We derive a closed, model space independent, expression for the
electromagnetic correction factor $\delta $ to the scattering length $a$
extracted from a hydrogenic atom with an extended charge to order $\alpha ^2$
and $a^3$ in the limit of a short ranged hadronic interaction.Comment: 4 pages; PANIC02, XVIth Conference on Particles and Nuclei, Osaka, to
  appear in Nuclear Physics 

B. Loiseau

Deser

Eiras

Ericson

Gasser

Hauser

Höhler

Lipartia

Lyuboviitskij

S. Wycech

Schröder

Sigg

T.E.O. Ericson

Trueman

English

arXiv

4 pages; PANIC02, XVIth Conference on Particles and Nuclei, Osaka, to appear in Nuclear Physics AInternational audienceWe derive a closed, model space independent, expression for the electromagnetic correction factor $\delta $ to the scattering length $a$ extracted from a hydrogenic atom with an extended charge to order $\alpha ^2$ and $a^3$ in the limit of a short ranged hadronic interaction

Ericson, T. E. O.

Loiseau, B.

Wycech, S.

HAL-IN2P3

Electromagnetic corrections to $\pi^-$p scattering length from pionic hydrogen

We derive a closed, model space independent, expression for the electromagnetic correction factor $\delta $ to the scattering length $a$ extracted from a hydrogenic atom with an extended charge to order $\alpha ^2$ and $a^3$ in the limit of a short ranged hadronic interaction.We derive a closed, model space independent, expression for the electromagnetic correction factor $\delta $ to the scattering length $a$ extracted from a hydrogenic atom with an extended charge to order $\alpha ^2$ and $a^3$ in the limit of a short ranged hadronic interaction

