Pseudoscalar Dominance in QCD

• Pseudoscalar dominance is a QCD concept where the pion pole provides the leading contribution to nucleon pseudoscalar amplitudes, with minimal corrections from excited states.
• Dispersive formulations and PCAC identities are employed to relate axial and pseudoscalar form factors, ensuring compatibility with chiral symmetry and pQCD asymptotics.
• Lattice QCD and continuum analyses quantitatively reconcile the Goldberger-Treiman discrepancy, confirming pion-pole dominance while accounting for corrections from higher pseudoscalar modes.

Pseudoscalar dominance denotes a family of saturation hypotheses for amplitudes, form factors, and susceptibilities carrying pseudoscalar quantum numbers. In its most developed contemporary form, it is a statement about the nucleon pseudoscalar density: the pion pole supplies the dominant contribution, while the lowest excited pseudoscalar states provide the minimal corrections required by analyticity, chiral symmetry, and pQCD short-distance behavior. In related settings, the same phrase can denote pion-pole saturation of the induced pseudoscalar response, low-temperature saturation of pseudoscalar susceptibilities by the pion, or more loosely the direct dynamical involvement of the pseudoscalar sector in effective interactions (Arriola et al., 2023).

1. Formal meaning and basic relations

In the nucleon axial sector, the standard decomposition of the isovector axial current introduces the axial and induced pseudoscalar form factors,

$$\langle N(p',s') \vert A_\mu\vert N(p,s) \rangle = \bar{u}_N(p',s') \bigg[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \bigg]\gamma_5 u_N(p,s),$$

while the pseudoscalar density defines

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$

These quantities are related by the nucleon-level PCAC identity

$$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$

so any claim of pseudoscalar dominance is a claim about how the pseudoscalar channel saturates this relation and the associated pole structure (Alexandrou et al., 11 Feb 2025).

Two related but distinct notions are standard. “Pion-pole dominance” (PPD) is the approximation

$G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$

where the longitudinal axial response is controlled by a single pion pole. A broader pseudoscalar-dominance statement uses

$$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$

so that the pion-nucleon form factor $G_{\pi NN}$, the residue of the pseudoscalar channel at the pion pole, and the Goldberger-Treiman relation are treated within one common framework. In this usage, the pion pole is dominant but not necessarily exhaustive, because regular non-pole pieces and excited pseudoscalar states can generate controlled deviations (Chen et al., 2021).

2. Dispersive nucleon formulation and Extended PCAC

The 2023 dispersive formulation starts from the nucleon matrix element of the pseudoscalar quark density,

$$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$

and emphasizes the analytic structure of $F_P(t)$: it is real in the spacelike region, has a pion pole at $t=m_\pi^2$, branch cuts beginning at $3\pi,5\pi,\dots$, and must fall at large Euclidean momentum in a way compatible with pQCD. The corresponding dispersion relation makes the pion pole explicit,

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$0

with the residue defining $$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$1. Neglect of the continuum yields the Goldberger-Treiman relation $$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$2, while the observed nonzero discrepancy motivates a structured contribution beyond the pion alone (Arriola et al., 2023).

The 2025 dispersive analysis recasts the same content in terms of

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$3

with

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$4

so that

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$5

This is presented as a modern realization of extended PCAC: the divergence of the axial current contains not only the pion field but also higher pseudoscalar states,

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$6

which in the large-$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$7 limit becomes an infinite tower of narrow poles. The practical claim is not unrestricted pole proliferation, but minimal pseudoscalar saturation subject to analyticity, PCAC, and pQCD asymptotics (Arriola et al., 30 Jul 2025).

3. Spectral constraints and the Goldberger-Treiman discrepancy

A central result of the modern dispersive program is that pure pion dominance is insufficient once the Goldberger-Treiman discrepancy and QCD asymptotics are imposed simultaneously. The 2025 analysis derives superconvergent sum rules from the deep-Euclidean behavior

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$8

and concludes that the spectral density must have at least one zero. Low-energy ChPT fixes the threshold behavior but contributes only

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$9

far below the observed discrepancy. The dominant correction is instead assigned to the first excited isovector pseudoscalar, $$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$0, with a negative residue $$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$1, while a Regge-like tail provides the intermediate/high-energy continuation. In this framework,

$$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$2

and the approximate relations

$$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$3

make explicit that the correction is controlled primarily by the first excited pseudoscalar state (Arriola et al., 30 Jul 2025).

