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Peter Higgs passes away

What he worked on was a way for elementary particles in the Standard Model of particle physics to get masses, the  Higgs mechanism

Peter Higgs was far from alone in working on this mechanism, and he, a group of Robert Brout and François Englert, and a group of Gerald Guralnik, C. R. Hagen, and Tom Kibble separately published papers on this mechanism almost simultaneously back in 1964.

Here are some alternative names:
  • Brout–Englert–Higgs - BEH
  • Higgs–Kibble - HK
  • Englert–Brout–Higgs–Guralnik–Hagen–Kibble - EBHGHK
  • Anderson–Higgs - AH
  • Anderson–Higgs–Kibble - AHK
  • Anderson-Brout-Englert-Guralnik-Hagen-Higgs-Kibble-'t-Hooft - ABEGHHKtH

What is the difficulty? Let us start with electromagnetism. It has two fields, the electric and the magnetic fields, and they both are made from two potentials, a scalar one and a vector one, related like time and space, and like energy and momentum. If one adds to this set of potentials a gradient of some arbitrary quantity, then the fields will remain unchanged. This is a "gauge symmetry".

If one gives a photonlike particle a mass in a naive way, that will break the gauge symmetry. But PH and his numerous colleagues discovered a way of keeping that breaking from happening: a Higgs particle, a particle that will preserve that symmetry, though in hidden form.

But it gives a mass to that photonlike particle by always having a nonzero ground-state or vacuum value. So in a sense, the Higgs particle is everywhere, giving mass to most other Standard-Model particles in proportion to how much those particles interact with it, complete with no interaction giving no mass.

It gets a nonzero ground state by having a potential with a hump in the middle, for zero field value. The ground state is for the field being in a trough around that hump, the trough's shape being involved with hiding that gauge symmetry.
 
Now for the weak interactions.

The first theory, Enrico Fermi's four-fermion theory, featured a contact interaction between all the particles involved. It involved two "charged currents": electron <-> neutrino and neutron <-> proton, both much like electromagnetic currents but changing what we now call flavors. In weak-interaction terminology, EM currents are "neutral currents", because they don't change the charges of particles involved in them.

That theory had a problem. If one goes up to high energies, like a few hundred proton masses, that theory gives an interaction strength that is too large to fit into quantum field theory.

A way out is for a particle to go between the various weak-interaction charged currents, just like the photon going between electromagnetic currents. This particle, the W, must be charged, to carry electric charge between one current and the other. It also must be much more massive than most known particles in the early 1960's to make its interaction close to a contact interaction. It flits into existence and then out of existence very fast to make this interaction, without a chance to travel very far.

The W's are better-behaved at high energies, but still not very good. That led to the effort to find a "nonabelian gauge field" that contains the W+ and W- and a third, neutral particle, one that interacts with neutral currents instead of charged ones, a W0. But is the W0 a photon? Let's see.

Let's look at the charges of each member of a weak doublet that it interacts with.

W0: some multiple of -1, 1
Up quark - down quark: +2/3, -1/3 -- +1/6 + (1/2, -1/2)
Neutrino - electron: 0, -1 -- -1/2 + (1/2, -1/2)

A further difference is that each charged elementary fermion field has two parts, parts that may be called left and right handed. Charged W's interact with only the left-hand part and neutral W's would do the same, but photons interact with both parts equally.

We therefore need another particle, a B, another one that interacts with neutral currents, one that will produce the missing charges. It interacts the same with both left-handed parts of a weak doublet, and differently with both right-handed parts of it.
 
At this point the W's and the B are still all massless, as are the elementary fermions.

Now for the Higgs particle. In unbroken electroweak symmetry, this particle is something like the neutrino-electron doublet: a H+, a H-, and two H0's. But this particle has a nonzero ground-state or vacuum field value, and this interacts with the W and the B to make the W+ and the W- have the same mass, one mixture of the W0 and the B, the Z, to have a somewhat higher mass, and another mixture of the W0 and the B, the photon, to continue to have zero mass. The H+ gets "eaten" by the massive W+, the H- by the massive W-, one H0 by the Z, with only remaining H0.