Ericson, Torleif Erik Oskar

CERN Document Server

1Electromagnetic corrections to −p scattering length from pionichydrogenT.E.O. Ericson a, B. Loiseau, b and S. Wycech ca CERN, CH-1211 Geneva 23, Switzerland, and TSL, Box 533, S-75121 Uppsala, SwedenbLPNHE, Universite P. & M. Curie, 4 Place Jussieu, F-75252 Paris, FrancecSoltan Institute for Nuclear Studies, PL-00681 Warszawa, PolandWe derive a closed, model space independent, expression for the electromagnetic cor-rection factor  to the scattering length a extracted from a hydrogenic atom with anextended charge to order 2 and a3 in the limit of a short ranged hadronic interaction.The strong interaction energy shifts 1s in pionic hydrogen and deuterium have recentlybeen measured to a remarkable precision [1,2].1s(−p) = [−7:108 0:013(stat) 0:034(syst)] eV; 1s(−d) = [2:460 0:048] eV:It is well known[3,4] that the strong interaction shift is intimately linked to the (com-plex) scattering length a in the absence of the Coulomb eld:1s = 4=2m 2B(0) a (1 + ): (1)Here B(r) is the Bohr wave function and m the −p reduced mass. It is important tounderstand the small correction  transparently and reliably to an accuracy matchingthe high experimental precision. This is desirable not only because the hadronic Nscattering lengths are key testing quantities for chiral physics, but also because they areneeded phenomenologically to a precision of about 1 % [5].The present standard conversion of experimental data to a scattering length uses thepotential approach of Sigg et al. [6], which describes the N interaction in terms of coupledequations using physical pion masses and an isospin invariant non-diagonal potentialmatched to scattering lengths. This gives (Sigg) = −2:1 0:5%. This scattering lengthrefers to a calculated ’isospin invariant’ amplitude obtained by setting the neutral pionmass equal to the charged one in the model.The approach has some weaknesses. First, the corresponding hadronic scattering am-plitude does not reproduce properly the low energy s-wave scattering expansion as wehave pointed out [5] and it is then not consistent with gauge invariance. Further, in theabsence of a description for the dispersive correction from the radiative channel −p! nγthey simulate it in an analogous way to that produced by charge exchange. The correctionto isospin invariance is model dependent.2Recently two calculations of the corrections inside Chiral Perturbation Theory (ChPT)have appeared with shockingly larger uncertainties. The next to leading order [7] gives(chiral) = −4:3  2:8 % while the next higher order [8] give (chiral) = −7:2  2:9 %.The main origin of the uncertainty is an anomalous photon-quark coupling. This largedierence with Ref.[6] is mainly due to the denition of the scattering length in the chirallimit with the charged pion mass equal to the neutral one. While this also gives an isospininvariant amplitude, it is now dened quite dierently.In view of this situation we have carefully examined the extraction of the scatteringlength from the energy shift, its denition and corrections. The physics is largely governedby the low energy expansion of the N amplitudes in the limit of isospin symmetry. Inunits of the charged pion mass mc [9][tan (0+)=q]pi−p!pi−p = (a+0+ + a−0+) + (b+0+ + b−0+)q2 + ::: (2)a−0+ ’ 0:09m−1c ; ja+0+j ’ (few %) a−0+; b−0+ = 0:013(6)m−3c ; b+0+ = −0:044(7)m−3c : (3)To elucidate the physics we use both potential approaches and multiple scattering ap-proaches together with analytical methods provided these reproduce the low energy ex-pansion consistently. Here our philosophy is close to that of Ref. [10], which takes thesame attitude with respect to the ChPT expansion, but we consider the phenomenologi-cal consistency with data to be more important. One notes that a) none of the previousapproaches appears to be fully consistent in this perspective; furthermore none of themconverts the energy shift 1s into a physical −p scattering length without the exter-nal Coulomb eld; b) the nite size shift is very small compared to the hadronic shift:fs  −1:5% of strong; c) this atomic problem in most aspects is highly non-relativisticand close to a long wave-length limit for the interaction.Corrections. A number of the corrections are associated with the behavior of the un-perturbed Bohr wave function at the origin: B(~r) = B(0)[1 − mr + ::] where ther-dependent term is of the order of 10−2 at distances of the size of a hadron.Vacuum polarisation correction. The Uehling potential is of order 2 but it givesa rather large contribution independent of the hadronic interaction such that vac 50% of − 1s. It has a long range and can be seen as a modication of the Coulomb eldwith a slight change of the wave function vac(0):vac = 2vac(0)=B(0) = +0:48%: (4)This correction is very accurate. It has been given explicitly by Eiras and Soto [11], butis implicitly included previously [6]. It is included in the most recent ChPT eective eldtheory (EFT) approach [8], but not in earlier approaches.Leading corrections from the extended charge. Previously Sigg et al.