The sigma -> gamma gamma Width from Nucleon Electromagnetic Polarizabilities
aa r X i v : . [ h e p - ph ] M a y CAFPE-92/08FTUV-08-0130UG-FT-222/08
The σ → γγ Width from the Nucleon ElectromagneticPolarizabilities
Jos´e Bernab´eu a and Joaquim Prades ba Departament de F´ısica Te`orica and IFIC, Universitat de Val`encia - CSIC,E-46100 Burjassot, Val`encia, Spain b CAFPE and Departamento de F´ısica Te´orica y del Cosmos, Universidad deGranada, Campus de Fuente Nueva, E-18002 Granada, Spain
Abstract
The lightest QCD resonance, the σ , has been recently fixed in the ππ scat-tering amplitude. The nature of this state remains nowadays one of themost intriguing and difficult issues in particle physics. Its coupling to pho-tons is crucial for discriminating its structure. We propose a new methodthat fixes this coupling using only available precise experimental data onthe proton electromagnetic polarizabilities together with analyticity andunitarity. Taking into account the uncertainties in the analysis and in theparameter values, our result is Γ pole ( σ → γγ ) = (1 . ± .
4) KeV.
Revised May 2008he lowest resonance in the QCD spectrum has the quantum numbers of thevacuum and is usually called the σ . The mass and width of this state has beenrecently fixed with a precision of just tens of MeV in [1] using an analytic contin-uation into the complex energy plane of the isopin I = 0 and angular momentumJ = 0 ππ partial wave scattering amplitude. On the first Riemann sheet of theenergy plane, the S -matrix has a zero at E = [(441 +16 − ) − i (272 +9 − )] MeV, whichreflects the σ pole on the second sheet at the same position. This is also a zeroat E ∗ in the inverse of the ππ partial wave S -matrix S = 1 + 2 i β ( t ) T ( t ) on thefirst Riemann sheet. Here, T ( t ) = 1 β ( t ) cot( δ ( t )) + iβ ( t ) (1)where δ ( t ) is the scalar-isoscalar ππ phase-shift, β ( t ) = q − m π /t and t = E .This result has been confirmed in Ref. [2] with the position of the σ pole at E = [(484 ± − i (255 ± σ in the QCD dynamics andin the QCD non-perturbative vacuum structure.Although the pole-dominance of the σ in the scalar-isoscalar ππ amplitude isapparent in a wide energy region around its position, its existence is somewhatmasked by the effects of its large width. For a narrow resonance, there is anobservable connection between the phase dependence of the physical amplitudeon the real axis and the one in the complex plane, as one crosses the pole position.This connection is, however, lost in the case of the σ with such a large width: onedoes not observe either a rapid variation of the amplitude phase [3] nor a Breit-Wigner type behavior around the resonance position. This enormous differencein the behavior of the amplitude as one moves away from the real axis is whathas made the σ existence and location so uncertain for so long.Yet the important question about what is the nature of the σ remains unan-swered [4–12]. What is its rˆole in the chiral dynamics of QCD? Is it a q - q state?Is it a π - π molecule? Is it a ( qq )-( qq ) tetraquark? Is it a glueball state? How isit possible to distinguish these different substructures? Two photon interactionscan shed some light on this question from the size of the σ → γγ width [13]. Thisis because this width is proportional to the square of the average electromag-netic charge of their constituents while its absolute scale depends on how theseconstituents form the σ . Recently, the authors of [14, 15] have calculated the γγ → ( ππ ) I =0 , amplitudes using twice-subtracted dispersion relations, in orderto weigh the low energy region in the dispersive integrand. Their results take intoaccount the now well known ππ final state interactions which contain the σ polein the scalar-isoscalar contribution. For the width of the σ into two-photons, theyobtain (4 . ± .
29) KeV in [14] and (1 . ± .