The Higgs particle also bridges the left-handed and the right-handed parts of the charged elementary fermions, thus also giving those particles their masses. Neutrinos are a puzzle, since they are close to massless, and they don't seem to have right-handed parts, only left-handed parts.

This was worked out by Steven Weinberg, Abdus Salam, Sheldon Glashow, and others in the late 1960's.


The first evidence of the Z was discovered in 1973, by bombarding a liquid-hydrogen target with neutrinos. These were made by smashing accelerated protons into a target, making a lot of particles focused forward by the protons' impacts. The unstable ones then decayed, with most of them making neutrinos. Some of them hit protons and electrons without turning into electrons or muons (heavy electrons), and that was evidence of weak neutral currents, and thus Z particles.

The W and the Z themselves were discovered in 1983, with an accelerator that smashed together protons and antiprotons with enough energy to make them real particles instead of evanescent virtual ones. They quickly decayed, too fast to make it into a detector, but from their decay products one could recognize them.
 
The Higgs Particle itself was discovered some 20 years after the discovery of the W and the Z, and nearly 10 years after the discovery of the top quark, the heaviest elementary fermion in the Standard Model.  Search for the Higgs boson

First a note on units. In elementary particle physics, masses and energies are reported in units of electron volts (eV) with prefixed forms like kev (thousand eV's) and MeV (million eV's) and GeV (billion EV's) and TeV (trillion EV's). 1 electron volt is the energy that an electron gets from dropping through a potential difference of 1 volt. Not just an electron, I must add, but anything with one elementary unit of electric charge.

An electron weighs in at 511 keV (0.511 MeV), a proton at 938.3 MeV (0.9383 GeV), and a neutron at 939.6 MeV (0.9396 GeV).

Search timeline: Large Hadron Collider unless stated otherwise
  • 2004: m > 114.4 GeV
  • 2010 Jul: m < 158 GeV & m > 175 GeV
  • 2011 Aug 11: m < 145 GeV & m > 466 GeV
  • 2011 Nov 18: 114 GeV < m < 141 GeV
  • 2011 Dec 13: 116 GeV < m < 127 GeV -- some excesses observed, low significance (1.9, 2.6 stdevs)
  • 2012 Mar 7: some excesses observed at the Tevatron (2.2 stdevs)
  • 2012 Jul 2: 116 GeV < m < 119 GeV & 122 GeV < m < 129 GeV -- excess at 126 GeV at 2.9 stdevs
  • 2012 Jul 4: discovery at reported masses 126.5 GeV, 125.3 +- 0.3 GeV with significance 4.9 stdevs
Since that discovery, this particle has been researched to see how well it fits the expected features of the Higgs particle.
  • Spin 0 -- spin 2 excluded at 99.9%
  • Positive parity -- negative parity excluded at 99.9%
  • Decays: rates expected from masses? -- successfully tested for the W, Z, top quark, bottom quark, tau lepton, muon, and a borderline observation for the charm quark.
 
Here is how the masses of the Standard-Model particles translate into Higgs-particle couplings. From  Vacuum expectation value the Higgs field has vacuum strength 246.22 GeV.

From  Standard Model and  W and Z bosons and calculating Higgs-particle couples as (mass) / (Higgs vacuum field strength):
  • Higgs - 125.09 GeV - 0.508
  • Z - 91.1876 GeV - 0.370
  • W - 80.377 GeV - 0.326
  • top quark - 173.5 GeV - 0.705
  • bottom quark - 4.24 GeV - 1.72*10-2
  • charm quark - 1.32 GeV - 5.36*10-3
  • strange quark - 87 MeV - 3.53*10-4
  • up quark - 1.9 MeV - 7.72*10-6
  • down quark - 4.4 MeV - 1.79*10-5
  • tau lepton - 1.78 GeV - 7.23*10-3
  • muon - 105.7 MeV - 4.29*10-4
  • electron - 0.511 MeV - 2.08*10-6
  • neutrinos - <~ 0.05 eV - 2*10-13
What makes this great spread of values

What is the energy density of the zero Higgs field in SI units? - Physics Stack Exchange - this translates into a mass density of 2.72*1028 kg/m3. How dense is the Universe? - BBC Science Focus Magazine But the present average density of the Universe is around 9*10-27 kg/m3

So the actual density is smaller by a factor of 3*1054 relative to the expected Higgs-field density.

