to their electric charge. I thought that "Higgs mass" is basically a "Higgs charge". By having different "Higgs charges", particles experience different strength of Higgs force. I assumed that Higgs coupling constant is, indeed, constant, it's the same regardless of interacting particles. Are you saying that Higgs field, unlike EM and other SM fields (e.g. color force) has *different* coupling constants to different fermions?
It's usually written that way. For certain the couplings are different - widely different - but there's no theory to explain them. No rationale exists for factoring them into a charge and a coupling constant.
As I'd explained earlier, the observed coupling-constant values are essentially eigenvalues. Let's approach this in another way, one that will relate masses more directly to Higgs-particle interactions. Treating up-like quarks as a group, and likewise for down-like ones and for charged leptons, one can write the interaction term as I = y_{ij}ψ_{Li}ψ_{Rj}φ y_{ij} = coupling constants - can be an arbitrary complex matrix ψ_{Li} = left-handed elementary fermions ψ_{Rj} = right-handed elementary fermions φ = Higgs particle (the one that survives electroweak symmetry breaking) Diagonalize y: y = V_{L}.Y.V_{R}^{+} where Y is diagonal and V_{L} and V_{R} are unitary. Taking ψ_{Li}V_{Lij} -> ψ_{M,Lj} ψ_{Rj}V_{Rij}^{+} -> ψ_{M,Ri} we get I = Ʃ_{i} Y_{i}ψ_{M,Li}ψ_{M,Ri}φ The Y elements may be complex, but we can absorb their phases into the ψ's: I = Ʃ_{i} |Y_{i}|ψ_{M,Li}ψ_{M,Ri}φ So what one finds from observations is the absolute values of the eigenvalues of the coupling matrices. U: 0.000008, 0.0054, 0.705 D: 0.000018, 0.00036, 0.017 E: 0.0000021, 0.000431, 0.00727 (mass / Higgs vacuum field: 245 GeV) U: up, charm, top D: down, strange, bottom E: electron, muon, tau Source: Standard Model (mathematical formulation) - Wikipedia Due to renormalization effects, one has to be careful of which energy scale one measures the masses at. That's not much of a problem for leptons, but it is a serious problem for quarks, because of the strength of the QCD interaction. The up, down, and strange quarks are measured at 2 GeV here, but the masses of the other quarks and the leptons are on-shell masses, measured at their own masses. Does anyone know what these particles' masses are when renormalized to (say) the top quark's mass?
So there are several complications in the way of getting from the bare Higgs-particle couplings to the observed masses. For electromagnetism, there is the interesting curiosity that all known elementary particles' electric charges are small-integer multiples of some elementary charge. This curiosity carries over into the weak-hypercharge values of the unbroken Standard Model. There does not seem to be any simple theoretical reason why that ought to be the case. A commonly-cited constraint is the cancellation of quantum-mechanical anomalies like the triangle anomaly, where three gauge particles couple to a loop of chiral fermions (Anomalies and the Standard Model, Adel Bilal's Lectures on Anomalies, etc.). The gauge part of the anomaly value is A_{abc} = Tr({T_{a},T_{b}}T_{c})_{L - R} where the T's are gauge operators on those fermions and the Tr is a sum over all the fermions' states. I'll use 3 = QCD, 2 = weak isospin, 1 = weak hypercharge, g = gravity In the Standard Model, several anomalies automatically cancel, like those with only one 3 or only one 2. This leaves nontrivial ones 111, 122, 133, 1gg, 333 The SM multiplets are, using notation (QCD, WIS, WHC) : Left: (3,2,yq), (1,2,yl) Right: (3,1,yq+1/2), (3,1,yq-1/2), (1,1,yl+1/2), (1,1,yl-1/2) The WHC values are intended to make electromagnetism non-chiral: left-handed and right-handed parts having the same electric charge: Q = I3 + Y. This leaves two free parameters that I call yq and yl. The 333, 133, and 1gg anomalies all cancel, and the 111 and 112 ones yield the constraint 3*yq + yl = 0 or yq, yl = yql * (B - L) baryon number - lepton number The Standard Model has yql = 1/2, but anomalies do not constrain yql's value. We must turn to GUT's. Of the simpler ones, Georgi-Glashow SU(5) constrains yql to its Standard-Model value, while SO(10) leaves it unconstrained: SO(10) -> SU(3) * SU(2) * U(1)_{Y} * U(1)_{B-L}
Let's try again, but without the constraint of electromagnetism being non-chiral. The right-handed quarks get coupling constants yu and yd, and the right-handed leptons yn and ye. The constraint 133 gives 2*yq - (yu+yd) = 0 The constraint 122 gives 3*yq + yl = 0 The constraint 1gg gives 3(2*yg - (yu+yd)) + (2*yl - (yn+ye)) = 0 So we get yu = yq + yqx, yd = yq - yqx, yn = yl + ylx, ye = yl - ylx The constraint 111 gives |yqx| = |ylx| (EM non-chiral value: 1/2) Essentially what I'd derived earlier, but with arbitrary overall scaling. Electroweak symmetry breaking does not constrain these y's, it must be said. That's because it only involves y(Higgs). So the Standard Model has plenty of loose ends.
I must concede that in the Standard Model, the interactions of the Higgs particle and the elementary fermions provide additional constraints. The interactions: (left quark) . (up Higgs) . (right up quark) (left quark) . (down Higgs) . (right down quark) (left lepton) . (up Higgs) . (right neutrino) (left lepton) . (down Higgs) . (right electron) where the up Higgs has U(1) strength yhu and the down Higgs has U(1) strength yhd. For the up and down Higgs in the MSSM, anomaly cancellation gives yhu + yhd = 0. This constraint one also gets from the SM Higgs. The EF-Higgs interactions give these constraints: yq + yhu - yu = 0, yq + yhd - yd = 0, yl + yhu - yn = 0, yl + yhd - ye = 0 Plugging in my previous post's results, yhu = yqx = ylx, yhd = - yqx = - ylx One gets from these yhu + yhd = 0 These values are the values that make electromagnetism non-chiral. Left-handed and right-handed EF's couple the same to the photon. Is this a coincidence? Or is there something deeper?