Physics 558 – Lecture 11

More comments about the Standard Model of the Electroweak Interactions:

Before proceeding to the issue of including more matter fields (other leptons and the quarks) there are a couple of comments that are relevant.

Cosmology: In Lecture 8 we allowed the Higgs to acquire a vacuum expectation value and rewrote the various parts of the Lagrangian in terms of fluctuations about that new minimum of the potential. Since we were concerned with defining the dynamics, we happily ignored any constant terms in the redefined Lagrangian density. In the context of particle physics alone, that attitude is presumably justified. However, since we also care about the implications for astrophysics, we should not be so careless. The uniformly distributed energy density implied by the vev of the Higgs field (and the corresponding constant term in the Lagrangian) can be argued to contribute to the background energy in the universe (and the Cosmological constant, although defining the 0 energy density level is always an issue in this game). Recall that last quarter (Lecture 3) we noted that the energy density in ordinary (visible) matter (protons in stars, etc.) seems to be of order 10^-7 protons/cc or ~ 10^-7 GeV/cc, which is just a few percent of the density required to close the universe (~ 10^-5 GeV/cc). On the other hand the vacuum energy associated with the Higgs vev is, at least naively, of order

Thus, for “natural” values of the coupling, , this single contribution is some 10⁵⁵ times too big to allow our universe to evolve as observed (if we all lived inside a basketball, it might work). Of course, if we allowed a negative cosmological constant (as people discuss this days), we could cancel this large positive contribution with a large negative one and arrive at a net result of the correct size. This is fine tuning, 1 part in 10⁵⁵, of a high order and considered to be utterly unacceptable (this concern is often called the cosmological problem). This is, in fact, a general problem for spontaneously broken symmetries and strongly urges us to consider something well beyond the Standard Model (where, hopefully, some larger symmetry structure constrains the properties of the vacuum) if we want to be able to understand particle physics and the astrophysics with the same set of tools.

Higgs Boson production and detection: In the HW we considered the coupling of the Higgs boson to a fermion (whose mass is obtained from its coupling to the vev of the Higgs). We found that the width describing the decay has the form (for )

Thus for certain fermions of interest, assuming all get their masses via a Yukawa coupling to the Higgs and assuming that (the lower limit suggested by the year-old CERN data), we find

Thus the fermionic decays of a Higgs, assuming it is too light to decay to a top quark pair, will be dominated by the bottom quark channel, and the branching ratio to electrons will be tiny (~ m_e²/m_b² ~ 10^-8). This tiny coupling to the electron also effects the production process. Consider the direct production process at LEP, . The general resonance form for the cross section is

To estimate whether we could see this signal, for example at LEP, in the channel (or better focus on ), we can use the results from Lecture 4,

Thus we conclude that the Higgs boson contribution will be only 1 part in 10³ to 10⁴, () and difficult to detect. In fact, the situation is worse due to the contribution of the Z in the s-channel, especially if we are near the Z pole. The coupling of the Higgs to the electron is just too small. Also note that this situation does not improve, i.e., the branching ratio does not increase, if m_h increases.

On the other hand, we have not yet discussed the role of the coupling between the Higgs and the vector bosons that appeared in the Feynman rules of Lecture 9,

and

To evaluate the decay , assuming m_h > 2 M_V, we need to know the result of multiplying the polarizations of the two vectors together and summing over the various polarization states. We start with the knowledge that the sum over the polarizations of a single vector particle is just the numerator in its propagator so that, with momentum q,

Thus we have

The variable x_V expresses the distance we are from threshold, i.e., x_V = 1 is the threshold (), while x_V < 1 means that the vector particles can be produced with nonzero kinetic energy in the decay. Thus the spin summed amplitude squared is

Since we started with a scalar particle, the amplitude can exhibit no angular dependence (i.e., no direction is defined by the initial state). Thus we can integrate the usual formula for the decay width into a two body channel to find (recall this result from Lecture 6 last quarter)

However, it is important to remember that, when we integrated, we assumed 2 distinct particles. If, instead, there are two identical particles in the final state, we have over-counted the phase space by a factor of 2. For example, if one of the 2 identical Z’s is going in the plus z direction, there is also one going in the minus z direction and the actual range of the polar angle to describe the distinct final states is 0 £ q £ p/2, rather than the usual 0 £ q £ p. Thus we need to include a factor of ½ in the Z result compared to the W result. We also note that

Finally, the two vector boson widths can be expressed as

In the same notation, the Higgs width to a fermion pair (including the fermion mass dependence) looks like (note the color factor for the sum over quarks)

Except for the final factors, these widths are very similar. Thus, above the vector boson production threshold, the widths for decay into these vector boson channels should be much larger than the lepton and quark channels (until we reach the top quark threshold and even that channel is smaller). The following table illustrates some possible relative values for these widths.