Does dark energy somehow compensate for that? I've seen the theory that dark energy somehow tracks the density of the rest of the Universe's constituents.
 
The low masses of neutrinos are just plain weird. Why don't they have masses that are greater than their corresponding charged leptons?

Here is a solution:  Seesaw mechanism

In it, neutrinos have a left-hand part and a right-hand part. It has a "Dirac mass" (m) that is like the charged elementary-fermion masses, bridging its left-hand and right-hand parts. But it also has a "Majorana mass" (M) of the right-hand part connected with itself, a mass much greater than the Standard-Model particles. These produce mixtures of mostly-left-handed and mostly-right-handed neutrinos with the opposite handedness (chirality) having fraction (m/M). The mostly-right-handed part has a Majorana mass of M - m2/M, and the mostly-left-handed part a Majorana mass of m2/M.

For m ~ 10 GeV and M ~ 1012 GeV the resulting neutrino mass m2/M ~ 0.1 eV -- close to what is observed in neutrino oscillations.

So neutrino-oscillation masses offer a glimpse into the world of grand unified theories, since the right-handed Majorana mass is almost at GUT masses, around 1016 GeV -  Grand unification energy
 
Phenomenology Beyond the Standard Model - Ellis.pdf - slides from a talk at a 2013 physics conference. In the last one,

"Let us be patient … If you have a problem, postulate a new particle"

Then giving lots of examples. I will mention them and add some details and some additional ones.

Known: problem, solution, when proposed, when detected
  • Electromagnetic-field equations -- electromagnetic waves -- 1861 (?) -- 1865, 1888
  • Special relativity and gravity -- gravitational waves -- 1916 -- 1974, 2015
  • Masses and charges of atomic nuclei -- neutron -- 1920 -- 1932
  • Quantium mechanics and special relativity -- antiparticles -- 1928 -- 1932
  • Missing energy, spin in beta decays -- neutrino -- 1930 -- 1956
  • Nucleon-nucleon interactions -- pion -- 1935 -- 1947
  • Lepton number conserved -- second neutrino -- late 1940's -- 1962
  • Flavor SU(3) symmetry -- omega baryon -- 1962 -- 1964
  • Flavor SU(3) symmetry -- quarks -- 1962 -- 1968, early 1970's
  • None of certain kinds of decay -- charm quark -- 1970 -- 1974
  • CP violation -- third generation of quarks -- 1973 -- 1977, 1995
  • Strong-interaction dynamics -- QCD, gluons -- 1972 -- 1979
  • Weak interactions -- W, Z -- 1968 -- 1983
  • Electroweak high-energy good behavior -- Higgs particle -- 1964 -- 2012
Hypothetical: problem, possible solution, when proposed
  • High-energy good behavior, avoiding fine tuning -- supersymmetry -- 1971
  • Strong CP problem (QCD doesn't violate it) -- axion -- 1977
  • Dark matter -- WIMP, ... --- ?
  • Dark energy -- quintessence -- ?
  • Cosmic inflation -- inflaton -- 1980
So it can be a long time between proposing something and detecting it.
 
A further complication of the weak interactions is cross-generation decays of quarks. Why do both down and strange quarks decay into up quarks? The solution is that the masses of the up-type quarks and the masses of the down-type quarks are not quite orthogonal, and this lack of orthogonality causes cross-generation decays.
  • Up-type: up, charm, top
  • Down-type: down, strange, bottom
  • Charged leptons: electron, muon, tau
For quarks, this generation mixing is specified in the  Cabibbo–Kobayashi–Maskawa matrix

Generation mixing also causes neutrino oscillations, and it is specified in the  Pontecorvo–Maki–Nakagawa–Sakata matrix

For some reason, quarks are close to orthogonal, while leptons are much less close.

