Remarks on Photon-Hadron Interactions

Scientific Paper
Title	Remarks on Photon-Hadron Interactions
Read in full	Link to paper
Author(s)	E Comay
Keywords	Photon, Hadron, VMD
Published	2003
Journal	Apeiron
Volume	10
Number	2
No. of pages	17

Read the full paper here

Abstract

Theoretical aspects of VMD and related approaches to real photon-hadron interaction are discussed. The work relies on special relativity, properties of linearly polarized photons, angular momentum conservation and relevant experiments. It is explained why VMD and similar approaches should not be regarded as part of a theory but, at most, as phenomenological models. A further experiment pertaining to this issue is suggested.

Overview

E. Comay's 2003 Apeiron paper is a critique of Vector Meson Dominance (VMD) and the family of related ideas he groups together as "Photon's Hadronic Structure Approaches" (PHSA). The experimental facts these approaches were built to explain are not in dispute and Comay states them plainly: the cross section for hard photons scattered from a proton target is practically the same as from a neutron target, and the number of hadrons emitted from a photon–proton interaction region exceeds the leptonic number by four orders of magnitude. Something about the photon's behaviour at high energy is strongly hadron-like and largely independent of the target's electric charge.

What Comay attacks is the theoretical construction erected on top of those facts — the claim that a physical photon is a superposition of a pure electromagnetic component and a hadronic component,

|γ> = c₀|γ₀> + c_h|h>,

with the fluctuation between the two treated as an inherent property of the photon, independent of its distance from any hadronic target, and the relative time spent in each state given by the squared moduli of the coefficients. In most versions |h> is a neutral vector meson carrying the photon's spin, parity and charge-conjugation quantum numbers; in Gribov's variant it may belong to a larger set of hadronic states.

The paper's thesis is a claim about status rather than about usefulness: this expression can be at most a phenomenological model, never part of a theory. Comay is careful to say that the practical merits of VMD as a model lie outside his scope, and he notes that the physics community's own usage — the name "Vector Dominance Models", and the classification of the topic under the phenomenological sections of PACS and of hep-ph@arXiv.org — already reflects that judgement. What he adds is a set of arguments showing that if the formula is promoted to theory, it fails refutation tests.

The argument

Theory versus model

Comay first sets out a working distinction, offered as useful for the case at hand rather than as a universal definition. Theories and models share three properties: both give a mathematically self-consistent scheme leading to formulas describing data; both are acceptable only within a domain of validity; both require experimentally determined constants. Three properties separate them. A theory's predictions must be very precise within its validity domain, while a model is acceptable if it is reasonably approximate. A theory's constants may be fixed by any experiment within its domain and then extrapolated into far regions — measure a body's mass once and Newtonian mechanics applies at all speeds well below c — whereas a model is generally useful only near where its constants were fitted, good for interpolation and deteriorating under extrapolation. Finally, a model is tested by practical benefit and may be patched with corrections (as the nuclear liquid drop model is supplemented by shell-model terms to account for magic numbers), while a theory is tested by correctness and can be refuted if it fails either experimental data or well-established theories confirmed by many experiments.

Wigner's analysis: the superposition is not a particle state

The first theoretical objection rests on Wigner's classification of the irreducible representations of the Poincaré group, standard input to quantum field theory. A massive particle corresponds to an irreducible representation characterized by its self-mass and spin; for massless particles such as the photon, spin is replaced by helicity. Since a quantum mechanical state of a particle is characterized by the eigenvalues of the self-mass and spin operators, every term of its wave function must carry the same eigenvalues of those operators. But in the PHSA expression the first term on the right-hand side is massless and the second is massive. Comay concludes that the superposition "cannot represent a quantum mechanical state of a particle" and that PHSA is therefore inconsistent with relativistic quantum field theory.

