Theta Vacua, QCD Sum Rules, and the Neutron Electric Dipole Moment

Adam Ritz; Maxim Pospelov

REVIEW

Theta Vacua, QCD Sum Rules, and the Neutron Electric Dipole Moment

Maxim Pospelov, Adam Ritz

High Energy Physics - Phenomenology · Submitted 30 Aug 1999 · 41 references

Pith's verdict: A real, independent sum-rule determination of d_n(theta) with an honest error budget; the soft spot is the phenomenological side, not the OPE.

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arXiv IDhep-ph/9908508 (version 4)

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CitationAdam Ritz, Maxim Pospelov. "Theta Vacua, QCD Sum Rules, and the Neutron Electric Dipole Moment." arXiv:hep-ph/9908508 (1999). https://pith.science/paper/LWTKX6FQ

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Editor's note

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This is the detailed version of the Pospelov-Ritz sum-rule calculation of the theta-induced neutron EDM, and it's worth your time if you care about the theta bound or about transferring the method to color EDMs. The core is genuinely new relative to the chiral-PT literature: they isolate the chirally invariant {F-tilde sigma, q-slash} structure, which is the one Lorentz channel whose coefficient is unambiguous under chiral rotations, and they carry the OPE to NNLO. The derivation of the (1 - m_pi^2/m_eta^2) factor in the chiral coupling, showing the EDM must vanish as U(1)_A is restored, is a nice piece of physics and, as far as I know, hadn't been written down. The large-Nc comparison in Section V is a clean parametric distinction between the two methods. Nothing here is fitted to the answer; chi and xi are external inputs from cited work, and the agreement with the chiral logarithm is a cross-check, not a calibration. That's the right structure for this kind of calculation, and the paper is candid about what it doesn't control: it says outright that it has 'no alternative but to neglect' the continuum because positivity is lost in a three-point function, and that consistency 'will require A ~ 0.' Those are acknowledged, not hidden. The stress-test's sharpening is fair, though: the two-sum-rule agreement at the extremum mostly tests the lambda extraction, and a common slope A can reconcile curves that differ in shape, so it doesn't directly bound A. That's a real gap in the argument, but it's a gap in a phenomenological-side assumption the authors flag, not a hole in the OPE. The other soft spots are proportionate: beta = 1 is chosen to kill the IR logarithm rather than by a principled criterion, the effective scale M ~ 0.7 GeV is low enough that OPE convergence is a legitimate worry, and the 40% is a heuristic budget. The v4 footnote correcting an overall factor of two is a reminder that the numerical extraction is delicate. None of this breaks the headline number, which is robust at the factor-of-two level and consistent with chiral PT and the quark model. Read it if you work on strong CP or on sum-rule methods for dimension-5 and -6 operators; the chirally invariant channel is the transferable idea. It deserves referees.

Referee report

Significance

If the extraction holds, this is an independent determination of d_n(θ) that does not rely on the chiral logarithm or its incalculable constant, and it is the first sum-rule treatment in a channel where the OPE is directly proportional to the electric dipole moment. The explicit demonstration that the unphysical θ_G − θ_q dependence cancels once mixing with the CP-conjugate currents is included is a genuine consistency check, and the derivation in §V.A of the U(1)-restoration factor in the chiral-loop result is a useful result in its own right. The numerical agreement with chiral perturbation theory and with the independent sum-rule calculation of Ref. [26] is a meaningful cross-check, and the |θ| bound is a falsifiable prediction. The paper is also candid about its soft spots, which is to its credit.

Major objections (4)

