Thompson's Renormalization Group Method Applied to QCD at High Energy Scale

Scientific Paper
Title	Thompson's Renormalization Group Method Applied to QCD at High Energy Scale
Read in full	Link to paper
Author(s)	Cláudio Nassif
Keywords	energy, magnetic field, Vacuum, action
Published	2007
Journal	ArXiv
No. of pages	13

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Abstract

We use a renormalization group method to treat QCD-vacuum behavior specially closer to the regime of asymptotic freedom. QCD-vacuum behaves effectively like a "paramagnetic system" of a classical theory in the sense that virtual color charges (gluons) emerges in it as a spin effect of a paramagnetic material when a magnetic field aligns their microscopic magnetic dipoles. Due to that strong classical analogy with the paramagnetism of Landau's theory,we will be able to use a certain Landau effective action without temperature and phase transition for just representing QCD-vacuum behavior at higher energies as being magnetization of a paramagnetic material in the presence of a magnetic field H. This reasoning will allow us to apply Thompson's approach to such an action in order to extract an "effective susceptibility" (χ > 0) of QCD-vacuum. It depends on logarithmic of energy scale u to investigate hadronic matter. Consequently we are able to get an "effective magnetic permeability" (μ > 1) of such a "paramagnetic vacuum". Actually,as QCD-vacuum must obey Lorentz invariance,the attainment of μ > 1 must simply require that the "effective electrical permissivity" is ε < 1 in such a way that με = 1 (c² = 1). This leads to the anti-screening effect where the asymptotic freedom takes place. We will also be able to extend our investigation to include both the diamagnetic fermionic properties of QED-vacuum (screening) and the paramagnetic bosonic properties of QCD-vacuum (anti-screening) into the same formalism by obtaining a β-function at 1 loop,where both the bosonic and fermionic contributions are considered.

Overview

This is a short arXiv preprint (arXiv:0708.2167, August 2007) by Cláudio Nassif with J. A. Helayel-Neto (CBPF, Rio de Janeiro) and P. R. Silva (UFMG, Belo Horizonte). Its subject is asymptotic freedom — the fact that the strong coupling between quarks weakens as the energy of the probe rises — and its aim is to recover that behaviour, and the one-loop β-function that describes it, without doing a loop calculation at all. Instead the authors use a heuristic scaling procedure due to C. J. Thompson (1976), originally devised to extract critical exponents from the Landau–Ginzburg–Wilson free energy by dimensional analysis on a coherence volume. The whole apparatus of Feynman diagrams is replaced by the rule that each term of the action, integrated over a coherence volume L^d, is "separately of the order of unity".

The physical picture the authors build on is N. K. Nielsen's semiclassical account of anti-screening: the QCD vacuum behaves like a paramagnetic medium, in which the virtual gluon cloud plays the role of magnetic dipoles aligning with an applied field, whereas the QED vacuum, whose virtual electron–positron pairs act as electric dipoles, behaves like a dielectric medium and screens. Since a relativistic vacuum must satisfy με = 1 with c = 1, a permeability μ > 1 forces a permittivity ε < 1 — which is precisely anti-screening. That inversion, which has no counterpart in an ordinary polarizable medium, is what the paper exploits: it converts a statement about magnetism into a statement about the effective strong charge. The departure from the standard account is therefore methodological rather than doctrinal — the authors are not disputing QCD but arguing that its running coupling can be obtained from scaling and a classical magnetic analogy.

The argument

The Lagrangian and the two quantum contributions to the color field

The paper starts from the ordinary QCD Lagrangian density, L = Σ_j ψ_j(iγ^μD_μ − m_j)ψ_j − ¼G^a_μνG^μν_a, noting that the gluon kinetic term has the same form as the photon term of QED but that the non-abelian field strength adds a gluon self-interaction. Gluons carry colour, photons carry no charge, and that difference is identified as the origin of asymptotic freedom.