15) KeV in the improved approachof [15]. Although the approach and methodology [16] are very similar in thesetwo calculations, there is an apparent discrepancy. Its origin is discussed in [15].1he different input used for the dispersive calculation of the production ampli-tudes of γγ → ππ and the use of different values for the position of the σ poleon the second Riemann sheet t σ and its coupling to two pions g σππ are equallyresponsible. Notice that although these last two inputs are not required in thedispersive calculation, the σ → γγ width obtained in [14, 15] depends criticallyon them [15].The experimental results on the γγ → ππ process are scarce and, in orderto extract information on the σ , unfortunately theoretically contaminated bythe Born term in the charged pion channel and by the isospin I = 2 amplitudein all cases, interfering with the I = 0 amplitude in the cross section [3]. Thepurpose of this paper is to point out that the coupling g σγγ of the σ mesonfound in the ππ scattering amplitude [1, 2] is a measurable quantity, directlyobtainable from the nucleon electromagnetic polarizabilities, and that it can beextracted with good precision from existing experimental values. This differsfrom the analysis in [17] where the properties of the σ meson of a Nambu–Jona-Lasinio model are used. The argument proceeds as follows. Besides the mass,electromagnetic charge and magnetic moment, the electric α and magnetic β polarizabilities structure constants determine the Compton scattering amplitude[18,19] and the differential cross section up to second and third order in the energyof the photon, respectively. The available experiments of Compton scattering onprotons and neutrons at low energies can be analyzed [20, 21] in terms of α and β , with the sum α + β constrained by the sum rule obtained from the forwarddispersion relation [22]. The results are α exp = 12 . ± . β exp = 1 . ∓ . α exp = 11 . ± . β exp = 3 . ∓ . − fm units.A separate theoretical determination of α and β needs more ingredients thanthe ones present in the forward sum rule. The authors of [23] investigated thisproblem using a backward dispersion relation for the physical spin averaged am-plitude. The corresponding sum rule for α − β contains contributions from ans-channel part and a t-channel part. The first is related to the multipole contentof the total photo-absorption cross section, whereas the t-channel part is relatedwith the imaginary part of the amplitude through a dispersion relation for t , asshown in [24]. This imaginary part of the amplitude is given by the processes γγ → ππ and ππ → N N via a unitarity relation. The result is the BEFT sumrule [23, 24], α − β =12 π Z ∞ ν th d νν s νM p [ σ (∆ π = yes) − σ (∆ π = no)]+ 1 π Z ∞ m π d t M p − t β ( t ) t ( (cid:12)(cid:12)(cid:12) f ( t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) F ( t ) (cid:12)(cid:12)(cid:12) (4 M p − t ) ( t − m π )16 (cid:12)(cid:12)(cid:12) f ( t ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) F ( t ) (cid:12)(cid:12)(cid:12)) (2)where M p is the proton mass, the partial wave helicity amplitudes f ( t ) and f ( t ) for N N → ππ are Frazer and Fulco’s [25] and the partial wave helicityamplitudes F ( t ) and F ( t ) for γγ → ππ are defined as in [26]. The absorptivepart in the s-channel contribution is obtained from that of the forward physicalamplitude by changing the sign of the non parity flip multipoles (∆ π = no). Areliable evaluation of this s-channel integrand [21] gives ( α − β ) s = − (5 . ± . α − β ) t to the isoscalar s-wave γγ → ππ amplitude F ( t ) pointed out. The “experimental” ( α − β ) t is thus15 . ± . . ± . ππ phase-shift in an odd number of π ’s. The d-wave contribution is muchsmaller than the s-wave one, so that we take it to be fixed by the Born term in thecrossed channel [28], this leads to ( α − β ) t = − .
7. Therefore, the “experimental”quantity to be compared with the result of the integral term containing F ( t ) inEq. (2) is ( α − β ) t = (16 . ± . | F ( t ) | amplitude in that integral iswhat we want to fix from this “experimental” value. The Frazer-Fulco’s | f ( t ) | amplitude is known with sufficient accuracy for our purposes from [27] and wehave assigned a 20 % uncertainty to the theoretical ( α − β ) t determination fromthe uncertainty of | f ( t ) | . Notice that the 1 /t factor in the integrand of Eq.(2) makes the well known low energy and, to a lesser extent, intermediate energycontributions, to be the dominant ones.On the physical sheet, we use the twice-subtracted dispersion relation [16] F ( t ) = L ( t ) − Ω( t ) " c t + t π Z ∞ m π d t ′ t ′ L ( t ′ )Im Ω − ( t ′ ) t ′ − t − iε (3)where c is a subtraction constant fixed by chiral perturbation theory (CHPT)[16, 29], c = α/ πf π with α ≃ /
