Proton polarisabilities from Compton data using Covariant Chiral EFT
PProton polarisabilities from Compton data using Covariant Chiral EFT
Vadim Lensky
1, 2, ∗ and Judith McGovern † Theoretical Physics Group, School of Physics and Astronomy,University of Manchester, Manchester, M13 9PL, United Kingdom Institute for Theoretical and Experimental Physics,B. Cheremushkinskaya 25, 117218 Moscow, Russia (Dated: October 25, 2018)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 α E = [10 . ± . ± . ± . × − fm and β M = [3 . ∓ . ± . ∓ . × − fm , inexcellent agreement with other chiral extractions of the same quantities. PACS numbers: 12.39.Fe, 13.60.Fz, 14.20.Dh
The electromagnetic polarisabilities of the proton havebeen a subject of investigation for many years; the earli-est extractions from low-energy Compton scattering datawere carried out in the 1950s, and the relevant databasewas greatly expanded in the 1990s. In the very low-energy regime one can make an expansion of the crosssection which deviates from the Thomson scattering onlythrough the inclusion of the two spin-independent polar-isabilities α E and β M . However, since very few mea-surements have been taken below 80 MeV, almost allextractions require theoretical input to describe the evo-lution of the cross section with energy. Historically, dis-persion relation (DR) approaches were used, with inputfrom pion photoproduction data. A model-independentconstraint can be obtained from the Baldin sum rule,most recently evaluated to give α E + β M = 13 . ± . − fm [1], so typically the parameter ex-tracted is α E − β M . In 2001 Olmos de Le´on et al. published the most comprehensive data set yet, obtainedwith the TAPS detector at MAMI, and in a DR frame-work analysed it together with other “modern” data togive α E − β M = 10 . ± . ± . χ EFT) canalso be used to describe Compton scattering amplitudes.These are field theories in which the interactions of low-energy degrees of freedom are governed by the symme-tries of QCD, and scattering amplitudes can be system-atically expanded in powers of the ratio of light to heavyscales. The former are typically external particle mo-menta of the order of the pion mass, and the latter aregoverned by those particles such as the ρ meson whichare not included explicitly in the theory but whose ef-fects, along with other short distance physics, are en-coded in low energy constants. At leading one-loop orderin the theory with pion and nucleons these predictions areparameter-free, but beyond leading order α E and β M are free parameters which can be fit to data. The firstattempt to do this was the work of Beane et al. [2, 3],working in heavy baryon (HB) chiral perturbation the- ory. The absence of a dynamical Delta isobar restrictedthe fit to relatively low momentum transfer, and as aresult the statistical errors were large. However, the in-clusion of the Delta followed shortly [4–6]. Most recently,the result α E − β M = 7 . ± . ± . et al. in ref. [7]. Although this value iscompatible with the previous chiral extractions [2, 3, 6],the calculation was carried out to a sufficiently high or-der, and fit a sufficiently large set of experimental data,that the results were precise enough to demonstrate atension with DR-based results at, roughly, the 2 σ level(combining all errors in quadrature). In addition to α E and β M there are also four spinpolarisabilities of the proton. One combination of these, γ , satisfies a Baldin-like sum rule and so is reasonablywell known: γ = − . ± . ± . − fm , [9]. In looking for sources of the discrepancy inthe extracted values of α E − β M , it has been suggestedthat the fact that the EFT and DR values of the spinpolarisabilities are quite discrepant is the problem; sincethose obtained at this order in the EFT are not in goodagreement with the sum-rule determination of γ , thishas been used to suggest that the EFT extraction is lessreliable, or at least that the errors are underestimated.In this paper we consider the situation in a differentvariant of χ EFT, namely one which does not use theheavy-baryon expansion but treats the nucleon fields asDirac spinors [10]. Compton scattering amplitudes werefirst calculated in this approach by Lensky and Pasca-lutsa in ref. [11], using (a modification of) the EOMS re-mormalisation scheme [12]. The power-counting schemewe use is the so-called “ δ -counting”, with the small pa-rameter δ ∼ m π / ∆ ∼ ∆ / Λ χ ∼ .