These mass-matrix misalignments have led to a lot of speculation about their underlying forms -- mass-matrix "textures" -- but without much success.
 
Let's now look at the low-energy Standard Model - low-energy relative to electroweak unification, of course.

ParticleSpinQCDQMass(es) in GeV
Z11091.1876
W11+-180.377
photon1100
gluon1800
Higgs010125.09
up-type quarks (3)1/23+2/30.0019, 1.32, 173.5
(antiparticles)1/23*-2/3(same)
down-type quarks (3)1/23-1/30.0044, 0.087, 4.24
(antiparticles)1/23*+1/3(same)
neutrinos (3)1/210<~ 10^(-10)
(antiparticles)1/210(same)
charged leptons (3)1/21-10.000510998950, 105.65838, 1.78
(antiparticles)1/21+1(same)

QCD: Quantum Chromodynamics. 1 = colorless, 3 = quarklike (3 colors), 3* = antiquarklike (3 anticolors), 8 = gluonlike (colors + anticolors - colorless combination)

Q: electric charge as a multiple of the elementary charge

Seems like a big zoo. But we've had zoos before: the chemical elements, atomic nuclei, and hadrons, and they were all resolved.
 
A proposed extension of the Standard Model has "supersymmetry" (SUSY), a kind of symmetry that relates different spins of particles. But SUSY-related particles share all the other quantum numbers. Here are all the expected SUSY partners of Standard-Model particles, with the extra Higgs particles needed to make it work. Since we have yet to observe any of these particles, it is evident that SUSY is broken, forcing up these particles' masses. This forcing up depends on the details of the SUSY breaking, so I won't give any estimates.

ParticleSpinQCDQ
Charged Higgs01+-1
Neutral Higgs (2)010
Charginos (2)1/21+-1
Neutralinos (4)1/210
Gluino1/280
Up-type squarks (6)03+2/3
(antiparticles)03*-2/3
Down-type squarks (6)03-1/3
(antiparticles)03*+1/3
sneutrinos (3)010
(antiparticles)010
selectrons (6)01-1
(antiparticles)01+1

Squarks = scalar quarks, etc. Charginos and neutralinos are mixtures of the SUSY partners of the photon, W, Z, and Higgs particles.

An even bigger zoo.
 
What do these particles look like in electroweak unification? This is with SUSY, and I've added a singlet Higgs and a right-handed neutrino for completeness. The Standard Model has neither, and it has only one Higgs - Hd with "Hu" being a flipped version of it.

ParticleHandSpinSuSpQCDWISWHC
Up Higgs HuL01/211/2+1/2
(antiparticle)R01/211/2-1/2
Down Higgs HdL01/211/2-1/2
(antiparticle)R01/211/2+1/2
Scalar Higgs HsL01/2100
(antiparticle)R01/2100
B-11/2100
W-11/2110
gluon-11/2800
Quark Q (3)L1/2031/2+1/6
(antiparticle)R1/203*1/2-1/6
Up-type U (3)R1/2030+2/3
(antiparticle)L1/203*0-2/3
Down-type D (3)R1/2030-1/3
(antiparticle)L1/203*0+1/3
Lepton (3)L1/2011/2-1/2
(antiparticle)R1/2011/2+1/2
Neutrino (3)R1/20100
(antiparticle)L1/20100
Electron (3)R1/2010-1
(antiparticle)L1/20101

Hand (chirality, handedness): L = left-handed, R = right-handed
SuSp = superpartner spin
WIS = weak isospin, a spin-like quantum number
WHC = weak hypercharge, a charge-like quantum number
 
There are some interrelationships.

One of them is the Gell-Mann-Nishijima one for electric charge in terms of weak isospin and weak hypercharge.

Weak isospin behaves just like spin, quantum-mechanical angular momentum. Its projected or third component WIS3 goes in integer steps from -WIS to +WIS.

