We discuss the application of an effective field theory (EFT) which
incorporates the chiral symmetry of QCD to Compton scattering from the proton
and deuteron. We describe the chiral EFT analysis of the proton Compton
scattering database presented in our recent review (arXiv:1203.6834), which
gives: alpha^{(p)}=10.5 +/- 0.5(stat) +/- 0.8(theory); beta^{(p)}= 2.7 +/-
0.5(stat) +/- 0.8(theory), for the electric and magnetic dipole polarizability
of the proton. We also summarize the chiral EFT analysis of the world data on
coherent Compton scattering from deuterium presented in arXiv:1203.6834. That
yields: alpha^{(s)}=10.5 +/- 2.0(stat) +/- 0.8(theory); beta^{(s)}=3.6 +/-
1.0(stat) +/- 0.8(theory).Comment: 5 pages. Invited talk, presented by Phillips at the 11th Conference
  on the Intersections of Nuclear and Particle Physics (CIPANP 2012), St.
  Petersburg, FL, May 201

Griesshammer, Harald W.

McGovern, Judith A.

Phillips, Daniel R.

English

arXiv

We discuss the application of an effective field theory (EFT) whichincorporates the chiral symmetry of QCD to Compton scattering from the protonand deuteron. We describe the chiral EFT analysis of the proton Comptonscattering database presented in our recent review (arXiv:1203.6834), whichgives: alpha^{(p)}=10.5 +/- 0.5(stat) +/- 0.8(theory); beta^{(p)}= 2.7 +/-0.5(stat) +/- 0.8(theory), for the electric and magnetic dipole polarizabilityof the proton. We also summarize the chiral EFT analysis of the world data oncoherent Compton scattering from deuterium presented in arXiv:1203.6834. Thatyields: alpha^{(s)}=10.5 +/- 2.0(stat) +/- 0.8(theory); beta^{(s)}=3.6 +/-1.0(stat) +/- 0.8(theory)

