Strong CP and the Axion Interface

The physical strong-$CP$ parameter is not the coefficient $\theta$ of the QCD topological term by itself, but the basis-invariant combination $\bar\theta=\theta+\arg\det(M_uM_d)$ modulo $2\pi$. A QCD axion replaces this fixed angle by a dynamical field: the QCD vacuum energy then drives the effective angle to a $CP$-conserving minimum, provided the Peccei–Quinn symmetry has sufficient quality and the usual vacuum assumptions hold. That mechanism is narrower than generic axionlike-particle phenomenology.

Required background. The $U(1)_A$ problem and QCD topology supplies the anomalous axial rotation, topological charge, and $\theta$-dependent vacuum energy. Helpful background. Explicit breaking and pseudo-Goldstone modes supplies the relation between an approximate shift symmetry, a potential, and a pseudo-Goldstone mass.

The invariant QCD vacuum angle

Use the inherited $\epsilon^{0123}=+1$ and $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$ conventions, and define

$$\widetilde G^{a\mu\nu} =\frac12\epsilon^{\mu\nu\rho\sigma}G^a_{\rho\sigma}, \qquad q(x)=\frac{g_s^2}{32\pi^2} G^a_{\mu\nu}\widetilde G^{a\mu\nu}.$$

The relevant QCD terms are

$$\mathcal L \supset \theta q(x) -\overline q_LM_q q_R -\overline q_RM_q^\dagger q_L, \qquad M_q=\operatorname{diag}(M_u,M_d).$$

This page fixes the sign of $\theta$ by this equation. A source that chooses the opposite sign for $q(x)$ or for the complex mass term will write a corresponding minus sign in $\bar\theta$; observables are unchanged after translating the whole convention.

To see which phase is physical, make a passive flavor-basis change

$$q_L'=Lq_L, \qquad q_R'=Rq_R,$$

where $L$ and $R$ are unitary. The mass matrix in the primed coordinates is

$$M_q'=LM_qR^\dagger,$$

so

$$\arg\det M_q' =\arg\det M_q-\arg\det(RL^\dagger).$$

The singlet axial part of this change of variables is anomalous. In the convention above its Jacobian shifts

$$\theta'=\theta+\arg\det(RL^\dagger).$$

Therefore

$$\boxed{ \bar\theta =\theta+\arg\det(M_uM_d) } \pmod{2\pi}$$

is invariant. For the simple axial redefinition $q'=e^{i\alpha\gamma^5}q$, one has $L=e^{-i\alpha}\mathbf 1$ and $R=e^{i\alpha}\mathbf 1$: $\arg\det M_q$ decreases by $2N_f\alpha$ while $\theta$ increases by the same amount. This anomaly–mass-phase cancellation is the essential derivation; keeping only one shift gives a basis-dependent answer. The same physics is presented with an explicitly translated sign convention in Schwartz 2014, § 29.5.3, pp. 609–613.

Finite-action gauge fields have integer topological charge

$$Q=\int d^4x\,q(x)\in\mathbb Z$$

under the standard boundary conditions. The quantum theory is consequently periodic in $\bar\theta$ with period $2\pi$. The density $G\widetilde G$ is $C$ even and $P$ and $T$ odd, so a generic $\bar\theta$ violates $CP$. The points $\bar\theta=0$ and $\pi$ modulo $2\pi$ require separate care: zero is $CP$ conserving, while at $\pi$ the action is $CP$ symmetric but the vacuum can have branch structure or spontaneous $CP$ breaking.

An exactly massless quark would provide an anomalous axial redefinition with no mass phase to reintroduce, making $\bar\theta$ unobservable. This is a useful structural limit, not an assumption made on this page.

Why the parameter is a problem

Near $\bar\theta=0$, a hadronic $CP$-odd observable has the schematic form

$$\mathcal O_{CP} =\bar\theta\,K_{\mathcal O} +O(\bar\theta^3),$$

where $K_{\mathcal O}$ is a nonperturbative QCD response. Examples include $CP$-odd pion–nucleon interactions and permanent hadronic electric dipole moments. The absence of an even power follows when the observable is odd under $CP$ and the vacuum is analytic about zero. The coefficient is not fixed by dimensional analysis: it must be matched through chiral methods, lattice QCD, sum rules, or another controlled nonperturbative calculation.

The strong-$CP$ problem is the coexistence of three facts:

1. $\bar\theta$ is allowed by QCD symmetries and is dimensionless, so it is not suppressed by a heavy mass scale.
2. diagonalizing complex Yukawa matrices does not remove it; the anomaly transfers their determinant phase into $\theta$.
3. hadronic $CP$ tests require its observable effects to be very small, yet the Standard Model does not supply a symmetry that sets the invariant to zero.

No current numerical bound is needed to state that problem. A quantitative extraction would additionally require the dated experimental likelihood, the hadronic response calculation, its renormalization convention, and correlated uncertainties.

