Physics 558 – Lecture 8

The Standard Model of the Electroweak Interactions – The Weinberg-Salam Model: (see, for example Chapter 20 in Peskin and Schroeder or Chapter 11 in Rolnick)

To begin let us focus on the lepton sector of the Standard Model so that we are only interested in the electroweak interactions (and gravity). Based on experimental results, we know that we want a theory with two, somewhat special properties. First we want the charge changing interactions, the charged currents, to couple only to the left-handed leptons (and righted-handed antileptons). So we consider a theory with massless fermion matter fields with specific chiral structure. Further we expect the weak interactions to arise from an SU(2) symmetric theory of weak isospin. This isospin is in analogy to the strong isospin we considered last quarter except that here we will require a local gauge symmetry. All generations of the leptons will have the same symmetry structure.

Looking at the first generation, we start with a left-handed doublet as our basic matter (chiral or Weyl) field, including the electron and its neutrino,

For the moment we will ignore the possibility of (and thus a right-handed neutrino), but we do need to include the right-handed electron. Since the field does not participate in the charged current interactions, it must be a singlet under the weak isospin,

(We can add a corresponding right-handed neutrino when the need arises.)

The other phenomenological feature of the electroweak interactions that we want to build into our theory is the fact that there are 3 massive vector bosons, after the spontaneous symmetry breaking, along with the remaining unbroken U(1) of E&M (with its massless photon). Thus we need a slightly more complicated theory than the last example of the previous lecture. To motivate the following theory, recall the Gell-Mann-Nishijima relation from last quarter. In the context of strong isospin and strong hypercharge, we expressed the electric charge of hadrons (and quarks) as a sum of the 3^rd component of isospin and the hypercharge (Y = B + S),

Here we are going to take over exactly the same formula but redefine I and Y to correspond to weak versions of isospin and hypercharge. Then we will apply it to leptons. Apparently the appropriate assignment of quantum numbers is (i.e., we know what values of Q and I₃ we want)

Thus our starting point assumes two distinct local gauge symmetries: SU(2)_L of weak isospin and U(1)_Y of weak hypercharge. By construction these are distinct groups, , and the overall group is an outer product,

Thus the starting point includes the following set of gauge bosons

By our usual dof counting, we have 3x2 + 2 = 8 degrees of freedom in the initial (unbroken) gauge fields. The part of the Lagrangian describing the pure gauge interactions looks like

where we have the usual nonAbelian and Abelian field strength tensors

The structure constants in the nonAbelian expression are those for the required SU(2) symmetry. The corresponding lepton part of the Lagrangian has the form of the usual covariant derivatives

where the are the Pauli matrices (divided by 2) representing the SU(2) generators and we have introduced 2 coupling constants, g for SU(2)_L and g¢ for U(1)_Y. We have also explicitly used the fact that the g^m coupling is diagonal in the helicity basis and that the R lepton is a SU(2) singlet. The total Lagrangian,

describes 4 massless gauge bosons and 3 massless fermions of definite helicity (and their antiparticles) or 4x2 + 3x2 = 14 degrees of freedom in total. The next step is to spontaneously break (most of) the symmetry via the Higgs mechanism as in the previous lecture. To this end we introduce a complex (to couple to the U(1)) doublet (for the SU(2)) of scalar fields

with 4 degrees of freedom. We assign the following quantum numbers

yielding 2 charged and 2 neutral scalar bosons. The components of the operator j can be thought of as annihilating particles of electric charge 1 and 0 (or creating the antiparticles) while j* does the same for the antiparticles (charge –1 and 0), as indicated by the labels above. The Lagrangian for the scalars has the expected form

with

and

We also include a new interaction to provide a mass for the charged lepton, the Yukawa term,

where the multiplication of the spinor and SU(2) indices is implicit (but relevant). Note in particular that the quantities in the ( )’s are SU(2) singlets but Lorenz spinors (or barred spinors). Thus the overall expression is invariant under both SU(2) and Lorenz transformations (i.e., there are no left over indices). The Yukawa coupling constant, G_e, is arbitrary, i.e., unconstrained by the symmetries. So at this point, prior to breaking the symmetries, we have a total of 14 + 4 = 18 degrees of freedom.

As we have practiced in the previous lecture, we now assume that and the scalar field acquires a vacuum expectation value. To match the bias built into our definitions of the conserved electric charge, we assume that it is the neutral component that gets the vev, i.e., j⁰. So, as in the previous lecture, we have

Recall that the remaining unbroken symmetry should correspond to the generator that annihilates the vacuum,

To determine the form of G we note that

but (as suggested earlier) we have

The generator that annihilates the vacuum, and thus defines the remaining conserved charge, is exactly the electric charge operator (modulo an overall factor). The corresponding gauge boson is expected to remain massless. To determine the structure of the small oscillations in the broken symmetry state we define a new representation for the scalar field just as we did in the last lecture,

The new degrees of freedom are described by the fields h and . Strictly speaking, the generator in the third component in the exponent should be replaced by , i.e., the generator orthogonal to the Q. However, since Q annihilates the vacuum, the expression above is operationally identical to the formally correct one. To see the particle content we transform to the U-gauge,

Note that, since the components of L have definite electric charge, which is still conserved, the gauge transformation of L only produces phases.

