Physics 557 – Lecture 12

Even More particles – heavy flavors:

The excitement of the 1970’s continued with the observation in 1974 (continuing through 1976) of a new quark flavor (recall that the same period also saw the observation of the third lepton generation, the t). The recognition of a new flavor was triggered by the observation of a new 1^{- -} state in the same channel as the photon at both an e⁺e^- machine SPEAR (at SLAC), where the state was called the Y, and a proton accelerator (at Brookhaven), where it was called the J. In the former case the process is direction formation

In the latter case the state was observed in production in a “Drell-Yan-like” channel

The actual production process involves the direct Drell-Yan electromagnetic channel seen at SLAC and various hadronic modes that we will discuss in more detail below. These discoveries led to Nobel Prizes for Burt Richter at SLAC and Sam Ting at Brookhaven. The observed states was quickly recognized to have quantum numbers I^G = 0^-, J^PC = 1^{- -}, i.e., a f-like state but with the previously predicted (see below) charm quark replacing the strange quark,

This particle has mass and width

The distinctive feature is the very narrow width. Recall that the decay width of the f is greater than 4 MeV, corresponding dominantly to decays to , i.e., to strange quark conserving channels. It was quickly learned that the 0^- charmed mesons, which correspond to the isospin doublet K’s and which are labeled the D’s, have masses and lifetimes

Thus the charmed quark conserving decay channel into is not kinematically available to the Y/J, i.e., the mass of the Y/J is less than 2 m_D. The only allowed strong decays occur through the OZI violating channels where the quarks annihilate into 3 (or more) gluons. These channels are quite suppressed (recall that the strong coupling a_s is a “running” coupling that becomes smaller at larger mass scales) and electromagnetic decay modes are competitive. This point is illustrated by the following approximate branching ratios,

Note the approximate factor of three between the lepton pair electromagnetic channels and the electromagnetic hadronic channel. The naïve theoretical expectation for this factor includes the same color factor of three that we observed in the weak decays of the t times the electric charges squared of the quarks involved () resulting in a final factor of 2. We will discuss this factor again in the context of e⁺e^- physics.

The purely hadronic decays of the Y/J involve a vast number of channels, most of which contain an odd number of pions (recall that the Y/J has G = -1). At the 1 % level there are also decays where a photon replaces one of the (3) gluons and one observes final states with a photon and an even number of pions.

The other expected ground states for the mesons containing a charm quark are also observed. These include the 1^- states,

For all charge states the primary decay modes are pD as we would expect, although there is little phase space for this decay and hence small widths. We also expect two isosinglet charmed and strange mesons, corresponding to the quark states and , and they are observed,

The corresponding 1^- states also seem to have been observed at 2112 MeV but the data is less complete.

Although these data on masses imply that the charm quark has a mass of order 1.5 GeV, we can still consider the full range of states implied by the assumption of a flavor SU(4) symmetry. The corresponding SU(4) representations can be found at the PDG web site (see the quark model section) and in the appendix to these lectures. Note that the presentations are now 3-dimensional (they were 2-D for SU(3) and 1-D for SU(2)). Candidates for all of the predicted states have been observed but the quantum numbers (J^P) have not been confirmed in all cases. Consult the PDG tables for more details.

As noted above, in another triumph for symmetries and theoretical physics, the existence of the charm quark had, in fact, been predicted in 1970. The context of this prediction is now called the GIM mechanism after the authors, S. Glashow, J. Iliopoulos and L. Maiani (the latter is currently the Director General of CERN, who made the decision two years ago to shut down LEP II even though there was some evidence of the Higgs boson). In 1970 it was already thought that the weak interactions involved not only the common charged current interactions, e.g.,

involving the then known weak interaction doublets

but also neutral current interactions which are diagonal in the individual states of each doublet. This expectation was confirmed in 1973 using neutrino beams at CERN to observe events of the form

although it was not until 1983 that the existence of both the W^± and Z⁰ particles were directly confirmed. Still, back in 1970, it was a theoretical conundrum to believe in a neutral weak current but to observe no flavor changing neutral currents (FCNC’s). In particular, one would expect, based on the weak isodoublet structure above, to observe neutral current coupling of the form (just the squares of the elements of the doublet)

However, the flavor changing decays of strange particles were observed to always occur via the charged current, e.g.,

This unacceptable situation was rescued by GIM, who (in keeping with our usual technique) postulated a new particle. They noticed that good things happened if a new Q = 2/3 quark, the c quark, formed a second weak doublet with the Cabibbo-rotated strange quark. Thus we would have

With this second weak isodoublet, the neutral couplings look like

Thus, as long as the quarks occur in complete multiplets, i.e., complete generations, the neutral current is diagonal and there are no FCNC’s, in agreement with data.

The insight of symmetries had again led the way!

