- Email: [email protected]
Determination of the πN scattering lengths from pionic hydrogen
9 December 1999
Physics Letters B 469 Ž1999. 25–29
Determination of the p N scattering lengths from pionic hydrogen a H.-Ch. Schroder , A. Badertscher a,1, P.F.A. Goudsmit a , M. Janousch a , H.J. Leisi a , ¨ E. Matsinos a , D. Sigg a , Z.G. Zhao a , D. Chatellard b, J.-P. Egger b, E. Jeannet b, K. Gabathuler c , P. Hauser c , L.M. Simons c , A.J. Rusi El Hassani d b
a ETH Zurich, Institute for Particle Physics, CH-8093 Zurich, Switzerland Institut de Physique de l’ UniÕersite´ de Neuchatel, ˆ CH-2000 Neuchatel, ˆ Switzerland c Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland d Dept. de Physique, Faculte´ des Sciences et Technique, Tanger, Morocco ´
Received 21 July 1999; received in revised form 15 October 1999; accepted 15 October 1999 Editor: J.P. Schiffer
Abstract The final results from the pionic hydrogen X-ray experiment at PSI are presented. With this experiment the strong interaction energy level shift ´ 1s and the total decay width G 1s of the 1s state of pionic hydrogen were obtained from a precise measurement of the 3p–1s X-ray line: ´ 1s s y7.108 " 0.013Žstat.. " 0.034Žsyst.. eV Žattractive. and G 1s s 0.868 " 0.040Žstat.. " 0.038Žsyst.. eV. The corresponding hadronic p N s-wave scattering lengths for elastic scattering and single charge exchange are: apy p p y p s 0.0883 " 0.0008 mpy1 and ap y p p 0 n s y0.128 " 0.006 mpy1. Combining the results of the pionic hydrogen and deuterium measurements, the isoscalar Ž b 0 . and isovector Ž b 1 . scattering lengths, corresponding to an isospin symmetric interaction, can be calculated: b 0 s 0.0016 " 0.0013 mpy1 and b 1 s y0.0868 " 0.0014 mpy1. q 1999 Elsevier Science B.V. All rights reserved.
™
The energy and line shape of the 3p–1s X-ray transitions in pionic hydrogen and deuterium were measured by the ETH Zurich-Neuchatel-PSI collaboˆ ration at the Paul Scherrer Institute ŽPSI, Villigen, Switzerland. with a high resolution bent crystal spectrometer. First results from pionic hydrogen were published in Ref. w1x, and the results of the deuterium measurements are given in Ref. w2x. Experimental details are described in Refs. w1,2x. In this paper we report the final results of this collaboration on pionic hydrogen, based on the data of Refs. w1,3,4x. Combining the pionic hydrogen result with that from
1
E-mail: [email protected]
™
pionic deuterium ŽRef. w2x., precise information on the p N scattering lengths is obtained. A unique feature of the pionic atom measurements is the possibility to study the p N interaction at threshold, and thus to be able to determine the p N scattering lengths directly, i.e. without extrapolations to threshold. Subtracting the measured 3p–1s transition energy from the calculated electromagnetic energy gives the strong interaction level shift ´ 1s of the 1s state Žshift and width of the 3p state are negligible., and from the measured line shape the decay width of the 1s state G 1s can be calculated. The total decay width of the 1s state is determined by the rate of the single charge exchange ŽSCX. reaction Žpy p .1s p 0 n and the radiative decay
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 2 3 7 - X
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H.-Ch. Schroder ¨ et al.r Physics Letters B 469 (1999) 25–29
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Žpy p .1s g n. The ratio of the two decay rates is the well known Panofsky ratio P s 1.546 " 0.009 w5x. The two quantities ´ 1s and G 1s are related to the s-wave scattering lengths for elastic Ž apy p ™ p y p . and SCX Ž apy p ™ p 0 n . scattering. We have established these relations numerically w6x taking into account various electromagnetic effects and using an attractive strong interaction potential for isospin 1r2 and a repulsive potential for isospin 3r2. The result can be expressed as electromagnetic correction factors d´ and dG to Deser-type formulae w6,7x: ´ 1s ap y p ™ p y p s y4 Ž 1 q d´ . , Ž 1. E1s rB G 1s q 1 2 s8 1q apy p ™ p 0 n Ž 1 q dG . 4 , Ž 2. E1s rB P with d´ s Žy2.1 " 0.5.% and dG s Žy1.3 " 0.5.%; E1s is the point-Coulomb electromagnetic binding energy of the 1s level, r B s 222.6 fm the Bohr radius and q s 28.04 MeVrc s 0.1421 fmy1 the c.m. momentum of the p 0 in the SCX reaction; E1srr B s 14.53 eVrfm.