Higgs mass =	115 GeV	300 GeV	500 GeV
	---	5.8 GeV	35 GeV
	---	2.6 GeV	17 GeV
	0.24 MeV	0.62 MeV	1.0 MeV
	4.5 MeV	12 MeV	20 MeV
	---	---	11 GeV

So, if allowed by phase space, the vector boson modes are much more important. However, this is clearly not the mode that suggested the 115 GeV mass for the Higgs, which is below the W production threshold. Instead the process studied at LEP involves 2 virtual Z’s as indicated in the figure. This process takes advantage of the large coupling of the Higgs to the vector bosons while exhibiting a typical gauge strength coupling at the electron vertex. Both the second Z and the Higgs decay into fermion-antifermion pairs, with bottom quarks the much-preferred mode for the Higgs. Thus the events of interest involve a lepton pair and two (bottom quark) jets, or 4-jets, two of which should exhibit the vertex corresponding to the bottom decay. In describing this scenario we have jumped ahead to the next topic – adding more matter fields to the Electroweak interactions.

More matter: Adding more leptons to our theory is, in fact, quite simple. We simply reproduce the original electron structure for the heavier leptons (the 2^nd and 3^rd generations).

Each generation consists of a left-handed doublet and a right-handed singlet (plus the antiparticles). The couplings of each generation are, by construction, identical to those we have already discussed for the electron. This is clearly not only the easiest way to introduce the 3 generations but guarantees agreement with the experimentally observed universal coupling of the 3 generations. Recall, for example, that the decay channels of both W and Z into each of the 3 generations,

are equal to within the experimental uncertainties. One, of course, does expect small deviations in these quantities due to the differences in the masses of the charged leptons. These mass differences are parameterized (but not explained) by introducing 3 Yukawa couplings – G_e, G_m, G_t, which are, a priori, unspecified.

Note that, in this limit where the neutrinos are massless (a feature we will correct latter), this is a complete specification. The charged leptons are labeled in terms of the mass eigenstates (independent of the actual source of the mass) and the neutrinos are labeled in terms of their partners in the charged current interactions. This is always possible as long as the neutrinos are degenerate in mass, even if they are not massless, i.e., if the only interaction that distinguishes the different neutrinos is their charged current interaction. If, instead, this degeneracy is broken (e.g., at least 1 neutrino is massive), one can define two different basis sets, one based on the masses and one based on the charged currents. The fact that these different basis sets need not coincide leads to the phenomena of neutrino oscillations. We will return to this point when we discuss neutrino masses in more detail.

Next we want to bring quarks into the discussion. For now we simply ignore the fact that they also participate in the strong interactions. Recall from last quarter that the small size of the two lowest mass quarks, u and d, compared to the “natural” QCD scale L_QCD provides a natural explanation of the approximately conserved (and global) strong isospin quantum number in nuclear physics. Here the issue is (local but spontaneously broken) weak isospin and, as suggested by the lepton sector, the left-handed quarks are taken to be in doublets under this gauge symmetry, while the righted-handed components are singlets. As we did for the leptons, we repeat the same structure for each generation

where we include left-handed and right-handed components for all flavors because we know that we want to include Dirac masses from the outset. For example, we imagine defining 6 different Yukawa couplings – G_u, G_d, G_c, G_s, G_t, G_b. Unlike the lepton case, we must face the ambiguity about the basis choice immediately. There is, a priori, no reason that the states that correspond to the mass eigenstates, as defined, for example, by the Yukawa couplings, are identical to the states that appear in the weak interactions. We can always pick one component of the SU(2) doublet to be a mass eigenstate, but the other component will, in general, be a linear combination of the 3 possible states. By convention, we choose the upper states in the weak isospin doublets (I₃ = +1/2 – u,c,t) to be the mass eigenstates. The observed lower components then will be linear combinations of the mass (flavor) eigenstates d,s,b. The electroweak theory has nothing to say about how these combinations arise and, for now, we simply parameterize the mixing. We write the “true” weak interaction isomultiplets as

