Flavor-Singlet Axial Charge in Baryons

• Flavor-singlet axial charge is defined as the net quark-spin contribution to baryon spin, combining Δu, Δd, and Δs (and Δc in SU(4)).
• Its extraction requires rigorous treatment of disconnected diagrams and careful renormalization due to the axial anomaly in QCD.
• Lattice-QCD and chiral model analyses consistently indicate a nucleon quark-spin contribution (ΔΣ) of roughly 0.29–0.36 at 2 GeV.

The flavor-singlet axial charge is the forward matrix element of the flavor-singlet axial current and, in the standard spin decomposition, measures the net quark-spin contribution carried by the relevant flavor set. For the nucleon in an $SU(3)$ basis it is $g_A^{(0)}=\Delta u+\Delta d+\Delta s$, while in $SU(4)$ baryon analyses it is extended to $g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$. In QCD the singlet current is distinguished from non-singlet axial currents by the axial anomaly, so the corresponding charge is scheme- and scale-dependent; in hadron models it is often treated instead as the constituent-level quark-spin content of the baryon (Gupta et al., 2018, Ahmed et al., 2021, Dahiya et al., 2023).

1. Operator definition and normalization

In continuum notation, the flavor-diagonal axial current for quark flavor $q$ is

$A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$

and its forward nucleon matrix element defines the flavor-diagonal axial charge through

$$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$

The flavor-singlet axial charge is then

$g_A^{(0)}=\Delta u+\Delta d+\Delta s,$

and the quark-spin contribution to the proton spin is $\frac12\Delta\Sigma=\frac12 g_A^{(0)}$ (Gupta et al., 2018).

The same object can be expressed as the zero-momentum-transfer limit of an axial form factor. In baryon form-factor notation,

$$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$

with $g_A^{(0)}=\Delta u+\Delta d+\Delta s$0, and the axial charge is $g_A^{(0)}=\Delta u+\Delta d+\Delta s$1. For the singlet channel this gives $g_A^{(0)}=\Delta u+\Delta d+\Delta s$2 (Dahiya et al., 2023).

In $g_A^{(0)}=\Delta u+\Delta d+\Delta s$3 treatments of light and charmed baryons, the singlet current is associated with the flavor-space identity, written through $g_A^{(0)}=\Delta u+\Delta d+\Delta s$4, and the diagonal axial combinations are

$g_A^{(0)}=\Delta u+\Delta d+\Delta s$5

Within that convention, the flavor-singlet axial charge is the total quark-spin content of the baryon, now including charm (Dahiya et al., 2023).

2. Axial anomaly, renormalization, and scheme dependence

The defining complication of the flavor-singlet axial current in QCD is the axial anomaly. In dimensional regularization with a non-anticommuting $g_A^{(0)}=\Delta u+\Delta d+\Delta s$6, the renormalized singlet current is written as

$g_A^{(0)}=\Delta u+\Delta d+\Delta s$7

with $g_A^{(0)}=\Delta u+\Delta d+\Delta s$8. The renormalized anomaly equation is

$g_A^{(0)}=\Delta u+\Delta d+\Delta s$9

and the equality $SU(4)$0 was verified explicitly to 4-loop order (Ahmed et al., 2021).

Because of this anomaly, the singlet current has a non-zero anomalous dimension and the singlet axial charge acquires scale dependence. This is why lattice determinations quote $SU(4)$1 in a definite scheme and at a definite scale, typically $SU(4)$2 at $SU(4)$3 GeV (Gupta et al., 2018). A modern singlet-form-factor calculation emphasizes that the singlet axial current is anomalous, that $SU(4)$4 is scale dependent and differs numerically from the non-singlet axial renormalization factor, and that the conversion to $SU(4)$5 uses the known singlet anomalous dimensions from Larin (Barone et al., 7 May 2026).

Different lattice programs have handled this issue at different levels of precision. One high-statistics $SU(4)$6-flavor calculation renormalized flavor-diagonal axial charges with the isovector $SU(4)$7 under the assumption that the singlet–nonsinglet difference is negligible at its precision, noting that the difference vanishes at 2-loop order (Gupta et al., 2018). A later clover-on-HISQ update implemented a full $SU(4)$8 flavor-mixing renormalization matrix in the $SU(4)$9 basis and found the off-diagonal $g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$0 entries smaller than $g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$1, so flavor mixing is present but numerically small (Park et al., 2024).

A common misconception is to treat the flavor-singlet axial charge as if it were on the same footing as $g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$2. The published renormalization analyses show that this is not correct: the non-singlet axial current is conserved in the massless limit, whereas the singlet current is anomalous and therefore requires separate renormalization logic (Ahmed et al., 2021).

3. Proton spin decomposition and lattice-QCD determinations

For the proton, the flavor-singlet axial charge enters the spin sum rule through

$g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$3

A direct lattice determination with $g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$4 flavor HISQ ensembles and clover valence quarks found

$g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$5

which implies

$g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$6

in the $g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$7 scheme at $g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$8 GeV. That result lies within the COMPASS interval $g_A^{0}=\Delta u+\Delta d+\Delta s+\Delta c$9 (Gupta et al., 2018).