ž
/
In a similar way the Žcomplex. scattering length of pionic deuterium ap d can be calculated from the shift and width of the deuterium 1s state w2x. ŽA recent measurement of the 1s shift of pionic deuterium w8x agrees with our value.. The real part Re ap d can be expressed in terms of the p N scattering lengths w9,2x. The constraints on the scattering lengths from the shift and width of hydrogen and the shift of deuterium is investigated, assuming isospin symmetry of the strong p N interaction. Provided the constraints of the three observables are compatible with the assumption of isospin symmetry, the isoscalar and isovector p N scattering lengths b 0 and b 1 can be calculated. Fig. 1 shows measured X-ray lines corresponding to 75% of our py p data w3x Žthe first part of the data is published in Ref. w1x.. The electronic argon K a 1 line served as a wavelength standard; from the Bragg angle difference between the argon K a 1 and the meas. pionic hydrogen line the transition energy E3py1s is obtained. Table 1 summarizes the measured and the calculated electromagnetic transition energies for the
Fig. 1. Measured X-ray lines: Ža. Pionic hydrogen line, Žb. electronic argon K a line, used as a wavelength standard, Žc. pionic beryllium line, used to determine the instrumental resolution function. One channel corresponds to five CCD pixels Žpixel size: 22.5 mm.. The energy dispersion of the spectrometer was 0.088, 0.101 and 0.080 eVrchannel at the Bragg angles of measurements Ža., Žb. and Žc., respectively.
H.-Ch. Schroder ¨ et al.r Physics Letters B 469 (1999) 25–29
27
Table 1 Measured and calculated Želectromagnetic. 3p–1s transition energy w6x and deduced strong interaction shift of the 1s state Energy ŽeV. me as. E3py1s
2885.916 " 0.013Žstat.. " 0.033Žsyst.. 2875.715 " 0.007 y0.102 " 0.003 3.235 " 0.001
Point-Coulomb, Klein-Gordon equation Coulomb finite size effect Žproton and pion. Vacuum polarization, order a 2 ŽUehling potential, finite size. Higher order corrections ŽHOC.: – Relativistic recoil and proton spin and anomalous magnetic moment of the proton – Vacuum polarization order a 3 – Vertex correction Pionic atom recoil energy
y0.047 " 0.000 0.018 " 0.000 y0.007 " 0.003 y0.004
el .mag . E3py1s Strong interaction shift ´ 1s
2878.808 " 0.008 y7.108 " 0.013Žstat.. " 0.034Žsyst..
3p–1s transition and gives the resulting value for the strong interaction shift:
´ 1s s y7.108 " 0.013 Ž stat. . " 0.034 Ž syst. . eV.
Ž 3. The pionic beryllium transition has a very small intrinsic width so that the 4f-3d line could be used to determine the instrumental resolution function. The spectrometer resolution was about 0.6 eV ŽFWHM., and the deconvolution of the measured line profile with the instrumental resolution function yields the line width w3x Žwe have added 9 meV to the width given in Ref. w3x to account for the Auger width of the Be line. meas . G 3py1s s 0.969 " 0.045 Ž stat. . " 0.010 Ž syst. . eV.
Ž 4. This line width has to be corrected for the Doppler broadening to obtain the decay width of the 1s state G 1s . The Doppler broadening of the X-ray lines occurs mainly as a consequence of an acceleration mechanism of the pionic atoms during the cascade, the so called Coulomb de-excitation w10x. In this process the excited pyp-atom penetrates the electron cloud of a normal hydrogen atom. In the atomic electric field a de-excitation of the pionic atom to a lower level takes place, in which no X-ray is emitted. The transition energy is shared as kinetic energy between the two collision partners. To study the kinetic energy distribution of pionic hydrogen atoms formed in gaseous and liquid hydrogen, the time-of-
™
flight of neutrons ŽnTOF. from the SCX reaction py p p 0 n was measured at PSI w11x. From the nTOF data and the absolute X-ray yields in pionic hydrogen from Ref. w12x the kinetic energy distribution of pionic atoms undergoing the 3p–1s radiative transition was obtained w4x. This allows the calculation of the Doppler correction factor D of the pionic X-ray lines
D'
meas . G 3py1s y G 1s
G 1s
s 0.12 " 0.05.
Ž 5.
Inserting the measured line width Ž4. into Eq. Ž5., the decay width of the 1s state is obtained:
G 1s s 0.868 " 0.040 Ž stat. . " 0.038 Ž syst. . eV.
Ž 6.
With the values Ž3. and Ž6. and adding the statistical and systematic errors linearly, the hadronic s-wave scattering lengths are obtained from Eqs. Ž1. and Ž2.: apy p ™ p y p s Ž 0.0883 " 0.0008 . mpy1
apy p ™ p 0 n s Ž y0.128 " 0.006 . mpy1 .
Ž 7. Ž 8.
