The isovector GMO sum rule for zero energy forward pion-nucleon scattering iscritically studied to obtain the charged pion-nucleon coupling constant usingthe precise negatively charged pion-proton and pion-deuteron scattering lengthsdeduced recently from pionic atom experiments. This direct determination leadsto a pseudoscalar charged pion-nucleon coupling constant of 14.23 +- 0.09(statistic) +- 0.17 (systematic). We obtain also accurate values for thepion-nucleon scattering lengths

Ericson, Torleif Eric Oskar

Loiseau, B

Thomas, A W

The isovector GMO sum rule for zero energy forward pion-nucleon scattering iscritically studied to obtain the charged pion-nucleon coupling constant usingthe precise negatively charged pion-proton and pion-deuteron scattering lengthsdeduced recently from pionic atom experiments. This direct determination leadsto a pseudoscalar charged pion-nucleon coupling constant of 14.23 +- 0.09(statistic) +- 0.17 (systematic). We obtain also accurate values for thepion-nucleon scattering lengths.The isovector GMO sum rule for zero energy forward pion-nucleon scattering is critically studied to obtain the charged pion-nucleon coupling constant using the precise negatively charged pion-proton and pion-deuteron scattering lengths deduced recently from pionic atom experiments. This direct determination leads to a pseudoscalar charged pion-nucleon coupling constant of 14.23 +- 0.09 (statistic) +- 0.17 (systematic). We obtain also accurate values for the pion-nucleon scattering lengths

