Effective Field Theory Description of the Pion Form Factor

A. Pich; F. Guerrero

REVIEW

Effective Field Theory Description of the Pion Form Factor

F. Guerrero, A. Pich

High Energy Physics - Phenomenology · Submitted 14 Jul 1997 · 24 references

Pith's verdict: A clean, genuinely parameter-free closed form for F(s); the only real soft spot is that the step fixing the rho peak is justified by an unpublished note.

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arXiv IDhep-ph/9707347 (version 1)

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CitationA. Pich, F. Guerrero. "Effective Field Theory Description of the Pion Form Factor." arXiv:hep-ph/9707347 (1997). https://pith.science/paper/DBDFSFEM

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Worth your time, with one caveat about where the peak normalization comes from. The result is Eq. (17): a Dyson-summed rho propagator times an Omnes exponential of the real part of the one-loop function, with m_pi, m_K, M_rho and f_pi as the only inputs. L9 is predicted from VMD in Eq. (7), not fitted, and the four inputs are measured constants. That is a real prediction, and it lands on the data in both modulus and phase up to about 1 GeV. The individual pieces are all in the literature and the authors say so in the Summary; what is new is the specific assembly, and the assembly is the useful part. The 17% enhancement of |F| at the peak from the exponential is a nice, checkable statement, and the phase-shift prediction in Eq. (18) falling out of the same formula is a good consistency test that the construction is not arbitrary. The O(p^4) matching to ChPT is done explicitly and correctly, and the rho width in Eq. (16) coming out at 144 MeV against 150.7 is a genuine success, not a fit. Where I would push: the step from Eq. (15) to Eq. (17). Eq. (15) is singular at s = M_rho^2, so it cannot describe the peak at all; the fix is introduced with 'This suggests to shift...' and the only quantitative support beyond leading order is delegated to Ref. [14], 'F. Guerrero, in preparation'. That is an unpublished single-author note, and it is load-bearing for exactly the region the paper cares about. The matching of the Omnes subtraction polynomial to Eq. (8) has the same character: Eq. (8) is additive, Eq. (15) multiplicative, so the matching holds at leading order in s/M_rho^2 only. Neither step is fitted to form-factor data, so I would not call the result tuned, but the sensitivity to those choices is not quantified anywhere. Minor: the data comparison is graphical, no chi-squared, no truncation band. The authors themselves flag the O(p^2) phase shift as poor at high energy, so that one is acknowledged, not hidden. Read it as a clean organizing formula, not a precision calculation. It deserves referees, and the referee report should ask for the Ref. [14] check to be made public or reproduced in the letter.

Referee report

Significance

If it holds, the result is a genuinely predictive statement: a single analytic expression built only from measured constants reproduces both the modulus and the phase of the pion form factor through the rho resonance region, where ChPT alone is useless. The ingredients are standard, but their combination into one formula with no fitted form-factor parameter is a useful and falsifiable benchmark, testable against e+e- -> pi+pi-, tau -> nu_tau pi pi0 and the phase-shift analyses, and readily extendable to other current matrix elements. The explicit prediction for delta_11(s) and for its asymptotic limit is a further checkable output. The paper is honest about the leading-order-in-1/N_C character of the VMD input and about the poor quality of the O(p^2) phase shift used in the Omnes integral.

Major objections (3)