The scattering lengths for elastic and SCX scattering can be expressed in terms of the scattering lengths of total isospin 1r2 Ž a1 . and 3r2 Ž a3 ., or, alternatively by the isoscalar and isovector scattering lengths b 0 and b 1: apy p ™ p y p s b 0 y b 1 s 13 Ž 2 a1 q a 3 . , apy p ™ p 0 n s '2 b 1 s
'2 3
Ž a3 y a1 . .
Ž 9. Ž 10 .
H.-Ch. Schroder ¨ et al.r Physics Letters B 469 (1999) 25–29
28
Combining the constraints from the hydrogen shift and width with Re apy d from the deuterium shift, the final result for b 0 and b 1 is obtained, with a correlation coefficient r 01 s 0.83:
Fig. 2. Constraints on the scattering lengths b 0 and b1 imposed by the measured strong interaction shift and width of the 1s state of pionic hydrogen and the 1s shift of pionic deuterium. The ellipse represents the one standard deviation contour for an isospin symmetric strong interaction. Also indicated are comparisons with theoretical calculations Žsee text..
Fig. 2 shows the constraints on b 0 and b 1 imposed by the values Ž7. and Ž8.. Also shown in Fig. 2 is the Žnearly horizontal. band derived from the real part of the py d scattering length w2x and the calculations of Thomas and Landau w9x, which relate Re apy d to the elementary p N scattering lengths 2 . Using only the two constraints from the hydrogen shift and width leads, with Eqs. Ž7. – Ž10., to b 0 s Žy2.2 " 4.3. = 10y3 my1 and b 1 s Žy90.5 " 4.2. = 10y3 my1 with p p a strong correlation r 01 s 0.98 due to the narrow diagonal band from the hydrogen shift. If isospin would be an exact symmetry, the constraints from all p N reactions would have a common intersection in the b 0 y b 1 plot w15x. Fig. 2 shows that with the present errors the three constraints derived from the pionic hydrogen and deuterium measurements are still compatible with isospin symmetry Žassuming that the calculations of Thomas and Landau are complete..
b 0 s Ž q1.6 " 1.3 . = 10y3 mpy1 ,
Ž 11 .
b 1 s Ž y86.8 " 1.4 . = 10y3 mpy1 .
Ž 12 .
The ellipse in Fig. 2 corresponds to this result. The black dot indicates the scattering lengths in the current-algebra limit, first obtained by Weinberg w16x, and the dashed rectangle ŽFMS. is the result of a recent third-order calculation within the framework of heavy baryon chiral perturbation theory w17x. We note that the isoscalar scattering length is compatible with the current-algebra limit Ž b 0 s 0. and the isovector scattering length differs by 9%. A more detailed comparison with other determinations of the scattering lengths, all of which are based on extrapolations to threshold, will be given in Ref. w4x. With the new b 1 value the Žpseudovector. p NN coupling constant can be calculated using the GMO sum rule w18x. f2 4p
s y0.571mp P b 1 y 0.0249 mby1 P J s 0.0757 " 0.0010.
For the dispersion integral J we use the average of the VPI and Karlsruhe-Helsinki ŽKH. values given in Ref. w19x: J s y1.0485 " 0.0075 mb; J VPI s y1.041 mb and J KH s y1.056 mb. With this coupling constant, the Goldberger-Treiman w20x discrepancy DGT , defined by the relation mN gp N N s g Ž 1 q DGT . Ž 14 . Fp A is obtained:
DGT s Ž 1.9 " 0.8 . = 10y2 .
The multiple scattering calculations in the py d system have since been repeated by Baru and Kudryavtsev w13x. Ericson et al. w14x are presently analysing in detail the various terms in the py d caculation.
Ž 15 .
The coupling constant gp N N in Ž14. is given by gp N N s
2
Ž 13 .
2 mN mp
P f s 13.11 " 0.09.
Ž 16 .
The nucleon axial vector coupling constant is g A s 1.2670 " 0.0035 w21x and the pion decay constant Fp s 92.42 " 0.26 MeV w22x. For m N in Ž13. and Ž14. the proton mass, and for mp in Ž16. the charged
H.-Ch. Schroder ¨ et al.r Physics Letters B 469 (1999) 25–29
29
pion mass were used. The implications of the result Ž15. are discussed in the context of chiral symmetry in Ref. w23x. Finally, the new b 0 value Ž11. results in a change of the p N sigma term, which is a measure for chiral symmetry breaking
for the success of the experiment. This research was supported in part by the Swiss National Science Foundation.