Loiseau, Benoit

Ericson, Torleif Erik Oskar

Thomas, Anthony William

CERN Document Server

How precisely can we determine the NN coupling constant from theisovector GMO sum rule?B. LoiseauLPNHE/LPTPE, Universite P. & M. Curie, 4 Place Jussieu, 75252 Paris, FranceT.E.O. EricsonCERN, CH-1211 Geneva 23, Switzerland, and TSL, Box 533, S-75121 Uppsala, SwedenA.W. ThomasCSSM, University of Adelaide, Adelaide 5005, AustraliaAbstractThe isovector GMO sum rule for zero energy forward N scattering is criticallystudied to obtain the charged NN coupling constant using the precise −p and−d scattering lengths deduced recently from pionic atom experiments. This directdetermination leads to g2c=4 = 14:230:09 (statistic) 0:17 (systematic) or f2c =4 =0:0786(11). We obtain also accurate values for the N scattering lengthsINTRODUCTION: ROBUST FORM OF THE GMO RELATIONThe analysis to determine the NN coupling constant should be clear and easily re-producible. One should do a detailed study for the statistical and systematic errors. Theprecise determination is an absolute statement, it could be erroneous and it should be im-provable. In this perspective the Goldberger-Miyazawa-Oehme (GMO) sum rule[1] mightbe a good candidate. It is a forward dispersion relation at zero energy for N scattering.It assumes scattering amplitudes to be analytical functions satisfying crossing symmetry.At rst isospin symmetry does not have to be assumed and it reads (for more details seee.g.[2]) with its numerical coecients: g2c=4 = −4:50 J−+103:3 [(a−p−a+p)=2], whereJ− is in mb the weighted integral, J− = (1=42)R10 (dk=pk2 + m2)[Total−p (k)− Total+p (k)]and ap are the p scattering lengths in units of m−1 . All ingredients are physical ob-servables but so far the lack of precision in ap (contribution of 2/3 to g2=4) led toapplications of the GMO relation as consistency check or constraint[3]. The 1s width ofthe −p atom[4] determines a−p!0n = −0:128(6) m−1 and assuming isospin symmetrythis gives a− = (a−p − a+p)=2 and g2c =4 = 14:2(4) using[5] J−= -1.077(47) mb. This isnot accurate enough although improvements will come[6].We here report on a possible way to improve the precision on g2c=4[7]. As a−p isprecisely known (0:0883(8) m−1 ) from energy shift in pionic hydrogen[8] one can write:g2c=4 = −4:50 J− + 103:3 a−p − 103:3 (a−p + a+p2): (1)and using the above cited J− (to be calculated later), g2c = 4:85(22)+9:12(8)−103:3[(a−p+a+p)=2]. This (not our nal result) shows that all the action is in the term 1=2(a−p +a+p),which, assuming isospin symmetry, is a+. If this quantity is positive g2c=4 is smallerthan 14, if it is negative it is larger. One way to determine the small a+ is to use theaccurate −d scattering length a−d = −0:0261(5) m−1 ; from the pionic deuterium 1senergy level[9]. To leading order this is the coherent sum of the − scattering lengthsfrom the proton and neutron, which, assuming charge symmetry (viz, a+p = a−n) is theterm required in our ’robust’ relation (1) The strong cancellation between the two termsis then done by the physics. In order to match the precision using the width, we only needa theoretical precision in the description of the deuteron scattering length to about 30%.ZERO-ENERGY −-DEUTERON SCATTERING AND a+In multiple scattering theory of zero-energy s-wave pion scattering from point-like nu-cleons and in the xed scattering-center approximation, the leading contribution is[10]:1astatic−d = S + D ::: with S = [(1+m=M)=(1+m=Md)](a−p +a−n); M and Md beingthe nucleon and deuteron masses respectively. The double scattering term D is:D = 2(1 + m=M)2(1 + m=Md)[(a−p + a−n2)2 − 2(a−p − a−n2)2] < 1=r >; (2)and with our nal scattering lengths D = −0:0256 m−1 quite close to a−d experimental.Fig. 1. Our graphical determi-nation of the N scatteringlengths in excellent agreementwith the central values of the ex-perimental PSI group[4], a+ =−22(43) 10−4m−1 ; a− = 905(42) 10−4m−1 .We shall here follow the recent theoretical multiple scattering investigation of Baruand Kudryatsev (B-K)[11]. The comparison of typical contributions is listed in Table 1.1)Fermi motion: the nucleons have a momentum distribution which produces an attractiveTable 1. Typical contributions to ad in units of 10−4m−1 , recall aexp:d = −261(5)[9].Contri D Fermi Absorption s-p (−p; γn) Form-butions motion 1) corr.[16] 2) interf. 3) double scatt. factor 4)Present -256(7) 61(7) -56(14) small -2 17(9)B-K -252 50 X -44 X 29(7)Contri- Non-static Isospin Higher p-wave Virtualbutions eects 5) violation 6) order 6) double scatt. 6) pion 6)Present 10(6) 3.5 4(1) -3 -7(2)B-K 10 3.5 6 -3 Xcontribution, calculable to leading order from < p2 > of the nucleon momenta in thedeuteron. The uncertainty of 7 comes from the D-state percentage in the deuteron, PD=4.3% vs. 5.7% for the Machleidt1[12] vs. the Paris[13] wave functions. 2) Absorptioncorrection: the absorption reaction −d !nn, using 3-body Faddeev approaches[14{16]produces a repulsive (−20%) contribution (not included in B-K). These studies were donecarefully but a modern reinvestigation of this term is highly desirable. 3) "s-p' interference:a −15% correction was obtained by B-K. We nd that it is a model dependent contributiondue nearly entirely to the Born term the contribution of which vanishes exactly. We havethen not considered this contribution. 4)Form factor: this non-local eect enters mainly viathe dominant isovector N s-wave interaction, closely linked to  exchange. It representsonly a correction of about −10%. 5) Non-static eects: these produce only a rather smallcorrection of about 4%. There are systematic cancellations between single and doublescattering as was rst demonstrated by Fa¨ldt[17]. It has been numerically investigatedby B-K and we have adopted their value, the error of 6 reflects a lack of independentverication. 