4, where ∆ = M ∆ − M N Because the paper is not easily available, the rather better agree-ment with the work of Baranov et al. [8] which, using a DR-basedfit to world data, finds α E − β M = 9 . ± . ± .
7, has beenless noted. In fact for “modern” pre-TAPS data alone, Baranov’sresult is 7 . ± . a r X i v : . [ nu c l - t h ] J a n γ E E γ M M γ E M γ M E HB δ [13] − . . . δ [7] − . . ∗ − . δ [14] − . . . − . ± .
45 2 . ± . − . ± .
15 2 . ± . χ PT and in DR, in units of 10 − fm . *This valuewas fit in ref. [7]; the predicted value would be γ M M = 6 . and Λ χ ∼ m ρ is the chiral scale [5]. In this counting, theHB work of ref. [7] is O ( e δ ) in the low-energy region.The covariant work of ref. [11] is O ( e δ ), at which or-der α E and β M are not free parameters but predicted.Substantial contributions to one or both come from πN and π ∆ loops and from ∆-pole graphs, and the final val-ues, which are the result of significant cancellations, are α E = 10 . ± . β M = 4 . ± .
7. The errors aretheory only. Though no fit to data is involved, plots werepresented in ref. [11] to show that the trend of world dataup to around 170 MeV is well reproduced in this calcula-tion. The spin polarisabilities are also predictions of thetheory at this order, and interestingly they are in goodagreement with the DR results, including for instance γ = − .
9; see table I.As yet no full calculation has been carried out in thecovariant theory at O ( e δ ). The extra graphs requiredat this order are not only πN loop graphs with inser-tions of second-order LECs, namely the proton and neu-tron anomalous magnetic moments and the πN scatter-ing LECs c i , but also photon-nucleon seagull graphs withfourth-order LECs which contribute directly to α E and β M . All of these were included in the heavy baryon cal-culations of ref. [7]. There it was shown that the contri-bution of the extra loop graphs was quite modest. How-ever the new counter-terms δα E and δβ M must be fit toCompton scattering data, and seem to be the principalnew effect at this order. In view of the interest in the ap-parent discrepancy between DR and χ EFT extractionsof α E and β M , we consider a partial O ( e δ ) covari-ant result to be of interest, and so we supplement theLagrangian for proton fields used in [11] with the term[16, 17] L (4) πN = πe (cid:0) ψδβ M F µρ F µρ ψ − M ( δα E + δβ M )( ∂ µ ψ ) F µρ F νρ ∂ ν ψ (cid:1) . (1)For a review of the power counting, and of the principlesunderlying the application of χ EFT to Compton scatter-ing, the reader is referred to ref. [13].The database of experimental Compton scattering re-sults for energies below 170 MeV, and its treatment, isthe same one as was used in ref. [7, 13]; the followingis only the briefest of summaries and a thorough discus-sion may be found in ref. [13]. The largest single dataset Α E1 Β M1 FIG. 1. (Colour online) One sigma contours for the oneparameter (Baldin-constrained) and two parameter (uncon-strained) fits, for three variants of the data set: the solid(black) line is our final result, while the dashed (blue) linesexclude all Hallin data and the dotted (green) line uses anupper cut-off of 150 MeV. These are sets (III), (II) and (I)respectively of ref. [7]. is from TAPS [1], for which we allow a point-to-pointsystematic error of 4% as advocated by Wissmann [18];other modern data is from [19–22], and a number of olderexperiments also contribute some points. Normalisationuncertainties are incorporated into the χ function in theusual way. As in ref. [7, 13] we used the data of Hallin etal. [21] below 150 MeV only.We do not fit to any higher-energy data but check byeye that the agreement continues to be good at higherenergies (at least as far as the deficiencies of the datasets allow such a judgement). We take the γN ∆ cou-pling constants obtained in a fit to photoproduction data: g M = 2 .