S s Fp2 Dq Ž n s 0,t s 2 mp2 . s Sd q DD ,
w1x D. Sigg et al., Phys. Rev. Lett. 75 Ž1995. 3245; D. Sigg et al., Nucl. Phys. A 609 Ž1996. 269. w2x D. Chatellard et al., Phys. Rev. Lett. 74 Ž1995.; D. Chatellard et al., Nucl. Phys. A 625 Ž1997. 855. w3x H.-Ch. Schroder, Ph.D. thesis, ETH Zurich, No. 11760, ¨ 1996, unpublished. w4x H.-Ch. Schroder ¨ et al., in preparation. w5x J. Spuller et al., Phys. Lett. B 67 Ž1977. 479. w6x D. Sigg, A. Badertscher, P.F.A. Goudsmit, H.J. Leisi, G.C. Oades, Nucl. Phys. A 609 Ž1996. 310. w7x S. Deser, M.L. Goldberger, K. Baumann, W. Thirring, Phys. Rev. 96 Ž1954. 774. w8x P. Hauser et al., Phys. Rev. C 58 Ž1998. R1869. w9x A.W. Thomas, R.H. Landau, Phys. Rep. C 58 Ž1980. 122. w10x L. Bracci, G. Fiorentini, Nuovo Cim. A 43 Ž1978. 9. w11x A. Badertscher et al., Phys. Lett. B 392 Ž1997. 278; A. Badertscher et al., PSI preprint PSI-PR-99-08. w12x A.J. Rusi El Hassani et al., Z. Phys. A 351 Ž1995. 113. w13x V.V. Baru, A.E. Kudryavtsev, Phys. Atom. Nucl. 60 Ž1997. 1475. w14x T.E.O. Ericson, B. Loiseau, A.W. Thomas, contribution to the PANIC99, XVth Particle and Nuclei International Conference, June 1999, Uppsala, Sweden. w15x H.J. Leisi et al., in: A. M Bernstein, B.R. Holstein ŽEds.., Chiral Dynamics: Theory and Experiment, Springer, 1995, p. 241. w16x S. Weinberg, Phys. Rev. Lett. 17 Ž1966. 616. w17x N. Fettes, U.-G. Meissner, S. Steininger, Nucl. Phys. A 640 Ž1998. 199. w18x M.L. Goldberger, H. Miyazawa, R. Oehme, Phys. Rev. 99 Ž1955. 986. w19x R.L. Workman, R.A. Arndt, M.M. Pavan, Phys. Rev. Lett. 68 Ž1992. 1653. w20x M.L. Goldberger, S.B. Treiman, Phys. Rev. 110 Ž1958. 1178. w21x Particle Data Group, Eur. Phys. J. C 3 Ž1998. 622. w22x Particle Data Group, Eur. Phys. J. C 3 Ž1998. 353. w23x H.J. Leisi, Experimental results in pion-nucleon scattering: QCD symmetry tests, in: D. Graudenz ŽEd.., PSI Proceedings 98-02, December 1998, ISSN 1019-6447, p. 33; ETHZ-IPP preprint PR-98-11, December 1998. w24x G. Hohler, in: H. Schopper ŽEd.., Landolt-Bornstein, ¨ ¨ Springer, Berlin, 1983, vol. 9b2. w25x J. Gasser, H. Leutwyler, M.P. Locher, M.E. Sainio, Phys. Lett. B 213 Ž1988. 85; J. Gasser, H. Leutwyler, M.E. Sainio, Phys. Lett. B 253 Ž1991. 252. w26x R. Koch, E. Pietarinen, Nucl. Phys. A 336 Ž1980. 331; R. Koch, Nucl. Phys. A 448 Ž1986. 707.
Ž 17 .
where Dq is the isospin-even p N scattering D amplitude w24x from which the pseudovector Born term is subtracted, taken at the Cheng-Dashen point Ž n s 0, t s 2 mp2 .. The term DD f 12 MeV w25x is due to the higher order contributions in the subthreshold expansion. In Ref. w25x the parametrization of Sd in terms of the two subtraction constants q Ž . aq 0q ' b 0 and a1q, the isospin even s- and p-wave q threshold parameters, is given. Using for a1q the q KH value w26x and for a 0q our value Ž11., we obtain
Sd s 57 MeV,
Ž 18 .
to be compared to Sd s 50 MeV w25x, the result one obtains when both subtraction constants are taken from the KH solution. The sigma term is thus increased by 7 MeV. More details on the sigma term can be found in Ref. w23x. Summarizing, we report the final results of the ETH Zurich–Neuchatel–PSI collaboration on the piˆ onic hydrogen experiments. We show that the constraints on the strong interaction scattering lengths b 0 and b 1 obtained from shift and width of the 1s state of pionic hydrogen and the shift of the 1s state of pionic deuterium are still compatible with isospin symmetry; precise values for b 0 and b 1 are obtained. From b 1 a new value for the p NN coupling constant is derived. The new b 0 value leads to an increase of the p N sigma term of 7 MeV.