The University of Manchester - Institutional Repository

The University of Manchester ResearchUsing EFT to analyze low-energy Compton scattering fromprotons and light nucleiLink to publication record in Manchester Research ExplorerCitation for published version (APA):Phillips, D. R., McGovern, J. A., & Griesshammer, H. W. (2012). Using EFT to analyze low-energy Comptonscattering from protons and light nuclei. In 11th Conference on the Intersections of Nuclear and Particle Physics(CIPANP 2012)Published in:11th Conference on the Intersections of Nuclear and Particle Physics (CIPANP 2012)Citing this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:12. Nov. 2022arXiv:1210.3577v1  [nucl-th]  12 Oct 2012Using EFT to analyze low-energy Comptonscattering from protons and light nucleiDaniel R. Phillips∗, Judith McGovern† and Harald W. Grießhammer∗∗∗Institute for Nuclear and Particle Physics and Department of Physics and Astronomy, OhioUniversity, Athens, OH 45701, USA†Theoretical Physics Group, School of Physics and Astronomy, University of Manchester,Manchester, M13 9PL, United Kingdom∗∗Institute for Nuclear Studies, Department of Physics, The George Washington University,Washington, DC 20052, USAAbstract. We discuss the application of an effective field theory (EFT)which incorporates thechiral symmetry of QCD to Compton scattering from the protonand deuteron. We describe thechiral EFT analysis of theγp scattering database presented in our recent review [1], which gives:α(p)E1 = 10.5±0.5(stat)±0.8(theory) , β(p)M1 = 2.7±0.5(stat)±0.8(theory),for the electric and magnetic dipole polarizability of the proton. We also summarize Ref. [1]’s chiralEFT analysis of the world data on coherent Compton scattering f om deuterium, which yields:α(s)E1 = 10.5±2.0(stat)±0.8(theory) , β(s)M1 = 3.6±1.0(stat)±0.8(theory).Keywords: Compton scattering, nucleon polarizabilitiesPACS: 12.39.Fe, 13.60.Fz, 14.20.DhTHEORETICAL BACKGROUNDExperiments to measure Compton scattering from the proton and deuteron are presentlybeing pursued at a number of facilities around the world, including MAMI (Mainz) [2,3], HIγS at TUNL [4], and MAX-Lab at Lund [5]. Chiral effective field theory (χEFT) isone of the main theoretical techniques used to analyze theseexperiments.χEFT gener-ates the most general Compton amplitude that is consistent with electromagnetic gaugeinvariance, the pattern of chiral-symmetry breaking in QCD, and Lorentz covariance, toany given order in the small parameterP ≡ {ω,mπ}/Λ, with ω the photon energy,mπthe pion mass andΛ the breakdown scale of the theory.The pioneering calculations ofγp scattering inχEFT [6, 7] were performed in atheory with only nucleons and pions as explicit degrees of freedom. This reduces thebreakdown scaleΛ to the energy at which the∆(1232) is excited, i.e.M∆ −MN ≈ 300MeV. In Refs. [6, 7] nucleon scalar dipole polarizabilitiesare predicted to be, atO(P3),αE1 = 10βM1 =10αEMg2A192πmπ f 2π= 12.6×10−4 fm3. (1)The corresponding cross sections agree well with experiment up to at leastω ∼ mπ , butdo not capture the rise of the data towards the∆(1232) peak. In this variant ofχEFTeven anO(P4) calculation cannot describe the data at backward angles once ω >∼ 180MeV [8, 9, 10].The inclusion of the Delta as an active degree of freedom in the theory is thereforeessential if the full power of the world’s Compton data to shed light on fundamentalhadron-structure parameters, e.g. polarizabilities, is to be realized. With the∆(1232)included inχEFT [11, 12, 13] the ratio(M∆ −MN)/Λ becomes one of the theory’sexpansion parameters. Ref. [14] pointed out that(M∆ −MN)/Λ is numerically rathersimilar to the expansion parameter in∆-less calculations,mπ/(M∆ −MN), and denotedboth asδ . In the “δ -counting” adopted in