The GMNJ formula is Q = WIS3 + WHC, and here are some worked examples:
B: {0} + 0 = {0}
W: {1, 0, -1} + 0 = {1, 0, -1}
Left-handed quarks: {1/2, -1/2} + 1/6 = {2/3, -1/3}
Right-handed up quarks: {0} + 2/3 = {2/3}
RIght-handed down quarks: {0} + -1/3 = {-1/3}
Left-handed leptons: {1/2, -1/2} + -1/2 = {0, -1}
Right-handed neutrinos: {0} + 0 = {0}
RIght-handed electrons: {0} + -1 = {-1}

The next one requires some introduction. Spins are additive, but vector additive, meaning that for spins S1 and S2, their total goes from |S1-S2| + (S1+S2) in integer steps. This means that there is a conserved quantity, spin parity, that is the fractional part of the spin: 0 or 1/2. It is additive modulo 1.

There is something similar for QCD called "triality". Colorless states have triality 0, quarks have triality 1/3, antiquarks triality -1/3, and gluons triality 0. With this size convention, triality is additve modulo 1.

Looking at hadrons, mesons have triality +1/3 - 1/3 = 0 and baryons triality 3 * (+1/3) = 0 modulo 1. These quark combinations are thus colorless.

This relates WIS and WHC:

WHC = WIS - (triality) + (integer)

I'll work it out again.
B: 0 = 0 + 0 + 0
W: 0 = 1 + 0 - 1
Left-handed quarks: +1/6 = 1/2 - 1/3 + 0
Right-handed up quarks: +2/3 = 0 - 1/3 + 1
RIght-handed down quarks: -1/3 = 0 - 1/3 + 0
Left-handed leptons: -1/2 = 1/2 + 0 - 1
Right-handed neutrinos: 0 = 0 + 0 + 0
RIght-handed electrons: -1 = 0 + 0 - 1

So QCD and electroweak unification are somehow related.
 
Another consequence of adding SUSY to the Standard Model is that the elementary fermions (quarks, leptons) and the Higgs particles are both "Wess-Zumino multiplets", containing spin-0 and spin-1/2 particles.

Likewise, the "gauge particles" (electroweak ones, gluon) are in SUSY multiplets spin-1 and spin-1/2 particles.

Applying SUSY to gravity gives "supergravity" (SUGRA), not superstrong gravity but supersymmetric gravity. The simplest forms have not only a graviton (spin 2) but also a gravitino (spin 3/2), and the more extended ones also have particles with spin 1, 1/2, and/or 0.

Turning to Particle Data Group's particle listings I find a lot of mass limits, typically around a few hundred GeV for colorless SUSY partners, and at least 1 TeV = 1,000 GeV for an extra Higgs particle and for colored SUSY partners.

These searches were likely done at the Large Hadron Collider, and at its energies, electromagnetic and weak interactions are roughly at the same strength, what one would expect from unbroken electroweak unification.


BTW, the simplest addition of SUSY to the Standard Model is the Minimal Supersymmetric Standard Model (MSSM).
 
The interactions between the elementary fermions (quarks, leptons) and the Higgs particles can be represented in a simplified way as

gu,ij.(U*)i.Hu.Qj + gd,ij.(D*)i.Hd.Qj + gn,ij.(N*)i.Hu.Lj + ge,ij.(E*)i.Hd.Lj
+ its "Hermitian conjugate", a version with particles and antiparticles reversed

The g's are the Higgs coupling constants, and the expression is summed over generations i and j with ranges 1, 2, 3.

In the Standard Model, the Higgs particle H has self-interaction
- m2 |H|2 + g |H|4

In the MSSM, the self-interaction (mu).Hu.Hd with SUSY-breaking terms.

The mass term (mu).Hu.Hd is an oddity, and a fix is the NMSSM, the Next-to-MSSM, with an additional Higgs particle, Hs:

g.Hu.Hd.Hs + g'.(Hs)3 + SUSY-breaking terms.

A big problem is that if electroweak symmetry breaking is due to SUSY breaking, then SUSY partners ought to also acquire masses from this same symmetry breaking, at around the EWSB mass scale, roughly the Higgs-particle field strength of 246 GeV. But the easiest to detect SUSY partners, the gluino and the squarks, have masses several times larger.