6) Isospin violation, higher order terms, p-wave double scattering,virtual pionscattering: these corrections are all small and controllable. The isospin violation in the2N interaction comes in part from the  − 0 mass dierence where an additional checkcomes from the chiral approach[18]. Based on this, we obtain the preliminary, thoughnearly nal, values (a−p + a−n)=2 = (−17 3(statistic) 9(systematic))10−4m−1 and(a−p − a−n)=2 = (900 12) 10−4m−1 : Our values represent a substantial improvementin accuracy as seen in Fig. 1. The contribution of the scattering lengths to g2c=4 has herea precision of about 1%.TOTAL CROSS SECTION INTEGRAL J−The cross section integral contributes only one third to the GMO relation. Total crosssections are inherently accurate and their contribution is calculated with accuracy, butfor the high energy region. The possibility of systematic eects in the dierence mustbe considered, particularly since Coulomb corrections have opposite sign for p. Theonly previous evaluation with a detailed discussion of errors is that by Koch[5]. Laterevaluations given in Table 2 nd values within the errors, but the uncertainties are notstated and analyzed. In view of obtaining a clear picture of the origin of uncertainties weTable 2. Some values of J− in mb, Ref.[19] uses 2 dierent PWA: K-H[20] and their own, VPI.Ref. Koch Workman et al. Workman et al. Arndt et al. Gibbs et al. Present1985[5] 1992; K-H[19] 1992; VPI[19] 1995[21] 1998[22] workJ− -1.077(47) -1.056 -1.072 -1.05 -1.051 -1.095(31)have reexamined this problem in spite of the consensus. We limit the discussion to thecritical features. The typical shape of the integrands is shown in Fig. 2 up to 2 Gev/c.Fig. 2. The J− separate integrandsfor p as well as their dier-ence as function of klab togetherwith the cumulative value of theintegral J−(klab) from the region0 < k < klab. The integrands arein units of mb GeV/c.There are no total cross-section measurements below 160 MeV/c, but the accuratelyknown ap give a strong constraint assuming isospin symmetry. The s- and p-wave con-tributions nearly cancels. A tiny correction occurs, since isospin is broken by the 3.3 MeVlower threshold for the 0n channel below the physical −p threshold. The main contri-butions come from the region of the  resonance and just above. There are no strongcancellations in the dierence between p cross sections in that region and the cross sec-tions have been very carefully analyzed. We have rst evaluated the hadronic cross sectionsup to 2 GeV/c based on the VPI phase-shift solution[23]. In doing so Coulomb correctionsand penetration factors have been taken into account in the adjustment to experimentaldata even if the treatment may not be optimal. It also allows for some isospin breaking,since the  mass splitting is parameterized[24]. In view of the not so high accuracy we aimfor, this should be adequate. Bugg[25,26] has emphasized that in the +p scattering thetotal cross sections are systematically reduced at all energies by the Coulomb repulsionbetween the particles and, conversely, enhanced in −p scattering. One must correct for3this eect, which gives a negative contribution to J−. Having made no correction for itat higher energies means that we will underestimate the coupling constant somewhat. Inthe region around 500 MeV/c there are long-standing problems with the experimentaltotal cross section data[24]. This uncertainty, larger than the Coulomb penetrability ef-fects, should be resolved. So we have preferred to use the SM95 PWA solution as the bestguide. The real uncertainty in J− comes from the high energy region and is linked to therelatively slow convergence of the integral.Table 3. Contributions to J− in mb according to interval of integration and to the total cross-section input. ‘Selected" is for the world data[28] with statistical and systematic errors 1%. Herethe rst given error is statistical and the second one systematic.k(GeV/c) 0 to 2 2 to 4.03 4.03 to 240 240 to 1Input SM95[23] Selected[28] Selected[28] Regge(94)[27]J−(mb) (-1.3020:006)(17) (0.0640:002)(7) (0.1330:005)(24) (0.030)k(GeV/c) 0 to 2 240 to 1 0 to 1 0 to 1Input Arndt98)[23] Regge(98)[28] SM95+Regge94 Arndt98+Regge98J−(mb) (-1.3290:006)(17) (0.018)(3) (-1.0750:008)(30) (-1.1140:008)(30)We have evaluated the dierent contributions (see Table 3) with no other Coulomband penetration corrections than those introduced by the experimental authors above2 GeV or by the theoretical analysis below 2 GeV. We nd, based on (integration ofhadronic cross section) the SM95 and Arndt 12/98 analysis below 2 GeV/c[23], and on theRegge pole PDG94[27] and PDG98[28] extrapolation beyond 240 GeV/c, the values J− =(−1:075 0:008)(30) mb and (−1:114 0:008)(30) mb respectively. We have adopted themean value J− = (−1:0950:008)(30) mb. In our calculation we have added a systematicuncertainty from Coulomb penetration eect of 0:017 from the region less than 2 GeV/c.RESULTS AND CONCLUSIONSWe have derived rst new values for the N scattering lengths from the −d one:a+ ’ a−p + a−n2= (−17 3)(9))  10−4m−1 ; a− ’a−p − a−n2= 900(12)  10−4m−1 :Our second conclusion concerns the charged NN coupling constant using these new ac-curate values in (1) with J− = (−1:095  0:008)(30) and charge symmetry:g2c=4 = (4:93  0:04)(14) + (9:12  0:08) + (0:18  0:03)(9) = (14:23  0:09)(17): (3)The uncertainty comes mainly from J−. This coupling constant which agrees quite wellwith the text book value, 14.28(18)[20] is intermediate between the low value deduced fromthe large data banks of NN and N scattering data[24,29] and the high value from np chargeexchange cross sections[30]. It is fully compatible with the latter, diering statistically byonly about one standard deviation.REFERENCES1. M. L. Goldberger, H. Miyazawa and R. Oehme, Application of Dispersion Relationsto Pion-Nucleon Scattering, Phys. Rev. 99, 986 (1955).2. G. Ho¨hler in Pion-Nucleon Scattering, Ed. H. Schopper, Landolt-Bo¨rnstein, New Se-ries, Vol. 9b (Springer, New York 1983); G. Ho¨hler, Determination of the NN Cou-pling Constant, contribution to this symposium.3. R. A. Arndt, R. L. Workman and M. M. Pavan, Pion-nucleon partial-wave analysiswith xed-t dispersion relation constraints, Phys. Rev. C49, 2729 (1994).4. H.-Ch. Schro¨der, A. Badertscher, P. F. A. Goudsmit, M. Janousch, H. J. 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How Precisely can we Determine the \piNN Coupling Constant from the Isovector GMO Sum Rule?

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