[6] numericallyinvestigated the shift in the presence of the combined charge distribution of the − andthe proton with a charge radius (hr2pii+hr2pi)1/2 ’ 1:07 fm. The poorly known range of thehadron interaction was varied. The EFT approaches do not clearly include the accuratelyknown electromagnetic form factors and appear to replace its eects by a dimensionalregularisation procedure and eective constants.Since the combined charge radius is larger than a typical hadronic interaction range itis natural to approach the problem from the limit of a zero range hadron interaction of3scattering length a as a leading approximation. The relation between scattering lengthand energy shift can then be obtained by matching the wave function to the scatteringlength condition at the origin. We rst observe that this problem has an exact solutionfor a charge placed on a sphere of radius R. The Coulomb potential inside this radius isconstant, i.e., a square well, and consequently the wave function is explicitly known. Bymatching to the outside Coulomb wave function, the well known Whittaker function, theexact condition for the binding is obtained and we retain the solution to order 2.We then consider the eect of the dierence in the Coulomb potential between thesoluble case and an actual charge distribution. This interaction potential is perfectlyregular and can be treated perturbatively. The result is the following closed expressionfor the scattering length correction, where C is the Euler constant: = −2m hriem + 2m a [2− C − ln (2)− hln (mr)iem] : (5)This relation ressembles supercially the Trueman formula, but the physics is quitedierent. The Trueman relation is expressed in terms of hadronic eective range anddoes not describe the nite size charge distribution which is the essential point here. Thereis no immediate relation between Trueman’s eective range parameter and the accuratelyknown electromagnetic expectation values hriem and hln riem above. The chiral approachis also dierent at this point since it does not include the eect of the extended charge.The terms in eq. (5) have a clear physical interpretation and are generated by therenormalization of the wave function at the origin. The rst term appears since the wavefunction of the extended charge at the origin isfs(0)=B(0) = 1−mhriem + ::: (6)This value for the wave function could have been used directly in the leading expressionfor the energy shift as a better rst approximation than the wave function in the pointCoulomb eld. The second term proportional to the hadronic scattering length is due tothe eective incident eld producing the hadron scattering and may be viewed as a cuspeect by the Coulomb eld. We can show that the hadronic modication of the regularwave function at the origin is exactly the one in the expression (5) above: reg(0)=fs(0) = 1 + 2m a [2− C − ln (2)− hln (mr)iem] + ::::: (7)Consequently to the order stated above the relation between the energy shift and thehadronic scattering length is fully understood in the limit of a zero range interaction.Correction for the q2 dependence of the N amplitude. In the zero range limit thescattering amplitude depends but trivially on the energy. The actual scattering occursin an attractive (Coulomb) potential of -2.5 MeV at the origin. The amplitude (2) hasa momentum (energy) dependence which must included consistently respecting gaugeinvariance. The corresponding expansion term b is closely linked to the -term [12]. Thisterm can be included [13] in an energy representation using gauge invariance or as amomentum dependent interaction to leading order. Substituting ! ! ! − eVC(0) in thehadronic amplitude (2) gives an additional terma q2 = b 2m h1=riem: (8)4This term expresses that the hadronic amplitude is evaluated at the local energy in theCoulomb eld (but with the correct, nearly vanishing, physical phase space for scattering).With the parameters (3) for the −p! −p scattering the correction is−1:20:4%, wherethe uncertainty is due to the experimental uncertainty of the phenomenological constantsof Eq. (2). The extended charge is essential here, since this contribution otherwise will bedivergent. This correction appears to correspond to the anomalous photon-quark couplingin the chiral EFT description, which represents a large source of uncertainty there.The total correction in the determination of the scattering length with the externalCoulomb eld switched o, is therefore total = −1:1 0:4%. This is a small, well denedcorrection. The scattering length contains isospin violating terms and is complex sinceboth the physical 0n and the γn channels are open.We can thus unambiguously convert the strong interaction energy shift into a corre-sponding scattering length. In the limit we consider, no model assumptions are neededand the resulting scattering length preserves the initial precision of the experiment. Itis a separate problem to convert this physical quantity into into an isospin symmetric orchiral scattering length.The −p scattering length deduced here refers to scattering from a neutral system. Itdenition is closely analogous to the one in the charge symmetric +n channel, which is inprinciple directly measurable. The open decay channel and mass dierences correspondexactly in the charge symmetry limit. This denes a natural starting point for the furtherdiscussion of its relation to isospin breaking and the chiral limit.We are presently [14] investigating the sensitivity of our procedure to plausible as-sumptions of the dependence of the physical hadronic amplitude on range etc. as wellas the dispersive eects produced by the open decay channels. We recall that the scat-tering length a with the external Coulomb eld removed still contains internal Coulombcorrections such as mass dierences and polarizability eects.The present method applies with minor modications to other pionic atoms, such asthe +−, the −K+ and −d systems.REFERENCES1. H. C. Schro¨der, et al., Eur. Phys. J. C 21 (2001) 178.2. P. Hauser , K. Kirch, L. M. Simons, et al., Rev. C 58 (1998) R1869.3. S. Deser et al., Phys. Rev. 96 (1954) 774.4. T.L. Trueman, Nucl.Phys. 26 (1961) 57.5. T. E. O. Ericson, B. Loiseau and A. Thomas, Phys.Rev. C66 (2002) 014005.6. D. Sigg, A. Badertscher, P. F. A. Goudsmit, et al., Nucl. Phys. A609, 310 (1996).7. V. E. Lyuboviitskij and A. Rusetsky, Phys.Lett. B494 (2000) 9.8. J. Gasser, M. A. Ivanov, E. Lipartia, et al., hep-ph/0206068 (2002).9. G. Ho¨hler in  N scattering, in Landolt-Bo¨rnstein, New Series, Vol. 9b (1983).10. E. Lipartia, V. E. Lyuboviitskij and A. Rusetsky, Phys.Lett. B533 (2002) 285.11. D. Eiras and J. Soto, Phys. Lett. B491 (2000) 101.12. T. E. O. Ericson Phys. Lett. 195B (1987) 116.13. T. E. O. Ericson, L. Tauscher, Phys. Lett. 112B (1982) 425.14. T. E. O. Ericson, B. Loiseau, S. Wycech, in preparation.

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