[§IV, Eq. (39) and discussion following Eq. (40)] The extraction of d_n requires the single-pole term A M^2 to be negligible against λ^2 m_n d_n across the Borel window. The paper argues for A ~ 0 from the observation that the two sum rules ν_a and ν_b (Eqs. 42 and 44) have extrema agreeing to about 10% and that their differing behaviour away from the extrema 'implies that for consistency we must assume A to be small.' That comparison is partly degenerate: ν_a and ν_b differ primarily in the determination of λ (Eq. 41 versus Eq. 43), so their agreement tests the λ extraction more than it bounds A, and a common slope A can reconcile curves that differ in shape. Please quantify the constraint directly, for example by fitting ν(M^2) to a + b M^2 over the Borel window and quoting b with its error, or by stating the range of A consistent with the curves and the corresponding shift in the extracted d_n.
[§IV, Eq. (48) and abstract] The quoted 40% precision is attributed to three sources: the sum-rule analysis, the error in χ, and an estimated ±20% for higher-dimensional operators and anomalous dimensions. The result is, as §VI states, 'to a large extent proportional to χ', and χ = −5.7 ± 0.6 GeV^{-2} is only a 10% input, so the 40% must be dominated by the first and third items. The single-pole contamination A, which is the least controlled ingredient, does not appear in the budget at all. Please give an explicit numerical breakdown of the 40% and include the A-induced uncertainty, so that the reader can see which systematic dominates and whether the quoted precision is meaningful.
[§III.B, Eq. (35), and §III.A] Section III advertises the cancellation of the unphysical combination θ_G − θ_q as 'a nontrivial check on the consistency of our approach.' That check is carried out for the logarithmic terms of Eqs. (29) and (31), but the O(1/Q^2) terms of Eq. (35) are evaluated by 'simply us[ing] the gauge θ_G = θ_q.' Since the ξ-dependent term survives at β = 1 and enters the sum rule in Eq. (39), the check should be completed for these terms as well, or the residual θ_G − θ_q dependence should be bounded and shown to be numerically negligible. As written, the consistency check is only partial at the order that actually contributes to the final number.
[§III and §IV, Eq. (29) and Eq. (39)] The leading OPE coefficient in Eq. (29) is computed at tree level, and the sum rule is evaluated at an effective scale M ~ 0.7 GeV, which the authors themselves flag as a convergence concern. The one-loop QCD correction to this coefficient is not estimated. With α_s(0.7 GeV) of order 0.5, a coefficient of a few in α_s/π would produce a 15–50% correction, comparable to the quoted 40% total error. The paper states that leading-log anomalous dimensions are omitted and folded into the error estimate, but a finite radiative correction to the leading Wilson coefficient is a different object and is not discussed. Please estimate this correction, or give an argument for why it is small in this channel.

Minor objections (8)

[§II, Eq. (9)] The transformation law is written as U_α P U_α; it should presumably read U_α P U_α^{-1} (or U_α P U_α^†), and the right-hand side appears to be missing a denominator or a term in the second line. Please check this equation, since it is the basis for the chirally invariant channel selection.
[§IV, paragraph following Eq. (44)] The sentence 'Extracting a numerical estimate for dn from Fig. 2' should refer to Fig. 4.
[Footnote 3] The note records that Eq. (48) corrects an overall factor-of-two error in previous versions. Please state explicitly which other numbers were affected by that correction (the abstract value 2.4 × 10^{-16}, the curves in Fig. 4, and the bound in Eq. (55)) and confirm that they have all been updated consistently.
[§V.A, sentence following Eq. (53)] The text says the correction from diagrams (b) and (c) is negligible 'as it is second order in the light quark masses.' The factor 1 − m_π^2/m_η^2 deviates from unity at first order in m_q (second order in the chiral counting, since m_q ~ p^2). Please clarify the power-counting statement.
[Reference [26]] Ref. [26] is cited as an unpublished preprint (hep-ph/9905317) without a journal reference, yet the numerical comparison in §III.B and §VI rests on it. Please supply the publication details.
[§III.A] The choice θ_q = θ_G is described as a 'gauge.' It is a choice of basis for the interpolating currents, not a gauge transformation, and the wording may mislead readers into thinking the result is gauge dependent.
[§IV, numerical inputs] The value χ = −5.7 ± 0.6 GeV^{-2} is taken from Ref. [37]. Since the final result is largely proportional to χ, a brief discussion of the spread among independent determinations of the magnetic susceptibility would strengthen the error estimate.
[Fig. 4 caption] The caption should state the Borel window plotted and the value of the continuum threshold s_0 used, so that the figure can be reproduced.

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The paper

The paper claims

A sum-rule calculation ties the neutron's theta-induced dipole to a vacuum susceptibility

The reading

If the QCD vacuum carries a nonzero $\theta$ angle, the neutron should acquire an electric dipole moment. This paper computes that moment with QCD sum rules — an expansion of a neutron-current correlator in vacuum condensates — rather than with the chiral-loop estimate that had stood since 1979. Working to next-to-next-to-leading order in the operator product expansion, the authors obtain $d_n(1.0)$ and quote $d_n = 2.$4^{{+1.0}}$_{-1.0}

What carries the argument

The load-bearing object is the correlator of neutron interpolating currents taken in a background electromagnetic field and a topologically nontrivial vacuum, projected onto the Dirac structure $d_n(1.0)$.