The theory-to-observable chain must preserve scheme cancellation:

$$\bar\theta \longrightarrow \sum_i c_i(\mu)\,\mathcal O_i(\mu) \longrightarrow \mathcal O_{CP}^{\rm hadron}.$$

Under a finite operator-basis change $\mathcal O'=R\mathcal O$, coefficients transform as $c'=R^{-\mathsf T}c$. Hence $c^{\mathsf T}\langle\mathcal O\rangle$ is unchanged, although an individual coefficient or matrix element is not. Chiral rotations also redistribute phases between $\theta$, masses, and higher-dimensional $CP$-odd operators. A quoted “induced $\theta$” is therefore meaningful only with the complete operator and phase convention; the observable amplitude is the invariant object.

Stage	Quantity carried forward	Method-dependent input	Failure if omitted
quark and gluon theory	$\bar\theta$ and any other $CP$-odd coefficients	field basis, operator normalization, matching scale	a mass phase is counted twice or not at all
hadronic effective theory	renormalized $CP$-odd couplings	nonperturbative matrix elements and correlations	coefficient–matrix-element scheme mismatch
observable	energy shift, form factor, or decay amplitude	kinematics, external-field and sign conventions	the reported sign or normalization is ambiguous
inference	constraint on the common parameters	dataset identity, likelihood, nuisance model	a mutable result is mistaken for a timeless constant

A dynamical angle

The Peccei–Quinn mechanism introduces a spontaneously broken anomalous global symmetry. Its angular mode $a(x)$ has an approximate shift symmetry and, after heavy fields are integrated out, couples to QCD. Normalize the low-energy field so that

$$\mathcal L_a =\frac12\partial_\mu a\,\partial^\mu a +\left(\bar\theta+\frac{a}{f_a}\right)q(x) +\mathcal L_{\rm derivative} +\mathcal L_{\rm model\text{-}dependent}.$$

Changing the sign of $a$ changes the sign written in the anomalous coupling and has no physical effect if all axion couplings are changed with it. More generally the ultraviolet theory gives $C_Ga/f_{\rm PQ}$; defining $f_a=f_{\rm PQ}/|C_G|$ produces the local normalization above, while the integer anomaly coefficient still controls the field’s global periodicity and domain-wall structure.

QCD generates a vacuum energy $E_{\rm QCD}(\vartheta)$ for

$$\vartheta=\bar\theta+\frac{a}{f_a}.$$

Thus the axion potential is

$$V_{\mathrm{QCD}}(a)=E_{\mathrm{QCD}}\!\left( \bar\theta+\frac{a}{f_a} \right).$$

At a regular $CP$-conserving minimum,

$$\frac{dV_{\mathrm{QCD}}}{da} =\frac1{f_a} \frac{dE_{\mathrm{QCD}}}{d\vartheta}=0, \qquad \frac{\langle a\rangle}{f_a} =-\bar\theta\pmod{2\pi}.$$

The effective angle therefore relaxes to zero. Expanding around that minimum defines the topological susceptibility

$$\chi_{\mathrm{QCD}} =\left. \frac{d^2E_{\mathrm{QCD}}}{d\vartheta^2} \right|_{\vartheta=0},$$

and gives the model-independent QCD contribution

$$m_a^2f_a^2=\chi_{\mathrm{QCD}}.$$

This relation, including controlled low-energy corrections and the role of quark-mass ratios, is derived in Grilli di Cortona et al. 2016, § 2.1, pp. 4–7. The dynamical cancellation follows the original Peccei–Quinn symmetry construction Peccei and Quinn 1977, pp. 1440–1443.

The argument assumes that QCD selects the relevant global minimum, cosmological evolution reaches it in the intended branch, and additional explicit breaking is negligible. It does not by itself solve the hierarchy, dark-matter abundance, isocurvature, domain-wall, or ultraviolet-completion questions.

QCD axion versus a generic axionlike particle

The name “axionlike” describes field content and approximate shift symmetry, not automatically a solution of strong $CP$.

Property	QCD axion	Generic axionlike particle
anomalous QCD coupling	required and normalized into $a/f_a$	optional
dominant potential	includes the QCD vacuum energy	may be set mainly by unrelated explicit breaking
mass–coupling relation	$m_a^2f_a^2=\chi_{\rm QCD}$ up to declared additional breaking	$m_a$ and couplings can be independent
strong-$CP$ cancellation	follows if the total minimum is at $\vartheta=0$	not implied by pseudoscalar couplings alone
photon, lepton, and flavor couplings	model dependent around the QCD relation	broadly model dependent

This distinction is the typed boundary to axionlike pseudoscalar portals: that page may vary masses and portal couplings independently, whereas a strong-$CP$ solution must retain the anomalous QCD potential and show that its true minimum suppresses $\vartheta$.