The 2023 large-$$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$4-inspired minimal ansatz implements the same idea with the pion and the two lightest excited isovector pseudoscalars,

$$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$5

From the pion-pole residue it obtains

$$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$6

to be compared with the Granada-2013 scattering benchmark $$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$7, and equivalently

$$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$8

The result is interpreted as evidence that the strong pion-nucleon-nucleon vertex is almost flat: the pion pole dominates the form factor, while higher pseudoscalar resonances generate the small but necessary correction (Arriola et al., 2023).

4. Lattice QCD and continuum tests

Direct physical-point lattice calculations have produced both tension and support for pseudoscalar dominance, depending on how cutoff effects and excited states are controlled. A 2020 study found that $$G_A(Q^2)-\frac{Q^2}{4m_N^2}G_P(Q^2)=\frac{m_q}{m_N}G_5(Q^2),$$9 and $G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$0 show the expected strong low-$G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$1 enhancement, with $G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$2 and $G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$3 on the main ensemble, but the diagnostic ratios

$G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$4

show sizeable deviations from unity at small $G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$5, even after an improved excited-state treatment. The interpretation given there is that low-energy $G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$6 states enter the three-point functions in a manner not fully captured by simpler two-state analyses. At the same time, indirect use of $G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$7 together with PCAC/PPD gives phenomenologically reasonable values, $G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$8, $G_P(Q^2)\simeq \frac{4m_N^2G_A(Q^2)}{Q^2+m_\pi^2},$9, and $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$0 (Alexandrou et al., 2020).

A later continuum-limit calculation with three $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$1 twisted-mass ensembles at the physical point identifies the dominant obstruction as a lattice artifact. Using lattice spacings $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$2, multiple source-sink separations from about $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$3 fm to $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$4 fm, two-state and three-state fits, and AIC model averaging, it finds significant $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$5 effects in both $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$6 and $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$7 at finite lattice spacing, but restoration of both relations in the continuum limit: $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$8 and $$G_5(Q^2)=\frac{m_\pi^2}{Q^2+m_\pi^2}\frac{f_\pi}{m_q}G_{\pi NN}(Q^2),$$9. The quoted continuum results are

$G_{\pi NN}$0

$G_{\pi NN}$1

which align the lattice data with the standard pseudoscalar-dominance picture once discretization effects are removed (Alexandrou et al., 11 Feb 2025).

A symmetry-preserving continuum quark+diquark calculation reaches a similar conclusion from a different direction. There, exact satisfaction of PCAC requires consistent construction of the external axial and pseudoscalar currents, including the axial and pseudoscalar seagull couplings. The directly computed $G_{\pi NN}$2 is then extremely close to the PPD estimate, with the ratio $G_{\pi NN}$3 staying near unity and differing by only about $G_{\pi NN}$4 near $G_{\pi NN}$5. The same framework gives $G_{\pi NN}$6, $G_{\pi NN}$7, $G_{\pi NN}$8, and $G_{\pi NN}$9, supporting the view that pion-pole dominance is an excellent approximation when current identities are implemented exactly (Chen et al., 2021).

5. Other QCD realizations of pseudoscalar-channel saturation

In finite-temperature QCD, pseudoscalar dominance appears in the low-temperature saturation of the pseudoscalar susceptibility by the pion pole. One-loop $$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$0 ChPT gives

$$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$1

so the pseudoscalar susceptibility tracks the chiral order parameter throughout the low-$$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$2 regime. This behavior is then linked to scalar-pseudoscalar partner degeneration near the chiral transition, with $$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$3 around restoration and the scalar channel described by saturation with the thermal $$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$4 resonance (Nicola et al., 2013).

