Photon Coupling with Matter, u R

Transcription

1 1 / 16 Photon Coupling with Matter, u R Consider the up quark. We know that the u R has electric charge 2 3 e (where e is the proton charge), and that the photon A is a linear combination of the B and W 3 vectors. However, the W 3 does not interact with right-handed particles, so the interaction of the photon with the u R must be the same as the interaction of the B with the u R : The factor cos θ w counts the amount of the B in the photon, while g is the U(1) coupling constant, and Y (u R ) is the weak hypercharge of the u R, which governs how much the u R couples to the B. This leads to an equation: Y (u R )g g g cos(θ w ) = Y (u R ) g 2 + g = e. We used the definition of the weak mixing angle in terms of the SU(2) and U(1) coupling constants g and g in the middle equation.

2 2 / 16 Photon Coupling with Matter, u R It turns out that the factor g g appears as on overall common factor each time the photon g 2 +g 2 couples to an electrically charged particle (we will see another example of this in a moment.) So we can define it to be the electric charge: e = g g g 2 + g 2 = g cos(θ w ) = g sin(θ w ) Then we have to choose the weak hypercharge of the u R to be Y (u r ) = 2 3. Notice that the electric charge of the proton is now understood in terms of two more fundamental parameters: the U(1) and SU(2) couplings. This is one way to see that the familiar electromagnetic interaction is unified with the weak forces. The unified forces are known as the electroweak force.

3 3 / 16 Photon Coupling with Matter, u L In just the same way, we can understand how the left-handed up quark couples to the photon. One thing is different: now the W 3 does play a role. So the coupling of the u L to the photon is made up of two parts: the coupling of the u L to the B (times cos θ W, the amount of the B that is mixed into the photon) plus the coupling of the u L to the W 3 (times sin θ W, which is the amount of the W 3 in the photon.) In terms of diagrams:

4 4 / 16 Photon Coupling with Matter, u L These diagrams lead to an equation: 2 3 e = Y (u L)g cos θ w + 1 g sin θw 2 Last time, we learned that the weak mixing angle g2 + g 2 is determined by couplings g and g as shown in g this triangle: θ g Using this information, we can fill in the value of cos θ w and sin θ w in our equation at the top of the slide. If we also use the definition of the electric charge, the equation becomes 2 3 e = Y (u L)e e so we find that the weak hypercharge of the left-handed up, Y (u L ) is =

5 5 / 16 Left- and right-handed weak hypercharges We see now that the weak hypercharge of the u R is while the weak hypercharge of the u L is Y (u r ) = 2 3, Y (u L ) = 1 6. So the left-handed and right-handed up quarks have completely different fundamental interactions in the standard model. It does not make sense to consider the left-handed and right-handed up quarks to be parts of the up quark when they have completely different interactions. They are just different particles. The Higgs breaks electroweak symmetry when it takes on its vacuum expectation value. After this happens, the left- and right-handed up quarks do have the same interactions with respect to the electromagnetic force. So it is only after electroweak symmetry that the up quark makes sense as a single particle. Mass terms for charged fermions mix left-handed and right-handed particles together. So before electroweak symmetry breaking, it is not possible to give mass to the up quark, or any other fermion in the standard model. But after electroweak symmetry is broken, then the fermions can get a mass. In the standard model, the mass of the up quark comes about because of its interaction with the Higgs field. This is also true of all fermions in the standard model.

6 6 / 16 Quark Interactions with the Higgs Now we ll discuss how the Higgs interacts with the fermions in the standard model. We ll focus on the examples of the up and down quark. In terms of diagrams, the fundamental interactions are: There are a few things to notice about this diagram. First, the strength of the interaction is given by m u/v (in the case of the up quark.) This ratio y u = m u/v is a dimensionless number, known as a Yukawa coupling, representing the strength of the coupling between the Higgs field H and the up quark. These ratios are fundamental parameters of the standard model; after symmetry breaking, they are related to the particle masses. The second point to notice is that the interaction between the up quark and the Higgs, for example, involves both the left-handed and the right-handed up quarks.

7 Quark Interactions with the Higgs After symmetry breaking, the fundamental interaction among the Higgs and quarks lead to two phenomena. The first is a coupling between the quarks and the physical Higgs scalar: This differs from the interaction on the previous slide only in that we have replaced the full Higgs field by the Higgs scalar. So the Higgs scalar interacts with the quarks; the strength of the interaction is given by the Yukawa coupling between the Higgs and the particular quark in question. After symmetry breaking, the mass of the quark ends up being proportional to the Yukawa coupling, so the Higgs scalar couples to a quark with a strength proportional to the quark mass. The second phenomenon is obtained by replacing the Higgs field in the fundamental interaction with its vacuum expectation value, v (which, of course, is just a constant): This diagram describes a process in which a single left-handed quark turns into a right-handed quark. The strength of this process is given by the quark mass. So after symmetry breaking, left-handed and right-handed quarks are tied together into one particle. Notice that before electroweak symmetry breaking it would not make sense to draw a diagram like this, because the weak hypercharges of the left- and right-handed particles is different. Then, a diagram like this would involve a particle of (weak hyper)charge 1 6 transforming into a particle of (weak hyper)charge 2 3. That would not be allowed, since charge must be conserved. 7 / 16

8 8 / 16 Particle Structure of the Standard Model Before symmetry breaking, the standard model has left- and right-handed particles. In the quark sector, as we ve seen the u L and u R quarks are tied together by the Higgs vacuum expectation value into the up quark. Similiarly, the d L and d R quarks are components of the down quark. The electron is also described as a pair of particles e L and e R which join together after electroweak symmetry breaking. e R does not interact with the W gauge bosons, but e L does. So it must be part of a pair of left-handed particles; this pair is ( νl where ν L is the (electron) neutrino. As far as we know, there are no right-handed neutrinos. e L )