A later lattice-QCD determination of the singlet axial form factor on $q$0 CLS ensembles obtained

$q$1

again in $q$2, together with

$q$3

That analysis states that the result suggests the intrinsic quark spin contributes roughly $q$4 to the proton spin, with the remaining fraction supplied by gluon and orbital angular momentum contributions (Barone et al., 7 May 2026).

A clover-fermion update reported preliminary flavor-diagonal charges whose singlet sum falls in the range

$q$5

across four analysis strategies, with dominant systematics attributed to excited-state contamination and chiral/continuum extrapolation rather than renormalization or flavor mixing (Park et al., 2024).

This body of work suggests that current lattice determinations cluster around $q$6 at $q$7 GeV, while consistently finding a small negative strange contribution.

Determination	Singlet result	Context
(Gupta et al., 2018)	$q$8	$q$9 flavor QCD, $A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$0
(Barone et al., 7 May 2026)	$A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$1	Full singlet form factor, $A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$2 lattice QCD
(Park et al., 2024)	$A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$3	Preliminary clover-on-HISQ update

4. Lattice extraction: disconnected diagrams, extrapolations, and form factors

A distinctive technical feature of the flavor-singlet channel is the necessity of disconnected diagrams. For flavor-diagonal operators, $A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$4 and $A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$5 each receive connected and disconnected contributions, whereas $A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$6 is purely disconnected in the proton because there are no strange valence quarks in the interpolating field (Gupta et al., 2018). Since disconnected contributions do not cancel in the singlet combination, a reliable determination of $A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$7 requires explicit control of those diagrams.

In the $A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$8-flavor HISQ calculation, disconnected loops were computed using stochastic estimators with the truncated solver method and coherent source techniques, while excited-state contamination was controlled by using $A_\mu^q(x)=\bar q(x)\gamma_\mu\gamma_5 q(x),$9–$$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$0 source–sink separations and multistate fits. The continuum extrapolation employed the CCFV ansatz

$$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$1

with the disconnected terms analyzed on fewer ensembles and with finite-volume effects neglected for those pieces after being found small in the connected sector (Gupta et al., 2018).

The clover-on-HISQ update used an empirical chiral–continuum–finite-volume form

$$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$2

and compared two renormalization strategies, $$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$3 and $$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$4, together with both standard and $$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$5-augmented excited-state fits. The off-diagonal flavor-mixing terms were found to be small, and the continuum-limit singlet results from $$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$6 and $$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$7 were consistent within errors (Park et al., 2024).

The singlet-form-factor calculation on CLS ensembles combined several ingredients that are now standard for this channel: the summation method for connected insertions, plateau fits with window averaging for disconnected insertions, a truncated $$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$8-expansion in $$\langle p,s|A_\mu^q(0)|p,s\rangle = 2\,g_A^q\,s_\mu, \qquad g_A^q\equiv \Delta q.$$9, and AIC model averaging over chiral, continuum, and finite-volume fit variants. Ensemble by ensemble, the singlet form factor was obtained through

$g_A^{(0)}=\Delta u+\Delta d+\Delta s,$0

because that route was less noisy than a fully direct singlet extraction (Barone et al., 7 May 2026).

Methodologically, the central point is that the flavor-singlet axial charge is not a simple by-product of isovector analyses. It is a disconnected-dominated observable whose extraction is coupled to renormalization in the anomalous singlet channel.

5. Constituent-quark and chiral models

Low-energy quark models usually identify the singlet axial charge with the quark-spin content of the baryon at a constituent scale and do not include explicit gluon-spin contributions or an explicit treatment of the axial $g_A^{(0)}=\Delta u+\Delta d+\Delta s,$1 anomaly. This yields a framework-dependent quantity that is useful for interpreting spin–flavor structure but should not be identified uncritically with the $g_A^{(0)}=\Delta u+\Delta d+\Delta s,$2 lattice quantity (Wang et al., 2021).

In an extended chiral constituent quark model for the proton with explicit $g_A^{(0)}=\Delta u+\Delta d+\Delta s,$3 Fock components, two parameter sets were studied. Set I gives

$g_A^{(0)}=\Delta u+\Delta d+\Delta s,$4

hence

$g_A^{(0)}=\Delta u+\Delta d+\Delta s,$5

Set II gives

$g_A^{(0)}=\Delta u+\Delta d+\Delta s,$6

hence

$g_A^{(0)}=\Delta u+\Delta d+\Delta s,$7

The same study states that the probabilities of the intrinsic five-quark Fock components in the proton wave function should be $g_A^{(0)}=\Delta u+\Delta d+\Delta s,$8, and that Set II is much closer to lattice-QCD results for $g_A^{(0)}=\Delta u+\Delta d+\Delta s,$9 (Wang et al., 2021).