At the quark level, a Landau-gauge Schwinger-Dyson analysis finds the dressed pseudoscalar density to be large for light quarks and nonconvergent in the chiral limit. The quoted renormalized zero-momentum values are $$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$5 for $$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$6 MeV, $$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$7 for $$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$8 MeV, and $$\langle N(p')|\bar q\,\{\tau,\hat m\}\, i\gamma_5\, q|N(p)\rangle = \bar u(p')\,\{\tau,\hat m\}\, i\gamma_5\, u(p)\, F_P(t), \qquad t=q^2,$$9 for $F_P(t)$0 MeV. This behavior is explicitly interpreted as pion-pole dominance: the pseudoscalar vertex is enhanced by the ladder-generated Nambu-Goldstone mode and inherits the $F_P(t)$1 singularity expected from current algebra and PCAC (Yamanaka et al., 2014).

In the $F_P(t)$2 sector, lattice QCD yields an analogous but richer structure because spin-$F_P(t)$3 matrix elements involve four axial and two pseudoscalar form factors. Two Goldberger-Treiman-type relations are derived,

$F_P(t)$4

and the form factors $F_P(t)$5 are tested against pion-pole-dominance expectations. The reported outcome is qualitative and often quantitative support for PCAC and pion-pole dominance in the $F_P(t)$6 sector, although subdominant form factors remain statistically challenging (Alexandrou et al., 2013).

The excited pseudoscalar states that enter nucleon-dominance analyses also admit hadronic dynamical realizations. A unitarized scalar-pseudoscalar scattering study generates resonances associated with $F_P(t)$7, $F_P(t)$8, $F_P(t)$9, $t=m_\pi^2$0, and $t=m_\pi^2$1 from rescattering involving the dynamically generated $t=m_\pi^2$2 and $t=m_\pi^2$3. This suggests a concrete hadronic mechanism for the intermediate pseudoscalar spectrum that appears as $t=m_\pi^2$4, or specifically $t=m_\pi^2$5, in dispersive saturation schemes (Albaladejo et al., 2010).

6. Boundaries of the concept and relation to other dominance models

The term is not interchangeable with every departure from vector meson dominance. In an NJL-type analysis with gauge-covariant diagonalization of axial-vector–pseudoscalar mixing, the correct treatment generates direct photon–pion–quark vertices and therefore departures from naive VMD. However, the authors explicitly reject a simple “pseudoscalar-dominance” replacement of VMD: some amplitudes acquire essential non-VMD contributions, but the pion electromagnetic form factor remains unchanged at the order considered, and anomalous decays such as $t=m_\pi^2$6 and $t=m_\pi^2$7 retain a surface-term ambiguity that must be fixed experimentally (Osipov et al., 2018).

Electromagnetic form factors of pseudoscalar mesons also illustrate the limited domain of hadronic pole dominance. A full-lattice study of a light pseudoscalar meson finds that low-$t=m_\pi^2$8 behavior is well approximated by a vector-meson pole, but the form factor peels away from that pole for $t=m_\pi^2$9, and $3\pi,5\pi,\dots$0 is nearly flat between $3\pi,5\pi,\dots$1 and $3\pi,5\pi,\dots$2, with slope $3\pi,5\pi,\dots$3 at $3\pi,5\pi,\dots$4, indicating crossover toward partonic scaling. By contrast, matched simulations with $3\pi,5\pi,\dots$5 show that the spacelike pseudoscalar meson vector form factor is nearly independent of $3\pi,5\pi,\dots$6 in shape and is consistent with VMD across those color numbers. These results delimit dominance ideas: they work well in specific low-energy domains, but not as universal descriptions across all momentum scales (Koponen et al., 2017, DeGrand, 2024).

The same caution appears in proposed experimental extractions of the nucleon pseudoscalar form factor. A study of charged-current quasielastic neutrino scattering argues that polarization observables could provide the first extraction of $3\pi,5\pi,\dots$7 from neutrino data without imposing PCAC or pion-pole dominance, especially with low-energy $3\pi,5\pi,\dots$8 beams or with $3\pi,5\pi,\dots$9. In a broader conceptual sense, reviews of emergent hadron mass treat pseudoscalar mesons as the most revealing laboratory for QCD mass generation, with the identity

$$\langle N(p',s') \vert P \vert N(p,s) \rangle = G_5(Q^2)\,\bar{u}_N(p',s')\gamma_5 u_N(p,s).$$00

expressing the connection between one-body quark dressing and the two-body Goldstone bound state. This usage elevates the pseudoscalar sector because of its symmetry content, not because a single universal dominance ansatz applies to every observable (Tomalak, 2020, Raya et al., 2024).