Parity Projected Sum Rules in QCD

• Parity projected sum rules are techniques that separate positive- and negative-parity contributions in baryonic QCD correlators using specific chiral-even and chiral-odd Dirac structures.
• The methods utilize rest-frame projectors, covariant combinations like √sρ₁±ρ₀, and Lorentz structure selection to isolate desired pole terms while eliminating unwanted parity contamination.
• These approaches have been applied to diverse systems—including nucleons, heavy baryons, pentaquarks, and hybrid baryons—and involve detailed procedures such as Borel transformation and continuum subtraction.

Parity projected sum rules are sum-rule constructions that separate positive- and negative-parity contributions carried by the same interpolating current. In baryonic QCD sum rules, this separation is needed because local or composite interpolating currents generically couple to both parities, with the opposite-parity coupling differing by a $\gamma_5$ insertion. The central mechanism is to exploit the different sign patterns of chiral-even and chiral-odd Dirac structures, or of selected Lorentz structures, so that linear combinations such as $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$, rest-frame projectors $P_\pm=(1\pm \gamma_0)/2$, or specially chosen combinations of invariant amplitudes eliminate unwanted parity contamination and isolate a definite pole contribution (Ohtani et al., 2012, Su et al., 2024, Wang, 2018). In light-cone QCD sum rules the same logic can be implemented through structure selection and linear-algebraic elimination of unwanted pole terms, including opposite-parity transitions and diagonal contributions from the wrong parity sector (Aliev et al., 2015, Aliev et al., 2014).

1. Formal basis of parity separation

In spin-$1/2$ channels, the starting point is the standard Dirac decomposition of a two-point correlator,

$\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$

or, in the rest frame, $\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$ (Su et al., 2024, Ohtani et al., 2012). The hadronic pole expansion then exhibits the characteristic parity sign flip in the scalar Dirac structure. For example, the standard pole model used in several analyses is

$\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$

or equivalent sign conventions depending on the current choice and the targeted parity channel (Su et al., 2024, Wang, 2018). The same current therefore couples to both parities, with matrix elements of the form

$$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$

or, in another convention,

$$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$

(Duan et al., 2024, Wang, 2018).

The practical consequence is that the chiral-even structure, proportional to $\slashed{q}$, receives same-sign contributions from both parities, whereas the chiral-odd structure, proportional to the identity, receives opposite-sign contributions (Su et al., 2024). This makes the combinations

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$0

the canonical parity-projected spectral densities in many baryonic analyses (Su et al., 2024, Wang, 2018). In forward-correlation formulations one may instead use the explicit rest-frame projectors

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$1

so that

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$2

with parity-separated spectral functions $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$3 (Ohtani et al., 2012). In HQET, parity projection becomes especially clean because anti-heavy-baryon poles at $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$4 are absent; one defines

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$5

with $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$6 in the rest frame (Nishikawa et al., 2024).

This sign structure is the common algebraic core of parity projected sum rules across conventional baryons, heavy baryons, pentaquarks, and hybrid baryons (Duan et al., 2024, Chen et al., 24 Jul 2025).

2. Structure selection and Lorentz projection

Beyond the simple $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$7 construction, parity projection can also be realized by choosing a set of Lorentz structures whose hadronic coefficients differ between parity channels. In light-cone QCD sum rules for negative-parity heavy-baryon magnetic moments, the correlator in an external electromagnetic field is decomposed over

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$8

and the corresponding invariant amplitudes are combined so that the negative-parity diagonal contribution is isolated while the positive-parity diagonal and opposite-parity transition terms are eliminated algebraically (Aliev et al., 2015). In that construction, the hadronic representation contains coefficients $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$9, $P_\pm=(1\pm \gamma_0)/2$0, $P_\pm=(1\pm \gamma_0)/2$1, and $P_\pm=(1\pm \gamma_0)/2$2 for positive-to-positive, negative-to-negative, and opposite-parity transitions, and the chosen four-structure basis yields a closed linear system from which the negative-parity diagonal term $P_\pm=(1\pm \gamma_0)/2$3 can be solved unambiguously (Aliev et al., 2015).