Acknowledgements We would like to thank L. Knecht, B. Leoni and H. Obermeier for their contributions to the mechanical design and construction of the spectrometer, and D. Varidel and P. Pollet for designing and building the CCD electronics. The excellent beam delivered by the PSI accelerator and technical staff was crucial
References
Physics Letters B 469 Ž1999. 25–29
Determination of the p N scattering lengths from pionic hydrogen a H.-Ch. Schroder , A. Badertscher a,1, P.F.A. Goudsmit a , M. Janousch a , H.J. Leisi a , ¨ E. Matsinos a , D. Sigg a , Z.G. Zhao a , D. Chatellard b, J.-P. Egger b, E. Jeannet b, K. Gabathuler c , P. Hauser c , L.M. Simons c , A.J. Rusi El Hassani d b
a ETH Zurich, Institute for Particle Physics, CH-8093 Zurich, Switzerland Institut de Physique de l’ UniÕersite´ de Neuchatel, ˆ CH-2000 Neuchatel, ˆ Switzerland c Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland d Dept. de Physique, Faculte´ des Sciences et Technique, Tanger, Morocco ´
Received 21 July 1999; received in revised form 15 October 1999; accepted 15 October 1999 Editor: J.P. Schiffer
Abstract The final results from the pionic hydrogen X-ray experiment at PSI are presented. With this experiment the strong interaction energy level shift ´ 1s and the total decay width G 1s of the 1s state of pionic hydrogen were obtained from a precise measurement of the 3p–1s X-ray line: ´ 1s s y7.108 " 0.013Žstat.. " 0.034Žsyst.. eV Žattractive. and G 1s s 0.868 " 0.040Žstat.. " 0.038Žsyst.. eV. The corresponding hadronic p N s-wave scattering lengths for elastic scattering and single charge exchange are: apy p p y p s 0.0883 " 0.0008 mpy1 and ap y p p 0 n s y0.128 " 0.006 mpy1. Combining the results of the pionic hydrogen and deuterium measurements, the isoscalar Ž b 0 . and isovector Ž b 1 . scattering lengths, corresponding to an isospin symmetric interaction, can be calculated: b 0 s 0.0016 " 0.0013 mpy1 and b 1 s y0.0868 " 0.0014 mpy1. q 1999 Elsevier Science B.V. All rights reserved.
™
The energy and line shape of the 3p–1s X-ray transitions in pionic hydrogen and deuterium were measured by the ETH Zurich-Neuchatel-PSI collaboˆ ration at the Paul Scherrer Institute ŽPSI, Villigen, Switzerland. with a high resolution bent crystal spectrometer. First results from pionic hydrogen were published in Ref. w1x, and the results of the deuterium measurements are given in Ref. w2x. Experimental details are described in Refs. w1,2x. In this paper we report the final results of this collaboration on pionic hydrogen, based on the data of Refs. w1,3,4x. Combining the pionic hydrogen result with that from
1
E-mail: [email protected]
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pionic deuterium ŽRef. w2x., precise information on the p N scattering lengths is obtained. A unique feature of the pionic atom measurements is the possibility to study the p N interaction at threshold, and thus to be able to determine the p N scattering lengths directly, i.e. without extrapolations to threshold. Subtracting the measured 3p–1s transition energy from the calculated electromagnetic energy gives the strong interaction level shift ´ 1s of the 1s state Žshift and width of the 3p state are negligible., and from the measured line shape the decay width of the 1s state G 1s can be calculated. The total decay width of the 1s state is determined by the rate of the single charge exchange ŽSCX. reaction Žpy p .1s p 0 n and the radiative decay
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 2 3 7 - X
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26
H.-Ch. Schroder ¨ et al.r Physics Letters B 469 (1999) 25–29
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Žpy p .1s g n. The ratio of the two decay rates is the well known Panofsky ratio P s 1.546 " 0.009 w5x. The two quantities ´ 1s and G 1s are related to the s-wave scattering lengths for elastic Ž apy p ™ p y p . and SCX Ž apy p ™ p 0 n . scattering. We have established these relations numerically w6x taking into account various electromagnetic effects and using an attractive strong interaction potential for isospin 1r2 and a repulsive potential for isospin 3r2. The result can be expressed as electromagnetic correction factors d´ and dG to Deser-type formulae w6,7x: ´ 1s ap y p ™ p y p s y4 Ž 1 q d´ . , Ž 1. E1s rB G 1s q 1 2 s8 1q apy p ™ p 0 n Ž 1 q dG . 4 , Ž 2. E1s rB P with d´ s Žy2.1 " 0.5.% and dG s Žy1.3 " 0.5.%; E1s is the point-Coulomb electromagnetic binding energy of the 1s level, r B s 222.6 fm the Bohr radius and q s 28.04 MeVrc s 0.1421 fmy1 the c.m. momentum of the p 0 in the SCX reaction; E1srr B s 14.53 eVrfm.