The mixing of the s and d quarks now assumes the canonical form

where the states on the left are in the basis of the weak interactions (the lower members of the weak isodoublets) and those on the right are in the mass eigenstate basis (the lower members of the strong isodoublets). This 2 x 2 mixing structure will soon assume a 3 x 3 form when we introduce the third generation of quarks. [Note that the choice to characterize the mixing in terms of the down-type quarks is purely by convention (and history). It could as well be performed in terms of the up-type quarks. Note also, as mentioned previously, that to be able to define mixing we need two distinct basis sets. In this case the two are provided by the mass (flavor) eigenstates and the charged weak current interaction states.]

The second weak isodoublet explains the observed structure of the weak decays of the D mesons. The Cabibbo favored charm changing decays are and . Thus it is no surprise that approximately 90 % of all final states in D decay have a K present. The study of these decays has provided a warehouse of information about the weak interactions that we will address in a later lecture.

The end of the golden decade of the 1970’s saw the detection of the 5^th quark, the bottom (sometimes beauty) or b quark in 1977. Again it was the vector state that was seen in the Drell-Yan process at Fermilab, and subsequently in e⁺e^- physics, that pointed to the new quark. The vector state is the(the upsilon), I^G (J^PC) = 0^- (1^{- -}),

Note the interesting feature that, although the mass is 3 times that of the Y/J, the width of this state is actually smaller than the state. The pseudoscalar states with the bottom quantum number include the doublets

As with the Y/J, these states are too massive to participate in a bottom conserving strong interaction decay of the ¡. Hence the strong decays of the ¡ must violate the OZI rule and involve the emission of at least 3 gluons. Thus we expect the rate of strong decays to scale as a_s³, the running coupling of the strong interaction, evaluated here at the scale set by m_¡, times any further shrinkage of the wave function itself (see the HW). Thus the appropriate coupling for these strong decays of the ¡ is smaller than for the Y/J. The ¡ is observed to have an electro-weak branching ratio of about 2.5 % into each of the channels e⁺e^-, m⁺m^- and t⁺t^-. There is a branching ratio of approximately 0.3 % into radiative decays, where a photon replaces one of the gluons mentioned above, and approximately 0.1 % into Y/J + X.

The weak decays for the B’s are determined by the expanded mixing described by the CKM (Cabibbo, Kobayashi and Maskawa) mixing matrix

where, as above, the basis on the left is weak isospin and on the right is strong isospin, the mass eigenstates. The indices label the relevant flavor changing decays. For the case of b, i.e., B decay, the dominant term is V_bc and there are sizeable branching ratios to Dp states.

The cast of bottom pseudoscalar states is completed by the isoscalar charm and strange states

There is also evidence for the corresponding vector states, the B*’s, with a mass of 5325.0 ± 0.6 MeV but confirmation of the I, J and P quantum numbers of these states is still incomplete. The dominant decay is to g B, as there is no phase space for a decay channel with a pion. Recall that the B* - B mass splitting (like the D* - D, K* - K and p - r) is thought to be a (strong) hyperfine effect, which depends on the quark mass like 1/m_Q. Thus, with each successively larger mass quark, we should observe smaller splitting.

From the standpoint of the strong interactions, one of the most interesting features of the and systems is how closely the spectra of states match those observed in the QED system of positronium. This point is illustrated in the following figures, which illustrate the levels of the three systems in typical nuclear physics (as opposed to atomic physics) spectroscopic notation – n ^Sl_J, with n the principal quantum number. First, the spectrum of the QED systems looks as follows.

Note the broken energy scale representing the fact that principal quantum number splitting is much larger than the other splittings. The scale for the other splittings is of order 10^-5 eV.

Next we turn to the charm quark system.

Note in this case the broad states (strong interaction decays) for the states above the threshold, which is not present in the QED case. Note also the presence of the 0^- state. Like the corresponding positronium state, this state can decay into a 2-vector boson state (gluons in this case), unlike the 1^- state that can decay only into a minimum of 3 gluons. Thus it is no surprise that the h(2980) (G_h ~ 13 MeV) is much broader than the Y(3097) (G_Y ~ 87 keV).

Finally the spectrum for the bottom quark case looks as follows.

So the question arises, why do these spectra look so similar? First we understand that all 3 systems describe nonrelativistic fermion – antifermion pairs. Further, in the QED case we know that the basic binding comes from the 1/r potential of electromagnetism, which yields energies for the states that depend on the principal quantum number (like 1/n²). The states are further separated by the spin-orbit interaction (the fine splitting, separating states of different l) and the spin-spin interaction (the hyperfine splitting, separating states of different total S). For the QCD case with heavy quarks we must consider compact wave functions with characteristic radius of order 1/m_Q << 1 fm. In this short distance limit the strong coupling is rather small (although still much larger than a_EM) and we expect the QCD potential to also have a coulomb, 1/r form. A useful phenomenological form for the full QCD potential is

The first term is the short distance limit of the color interaction with the appropriate prefactor for in a color singlet (as we have noted earlier). The second, linear term represents the confinement facet of QCD in a fashion that is suggested by the linear Regge trajectories of the previous lecture. The naive picture of the self-interacting flux (the gluons) that connects the quark and antiquark is provided in the following figure.