The energy dependence assumption and Lorentz invariance

The second objection concerns how c₀ and c_h are supposed to behave under Lorentz transformations. Practitioners assume the hadronic part of soft photons is negligible — otherwise optical or blackbody photons would interact strongly with hadrons — which Comay labels the energy dependence assumption. He argues it cannot sit inside a relativistic theory. Suppose energetic photons are found to interact hadronically 10% of the time and electromagnetically 90%, and suppose, as PHSA insists, that this ratio is an inherent property of the photon independent of any target. Relativity then requires the ratio to be preserved under all Lorentz transformations, including one into a frame where the photon is soft. Since by construction c₀ and c_h are transition probabilities from the physical photon state to the pure electromagnetic and hadronic states, and, quoting Wigner, "the transition probability has an invariant physical sense," the energy dependence assumption is inconsistent with the invariance it must respect.

Linear polarization and cylindrical symmetry

The third and most detailed objection turns on the transverse structure of a linearly polarized photon. Take a photon moving along the z-axis with its electric field along x. Such a photon manifestly lacks cylindrical symmetry about z, since rotating about that axis changes the field directions; and because the vector potential A is parallel to the electric field and the electromagnetic interaction term of the Lagrangian density is

L_int = -j_μA^μ,

a linearly polarized photon interacts with matter in a way that breaks cylindrical symmetry about z.

Comay then shows that the assumed hadronic part does not. Angular momentum conservation forces the hadronic state to carry the same angular momentum content as the photon's helicity state, so for linear polarization it is the equal superposition of the two helicities

|SM> = (|11> + |1 -1>)/√2.

Rotating by π/2 about z — the rotation that exchanges the directions of the undulating electric and magnetic fields — multiplies each term by e^-imφ, giving factors ∓i and hence |SM>_rot = -i(|11> - |1 -1>)/√2. Building the full wave function with a target proton of spin projection ±1/2 and evaluating the Hamiltonian yields four terms for the unrotated state and four for the rotated one, differing only in the signs of the second and third. Those two cross terms are then shown to vanish. Three angular momenta are in play — the assumed hadronic part of the photon, the target proton's spin, and the spatial angular momentum between the incoming vector meson and the proton — and the last has zero projection on z because the photon's linear momentum is parallel to that axis, (r × p)·p = 0. The bra of the second term then has total M = 1 ± 1/2 while the ket has M = -1 ± 1/2; since H is a scalar in three-dimensional space the M-values do not match and the term vanishes, and the same argument disposes of the third.

With the cross terms gone the rotated and unrotated expressions are identical, so a hadronic component would scatter from an unpolarized proton target in a cylindrically symmetric way. The conclusion Comay draws is that the photon's transverse information — its linear polarization — would have to disappear whenever the physical photon fluctuates into a hadronic state, which "clearly reduces the theoretical appeal of the VMD hypothesis" and, more usefully, makes it testable.

Experimental considerations

Two experimental lines follow. The first is a thought experiment on photon–photon scattering, chosen because photons have no rest frame and so no frame can be called preferential. Two sources S₁ and S₂ at x = ±1 emit soft rays that cross at a point O in the (x,y) plane. In that frame the interaction is purely electromagnetic and, so long as Maxwell's equations remain linear, vanishes. Viewed from a frame moving along the negative y-axis at close to the speed of light the same photons are very energetic and, under VMD, must carry hadronic parts, so a hadron–hadron process should occur on top of the (still null) Lorentz-transformed electromagnetic one. Since the fraction of events in which photons exchange energy-momentum must be the same in every inertial frame, this is a contradiction, and it rules out the option that c_h tends to zero in frames where the photon is soft.

