Heavy Quarks and e + e - Annihilation Published By: Thomas Appelquist and H. David Politzer

1. Heavy Quarks ande+e- Annihilation Published By: Thomas Appelquist and H. David Politzer (6 January 1975) Presented By: Stephen Bello December 2013

2. Let’s begin shall we… • While there are three light quarks that compose ordinary hadrons (Up, Down, and Strange), there may also be a heavy quark. We shall dub this heavy quark - charmed P’. • A heavy quark had been suggested in earlier papers and was consistent with known observed scaling and successful sum rules for inelastic lepton-hadron scattering. • At energy scalesgreater thanthe P’P’antithreshold the hadronic cross section scales as if it were part of the free-quark model. This should occur due to the smallness of the asymptotic effective coupling. • Energy scaling also holds in the region above both the λλanti and below the P’P’anti threshold. The magnitude of this scaling is determined by the charges held by the three light quarks. • The problem involves large enhancements within a finite region both above and below the P’P’anti threshold. I will examine the behavior in this region and the way we approach the asymptotic region above it.

3. Consider the Lagrangian: • Fμυ is the non-Abelian gauge-covariant curl, Ψ is several quark color multiplets, i runs over the various colors, Dμ is what is known as the gauge-covarient derivative, and m is the quark mass matrix. • If we allow the color gauge symmetry to be exact strong forces at large distances comes into play. From this, the quark color multipletsobtain specific masses: • mP, mR, and mλ have small masses (< 1 GeV) • mP’ has a bigger mass (> 1 GeV) • When we renormalize, we introduce ‘g’ and say it’s a function of some Euclidean momentum configurationthat hasscale M. • We needasymptotic freedom to help explain Bjorken scaling. So for M = 2 GeV, αs = g2/4π must be quite small.

4. Renormalization implies that in the one-photon approximation, σ(e+e- -> hadrons) is of the form σ(s, g, m, M) = σ(s, gbar(s), m(s), s1/2). • s is the square of the center-of-mass energy. • gbar(s) ≈ g[1 + g2 bln(s/M2)]-1/2 • m(s)≈ m[1 + g2 bln[s/M2)]d……(but only for small gbar) • b and dare positive constants of the system • The total cross section is a function of only a singleenergy and is completely governed by gbar(s). • Say we are interested in a range of values for s. If we are interested in such a range, then we need to also say that ln(s/M2) = O(1) as well as g and gbarbeing roughly equivalent in perturbation and m(s) = m.

5. Figure 1: All the contributions to the complete e+e- cross section (summed over both light and heavy quarks). • The first two orders in αs that contribute to σee involve the graphs above. A nice thing about light quarks is that their calculation can be accomplished with the masses scaled to zero. • Kinoshita’s mass-singularity theorem tells us that no singularities exist to any perturbation theory order because what we are calculating is just the total transition probability of the system. • Only the total cross section is able to be calculated this way. • Partial rates, final state details, and the debate about whether quarks exist as physical particles (not important) involves Logarithmic Singularities (which cancel in the cross section and are therefore ignored).

6. O(√s) is defined as the mass of P’ so we have to keep it. Making use of identical electrodynamic calculations and exhibiting their contributions with the light quarks we obtain the following equation: • The two sums over squares (of quark charges) are 2 and 4/3 in the three-quartet model. We can approximate f(υ) above to: • As υ goes to 0, f(υ)≅1/υ. This behavior is a result of the (b) diagram in Figure 1as well asa Coulomb-like final-state interaction. • n gluon ladder exchanges in nth order gives n factors of 1/υ. This breakdown for small values of υ is related to the breakdown below 4(mP’)2. • If the value of ‘s’ is large enough that (4/3)αsf(υ) ≤ ½, we can use 2nd order perturbation theory to find the approach to free-quark model scaling. A reasonable value of the strong constant suggests the final-state interaction should produce a 15-20% drop in R from s = 25 GeV2 to s = 81 GeV2.

7. There is another breakdown of the perturbation expansion. This time, it comes from the non-Abelian structure. As s -> 4(mP’)2 the momentum through the gluon lines in Figure 1 quickly drops to zero. When s becomes less than ≈1 GeV2, the higher order terms will start to have a larger and larger effect on the overall system. • The effect of P’ below 4(mP’)2 can be seen above (which is of order αs3).The figure will remain as a very small correction to the R(s) equation until4(mP’)2 – s ≤ 4 GeV2. • At approximately 8 GeV2 centered upon 4(mP’)2, perturbation theory breaks due to non-Abelian effects.

8. Orthocharmonium and Balmer • When we look at values below 4(mP’)2 ladder exchanges between quarks gives us the same type of answer as when we looked above 4(mP’)2 – which is a complete breakdown of the overall exchanges. • “orthocharmonium” bound states are created as a result of this. • The Balmer Formula informs us that the ground state of such a system is located at the point: s = 4(mP’)2 – (4/3αs)2(mP’)2. It is very likely that is is well inside the 8 GeV2 region mentioned on my previous slide. • If we try to estimate the size of a charmonium atom we can see that a Coulomb-like picture is not completely correct: • Bohr Radius: (⅔αsmP’)-1 • Setting αs ≤ 0.3 and mP’ ≤ 2 GeVsuggests this radius is larger than 3 GeV-1. • Observations say this is too large to use a Coulomb-like potential.

9. The Hadron and Lepton Ratios • For the moment ignore the Coulomb-like potential problem. • The three-gluon discontinuity will give us the hadron width. The leptonic width (from one photon interaction) can also be calculated using a similar process • Γh • As always, α = 1/137. • The ratio of these two equations is independent of any wave-function effects. • Around this time, Augustin and Aubert discovered a resonance. Appelquist and Politzersuggested this resonance with a mass of around 3 GeV was actually orthocharmonium. Doing so allows the ratio above to ‘fix’ the value of the strong coupling constant.

10. The Subtle Existence of Paracharmonium • The bound states may not just be limited to orthocharmonium. There should also exist a paracharmonium (0-) bound state that had a mass only slightly smaller than orthocharmonium’s.. • In the Coulomb approximation, the ground-state width of paracharmonium is: • The values of the strong coupling constant and mP’are what I defined in the last slide. The value of Γh(para) is about 1.3 MeV. • If we take the ratio of Γh(ortho) /Γh(para) with Γh(para) set more reliably to 6 MeV (from a different source): ~0.013

11. Conclusions • The existence of paracharmonium with the width found and a mass around the order of 3 GeV is important to look for experimentally. • Any two-quark system where the sum of the masses corresponds to a small gbar can be studied analogously. Other aspects of charm phenomenology will come from the small gbar. • From this we can identify charmed hadrons as well as a light quark mass minus a binding energy. • The existence of other heavy quarks will not complicate matters in principle since their effects are calculable.

12. S.L. Glashow, J. Illiopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970) T. Kinoshita, J. Math. Phys. (N.Y.) 3, 650 (1962) 4. Jost and J.M. Luttinger, Helv. Phys. Acta 23, 201 (1950; G. Kallen and A. Sabry, Kgl. Dan. Vidensk. Selsk., Mat.-Fys. Medd. 29, No. 17 (1955). J. Schwinger, Particles, Sources, and Fields (Addison-Wesley, New York, 1973), Vol. II, Chap. 5-4. J.-E. Augustinet al., Phys. Rev. Lett. 33, 1466 (1974); J.J. Aubertet al., Phys Rev. Lett. 33, 1404 (1974). T. Goldman, private communication