We present a fit of the spin-independent electromagnetic polarisabilities of
the proton to low-energy Compton scattering data in the framework of covariant
baryon chiral effective field theory. Using the Baldin sum rule to constrain
their sum, we obtain
$\alpha=[10.6\pm0.25$(stat)$\pm0.2$(Baldin)$\pm0.4$(theory)$]\times10^{-4}$fm$^3$
and $\beta
=[3.2\mp0.25$(stat)$\pm0.2$(Baldin)$\mp0.4$(theory)$]\times10^{-4}$fm$^3$, in
excellent agreement with other chiral extractions of the same quantities.Comment: 4 pages, 2 figure

Lensky, Vadim

McGovern, Judith A.

English

arXiv

We present a fit of the spin-independent electromagnetic polarizabilities of the proton to low-energy Compton scattering data in the framework of covariant baryon chiral effective field theory. Using the Baldin sum rule to constrain their sum, we obtain αE1=[10.6±0.25(stat)±0. 2(Baldin)±0.4(theory)]×10-4fm3 and βM1=[3.2-0.25(stat) ±0.2(Baldin)-0.4(theory)]×10-4fm3, in excellent agreement with other chiral extractions of the same quantities. © 2014 American Physical Society

McGovern, Judith

The Egyptian Cardiothoracic Surgeon (ECTS - E-Journal)

Proton polarizabilities from Compton data using covariant chiral effective field theory