Extended reading notes

Core claim

The paper's claim is that the theta-induced neutron EDM can be extracted cleanly from QCD sum rules, and that the answer is $d_n(1.0)$.

Load-bearing premise

Reading the EDM off the sum rule requires assuming that the single-pole contamination term is negligible and independent of the Borel mass, and that continuum contributions can be dropped entirely because positivity constraints are lost in a three-point correlator.

Editorial extensions

If this is right

The theta parameter is bounded at $|\bar\theta| < 3\times 10^{-10}$ from the then-current experimental neutron EDM limit, close to but slightly weaker than the chiral-logarithm bound.
The theta-induced EDM is proportional to the vacuum's electromagnetic susceptibility $\chi$, so the prediction inherits that quantity's roughly ten percent uncertainty as its dominant numerical error.
Because the result vanishes as $m_\eta \to m_\pi$, any model in which the $U(1)$ anomaly is switched off predicts no theta-induced neutron EDM at leading order.
The same chirally invariant channel can be reused for dimension-5 and dimension-6 CP-odd operators, in particular quark colour EDMs, where existing methods disagree by more than an order of magnitude.
In models where the theta angle is generated radiatively rather than postulated, the induced EDM can be computed once the model's CP-violating phases are specified, with this number supplying the hadronic conversion factor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's insistence that the EDM must vanish when $m_\eta = m_\pi$ is a sharper consistency test than the numerical value itself; a lattice calculation of the same correlator at unphysical pion masses could check the $m_\eta$ dependence directly rather than only the endpoint.
The disagreement between sum-rule and chiral-loop treatments of colour-EDM-induced EDMs may partly trace to the choice of Dirac structure, since only the chirally invariant channel avoids uncalculable mixed condensates at leading order.
Because the extracted value sits at a low effective Borel scale of roughly $0.7\,\mathrm{GeV}$, the quoted forty percent error is plausibly an underestimate; a dedicated study of the continuum and single-pole contributions would sharpen or widen the theta bound accordingly.
The large-$N_c$ scaling distinction the authors draw between the tree-level OPE and the chiral loop suggests that meson-sector observables, where $N_c$ systematics are better controlled, could arbitrate between the two pictures.

Original abstract

We present a detailed study of the electric dipole moment of the neutron induced by a vacuum theta angle within the framework of QCD sum rules. At next-to-next-to leading order in the operator product expansion, we find the result d_n(\theta) = 2.4 x 10^{-16} \theta e cm, to approximately 40% precision. With the current experimental bound this translates into a limit on the theta parameter of |\theta| < 3 x 10^{-10}. We compare this result with the long-standing estimates obtained within chiral perturbation theory, and observe a numerical similarity, but also significant differences in the source of the dominant contribution.

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The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

We present a detailed study of the electric dipole moment of the neutron induced by a vacuum theta angle within the framework of QCD sum rules. At next-to-next-to leading order in the operator product expansion, we find the result d_n(\theta) = 2.4 x 10^{-16} \theta e cm, to approximately 40% precision. With the current experimental bound this translates into a limit on the theta parameter of |\theta| < 3 x 10^{-10}. We compare this result with the long-standing estimates obtained within chiral perturbation theory, and observe a numerical similarity, but also significant differences in the source of the dominant contribution.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a detailed study of the electric dipole moment of the neutron induced by a vacuum theta angle within the framework of QCD sum rules.	contribution 1.00	0.92	0.08	asserted 0.96
s1	At next-to-next-to leading order in the operator product expansion, we find the result d_n(\theta) = 2.4 x 10^{-16} \theta e cm, to approximately 40% precision.	result 1.00	0.97	0.61	asserted 1.00
s2	With the current experimental bound this translates into a limit on the theta parameter of \|\theta\| < 3 x 10^{-10}.	interpretation 0.75	0.83	0.45	asserted 0.99
s3	We compare this result with the long-standing estimates obtained within chiral perturbation theory, and observe a numerical similarity, but also significant differences in the source of the dominant contribution.	interpretation 0.62	0.78	0.08	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.77.

