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129A Lecture Notes
Standard Model
1 Issues of Mass
We have learned that the weak interaction can be explained by the exchange of W and Z bosons, which arise in SU (2) × U (1) gauge theory together with the photon. In particular, the Z boson and the photon are linear combina- tions of neutral SU (2) gauge boson W 3 and the hypercharge gauge boson B,
Z = W 3 cos θW − B sin θW , (1) A = W 3 sin θW + B cos θW. (2)
This raises a very naive question: how come that out of four gauge bosons three of them become massive while a particular combination (photon) re- mains massless? This issue is also coupled to the number of degrees of freedom. A massless spin one boson such as the photon has only two degrees of freedom: helicity ±1. They correspond to two circular polarizations in classical electromag- netic wave. On the other hand, a massive spin one boson can be looked at in its rest frame, and it forms the usual spin one representation J = 1 of the angular momentum. Therefore there are three degrees of freedom, Jz = 1, 0 , −1. Where does the additional degree of freedom come from? The problem actually does not stop with the gauge bosons. Look at the quantum number assignments under SU (2) × U (1) of quarks and leptons,
( uL d′ L
)+1/ 6 , u+2 R /^3 , d− R 1 /^3 ,
( νe eL
)− 1 / 2 , e− R 1. (3)
(Here only the first generation particles are shown, but the second and the third generation particles have the same quantum numbers.) Because of the V − A nature of the charged-current weak interaction, only the left-handed particles are weak isodoublets and hence couple to the W -boson, while the right-handed particles are singlets. They have different hypercharges. If you see two particles with different electric charges, you would say they are different particles. In the same way, we have to admit that the right-handed
and left-handed electrons are different particles. No matter how bizzarre it sounds, we have to admit it is true. If the electron were massless, this poses no problem. Because they would zoom around at the speed of light, the left-handed electrons would be left- handed in all frames of reference, and this distinction is Lorentz invariant. You have the left-handed electron state and the right-handed positron state in the doublet. But once it has a mass, you can stop it, and observe that a spin 1/2 particle must have two states, spin up and down. Somehow the right- handed electron state must come in, together with the left-handed positron state, and they are mixed up. What is going on? Overall, the issue of mass is what we used to take for granted in Newtonian mechanics and even in quantum mechanics, but we can’t ignore it anymore in the world of elementary particles. We somehow have to think about the origin of mass.
2 Superconductor
This is the point where particle physicists learned a great deal from condensed matter colleagues. The problem we are facing is that we somehow need to understand how a gauge boson, which is supposed to be massless and has only two degrees of freedom such as the photon, can acquire a mass and make the force short-ranged. It turns out that we have seen such a system in the laboratory: superconductors. The famous Meißner effect of a superconductor is an effect that the mag- netic field is repelled out from a superconductor. As a result, a piece of superconductor can float in a magnetic field. You may have seen a demo of this effect. If you look more closely at the magnetic field at the edge of the superconductor, you find that the magnetic field is not completely repelled, but penetrates into the superconductor over a characteristic distance scale called the penetration length λp. The magnetic field is damped exponentially as e−r/λp^ into the superconductor. The magnetic field is short-ranged! Of course the magnetic field is a long-ranged force, but somehow managed to become short-ranged in a superconductor. This is the model we would like to learn from in order to understand the short-ranged weak interactions. Most superconductors are metal above the phase transition temperature. (There are, however, polymers and ceramic that become superconductors as well.) In a metal, “free electrons” move around in the lattice made of
The Maxwell’s equation (or Amp`ere’s law) for the magnetic field is
∇ ×^ ~ B~ = μ 0 ~j. (5)
(Here I ignored the time derivative of the electric field.) In the presence of an external field, the condensate has the constant number density ψ∗ψ(~x) = ρ, and hence we can write ψ(~x) =
ρ apart from the phase ambiguity. Putting them together, we find
∇ ×^ ~ B~ = μ 04 e
2 2 m
ρ A.~ (6)
In the Coulomb gauge ∇ ·~ A~ = 0, the l.h.s. simplifies to (∇ ×~ B~)i =
ijk∇j (klm∇lAm) = ∇i∇j Aj − ∇j ∇j Ai = ∇i(∇ ·~ A~) − ∆Ai = −∆Ai. There- fore, we find
− ∆ A~ = −μ 0
2 e^2 m
ρ A.~ (7)
If you specialize to the z direction and one component of the vector potential it is easy to see that the equation is
d^2 dz^2
A =
2 e^2 μ 0 ρ m
A, (8)
and hence is exponentially damped
A ∝ e−z/λp^ , λp =
√ m 2 e^2 μ 0 ρ
An intuitive way to see what made the magnetic field short-ranged is to picture the photon getting bumped around by the condensate. Because the photon couples to anything that is electrically charged, it bumps on the condensate. Then it gets bounced around, and becomes short-ranged. And as Yukawa said, a short-ranged force means a massive particle.