The University of Manchester - Institutional Repository

RAPID COMMUNICATIONSPHYSICAL REVIEW C 89, 032202(R) (2014)Proton polarizabilities from Compton data using covariant chiral effective field theoryVadim Lensky1,2,* and Judith McGovern1,†1Theoretical Physics Group, School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, United Kingdom2Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia(Received 16 January 2014; published 10 March 2014)We present a fit of the spin-independent electromagnetic polarizabilities of the proton to low-energy Comptonscattering data in the framework of covariant baryon chiral effective field theory. Using the Baldin sum ruleto constrain their sum, we obtain αE1 = [10.6 ± 0.25(stat) ± 0.2(Baldin) ± 0.4(theory)] × 10−4 fm3 and βM1 =[3.2 ∓ 0.25(stat) ± 0.2(Baldin) ∓ 0.4(theory)] × 10−4 fm3, in excellent agreement with other chiral extractionsof the same quantities.DOI: 10.1103/PhysRevC.89.032202 PACS number(s): 12.39.Fe, 13.60.Fz, 14.20.DhThe electromagnetic polarizabilities of the proton havebeen a subject of investigation for many years; the earliestextractions from low-energy Compton scattering data werecarried out in the 1950s, and the relevant database was greatlyexpanded in the 1990s. In the very-low-energy regime one canmake an expansion of the cross section which deviates fromthe Powell one only through the inclusion of the two spin-independent polarizabilities αE1 and βM1. However, since veryfew measurements have been taken below 80 MeV, almost allextractions require theoretical input to describe the evolutionof the cross section with energy. Historically, dispersionrelation (DR) approaches were used, with input from pionphotoproduction data. A model-independent constraint can beobtained from the Baldin sum rule, most recently evaluatedto give αE1 + βM1 = 13.8 ± 0.4 in units of 10−4 fm3 [1], sotypically the parameter extracted is αE1 − βM1. In 2001 Olmosde León et al. published the most comprehensive data setyet, obtained with the TAPS detector at the Mainz Microtron,and in a DR framework they analyzed it together with other“modern” data to give αE1 − βM1 = 10.5 ± 0.9 ± 0.7 in thesame units [1]. For some time this was regarded as the definitiveresult.However, chiral effective field theories (χEFTs) can alsobe used to describe Compton scattering amplitudes. Theseare field theories in which the interactions of low-energydegrees of freedom are governed by the symmetries of QCD,and scattering amplitudes can be systematically expanded inpowers of the ratio of light to heavy scales. The former aretypically external particle momenta of the order of the pionmass, and the latter are governed by those particles such asthe ρ meson which are not included explicitly in the theorybut whose effects, along with other short distance physics, areencoded in low-energy constants. At leading one-loop orderin the theory with pions and nucleons these predictions areparameter free, but beyond leading order αE1 and βM1 are freeparameters which can be fit to data. The first attempt to do thiswas the work of Beane et al. [2,3], working in heavy baryon(HB) chiral perturbation theory. The absence of a dynamical isobar restricted the fit to relatively low momentum transfer,*lensky@itep.ru†Judith.McGovern@manchester.ac.ukand as a result the statistical errors were large. However,the inclusion of the  followed shortly [4–6]. Most recently,the result αE1 − βM1 = 7.5 ± 0.7 ± 0.6 has been obtained byMcGovern et al. in Ref. [7]. Although this value is compatiblewith the previous chiral extractions [2,3,6], the calculation wascarried out to a sufficiently high order, and fit a sufficientlylarge set of experimental data, that the results were preciseenough to demonstrate a tension with DR-based results at,roughly, the 2σ level (combining all errors in quadrature).1In addition to αE1 and βM1 there are also four spinpolarizabilities of the proton. One combination of these,γ0, satisfies a Baldin-like sum rule and so is reasonablywell known: γ0 = −0.90 ± 0.08(stat) ± 0.11(sys) in units of10−4 fm4 [9]. In looking for sources of the discrepancy inthe extracted values of αE1 − βM1, it has been suggestedthat the problem is that the EFT and DR values of the spinpolarizabilities are quite discrepant; since those obtained atthis order in the EFT are not in good agreement with thesum-rule determination of γ0, this has been used to suggestthat the EFT extraction is less reliable, or at least that theerrors are underestimated.In this paper we consider the situation in a different variantof χEFT, namely, one which does not use the heavy baryonexpansion but treats the nucleon fields as Dirac spinors [10].Compton scattering amplitudes were first calculated in thisapproach by Lensky and Pascalutsa in Ref. [11], using (amodification of) the extended on-mass-shell (EOMS) renor-malization scheme [12]. The power-counting scheme we useis the so-called δ counting, with the small parameter δ ∼mπ/ ∼ /χ ∼ 0.4, where  = M − MN and χ ∼ mρis the chiral scale [5]. In this counting, the