and the mixing as

where the indices correspond to the underlying charged current transition (between mass eigenstates, i.e., the unprimed states). The mixing matrix is called the CKM (Cabibbo-Kobayashi-Maskawa) matrix. It leads to the enormous richness of the weak interactions. The fact that the matrix has non-zero off-diagonal terms means that heavy flavors can decay into less massive flavors and explains why we see no stable matter composed of the heavier quarks. While the Standard Model does not predict the elements of V, it does allow us to determine them from experiment. An important question is – how many parameters are there in V? We know from our discussions of group theory that an N x N unitary matrix (and V is necessarily unitary), where we keep the trace, has N² real parameters (recall if we remove the trace we find N² – 1, the number of generators for SU(N)). Now we ask, how many of these parameters are physically relevant? Since we know that quantum mechanical matrix elements are insensitive to the overall phase of a given wavefunction, we can absorb (2N – 1) phases into the definition of the 2N quark states without changing the quantum physics. The –1 arises from the fact that V is invariant to an overall change in phase of all 2N states. Thus we have N² – (2N – 1) = (N - 1)² parameters. To interpret this number recall that an orthogonal (real) N x N matrix has N(N –1)/2 real parameters (see our discussion of SO(N)). Thus for N > 2 it is not possible to make V a real matrix by redefining the phases of the quark states. In general, V must contain

true (i.e., physically relevant) phases. For the minimal case, N = 2, there is just one real parameter (no phases!), which we remember is the Cabibbo angle q_C that we discussed last quarter and which describes the mixing of the down and strange quarks. Of more interest to us in our version of the universe is the case N = 3 (the 3 generations). Now we have 1 phase and 3 real mixing angles. As parameterized by the PDG in terms of 3 angles, q_ij , characterizing the mixing of generations i with j (c_ij = cosq_ij , s_ij = sinq_ij) and the phase d_ij, the matrix looks like

In the limit q₁₃, q₂₃ ® 0, we recognize q₁₂ as the Cabibbo angle. Our experimental knowledge of the magnitudes of the various components in this matrix is characterized (in terms of ranges quoted by the PDG) as

We see that the matrix is not far from being diagonal but, as noted above, there is considerable experimental impact due to the off-diagonal terms in the form of heavy flavor decays. The existence of the largely undetermined phase is also important. In a world with only 2 generations, and thus no phase, the electroweak interactions could not result in CP (or T) violating interactions. In our case, with 3 generations, such effects are expected (although the magnitude is not predicted) and are being searched for enthusiastically. We will pursue this subject when we study the K-meson system in Lecture 13.

One nearly final issue is that of the neutral weak currents. Recall that the Z boson exhibited diagonal couplings to both the electron and the neutrino. Thus we expect similar couplings, individually, to the upper and lower components of the doublets above. In a world with only 3 quarks, i.e., the world of Cabibbo before the discovery of the c quark, this was a serious problem. The diagonal couplings to the u and the d¢ quarks yield both strangeness preserving and strangeness changing neutral current interactions,

The second line implies that strange mesons should be able to decay to ordinary mesons via the neutral current, but experimentally flavor changing neutral currents (FCNC) are vanishingly small. For example, we can compare the following fractional rates

These severe constraints on FCNC offer severe constraints on any models with larger symmetries and more interactions. In the Standard Model the problem vanishes when we include quarks as complete doublets of the weak isospin (recall we did not do this for strong isospin). The diagonal couplings for u, d¢, c and s¢ quarks yield the following structure,

The flavor structure is clearly diagonal with no FCNC! This cancellation of offending terms was, in fact, invoked to predict the existence of the c quark (to complete the second doublet) by Glashow, Iliopoulos and Maiani before its first experimental observation and is called the GIM mechanism. It clearly generalizes to the case of 3 generations, all in doublets. In processes involving virtual loops the cancellation may not be exact because the quarks are not exactly degenerate in mass, but FCNC are tiny.