The squared colour field is then split into three pieces, E²_a = E²_cl.a + ΔE²_F.a + ΔE²_B.a: a classical part, a fermionic quantum part analogous to the QED vacuum-polarization term treated in the authors' earlier work, and a purely bosonic part with no QED analogue. Correspondingly the energy density is u = u_cl. + u_F + u_B, and the claim is that u_B enters with the opposite sign to u_F. Applying Thompson's first prescription to the fermion kinetic term over a four-volume with dV₄ ~ r³dr gives the scaling of the quark condensate, ⟨ψ̅_jψ_j⟩_r ~ 1/r³, hence ΔE²_F.a ∝ 1/r³ — a logarithmic correction on top of the inverse-square classical law.

The Landau Hamiltonian for "color paramagnetism"

The colour magnetization M_a = χ_eff.H_a is promoted to a scalar field Φ_a(r), and an effective Landau functional is written, F = ∫[(∇Φ_a)² + R(L)Φ_a² + K(L)Φ_a⁴]dr — with, the authors stress, no temperature and no phase transition, so only the first two of Thompson's three assumptions are needed. Demanding each term be of order unity over L^d yields Φ_a² ~ L^2−d, R(L)L² ~ 1, and K(L) = L^d−4 for d ≤ 4 with K constant for d ≥ 4. The upper critical dimension is therefore d_c = 4, which the authors emphasise coincides with the dimension of space-time.

Exactly at d = 4 the naive scaling gives no dependence, so the calculation is refined by carrying the r-dependence inside the integral: ∫K(r)r^3−ddr ~ 1 becomes ∫Kr⁻¹dr ~ 1, whence K ~ 1/ln(L₀/L) = 1/ln(u/u₀) with u = 1/L. Setting Φ_a² = −R/K in analogy with the Landau equilibrium magnetization and matching to the paramagnetic energy density −½·4πχH² fixes the constants and delivers the paper's central result for the susceptibility:

χ_eff. = χ(u) = ln(u/u₀)

so that μ(u) = 1 + 4π ln(u/u₀) and, by με = 1, ε(u) = 1/[1 + 4π ln(u/u₀)]. With α_S = εα_0S this gives a coupling falling to zero as u → ∞ and diverging as u → u₀: asymptotic freedom at short distance, confinement-like strong coupling in the infrared. Since K(u) ~ χ(u)⁻¹, the quartic coupling of the Landau functional is read directly as the strong coupling strength.

Counting bosonic and fermionic species: a unified β-function

The last section derives a running equation from quantum fluctuations of the field energy. Writing the quantum-regime energy density as u_q = (1/8π)αℏc/(λ_cr³) below a cut-off wavelength λ_c0 = ℏ/m₀c (and recovering the classical 1/r⁴ above it), integrating over a spherical coherence volume, and assuming that the fractional shift in coupling tracks the fractional shift in mass, Δα/α = Δm/m, the authors obtain

α(u) = α₀ / [1 + (α_ref/2) ln(u/u₀)], u dα/du = −(α_ref/α₀)α²

The sign of α_ref decides everything: positive gives asymptotic freedom, negative reproduces the Landau singularity characteristic of QED screening. The coefficient is then written as a species count, α_ref/α₀ = (N_B − N_F)/Q, with N_F = 2n_F (a factor 2 for spin degeneracy 2s+1 at s = ½, each fermion–antifermion pair counted as one "flavor electric dipole") and N_B = 2(n_g + n_g̅) (a factor 2 for the two gluon polarizations, gluon and anti-gluon counted separately as independent magnetic dipoles).

