Nucleon electric dipole form factor in QCD vacuum

The resolution of the CP problem in baryogenesis is central to our understanding of baryon asymmetry in the universe Sakharov (1967). CP violation in the weak sector of the standard model, proves to be orders of magnitude away from its resolution, while CP violation in the strong sector, puts its resolution within range Kharzeev et al. (2020).

The QCD vacuum is rich with topologically active pseudoparticles which are CP conjugated pairs (instantons and anti-instantons). They are natural sources of local CP violation effects. CP is strongly violated in QCD at finite theta angle. A reliable description of ensembles of these pseudoparticles in the semi-classical approximation, is provided by the instanton liquid model (ILM) Diakonov et al. (1996); Schäfer and Shuryak (1998); Nowak et al. (1996); Liu (2025) (and references therein). The model has proven to be very useful in capturing many aspects of most hadronic correlations both in vacuum, in hadronic states and in matter. Here, we will focus on understanding the role of these pseudoparticles in the composition of the hadronic electric dipole moment.

For many decades, the nucleon electric dipole moment has been used as a measure of the strong CP violation caused by a finite theta angle in QCD. The current empirical estimate puts its upper bound at about $10^{-26}\,e$-cm Abel et al. (2020). This is an ideal task for ab-initio lattice simulations, yet the smallness of the observable makes the task daunting in light of the signal-to-noise ratio Syritsyn et al. (2019); Alexandrou et al. (2021). Notwithstanding this, the latest lattice estimate puts it at about $10^{-16}\,\theta\,e$-cm for a finite theta angle Liang et al. (2023), which would limit $\theta$ to about $10^{-10}$ by the empirical bound.

The QCD vacuum breaks conformal symmetry, a mechanism at the origin of most hadronic masses. Detailed gradient flow (cooling) techniques have revealed a striking semi-classical landscape made of instantons and anti-instantons, the vacuum tunneling pseudoparticles with unit topological charges Leinweber (1999); Michael and Spencer (1995a, b); Biddle et al. (2018); Athenodorou et al. (2018); Ringwald and Schrempp (1999). These pseudoparticles break chiral symmetry through fermionic zero modes with fixed chirality (left or right). The key features of this landscape are Shuryak (1982)

$$n_{I+A}\equiv\frac{1}{R^{4}}\approx\frac{1}{{\rm fm}^{4}}\qquad\qquad\frac{\rho}{R}\approx\frac{1}{3}$$

(1)

for the instanton plus anti-instanton density and size, respectively. The hadronic scale $R=1\,{\rm fm}$ emerges as the mean quantum tunneling rate of the pseudoparticles. Deep in the cooling time the tunnelings are sparse, well described by the instanton liquid model (ILM) with a packing fraction

$\displaystyle\kappa\equiv\pi^{2}\rho^{4}n_{I+A}\approx 0.1$

(2)

The main purpose of this work is to evaluate the C-odd nucleon electric dipole form factor, and provide an estimate of the proton and neutron electric dipole moments in the ILM. In section II we briefly review the salient features of the pseudoparticle fluctuations in the ILM, with a comparison to some lattice simulations. In section III we define the four C-even or odd form factors associated to the electric current in the nucleon. The C-odd electric dipole form factor for each flavor contribution is evaluated in the ILM. This form factor is found to be related to the Pauli form factor. In particular, the finite proton and neutron electric dipole moments in the ILM are shown to be comparable to some recent lattice simulations. Our conclusions are in section IV. Further details can be found in the appendices.

The size distribution of the instantons and anti-instantons density (their tunneling rate) in the vacuum is well captured semi-empirically by

$$n(\rho)\sim{1\over\rho^{5}}\big{(}\rho\Lambda_{QCD}\big{)}^{b}\,e^{-C\rho^{2}/R^{2}}$$

(3)

with the mean value (1). Here $b=11N_{c}/3-2N_{f}/3$ (one loop), and $C$ is a number of order 1 fixed by the binary gauge interactions among pseudoparticles in the vacuum Diakonov (1996); Shuryak (1999).

To capture the fluctuations of the number of pseudoparticles $N_{\pm}$, we identify them with the scalar and pseudoscalar gluonic densities in the vacuum

	$\displaystyle\frac{1}{32\pi^{2}}\int d^{4}x\,F^{2}(x)$	$\displaystyle=$	$\displaystyle(N_{+}+N_{-})=N$
	$\displaystyle\frac{1}{32\pi^{2}}\int d^{4}x\,F\tilde{F}(x)$	$\displaystyle=$	$\displaystyle(N_{+}-N_{-})=\Delta$		(4)

with $gF\rightarrow F$ assumed. For a random quenched vacuum with Poissonian $N_{\pm}$ at finite angle $\theta$,

	$\displaystyle\frac{\langle F^{2}\rangle_{\theta}}{32\pi^{2}}$	$\displaystyle\approx$	$\displaystyle n_{I+A}\,{\rm cos\,\theta}$
	$\displaystyle\frac{\langle F\tilde{F}\rangle_{\theta}}{32\pi^{2}}$	$\displaystyle\approx$	$\displaystyle in_{I+A}\,{\rm sin\,\theta}$		(5)

with the breaking of scale or conformal symmetry manifested for vanishing theta. However, in the ILM the distributions of $N_{\pm}$ are not Poissonian. The fluctuations in the sum $N$ follow from low-energy theorems Novikov et al. (1981)

$$\mathbb{P}(N)=e^{-\frac{b\bar{N}}{4}}\bigg{(}\frac{\overline{N}}{N}\bigg{)}^{\frac{bN}{4}}$$