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sources read

arXiv listing abstract v4

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references and citing sentences
relations to other papers

levels run

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S sections
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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States the same result 1 paper

Combined with the new experimental bound, this translates into the limit |\theta|< 3 x 10^{-10}.

Theta-Induced Electric Dipole Moment of the Neutron via QCD Sum Rules · 1.00

Papers citing it · 31 sentences without a confident label

Contests it 1 paper

The procedure advocated in [ref] where $β$ is chosen such that subleading IR logarithms are cancelled in the QCD $θ$-term and CEDM contributions to the nEDM is also not useful, since in the OPE side ([ref]) of our sum rule IR logarithms appear already at LO.

Sum rules for CP-violating operators of Weinberg type · Matching and discussion · contest 0.71

Uses its method or tool 2 papers

In this section, we derive the phenomenological form $Π_phen$ of the correlator ([ref]) following the argument presented in [ref] .

Sum rules for CP-violating operators of Weinberg type · Hadronic representation · 0.95
The contribution of quark EDMs and CEDMs to nucleon EDMs can be computed on an equal footing by means of the QCD sum-rule technique [ref] , as described in Refs. [ref] .

Quantifying Limits on CP Violating Phases from EDMs in Supersymmetry · Nucleon · 0.95

Uses its data or values 19 papers

We have used a recent lattice QCD calculation [ref] of the $θ_eff$ parameter from the neutron electric dipole moment (as opposed to the usual QCD sum rules [ref] ), which results in a bound that accounts for both theoretical and experimental uncertainties.

Cornering the axion with CP-violating interactions · Axion couplings · 0.85
Upper bounds on the neutron EDM require $δθ̅≤10^-10$ [ref] , which translate into an upper limit on the sparticle mass scale m_SUSY<1.3×10^8/√(L)( Λ/10^12 GeV) GeV.

CPon Dark Matter · Constraints from the neutron EDM · 0.87
We used the result in [ref] , obtained with QCD sum rules; for other evaluations see e.g. [ref] .[Current lattice QCD results on $g_aD$ do not show a statistically significant non-zero signal [ref] .] We infer from Fig. [ref], that the preferred axion mass window [ref] could definitely be probed in phase III of CASPEr-Electric.[The sensitivity in $g_aD$ improves with the scanning time as $t^1/4$.

Axion mass prediction from minimal grand unification · Introduction · 0.99

Compares with it 3 papers

We report new results for the $β=-1$ choice finding that $d_n(θ)$ is consistent, both in sign and magnitude, with earlier estimates of $d_n(θ)$ using $β=+1$ [ref] .

Chiral properties of the nucleon interpolating current and $\theta$-dependent observables · Introduction · 0.98
Sum-rule studies of the $θ$-term and CEDM contributions to the nEDM [ref] have also found uncertainties of a similar magnitude.

Sum rules for CP-violating operators of Weinberg type · Conclusions · 0.93
Effective theories like chiral perturbation theory [ref] and several models in QCD sum rules [ref] have found $d_N^p(n)=(-)(1$–$4)×10^-3$ e$·$fm (the minus sign is for the neutron), about one order of magnitude smaller than the central value of lattice QCD results computed at unphysically large pion mass.

Neutron and proton electric dipole moments from $N_f=2+1$ domain-wall fermion lattice QCD · Lattice results for the neutron and proton EDM · 0.93

Cites it as background 5 papers

For QCD axions [ref] , which relax the $θ̅$ parameter of the Standard Model (SM) to zero (see [ref] for recent reviews), the restricting power of this statement is especially clear [ref] : even a slight breaking of the Peccei-Quinn (PQ) symmetry can be detected given the sharp bound on strong CP violation, coming for instance from EDM measurements [ref] , θ̅≲10^-10 .

Heavy fields and the axion quality problem · Introduction · 0.84
A detailed expression for the resulting correlator $Π(Q^2)^OPE$ goes beyond the scope of this review but can be found in, e.g., Refs. [ref] .

Electric Dipole Moments of Nucleons, Nuclei, and Atoms: The Standard Model and Beyond · QCD Sum Rules · 0.84
This so-called strong CP problem has no satisfactory explanation up to now (for detailed reviews, see [ref] ).

Theta term in a bounded region · Introduction · 0.98

Discussion

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