Goldberger-Treiman Discrepancy

• Goldberger–Treiman discrepancy is defined as the deviation between the chiral-limit Goldberger–Treiman relation and its physical realization in hadron dynamics.
• Modern lattice QCD, covariant baryon chiral perturbation theory, and dispersive analyses consistently find the nucleon discrepancy at the few-percent level, contrasting with larger SU(3) violations.
• The discrepancy serves as a diagnostic tool to test chiral symmetry breaking and renormalization-scheme effects in effective field theories of strong interactions.

Searching arXiv for relevant papers on the Goldberger–Treiman discrepancy and closely related lattice/ChPT analyses. {"query":"Goldberger-Treiman discrepancy nucleon lattice chiral perturbation theory arXiv", "max_results": 10} Relevant arXiv papers include recent lattice and EFT analyses of the Goldberger–Treiman relation and discrepancy, notably (Alexandrou et al., 2023, Arriola et al., 30 Jul 2025, Alexandrou et al., 2021, Alarcon et al., 2011), and (Alarcón et al., 2011). The Goldberger–Treiman discrepancy is the measure of the deviation between the exact chiral-limit Goldberger–Treiman relation and its realization for physical hadrons. In its canonical nucleon form, the relation connects the nucleon axial charge, the pion–nucleon coupling, the pion decay constant, and the nucleon mass, and is exact only in the chiral limit under PCAC and pion-pole dominance. In contemporary usage, however, the term also covers channel-dependent generalizations in the nucleon isovector, SU(3) octet, hyperon, $N\!\to\!\Delta$, and decuplet sectors. Modern lattice QCD, covariant baryon chiral perturbation theory, and dispersive analyses all agree that the nucleon discrepancy is small, at the few-percent level, while some extensions—most notably the SU(3) octet $\eta$-channel relation—can exhibit much larger violations (Alexandrou et al., 2023, Alexandrou et al., 2021).

1. Canonical definition and basic formalism

For the standard isovector nucleon case, the Goldberger–Treiman relation is

$g_A\,M_N \simeq F_\pi\,g_{\pi NN},$

with $g_A \equiv G_A(0)$, $M_N$ the nucleon mass, $F_\pi$ the pion decay constant, and $g_{\pi NN}$ the pion–nucleon coupling. A standard definition of the discrepancy is

$\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$

If the physical-point relation were exact, $\Delta_{\rm GT}=0$. In the nucleon form-factor language, the axial current matrix element is decomposed as

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

and PCAC implies the finite-$\eta$0 relation

$\eta$1

At the pion pole this gives $\eta$2, while the chiral limit $\eta$3 recovers the familiar low-energy theorem (Alexandrou et al., 2023).

A second, equivalent viewpoint uses the pseudoscalar form factor. In dispersive formulations one writes

$\eta$4

where $\eta$5 is a pion-pole–removed pseudoscalar form factor. This representation is useful because it expresses the discrepancy directly in terms of analyticity and spectral information in the pseudoscalar channel (Arriola et al., 30 Jul 2025).

The literature also contains sign-convention differences. In some covariant baryon ChPT analyses the strong coupling is parameterized as

$\eta$6

while elsewhere the discrepancy is defined with the minus sign convention above. The physical content is the same: $\eta$7 measures the relative mismatch between the axial and strong couplings once explicit chiral symmetry breaking is included (Alarcon et al., 2011).

2. Chiral perturbation theory and $\eta$8 scattering extractions

In baryon ChPT the discrepancy first appears beyond leading order and is controlled, at $\eta$9, by the low-energy constant $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$0: $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$1 This makes the discrepancy a sensitive diagnostic of how covariant formulations handle analytic loop contributions and power-counting–breaking terms (Alarcón et al., 2011).

A central development was the comparison between Infrared Regularization (IR) and Extended-On-Mass-Shell (EOMS) in relativistic $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$2 scattering analyses. Using phase-shift fits, a tree-level extraction gave

$g_A\,M_N \simeq F_\pi\,g_{\pi NN},$3

When the full $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$4 loop contributions were implemented in IR, the effective GT violation became very large: $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$5 at $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$6 GeV and $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$7 at $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$8 GeV for a representative fit. EOMS, by contrast, yielded a small GT deviation compatible with phenomenology, around $g_A\,M_N \simeq F_\pi\,g_{\pi NN},$9, while remaining scale independent and free of the unphysical cut that limits IR (Alarcón et al., 2011).

