Chiral-Scale Perturbation Theory About an Infrared Fixed Point

We review the failure of lowest order chiral $SU(3)_L \times SU(3)_R$ perturbation theory $\chi$PT$_3$ to account for amplitudes involving the $f_0(500)$ resonance and $O(m_K)$ extrapolations in momenta. We summarize our proposal to replace $\chi$PT$_3$ with a new effective theory $\chi$PT$_\sigma$ based on a low-energy expansion about an infrared fixed point in 3-flavour QCD. At the fixed point, the quark condensate $\langle\bar{q}q\rangle_\mathrm{vac}\neq 0$ induces nine Nambu-Goldstone bosons: $\pi, K, \eta$ and a QCD dilaton $\sigma$ which we identify with the $f_0(500)$ resonance. We discuss the construction of the $\chi$PT$_\sigma$ Lagrangian and its implications for meson phenomenology at low-energies. Our main results include a simple explanation for the $\Delta I = 1/2$ rule in $K$-decays and an estimate for the Drell-Yan ratio in the infrared limit.


Three-flavor chiral expansions: problems in the scalar-isoscalar channel
Chiral SU(3) L × SU(3) R perturbation theory χPT 3 is nowadays well established as the framework to systematically analyze the low-energy interactions of π, K, η mesons -the pseudo Nambu-Goldstone (NG) bosons of approximate chiral symmetry. The method relies on expansions about a NG-symmetry, viz., low-energy scattering amplitudes and matrix elements can be described by an asymptotic series in powers and logarithms of O(m K ) momentum and quark masses m u,d,s = O(m 2 K ), with m u,d /m s held fixed. The scheme works provided that contributions from the NG sector {π, K, η} dominate those from the non-NG sector {ρ, ω, . . .}; an assumption known as the partial conservation of axial current (PCAC) hypothesis.
It has been observed [1], however, that the χPT 3 expansion (1) is afflicted with a peculiar malady: it typically diverges for amplitudes which involve both a 0 ++ channel and O(m K ) extrapolations in momenta. The origin of this phenomenon can be traced to the f 0 (500) resonance, a broad 0 ++ state whose complex pole mass and residue [2] m f 0 = 441 − i 272 MeV and |g f 0 ππ | = 3.31 GeV (2) a e-mail: rcrewthe@physics.adelaide.edu.au b e-mail: tunstall@itp.unibe.ch c Speaker.

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Not NG bosons scale separation (a) Scale separations between Nambu-Goldstone (NG) sectors and other hadrons for each type of chiral perturbation theory χPT discussed in this proceeding. In conventional three-flavor theory χPT 3 (top diagram), there is no scale separation: the non-NG boson f 0 (500) sits in the middle of the NG sector {π, K, η}. Our three-flavor proposal χPT σ (bottom diagram) for O(m K ) extrapolations in momenta implies a clear scale separation between the NG sector {π, K, η, σ = f 0 } and the non-NG sector [6] for continued growth in α s with decreasing scale µ. Despite extensive literature [7] concerning the existence of α IR , there is currently no consensus which of the above two, physically distinct, scenarios is actually realized in QCD. In particular, it is unclear how sensitive existing results are to variations in N f . This is perhaps unsurprising, since modern calculations utilize different, nonperturbative definitions of α s , thereby making comparisons between various analyses difficult. Figure 1 have been determined to remarkable precision. Since χPT 3 classes f 0 pole terms as next-to-leading order (NLO), figure 1a shows why the low-energy expansion (1) fails: the location of f 0 and its strong coupling to π, K, η mesons invalidates the requirements of PCAC.