[Section 4, Eqs. (15)-(17)] The step from Eq. (15) to Eq. (17) is load-bearing for the paper's headline claim and is not derived in the manuscript. Eq. (15) is singular at s = M_rho^2, so it cannot describe the rho peak at all; the peak description rests entirely on moving Im A into the propagator as -i M_rho Gamma_rho(s). The text introduces this with 'This suggests to shift the imaginary part...', and the only quantitative support that Eq. (17) is the correct choice beyond O(p^4) is delegated to Ref. [14], 'F. Guerrero, in preparation'. Since the peak region is precisely where O(p^4) is insufficient, the reader cannot verify the central claim from the published literature. Please either include the O(p^6) comparison in this letter (a short table of the leading and subleading chiral-log coefficients of Eq. (17) versus the two-loop result of Ref. [10] would suffice), or replace Ref. [14] with a published source, or explicitly soften the claim that Eq. (17) is derived rather than motivated.
[Section 3, Eqs. (8), (12) and (15)] The Omnes subtraction polynomial is fixed by matching the multiplicative form (VMD times exponential) to the additive form of Eq. (8) (VMD plus one-loop), a matching that can only hold at leading order in s/M_rho^2. The authors acknowledge the residual polynomial ambiguity ('it remains, however, a polynomial ambiguity at higher orders'), but the consequence is not quantified: the 17% enhancement of |F(s)| at s = M_rho^2, which is what brings the curve onto the data at the peak, follows from this matching choice. Please give a sensitivity estimate, for example by varying the matching point or the subtraction order, by dropping the K-bar K term in Eq. (16), or by using the O(p^4) phase shift in the Omnes integral, and state how much the peak normalization moves. A statement such as 'the peak normalization is stable at the x% level under these variations' would convert a scheme choice into a controlled approximation.
[Section 5, Figures 1 and 2] The sole evidence offered for the claim of a 'surprisingly good description of the experimental data' is the graphical comparison in Figures 1 and 2. There is no chi-squared, no residual plot, no table of F(s) at selected energies, and no theoretical uncertainty band from the truncation. For a parameter-free prediction this is the natural quantitative test and it is inexpensive to provide. Please add at least a small table of |F(s)|^2 (or F(s)) at a handful of energies across the quoted range with the experimental errors and the corresponding residuals, and an estimate of the theoretical uncertainty obtained by varying M_rho within its error bar and by the variations listed in the previous comment. The same applies to delta_11 in Figure 2.

Minor objections (6)

[Eqs. (3) and (16)] Please check the sign convention in Eq. (3). As printed, the term sigma_P^3 ln[(sigma_P+1)/(sigma_P-1)] gives Im A = +pi sigma_P^3 above threshold, which combined with the explicit minus sign in Eq. (16) would make Gamma_rho(s) negative. The physical requirement Im F > 0 above threshold, together with Eq. (14), fixes the sign; a clarifying sentence on the branch of the logarithm would remove the ambiguity.
[Figure 1] The horizontal axis is labelled 's/sqrt(s)', which is just sqrt(s); please label it directly as sqrt(s) in GeV, and state the binning or smoothing applied to the data points.
[Section 5, omega contamination] The isoscalar omega contamination near the rho peak is mentioned and dismissed as invisible at the scale of the figure. A one-line numerical estimate of its size, with the reference to Ref. [16], would let the reader judge whether it can affect the peak comparison.
[Section 5, data selection] The introduction motivates the work by the new hadronic tau data, but the numerical comparison uses only e+e- -> pi+pi-. Since tau -> nu_tau pi- pi0 gives an independent and statistically superior determination of the same isovector form factor, a brief remark on whether the prediction also describes those data, or why they were not used, would strengthen the phenomenological case.
[Section 4, O(p^6) statement] The assertion that Eq. (17) 'does a quite good job in generating the leading O(p^6) contributions' and reproduces the linear and quadratic chiral logarithms is stated qualitatively and supported only by Ref. [14]. Even a single sentence quoting the numerical size of the discrepancy in the polynomial part would make this claim assessable within the letter.
[Text, Sections 3-4] There are several typographical and grammatical slips that should be corrected: 'Eqs. (15) has obvious shortcomings' (Section 4), 'makes the phase shift pass trough pi/2' (Section 4), and 'the resulting parameter-free prediction' hyphenation in the abstract. None affects the physics.

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The paper

The paper claims

The pion form factor is written down as a closed expression built only from measured constants, and it matches the data up to about 1 GeV.

The reading

This paper asks how much of the pion electromagnetic form factor — the function that says how likely a photon is to turn into a pair of pions at a given energy — can be written down without fitting anything to form-factor data. The authors combine four ingredients: the $rho$ meson as the dominant resonance in the large-$N_C$ limit of QCD, the one-loop chiral corrections that carry the pion-pion final-state interaction, the Omnès dispersion solution that exponentiates those corrections, and a Dyson summation of the $rho$ self-energy that turns the $rho$ width into an energy-dependent width. The output is a single closed expression, Eq. (17), containing only $m_pi$, $m_K$, $M_rho$ and $f_pi$. If it holds, both the shape and the normalization of the $rho$ peak, and the pion-pion P-wave phase shift, follow from constants measured elsewhere rather than from tuning to the form factor itself. The authors report agreement with $e^+e^- to pi^+pi^-$ data and with phase-shift data up to energies of order 1 GeV.