Q is fixed by a boundary condition: setting n_g = 0 and one flavour must reproduce the QED result [u dα/du]_QED = 2α²/3π, giving Q = 3π. Restoring the eight gluon states of SU(3) — counted twice, as g and g̅ — gives N_B = 32 and the paper's final result

[u dα/du]_QCD = −[(32 − 2n_F)/3π]α²

which the authors set beside the textbook one-loop expression they quote as −[(33 − 2n_F)/3π]α². Taking α₀ = α(1 GeV) ≈ 0.43 from a published plot and n_F = 6 gives α_ref ≈ 0.91 and, from α_ref ≡ α(u_ref), a scale u_ref ≈ 383 MeV; with n_F = 5 the numbers become α_ref ≈ 1 and u_ref ≈ 320 MeV, which the authors note is close to a constituent quark mass if each valence quark carries a third of the nucleon mass.

Assessment

What is genuinely attractive here is economy. Thompson's method is dimensional analysis with a coherence volume and nothing else, and it is remarkable that so crude a procedure reproduces the qualitative structure of the running coupling — the logarithm, the upper critical dimension d_c = 4 falling on the physical space-time dimension, the sign flip between screening and anti-screening, and a one-loop numerator within a unit of the textbook 33. The magnetic reading of anti-screening is also pedagogically real, not the authors' invention: it goes back to Nielsen's 1981 American Journal of Physics treatment, and the paper's contribution is to attach a scaling calculation to it. The species-counting step, in which the numerator 33 is decomposed into gluon polarizations and quark spins, is the most interesting part of the paper, because it makes an otherwise opaque group-theory factor look like a dipole census.

The difficulties are also real. Most of them are the usual price of a heuristic: the "~ 1" of Thompson's prescription discards every numerical factor, so agreement to within a unit in the numerator is not something the method can honestly claim to have earned — a different choice of coherence volume or of what counts as "order unity" would move the answer. Several steps are identifications rather than derivations. The constants in equation (24) are fixed by declaring c₁ ≡ 2π and R(r) ≡ H_a(r)², yet equation (16) had already required R(L)L² ~ 1, i.e. R ~ u²; the two statements are compatible only if the colour magnetic field is itself scale-dependent in exactly the required way, which is asserted, not shown. Likewise Δα/α = Δm/m is justified only by the remark that both increments are logarithmic in scale. And the decomposition (α)² = αα_ref introduces the parameter that carries the entire physical content of the result; α_ref is defined into existence and only afterwards interpreted.

Two of the paper's own numbers do not hold up. First, the comparison in equations (47)–(48) is not like with like. The authors' QED normalization, u dα/du = 2α²/3π, is the standard one-loop QED result in the d/d ln μ convention; but in that same convention the standard QCD one-loop β-function is −(33 − 2n_F)α²/6π, not /3π as quoted in equation (48). With the denominator corrected, the paper's coefficient (32 − 2n_F)/3π = (64 − 4n_F)/6π is roughly twice the accepted value — at n_F = 6, 40 against 21. The celebrated "32 versus 33" is then an agreement of numerators across mismatched conventions rather than a near-reproduction of the standard result, and the factor 2 plausibly traces to the decision to count g and g̅ as separate dipoles, which is what doubles N_B from 16 to 32. Second, the quoted u_ref ≈ 383 MeV for n_F = 6 does not follow from the paper's own equation (40): with α₀ = 0.43 and α_ref = 0.9125, solving α_ref = α₀/[1 + (α_ref/2)ln(u_ref/u₀)] gives u_ref ≈ 314 MeV. The n_F = 5 figures reproduce exactly (α_ref = 1.004, u_ref = 320 MeV), which suggests an arithmetic slip in the six-flavour line rather than a different formula. The correction is not damaging — 314 MeV sits just as comfortably beside a third of the nucleon mass — but it removes the paper's remark that the value is "slightly greater than the quark constituent mass of the nucleon."

Finally, a point of scope. The paper does not challenge QCD; it reconstructs a known result by an unconventional route, and its value stands or falls on whether that route is illuminating rather than on whether it is right about nature. Readers looking here for a critique of the standard model of strong interactions will not find one. Readers interested in how far pure scaling arguments can substitute for loop integrals — the same question the authors pursued earlier for diffusion-limited reactions and for QED — will find a compact and honest example, provided the two numerical points above are kept in mind.