137 the fine-structure constant, f π = 92 . t ) = exp " tπ Z ∞ m π d t ′ t ′ δ ( t ′ ) t ′ − t − iε (4)is the scalar-isoscalar ππ Omn`es function [30] which gives the correct right-handcut contribution and L ( t ) is the left-hand cut contribution. In this way we ensureunitarity, the correct analytic structure of F ( t ) and that the σ pole propertiesenter through the scalar-isoscalar phase-shift δ ( t ) from T ( t ) in (1). Here we shalluse a simple analytic expression for T ( t ), compatible with Roy’s equations, which3akes a three parameter fit from [2] including both low energy kaon data and highenergy data. This fit is valid up to values of t of the order of 1 GeV , which isenough in the integrand of the polarizability sum rule in Eq. (2).At the σ pole position on the first Riemann sheet [3, 14, 15] F ( t σ ) = e √ g σγγ g σππ iβ ( t σ ) , (5)where e is the electron charge, g σππ is the residue of the ππ scattering amplitudeat the σ pole on the second Riemann sheet and g σγγ g σππ is proportional to theresidue of the γγ → ππ scalar-isoscalar scattering amplitude on the second Rie-mann sheet. The proportionality factors are such that g σππ and g σγγ agree withthose used in [3, 14]. The pole width is given by [3, 14]Γ pole ( σ → γγ ) = α | β ( t σ ) g σγγ | M σ (6)that agrees, modulo normalizations, with that of Ref. [15]. This is not the ob-servable radiative width that would be associated with a possible Breit-Wignerresonance in the physical γγ → ( ππ ) I =0 amplitude. However, in order to discussthe structure of the σ , one has to move around the pole and Γ pole ( σ → γγ ) is theappropriate one.Due to Low’s low energy theorem [18], the amplitude F ( t ) is given by theBorn term at low energies. Then, as a first approximation, we consider the left-hand cut contribution L ( t ) in (3) to be the Born contribution L B ( t ) to the crossedchannel describing the pion Compton scattering γπ → γπL B ( t ) = e − β ( t ) β ( t ) log β ( t )1 − β ( t ) ! . (7)Inserted into the dispersion relation in Eq. (3), this contribution leads [16, 28] toa Born amplitude F ( t ) | B for the annihilation channel γγ → ππ dressed with ππ final state interactions. Thus, this F ( t ) | B includes the σ and is compatible withunitarity and analyticity.The evaluation of the sum rule in Eq. (2) with this F ( t ) | B results in avalue ( α − β ) t | B = 6 . ± .
2, much smaller than the “experiment”. The quoteduncertainty stems from the uncertainties in the input data needed for the sumrule in Eq. (2). The main reason for this small value is the presence of a zero inthe integrand of that sum rule at a moderate t -value t ≃ .
30 GeV , as shown inFig. 1. The amplitude F ( t ) | B , when analytically continued to complex t , hasthe σ pole in the second Riemann sheet at t = t σ = ([(474 ± − i (254 ± with g σππ = [(452 ±
4) + i (224 ± g σγγ /g σππ | B = (0 . +0 . − . ) − i (0 . ± .
03) and Γ pole ( σ → γγ ) | B = (2 . ± .
10 15 20 25 t €€€€€€€€€€€€ m Π - fm €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ m Π Figure 1:
The integrand of ( α − β ) t in (2). The broken line is when using L ( t ) = L B ( t )in (3) and the continuous line is when using L ( t ) = L B ( t ) + L A ( t ) + L V ( t ) in (3) asexplained in the text. nucleon electromagnetic polarizabilities, as we have seen, and the pion Comptonscattering description has to go beyond the Born approximation L B ( t ), with amodification of the left-hand cut L ( t ) contribution in Eq. (3).At intermediate energies, this modification is due to resonance exchanges withthe leading ones being γπ → a , ρ, ω → γπ [15,16]. The a exchange contributionto L ( t ) is L A ( t ) = e C πf π " t + M a β ( t ) log β ( t ) + t A /t − β ( t ) + t A /t ! (8)while the ρ and ω resonances exchange contribution to L ( t ) in nonet symmetry( M ρ = M ω = M V ≃
782 MeV) is L V ( t ) = e R V " t − M V β ( t ) log β ( t ) + t V /t − β ( t ) + t V /t ! (9)with t R = 2( M R − m π ). The low energy limit of L V ( t ) goes as t and we fix R V =1.49 GeV − by using the well known ω → πγ decay. Though the low energy limitof L A ( t ) goes as t and corresponds to the pion electromagnetic polarizability( α − β ) π ± or equivalently to L + L = (1 . ± . · − in CHPT [31], weconsider L A ( t ) as an effective contribution for moderate higher values of t with C a real constant to be determined phenomenologically and not connected tothe pion polarizability. This is supported by the fact that the a → πγ couplingis not so well known at intermediate energies. We fix C by requiring that the“experimental” value of ( α − β ) t is reproduced within 1.5 standard deviations ofthe total uncertainty when L ( t ) in (3) is given by L ( t ) = L B ( t ) + L A ( t ) + L V ( t ).This procedure leads to C = 0 . ± .