ž
/
In a similar way the Žcomplex. scattering length of pionic deuterium ap d can be calculated from the shift and width of the deuterium 1s state w2x. ŽA recent measurement of the 1s shift of pionic deuterium w8x agrees with our value.. The real part Re ap d can be expressed in terms of the p N scattering lengths w9,2x. The constraints on the scattering lengths from the shift and width of hydrogen and the shift of deuterium is investigated, assuming isospin symmetry of the strong p N interaction. Provided the constraints of the three observables are compatible with the assumption of isospin symmetry, the isoscalar and isovector p N scattering lengths b 0 and b 1 can be calculated. Fig. 1 shows measured X-ray lines corresponding to 75% of our py p data w3x Žthe first part of the data is published in Ref. w1x.. The electronic argon K a 1 line served as a wavelength standard; from the Bragg angle difference between the argon K a 1 and the meas. pionic hydrogen line the transition energy E3py1s is obtained. Table 1 summarizes the measured and the calculated electromagnetic transition energies for the
Fig. 1. Measured X-ray lines: Ža. Pionic hydrogen line, Žb. electronic argon K a line, used as a wavelength standard, Žc. pionic beryllium line, used to determine the instrumental resolution function. One channel corresponds to five CCD pixels Žpixel size: 22.5 mm.. The energy dispersion of the spectrometer was 0.088, 0.101 and 0.080 eVrchannel at the Bragg angles of measurements Ža., Žb. and Žc., respectively.
H.-Ch. Schroder ¨ et al.r Physics Letters B 469 (1999) 25–29
27
Table 1 Measured and calculated Želectromagnetic. 3p–1s transition energy w6x and deduced strong interaction shift of the 1s state Energy ŽeV. me as. E3py1s
2885.916 " 0.013Žstat.. " 0.033Žsyst.. 2875.715 " 0.007 y0.102 " 0.003 3.235 " 0.001
Point-Coulomb, Klein-Gordon equation Coulomb finite size effect Žproton and pion. Vacuum polarization, order a 2 ŽUehling potential, finite size. Higher order corrections ŽHOC.: – Relativistic recoil and proton spin and anomalous magnetic moment of the proton – Vacuum polarization order a 3 – Vertex correction Pionic atom recoil energy
y0.047 " 0.000 0.018 " 0.000 y0.007 " 0.003 y0.004
el .mag . E3py1s Strong interaction shift ´ 1s
2878.808 " 0.008 y7.108 " 0.013Žstat.. " 0.034Žsyst..
3p–1s transition and gives the resulting value for the strong interaction shift:
´ 1s s y7.108 " 0.013 Ž stat. . " 0.034 Ž syst. . eV.
Ž 3. The pionic beryllium transition has a very small intrinsic width so that the 4f-3d line could be used to determine the instrumental resolution function. The spectrometer resolution was about 0.6 eV ŽFWHM., and the deconvolution of the measured line profile with the instrumental resolution function yields the line width w3x Žwe have added 9 meV to the width given in Ref. w3x to account for the Auger width of the Be line. meas . G 3py1s s 0.969 " 0.045 Ž stat. . " 0.010 Ž syst. . eV.
Ž 4. This line width has to be corrected for the Doppler broadening to obtain the decay width of the 1s state G 1s . The Doppler broadening of the X-ray lines occurs mainly as a consequence of an acceleration mechanism of the pionic atoms during the cascade, the so called Coulomb de-excitation w10x. In this process the excited pyp-atom penetrates the electron cloud of a normal hydrogen atom. In the atomic electric field a de-excitation of the pionic atom to a lower level takes place, in which no X-ray is emitted. The transition energy is shared as kinetic energy between the two collision partners. To study the kinetic energy distribution of pionic hydrogen atoms formed in gaseous and liquid hydrogen, the time-of-
™
flight of neutrons ŽnTOF. from the SCX reaction py p p 0 n was measured at PSI w11x. From the nTOF data and the absolute X-ray yields in pionic hydrogen from Ref. w12x the kinetic energy distribution of pionic atoms undergoing the 3p–1s radiative transition was obtained w4x. This allows the calculation of the Doppler correction factor D of the pionic X-ray lines
D'
meas . G 3py1s y G 1s
G 1s
s 0.12 " 0.05.
Ž 5.
Inserting the measured line width Ž4. into Eq. Ž5., the decay width of the 1s state is obtained:
G 1s s 0.868 " 0.040 Ž stat. . " 0.038 Ž syst. . eV.
Ž 6.
With the values Ž3. and Ž6. and adding the statistical and systematic errors linearly, the hadronic s-wave scattering lengths are obtained from Eqs. Ž1. and Ž2.: apy p ™ p y p s Ž 0.0883 " 0.0008 . mpy1
apy p ™ p 0 n s Ž y0.128 " 0.006 . mpy1 .
Ž 7. Ž 8.