The EOMS reformulation was then presented as solving the long-standing covariant-ChPT GT problem. In that framework the loop contribution to the discrepancy was reported to be very small, about $g_A \equiv G_A(0)$0, with fitted total deviations of $g_A \equiv G_A(0)$1 from KA85 and $g_A \equiv G_A(0)$2 from WI08 at $g_A \equiv G_A(0)$3. The broader conclusion was that the large covariant discrepancy was not a problem of baryon ChPT itself, but of the IR prescription (Alarcon et al., 2011).

Within this EFT perspective, the Goldberger–Treiman discrepancy is therefore not merely a phenomenological number. It is a renormalization-scheme stress test for relativistic baryon EFT, particularly for the treatment of analytic loop terms, scale dependence, and the matching between axial and strong vertices.

3. Lattice QCD determination in the nucleon channel

A continuum-limit lattice determination using three $g_A \equiv G_A(0)$4 twisted-mass Clover ensembles at approximately physical quark masses found that both the PCAC and pion-pole-dominance relations are satisfied in the continuum limit. The same study reported

$g_A \equiv G_A(0)$5

$g_A \equiv G_A(0)$6

The Goldberger–Treiman discrepancy was extracted primarily from the continuum-extrapolated ratio $g_A \equiv G_A(0)$7, yielding

$g_A \equiv G_A(0)$8

summarized as a discrepancy of about $g_A \equiv G_A(0)$9, together with

$M_N$0

The same calculation found $M_N$1, so pion-pole dominance holds at the pole within about $M_N$2 (Alexandrou et al., 2023).

This result places the nucleon discrepancy squarely in the few-percent regime expected from ChPT. It also sharpens the conceptual point that the discrepancy is best interpreted as a finite-mass correction to a chiral low-energy theorem rather than as evidence for a failure of PCAC. In the continuum-extrapolated lattice data, PCAC is restored at the form-factor level; the nonzero discrepancy is then the residual physical effect of explicit chiral symmetry breaking.

The lattice result also aligns with the isovector comparison quoted in later SU(3) analyses, where the isovector violation was described as “about 2%.” That comparison has become important because it provides a controlled baseline against which larger discrepancies in other flavor channels can be assessed (Alexandrou et al., 2021).

4. Dispersive, pseudoscalar-dominance, and QCD sum-rule formulations

A different line of work formulates the discrepancy through analyticity and dispersion relations for the nucleon pseudoscalar form factor. In the pseudoscalar-meson-dominance approach based on an Extended PCAC, the spectral function in the $M_N$3 channel must change sign and hence must have at least one zero. In a minimal resonance-saturation picture with $M_N$4, $M_N$5, and $M_N$6, one obtains

$M_N$7

The same framework emphasizes that the discrepancy is driven primarily by the resonance region and is consistent with almost flat strong pion–nucleon–nucleon vertices (Arriola et al., 2023).

A subsequent dispersive analysis of the pseudoscalar density matrix element combined ChPT at low energies, resonances at intermediate energies, Regge power-like behavior, and pQCD asymptotics. It derived three sum rules for the pseudoscalar spectral function and again showed that the spectral density must have at least one zero. With a model built around the $M_N$8 plus a Regge tail, it reported

$M_N$9

The same work stressed the identity

$F_\pi$0

which is particularly convenient for lattice QCD analyses of the pseudoscalar channel (Arriola et al., 30 Jul 2025).

QCD sum rules supply a third formulation. Replacing Borel kernels by polynomial kernels tailored to suppress baryonic resonance contributions, a meson–baryon coupling analysis obtained

$F_\pi$1

and used the standard phenomenological estimate

$F_\pi$2

In that framework the discrepancy remains small in the nucleon sector but becomes substantially larger in strange channels, while still satisfying the Dashen–Weinstein relation rather well (Nasrallah, 2021).

Taken together, these dispersive and sum-rule approaches reinforce a consistent picture: in the nucleon channel the discrepancy is positive and small, typically at the $F_\pi$3 level in dispersive analyses and at the few-percent level in lattice and phenomenological determinations. This suggests that the canonical GT relation is numerically robust, but not exact, in physical QCD.