The idea is that, at separations short compared to 1 fm, we see coulomb like flux structure, i.e., a flux density that falls off like 1/r². At separations large compared to 1 fm, the self-interactions of the flux (the glue) confines it to a flux tube with approximately fixed (transverse) size of order 1 fm, i.e., a fixed flux density. The former situation yields a potential energy that behaves as 1/r at small r while the “flux tube” leads a potential that increases linearly with r at large r.

The similar 1/r potential for all three cases leads to the similar spectra of states. The larger magnitude of a_s (> a_EM) and the effect of the linear term lead to somewhat different relative roles for the principal quantum number, fine splitting and hyperfine splitting. The relative role of the first is reduced while the role of the last two varies with the heavy quark mass. The linear term in the potential also leads to an increasing number of states below the threshold for unsuppressed strong decays. A reasonable description of the meson spectra is supplied with the above potential using a_s ~ 0.2 and k ~ 1 GeV/fm, as suggested by the Regge behavior discussion in the previous lecture.

By 1980 we were again back in the “soup” with 3 observed down-like quarks and only 2 up-like. As noted in our discussion of FCNC’s (and also in the context of the cancellations of certain “triangle anomalies” that we will discuss soon) it seemed necessary that there be a sixth quark, the top quark (also once called truth to go with the beauty quark). For reasons yet to be explained, but almost surely very important, the top quark turns out to be very heavy,

and its existence was not confirmed until 1995 at the Tevatron Collider at Fermilab. The top quark was detected not via the vector bound state, as with charm and bottom (see below and note that we have no e⁺e^- machines of 350 GeV anyway), but directly from its weak decays. In the proton-antiproton collisions at Fermilab top-anti-top quark pairs are produced either via the annihilation of a valence quark-antiquark pair into a virtual gluon, which couples to the top pair, or from the coupling of a gluon from each beam hadron directly to the top pair (called the Einhorn-Ellis mechanism in some quarters). [We will study such processes in more detail when we get to the strong interactions.] The top quarks then decay weakly into real W’s and (primarily) b quarks. The W’s then decay into either lepton pairs or quark pairs, which produce jets in the final states. The b quarks form B mesons, whose decays close to the interaction region (i.e., after short times) are detected in “vertex detectors”, typically strips of silicon. Thus we see

Reconstructing such events is clearly a challenge.

One of the important features of the systematics of top quark studies arises directly from its huge mass. Since it is massive enough to decay into a real W, its decay width is proportional to just a single factor of the Fermi constant, and not G_F² as we had for all lighter particles. Keeping effects of the W mass in the phase space integration but ignoring both the mass of the bottom quark and higher order terms in the weak and strong couplings we have (we will do the details shortly)

While this is unambiguously a weak interaction decay, the width is that of a “super strong” decay (recall the r has a width of only 150 MeV). The corresponding time scale is of order 0.5 x 10^-24 s, an order of magnitude smaller than the time for a photon, or a gluon, to travel a distance of 1 fm. Thus the top and anti-top quarks produced at Fermilab decay long before they have an opportunity to interact strongly and form bound states (i.e., they decay before they “hadronize”). The top quark system in the laboratory is not characterized by the spectra of bound states that we saw in charmonium and bottomonium (even though, as we saw above, the linear, confining potential would predict that ~ 10 such states would exist if we turned off the weak interactions).

In summary, we now have all the fermions that we “need” (modulo the RH neutrinos).

Particle	Spin	Q EM	SU(2)L Weak	SU(3) Strong	Mass
Electron	½	-1	LH-2, RH-1	1	0.511 MeV
ne	½ LH	0	2	1	< 3 eV
Muon	½	-1	LH-2, RH-1	1	105.66 MeV
nm	½ LH	0	2	1	< 0.19 MeV
Tau	½	-1	LH-2, RH-1	1	1777.0 ± 0.3 MeV
nt	½ LH	0	2	1	< 18.2 MeV
u quark	½	2/3	LH-2, RH-1	3	1 to 5 MeV
d quark	½	-1/3	LH-2, RH-1	3	3 to 9 MeV
c quark	½	2/3	LH-2, RH-1	3	1.15 to 1.35 GeV
s quark	½	-1/3	LH-2, RH-1	3	75 to 170 MeV
t quark	½	2/3	LH-2, RH-1	3	174.3 ± 5.1 GeV
b quark	½	-1/3	LH-2, RH-1	3	4.0 to 4.4 GeV

The masses indicated in the table are the “current” quark masses. These particles constitute 3 complete “generations”, differing only in their masses. The question remains, why 3 generations and not 1 or 2? (Do we really need CP violation?)

Next we will discuss the symmetries of the interactions and the required vector (gauge) particles.