For the octet baryons more generally, an extended chiral constituent quark model with compact five-quark components reports that the singlet axial charges of the octet baryons fall in the range $\frac12\Delta\Sigma=\frac12 g_A^{(0)}$0, and emphasizes that the light-quark spins $\frac12\Delta\Sigma=\frac12 g_A^{(0)}$1 and $\frac12\Delta\Sigma=\frac12 g_A^{(0)}$2 in the $\frac12\Delta\Sigma=\frac12 g_A^{(0)}$3 baryon can be small but negative, whereas they exactly vanish in the traditional three-quark model (Qi et al., 2022).

In an $\frac12\Delta\Sigma=\frac12 g_A^{(0)}$4 chiral constituent quark model including explicit charm, the proton result at $\frac12\Delta\Sigma=\frac12 g_A^{(0)}$5 is

$\frac12\Delta\Sigma=\frac12 g_A^{(0)}$6

so

$\frac12\Delta\Sigma=\frac12 g_A^{(0)}$7

The same framework predicts that singlet axial charges in charmed baryons can become substantially larger and often charm dominated; for example the doubly charmed triplet gives $\frac12\Delta\Sigma=\frac12 g_A^{(0)}$8, while the spin-$\frac12\Delta\Sigma=\frac12 g_A^{(0)}$9 $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$0 gives $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$1 with $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$2 (Dahiya et al., 2023).

These model results do not contradict the lower lattice-QCD nucleon values at $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$3 GeV; rather, they encode different physics input. A plausible implication is that the size of $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$4 is highly sensitive to whether sea-quark dynamics are represented by explicit five-quark or Goldstone-boson fluctuations, and to whether anomalous gluonic effects are treated explicitly or absorbed into model parameters.

6. Extensions to charmed baryons, decuplet states, and lattice anomaly constructions

Beyond the nucleon, the flavor-singlet axial charge has been studied as a full form factor in heavy-flavor baryon multiplets. In the $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$5 chiral constituent quark model, the singlet axial form factor is parameterized by the standard dipole form

$$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$6

and all axial-vector charges decrease with increasing $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$7. For singly charmed baryons, the singlet and charmed charges differ because the charm quark in the constituent structure leads to a significant charm spin polarization; for doubly charmed baryons, the charmed charge dominates over singlet, triplet, and octet charges because the constituent structure includes two charm quarks (Dahiya et al., 2023).

A different pattern appears in the chiral quark-soliton model for the baryon decuplet. There the singlet axial-vector form factors $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$8 and $$\langle B(p')|A_\mu^a|B(p)\rangle = \bar u(p') \left[ \gamma_\mu\gamma_5\,G_A^a(Q^2) + \frac{q_\mu}{2M_B}\gamma_5\,G_P^a(Q^2) \right] u(p),$$9 receive no leading $g_A^{(0)}=\Delta u+\Delta d+\Delta s$00 contribution; the nonzero singlet charge arises from rotational $g_A^{(0)}=\Delta u+\Delta d+\Delta s$01 corrections and from operator $g_A^{(0)}=\Delta u+\Delta d+\Delta s$02-breaking terms proportional to $g_A^{(0)}=\Delta u+\Delta d+\Delta s$03, while the wave-function $g_A^{(0)}=\Delta u+\Delta d+\Delta s$04-breaking contribution vanishes. The resulting $g_A^{(0)}=\Delta u+\Delta d+\Delta s$05 values are almost the same for all decuplet baryons, and the linear $g_A^{(0)}=\Delta u+\Delta d+\Delta s$06 effects on $g_A^{(0)}=\Delta u+\Delta d+\Delta s$07 are almost negligible for both $g_A^{(0)}=\Delta u+\Delta d+\Delta s$08 and $g_A^{(0)}=\Delta u+\Delta d+\Delta s$09 (Jun et al., 2020).

The anomaly aspect of the flavor-singlet axial charge has also been examined directly in lattice field theory. A study of minimally doubled fermions shows that the true flavor-singlet axial symmetry is non-local in time and is broken by gauge interactions, so the associated singlet axial current is not conserved; recovering the correct anomaly requires fine tuning of both the lattice action and the axial current (Tiburzi, 2010). A more recent flavored lattice Schwinger model constructs an exact, gauge-invariant lattice axial charge $g_A^{(0)}=\Delta u+\Delta d+\Delta s$10 and derives the continuum-limit anomaly equation

$g_A^{(0)}=\Delta u+\Delta d+\Delta s$11

with the factor of $g_A^{(0)}=\Delta u+\Delta d+\Delta s$12 reflecting the doubled fermionic content, while restriction to the $g_A^{(0)}=\Delta u+\Delta d+\Delta s$13 sector reproduces the standard single-flavor result (Bakircioglu, 14 Apr 2026).

Taken together, these extensions show that the flavor-singlet axial charge sits at the intersection of spin decomposition, flavor structure, and anomaly physics. In hadron phenomenology it tracks the total quark-spin content of a chosen flavor basis; in QCD proper it is an anomalous, renormalized observable whose precise value depends on disconnected contributions, flavor mixing, and the renormalization scheme.