20 and the integrand of the sum rule isgiven in Fig. 1 as a continuous line. Notice that C has to be positive in order tomatch the “experimental” value of ( α − β ) t and that the zero at t in the dressed5orn amplitude has clearly disappeared. Moreover, in spite of the fundamentaldynamics of the σ resonance in the t-channel polarizability sum rule, there is notrace of a resonant Breit-Wigner type behavior when going to the physical real t axis, see Fig. 1.The low-energy γγ → π π cross-sections obtained for the two cases studiedabove are similar [15]. The central values are compatible with the data for valuesof t below (450 MeV) and are above the data but compatible within two stan-dard deviations for larger values of t up to (600 MeV) and within one standarddeviation for t between (600 MeV) and (800 MeV) .When F ( t ) is analytically continued to the complex plane, at t σ on the firstRiemann sheet one gets g σγγ /g σππ = (0 . +0 . − . ) − i (0 . ± .
03) which has asmaller absolute value when compared with g σγγ /g σππ | B and leads to Γ pole ( σ → γγ ) = (1 . ± .
3) KeV. This is the main result of this paper. The error quotedhere is from the uncertainties in the “experimental” value of ( α − β ) t and theinputs of the sum rule (2) only.In order to obtain the rest of the uncertainty, we modify the σ propertiesin the pion scattering as follows. We still use the three parameter fit formulaincluding low energy kaon data and high energy data for cot( δ ( t )) in [2] as inputin the amplitude T ( t ) but with parameter values slightly modified in order toreproduce the σ pole position t σ = ([(441 ± − i (272 ± found in [1].In that case, we get g σππ = [(480 ±
7) + i (191 ± T ( t ) and thedressed Born amplitude in (3), one gets ( α − β ) t | B = 6 . ± . g σγγ /g σππ | B =(0 . ± . − i (0 . ± .
03) and Γ pole ( σ → γγ ) | B = (3 . ± .
4) KeV. Theintegrand of ( α − β ) t in (2) for this case is very similar to the broken line ofFig. 1. The effective value of C in (8) moves to C = 0 . ± .
20 when fixed toreproduce the “experimental” value of ( α − β ) t within 1.5 standard deviations ofthe total uncertainty. With this new C , the analytic continuation to the new t σ gives g σγγ /g σππ = (0 . +0 . − . ) − i (0 . ± .
03) and Γ pole ( σ → γγ ) = (1 . ± . α − β ) t in (2) for this case is very similar to thecontinuous curve of Fig. 1.As final result for the electromagnetic pole width of the σ found in the ππ scattering amplitude, we quoteΓ pole ( σ → γγ ) = (1 . ± .
4) KeV (10)which is the weighted average for the results of the σ → γγ width using the g σγγ coupling in (6) obtained when the F ( t ) amplitude in (3) is analytically continuedto σ pole position t σ on the first Riemann sheet (5) in two cases: first, when usingfor cot( δ ( t )) in (1) the three-parameter fit formula from [2] including both lowenergy kaon data and high energy data; second, when varying the parametersof the fit for cot( δ ( t )) found in [2] in order to mimic the pole position foundin [1]. In both cases, this F ( t ) reproduces within 1.5 standard deviations the“experimental” value of ( α − β ) t when inserted in the BEFT sum rule (2).6o conclude, we have shown that the scalar-isoscalar γγ → ππ amplitude F ( t ) may be fixed using analyticity, unitarity and experimental information onthe nucleon electromagnetic polarizabilities. This is possible and direct