Ref. [14] powers of the electronic charge eare shown explicitly, while Refs [6, 7] countede ∼ P. Thus the Thomson amplitudeis O(P2) ∼ O(e2δ 0) and structure effects start withπN loops atO(P3) ∼ O(e2δ 2) inthe low-energy region. Further, sinceM∆ −MN ∼ δ , whereasmπ ∼ δ 2, π∆ loops aresuppressed by an additional power ofδ , and do not enter the amplitude untilO(e2δ 3).The Delta-pole graph has a special role inδ -counting: it too isO(e2δ 3) for ω ∼ mπ ,but it becomes enhanced in the regionω ∼M∆−MN , because of proximity to the Delta’son-shell point [14]. In this domain the effects that generatthe resonance’s finite widthmust be resummed. The dominantγp scattering mechanism in this “medium-energy”region is then the excitation of a dressed∆(1232) by the magnetic transition from thenucleon state, followed by de-excitation via the same M1 transition. This effect occurs atO(e2δ−1). At O(e2δ 0) the E2N → ∆(1232) transition must also be considered. Furtherdiscussion ofχEFT, and this power counting, can be found in Ref. [1].Eq. (1) is also the prediction for theneutron polarizabilities in (Delta-less)χEFT atO(P3). To accessα(n)E1 andβ(n)M1 experimentally a nuclear target is required. The deuteronis the simplest nucleus, and an accurate description of its structure is obtained whenχEFT is applied to the NN problem (see, e.g., Refs. [15, 16]).γd scattering was firstcalculated in (∆-less)χEFT in Ref. [17] atO(P3), where it was demonstrated that thereare large isoscalarγNN → γNN mechanisms (“exchange currents”) which are crucial toobtaining reasonable agreement with the data. This calculation was extended to higherorders, and augmented with∆(1232) degrees of freedom, in Ref. [9, 10, 18]. However,it was not until the work of Ref. [19] that aχEFT treatment of Compton scattering fromdeuterium which respected the Thomson limit for theγd amplitude was formulated. Thiscomputation also included∆(1232) degrees of freedom, and demonstrated the ability ofχEFT to describe deuterium Compton data from threshold toω >∼ 100 MeV.A NEW ANALYSIS OF γp SCATTERING IN χEFTThe calculation of theγp differential cross section presented here includes the nucleonBorn graph and thet-channelπ0 pole graph (both calculated covariantly). The∆-polegraphs (s- andu-channel) are dealt with as described in Refs. [1, 14, 20]—covariantlyand with a finite width stemming fromπN loops. ComptonπN andπ∆ loop graphs arealso added, so the amplitude includes effects which are of leading or next-to-leading or-der throughout the kinematic region 0≤ωlab<∼ 350 MeV (apart from the loop correctionto theγN∆ vertices), and all effects up to next-to-next-to-leading order—O(e2δ 3)—inthe low-energy regionω ∼ mπ . In addition, we include the contact interactions whichencode the short-distance (r ≪ 1/mπ ) contributions to the scalar polarizabilities. Strictlyspeaking these do not occur untilO(e2δ 4) in the low-energy domain, but they are nec-essary for an accurate description ofγp data in HBχEFT (c.f. the case where pion loopsare calculated relativistically [20, 21]). The coefficients of these contact terms are fit totheγp database, which is equivalent to fittingα(p)E1 andβ(p)M1.The parameters used in theπN sector take standard values (see Ref. [1]). The∆(1232)parametersM∆−MN = 293 MeV andgπN∆ = 1.425 are obtained from the Breit-Wignerpeak and width, the latter via the relativistic formula. We adoptb2/b1 = −0.34 for theratio of E2 and M1 couplings (c.f. theχEFT study of pion photoproduction [22]).Three EFT parameters—b1, α(p)E1 , andβ(p)M1—remain to be determined. They are fit tothe γp data base discussed extensively in Ref. [1]. For the reasons explained there, inthe low-energy region we include data from Refs. [23, 24, 25,26, 27, 28, 29, 30, 31].We float the