What carries the argument

Two objects carry the argument. The first is the Omnès solution of the dispersion relation for $F(s)$, which turns the Watson final-state theorem — the statement that below the inelastic threshold the phase of the form factor equals the pion-pion P-wave phase shift — into an exponential of the phase shift integrated over energy. This is what converts a one-loop chiral logarithm into a resummed object valid further up in energy. The second is the $rho$ propagator with a running width, obtained by Dyson-summing the $rho$ self-energy computed in the resonance chiral effective theory. The authors' move is to shift the imaginary part of the loop function out of the Omnès exponential and into that propagator, which leaves the $O(p^4)$ chiral result unchanged but makes the phase shift pass through $pi/2$ at the resonance. The supporting relation is $F_V G_V = f_pi^2$, which follows from requiring that $F(s)$ obey an unsubtracted dispersion relation at leading order in $1/N_C$; it is what fixes the $rho$ coupling and hence the overall normalization.

Extended reading notes

Core claim

The central claim is Eq. (17): $F(s) = M_rho^2 / [M_rho^2 - s - i M_rho Gamma_rho(s)] times exp{ -s/(96 pi^2 f_pi^2) [ Re A(m_pi^2/s, m_pi^2/M_rho^2) + (1/2) Re A(m_K^2/s, m_K^2/M_rho^2) ] }$, with the running width $Gamma_rho(s)$ given in Eq. (16) by the same loop function's imaginary part. Every quantity in it is a measured constant, so the authors call it a parameter-free prediction. Two consequences are drawn. First, the exponential factor is not decorative: it raises $|F(s)|$ by about 17 percent at $s = M_rho^2$, and that enhancement is what brings the curve onto the data at the peak. Second, reading the phase off the same expression gives $delta_{11}(s) = arctan{ M_rho Gamma_rho(s) / (M_rho^2 - s) }$, which reduces to the lowest-order chiral result at small $s$, passes through $pi/2$ at the $rho$ mass, and approaches $161^circ$ asymptotically once the $K bar K$ channel is included ($167^circ$ with the elastic $2pi$ channel alone). The authors also note that the same logic reproduces the measured $O(p^4)$ chiral coupling, $L_9 = f_pi^2/(2 M_rho^2) = 7.2 times 10^{-3}$, in agreement with the value extracted from the pion charge radius.

Load-bearing premise

The subtraction polynomial in the Omnès representation is not derived: it is fixed by matching to the one-loop $rho$-improved expression of Eq. (8), and the imaginary part is then moved out of the exponential and into the $rho$ propagator because Dyson summation is said to suggest it, so if that matching is not the correct way to fix the subtraction, the 17 percent enhancement that puts the curve on the data at the $rho$ peak is a choice rather than a prediction.

Editorial extensions

If this is right

The $rho$-peak normalization of the pion form factor becomes a prediction rather than a fit: the 17 percent enhancement from the summed chiral logarithms must appear in the data at $s = M_rho^2$.
The pion-pion P-wave phase shift acquires a closed form that interpolates between the low-energy chiral result and a definite high-energy limit of $161^circ$ to $167^circ$.
The measured value of the chiral coupling $L_9$ is explained as a large-$N_C$ $rho$-exchange effect, $L_9 = f_pi^2/(2M_rho^2)$, rather than as an independent parameter.
The same construction — leading large-$N_C$ resonance exchange, Omnès exponentiation of final-state interactions, running width from Dyson summation — is proposed as a template for other current matrix elements where the underlying dynamics is messier.

Reading between the lines

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

If the shape and normalization of the $rho$ peak really follow from $m_pi$, $m_K$, $M_rho$ and $f_pi$ alone, then the same four constants should also pin down the two-pion spectral function measured in $tau^- to nu_tau pi^- pi^0$, giving a cross-check in a channel with different systematic uncertainties from $e^+e^-$.
Because the phase is predicted as well as the modulus, a simultaneous fit to $|F(s)|$ and to $delta_{11}$ is over-determined; the interesting test is whether the same parameters that describe the modulus also describe the phase near 1 GeV, where the $K bar K$ channel starts to matter.
The construction is a statement about how much of the resonance region is fixed by short-distance consistency conditions plus unitarity, so a natural extension is to ask which of the remaining $O(p^6)$ chiral couplings this approach leaves undetermined, and whether those are the ones that would shift the peak normalization.
The 'parameter-free' character depends on treating the Omnès subtraction polynomial as fixed by matching to the one-loop $rho$-improved result; an independent determination of those subtraction constants from data would convert the claim into a genuine test rather than a consistency check.