(6)

with the vacuum topological compressibility

$$\frac{\sigma_{T}}{V_{4}}=\frac{\langle(N-\bar{N})^{2}\rangle}{V_{4}}=\frac{4}{b}\,n_{I+A}$$

(7)

and the volume extensive mean $\bar{N}=n_{I+A}V_{4}$. The fluctuations in the difference $\Delta$ are fixed by the topological susceptibity Diakonov et al. (1996). At $\theta=0$, the distribution reads

$$\mathds{Q}(\Delta)=\frac{1}{\sqrt{2\pi\chi_{t}}}\exp\left(-\frac{\Delta^{2}}{2\chi_{t}}\right)$$

(8)

The topological susceptibility $\chi_{t}$ in gluodynamics is

$\displaystyle\frac{\chi_{t}}{V_{4}}=\frac{\langle\Delta^{2}\rangle}{V_{4}}\approx n_{I+A}$

(9)

which is large. However, in QCD it is substantially screened by the light quarks, as illustrated in Fig. 1 with current quark mass $m=10$ MeV. The width of the distribution reads

$\displaystyle\frac{\langle\Delta^{2}\rangle}{\bar{N}}\sim\bigg{(}1+N_{f}\frac{m^{*}}{m}\bigg{)}^{-1}$

(10)

Note that the topological susceptibility is sensitive to the determinantal mass $m^{*}$. The details about how to fix the determinantal mass are given in Appendix C and in Liu et al. (2023a, 2024) (and references therein).

In Fig. 1 we show the unquenched ILM results (10) (red-solid) and quenched ILM (9) (blue-solid), compared to the lattice results from the ETMC collaboration Alexandrou et al. (2021). The latters used twisted mass clover-improved fermions $N_{f}=2+1+1$, in a 4-volume $64^{3}\times 128~{}a^{4}$ with lattice spacing $a=0.0801(4)$ fm, and physical pion mass $m_{\pi}=139$ MeV. For the ILM parameters see below.

At finite vacuum angle $\theta$, the QCD action in Euclidean signature is supplemented by the topological term

$$\frac{\theta}{32\pi^{2}}\int d^{4}x\,F\tilde{F}\rightarrow\theta\Delta$$

which is at the origin of strong CP violation. In the ILM, this contribution acts as a topological chemical potential for $\Delta$, by enhancing instantons and depleting anti-instantons. This affects most observables, and in particular the electromagnetic (EM) form factors of hadrons, as we will detail in this section.

The coupling to photon can be described by Dirac $F_{1}$, Pauli $F_{2}$, electric dipole moment form factor $F_{3}$, and axial tensor form factor $F_{A}$.

		$\displaystyle\langle N^{\prime}|J^{\mu}_{\rm EM}=\sum_{f}Q_{f}\bar{\psi}_{f}\gamma^{\mu}\psi_{f}|N\rangle$		(11)
		$\displaystyle=\bar{u}_{s^{\prime}}(P^{\prime})\left[\gamma^{\mu}F_{1}(Q^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2M_{N}}\left(F_{2}(Q^{2})-i\gamma^{5}F_{3}(Q^{2})\right)+\frac{1}{M_{N}^{2}}\left(\not{q}q^{\mu}-q^{2}\gamma^{\mu}\right)\gamma^{5}F_{A}(Q^{2})\right]u_{s}(P)$

with the Sachs FFs $G_{C,M}$

	$\displaystyle G_{C}(Q^{2})$	$\displaystyle=$	$\displaystyle F_{1}(Q^{2})+\frac{Q^{2}}{4M^{2}_{N}}F_{2}(Q^{2})$
	$\displaystyle G_{M}(Q^{2})$	$\displaystyle=$	$\displaystyle F_{1}(Q^{2})+F_{2}(Q^{2})\,$		(12)

and the electric dipole and axial-tensor FFs $G_{D,A}$

	$\displaystyle G_{D}(Q^{2})$	$\displaystyle=$	$\displaystyle F_{3}(Q^{2})$
	$\displaystyle G_{A}(Q^{2})$	$\displaystyle=$	$\displaystyle F_{A}(Q^{2})$		(13)

with the two latters vanishing for $\theta=0$. The magnetic and electric dipole moment are defined respectively,

$\displaystyle\mu_{N}=G_{M}(0)e/2M_{N}$

$\displaystyle d_{N}=F_{3}(0)e/2M_{N}$

(14)

For finite theta, (LABEL:EM_form) will be organized through

$$J_{\mu}^{\mathrm{Dirac}}(q,\theta)+J_{\mu}^{\mathrm{Pauli}}(q,\theta)+J_{\mu}^{A}(q,\theta)$$

(15)

In QCD, the breaking of conformal symmetry puts stringent constraints on the bulk hadronic correlations in the form of low energy theorems Novikov et al. (1981). These constraints are enforced in the QCD instanton vacuum in the form of stronger than Poisson fluctuations in the number of pseudoparticles, with a vacuum compressibility indicative of a quantum liquid. The fluctuations in the difference of the pseudoparticles are peaked around neutral topological charge with the variance, which is fixed by the topological susceptibility which is large in gluodynamics, but substantially screened in QCD. These are essential features of the ILM.

In the ILM, the Pauli form factor of constituent quarks receive a large contribution in the ILM, correcting an early observation in Kochelev (2003). In the presence of a CP violation by a small vacuum angle $\theta$, we have shown that an axial contribution develops in the Pauli form factor, driven mostly by the vacuum topological susceptibility. We have used this result to estimate the induced the electric dipole moment in both the proton and neutron, as well as the semi-hard momentum dependence of the Pauli form factors $F_{2}$, $F_{3}$ induced by the pseudoparticles. Our results are in the range of recently reported lattice results extrapolated at the physical pion mass Liang et al. (2023).