The scattering lengths for elastic and SCX scattering can be expressed in terms of the scattering lengths of total isospin 1r2 Ž a1 . and 3r2 Ž a3 ., or, alternatively by the isoscalar and isovector scattering lengths b 0 and b 1: apy p ™ p y p s b 0 y b 1 s 13 Ž 2 a1 q a 3 . , apy p ™ p 0 n s '2 b 1 s
'2 3
Ž a3 y a1 . .
Ž 9. Ž 10 .
H.-Ch. Schroder ¨ et al.r Physics Letters B 469 (1999) 25–29
28
Combining the constraints from the hydrogen shift and width with Re apy d from the deuterium shift, the final result for b 0 and b 1 is obtained, with a correlation coefficient r 01 s 0.83:
Fig. 2. Constraints on the scattering lengths b 0 and b1 imposed by the measured strong interaction shift and width of the 1s state of pionic hydrogen and the 1s shift of pionic deuterium. The ellipse represents the one standard deviation contour for an isospin symmetric strong interaction. Also indicated are comparisons with theoretical calculations Žsee text..
Fig. 2 shows the constraints on b 0 and b 1 imposed by the values Ž7. and Ž8.. Also shown in Fig. 2 is the Žnearly horizontal. band derived from the real part of the py d scattering length w2x and the calculations of Thomas and Landau w9x, which relate Re apy d to the elementary p N scattering lengths 2 . Using only the two constraints from the hydrogen shift and width leads, with Eqs. Ž7. – Ž10., to b 0 s Žy2.2 " 4.3. = 10y3 my1 and b 1 s Žy90.5 " 4.2. = 10y3 my1 with p p a strong correlation r 01 s 0.98 due to the narrow diagonal band from the hydrogen shift. If isospin would be an exact symmetry, the constraints from all p N reactions would have a common intersection in the b 0 y b 1 plot w15x. Fig. 2 shows that with the present errors the three constraints derived from the pionic hydrogen and deuterium measurements are still compatible with isospin symmetry Žassuming that the calculations of Thomas and Landau are complete..
b 0 s Ž q1.6 " 1.3 . = 10y3 mpy1 ,
Ž 11 .
b 1 s Ž y86.8 " 1.4 . = 10y3 mpy1 .
Ž 12 .
The ellipse in Fig. 2 corresponds to this result. The black dot indicates the scattering lengths in the current-algebra limit, first obtained by Weinberg w16x, and the dashed rectangle ŽFMS. is the result of a recent third-order calculation within the framework of heavy baryon chiral perturbation theory w17x. We note that the isoscalar scattering length is compatible with the current-algebra limit Ž b 0 s 0. and the isovector scattering length differs by 9%. A more detailed comparison with other determinations of the scattering lengths, all of which are based on extrapolations to threshold, will be given in Ref. w4x. With the new b 1 value the Žpseudovector. p NN coupling constant can be calculated using the GMO sum rule w18x. f2 4p
s y0.571mp P b 1 y 0.0249 mby1 P J s 0.0757 " 0.0010.
For the dispersion integral J we use the average of the VPI and Karlsruhe-Helsinki ŽKH. values given in Ref. w19x: J s y1.0485 " 0.0075 mb; J VPI s y1.041 mb and J KH s y1.056 mb. With this coupling constant, the Goldberger-Treiman w20x discrepancy DGT , defined by the relation mN gp N N s g Ž 1 q DGT . Ž 14 . Fp A is obtained:
DGT s Ž 1.9 " 0.8 . = 10y2 .
The multiple scattering calculations in the py d system have since been repeated by Baru and Kudryavtsev w13x. Ericson et al. w14x are presently analysing in detail the various terms in the py d caculation.
Ž 15 .
The coupling constant gp N N in Ž14. is given by gp N N s
2
Ž 13 .
2 mN mp
P f s 13.11 " 0.09.
Ž 16 .
The nucleon axial vector coupling constant is g A s 1.2670 " 0.0035 w21x and the pion decay constant Fp s 92.42 " 0.26 MeV w22x. For m N in Ž13. and Ž14. the proton mass, and for mp in Ž16. the charged
H.-Ch. Schroder ¨ et al.r Physics Letters B 469 (1999) 25–29
29
pion mass were used. The implications of the result Ž15. are discussed in the context of chiral symmetry in Ref. w23x. Finally, the new b 0 value Ž11. results in a change of the p N sigma term, which is a measure for chiral symmetry breaking
for the success of the experiment. This research was supported in part by the Swiss National Science Foundation.