Original abstract

Using our present knowledge on effective hadronic theories, short-distance QCD information, the $1/N_C$ expansion, analyticity and unitarity, we derive an expression for the pion form factor, in terms of $m_\pi, m_K, M_\rho$ and $f_\pi$. This parameter-free prediction provides a surprisingly good description of the experimental data up to energies of the order of 1 GeV.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Using our present knowledge on effective hadronic theories, short-distance QCD information, the $1/N_C$ expansion, analyticity and unitarity, we derive an expression for the pion form factor, in terms of $m_\pi, m_K, M_\rho$ and $f_\pi$. This parameter-free prediction provides a surprisingly good description of the experimental data up to energies of the order of 1 GeV.

background contribution method result interpretation main claim

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s0	Using our present knowledge on effective hadronic theories, short-distance QCD information, the $1/N_C$ expansion, analyticity and unitarity, we derive an expression for the pion form factor, in terms of $m_\pi, m_K, M_\rho$ and $f_\pi$.	contribution 0.68	0.94	0.25	asserted 1.00
s1	This parameter-free prediction provides a surprisingly good description of the experimental data up to energies of the order of 1 GeV.	result 0.94	0.84	0.74	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

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Figures

**Figure 1.** F(s)| 2 (in logarithmic scale) versus s/√ s. The continuous curve shows the theoretical prediction in Eq. (17). The known coefficients for linear and quadratic logarithms of the ChPT result,10 are well reproduced in the chiral limit. While this is also true for Eq. (15), the expansion of Eq. (17) gives a better description14 of the polynomial part of the O(p 6 ) ChPT result. 5. Numerical Results We can see graphically in figure 1 how Eq. (17) provides a very good description of the data15 up to rather high energies. The only parameters appearing in the pion form factor formula

**Figure 2.** δ 1 1 (s) versus √ s. The continuous curve shows the theoretical prediction in Eq. (18). This contribution is well–known and generates a slight distortion of the ρ peak, which can easily be included in the theoretical formula.16 The effect cannot be appreciated at the scale of the figure. We also obtain a prediction for the phase shift δ 1 1 . From Eq. (17) we get: δ 1 1 (s) = arctan ( MρΓρ(s) M2 ρ − s ) . (18) For s << M2 ρ , this expression reduces to the O(p 2 ) ChPT result in Eq. (14). As shown in figure 2, the improvement obtained through Eq. (18) provides a qu

Figures reproduced from arXiv:hep-ph/9707347 by the authors.

References

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Importantly, we find for the sum rule <ref> a value of $1.003$ taking a cutoff of $Λ=3$ GeV, in line with our model expectations and ensuring relibale results for DR2.[Interesting enough, if fitting to the exponential parametrization in Ref. [ref] , values below $1$ are obtained, pointing to a violation of the sum rule and unreliable results.

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To estimate the coefficient $c_T$ multiplying $ϵ_T$ in pipi0, we use a relatively simple form of the vector form factor based on analyticity, unitarity, chiral symmetry, and the high-momentum asymptotic behavior of QCD [ref] , as well as a dispersive parameterization based on data (see Ref. [ref] and references therein).[At the precision needed we can ignore isospin-breaking and new physics contaminations.] We treat the tensor form factor as follows: * First, we assume that the proportionality of the tensor and vector form factors, which is exact in the elastic region [ref] , holds over the whole $s$ region allowed by kinematics, namely F_T^ππ (s) = F^ππ_T(0) F_V^ππ (s) .

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We also used the Guerrero-Pich (GP) [ref] parametrization.

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FSI are resummed by means of an Omnès function [ref] , see Section [ref].

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To make direct comparisons with Ref. [ref] , we use the same pion vector form factor as employed in the former reference, which is originally calculated within the resonance chiral theory by taking into account the constraints of analyticity and unitarity in Ref. [ref] .

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The description provided by $RχT$ complies with the low-energy properties of the underlying theory, at least up to next-to-leading order[Also the leading next-to-next-to-leading (NNLO) order terms [ref] are reproduced [ref] .].

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Regarding the form factors, it is noted that recently Ref. [ref] performs a systematic investigation of different form-factor parametrizations, including the Gounaris-Sakurai model [ref] , Kuhn-Santamaria model [ref] , Guerrero-Pich model [ref] and the dispersive form [ref] .

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The vector form factor at the next-to-leading order in 1/N(C): chiral couplings L9(mu) and C88(mu) - C90(mu) · The vector form factor of the pion · 0.92
Alternative dispersive analyses applied so far are based on representations either in terms of the phase, the so-called Omnès representation [ref] , or in terms of the modulus [ref] .

Spacelike pion form factor from analytic continuation and the onset of perturbative QCD · Analytic continuation · 0.93

Discussion

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