normalization of each of these data sets within the quoted normalizationuncertainty. In the medium-energy region the data sets of Refs. [29, 32] are in significantdisagreement with those from MAMI (most notably Refs. [33, 34]) and a consistent fitcannot be obtained. We have chosen to use the MAMI data for ourfits in this region [1].Our strategy is to determine theγN∆ M1 couplingb1 by considering the MAMI datafor ωlab =200–325 MeV, then fitα(p)E1 andβ(p)M1 to the low-energy data up to 170 MeV,and iterate until convergence is reached. Theχ2/d.o.f. of the low-energy Hallin datais hard to accept, and so we prefer to quote our best results without them. We thenobtain a solution with aχ2/d.o.f. = 106.1/124. This “modifiedO(e2δ 3)” fit yieldsb1 = 3.66±0.03 and the values ofα(p)E1 andβ(p)M1 quoted in the abstract. Since these sumto a value consistent with the Baldin sum-rule constraintα(p)E1 +β(p)M1 = 13.8±0.4 [31],one can impose this relation to findα(p)E1 −β(p)M1 = 7.7±0.6 with χ2/d.o.f.= 106.5/125,an unchangedb1, and the cross sections displayed in Fig. 1. These fits are stable againstreasonable variations in the procedure [1], and agree with data [33, 34, 35, 36, 37, 38, 39]well beyond the region in which the free parameters are determin d (see Fig. 1).A NEW ANALYSIS OF γd SCATTERING IN χEFTThe χEFT treatment ofγd scattering developed in Ref. [19] is valid from thresholdto ωlab>∼ 100 MeV, and represents a complete (modified)O(e2δ 3) calculation in boththe two- and one-nucleon sectors. It has the added virtue that the dependence of crosssections on the choice of deuteron wave function is< 1% (c.f. Ref. [10]). We nowemploy theχEFT deuteron wave function at NNLO [15] within this theory toextractα(s)E1 andβ(s)M1 from theγd elastic data of Refs. [40, 41, 42].The fit to these isoscalar combinations of scalar dipole polarizabilities yields theresults quoted in the abstract, withχ2/d.o.f. = 24.3/24 (see Fig. 2). In contrast to theproton case, this is a consistent data base: each experimentcontributes roughly equallyto theχ2, and the extracted polarizabilities are largely insensitive o the elimination ofany one data set. The isoscalar polarizabilities we obtain are close to the proton ones, soisovector effects inαE1 andβM1 are small, as predicted byχEFT atO(P3). We also usedthe (isoscalar) Baldin constraint to reduce statistical uncertainties in a one-parameter fit,Θ lab=60 °40 60 80 100 120 140 160051015202530350 50 100 150 200 250 300050100150Θ lab=85 °40 60 80 100 120 140 160051015202530350 50 100 150 200 250 300050100150Θ lab=110 °40 60 80 100 120 140 1600510152025300 50 100 150 200 250 300050100150Θ lab=133 °40 60 80 100 120 140 1600510152025300 50 100 150 200 250 300050100150FIGURE 1. Comparison of ourχEFT result forγp scattering with data. The lab cross section in nb/sr isshown in 10◦ bins forθlab as a function ofωlab in MeV, with insets showing the fit region. The grey bandshows the variation within the statistical error of the one-parameter fit. Adapted from Ref. [1], where thelegend for experimental data can be found.FIGURE 2. γd cross sections at 49 and 94.5 MeV in the two-parameter (dashe ) and one-parameter(solid) determinations of the isoscalar spin-independentipole polarisabilities. Bands: statistical error ofthe Baldin constrained fit. Data at 49 (94.5) MeV from Ref. [40] ([41]). Adapted from Ref. [1].and obtained very similar results (see Fig. 2). For further details, see Ref. [1].ACKNOWLEDGMENTSD. R. P. thanks the organizers of CIPANP 2012 for a stimulating meeting in a greatlocation, and, in particular, Vladimir Pascalutsa and Michel Guidal for co-ordinatingthe “Nucleon structure” session. This work was supported inpart by UK Scienceand Technology Facilities Council grants ST/F012047/1, ST/J000159/1 (JMcG) andST/F006861/1 (DRP), by the US Department of Energy under grants DE-FG02-95ER-40907 (HWG) and DE-FG02-93ER-40756 (DRP), by the US National Science Founda-tion CAREER award PHY-0645498 (HWG), and by University Facilitating Funds of theGeorge Washington University (HWG).REFERENCES1. H. W. Grießhammer, J. A. McGovern, D. R. Phillips, and G. Feldman, Prog. Part. Nucl. Phys.,67, 841(2012).2. D. Hornidge, these proceedings.3. R. Miskimen, these proceedings.4. H. R. Weller, M. W. Ahmed, S. Henshaw and S. Stave, AIP Conf.Proc.1182, 890 (2009).5. G. Feldmanet al., Few Body Syst.44, 325 (2008).6. V. Bernard, N. Kaiser, U. G. Meißner, Phys. Rev. Lett.67 , 1515 (1991).7. V. Bernard, N. Kaiser and U. G. Meißner, Int. J. Mod. Phys. E4, 193 (1995).8. J. A. McGovern, Phys. Rev.C63, 064608 (2001).9. S. R. Beane, M. Malheiro, J. A. McGovern, D. R. Phillips, U.van Kolck, Phys. Lett.B567, 200 (2003);Erratum-ibid. B607 320 (2005).10. S. R. Beane, M. Malheiro, J. A. McGovern, D. R. Phillips and U. van Kolck, Nucl. Phys. A747 311,(2005).11. M. N. Butler, M. J. Savage and R. P. Springer, Nucl. Phys. B399, 69 (1993).12. T. R. Hemmert, B. R. Holstein and J. Kambor, Phys. Lett. B395, 89 (1997).13. R. P. Hildebrandt, H. W. Grießhammer, T. R. Hemmert and B.Pasquini, Eur. Phys. J.A20, 293 (2004).14. V. Pascalutsa, D. R. Phillips, Phys. Rev. C67, 055202 (2003).15. E. Epelbaum, W. Glöckle and U.-G. Meißner, Nucl. Phys. A747, 362 (2005).16. D. R. Phillips, J. Phys. G34, 365 (2007).17. S. R. Beane, M. Malheiro, D. R. Phillips and U. van Kolck, Nucl. Phys. A656, 367 (1999).18. R. P. Hildebrandt, H. W. Grießhammer, T. R. Hemmert and D.R. Phillips, Nucl. Phys. A748, 573(2005).19. R. P. Hildebrandt, H. W. Grießhammer and T. R. Hemmert, Eur. Phys. J. A46, 111 (2010).20. V. Lensky, V. Pascalutsa, Eur. Phys. J. C65 , 195 (2010).21. V. Lensky, J. A. McGovern, D. R. Phillips and V. Pascalutsa, arXiv:1208.4559 [nucl-th].22. V. Pascalutsa and M. Vanderhaeghen, Phys. Rev. D73, 034003 (2006).23. L. G. Hyman, R. Ely, D. H. Frisch and M. A. Wahlig, Phys. Rev. Lett.3, 93 (1959).24. V. I. Goldanskyet al., Nucl. Phys.18 (1960) 473.25. G. Pughet al., Phys. Rev. 105, 982 (1957) and references therein; MIT Summer study 1967 p555.26. P. S. Baranovet al., Phys. Lett. B 52, 122 (1974); Sov. J. Nucl. Phys. 21, 355 (1975)27. F. J. Federspiel, R. A. Eisenstein, M. A. Lucaset al., Phys. Rev. Lett.67, 1511-1514 (1991).28. A. Ziegeret al., Phys. Lett. B278, 34 (1992).29. E. L. Hallinet al., Phys. Rev. C48, 1497 (1993).30. B. E. MacGibbonet al. Phys. Rev. C52, 2097 (1995).31. V. Olmos de Leónet al., Eur. Phys. J. A10, 207 (2001).32. G. Blanpiedet al., Phys. Rev. C64, 025203 (2001).33. S. Wolfet al., Eur. Phys. J. A12, 231 (2001).34. M. Camenet al., Phys. Rev. C65, 032202 (2002).35. G. Bernardiniet al. Nuovo Cimento 18, 1203 (1960)36. P. S. Baranovet al., Sov. J. Nucl. Phys. 3, 1083 (1966); Sov Phys JETP 23, 242 (1996).37. H. Genzel, M. Jung, R. Wedemeyeret al., Z. Phys.A279, 399-406 (1976).38. J. Peiset al., Phys. Lett., 384, 37 (1996); A. Hü ngert al., Nucl. Phys. A 620 385 (1997).39. F. Wissmannet al., Nucl. Phys. A660, 232 (1999).40. M. A. Lucas, Ph.D. thesis, University of Illinois (1994).41. D. L. Hornidgeet al., Phys. Rev. Lett.84 (2000) 2334.42. M. Lundinet al., Phys. Rev. Lett.90 (2003) 192501.

Using EFT to analyze low-energy Compton scattering from protons and light nuclei

Daniel R. Phillips

Judith McGovern

Harald W. Grießhammer

Crossref

Using EFT to analyze low-energy Compton scattering from protons and
  light nuclei

Using EFT to analyze low-energy Compton scattering from protons and light nuclei

Abstract

Similar works

Full text

Available Versions

The University of Manchester - Institutional Repository

Crossref