3 Higgs Condensate in Universe
Now it is pretty clear what we have to swallow: our Universe is filled with a Bose–Einstein condensate of something that is charged under the SU (2) × U (1). It has a name, Higgs condensate, even though we don’t quite know what it is yet.
in vacuum in superconductor
Figure 1: The photon gets bounced around by a Bose–Einstein condensate with an electric charge, and becomes short-ranged.
We know something, however. The condensate should not disturb pho- tons, while it should W and Z bosons. That fixes the quantum number of the condensate; it should be basically the same as the neutrinos. Neutrinos do not carry an electric charge, but does interact with W and Z bosons. This was possible because the neutrinos are in isodoublets with hypercharge − 1 /2, and the combination or W 3 and B that couples to this component is precisely the Z boson. Therefore, if the Higgs boson is an isodoublet and has hypercharge − 1 /2,
H =
( H^0 H−
) , (10)
it has exactly the same coupling as the lepton doublet has, and the neutral (upper) component behaves the same way as the neutrinos. Once this com- ponent acquires a condensate, it disturbs W and Z but not the photon. This is precisely what we need. In fact, this idea allows us to calculate the mass of the W and Z bosons given the condensate 〈H^0 〉 = v/
- The coupling is given by
g
~τ 2
· W~ + g′
( −
) B =
( gW 3 − g′B
√^2 gW^ + 2 gW −^ −gW 3 − g′B
)
( gZ Z
√^2 gW^ + 2 gW −^2 eA + (−1 + 2 sin^2 θW )Z
) .(11)
Therefore, the coupling of the W and Z to the condensate generates the masses
m^2 W =
g^2 v^2 , m^2 Z =
g Z^2 v^2. (12)
Recalling gZ = e/ cos θW sin θW and g = e/ sin θW , we find
m^2 Z cos^2 θW = m^2 W. (13)
dox between purely V − A nature of the charged-current weak interaction is reconciled with the finite mass of the electron. The generated electron mass is me = yev. (15)
There is no theoretical principle that determines the size of this Yukawa coupling. The Higgs boson is not a gauge boson, and it is not subject to the universality as the gauge interactions. We simply choose the size to reproduce the observed mass, ye ≈ 2 × 10 −^6.
e
t
eL
eR eL
eR
μ R μ R
μ L μ L μ R
μ L
tL
tR
tL
tR
ν L
Figure 2: The left-handed particles bump on the condensate and become right-handed, and vice versa. They mix quantum mechanically and the Hamiltonian eigenstates are their mixtures. On the other hand, neutrinos can’t bump on the condensate because there are no right-handed neutrinos.
We introduce different Yukawa couplings to all three generations of the charged leptons, yμ ≈ 4 × 10 −^4 , yτ ≈ 7 × 10 −^5 , as to reproduce the observed masses. What about quarks? There is an additional complication because there are both right-handed up- and down-type quarks, while there are no right- handed neutrinos in the lepton sector. Therefore there are two types of Yukawa couplings needed. Moreover, as you will see soon below, we can let any three generations of right-handed and left-handed quarks couple to the Higgs boson. We need to keep track of the generation index i = 1, 2 , 3 for the left-handed uLi, dLi and right-handed uRi, dRi quarks. The general Yukawa
couplings then become matrices Yd and Yu. The element (Yd)ij makes the j-th right-handed down quark dRj emit the neutral Higgs boson and transforms it to the i-th left-handed down quark dLi. Similarly, the element (Yu)ij makes uRj emit the anti-particle of the neutral Higgs boson and transforms it to the uLi. It has to be the anti-particle in order to conserve the hypercharge. Once the Higgs boson condenses, both up- and down-type quarks acquire mass matrices, Mu = Yuv, Md = Ydv. (16) Because the mass appears in the Hamiltonian, E =
~p^2 c^2 + m^2 c^4 , we need to diagonalize the mass matrix to obtain the Hamiltonian eigenstates. The point is that we need mass-squared, and it needs to be hermitean. There are two ways to construct hermitean mass-squared matrices, M (^) u†Mu and MuM (^) u†. Which one do we use? Well, both. Remember Mu acts on the right-handed up quarks on the right, and M (^) u† then acts on the left-handed up quarks because of the transposition. We in general need to diagonalize both matrices on the space of right-handed and left-handed up-quarks sep- arately. The same is true with the down quarks. Therefore, we need four independent unitary rotations,
M (^) u†Mu = VuR D^2 uV (^) u†R , MuM (^) u† = VuL D^2 uV (^) u†L , (17) M (^) d† Md = VdR D^2 dV (^) d†R , MdM (^) d† = VdL D^2 dV (^) d†L. (18)
Here, Du,d are diagonal matrices of mass eigenvalues,
Du =
mu 0 0 0 mc 0 0 0 mt
, D d =
md 0 0 0 ms 0 0 0 mb
. (19)
Four unitary matrices VuR , VuL , VdR , and VdL are all different in general. The important point is that dL and uL live in the doublets. Therefore, we you do VdL rotations on dL and VuL rotations on uL, there in general appears a mismatch. In other words, there is no basis in which both components in a given doublet are in the mass eigenstates. This is the origin of the Cabibbo– Kobayashi–Maskawa mixing matrix in the Standard Model. Using the mass eigenstates umL and dmL , the original doublets are given in
Qi =
( uLi dLi
)
( (VuL )ij umLj (VdL )ij dmLj
)
. (20)
LEP-II experiment has searched for the Higgs boson extensively. In the case of LEP-II, electron positron annihilation produces a virtual Z-boson, which in turn can become a real Z-boson and a Higgs boson. The coupling of ZZh is proportional to the Z mass and hence is large. It gradually increased the center-of-momentum energy up to 209 GeV. Higgs boson decays rapidly into the heaviest particle available. At this energy, it is b¯b. Thanks to the fact that Vcb is small, b-quarks form mesons that are relatively long-lived and we can tag their decays to look for such events.