Three-flavor chiral-scale expansions about an infrared fixed point
In this proceeding, we summarize our proposal [3] to solve the convergence problem of χPT 3 expansions (1) by modifying the leading order (LO) of the 3-flavor theory. In short, our solution involves extending the standard NG sector {π, K, η} to include f 0 (500) as a QCD dilaton σ associated with the spontaneous breaking of scale invariance. The scale symmetric counterpart of PCAC -partial conservation of dilatation current (PCDC) -then implies that amplitudes with σ/ f 0 pole terms dominate, compared with contributions from the non-NG sector {ρ, ω, K * , N, η ′ , . . .}. 1 This scenario can occur in QCD if at low energy scales µ ≪ m t,b,c , the strong coupling α s for the 3-flavor theory runs nonperturbatively to an infrared fixed point α IR (figure 1b). At the fixed point, the gluonic term in the strong trace anomaly [9] vanishes, which implies that in the chiral limit our proposal is to replace χPT 3 by chiral-scale perturbation theory χPT σ , where the strange quark mass m s in (4) sets the scale of m 2 f 0 as well as m 2 K and m 2 η ( figure 1a, bottom diagram). As a result, the rules for counting powers of m K are changed: f 0 pole amplitudes (NLO in χPT 3 ) are promoted to LO. That fixes the LO problem for amplitudes involving 0 ++ channels and O(m K ) extrapolations in momenta. Note that we achieve this without upsetting successful LO χPT 3 predictions for amplitudes which do not involve the f 0 ; that is because the χPT 3 Lagrangian equals the σ → 0 limit of the χPT σ Lagrangian.
In the physical region 0 < α s < α IR , the effective theory consists of operators constructed from the SU(3) field U=U(π, K, η) and chiral invariant dilaton σ, with terms classified by their scaling dimension d: Explicit formulas for the strong, weak, and electromagnetic interactions are obtained by scaling Lagrangian operators such as K U, U † = 1 4 F 2 π Tr(∂ µ U∂ µ U † ) and K σ = 1 2 ∂ µ σ∂ µ σ by appropriate powers of the d = 1 field e σ/F σ . For example, the LO strong Lagrangian reads where F σ ≈ 100 MeV is the dilaton decay constant, whose value is estimated by applying an analogue of the Goldberger-Treiman relation to analyses of NN-scattering [10]. Here the anomalous dimensions γ m = γ m (α IR ) and β ′ = β(α IR ) are evaluated at the fixed point because we expand in α s about α IR . The low-energy constants c 1 and c 2 are not fixed by symmetry arguments alone, while vacuum stability in the σ direction implies that both c 3 and c 4 are O(M). From (7), one obtains formulas for the dilaton mass m σ m 2 and σππ coupling Note that (9) is derivative, so an on-shell dilaton is O(m 2 σ ) and consistent with σ being the broad resonance f 0 (500).
Our proposed replacement for χPT 3 possesses some desirable features, the foremost being: 1. The ∆I = 1/2 rule for K-decays emerges as a consequence of χPT σ , with a dilaton pole diagram (figure 2a) accounting for the large I = 0 amplitude in K S → ππ. Here, vacuum alignment [13] of the effective potential induces an interaction L K S σ = g K S σ K S σ which mixes K S and σ in LO. The effective coupling g K S σ is fixed by data on γγ → π 0 π 0 and K S → γγ, with our estimate |g K S σ | ≈ 4.4 × 10 3 keV 2 accurate to a precision 30% expected from a 3-flavor expansion. Combined with data for the f 0 width (Eq. (2)), we find an amplitude A σ-pole ≈ 0.34 keV which accounts for the large magnitude |A 0 | expt. = 0.33 keV. Consequently, the LO of χPT σ explains the ∆I = 1/2 rule for kaon decays.
(a) Tree diagrams in the effective theory χPT σ for the decay K S → ππ. The vertex amplitudes due to 8 and 27 contact couplings g 8 and g 27 are dominated by the σ/ f 0 -pole amplitude. The magnitude of g K S σ is found by applying χPT σ to K S → γγ and γγ → ππ. σ π π γ γ g σγγ g σππ (b) Dilaton pole in γγ → ππ. Lowest order χPT σ includes other tree diagrams (for π + π − production) and also π ± , K ± loop diagrams (suppressed by a factor 1/N c ) coupled to both photons. Figure 2 2. Our analysis of γγ channels and the electromagnetic trace anomaly [11,12] yields a relation between the effective σγγ coupling and the nonperturbative Drell-Yan ratio R IR at α IR :