97 and g E = − . α E + β M = 13 . ± . α E = 10 . ± . ± . β M = 3 . ± . ± . χ of 111.8 for 135 d.o.f. For the Baldin-constrainedfit we obtain α E − β M = 7 . ± . ± . α E =10 . ± . ± . ± . β M =3 . ∓ . ∓ . ± . χ of 112.5 for 136 d.o.f. The theory errors havebeen conservatively calculated, based on the shift of α E − β M from e δ to partial e δ , multiplied by theparameter δ ∼ . Θ lab (cid:61)
60 °
200 220 240 260 280 300 320 340050100150200250300 Θ lab (cid:61)
85 °
200 220 240 260 280 300 320 340050100150200 Θ lab (cid:61)
133 °
200 220 240 260 280 300 320 340050100150 Θ lab (cid:61)
155 °
200 220 240 260 280 300 320 340050100150
FIG. 2. (Colour online) The predictions for the cross section in nb/sr as a function of incoming photon energy in MeV (both inlab frame), using the values from the Baldin-constrained fit α E = 10 . β M = 3 .
2, and with the band showing statisticalerrors only (solid line, red). For comparison the results of HB χ PT are shown: O ( e δ ) [13] (dotted, light blue) and O ( e δ )[7] (dashed, dark blue). Data points are shown without floating normalisation, and including those points which were excludedfrom the fit. The key is given in Table 3.1 of ref. [13]; in particular purple diamonds are Mainz data [1, 25]; red squares,MacGibbon [22]; black squares, Hallin [21]; and blue triangles, Federspiel [19]. These results are completely consistent with one an-other, as can be seen from Fig. 1 which also demonstratesthe sensitivity to variations of the choice of database.Furthermore they are consistent with the correspondingresults in the heavy baryon extraction [7], and in factthe central values of the Baldin-constrained fit are es-sentially equal to those of that work, namely α E =10 . ± . β M = 3 . ∓ . γ M M , was fitted in ref. [7].) It isalso interesting to recall that partial and full O ( e δ ) HBextractions of α E − β M agree very well [7].In Fig. 2 we show the fit along with a selection of data.The low-energy fit is excellent, and below the photopro-duction threshold the predictions of covariant and heavy-baryon χ PT are largely indistinguishable. (The similar-ity of the covariant and HB predictions for the unpo-larised cross sections in this region was already noted inref. [24].) The values of α E and β M in all three cases areextremely close, but the spin polarisabilities are quite dis-parate. The HB O ( e δ ) curves give a stronger cusp thanthe covariant version, but the HB O ( e δ ) is in very good agreement with the covariant calculation up to 200 MeVand beyond. Though only low-energy data is used in thefit, the good agreement with the Mainz data [25] con-tinues into the resonance region, though at most anglesthe Delta peak is somewhat too high. It should be notedthough that in this region the power counting changesand the EFT calculation is only NLO. Moreover the HBfits varied the γN ∆ coupling constant g M whereas in thepresent, covariant fit we have used the value obtained inthe covariant theory from photoproduction [23], whichis around 10% higher. The leading dependence of theheight of the peak on the coupling constant is g M , anda better fit in the resonance region could be obtainedby allowing a modest variation of this parameter withnegligible effect on the low-energy fit.In summary, we have shown that, in a fit to low-energyCompton scattering data, very similar results are ob-tained for the electromagnetic polarisabilities of the pro-ton, α E and β M , whether the covariant or heavy baryonversions of chiral effective field theory are used. In par-ticular if the Baldin sum-rule constraint is applied, theextracted values of α E − β M are essentially identical.This result is unexpected because some other predictionsof the two versions, notably the spin polarisabilities, arenot in good agreement. However these are not the dom-inant drivers of the energy evolution at photon energiescomparable to m π , and the two versions of the theorymake very similar predictions for the overall cross sec-tion. It should be noted that this energy dependence,including the cusp at photoproduction threshold gener-ated by chiral loops, is highly non-trivial. The excellentfit to data with only one free parameter demonstrates thepredictive power of χ EFT.It is still to be tested whether the tension betweenthe χ EFT extraction and the widely accepted dispersion-relation-based one of ref. [1] is due to the larger and morecarefully handled dataset used in the chiral extractions,or to some other feature of the predictions of the twotheories. More unpolarised data, particularly at energiesaround m π at backward angles, might be needed to re-solve the issue. To further explore spin polarisabilities,though, it is clear that polarised scattering measurementswill be required.We are grateful to Daniel Phillips and HaraldGrießhammer for discussions regarding aspects of thesecalculations, and to Mike Birse and Vladimir Pascalutsafor useful comments on the manuscript. This work hasbeen supported in part by UK Science and TechnologyFacilities Council grants ST/F012047/1, ST/J000159/1 ∗ [email protected] † [email protected][1] V. Olmos de Le´on et al. , Eur. Phys. J.