An analogous elimination mechanism is used in octet electromagnetic transition LCSR. There the unwanted diagonal positive-parity contributions appear in specific invariant amplitudes, and the combination of sum rules derived from different Lorentz structures removes them (Aliev et al., 2014). The paper explicitly emphasizes that “unwanted contributions of the diagonal transitions among positive parity octet baryons are eliminated by combining the sum rules derived from different Lorentz structures” (Aliev et al., 2014).

For higher-spin currents, parity projection is typically coupled to spin projection. In the $P_\pm=(1\pm \gamma_0)/2$4-wave $P_\pm=(1\pm \gamma_0)/2$5 analysis, the spin-$P_\pm=(1\pm \gamma_0)/2$6 correlator is projected onto the transverse tensor

$P_\pm=(1\pm \gamma_0)/2$7

and only then is the Dirac decomposition

$P_\pm=(1\pm \gamma_0)/2$8

used to form $P_\pm=(1\pm \gamma_0)/2$9 (Su et al., 2024). In doubly charmed pentaquarks, the spin-$1/2$0 and spin-$1/2$1 channels are isolated through the $1/2$2 and $1/2$3 structures, respectively, after which the same parity-projected combinations are applied to the invariant amplitudes (Duan et al., 2024). Hidden-charm $1/2$4 pentaquark analyses adopt the same strategy, with $1/2$5 for $1/2$6 and $1/2$7 for $1/2$8, combined with

$1/2$9

to isolate parity (Wang et al., 24 Aug 2025).

This suggests that parity projection is not a single formula but a family of algebraic procedures. The common requirement is that the selected structures must differentiate the sign pattern induced by $\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$0 or by parity-dependent pole terms strongly enough that the unwanted sector can be canceled.

3. Borel transformation, continuum subtraction, and OPE organization

Once the parity-separated spectral density has been constructed, the remaining machinery is the standard QCD sum-rule matching between hadronic and OPE sides. In the parity-projected $\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$1 study, the Borel-transformed sum rules are

$\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$2

with the associated mass extraction

$\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$3

(Su et al., 2024). The same form is used for triply charmed pentaquarks, with $\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$4 and an explicit factor $\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$5 multiplying the scalar QCD spectral density to homogenize dimensions: $\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$6 (Wang, 2018).

Continuum subtraction is performed by quark–hadron duality in all these constructions. In light-cone heavy-baryon magnetic-moment sum rules, the double Borel transform in $\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$7 and $\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$8 suppresses higher states, and continuum subtraction is implemented by replacing the upper integration limit by $\Pi(q)=\Pi_1(q^2)\slashed{q}+\Pi_0(q^2)\,,$9 in the spectral densities (Aliev et al., 2015). In the pentaquark studies, the working windows are fixed by two standard diagnostics: OPE convergence and pole contribution. For $\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$0, the criteria were

$\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$1

(Su et al., 2024). In doubly charmed pentaquarks the analogous quantities are denoted $\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$2 and $\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$3, with the lower edge of the Borel window set by small $\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$4 and the upper edge by sizable pole contribution (Duan et al., 2024). Hidden-charm $\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$5 pentaquarks additionally introduce a contamination metric

$\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$6

to quantify the positive-parity contamination that would be present in a non-projected treatment; the reported values $\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$7–$\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$8 indicate sizable contamination in traditional sum rules (Wang et al., 24 Aug 2025).

The OPE itself varies by application but follows a common pattern. Baryon and pentaquark analyses include perturbative terms, quark condensates, gluon condensates, mixed condensates, and higher-dimensional factorized products (Duan et al., 2024, Wang, 2018). The hidden-charm $\Pi(q_0)=\gamma_0\Pi_1(q_0)+\Pi_2(q_0)$9 analysis extends the OPE to dimension 13 and states that the $\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$0 terms, accompanied by powers of $\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$1, are crucial to stabilize the Borel window (Wang et al., 24 Aug 2025). In the nucleon forward-correlation treatment, first-order $\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$2 corrections are included, but a phase-rotated Gaussian kernel is used to suppress the large perturbative corrections and the continuum simultaneously (Ohtani et al., 2012).