becausethis component is projected out in the sum rule (2). When both F ( t ) in (3) and T ( t ) in (1) are analytically continued to the complex plane, the σ pole positionand its g σγγ /g σππ and g σππ residues become fixed.We thank the CERN Theory Unit where this work was initiated for warmhospitality. This work has been supported in part by the European Commission(EC) RTN network FLAVIAnet Contract No. MRTN-CT-2006-035482 (J.P.), byMEC, Spain and FEDER (EC) Grants No. FPA2005-01678 (J.B.) and FPA2006-05294 (J.P.), Sabbatical Grant No. PR2006-0369 (J.P.), by Junta de Andaluc´ıaGrants No. P05-FQM 101 (J.P.), P05-FQM 467 (J.P.) and P07-FQM 03048(J.P.) and by the Spanish Consolider-Ingenio 2010 Programme CPAN Grant No.CSD2007-00042. We also would like to thank Heiri Leutwyler, Jos´e A. Oller, Jos´eRam´on Pel´aez and Mike Pennington for useful discussions and sharing unpub-lished results. References [1] I. Caprini, G. Colangelo, H. Leutwyler, Phys. Rev. Lett. (2006) 132001.[2] R. Garc´ıa-Mart´ın, J.R. Pel´aez , F.J. Yndur´ain, Phys. Rev. D (2007)074034.[3] M.R. Pennington, Mod. Phys. Lett. A (2007) 1439.[4] R.L. Jaffe, F. Wilczek, Phys. Rev. Lett. (2003) 232003; R.L. Jaffe, Phys.Rev. D (1977) 267.[5] J.D. Weinstein, N. Isgur, Phys. Rev. D (1990) 2236; Phys. Rev. D (1983) 588; Phys. Rev. Lett. (1982) 659.[6] P. Minkowski, W. Ochs, Eur. Phys. J C (1999) 283; S. Narison, Nucl. Phys.B (1998) 312; A. Bramon, S. Narison, Mod. Phys. Lett. A (1989) 1113.[7] C. Amsler, N.A. Tornqvist, Phys. Rept. (2004) 61; N.A. Tornqvist, A.D.Polosa, Nucl. Phys. A (2001) 259.[8] N.A. Tornqvist, Acta Phys. Polon. B (2007) 2831.[9] J.A. Oller, E. Oset, Nucl. Phys. A (1997) 438; Erratum ibid. A (1999) 407; A. Dobado, J.R. Pel´aez, Phys. Rev. D (1997) 3057.[10] G. ’Hooft et al. , arXiv:0801.2288 [hep-ph].711] A.H. Fariborz, R. Jora, J. Schechter, arXiv:0801.2552 [hep-ph].[12] G. Mennessier, S. Narison, W. Ochs, arXiv:0804.4452 [hep-ph].[13] G. Mennessier et al. , arXiv:0707.4511 [hep-ph].[14] M.R. Pennington, Phys. Rev. Lett. (2006) 011601.[15] J.A. Oller, L. Roca, C. Schat, Phys. Lett. B (2008) 201; J.A. Oller, L.Roca, arXiv:0804.0309 [hep-ph] and private communication.[16] D. Morgan, M. Pennington, Phys. Lett. B (1991) 134; J.F. Donoghue,B.R. Holstein, Phys. Rev. D (1993) 137.[17] M. Schumacher, Eur. Phys. J. A (2007) 293; ibid. (2007) 327; ibid. (2006) 413.[18] F. Low, Phys. Rev. (1954) 1428; M. Gell-Mann, M.L. Goldberger, Phys.Rev. (1954) 1433; H.D.I. Abarbanel, M.L. Goldberger, Phys. Rev. (1968) 1594.[19] A. Klein, Phys. Rev. (1955) 998; A.M. Baldin, Nucl. Phys. (1960)310; V.A. Petrun’kin, Sov. Phys. JETP (1961) 808.[20] W.M. Yao et al. [Particle Data Group], J. Phys. G (2006) 1.[21] M. Schumacher, Prog. Part. Nucl. Phys. (2005) 567; M.I. Levchuk et al. ,arXiv:hep-ph/0511193.[22] M. Damashek, F.J. Gilman, Phys. Rev. D (1970) 1319.[23] J. Bernab´eu, T.E.O. Ericson, C. Ferro Fontan, Phys. Lett. B (1974) 381.[24] J. Bernab´eu, R. Tarrach, Phys. Lett. B (1977) 484.[25] W.R. Frazer, J.R. Fulco, Phys. Rev. (1960) 1603.[26] O. Babelon et al. , Nucl. Phys. B (1976) 252.[27] G.E. Bohannon, Phys. Rev. D (1976) 126; G.E. Bohannon, P. Signell,Phys. Rev. D (1974) 815.[28] B.R. Holstein, A.M. Nathan, Phys. Rev. D (1994) 6101.[29] J. Gasser, H. Leutwyler, Nucl. Phys. B (1985) 465.[30] R. Omn`es, Nuovo Cim. (1958) 316.[31] J. Bijnens, F. Cornet, Nucl. Phys B296