A10 (2001) 207.[2] S. R. Beane, M. Malheiro, J. A. McGovern,D. R. Phillips, U. van Kolck, Phys. Lett.
B567 (2003)200-206, [Erratum-ibid. (2005) 320] [arXiv:nucl-th/0209002].[3] S. R. Beane, M. Malheiro, J. A. McGovern, D. R. Phillipsand U. van Kolck, Nucl. Phys. A (2005) 311[arXiv:nucl-th/0403088]. [4] T. R. Hemmert, B. R. Holstein and J. Kambor, Phys.Rev. D (1997) 5598 [arXiv:hep-ph/9612374].[5] V. Pascalutsa, D. R. Phillips, Phys. Rev. C67 (2003)055202 [arXiv:nucl-th/0212024].[6] R. P. Hildebrandt, H. W. Grießhammer, T. R. Hem-mert and B. Pasquini, Eur. Phys. J. A (2004) 293[arXiv:nucl-th/0307070].[7] J. A. McGovern, D. R. Phillips and H. W. Grießhammer,Eur. Phys. J. A (2013) 12 [arXiv:1210.4104 [nucl-th]].[8] P. S. Baranov, A. I. Lvov, V. A. Petrunkin and N. L.Shtarkov, Phys. Part. Nucl. (2001) 376.[9] B. Pasquini, P. Pedroni, D. Drechsel, Phys. Lett. B687 (2010) 160. [arXiv:1001.4230 [hep-ph]].[10] J. Gasser, M. E. Sainio and A. Svarc, Nucl. Phys. B (1988) 779.[11] V. Lensky, V. Pascalutsa, Eur. Phys. J.
C65 (2010) 195.[arXiv:0907.0451 [hep-ph]].[12] T. Fuchs, J. Gegelia, G. Japaridze and S. Scherer,power counting,” Phys. Rev. D (2003) 056005 [hep-ph/0302117].[13] H. W. Grießhammer, J. A. McGovern, D. R. Phillipsand G. Feldman, Prog. Part. Nucl. Phys. (2012) 841[arXiv:1203.6834 [nucl-th]].[14] V. Lensky, V. Pascalutsa, J. McGovern in preparation.[15] D. Drechsel, B. Pasquini and M. Vanderhaeghen, Phys.Rept. (2003) 99 [arXiv:hep-ph/0212124].[16] N. Fettes, U. -G. Meißner, M. Mojˇziˇs and S. Steininger,Annals Phys. (2000) 273 [Erratum-ibid. (2001)249] [arXiv:hep-ph/0001308].[17] N. Krupina and V. Pascalutsa, Phys. Rev. Lett. (2013) 262001 [arXiv:1304.7404 [nucl-th]].[18] F. Wissmann, Compton Scattering: Investigating theStructure of the Nucleon with Real Photons , SpringerTracts in Modern Physics, Volume 200 (2004).[19] F. J. Federspiel et al. , Phys. Rev. Lett. (1991) 1511.[20] A. Zieger et al. , Phys. Lett. B (1992) 34.[21] E. L. Hallin et al. , Phys. Rev. C48 (1993) 1497.[22] B. E. MacGibbon et al. , Phys. Rev.
C52 (1995) 2097.[23] V. Pascalutsa and M. Vanderhaeghen, Phys. Rev. D (2006) 034003 [arXiv:hep-ph/0512244].[24] V. Lensky, J. A. McGovern, D. R. Phillips and V. Pasca-lutsa, Phys. Rev. C (2012) 048201 [arXiv:1208.4559[nucl-th]].[25] S. Wolf, V. Lisin, R. Kondratiev, A. M. Massone,G. Galler, J. Ahrens, H. J. Arends and R. Beck et al. ,Eur. Phys. J. A12