HB work of Ref. [7]is O(e2δ4) in the low-energy region. The covariant work ofRef. [11] is O(e2δ3), at which order αE1 and βM1 are notfree parameters but predicted. Substantial contributions toone or both come from πN and π loops and from -polegraphs, and the final values, which are the result of significant1Because the paper is not easily available, the rather betteragreement with the work of Baranov et al. [8] which, using aDR-based fit to world data, finds αE1 − βM1 = 9.5 ± 1.0 ± 0.7, hasbeen less noted. In fact for “modern” pre-TAPS data alone, Baranov’sresult is 7.7 ± 1.2.0556-2813/2014/89(3)/032202(4) 032202-1 ©2014 American Physical SocietyRAPID COMMUNICATIONSVADIM LENSKY AND JUDITH MCGOVERN PHYSICAL REVIEW C 89, 032202(R) (2014)TABLE I. Predictions for the spin polarizabilities in three variantsof χPT and in DR, in units of 10−4 fm4.γE1E1 γM1M1 γE1M2 γM1E2HB δ3 [13] −5.5 2.1 0.5 1.3HB δ4 [7] −1.1 2.2a −0.4 1.9Cov. δ3 [14] −3.3 3.0 0.2 1.1DR [13,15] −3.85 ± 0.45 2.8 ± 0.1 −0.15 ± 0.15 2.0 ± 0.1aThis value was fit in Ref. [7]; the predicted value would beγM1M1 = 6.4.cancellations, are αE1 = 10.8 ± 0.7 and βM1 = 4.0 ± 0.7. Theerrors are theory only. Though no fit to data is involved, plotswere presented in Ref. [11] to show that the trend of world dataup to around 170 MeV is well reproduced in this calculation.The spin polarizabilities are also predictions of the theory atthis order, and interestingly they are in good agreement withthe DR results, including for instance γ0 = −0.9; see Table I.As yet no full calculation has been carried out in thecovariant theory at O(e2δ4). The extra graphs required atthis order are not only πN loop graphs with insertions ofsecond-order LECs, namely, the proton and neutron anomalousmagnetic moments and the πN scattering LECs ci , but alsophoton-nucleon seagull graphs with fourth-order LECs whichcontribute directly to αE1 and βM1. All of these were includedin the heavy baryon calculations of Ref. [7]. There, it wasshown that the contribution of the extra loop graphs was quitemodest. However, the new counter-terms δαE1 and δβM1 mustbe fit to Compton scattering data, and seem to be the principalnew effect at this order. In view of the interest in the apparentdiscrepancy between DR and χEFT extractions of αE1 andβM1, we consider a partial O(e2δ4) covariant result to be ofinterest, and so we supplement the Lagrangian for proton fieldsused in [11] with the term [16,17]L(4)πN = πe2[ψδβM1FμρFμρψ− 2M2N(δαE1 + δβM1)(∂μψ)FμρF νρ∂νψ]. (1)For a review of the power counting, and of the principlesunderlying the application of χEFT to Compton scattering,the reader is referred to Ref. [13].The database of experimental Compton scattering resultsfor energies below 170 MeV, and its treatment, is the same oneas was used in Refs. [7,13]; the following is only the briefestof summaries and a thorough discussion may be found inRef. [13]. The largest single data set is from TAPS [1], forwhich we allow a point-to-point systematic error of 4% asadvocated by Wissmann [18]; other modern data are from [19–22], and a number of older experiments also contribute somepoints. Normalization uncertainties are incorporated into theχ2 function in the usual way. As in Refs. [7,13], for our finalresult we use the data of Hallin et al. [21] below 150 MeVonly.We do not fit to any higher-energy data but check by eyethat the agreement continues to be good at higher energies (atleast as far as the deficiencies of the data sets allow such ajudgment). We take the γN coupling constants obtained in10.0 10.5 11.0 11.5 12.02.53.03.54.04.5αE1βM1FIG. 1. (Color online) 1σ contours for the one-parameter(Baldin-constrained) and two-parameter (unconstrained) fits, forthree variants of the data set: the solid (black) line is our final result,while the dashed (blue) lines exclude all Hallin data, and the dotted(green) line uses an upper cutoff of 150 MeV. These are sets (III),(II), and (I), respectively, of Ref. [7].a fit to photoproduction data: gM = 2.97 and gE = −1.0 [23].Other parameters are given in Ref. [7]. We present results bothwith and without imposing the Baldin sum rule αE1 + βM1 =13.8 ± 0.4 [1] as follows:αE1 = 10.9 ± 0.45(stat) ± 0.4(theory),(2)βM1 = 3.6 ± 0.55(stat) ± 0.4(theory),with χ2 of 111.8 for 135 degrees of freedom (d.o.f.).For the Baldin-constrained fit we obtain αE1 − βM1 = 7.4 ±0.5(stat) ± 0.4(theory), givingαE1 = 10.6 ± 0.25(stat) ± 0.2(Baldin) ± 0.4(theory),(3)βM1 = 3.2 ∓ 0.25(stat) ∓ 0.2(Baldin) ± 0.4(theory),with χ2 of 112.5 for 136 d.o.f. The theory errors have beenconservatively calculated, based on the shift of αE1 − βM1from e2δ3 to partial e2δ4, multiplied by the parameter δ ∼ 0.4.As can be seen from Fig. 1 (which also demonstrates thesensitivity to variations of the choice of database), these resultsare completely consistent with one another. Furthermorethey are consistent with the corresponding results in theheavy baryon extraction [7], and in fact the central valuesof the Baldin-constrained fit are essentially equal to thoseof that work, namely, αE1 = 10.65 ± 0.35(stat), βM1 = 3.2 ∓0.35(stat). (The statistical error is higher than in the currentwork because an extra parameter, γM1M1, was fitted in Ref. [7].)It is also interesting to recall that partial and full O(e2δ4) HBextractions of αE1 − βM1 agree very well [7].In Fig. 2 we show the fit along with a selection of data. Thelow-energy fit is excellent, and below the photoproductionthreshold the predictions of covariant and heavy-baryon χPTare largely indistinguishable. (The similarity of the covariantand HB predictions for the unpolarized cross sections inthis region was already noted in Ref. [24].) The values ofαE1 and βM1 in all three cases are extremely close, but thespin polarizabilities are quite disparate. The HB O(e2δ3)curves give a stronger cusp than the covariant version, but032202-2RAPID COMMUNICATIONSPROTON POLARIZABILITIES FROM COMPTON DATA . . . PHYSICAL REVIEW C 89, 032202(R) (2014)FIG. 2. (Color online) Predictions for the cross section in nb/sr as a function of incoming photon energy in MeV (both in the laboratoryframe), using the values from the Baldin-constrained fit αE1 = 10.6 and βM1 = 3.2, and with the band showing statistical errors only (solid line,red). For comparison the results of HB χPT are shown: O(e2δ3) [13] (dotted, light blue) and O(e2δ4) [7] (dashed, dark blue). Data points areshown without floating normalization, and including those points which were excluded from the fit. The key is given in Table 3.1 of Ref. [13]; inparticular purple diamonds are Mainz data [1,25]; red squares, MacGibbon [22]; black squares, Hallin [21]; and blue triangles, Federspiel [19].The insets show the extended range beyond ωlab = 200 MeV.the HB O(e2δ4) is in very good agreement with the covariantcalculation up to 200 MeV and beyond. Though only low-energy data are used in the fit, the good agreement with theMainz data [25] continues into the resonance region, thoughat most angles the  peak is somewhat too high. It should benoted though that in this region the power counting changesand the EFT calculation is only leading order. Moreover theHB fits varied the γN coupling constant gM whereas inthe present, covariant fit we have used the value obtainedin the covariant theory from photoproduction [23], which isaround 10% higher. The leading dependence of the heightof the peak on the coupling constant is g4M , and a betterfit in the resonance region could be obtained by allowing amodest variation of this parameter with negligible effect onthe low-energy fit.In summary, we have shown that in a fit to low-energyCompton scattering data, very similar results are obtainedfor the electromagnetic polarizabilities of the proton, αE1and βM1, whether the covariant or heavy baryon versionsof chiral effective field theory are used. In particular if theBaldin sum-rule constraint is applied, the extracted values ofαE1 − βM1 are essentially identical. This result is unexpectedbecause some other predictions of the two versions, notablythe spin polarizabilities, are apparently not in good agreement.However, these are not the dominant drivers of the energyevolution at photon energies comparable to mπ , and the twoversions of the theory make very similar predictions for theoverall cross section. It should be noted that this energydependence, including the cusp at photoproduction thresholdgenerated by chiral loops, is highly nontrivial. The excellentfit to data with only one free parameter demonstrates thepredictive power of χEFT.It is still to be tested whether the tension between the χEFTextraction and the widely accepted dispersion-relation-basedone of Ref. [1] is due to the larger and more carefully handleddata set used in the chiral extractions, or to some other featureof the predictions of the two theories. More unpolarizeddata, particularly at energies around mπ at backward angles,might be needed to resolve the issue. To further explore spinpolarizabilities, though, it is clear that polarized scatteringmeasurements will be required.We are grateful to Daniel Phillips and Harald Grießhammerfor discussions regarding aspects of these calculations, and032202-3RAPID COMMUNICATIONSVADIM LENSKY AND JUDITH MCGOVERN PHYSICAL REVIEW C 89, 032202(R) (2014)to Mike Birse and Vladimir Pascalutsa for useful commentson the manuscript. This work has been supported in part byU.K. Science and Technology Facilities Council Grants No.ST/F012047/1 and No. ST/J000159/1.[1] V. Olmos de León et al., Eur. Phys. J. A 10, 207 (2001).[2] S. R. Beane, M. Malheiro, J. A. McGovern, D. R. Phillips,U. van Kolck, Phys. Lett. B 567, 200 (2003); ,607, 320(2005).[3] S. R. Beane, M. Malheiro, J. A. McGovern, D. R. Phillips, andU. van Kolck, Nucl. Phys. A 747, 311 (2005).[4] T. R. Hemmert, B. R. Holstein, and J. Kambor, Phys. Rev. D 55,5598 (1997).[5] V. Pascalutsa and D. R. Phillips, Phys. Rev. C 67, 055202(2003).[6] R. P. 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Proton polarisabilities from Compton data using Covariant Chiral EFT

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