In the beginning theory of electroweak interactions and weak isospin, all of the quarks, leptons, Ws and Bº have to be massless to have the weak isospin symmetry. But the quarks, leptons and bosons have different masses in the real world. A brilliant way out of this was found by Higgs, Steven Weinberg and Abdus Salam. The symmetry of the interactions would not have to be altered, if the vacuum of space provided the particles with masses by interactions with it. Something had to be put into the vacuum to cause this, and it was called the Higgs field. The Higgs had to be coupled or interact with the other fields, in what are called Yukawa couplings.
Higgs is one of six authors on this field in 1964 who were recognized by the award of the 2010 J. J. Sakurai Prize. Here are their pictures and names:
| Carl R. Hagen Francois Englert Gerald S. Guralnik Peter W. Higgs Robert Brout T.W.B. Kibble |
where Peter Higgs is shown in the following picture:
One problem remained, however. As we had discussed, a massless spin one photon field only needed two field components, Sz = +1, and Sz = -1, its two helicities or circular polarizations. When the Higgs field acquires a non-zero vacuum value and generates masses, the photon field has to stay massless as it is known to be. But in the case of the W fields of spin 1, there are only two components when massless, but the massive W+ and W¯ (80 GeV), need a third spin component Sz = 0, since they can exist at rest. Similarly, the initial massless Zº field with two helicities has to gain another Sz = 0 component when massive (91 GeV). So in addition to a neutral Higgs field to fill the vacuum, we need a positive and a negative charged spinless field to add components to W+ and W¯, and a neutral spinless field to add a component to Zº. Thus we need four Higgs fields or field components, of which two are neutral and one is positively charged, and one is negatively charged.
The four components can be supplied by a weak isospin doublet field I = 1/2, composed of an H+ and Hº, and its anti-particles, H¯ and anti-Hº. H+ should have Iz = +1 to make the Sz = 0 component of W+ which has Iz = +1, and its anti-particle, H-, will then have Iz = -1 to make the Sz = 0 component of W-. We crudely say that the H+ and H¯ fields have been “eaten” to make up the massive W+ and W¯ fields. Here is a cute illustration to help you remember by Philip Tanedo of the US LHC collaboration, and of the Quantum Diaries blog http://www.quantumdiaries.org/author/flip-tanedo/
Since Wº and Bº both have Iz = 0, a linear combination of Hº and anti-Hº, both with Iz = 0, can be used to make up the massive Sz = 0 component of Zº. An orthogonal or perpendicular linear combination is then left uneaten as the Higgs field called φ = (v + h)/√2, where v = 246 GeV is the constant part that fills the vacuum uniformly everywhere, and h is the part that can be excited or brought into existence in a collider experiment.
The mixing of Wº and Bº to form Zº and the photon γ is just like a rotation by the Weinberg angle θW (W_3 is the same as Wº), and the photon γ remains massless and zips away at the speed of light:
γ = cos(θW) Bº + sin(θW) Wº
Zº = -sin(θW) Bº + cos(θW) Wº
The value of θW is determined by cos(θW) = (mass of W) / (mass of Z) = 80 Gev / 91 GeV.
θW is about 30°.
The Zº is the weak neutral current boson, and can be exchanged at the same time as the photon is in an interaction:
The Zº coupling is partly with the Wº coupling with the left handed fermions and partly proportional to the charge Q of the fermion with the same vector interaction as the photon has.
The W and Zº weak interactions are short range because of their heavy masses of 80 GeV and 91 GeV, respectively. Their range is about 1/100 the size of the proton. So when two protons collide at high energy and short time, the probability of interacting is that of the quarks from each proton passing by each other within the distance of 1/100 the proton size. Furthermore, because of the unified electroweak theory above, they couple roughly with the same weak charge as the photon, whose dimensionless form is
α = e²/hbar c = 1/137.036,
where (hbar x c) = 197 MeV fm, MeV is a million electron volts, and a Fermi, fm = 10^(-13) centimeters, the size of a proton.
So the probability of producing weak bosons or having a weak interaction in the LHC is many orders of magnitude lower than a QCD interaction where the range is long, the quarks are usually light, and the coupling is strong.
Here we simplify and avoid mathematical complexities. The contribution to the energy E from or the interaction energy V of the neutral scalar Higgs field φ coupling to a fermion quark or lepton f is called a Yukawa coupling y:
E ← y φ (L ←R),
where (L←R) is a term that takes a right handed spin fermion f to a left handed one.
The mass m of the fermion f also contributes to the energy with the same (L←R) term as
E ← m (L←R). Here we are assuming no masses at the start to have perfect Weak Isospin Symmetry.
The Higgs trick is to have the Higgs field φ attain a vacuum expectation value v which is constant everywhere, plus an excitable part h:
φ = (v + h)/√2. Then from the Yukawa term we have
E = y (v + h)/√2 (L←R).
But the constant coefficient of the mass term (L←R) is the mass, which is now
m_f = y_f · v/√2.
We solve this for each fermion y_f /√2 = m_f / v, the interaction term or energy from the excitable part of the Higgs field h becomes
E = (m / v) h (L←R) ≡ g h (L←R),
where g is now the coupling strength of the Higgs field h to the fermion.
So the couplings of each fermion to the Higgs field h is
g_f = m_f / v,
and is proportional to the mass of any fermion f divided by the same constant, v = 246 GeV, which sets the mass scale of the Higgs interactions.
(As an aside on weak isospin, the appearance of the weak isospin 1/2 left handed fermion field L and the weak isospin zero fermion field R in the abbreviated term (L←R) means that this term has weak isospin 1/2 + 0 = 1/2. This is coupled to the Higgs field φ of weak isospin 1/2 in such a way to give a total weak isospin 0 for the energy E, making it weak isospin symmetric.)
The masses of the W and Z spin 1 (vector) bosons are given in terms of the Higgs vacuum value v and sin(θW) and cos(θW):
m_W = e v /(2 sin(θW);
m_Z = m_W/cos(θW).
The one thing not predicted in the theory is the mass of the Higgs. This has of course been a nightmare, since particle physicists didn’t know how far to push the previous accelerators in energy or run time to search for it, and how much energy would be needed to produce it if it turned out to be very heavy. If the new particle is the Standard Model Higgs at
m_H = 125 GeV,
a great sigh of relief is given that it is not hundreds of GeV. On the other hand, it was just a bit more massive than the LEP large electron positron collider at CERN in the same tunnel could reach at 114 GeV. It was also more massive than the Fermilab Tevatron outside of Chicago could reach with more than a few events.
From the present LHC runs of 6,000 trillion collisions, about 400 Higgs events have been detected.
The coupling of W’s and Z’s to the Higgs is proportional to their masses squared:
g_HWW = 2 (m_W)² / v ; and g_HZZ = 2 (m_Z)² / v.
The self coupling of the Higgs to itself is:
g_HHH = 3 (m_H)² / v; and their is a coupling of four Higgses at the same point given by
g_HHHH = 3 (m_H)² / v²
Now that the mass m_H of the presumed Higgs is known, all of its production channels and decay probabilities (called cross sections) can be compared to that predicted by the Standard Model couplings above to test whether it is indeed the Standard Model Higgs.
Well, that is the end of my simplified discussion of the Higgs boson. I will keep up with the new results from the analysis of already taken data that ended in 2012. A more complete set of lectures on the Higgs that is free on the web is by Matt Strassler at
http://profmattstrassler.com/ ,
and there is up-to-date coverage on Résonaances at
http://resonaances.blogspot.com/
There is also good coverage of particle physics and the Higgs by Flip Tanedo at