S s Fp2 Dq Ž n s 0,t s 2 mp2 . s Sd q DD ,
w1x D. Sigg et al., Phys. Rev. Lett. 75 Ž1995. 3245; D. Sigg et al., Nucl. Phys. A 609 Ž1996. 269. w2x D. Chatellard et al., Phys. Rev. Lett. 74 Ž1995.; D. Chatellard et al., Nucl. Phys. A 625 Ž1997. 855. w3x H.-Ch. Schroder, Ph.D. thesis, ETH Zurich, No. 11760, ¨ 1996, unpublished. w4x H.-Ch. Schroder ¨ et al., in preparation. w5x J. Spuller et al., Phys. Lett. B 67 Ž1977. 479. w6x D. Sigg, A. Badertscher, P.F.A. Goudsmit, H.J. Leisi, G.C. Oades, Nucl. Phys. A 609 Ž1996. 310. w7x S. Deser, M.L. Goldberger, K. Baumann, W. Thirring, Phys. Rev. 96 Ž1954. 774. w8x P. Hauser et al., Phys. Rev. C 58 Ž1998. R1869. w9x A.W. Thomas, R.H. Landau, Phys. Rep. C 58 Ž1980. 122. w10x L. Bracci, G. Fiorentini, Nuovo Cim. A 43 Ž1978. 9. w11x A. Badertscher et al., Phys. Lett. B 392 Ž1997. 278; A. Badertscher et al., PSI preprint PSI-PR-99-08. w12x A.J. Rusi El Hassani et al., Z. Phys. A 351 Ž1995. 113. w13x V.V. Baru, A.E. Kudryavtsev, Phys. Atom. Nucl. 60 Ž1997. 1475. w14x T.E.O. Ericson, B. Loiseau, A.W. Thomas, contribution to the PANIC99, XVth Particle and Nuclei International Conference, June 1999, Uppsala, Sweden. w15x H.J. Leisi et al., in: A. M Bernstein, B.R. Holstein ŽEds.., Chiral Dynamics: Theory and Experiment, Springer, 1995, p. 241. w16x S. Weinberg, Phys. Rev. Lett. 17 Ž1966. 616. w17x N. Fettes, U.-G. Meissner, S. Steininger, Nucl. Phys. A 640 Ž1998. 199. w18x M.L. Goldberger, H. Miyazawa, R. Oehme, Phys. Rev. 99 Ž1955. 986. w19x R.L. Workman, R.A. Arndt, M.M. Pavan, Phys. Rev. Lett. 68 Ž1992. 1653. w20x M.L. Goldberger, S.B. Treiman, Phys. Rev. 110 Ž1958. 1178. w21x Particle Data Group, Eur. Phys. J. C 3 Ž1998. 622. w22x Particle Data Group, Eur. Phys. J. C 3 Ž1998. 353. w23x H.J. Leisi, Experimental results in pion-nucleon scattering: QCD symmetry tests, in: D. Graudenz ŽEd.., PSI Proceedings 98-02, December 1998, ISSN 1019-6447, p. 33; ETHZ-IPP preprint PR-98-11, December 1998. w24x G. Hohler, in: H. Schopper ŽEd.., Landolt-Bornstein, ¨ ¨ Springer, Berlin, 1983, vol. 9b2. w25x J. Gasser, H. Leutwyler, M.P. Locher, M.E. Sainio, Phys. Lett. B 213 Ž1988. 85; J. Gasser, H. Leutwyler, M.E. Sainio, Phys. Lett. B 253 Ž1991. 252. w26x R. Koch, E. Pietarinen, Nucl. Phys. A 336 Ž1980. 331; R. Koch, Nucl. Phys. A 448 Ž1986. 707.
Ž 17 .
where Dq is the isospin-even p N scattering D amplitude w24x from which the pseudovector Born term is subtracted, taken at the Cheng-Dashen point Ž n s 0, t s 2 mp2 .. The term DD f 12 MeV w25x is due to the higher order contributions in the subthreshold expansion. In Ref. w25x the parametrization of Sd in terms of the two subtraction constants q Ž . aq 0q ' b 0 and a1q, the isospin even s- and p-wave q threshold parameters, is given. Using for a1q the q KH value w26x and for a 0q our value Ž11., we obtain
Sd s 57 MeV,
Ž 18 .
to be compared to Sd s 50 MeV w25x, the result one obtains when both subtraction constants are taken from the KH solution. The sigma term is thus increased by 7 MeV. More details on the sigma term can be found in Ref. w23x. Summarizing, we report the final results of the ETH Zurich–Neuchatel–PSI collaboration on the piˆ onic hydrogen experiments. We show that the constraints on the strong interaction scattering lengths b 0 and b 1 obtained from shift and width of the 1s state of pionic hydrogen and the shift of the 1s state of pionic deuterium are still compatible with isospin symmetry; precise values for b 0 and b 1 are obtained. From b 1 a new value for the p NN coupling constant is derived. The new b 0 value leads to an increase of the p N sigma term of 7 MeV.
Acknowledgements We would like to thank L. Knecht, B. Leoni and H. Obermeier for their contributions to the mechanical design and construction of the spectrometer, and D. Varidel and P. Pollet for designing and building the CCD electronics. The excellent beam delivered by the PSI accelerator and technical staff was crucial
References