In the new basis the Yukawa terms look like (i.e., involve only the lower component of L)

In the last line we recognize the first term as an ordinary Dirac mass for the electron, , while the second term is the coupling of the electron to the Higgs field, h. Note that, since the mass and the coupling of the electron are both proportional to the Yukawa coupling, G_e, the coupling of the Higgs boson to a fermion will, in general, be proportional to the mass of the fermion. Also note that the neutrino does NOT acquire a mass. As the neutrino has no right-handed component, it simply cannot get a mass (in this version of the Standard Model).

Next consider the form of the Lagrangian for the scalar field in the U-gauge. Explicitly using the forms of the Pauli matrices, we find (for the small oscillations, i.e., keeping only quadratic terms)

The first line tells us, as expected, that the remaining neutral (singlet) scalar particle (note that there are no remaining couplings to the vector bosons), the Higgs boson, has a mass given by . To see the particle content of this Lagrangian for the vector particles it is best to shift to basis states of definite electric charge. These states are the analogues of the ladder operators that we discussed last quarter. Written in terms of the field operators that create particles (and annihilate the antiparticles) we have

where the last form annihilates the corresponding particles (and creates the antiparticles). The corresponding generators are represented by

These generators obviously raise and lower, respectively, the isospin of the leptons.

Returning to the Lagrangian for the scalar sector, we can write the first term in the second line as

The last line is in symmetrized form to provide a familiar expression with one term for each charge state, from which we can identify the mass of the charged massive vector bosons, the usual W’s, as .

The last term in the above expression for the scalar Lagrangian corresponds to the resulting massive vector boson of charge zero. As expected it is a mixture of the third component of the original SU(2)_L gauge boson and the U(1)_Y gauge boson.

This mixing is conventionally expressed in terms of a weak (or Weinberg) mixing angle, q_W, defined in terms of the initial gauge couplings (recall that this is not a truly unified theory but rather this 1 parameter characterizes the ratio between the couplings in the 2 individual gauge theories)

Thus the properly normalized massive neutral vector boson can be expressed as

The final expression for the scalar Lagrangian at quadratic order is then

Thus we can identify the mass of the Z⁰ as

Note that there is a fourth vector field, which is neutral and orthogonal to the Z. This is the usual vector potential of E&M (the remaining U(1) symmetry) and the excitations of this field are photons. In the current notation it has the form

This field does NOT acquire a mass as a result of the spontaneous symmetry breaking.

Finally consider the lepton sector in the Lagrangian (the gauge sector is formally unchanged). The charged current term can be written

We can visualize the first term as describing an incoming electron absorbing an incoming W⁺ to become a neutrino while the second term describes an incoming neutrino absorbing an incoming W^- and becoming an electron.

In this form we can identify the coupling we know from our phenomenological model of the weak interactions and quantify the size of the Higgs vev,

Thus we can solve for the vev value,

It is this number that identifies the “weak scale” as about 1 TeV. We also see that

The last of the new terms in the lepton Lagrangian compose the neutral current sector and can be written

The first terms allows us to identify the electric charge (Q_e < 0) of the electron,

Thus we can rewrite the neutral current Lagrangian in various formats (recall that we have defined Q_e = -e)

where we have defined (as you will often see in the literature)

From these expressions we can determine many features of the neutral currents in this spontaneously broken theory.

· The coupling to the vector gauge field of the remaining U(1) symmetry is pure vector (no axial component) as is required by the phenomenology.

· The coupling of the Z to neutrinos is purely left-handed.

· The coupling of the Z to the electron has both left-handed and right-handed contributions but of unequal strength, i.e., the form of parity violation in the neutral current is complicated and depends on x_W = sin²q_W. The last expression shows the coupling in vector and axial vector format. For sin²q_W = ¼ (nearly true experimentally) the coupling is pure axial vector.

At tree level (essentially at the classical level) we can now write the following relations

Likewise we can write

This last expression is sometimes used as a definition of sin²q_W to all orders in perturbation theory. Typically at higher orders all of the relations above receive corrections. One will often see the definition for the ratio of the neutral current to the charged current coupling as

where in Weinberg-Salam (i.e., with a single complex doublet) this ratio is unity at lowest order. With precision measurements of the various quantities one can look for “physics beyond the Standard Model” by looking for unexpected deviations from unity. We will return to this subject after we have introduced quarks into the weak interactions.

Let us close this lecture by recounting the degrees of freedom.

particle	spin	mass	# dof
Higgs, h (neutral)	0		1
Electron, e	1/2		4
Neutrino, n	1/2		2
Charged Vector Boson, W^±	1		6
Neutral Vector Boson, Z	1		3
Photon, g	1		2
		Total	18

Thus the total number of degrees of freedom is unchanged from the result prior to the breaking of the symmetry. The experimental results tell us that (modulo radiative corrections):

The individual parameters m and l, and hence the mass of the Higgs boson, are not directly determined. The couplings are related by (e > 0)