Form follows function in QCD

9 March 2022
Hadron from factors

In the 1970s, the study of low-energy (few GeV) hadron–hadron collisions in bubble chambers was all the rage. It seemed that we understood very little. We had the SU(3) of flavour, Regge theory and the S-matrix to describe hadronic processes, but no overarching theory. Of course, theorists were already working on perturbative QCD and this started to gain traction when experimental results from the Big European Bubble Chamber at CERN showed signs of the scaling violations and made an early measurement of the QCD scale, ΛQCD. We have been living with the predictions of perturbative QCD ever since, at increasingly higher orders. But there have always been non-perturbative inputs, such as the parton distribution functions.

Hadron Form Factors: From Basic Phenomenology to QCD Sum Rules takes us back to low-energy hadron physics and shows us how much more we know about it today. In particular, it explores the formalism for heavy-flavour decays, which is particularly relevant at a time when it seems that the only anomalies we observe with respect to the Standard Model appear in various B-meson decays. It also explores the connections between space-like and time-like processes in terms of QCD sum rules connecting perturbative and non-perturbative behaviour.

The book takes us back to low-energy hadron physics and shows us how much more we know about it today

The general introduction reminds us of the formalism of form factors in the atomic case. This is generalised to mesons and baryons in chapters 2 and 3, after the introduction of QCD in chapter 1, with an emphasis on quark and gluon electroweak currents and their generalisation to effective currents. Hadron spectroscopy is reviewed from a modern perspective and heavy-quark effective theory is introduced. In chapter 2, the formalism for the pion form factor, which is related to the pion decay constant, is introduced via e-π scattering. Due emphasis is placed on how one may measure these quantities. I also appreciated the explanation of how a pseudoscalar particle such as the pion can decay via the axial vector current – a question


often raised by smart undergraduates. (Clue: the axial vector current is not conserved). Next, the πe3 decay is considered and generalised to K-, D- and B-meson semileptonic decays. Chapter 3 covers the baryon form factors and their decay constants, and chapter 4 considers hadronic radiative transitions. Chapter 5 relates the pion form factor in the space-like region to its counterpart in the time-like region in e+e → π+π, where one has to consider resonances and widths. Relationships are developed, whereby one can see that by measuring pion and kaon form factors in e+e scattering one can predict the widths of decays such as τ → ππν and τ → KKν. In chapter 6, non-local hadronic matrix elements are introduced to extend the formalism to deal with decays such as π → γγ and B → Kμμ.

The book shifts gears in chapters 7–10. Here, QCD is used to calculate hadronic matrix elements. Chapter 7 covers the calculation of the form factors in the infinite momentum frame, whereby the asymptotic form factor can be expressed in terms of the pion decay constant and a pion distribution amplitude describing the momentum distribution between two valence partons in the pion. In chapter 8, the QCD sum rules are introduced. The two-point correlation of quark current operators can be calculated in perturbative QCD at large space-like momenta, and the result is expressed in terms of perturbative contributions and the QCD vacuum condensates. This can then be related through the sum rule to the hadronic degrees of freedom in the time-like region. Such sum rules are used to gain information on both condensate densities or quark masses from accurate hadronic data and hadronic decay constants and masses from QCD calculations. The connection is made to parton–hadron duality and to the operator product expansion. Some illustrative examples of the technique, such as the calculation of the strange-quark mass and the pion decay constant, are also given. Chapter 9 concerns the light-cone expansion and light-cone dominance, which is then used to explain the role of light-cone sum rules in chapter 10. The use of these sum rules in calculating hadron form factors is illustrated with the pion form factor and also with the heavy-to-light form factors necessary for B → π, B → K, D → π, D → K and B → D decays.

Overall, this book is not an easy read, but there are many useful insights. This is essentially a textbook, and a valuable reference work that belongs in the libraries of particle-physics institutes around the world.

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