Let us now summarize these results with the Feynman rules for the quark weak interaction vertices. Note that, since the electric charges of the quarks are not identical to those of the leptons, the weak hypercharges of the quarks are also different (). These choices yield the expected results of Q_u = (2/3)e and Q_d = (-1/3)e. We can then specify the couplings of the quarks in terms of their electric charge and the weak isospin. The coupling to the photon has the familiar form

The Z coupling can be written (compare to our result for the lepton with Q_e = -e and Q_n = 0),

This form generalizes directly to the other generations, which have the same identifications for isospin and electric charge (i.e., identify the c and t with u and the s and b with d).

Finally we can write the charged current, W coupling, as

with a corresponding expression for the conjugate process, , . Again there are similar expressions for the couplings of the (mixed) members of the doublet in the other generations.

The question naturally arises, is there any connection between the structure outlined above for the quarks and for the leptons, or do they just represent two independent realizations of electroweak interactions? This brings us to the formal but important topic of anomalies. We have motivated the Weinberg-Salam model, at least partially, by claiming (largely without direct proof) that gauge theories are the preferred way to go because of the property of renormalizability. The underlying gauge invariance guarantees that various cancellations occur to all orders in the perturbative analysis and thus that the theory is well behaved. There is one potential problem with this approach. There exists a class of fermion loop contributions that violate the classical symmetries and conservation laws of the underlying gauge theory. These contributions are called anomalies for their anomalous quantum behavior. The simplest, and for us most important example, is the “triangle” diagram involving 2 vector currents and 1 axial vector current. Since such contributions can arise in the electroweak (but not E&M alone), we should briefly discuss them here. We will not prove that such diagrams violate the conservation laws (this is discussed in most of the texts in the book list) but we will note the sufficient conditions for the diagrams to cancel among themselves, leaving the theory renormalizable. Consider a gauge theory, like the electroweak interactions, with interactions expressed in terms of currents formed from chiral fermions,

where the are the appropriate representation of the generators of the symmetry group and we allow differences between the couplings to the left- and right-handed fermions, e.g., they are in different representations of the underlying symmetry. In terms of these quantities, the coefficient of the anomaly contains the factor

.

We recognize that the trace arises from the fermion loop, the anticommutator arises from summing the two orderings of the vector currents (the two possible directions of the fermion within the loop) and the difference of the R and L terms arises from the one axial current with involves right-handed minus left-handed couplings. We see immediately (as expected) that a purely vector theory, like E&M where the L and R generators are equal, will not have anomalies. In the more general case the question is whether, when summed over all of the fermions in a given representation or generation, the various contributions sum to zero,

In particular, we want to consider our SU(2)_L x U(1)_Y theory. If a, b and c are SU(2) indices (corresponding to the fields W¹, W² or W³), only the left-handed couplings to doublets are present and we have

(recall the generators are traceless) and the anomaly cancellation condition is satisfied for each doublet separately. If a is from SU(2) but b and c are U(1), we find

Things get interesting for 1 generator from U(1)_Y and 2 from the SU(2)_L. Then we require that (think about a = b = 3)

and, using , we require (since )

The final case is for all U(1) generators and we require

Again using the definitions and , we find that this expression is proportional to the previous one. Thus the sole constraint for the vanishing of the coefficient of the anomaly is that the sum of the charges of the fermion fields vanish when summed over an allowed generation. This clearly does not work for the leptons alone, 0 – 1 ¹ 0. If we add the quarks without color, we are still in trouble, 0 – 1 + 2/3 – 1/3 ¹ 0. Finally with color,

and all is right with the world. Thus, to make the theory of the electroweak interactions viable, we not only need the quarks to go with the leptons, but we also need them to come in 3 colors. This statement is true for each generation. In some sense the electroweak interactions predict the rest of the Standard Model, QCD! This cancellation requirement also offers another constraint (along with no large FCNC) on any theories with larger symmetries and more matter fields.

Before discussing the strong interactions in more detail, we will consider two of the more interesting sectors of the EW interactions, the neutrino sector and the kaon sector (and CP violations).