4. Implementations in heavy baryons, nucleons, and HQET

A compact way to compare representative realizations is the following.

System	Projection mechanism	Characteristic feature
Nucleon	Forward correlator with $\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$3	Phase-rotated Gaussian kernel suppresses large $\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$4 corrections (Ohtani et al., 2012)
Heavy baryons in HQET	$\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$5	Negative-parity sum rules receive no chiral-odd dimension-3 terms (Nishikawa et al., 2024)
Negative-parity heavy-baryon magnetic moments	Four Lorentz structures and linear elimination of $\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$6	Isolates negative-parity diagonal magnetic moment (Aliev et al., 2015)
$\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$7	Spin projection plus $\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$8	Derivative currents favor negative-parity $\Pi(q)\approx \lambda_+^2 \frac{\slashed{q}+M_+}{q^2-M_+^2} +\lambda_-^2 \frac{\slashed{q}-M_-}{q^2-M_-^2}+\cdots$9-wave states (Su et al., 2024)

In the nucleon case, parity projection is reformulated using the forward-time, or “old-fashioned,” correlator,

$$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$0

leading to

$$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$1

(Ohtani et al., 2012). The resulting sum rule, after phase rotation of the Gaussian kernel, is reported to be dominated by the term containing the chiral condensate of dimension 3 (Ohtani et al., 2012). This formulation was motivated by the need to calculate higher-order OPE terms unambiguously.

In HQET heavy-baryon sum rules, the parity projection is sharpened by the heavy-quark limit. The correlator

$$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$2

contains only heavy-baryon poles at positive $$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$3, and parity is separated with

$$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$4

A distinctive conclusion of this formulation is that in the heavy-quark limit the chiral odd condensates do not contribute to the negative parity states (Nishikawa et al., 2024). The paper attributes this to the fact that the non-diagonal correlators that generate chiral-odd condensates project purely onto positive parity because $$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$5 (Nishikawa et al., 2024). The explicit NLO $$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$6 corrections to the dimension-0 and dimension-3 terms are reported to reduce the predicted masses and improve stability, especially for negative-parity states (Nishikawa et al., 2024).

The light-cone heavy-baryon magnetic-moment analysis illustrates a different use of parity projection: not only to separate parity partners in masses, but to isolate definite-parity electromagnetic observables when the interpolating current couples to both positive- and negative-parity baryons (Aliev et al., 2015). The final parity-projected magnetic-moment expression is written in terms of the Borel-transformed OPE amplitudes $$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$7 and residues $$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$8, after eliminating all unwanted opposite-parity terms through the chosen Lorentz basis (Aliev et al., 2015).

5. Applications to pentaquarks, hybrid baryons, and $$\langle 0|J|X_{1/2}^-\rangle=f_-\,u(p),\qquad \langle 0|J|X_{1/2}^+\rangle=f_+\,\gamma_5\,u(p),$$9

Parity projected sum rules have been applied to multiquark channels with several spin assignments. In doubly charmed pentaquarks, the method was used for $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$0 and $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$1 channels with $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$2, $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$3, and $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$4 (Duan et al., 2024). The reported bound negative-parity channels are $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$5, $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$6, $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$7, $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$8, and $$\langle 0|J|P^-(p)\rangle=\lambda_-\,u(p),\qquad \langle 0|J|P^+(p)\rangle=\lambda_+\,i\gamma_5\,u(p)$$9, while the positive-parity bound channels are $\slashed{q}$0, $\slashed{q}$1, and $\slashed{q}$2 (Duan et al., 2024). The same paper emphasizes that the triply charged $\slashed{q}$3 and neutral $\slashed{q}$4 would definitely be pentaquark states due to their exotic charges (Duan et al., 2024).