Explicit breaking and axion quality

An extra Peccei–Quinn-breaking contribution $\Delta V(a)$ generally displaces the QCD minimum. Let $a_0=-f_a\bar\theta$ and suppose the displacement is small. Writing the residual angle as $\delta\vartheta=(a-a_0)/f_a$, the stationarity equation gives

$$0 =\frac{\chi_{\rm QCD}}{f_a}\delta\vartheta +\Delta V'(a_0)+O(\delta\vartheta^2),$$

and hence

$$\delta\vartheta \simeq -\frac{f_a\Delta V'(a_0)}{\chi_{\rm QCD}}.$$

This is the axion-quality test. A small coefficient in $\Delta V$ is not sufficient if a large harmonic number or an unfavorable phase produces a large slope at $a_0$. Conversely, an extra term aligned so that $\Delta V'(a_0)=0$ need not shift the minimum at first order, although it changes the mass and higher derivatives. Every claimed solution should therefore specify the full periodic potential, anomaly coefficient, phases, and which minimum is occupied.

Independent checks and limitations

• Anomalous rephasing: apply an arbitrary singlet axial basis change. The shifts of $\theta$ and $\arg\det M_q$ must cancel in $\bar\theta$.
• Vector rephasing: set $L=R$. Neither the mass determinant phase nor $\theta$ changes; a vector flavor convention cannot affect strong $CP$.
• Periodicity: replace $\bar\theta$ by $\bar\theta+2\pi$. The partition function and every observable must be unchanged under the stated topological boundary conditions.
• Massless-quark limit: if one quark mass is exactly zero, verify that its axial phase can remove $\bar\theta$ without introducing a mass phase.
• CP limit: at the ordinary QCD minimum, $\vartheta=0$ and all effects proportional to a single $\bar\theta$ insertion vanish.
• Dimensions: $[a]=[f_a]=1$, $[\chi_{\rm QCD}]=4$, so $m_a^2=\chi_{\mathrm{QCD}}/f_a^2$ has dimension two and $f_a\Delta V'/\chi_{\mathrm{QCD}}$ is dimensionless.
• Scheme cancellation: evolve coefficients and matrix elements in the same basis, including any finite chiral rotation. A residual scale dependence in the observable signals incomplete matching or truncation.
• Scope: present-day dipole limits, axion mass windows, dark-matter fractions, and search exclusions are dated evidence, not fixed facts of this derivation.

Common pitfalls

Setting $\theta=0$ and declaring strong $CP$ solved. A complex quark-mass determinant regenerates the invariant phase after diagonalization. The correct object is $\bar\theta$ in a fully stated sign convention.

Calling every light pseudoscalar an axion. A generic axionlike particle need not couple anomalously to QCD or minimize the effective vacuum angle. Demonstrate the QCD coupling and total potential before claiming the Peccei–Quinn solution.

Using $m_a\propto f_a^{-1}$ after adding an unrelated potential. The relation $m_a^2f_a^2=\chi_{\rm QCD}$ is the QCD contribution. Additional explicit breaking can change the mass and, more seriously, shift the $CP$ minimum.

Informal self-check

For one Dirac quark with mass $m e^{i\phi}$, perform the passive axial basis change $q'=e^{i\alpha\gamma^5}q$. What value of $\alpha$ makes the mass real, and what happens to $\bar\theta$?

Answer

Here $M'=e^{-2i\alpha}M$, so choose $2\alpha=\phi$ modulo $2\pi$ to make $M'$ real and positive. The anomaly gives $\theta'=\theta+2\alpha=\theta+\phi$, while $\arg M'=0$. Therefore $\bar\theta'=\theta'=\theta+\phi=\bar\theta$: diagonalizing the mass moves the phase into the topological term rather than removing it.

Handoffs

• Send the topological susceptibility, anomalous Ward identities, and vacuum-branch questions to the $U(1)_A$ and QCD-topology treatment.
• Send pseudo-Goldstone power counting and explicit-breaking potentials to explicit breaking and pseudo-Goldstone modes.
• Send independent mass and portal-coupling phenomenology to axionlike pseudoscalar portals, carrying the QCD anomaly coefficient when strong $CP$ remains in scope.
• Send any current limit, preferred region, or cosmological claim to a dated Research record with the dataset identity, likelihood, model assumptions, and covariance.

References

• Grilli di Cortona, Giovanni, Edward Hardy, Javier Pardo Vega, and Giovanni Villadoro. “The QCD Axion, Precisely.” Journal of High Energy Physics 2016, no. 1 (2016): 034. DOI · Open PDF
• Peccei, Roberto D., and Helen R. Quinn. “CP Conservation in the Presence of Pseudoparticles.” Physical Review Letters 38 (1977): 1440–1443. DOI
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014, § 29.5.3. DOI