Figure 3: Branching fractions of the Higgs boson into various final states. It basically decays into the heaviest particle kinematically allowed.
Unfortunately it did not find the Higgs boson; it has set a lower limit on its mass at 114.4 GeV at 95% confidence level. However, it did see a hint for a new particle at 116 GeV. The hint was only at two sigma level, and may be a statistical fluctuation. We don’t know. To get a better idea on where the Higgs boson should be, I’d like to recall a story how the top quark mass was known before it was discovered. The Z boson can split into a virtual pair of top and anti-top quarks, and come back. Similarly, the W boson can split into a virtual top-bottom pair. These
processes give a small correction to the mass of the Z and W bosons at a few percent level. Given amazing precision achieved in experiments, such a few percent correction can be extracted, and be used to determine the mass of the top quark. The relationship between the mZ and mW is modified to
m^2 W = m^2 Z ρ cos^2 θW , (23)
where θW is measured in the forward-backward asymmetries at the Z-pole, and the correction factor is
ρ = 1 + 3
GF m^2 t 8
2 π^2
Precise measurements of mZ , mW , and θW allowed us to extract mt before it was discovered. Indeed, the current data set without including the direct measurement of mt gives mt = 181+11 − 9 GeV, while the measurement from Tevatron gives mt = 174. 3 ± 5 .1 GeV. See http://lepewwg.web.cern.ch/ LEPEWWG/stanmod/. The direct measurement comes right in the middle of the range suggested by the indirect measurements which do not observe the top quark at all. The idea is to repeat this game on mh this time. Unfortunately, the observables depend only rather weakly on mh, and the dependence is loga- rithmic. It makes the extraction of mh from the precision measurements very hard. Nonetheless, the fit to the current data suggest mh = 85+54 − 34 GeV, and the 95% CL upper bound is mh < 196 GeV. It strongly suggests that the Higgs boson is just around the corner. The next experiment that has a chance of discoverying the Higgs boson is the new run of Tevatron pp¯ collider at Fermilab, Illinois. It has increased the center-of-momentum energy from 1.8 TeV to 2 TeV, and is running at much higher intensity. The production is due to the fusion of, say, an up quark inside the proton and an anti-down quark inside the anti-proton, going through a virtual W , to the final state of a real W and Higgs. Then the W decays into a lepton and a neutrino, which is a very clean signature, plus the Higgs boson decaying into b¯b. Beyond Tevatron, a new pp collider Large Hadron Collider (LHC) will be put in the LEP tunnel. The center-of-momentum energy is 14 TeV, and will run at even higher intensity than the current Tevatron. At this high energy, the collision of the gluons inside the proton dominates, and the annihilation of quarks and anti-quarks becomes subdominant. Because of this reason, pp
0
2
4
6
20 100 400 mH [GeV]
∆χ
2
Excluded Preliminary
∆α∆αhad(5) = 0.02761±0. 0.02747±0. Without NuTeV
theory uncertainty
Figure 5: ∆χ^2 = χ^2 − χ^2 min vs. mh curve. The line is the result of the fit using all data (last column of Table 13.2); the band represents an estimate of the theoretical error due to missing higher order corrections. The vertical band shows the 95% CL exclusion limit on mh from the direct search. Taken from the Winter 2001 data by LEP Electroweak Working Group, http:// lepewwg.web.cern.ch/LEPEWWG/stanmod/.
Figure 6: The necessary integrated luminosity for exclusion and discovery of the Higgs boson at Tevatron. The current (winter 2002) integrated luminosity is about 100 pb−^1 = 0.1 fb−^1. Much more luminosity is needed.