For triply charmed pentaquarks, parity projection is described as an exact separation at the level of spectral densities (Wang, 2018). The extracted masses are

$\slashed{q}$5

with the negative-parity sum rule dominated by the quark condensate of dimension 3 and more sensitive to the energy scale $\slashed{q}$6 than the positive-parity one (Wang, 2018).

In hidden-charm $\slashed{q}$7 pentaquarks with $\slashed{q}$8, the method is stated to distinguish contributions of the pentaquark states with negative parity from positive parity unambiguously and to obtain clean QCD sum rules for the pentaquark states with negative parity (Wang et al., 24 Aug 2025). The paper identifies a $\slashed{q}$9 assignment for $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$00 with the predicted mass

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$01

for one of the antisymmetrized diquark–diquark–antiquark currents, and finds several negative-parity assignments near $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$02–$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$03 GeV compatible with $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$04 (Wang et al., 24 Aug 2025).

Hybrid baryons provide another distinct application. Using parity-projected QCD sum rules, the lowest-lying hybrid baryons are predicted to be the negative-parity $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$05 and $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$06 states, with positive-parity ones much heavier (Chen et al., 24 Jul 2025). The lightest reported values are

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$07

for specific hybrid currents (Chen et al., 24 Jul 2025). In this case the parity splitting is tied to the opposite signs of the chiral-odd OPE terms in the projected combinations $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$08 (Chen et al., 24 Jul 2025).

For $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$09, derivative $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$10-wave currents with a covariant derivative and parity-projected QCD sum rules yield

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$11

while the positive-parity partners are reported near $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$12 and $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$13 GeV (Su et al., 2024). The analysis concludes that $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$14 is likely to be a negative parity $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$15-wave excited state, but that the spin remains undetermined (Su et al., 2024).

6. Generalizations, limitations, and broader context

A recurrent misconception is that parity projection is identical to inserting $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$16 and nothing more. The literature summarized here shows three distinct but related realizations: explicit rest-frame projectors in forward correlators (Ohtani et al., 2012), covariant combinations $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$17 in Borel sum rules (Su et al., 2024, Wang, 2018), and Lorentz-structure engineering that algebraically cancels unwanted parity sectors in light-cone correlators (Aliev et al., 2015, Aliev et al., 2014). These formulations are equivalent in spirit but not identical in implementation.

Another important point is that parity projection is often inseparable from spin projection. In spin-$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$18 and spin-$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$19 correlators, lower-spin contamination must be removed first by selecting transverse or symmetric tensor structures, such as

$\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$20

before the parity-separated spectral combinations become meaningful (Su et al., 2024, Duan et al., 2024, Wang et al., 24 Aug 2025).

The method also has limitations that are explicitly acknowledged in the source literature. Continuum modeling remains a source of uncertainty because the continuum does not necessarily respect parity separation as sharply as narrow poles do; this is why pole-dominance criteria and Borel-window stability are imposed (Wang, 2018). Near-threshold positive-parity states can be especially sensitive to the choice of $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$21 and to coupled-channel dynamics beyond a one-pole ansatz (Duan et al., 2024). In the nucleon case, large $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$22 corrections in the perturbative term motivated the use of a phase-rotated Gaussian kernel to suppress them (Ohtani et al., 2012). In HQET, the strict heavy-quark limit improves parity separation but leaves sizeable $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$23 effects for charm-sector phenomenology (Nishikawa et al., 2024).

A broader implication is stated explicitly in the heavy-baryon magnetic-moment work: the methodology exemplifies how to construct sum rules that isolate a specific parity channel in problems where interpolating currents couple to both parities, and it is applicable beyond magnetic moments, including masses, couplings, and transition form factors (Aliev et al., 2015). A mathematically distinct but conceptually related variant appears in quantum-mechanical sum rules for parity-related one-dimensional potentials, where parity projectors $\sqrt{s}\,\rho^1(s)\pm \rho^0(s)$24 resolve sum rules into even and odd sectors and generate exact identities for Airy zeros and half-line harmonic-oscillator matrix elements (Ayorinde et al., 2010). This suggests that parity projection is best understood as a spectral-resolution principle rather than a technique confined to any one correlator or any one version of QCD sum rules.