9 9 / 16 Particle Structure of the Standard Model One curious thing about nature is that this structure is repeated three times. We call these copies generations : there are three generations of quarks, for instance. The particles in each generation are as follows:

10 10 / 16 Higgs Couplings and Particle Masses We are almost in a position to understand how the Higgs boson behaves at the LHC. The key idea about the behaviour of the Higgs boson is simple to summarise: the Higgs couples to mass We have already seen this in the case of the quarks: the quark mass is given by the product of the Higgs vacuum expectation value, times the Yukawa coupling between the Higgs and the quark. So the Yukawa couplings are proportional to the Higgs mass. For fermions, the Higgs coupling is directly proportional to the Higgs mass. So to understand how the Higgs is produced at the LHC, we need to know a little about the masses of the fundamental particles. Particle Mass (GeV/c 2 ) Particle Mass (GeV/c 2 ) top quark 170 muon 0.11 Z boson 91 strange quark 0.1 W boson 80 down quark bottom quark 4 up quark tauon 1.7 electron charm quark 1.2 neutrinos 0 The gluons and the photon are massless. Notice that there is a big spread of masses in this table: the top quark couples the Higgs with a strength about 100,000 times the up quark coupling to the Higgs!

11 11 / 16 Producing the Higgs Boson at the LHC Now, let s move on to see how the Higgs is created at the LHC. The LHC is a proton-proton collider: protons are accelerated to very high energies in two beams, which are brought together so that the protons collide. In the same way that accelerating electrons radiate photons (electromagnetic waves) in your mobile phone antennae, it is possible for the accelerating particles inside the proton to radiate other kinds of particles during the collision. Any kind of particle may be created, provided: 1. there is enough energy to make it; 2. the components of the proton couple to the particle of interest. So to make a Higgs boson, we need enough energy. Since the LHC has run so far at energies of 7,000 GeV and 8,000 GeV, there is enough energy to make a Higgs boson which has a mass of about 125 GeV/c 2.

12 12 / 16 Producing the Higgs Boson at the LHC It is also the case that the components of the proton couple to the Higgs. What are these components? One picture of the proton is that it is made of two up quarks and a down quark, held together by gluons. But in more detail, there are also other quarks in the proton. In fact, top quarks are present in the proton via a splitting of gluons: t g g So we can create a Higgs boson from two gluons, in a process known as gluon fusion: g t h g This does not happen much, but the coupling of the Higgs to the top is 100,000 = 10 5 times the coupling of the Higgs to the up and down quarks. In addition, these diagrams describe the quantum mechanical amplitude for a process to occur. To get the probability, the amplitude has to be squared. So the gluon fusion process is enhanced relative to the coupling of the Higgs to the up quarks by a factor of which is 10,000 million. Even though you have to pay a price to get the top quarks from the gluons, this is the most important production process of the Higgs scalar at the LHC.

13 13 / 16 Higgs Decays The Higgs boson is not a stable particle, so the experimental teams at the LHC face the challenge of finding evidence of the fleeting existence of the Higgs boson. They do this by understanding how the Higgs decays, and distinguishing these decays from other processes that can occur at the LHC. So now we ll begin to think about the various possible ways that the Higgs can decay. Since the Higgs couples strongly to the top quark, you might think that the most important Higgs decay would be to two top quarks. But, in fact, this decay process is not allowed. The reason is simply conservation of energy. In its rest frame, the Higgs boson has a mass-energy of 125 GeV. But two moving top quarks have an energy of at least twice the mass-energy of the top quark, which is about 340 GeV. So there isn t enough energy for a Higgs scalar to decay to two top quarks.

14 14 / 16 Higgs Decays The next heaviest quark is the bottom quark. Since the mass of the bottom quark is around 4 GeV/c 2, there is certainly enough energy for the Higgs to decay to bottom quarks. In fact, this process h bb is the most likely way for the Higgs scalar to decay. But at the LHC, there are many other processes which result in creating various bottom quarks. So even though the Higgs likes to decay to bottom quarks, we have to consider other possible Higgs decay channels to find experimental evidence of the existence of the Higgs.

15 15 / 16 Higgs Decays We have seen that the Higgs is created via gluon fusion. In a very similar way, the Higgs can decay to gluons: g h But since the proton is full of gluons, again there are many processes at the LHC which create gluons. A very similar process, however, is a good way of finding the Higgs at the LHC: g h In this process, the Higgs decays, via a loop of some charged particles, into two photons. The photons have high energies and are a clean signature of the Higgs boson at the LHC.

16 16 / 16 Higgs Decays Another interesting decay of the Higgs is to two W bosons: h WW. The mass-energy of a W boson is about 80 GeV, so two W s have an energy of at least 160 GeV. Since the Higgs mass is 125 GeV/c 2, the decay of the Higgs scalar to two W bosons is somewhat disfavoured. But the Higgs couples quite strongly to the W, so the decay does happen quite often. Since there is not enough energy to create two W bosons, only one real W is created. The other W in the decay is really a short-lived disturbance in the W field which falls very quickly apart. But since this disturbance falls apart just like a real W would decay, the process looks experimentally as though two W s were created. In a similar manner, the Higgs can decay to two Z bosons. This decay is less important for two reasons: 1. the Z boson is heavier than the W, so there is more energy deficit; 2. there are two W bosons (after symmetry breaking) but only one Z. So there s always a factor of 2 enhancing the W decay process over the Z decay process. There are various possible decay channels. The most probable decay products in descending order are bb, WW, gg, ττ, ZZ, cc and γγ.