My position on SUSY is roughly that of Albert Einstein if deflection of starlight by the Sun did not match his predictions: "Then I would feel sorry for the good Lord. The theory is correct."
 
Seeing all these patterns in the elementary-particle zoo suggests some underlying order, some order much like those discovered for previous zoos: chemical elements, nuclei, and hadrons.

That underlying order is a grand unified theory, and numerous such theories have been proposed. I will discuss those that I think are especially simple.

Grand unified theories always include some superset of the gauge-particle symmetry group of the Standard Model, some superset that is typically simpler in structure. So let us look at that group.

Quantum chromodynamics (QCD) has one called SU(3) and electroweak (EW) unification has one called SU(2)*U(1) with the SU(2) being for the unified W and the U(1) being for the unified B. EW breaking gives a much-reduced symmetry group, U(1) for electromagnetism.

Thus, the Standard Model has SU(3)*SU(2)*U(1)

I will also be giving the multiplicity of each multiplet of particle, ignoring generations and spin states.

The unbroken (MS)SM multiplicities:
  • Gauge particles: B 1, W 3, gluon 8
  • Higgs particles: Hu 2, Hd 2, Hs 1
  • Quarks: (left) Q 6, (right) U 3, D 3
  • Leptons: (left) L 2, (right) N 1, E 1
 
The first grand unified theory (GUT) I will consider is the Georgi-Glashow SU(5) one, the smallest one that includes the Standard-Model symmetry group as a single group rather than a product of groups.

ParticleHandSpinMultComposition
Gauge-1, 1/224gluon + W + B + (3,2,-5/6) + (3*,2,5/6)
Up HiggsL0, 1/25Hu + (3,1,-1/3)
(antiparticle)R0, 1/25*Hu* + (3*,1,1/3)
Down HiggsL0, 1/25*Hd + (3*,1,1/3)
(antiparticle)R0, 1/25Hd* + (3,1,-1/3)
Scalar HiggsL0, 1/21Hs
(antiparticle)R0, 1/21Hs*
Elem. fm. 1L1/2, 01N*
(antiparticle)R1/2, 01N
Elem. fm. 2L1/2, 010Q + U* + E*
(antiparticle)R1/2, 010*Q* + U + E
Elem. fm. 3L1/2, 05*L + D*
(antiparticle)R1/2, 05L* + D

Mult is multiplet, denoted by its size and sometimes also for additional symbols that distinguish same-size ones, like one being the conjugate or antiparticle version of another one. I use a * for that distinction, though elementary particle physicists usually write a bar over the symbol.

The Higgs interactions become
  • Up quark: F(10).H(5).F(10)
  • Down quark: F(5*).H(5*).F(10)
  • Neutrino: F(1).H(5).F(5*)
  • Electron: F(10).H(5*).F(5*)
  • Self-interaction: H(5).H(5*)
H = Higgs, F = elementary fermion (quark, lepton)

Note that the down-quark and electron ones are mirror images of each other, thus predicting that the bottom quark and the tau lepton, at least, have the same mass at GUT energy scales.
 
A further consequence of this SU(5) GUT is some extra particles, the leptoquarks (3,2,-5/6) with antiparticles (3*,2,5/6) and the Higgs triplets (3,1,-1/3) with antiparticles (3,1,-1/3). At low energies, we have leptoquarks X (3,-4/3) and Y (3,-1/3) wtih antiparticles X* (3*,+4/3) and Y* (3*,+1/3) and Higgs triplets (3,-1/3) with antiparticle (3*,1/3). Three entries (QCD, WIS, WHC), two entries (QCD, Q): Q = electric charge.

These particles make proton decay, decay of isolated protons. That has been searched for, with an upper limit on the proton decay rate now low enough to rule out some GUT's.

This is not only a decay of protons, but also of neutrons and of every other kind of hadron. Neutrons should decay in GUT fashion close to the rate of protons.

But this SU(5) GUT has a conserved quantity: B - L = (baryon number) - (lepton number)

That means that protons can decay by (proton) -> (neutral pion) + (positron) but not by some other kinds of decays.
 
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