The second is a concrete proposal in the alternative case, where c₀ and c_h keep their absolute values under Lorentz transformation. Take Compton scattering of a 1 MeV photon on a hydrogen electron in the backward direction θ = π. The Compton relation

k_out = k_in/[1 + (2k_in/m)sin²(θ/2)]

gives k_out ≈ 0.2 MeV, and the unpolarized Compton cross section gives dσ/dΩ(θ = π) ≈ 6.5 mb. For the proton the Compton process is negligible — the proton/electron mass ratio is about 2000 and the cross section smaller by a factor of about 1/4000000 — but a hadronic photon component would allow meson–proton scattering, in which, since the proton's mass is 938 times the photon energy, the photon's energy is nearly conserved. With an interaction region of effective radius under 10 f and a photon momentum of 1 MeV ≈ 1/197 f^-1, the spatial angular momentum practically vanishes and only the S-wave contributes. Taking the low-energy π–proton cross section σ ≈ 7 mb and arguing on a quark count that vector meson–proton scattering is of the same order, division by 4π gives dσ/dΩ ≈ 0.6 mb — comparable to the 6.5 mb Compton figure. The signature is unmistakable: some fraction of backward-scattered photons should come out at 1 MeV rather than the 0.2 MeV required by the Compton relation. Comay notes that to his knowledge no such effect has ever been reported, and that a dedicated experiment could settle the matter, since the two photon energies are easy to tell apart.

Finally, the polarization test has already been done. Experiments using linearly polarized photons to measure outgoing pions in γp and γn collisions — Bellenger et al. (1969), Anderson et al. (1971) and Sherden et al. (1973) — show that cylindrical symmetry is not conserved, contrary to what the hadronic component would give.

Conclusion

Comay concludes that VMD can be no more than a phenomenological model; if its merits are extended so that it is treated as part of a theory, it must stand refutation tests, and by the theoretical and experimental arguments above it fails them. He places his conclusions alongside earlier criticism, citing J. I. Friedman's Nobel lecture, which shows an inconsistency of VMD with experimental data, and the "rather exceptional humoristic-sarcastic poster" reproduced in the Bauer–Spital–Yennie–Pipkin review, which he reads as reflecting both the theoretical difficulties and the disbelief surrounding VMD. His closing sentence is a statement of what remains open rather than of what has been replaced: "the hadronic features of real photon-hadron interaction await theoretical interpretation."

Assessment

This is a disciplined piece of criticism, and its most attractive feature is how little it asks the reader to concede. It advances no alternative model, disputes no measurement, and grants at the outset that the hadron-like behaviour of hard photons is a real and well-documented phenomenon. Its arguments are drawn from resources mainstream physics already accepts — Wigner's representation theory, the Lorentz invariance of transition probabilities, angular momentum conservation, the standard Compton formulas — and every criticism is aimed at internal consistency rather than at the data. The polarization argument is the strongest part: it produces a sharp qualitative prediction (a hadronic component erases linear polarization information, and hence cannot break cylindrical symmetry) and then points to published photoproduction asymmetry measurements that show the symmetry is broken. The proposed backward-Compton test is similarly well posed, with a numerical estimate and an unambiguous experimental signature.

The main limitation is one Comay flags himself in his opening pages: the target is largely a position few practitioners defend in the strong form he attacks. VMD has been understood as a phenomenological model since well before 2003, and the paper's own evidence for this — the "Models" in the name, the PACS and hep-ph classification — cuts both ways. If the community already treats VMD as a fitted low-energy effective description, superseded at high energy by the partonic and QCD account of the photon's hadronic structure, then showing that its wave-function notation cannot be read literally as an eigenstate of the mass operator establishes less than it appears to. A quantum superposition of states of different mass is not automatically illegitimate — neutrino flavour states are exactly that, and are experimentally established — so the Wigner argument shows that the expression is not a stationary one-particle state, which is not quite the same as showing it is incoherent. Similarly, the theory/model distinction on which the whole paper turns is offered as a suggestion rather than derived, and the numerical estimate in the Compton proposal leans on a chain of plausible but rough steps: a vector-meson–proton cross section inferred from π–proton data by quark counting, and a pure S-wave assumption. The paper's real contribution is therefore less a refutation of a working tool than a clear statement of what a genuine theory of photon–hadron interaction would have to supply and does not yet — a point Comay makes explicitly in his last line.