5. SU(3) extensions: octet, strange, and hyperon discrepancies

The best-known counterexample to the “few-percent” intuition is the SU(3) octet extension involving the $F_\pi$4 channel. For the octet current

$F_\pi$5

the analogue of the GT relation is

$F_\pi$6

and the corresponding discrepancy is defined as

$F_\pi$7

Using $F_\pi$8 twisted-mass fermions at the physical point, one lattice study found

$F_\pi$9

and

$g_{\pi NN}$0

summarized as

$g_{\pi NN}$1

The same calculation reported SU(3) breaking of up to $g_{\pi NN}$2 for the axial form factors and up to $g_{\pi NN}$3 for the induced pseudoscalar form factors, and explicitly contrasted the octet $g_{\pi NN}$4 violation with the $g_{\pi NN}$5 isovector result. It attributed the large octet discrepancy to the large $g_{\pi NN}$6-meson mass, sizable SU(3) breaking, and the enhanced role of disconnected strange-quark contributions in $g_{\pi NN}$7 (Alexandrou et al., 2021).

Hyperon GT discrepancies are also substantially larger than the nucleon one. In a QCD sum-rule analysis of meson–baryon couplings, the strange-channel GT discrepancies were defined as

$g_{\pi NN}$8

with quoted values

$g_{\pi NN}$9

Using

$\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$0

the same work emphasized that the Dashen–Weinstein relation is quite well satisfied, despite the much larger strange-channel discrepancies (Nasrallah, 2021).

A common misconception is therefore that the Goldberger–Treiman discrepancy is universally a few-percent quantity. That is correct only for the canonical SU(2) nucleon relation. Once the relation is extended to SU(3) octet or hyperon channels, the heavier pseudoscalar masses and stronger flavor breaking can generate violations at the $\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$1, $\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$2, or even $\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$3 level.

6. Off-diagonal, $\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$4-sector, and decuplet generalizations

The term is also used for off-diagonal axial transitions. In neutrino-induced weak pion production, the off-diagonal GT relation connects the $\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$5 transition form factor $\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$6 to the strong $\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$7 coupling: $\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$8 A fit to ANL and BNL neutrino–deuteron data including deuteron corrections and chiral non-resonant background terms gave

$\Delta_{\rm GT}=1-\frac{g_A M_N}{g_{\pi NN}F_\pi}.$9

which corresponds to a difference from the off-diagonal GT value at about the $\Delta_{\rm GT}=0$0 level and to a discrepancy of roughly $\Delta_{\rm GT}=0$1. The same analysis noted that background terms reduce $\Delta_{\rm GT}=0$2 by about $\Delta_{\rm GT}=0$3, whereas deuteron effects increase it by about $\Delta_{\rm GT}=0$4 (Hernandez et al., 2010).

In lattice QCD, the $\Delta_{\rm GT}=0$5 axial-vector and pseudoscalar form factors lead to two $\Delta_{\rm GT}=0$6-sector GT relations,

$\Delta_{\rm GT}=0$7

The corresponding GT ratios were found to be close to unity for the dominant channel at $\Delta_{\rm GT}=0$8, and the authors concluded that these $\Delta_{\rm GT}=0$9 GT relations are satisfied at the same level of accuracy as in the nucleon case (Alexandrou et al., 2011).

At the level of the baryon decuplet chiral Lagrangian, the GT relation connects the poorly known decuplet axial coupling $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$0 to decuplet–decuplet–Goldstone couplings. The sign and size of $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$1 remain under debate: large-$$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$2 and quark-model estimates suggest $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$3 with the same sign as $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$4 and $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$5, whereas some phenomenological analyses infer the opposite sign for the $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$6 coupling. A proposed “Wu-type” probe is the semileptonic decay $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$7. At leading order, a feasibility estimate gave branching ratios of $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$8 for $$\langle N(p',s') \vert A_\mu \vert N(p,s) \rangle = \bar u_N(p',s') \left[ \gamma_\mu G_A(Q^2)-\frac{Q_\mu}{2m_N}G_P(Q^2) \right]\gamma_5 u_N(p,s),$$9 and an electron forward–backward asymmetry of $\eta$00 for $\eta$01 and $\eta$02 for $\eta$03. In this sector the discrepancy is not yet a settled number; it is instead part of an active program to determine $\eta$04 directly and then test decuplet GT relations beyond leading order (Bertilsson et al., 2023).

A broader implication is that “Goldberger–Treiman discrepancy” no longer denotes a single universal observable. It refers to a family of symmetry-breaking diagnostics whose magnitude depends strongly on the current, flavor structure, hadronic channel, and extraction scheme. In the nucleon isovector channel the discrepancy is a precision few-percent quantity; in SU(3) octet and some transition channels it becomes a probe of heavy-meson kinematics, disconnected contributions, and the limits of naive symmetry extension.